EST]MATING VARIANCE COMPONENTS
FOR TWO-WAY DISPROPORTIONATE DATA
BY THE METHOD OF UNWEIGHTED .MEANS
by
Tyler D. Hartwell
Institute of Statistics
Mimeograph Series No. 755
June 1971 - Raleigh
·e
iv
TABLE OF CONTENTS
Page
LIST OF TABLES .
vi
1.
INTRODUCTION
1
2.
REVIEW OF LITERATURE
4
3.
ANALYTICAL RESULTS FOR ONE MISSING CELL
9
3.1
3.2
3.3
3.4
4.
ANALYTICAL RESULTS FOR MORE THAN ONE MISSING CELL
4.1
4.2
4.3
5.
Introduction
.
The Method of Unweighted Means
Expected Mean Squares with One Missing Cell
Variances of Estimated Variance Components with
One Missing Cell
. . . . . .
Introduction
Two or More Missing Cells in the Same Row (or
Column) . . . . .
Two or More Missing Cells in Different Rows
and Columns
Introduction
General Description of the Computer Program VC
Designs Investigated
Resul ts
. . . . . . . . . . .
5.4.1
5.4.2
5.4.3
6.
6.3
7.
Mean Square Errors of the Estimated
Variance Components . . . . . . .
Expected Mean Squares for the Method
of Unweighted Means . . . . .
Biases of the Estimated Variance
Components for Procedure M
UNWEIGHTED MEANS FOR DESIGNS WITH TWO LEVELS OF NESTING
6.1
6.2
...
.
......
..... . .
Introduction
.
Expected Mean Squares and Estimated Variance
Components
Variances of the Estimated Variance Components
..
SUMMARY AND SUGGESTIONS FOR FUTURE RESEARCH
7.1
7.2
14
32
38
38
38
55
82
COMPUTER STUDY . . .
5.1
5.2
5.3
5.4
9
10
Summary. . . . . . . . .
Suggested Future Research
82
83
86
92
93
101
108
116
116
119
129
138
138
144
v
TABLE OF CONTENTS (continued)
Page
8.
LIST OF REFERENCES • • • • • • •
145
9.
APPENDIX. MEAN SQUARE ERRORS OF ESTIMATES OF VARIANCE
COMPONENTS • • • • • • • • • • • • • • • • • • • • •
148
vi
LIST OF TABLES
Page
3.1
4.1
4.2
4.3
4.4
5.1
Expected mean squares for a two-way disproportionate
design with one missing cell using the method of
unweighted means . . . . . . . . . . . . . . . . .
28
Expected mean squares for a two-way disproportionate
design with two missing cells in the same row using
the method of unweighted means . . . . . . . . . .
50
Expected mean squares for a two-way disproportionate
design with M missing cells in the same row using
the method of unweighted means . . . . . . . . . .
53
Expected mean squares for a two-way disproportionate
design with two missing cells in different rows and
columns using the method of unweighted means . . .
65
Expected mean squares for a two-way disproportionate
design with M missing cells in different rows and
columns using the method of unweighted means . . .
73
The n .. arrangements for the 3 x 3 designs investigated
~J
87
and the base designs from which they were derived
5.2
The n .. arrangements for the 4 x 4 designs investigated
~J
88
and the base designs from which they were derived
5.3
5.4
5.5
5.6
The n .. arrangements for the 6 x 6 designs investigated
.
~J
and the base designs from which they were derived
89
General results using mean square errors to determine
the best estimating procedure . . . .
. . . .
96
Coefficients of the variance components in the
expected mean squares for the method of unweighted
means for several two-way disproportionate designs
with missing cells . . . • . . . . . . . . . .
Biases of estimated variance components obtained by
2
2
Procedure Mwhen ~
.25 and ~ = 1
.
rc =
e
...
111
Biases of estimated variance components obtained by
2
2
1 and ~ = 1
. .
Procedure Mwhen ~
rc =
e
...
112
.
5.7
.
5.8
103
.
Biases of estimated variance components obtained by
Procedure
Mwhen
~2rc = 4 and ~2e = 1 . . . . .
113
vii
LIST OF TABLES (continued)
Page
5.9
6.1
Percentage contribution of the bias of row and column
variance components estimated by Procedure
to the
mean square error) 3 x 3 - B designs . . . .
M
Expected mean squares for a two-way disproportionate
design with two levels of nesting within each cell
~sing the method of unweighted means with cell means
y. .
.
~J
6.2
.. . . . .
9.2
9.3
9.4
127
. . . .
Ratios of the variances of the estimated row variance
components obtained by using the analysis of variance
procedure based upon the method of unweighted means with
cell means -y .. and y-*
.
~J
9.1
1]
9.6
9.7
9.8
9.9
9.10
135
Mean square errors of estimates of variance components
for 3 x 3
A designs
· · ·
·
···
149
Mean square errors of estimates of variance components
for 3 x 3
B designs ·
··
151
Mean square errors of estimates of variance components
for 3 x 3
C designs
·
·
154
Mean square errors of estimates of variance components
for 3 x 3 - D designs
158
Mean square errors of estimates of variance components
for 3 x 3
E designs
· ·
·
···
161
Mean square errors of estimates of variance components
for 4 x 4
A designs
·
·
164
Mean square errors of estimates of variance components
for 4 x 4
B designs
·
167
Mean square errors of estimates of variance components
for 6 x 6
A designs
·
·
·
171
Mean square errors of estimates of variance components
B designs
for 6 x 6
· ·
··
173
Mean square errors of estimates of variance components
for 6 x 6
C designs ·
175
-
·
····
-
·
··
·
··
····
9.5
126
Expected mean squares for a two-way disproportionate
design with two levels of nesting within each cell
using the method of unweighted means with cell means
y-*ij
6.3
114
-
-
-
-
-
·
··
··
·
··
·
·
···
····
···
···
··
·
··
·
··
···
··
-
·
·
··
····
····
I
,
1.
INTRODUCTION
Crossed classifications with unequal numbers of observations in
the cells pose difficult computational problems for the method of
fitting constants, Yates [1934].
To avoid these computational
difficulties, Yates [1934], has suggested an alternative procedure,
the method of unweighted means, which is a simple analysis based on
the cell means.
This analysis is limited to cases for which all of
the cells have one or more observations.
To avoid this limitation,
the primary purpose of this dissertation is to study the method of
unweighted means with regard to estimating the variance components for
a two-way classification with unequal numbers of observations in the
cells and with one or more cells having no observations.
The model
used is the variance component Model II given by Eisenhart [1947].
In particular, two estimating procedures are examined which are
based upon the method of unweighted means.
The first (denoted by
Procedure M) estimates the variance components in the following manner:
1.
a missing value estimation procedure based on cell
means is used to estimate an observation (observations)
for the missing cell (cells);
2.
the mean squares and their expectations are then computed
for the method of unweighted means and
3.
the variance components are estimated by equatinp, the
expected mean squares to the computed mean squares in the
analysis of variance table and solving the resulting set
of equations for the estimates of the variance components.
Procedure M gives unbiased estimates for the variance components.
2
~
The second procedure (denoted by Procedure M) is the same as
Procedure M except that approximate expected mean squares are used
which are essentially those that would be obtained by using the method
of unweighted means for a design where there is no missing cell (cells).
Procedure
Mgives
biased estimates for the variance components.
In order to evaluate Procedures M and
M the
variances and mean
square errors of the estimated variance components are computed for
both procedures for several experimental designs with missing cells and
a number of parameter combinations (true values of the variance
components).
These variances and mean square errors are then compared
with corresponding variances obtained by a widely used estimating procedure based on the method of fitting constants.
The variances for
all three procedures were computed by using a computer program written
for the IBM/360 model 75.
The computer program uses a computational
method given by Bush [1962J to compute the variances for the procedure
based on the method of fitting constants.
In addition to the two-way classification with unequal numbers of
observations in the cells, the method of unweighted means is also
studied for estimating the variance components for a two-way classification with two levels of nesting within each of the cells (in this case
no missing cells were allowed).
Here the estimating procedure is
again based on computing mean squares and their expectations by the
method of unweighted means; but due to the two levels of nesting, the
estimating procedure is examined for two different estimates of the
cell means:
(a) the simple arithmetic means within cells and (b) the
unweighted means of class means within cells.
3
The format of this dissertation is the following:
a review of the
pertinent literature is given in Chapter 2; for two-way disporportionate
designs with one missing cell the (a) analytic formulas for the expected
mean squares using the method of unweighted means after having estimated
the missing cell and (b) the estimates of the variance components for
Procedures M and
~1
are given in Chapter 3: the results given in Chapter 3
are extended to designs with more than one missing cell in Chapter 4;
for a number of experimental designs and parameter values the computer
results for (a) the variances and mean square errors of the estimated
variance components using Procedures M,
Mand
the procedure based on
the method of fitting constants, (b) the numerical values of the expected
mean squares for the method of unweighted means and (c) the biases in
the estimated variance components obtained by Procedure M are presented
in Chapter 5; the method of unweighted means is examined for estimating
variance components for a two-way design with two levels of nesting within each cell in Chapter 6; a summary and suggestions for future research
are given in Chapter 7.
4
2.
REVIEW OF LITERATURE
There have been several articles written about the estimation of
variance components in a two-way classification with unequal numbers of
observations in the cells for the completely random model.
In most of
these articles the estimation procedure involves calculating mean squares,
computing the expectations of these mean squares under the variance
component model and then equating calculated mean squares to their
expected values and solving the set of equations for the variance components.
This procedure (usually referred to as the analysis of variance
procedure) gives unbiased estimators for the variance components under
the completely random model.
Using the analysis of variance procedure
Henderson [1953] describes and gives some examples of estimating the
variance components (a) by computing mean squares by the method of
fitting constants
(!.~.
mean squares for rows adjusted for columns
ignoring iteractions; mean squares for columns adjusted for rows ignoring
interac~ions;
mean squares for interactions adjusted for rows and columns)
and (b) by using unadjusted mean squares computed by the usual formulas
for the analysis of orthogonal data (these two methods are called Method
3 and Method 1, respectively). Searle [1956, 1958, 1961] studies
Henderson's Method 1 and develops matrix methods by which the variances
of the estimated variance components for this method can be obtained.
Searle found the variances of the estimated variance components by using
the fact that if
2
1
~)
= y'Gly
2 2 = y'G 2y
are functions of variables
having a multinormal distribution with means zero
and variance-covariance matrix
matrices, then
and
V and
G
l
and
and
G2
are square symmetric
5
Searle [1968] reviews and reformulates the methods given by Henderson
[1953J.
Searle's work on the variances of the Method 1 estimated
variance components is extended by Mahamunulu [1963] to the three-level
nested classification and by Blischke [1968] to the r-way crossed
classification.
Low [1964J gives algebraic expressions similar to
Searle's for the variances of the estimators obtained by Henderson's
Method 3 when there is no interaction in the two-way classification
model.
Harville [1969] extends this work by giving formulas for the
variances of Method 3 estimators for the two-way classification with
interaction and indicates how these formulas may be compared with those
of Searle for the variances of the Method 1 estimators.
Bush and Anderson [1963] compare three analysis of variance procedures for estimating variance components by developing an analytic
procedure for obtaining the variances of estimates of the variance
components and then numerically computing these variances for each
procedure for a number of experimental designs.
The three procedures
studied are a procedure based on the method of weighted squares of
means, Yates [1934]; and Henderson's Method 1 and Method 3.
Based on
their computations of the variances the authors reach the conclusion
that of the three estimating procedures studied the procedure based on
the method of fitting constants (Method 3) should be used.
Hirotsu
[1966] examines estimating the variance components by computing mean
squares by the method of unweighted means when there are no missing
cells in the design.
This method has the advantage that it requires
a minimum of computation.
Hirotsu gives analytic formulas for the
variances of his estimates and compares these variances for several
designs to those for the three procedures studied by Bush and
Anderson [1963].
He finds that his procedure compared quite favorably
6
with the other three procedures.
The estimating procedure used by
Hirotsu is examined in detail in Chapter 3 of this dissertation,
In many of the articles which examine the analysis of variance
procedure for estimating variance components for unbalanced designs the
authors find that the expectations of the computed mean squares and the
variances of the estimated variance components are difficult to obtain.
This is especially true of Henderson's Method 3.
Accordingly, suitable
numerical techniques which can be used on a computer for any unbalanced
design can be quite helpful.
Besides the work of Bush and Anderson [1963]
in this area, Hartley [1967] develops
a computer procedure called
'synthesis'which can be used to find the numerical values of the coefficients in the formulas for expected mean squares for any analysis of
variance.
Hartley's procedure can also be used to find numerical values
of the variance and covariances of the expected mean squares.
Similarly,
Gaylor et al. [1970] give a procedure which uses the forward solution of
the Abbreviated Doolittle Method to obtain the numerical values of the
coefficients in the expected mean squares for the method of fitting
constants.
In addition to the analysis of variance procedure, several other
methods have been examined for estimating variance components.
Koch
[1967, 1968] develops a procedure which does not require an analysis of
variance table as most other procedures do.
The method uses the fact
that the expected value of the squared difference of two observations is
a function of the variance components.
For example, for the usual two-
level nested design
=
i
j
= 1,
=
2,
1, 2,
... ,
. .. ,
a
n ..
1J
(2.1)
7
where
~
is a constant and
a o and
1
eo
c
~
are normally independently
distributed random variables with means zero and variances
2
0e ,respectively.
E
~Yij
a
2
a
and
Then
- Yk£)2J
2
e
=
20
=
2
2
2(0 + 0 )
a
e
if
= k,
i
j# £
(2.2)
if
~
i
k .
Koch forms sums of all the unbiased estimators given in (2.2) and
equates these sums to their expected values and solves the resulting
equations for the estimates of the variance components.
In his articles
Koch estimates variance components by the above method for nested andj
or crossed designs.
Koch's estimates are unbiased and consistent and
in the case of balanced designs his estimators coincide with those
obtained from the analysis of variance procedure.
Hartley and Rao [1967] develop an iterative maximum-likelihood
procedure for estimating the variance components from unbalanced
designs.
Their procedure which requires the use of a computer leads
to consistent and asymptotically efficient estimates and avoids the
occurrence of negative estimates of variance components.
In addition,
the authors also give small sample confidence regions for the parameters in their model.
Hartley and Rao point out that their technique
will permit Monte Carlo evaluations of small sample variances of
their estimators as was done by Bush and Anderson [1963].
Anderson
and Bancroft [1952] and Anderson [1961] also discuss the use and
underlying complexities of maximum likelihood or modified maximum
likelihood for estimating variance components.
Searle [1970] obtains
the large sample variance-covariance matrix for the maximum likelihood
estimators of variance components for unbalanced data from a general
8
mixed model.
Searle points out that even though the maximum likeli-
hood estimators themselves cannot be obtained the large sample variances
of these estimators are relatively easy to compute.
Thus, these large
sample variances may be compared with the variances from other methods
of estimation to obtain a measure of asymptotic efficiency of these
other methods.
Anderson [1961] gives an estimation procedure which lists certain
mean squares and their corresponding expectations and performs iterative
weighted least squares to estimate the variance components using prior
estimates of the variance components to determine the weights.
This
procedure uses the computed mean squares as the dependent variables
and the variance components as regression coefficients.
The method of unweighted means given by Yates [1934] has been
studied by many authors in addition to Hirotsu [1966]; for example,
Anderson and Bancroft [1952], Goss1ee [1956], Goss1ee and Lucas [1965],
Webster [1968] and Hirotsu [1968].
However, these publications do
not deal primarily with the estimation of variance components.
Instead,
most of these authors study the method of unweighted means from the
standpoint of testing hypothesis.
The method of unweighted means is appealing because of its
computational simplicity, but it has been limited to designs in which
observations exist in all cells.
In this dissertation the method of
unweighted means is extended to designs which contain empty cells.
After supplying estimates of the means for the missing cells, the method
of unweighted means is applied to provide estimates of the variance
components.
Properties of these estimates are studied.
9
3.
ANALYTICAL RESULTS FOR ONE MISSING CELL
3.1
Introduction
The purpose of this chapter is to examine the method of unweighted
means with regard to estimating the variance components when there is
one cell with no observations in a two-way classification with unequal
numbers of observations in the cells.
The mathematical model that will
be used is the variance components Model II given by Eisenhart [1947J.
It can be written as
y.~J'k
~
==
+ r. + c. + rc .. + e
J
~
i
k
ijk
~J
(3.1)
l,2, ... ,aj j
1,2, ... ,n ..
1,2, ... ,b;
~J
. ,~n t h e ~. th row
. teo
h
b serve d va 1 ue f or t
he
k th
were
y. 'k ~s
o b servat~on
h
~J
and the jth column.
e
ijk
~
In the model,
is a constant; r., c., rc .. and
~
J
q
are normally independently distributed random variables with means
.
zero an d
var~ances
~
2
r
~C2, ~2rc and ~2e respectively; ~.~.
,
2
e ijk - NID (0, ~e)'
Totals and means are denoted by dropping subscripts)
I: Y. 'k
k ~J
I:I:Y" k
j k q
Yij'
I: I: I: Y" k
. . k ~J
Y
,
I: I: n ..
i j ~J
==
1
-I:I:Y" k
n .. k
~J
J
~
n.
I: n ..
q
n
,
1
i
y ..
y. 'k
n .. I:
k q
~J
and
I: n ..
~
j
J
~
I:I:Y" k
i k
~J
~J
1
n .. k
-I:I:I:Y" k
~
J
~J
n.
~J
J
1
-I:I:Y" k
n . . k ~J
~
J
Y
~.~.,
10
A missing cell implies that one of the n ..
0) i.e. cell (ij) has no
~J
observations in it.
In the next section the method of unweighted means and the work
of Hirotsu [1966J will be reviewed for the case of all n., > O.
~J
3.2
The Method of Unweighted Means
The method of unweighted means for the two-way classification is
simply to compute the mean for each of the ab cells of the design and
then to run the usual analysis of variance on these means as if they
were the original observations.
1
for all i and j
l: y, 'k
n .. k
~J
That is) compute
~J
(3.2)
and compute the usual sum of squares for rows) columns and interaction
using the
y...
~J
Using the notation given by Hirotsu [1966J this implies
that the mean square for rows is
MSR
=
l: l:
i j
(Y i
y)2/(a - 1)
-
v'
L
(3.3)
G Y
R-
where X' is a vector of cell means = (yU) Y12)
...
)
... -Ylb) ...
)
)
Yal)
Yab) and
GR
Here {p/Q}
{p/Q}a)
a
P
=
J:...J
ab b·b
Q =
-1
ab(a-l) J b . b
denotes a matrix whose diagonal elements are all equal to P
and whose off diagonal elements are all equal to Q.
Note P and Q are
b x b matrices; and therefore) G is a ab x ab matrix.
R
b x b matrix whose every element is one.
columns is
J . denotes a
b b
Likewise) the mean square for
11
MSC
= ~ ~ (~.
i
j
_ ~)2 /
(b _ 1)
J
where
1
1
a (b-l) [I b - -b J b · b J
and the mean square for interaction is
~. +~)2
=
MSRC
J
(a-I) (b-l)
/
(3.4)
where
=
assuming no n ..
~J
= O.
The mean square for error is the usual one,
namely
MSE
=
-
~ ~ ~ (Y ' - y ,)
iJ
iJ k
i j k
2
/ (n-ab) .
Now Xl has a multivariate normal distribution with variance-covariance
matrix V which can be written as
(D/C}
V
where
a
+ diag
(o-~
)
n
(3.5)
ij
and
D
C
=
Thus, the results of Lancaster [1954] and Searle [1956] can be used
which state that if Z
= Xl
G X is a function of variables
X,
having a
\
,
\
12
mu1tinorma1 distribution with variance-covariance matrix V, then
E(Z)
=
1
(3.6.1)
tr (V· G)
and
Var(Z)
= 2 tr (V. G) 2 .
(3.6.2)
Using this Hirotsu showed for the case of all n .. > 0
q
tr(V· G )
R
=....!..
ab
_1_
. . n..
L: L:
~
J
~J
cle
+
i rc
+ bcr2
r
E(MSC) = E(X' GcX)
and
tr(V· G ) - 1 L: L: _1_ cr2 + cr2
rc
RC - ab . . n.. e
~
J
(3. 7)
~J
With these results and the fact that
E(MSE)
cr
2
e
it is then possible to estimate the variance components cr;,
cr
2
e
cr~, cr;c and
by setting the expected mean squares in (3. 7) equal to the mean
1
To be more precise
E(Z) = tr(V· G) +
Var(Z)
~I
= 2tr(V.G)2
G~
and
+ 4~' G V G ~
where~'
is a vector of ~'s. However as pointed out by Searle [1970J
analysis of variance mean squares are such that row sums of G are zero;
i.e., G 1 = 0 where 1 is a vector of l's. Therefore, G~ = G} ~ = 0
whIch implies
tr(V. G)
+~'
G~
=
tr(V· G)
2tr(V.G)2 + 4~' G V G ~
and
= 2tr(v.G)2
J
13
squares in the analysis of variance table (this method of estimation is
referred to as the analysis of variance procedure).
The results are
(3.8)
,,2
0-rc ==
.l... L: L: _1_ (MSE)
Xl G
RC X - ab . . n ..
1. J
1.J
,,2
0e
MSE
==
Hirotsu then showed by using (3.6.2) and (8) that
Var(o-"2 )
r
==
2tr (V .
==
2
2
b
b1
[ :-21
+ 20-
2
e
(
(G
R
- G ))
RC
2
4
b 0rc
4 +~ 2 2 +
00- 0a-1 r rc
r
(a-1) (b-1)
2
2
0- + 0-rc )
b-1
r
(3.9)
H1
a(a-1)
(H3 _ H2) + (H5 + H2 - H3 - H4)
(a_1)2
)]
where
H1
L: L: _1_
i j n ij
,
H2 == L: L: _1_
. . 2
1. J n ..
1.J
and
H4 ==
,,2
Var(o- )
c
==
,,2
r
Var(o-)
H3 == L: L: L: _1_ _1_
i j t n ij nit
,
H5 == L: L: L: L:
1
i j k t nijn kt
with a and b interchanged, 0and H4 and H3 interchanged.
2
r
replaced by 0-
2
c
14
,,2
Var(o-
rc
1 L: L: - 1 )2 Var(o-)
"2
2tr(V· G ) 2 T. ( -b
RC
a
. . n..
e
)
J
~
~J
2 [40-rc + ab2 (H1)
(a-1) (b-1)
0-
2 2
0rc e
4 (ab(a-2) (b-2)H2 + a(a-2)H3 + b(b-2)H4
+ o-e
2 2
a b (a-1) (b-1)
4
20e
=
3.3
n-ab
Expected Mean Squares with One Missing Cell
In order to extend the results of Hirotsu which were given above
to the case when one n
ij
=
0 (denote the missing cell by (i j 1»
l
necessary to estimate an observation for the missing cell.
it is
This is due
to the fact that the expected mean squares for the method of unweighted
means, see (3.7), include the term L: L: lin ...
..
~
J
~J
The formula that is used
to estimate this missing cell is the same one given by
Ya~es
[1933J for
the least squares estimate, X, of the missing value for a randomized
complete block (R.C.B.) design, namely
(a R.
X
~l
+ b C.
J1
- G)/(a-1) (b-l)
(3. 10)
where R.
is the total of the observations in row i , C. is the total
1
J1
of the observations in column jl and G is the grand total of the exist~l
ing observations.
This formula is used because of the fact that the
method of unweighted means takes a two-way disproportionate design and
converts it to a R.C.B. design by analyzing cell means.
In terms of
15
the model given in (3.1), (3.10)becomes
a
y- . . + b
~1 J
x
(3.11)
(a-1) (b-1)
Equation (3.11) can be derived directly by minimizing the least squares
equation from (3.1) for the sum of squares for interaction after having
replaced the observations in each cell by their cell means (see (3.2)).
That is, the formula given for the missing cell in (3.11) is that value
which minimizes the sum of squares for interaction for the method of
unweighted means with respect to X.
Section 4.2 below shows this
minimization procedure in detail for the case of two missing cells.
By replacing
y. j
~1
=
with
IJ. + r.
~1
+ c
etc.
j
(3.12)
in (3.11) we can then obtain X in terms of the model parameters,
rc ..
I:
ifi
+
X
~J1
1
(a-1)
--:--~-
I:
rc ..
I:
ifi
!.~.,
jfj 1
~J
1
(a-1) (b-1)
(3.13)
I: e .. k
k ~J1
n ..
~J1
(a-1)
I: e. 'k
k ~J
n .. (a-1) (b-1)
~J
Having estimated a value for the missing cell (i j 1) it is then
1
possible based upon the method of unweighted means to estimate the
variance components
~;, ~~, ~;c and ~~ using the same procedure
16
employed by Hirotsu.
That is, equation (3.6.1) is used to compute the
expected mean squares for rows, columns, and interaction.
Then the
estimates of the variance components are obtained by equating the expected mean squares to the computed mean squares in the analysis of
variance table.
However, this procedure is complicated somewhat by the
fact that the vector of cell means Z' (assuming for example that cell
(1, 1) is missing) now is equal to (X, Y , .. "
12
Yal ,
)lIb' .... ,
... ,
Thus, the variance-covariance matrix, call it V , of Xl no
l
Yab)'
longer has the simple form shown in (3.5) but will have in its first
row and column the variance and covariances of X.
That is,
-
-
Var(X)
r,
Cov(X, Y12)
-
vov(X'Y12)
IJ
2
2 + IJ2
+ IJ rc
c
r
cov(X,Y
13
)
2
IJ
r
.,.J." "
Cov(X'Yab)
"Q" 0"
0
.'
0
2
+ IJ/n 12
Cov(X, Y13)
IJ2
r
IJ
2
2
2
+ IJ + IJ
rc
c
r
..
<>
••
2
+ IJ/n 13
.
.
.
r,
~ov(X'Yab)
0
0
.
.....
0'>
..
IJ
2
2
2
+ IJ + IJ
r
rc
c
2
+ IJe In a b
>-
-
(3.14.1)
17
V
ll
V
12
V
13
V
14
Vla
Vi2
V
22
V
23
V
23
V
23
Vi3
V
23
V
33
V
23
V
23
V
14
V
23
V
23
V
44
V
23
=
VI
V
aa
la
where V ) V '"
ll
22
V ; V ) V ... V ; and V are all b x b matrices
la
23
aa
12
13
which are defined as
Var(X)
cov(X)Y12)
Cov(X'Y 12
(J2
r
.0'
(J2
+
(J2
+
(J;/n 12
c
+
...
rc
Cov (X) y1b)
(J2
r
(3.14.2)
.
.
Cov(X'Ylb)
(J2
...
r
(J 2
r
+
(Jc2
+
(J2rc
2
+ (J/n lb
= {i
r
+
ic
+
iii}
rc r b
+ diag
((J;
)
n
2j
etc. for V ,
33
.0.)
V
aa
(Recall the notation
)
b
was defined in (3.3) above.)
18
COv(X'Y21)
Cov(X,y
22
...
)
...
(J2
c
0
Cov(X,y
2b
0
.
.
.
etc. for V13'
V14 '
... , V1a
(J2
c
...
0
0
)
(3.14.3)
and
It is easy to show using (3. 13) that
Var(X) =
ir
+ (J2 +
(a+b-1) (J2 + (J2 [
1
c
(a-1) (b-1)
rc
e
(a_1)2
1
(3. 15)
Cov(X,Y .. )
1.J 1
Cov(X,y. .)
1. J
1
=
(J2
(J2
e
(J2 + rc +
a-1
(a-1) n ..
c
1.J 1
=
(J2
(J 2
rc
e
2
(J
r + b-1 + (b-1) n. .
1. 1 J
and
Cov(X,Y .. )
1.J
=
-(J2
rc
(a-1) (b-1)
(J 2
e
(a-1) (b-1)n ..
1.J
-
+
19
To illustrate the above notation, suppose that a 3 x 3 design with
cell (1, 1) missing is to be used to estimate the variance components,
2:.;:. ,
the n." arrangement is given by
~J
n
X
n
n
21
31
n
n
n
12
n
22
n
32
13
23
33
where an X indicates that the cell has no observations.
Here, using
(3.13) we have
L:
X
= IJ. + r 1 + c 1 +
itl
-
rC~l
2
...
L: rc "
lJ
+ jf1
2
rc ..
~J
4
(3.16)
L: e. Ok
k
~J
4n ..
~J
and the vector of cell means is
Then using (3.14.1) and (3.15) the variance-covariance matrix of Xl
is given by
v1
=
2
+ -5
r+c
4
CT
2
[lJ
e
+
CT
2
rc
CT
CT
2
r
2
12
+-
2
0"
r
CT
2
+ 0"12
2
2
CT
13
+2
2
O"r+c
CT
CT
CT
2
r
2
r
2
r
2
CTr
+c
+
CT
CT
2
Sym.
CT
2
CT
2
22
-4
CT
2
23
-4
2l
+2
2
c
0
CT
0
0
0
CT
CT
2
c
2
CT
31
+2
CT
2
32
-4
0
CT
0
0
2
c
2
0"33
-4
0
CT
CT
2
13
+-
2
c
CT
2
12
+
2
2
r
2
c
2
c
O'
2
13
(Djc)Z + diag (
::j)
N
o
e
e
e
21
where
D
{0'"2 + 0'"2 + 2 / 2}
0'"rc 0'"r 3
r
c
0'"2
2
2 +~
0'" .. = 0'"rc
n ..
~J
~J
and
C = 0'"
2
2
2
0'"r+c = 0'"r + 0'"c
2
1
c 3
as in (3.5»)
and
1
1 +_1_
1
1
L: _1_ +
L:
L:
[ IJ = - - L: n
..
(2) 2 jfl n lj
(2) 2 ifl nil
(2)2(2)Z ifl jfl ~J
Having set up VI) the expected mean square for rows in the
general a x b design (assuming cell (1) 1) is the missing cell) is
calculated as follows:
E(MSR)
= tr(v( GR)
Vll
V12
V
l3
VIa
p
Q
Q
Q
Viz
V
Z2
V
Z3
V
Z3
Q
P
Q
Q
= tr Vi3
Vi3
V
33
V
23
Q
Q
p
Q
V
Q
Q
Q
p
aa
+ tr[(2(V 12 + V13 + ... VIa) + (a-I) (a-Z) V ) . QJ
Z3
(3. 17)
22
where P and Q were defined in (303).
~
2
a O"r+e
+ 0"
+ire
[ 1J
a 0"
2
r
a 0"
2
(J12
2
a (J +
(b-1)
r
a 0"
2
r
0
2
r+e
a 0"
a 0"
-
2
r
2
(J1b
+ (b-1)
2
0"13
+
(b-1)
2
0"12
+
(b-1)
2
[2J
e
Now from (3.14.2) and (3,14.3)
2
r
0
.
0
a (J2
r
2
+ a 0"
re
+ (J2 L: _1_
e i n i2
2
(J13
2
a 0" + - (b-1)
r
a (J2
r+e
+ a (J2
re
2 L: _1_
+ (J
e i nU
2
a 0"
r
0
"
0
a (J 2
r
..
2
O"lb
2
a (J +
(b-1)
r
where
a (J2
r
a (J
a + b - 1
[ 1J = (a-1) (b-1) + (a-1)
[2J
2
r
and
(3.18)
,
a 2 _ 2a + 2
1
L: _1_+
2
ir 1 nil (b-1) 2
(a-1)
(J2
2
2 +~
(J .. = 0"
re
n ..
1J
1J
a ao2
r+e
2
+ a (J
re
+ (J2 L: _1_
e i nib
2
O"r+e
L:
jr
1
1
-+
L:
2
2
1 n 1j (a-1) (b-1) ir 1
ao 2 + (J 2
r
e
jr 1 n ij ,
L: _1_
23
,---,
~I;
h1~
•..-1
,--..
aJ
..0
N
b
~
I
'-'
. .
,--..
+
~
0
I
U
Nbl-l
,--..
ctl
N
U
b
,--..
~
I
ctl
'-'
'-'
~
I
ctl
'-'
~
I
.
.
.
..
,---,
I
~ cic;j
h1~
•..-1
,--..
~
I
N
..0
NbaJ
'-'
+
U
Nbl-l
b
U
,--..
,--..
~
~
I
I
.
0
. .
0
ctl
'-'
ctl
'-'
,--..
~
I
ctl
'-'
'----'
I
N
~I ci~
U
I-l
~
b
h1'ik..
+
+
NbU
,--..
~
I
ctl
'-'
'..-1
Nb
aJl~ctl
I
'-'
+
0
24
tr [ (V 11 + V22 +
=
J:..
ab
[ab
2
ir
0
0
0
+ V )
aa
2
+ ab cr- +
c
0
irc
+ cr-;([2] + L L
i jfl
p]
I
a+b-l
+ a-I +
(a-I) (b-l)
1 + _2_ L _1_)
n ij
b-l jfl n lj
tr [ 2 (V12 + V13 + .•. + Vta) • Q
2
2
- -a=-b-7(a2;:"'_-=1""-) [(a-I) bcr- + crc
e
J
2~~~1)
,
J
1-(
11) L _1_
aif1 nil
and
tr [(a-I) (a-2) V23 Q]
o
=
- (a- 1) (a- 2) b cr-2
ab(a-l)
c
Accordingly,
I
a+b-l
)
(a-I) (b-l) + ab + 1
+
1
2
2 L.
L _1_
(a-I) (b-l) if1 jfl n ij
1
n ..
1.J
+ a(b-l»)
25
+
2
_1_)
L:
(a-1) 2 (b-1) if1 jf1 n ij
= bo-2 +
r
0-2
L:
(1 +
rc
( + L :1- 1
if1 nil
1
(a-1) (b-1)
(3.19)
)
1 ) + L :1(a_1)2
jf1 n 1j
2b-1) ]
( 1 + (b-1) 2
Similarly, the expected mean square for columns is
= tr[(V 11 + V22 +
+ tr[(2(V
12
+ V + ... + V ) + (a-1) (a-2)V )·R]
1a
23
13
where R was defined in (3.4).
=
+ V ). R]
aa
Here
2
1
[ 0-r ( a (b-1) - a (b-1)
a(b-1)
2 ( a+b-1
re (a-1)b
+
0-
+
0-
+
(a-1) (b-1)
b
I
+ a (b-1)
0-
2
c
2 (b-1) + a (b~ 1) 2 )
b(b-1)
1
2 ( (b-1) [2] + b-1 L L
e b b i jf1 n q..
26
1
= a (b-1)
2
[ (a-1) (b-1) 0c2 + 02 ( b-b +-b1)
rc
and
tr[(a-1) (a-2) V 'R]
23
=
(a-1) (a-2) (J2
a
c
Thus, the expected mean square for columns can be shown to be
E(MSC)
(
+
1 +
02 [
1
+ ~
2:
2:
(a-1) (b-1) )
ab if 1 ]fl n.~J
1
>
2a-1
)
2
2
(a-1) (b-l)
(1 +
2a-,1
2
(a-1)
I
(3.20)
Finally, the expected mean square for interaction is
where
(a-1) (b-1)
GRC1
= (a-1) (b-1)-1 GRC .
G
must be used in the present
RCI
case instead of G
(defined in (3.4»
RC
because the missing cell takes
one degree of freedom away from interaction.
Proceeding in the same
manner as for the expected mean squares of rows and columns it can be
shown that
27
1
E(MSRC)
1
2
2
(a-I) (b-l)
n ..
~J
)
(3.21)
+
L:
- 1
if1 nil
I
1-
1 2)+
(a-1)
1 (1
L:
1
(b_
l)
-
jf1 n 1j
2) ] rr~ + rr~c
Table 3.1 gives a summary of the preceding work on expected mean
squares for the case of one missing cell.
Examination of Table 3.1
shows that as a and b become large
K1
,.
,.
,,1 [
=
K3 = K5 = -ab
and K2 = K4 = 1,
i.~.
L:
+ L: + L:
2
3
1
]
1
ab
1
= --
n ..
~J
the coefficients of the expected mean squares
approach those shown in (3.7) above for the case of no missing cells.
Also, if the number of observations in each cell of the design are
equal (i.e. all n ..
- -
L:
=
~J
(a-1) (b-1)
1
n
= n) then
L:
= (a-I)
2
n
and
L:
3
=b-1
n
In this case the expected mean squares in Table 3.1 reduce to
E(MSR)
=
E(MSC)
=
E(MSRC)
E(MSE)
~[
1
1 + (a-1) (b-1)
~[ 1
1
n
= cr2e
~
2
e
1
+ (a-1) (b-1)
]~;
1
+[ 1 + (a-1) (b-1)
]rr; +[ +
1
+
1
(a-1) (b-1)
]~;c
+bi
r
]~;c
+
~
2
rc
a~
2
c
28
Table 3.1
Expected mean squares for a two-way disproportionate design
with one missing cell using the method of unweighted means
Coefficients of Variance Components in E(MS)
2
2
2
cr
cr
cr
rc
c
e
Source
=
K1
1
ab [ (1 +
Rows
(1 - (a_II) 2 ) .2: 2
+
(1
1 +
_.
b
1
(a,- 1) (b- 1)
+ 2b-1 ) 2: ]
(b-1) 2 . 3
=
1
+
ab [ (1
Interaction
2b-1
) .2:
1
2
2
(a-I) (b--1)
+
K3
Columns
K2
K4
2a-1
) .2:
1
2
2
(a-I) (b-1)
+
(1 +
+
(1 - (b_ 1) 2)' "3 ]
1 +
=
a
1
(a-I) (b-1)
2a-1 ).2:
(a_1)2
2
1
K5 =
C
1
.2:
ab [ (1 - (a-I) 2\b_1) 2 )
1
+
(1 - (a_II) 2 ) .2:2
+
(1 - (b_ l) 2 ) . "3 ]
1
1
Error
1
where the missing cell is (i j 1) and
1
_1_
n
i r i 1 jd 1 ij
2: = 2:
2:
.L
.L
1
J
2: = 2:
2.L
_1_
n
iri1 ij1
J
1 and C =
(a-1) (b-1)
1 (a-I) (b-1) - 1
jrj1 i 1 j
2: = 2:
3
.L
n
cr
2
r
29
As mentioned previously) the results given in Table 3.1 can be
used to estimate the variance components by setting the expected mean
squares equal to the computed mean squares.
"2
r
CT
This gives
1
=1)" [MSR - K2·MSRC + (K2·K5 - Kl) MSE]
1
K2· X' GRC1X]
=1)" [X' GRX
1
=1)" [X'
(G
R
. MSE
+ (K2·K5 - Kl)
b
) X] +
- K2.G
RC1
(K2· K5 - K1)
b
.
MSE
(3.22)
"2
c
1
(K4.K5 - K3)
a [X' (GC - K4· GRC1 ) 1] +
a
. MSE
CT
"2
CT
e
=
MSE
In the case of all n ..
~J
MSR -
=n
the equations in (3.22) will reduce to
[1 + (a-l)\b-l) ] MSRC
b
MSC -
[1 + (a-l)\b-l) ] MSRC
a
MSRC _ MSE
n
"2
e
CT
=
and
MSE
As a notational convenience) the estimating procedure used to obtain
the estimates in (3.22) will be called Procedure M.
The estimates of the variance components given in (3.22) have the
property that they are unbiased estimates.
Another set of estimates
30
for the variance components which are biased but have relatively simple
computational formulas can be obtained by essentially ignoring the
missing cell and using the following approximate expected mean squares:
"..-........
E(MSR)
---..
E(MSC)
=
----
=
E(MSE)
=
E(MSRC)
(~
=
I~
(~
L: L:
i j
f(i 1 j 1)
L: L:
i j
f(i 1 j 1)
L: L:
i j
f(i 1 j 1)
n~. ) 0-e + 0-rc
2
2
~J
~ o-~
n .)
+
~J
+ be,z
r
2
2
+ a0rc
c
a-
(3.23)
~
2
n .) 0-; + 0-rc
~J
2
0e
where s is the number of occupied cells in the design.
These approxi-
mate expected mean squares are the same as those given in (3. 7) for
the case of no missing cells except that in the present case (harmonic
mean)
-1
,
..!.
I: I: _l_
ab . . n .. '
~
J
is replaced by
~J
1
s
1
I: I:
n ..
i j
~J
j
r(i 1 1)
(denoted by
K).
The term '"K is used because i t gives a better approximation to K5, the
correct coefficient for 0-2 in E(MSRC) , than does (harmonic mean) -1 (see
e
Chapter 5) j and therefore, reduces the bias in the estimate of 0-2 (see
rc
(3.25) below).
mean)
-1
That
K is
a better approximation to K5 than (harmonic
is easy to see in the case of all n .. = n where
~J
31
1
1
1 [ -+
1
-,L:,L:--:=ab . . n..
ab 1
~
J
'"
:=
n. .
~J
1 + J:..
n
ab
K
_1
1
s
~J
]:=.1-[
1: + ~bc-l ]
ab 1
n
(1 ~)
ab~ 11
1
n ..
~J
and
ab;
1) :=
1
n
KS
Setting the approximate expected mean squares in (3.23) equal to
the computed mean squares gives the following biased estimates of the
variance components (the estimating procedure used to obtain these
A
estimates will be called Procedure M):
~2
MSR - MSRC
b
~2
MSC - MSRC
a
cr
r
cr
c
(3.24)
~2
cr
rc
~2
cr
e
:=
MSRC
-I ~
L: L:
i j
1(i1 j 1)
n~ . ) MSE
and
~J
MSE
'" estimates given in (3.24) are found
The biases for the Procedure M
by taking expectations and using the results given in Table 3.1,
(J
2
rc
~'~'J
32
E(;2) - 0-2
c
c
K~-l
;::: (
E(~2 ) - 0-2
rc
re
2
2
2
K5·0-e _ 0-rc ] - 0-c
;::: -l[ K3· 0-2 + K4. 0-2 + ao- 2
rc
a
e
c
I
2
0(
rc +
2
2
K5·0- + 0rc
e
;:::
(KS -
K)
0-
K3~K51
0-
2
K·o- e
2
0rc
A
(3.25)
2
e
and
2
e
In general it does not appear possible to state that the biases
given in (3.25) are always positive or negative.
However, if
2
2
~2
~2
0»0- then the biases in 0- and 0- will be positive since
rc
e r e
(K2-1)/b and (K4-1)/a are always greater than zero.
~2
is usually very close to K5 the bias in 0-
rc
Also, since K
will be quite small.
In
Chapter 5 a detailed numerical study is made of the biases in (3.25)
for several different designs.
3.4
Variances of Estimated Variance Components with One Missing
Cell
The variances of the estimated variance components for Procedure M
given in (3.22) can be computed by using
equation~(3.6.2).
This
computation is simplified by the fact that the error sum of squares is
independent of the sum of squares for rows, columns and interaction.
That is, the within cells sum of squares is independent of all between
cells sum of squares.
Also, the sum of squares for error divided by
2
2
2
0- (i.e. SSE/0- ) has a X distribution with N-s degrees of freedom
e
- -
e
(where again s is equal to the number of occupied cells in the design).
Thus, the variance of the error mean square is 20-4 /(rt_s).
e
Accordingly,
33
,,2
Var(cr )
r
(K2,KS - Kl) 2
2
2 tr [ VI' (G R - K2' GRCl ) ] 2 +
2
b
b
=
,,2
Var(cr )
c
1J<4, KS - K3)
2
= tr[ Vl,(G - K4'G
)] 2 +
2
RCl
C
a2
a
,,2
Var(cr )
rc
=
2tr
2
2cr4
e
i:l-s
2cr4
e
n-s
[ Vl,GRCl ]2 + (KS) 2• n-=
2;
(3.26)
In attempting to reduce the equations given in (3.26) to an
analytic form such as that obtained by Hirotsu (see (309»
the variance
of ~2 was looked at initially since it had the easiest computational
rc
formula.
However, the matrix algebra involved in reducing this equa-
tion was quite tedious and only the result of the computation will be
given here, namely:
,,2
Var(lY )
rc
Ci 12cr;c [ (b-l)Tl + (a-l)T2 + (a-I) (b-l)T3]
+ 4cr;ccr; [ Tl. L:
3 + T2· L:. 2 + T3· L: l ]
(3027)
+ 2cr: [ Tl· L:
+ (KS)
2
+ T2 L: + T3,L: ] )
4
S
0
6
2cr4
~
n-s
where
=
(b-2)
a (a-I) (b-l)
3 '
T2
=
(a-2)
b(b-l) (a-I) 3 '
and
34
T3
=
(b-2)
+ _ _-:(>-a_-.;::.2)<-------o223
b 2 (a_l)3(b_l)2
a (a-l) (b-l)
21
2) 2 +
a-2
2
(a-l) (b-l)
(a-l)
+
+
~l)
~4
(a-2) (b-2)
(a_l)2(b_l)2
~3
1)
2
2
(a-l) (b-l)
(1+ (a-l) ~ (b-l) )2
(b-2)
(b-l) 2
~2}
I
1 +
I
1
1- (a-l) (b-l)
are defined in Table 3.1; and
1
2
=
~5
n ..
and
=
~J
~6
=
~
1
-2-
=
and
jfjl n . .
(a-l) (b-l)
(a-l) (b-l) - 1 .
~l J
Unfortunately} the matrix algebra that would have been involved in
~2
~2
r
c
solving analytically for Var(cr) and Var(cr) in (3.26) was enormous
and it was not felt that it was worthwhile to carry it out in view of
~2
the complex formula obtained in (3.27) for Var(cr ).
rc
Instead} it was
decided to write a computer program that would solve the equations
given in (3.26) numerically for the desired variances given a
particular design
(i.~.
a particular n .. pattern).
~J
Using this computer
program it would then be possible to calculate the variances of estimated variance components obtained by Procedure M for any desired
design with one missing cell.
In addition, the computer program could
also be used to obtain the variances of estimated variance components
35
for designs which had more than one missing cell (see Chapter 4).
Chapter 5 below describes the computer program (called VC) that was
written to estimate these variances and the numerical results
obtained by running VC with several different design patterns.
In addition to deriving the variance equations in (3.26»)
equation (3.6.2) can also be used to obtain the variances of the
A
estimated variance components for Procedure M gi.ven in (3.24).
The
result of this computation is
~2
Var(o- )
r
~2
and
Var(o- )
c
(3.28)
~2
Var(o- )
rc
~2
Here the Var(o- ) can be reduced to the analytic form given for
rc
A2
Var(o- ) in (3.27) with K5 replaced with K. As was the case in
rc
A
~2
~2
r
c
(3.26) theVar(o-) and Var(o- ) were not reduced due to the matrix
algebra involved in these calculations.
Using (3.25») (3.26) and (3.28) the differences between the
variances of the unbiased estimates for Procedure M given in (3.22) and
the mean square errors
2
of the biased estimates for Procedure M in
2
A2
The mean square error (M.S. E.) of an estimator 0- of the parameter
2
0- is defined as
A2
M.S.E.(o-)
:::::
A2
2 A2
Var(o-) + bias (0-)
36
(3.24) can be shown to be
"2
~2
r
r
3
Var(o-) - M.S.E. (0-)
=
2
b12 [ (K2 -1) Va r (MSRC) + (K2' K5- K1) 2 Var (MSE)
4
- 2(K2-1) Cov(MSR, MSRC) - (K2_1)2 0rc
- (K1-K5) 2 0-4 - 2 (K2-1) (K1-K5)
e
0-2 0-2]
rc e
(3.29)
(K2-1) 2
4
- 2 (K2-1) (K1-K5)
rc
0-
2
re
0-
0-
2
e
3rt should be noted here that in general
However, as for the case of E(Z)
G~ = GRC1~ = 0 which implies
= E(X'GX)
A similar result holds for Cov(MSC, MSRC) .
given in (3.6.1)
37
~2
~2
Var(a- ) - M, S, E, (a )
c
c'
2
a12 [ (K4 _l) Var(MSRC) + (K4,K5-K3)2 Var(MSE)
- 2(K4-l) Cov(MSC) MSRC) - (K4-l)
2
a
4
rc
_ (K4-l) 2 a-4 _ 2 (K4-l) (K3-K5) a-2 a- 2
rc
rc e
4
~2
Var(a- )
rc
~2
M.S,E. (a-
rc
)
=
2 2 2a2 4
(K5 -K ) ~ - (K5-K) an-s
e
Again) the matrix algebra involved in reducing the first two of
these equations was not considered worthwhile,
Instead) the computer
program VC was modified to solve the equations in (3.28) numerically
for a particular design,
It was then possible to compare the mean
A
square errors of the estimates for Procedure M with the variances of
the estimates for Procedure M for a given design.
The results of this
comparison for several different designs are also discussed in
Chapter 5.
38
4.
ANALYTICAL RESULTS FOR MORE THAN ONE MISSING CELL
4.1
Introduction
In this chapter the results of Chapter 3 for computing expected
mean squares and estimating variance components based upon the method
of unweighted means will be extended to the case of two or more cells
missing from a two-way disproportionate design.
The method of
analysis is essentially the same as that used in Chapter 3 but the
results obtained are more complex especially those for missing cells
in different rows and columns.
No attempt is made in this chapter to
determine analytic formulas for the variances of the estimated variance
components due to their complexity; instead, Chapter 5 will examine
these variances numerically.
Initially, in Section 4.2 designs with
missing cells in the same row (or column) will be studied.
Then in
Section 4.3 designs with missing cells in different rows and columns
are examined.
4.2
Two or More Missing Cells in the Same Row (or Column)
First, it will be assumed that we have a design with two missing
cells in the same row.
(i j 2)'
l
Denote the two missing cells by (iljl) and
As discussed in Chapter 3 a missing value formula must first
be used to estimate an observation for each of these missing cells
before the method of unweighted means can be applied.
Burrows [1966J
gives formulas for obtaining one or more missing values.
The formulas
for missing values can be derived by minimizing the sum of squares for
interaction with respect to the missing values.
That is, assume the
usual model for a randomized complete block design,
39
Y;J'
=
~ +
.L
i
=
t. + b. + tb ..
~
J
~J
1) 2) ...
J
a;
j
=
1) 2) , .•
J
b
with the usual definitions for 1I.)
t.,
b.J and tb ~J
...
,..
~.
Let two cells in
the same row be missing from this design and denote the missing values
Then the sum of squares for
for these cells by X. .
~l J 1
interaction is given by
(4. 1)
55 (TB)
where
T.
B.
=
T.
=
J
L:
y ..
i
f
i
L:
y ..
j
f
jp j 2
.
J
~
i
~J
~J
~l
=
=
and
G
=
L:
L:
i
j
f(i j 1)
l
(i l j 2)
l
40
Taking the partial derivatives of (4.1) with respect to X. .
1
1J 1
and X. . gives
1 1J 2
(4.2)
and
=
(4.3)
+
Note, that the second derivatives of (4.1) with respect to X. .
1
1J 1
and X. . are positive. Then setting (4.2) and (4.3) equal to zero
1 J
1 2
and solving for X. . and X. . we obtain the formulas for the missing
1 J
1 J
1 2
1 1
values given by Burrows for minimizing SSTB
X. .
1
1J 1
=
a T.1 0 + (b-1) B. 0 + B. 0 - G0
J2
J1
1
(a-1) (b- 2)
(4.4)
and
X••
1
1J 2
=
X••
1
1J 1
+
B. 0
J2
- B.J10
a-1
Now (4.4) can also be used to estimate two missing values in the
same row for a two-way disproportionate design after having replaced
41
the observations in each cell by the cell mean (equation (3.2)).
This
gives
a
. .1.
L:
y..
.
JrJl,J2
~lJ
+ (b- 1) L:
. .1.
y..
~r~l
~Jl
+ L:
. .1.
~r~l
y.. q2
(a-l) (b- 2)
and
(4.5)
Thus, (4.5) gives those values that minimize the sum of squares for
interaction, using the method of unweighted means, with respect to
X. .
~lJl
and X. .
~lJ2
for a two-way disproportionate design.
By substituting (3.12) into (4.5) we obtain X. .
~lJl
terms of the model parameters,
ili l
X. .
~lJ2
L:
jh 1j 2
~J
in
L: e. .k
k ~l J
n . . (b-2)
~lJ
L: e .. k
k ~J
n ..
~lJ2
i.~.
L:
L:
rc ..
ili l jh j 2 ~J
l
(a-l) (b-2)
L:
and X. .
(a-l) (b-2)
(4.6)
has the same formula with jl replaced by j2.
With the above estimates for the two missing cells Procedures M
"-
and M can again be used to estimate the variance components as in
42
Chapter 3.
First, for Procedure M the expected mean squares for rows)
columns and interaction are derived by using (3,6.1) which implies that
the variance-covariance matrix of Xi (the vector of cell means inc1uding the missing cells) must be computed.
The elements of this matrix
are the same as those given in (3.14,1) except for the variances and
Using (4.6) we obtain for X. .
~1 J
1
2 + (J2 +
a+b-2
(J2
(Jr
c
(a-1) (b-2) rc
+
2[ 1 L
(Je (a-1) 2 ifi
1
1
+
1
f-:]
+
~J
=
a(J2
2 +
rc
(Jr
(a-l) (b-2)
+
+
=
2[
(Je
1
(b_2)2
L
jfj 1j 2
1
n ..
~J
1
L
L
2
2
(a-1) (b-2) ifi jf] 1 j 2
1
(J2
(J 2
2 + rc +
e
(Jc
a- 1 -(-a---l-)-n-.-.1J 1
n~J
(4. 7)
43
2
2
(J
(J
+ ~ + -::---::-.,;;.e_ _
r
b-2
(b-2) n. .
(J2
=
~1 J
(J
2
rc
(a-1) (b-2)
=
(J
2
e
iri1
(a-1) (b-2)n ..
~J
The variances and covariances for X. .
~lJ2
jrj1,j2
are the same as those in
(4.7) with j1 replaced by j2'
By using (4.7) the variance-covariance matrix for Xl, call it V '
2
can then be written down.
For example, for a 3 x 3 design with cells
(1, 1) and (1, 3) missing, i.e.
X
n
n
21
31
n
n
n
X
12
22
32
the variance-covariance matrix of Xl
n
n
23
33
V
..-
=
2
2
(Tr+c + 2eirc
+ i[l]
e
(T2
r
+
2
(T12
232
(T + - (T
r
2 rc
+ (T;[2]
(T2 + -3 (T2
r
2 rc
(T2
c
2
+ (T12
+ (T2[ 2]
2
(T I
+_2_
2
2
(Tr+c
(T2
r
(T2
r
e
+
(T2
(T2
+2i
r+c
rc
r
+
2
(T12
Sym .
(T22
2
(T12
a
--y
(T2
c
a
2
+ (T12
(T2
c
2
2
2
(T3I
+2
a
a
(T32
- -2-
(T2
c
a
a
-
2
a
(T22
-2
+ (T2[ 3]
(T2
c
2
2
a
(T23
+2
e
(n/el 2 +
dia g (
(T32
-2
(T2
c
2
(T3 3
+2
(4.8)
::J
......
~
~
e
e
e
45
=
where
D
cr,2 ,
~J
V
13
vi2
V
22
V
23
vi3
V
Z3
V
33
+ cr
2
c +
i rc /crr2 }3
and
cr2 1
c 3
C
as in (3.5),
1 +...!...
1
_1_ +
_1 ]
L:
12 L: -L:
L:
2
n
[ 2 ill nil
(2)2(1 2) ill jll,3 n ij ,
1 j/1,3 1j
= [ 12
1
[3J
V12
= (clr
[lJ
[ 2J
Vn
1
1
L:
-- +
1]
L:
2
2
n
(2 ) (1 ) ill j/1,3 nij
J'/1 , 3 1j
L:
=
[1J with
=
cr2
2 +~
cr
rc
n, ,
j1
=
1
and
~J
replaced by
2
cr
r+c
=
j2
cr
3
)
.'
2
+ cr2
c
r
Given V the expected mean square for rows in the general a x b
2
disproportionate design (assuming that cells (1, 1) and (1, 3) are
missing) is given by
E(MSR)
=
+ tr[ (2' (V 12 + V13 + '"
+ V1a ) + (a-1)(a-2)V ) 'QJ
23
which is the same as equation (3.17) for the case of one missing cell
except that in the present case the equations in (4. 7) are used to
define the elements of V ' V ' etc. instead of the equations in
11
22
(3.15).
Thus, in the present case it can be shown that
VII + V
+ ••. + V
aa
22
2
2
a(J + + (J [IJ
r c
rc
=
2
2
(J12
a(Jr + (b-2)
+ (J2[2J
e
2
a(Jrc
a(J +
r
(a-I) (b-2)
2
·
a(J2
r
·.
2
(JIb
2
a(J + - r
(b-2)
+ i[3J
e
(J2
2 +
12
a(Jr
(b-2)
a(J2
+ a(J2
r+c
rc
2
2
(J12
a(Jr + (b-2)
2
2
a(Jrc
a(J +
r
(a-I) (b-2)
2
1
+ (J L : e i n i2
2
2
(J12
a(Jr + (b-2)
2
2
a(J
+ (J [IJ
r+c
rc
+ (J2[3J
e
+ i[4J
e
.
.
2
2
(JIb
a(Jr + (b-2)
· . .
2
2
(JIb
a(Jr + (b-2)
a(J2
r
.
.
2
2
(JIb
a(Jr + (b-2)
.
.
·
.
a(J2
+ a(J2
r+c
rc
2
1
+ (J L : e i nib
-
.pO'
e
e
e
47
where
[ 1]
[2]
=
=
a+b-2
+ (a-1)
(a-1) (b-2)
a
+
J
2
1
1
- 2a + 2
L: _1_ +
L:
2
ir1 nil
(b-2) 2 J'r 1J 3 n 1 J,
(a-1)
-
1
2
2 L:
L:
(a-1) (b-2) ir1 jr1,3
1
1
n, ,
q
)
[ 3]
=
1
1
-+
L:
-1
2
2 L:
(b-2) 2 J'r 1J 3 n 1 J,
(a-1) (b-2) ir1 jr 1J 3 n ij
[4]
=
[1] with
L:
j1
=
1
replaced by
2
cr
+~
n, ,
1J
and
cr
2
r+c
j2
=3
,
J
V12 + V13 + ... + VIa
=
,.-
2
2
(a-I) rr + rr
c
rc
1]
- [ (a-1)rr2 + rr2 E--rc
e it1 n i2
(a-I) (b-2)
- [ (a-I) rr2
a
· ·
a
· ·
+ rr2
1 ]
E--e it1 nib
(a-I) (b-2)
rc
2
rr
+~ E _l_
a-I if1 nil
a
2
c
(a-I) rr
1]
+ rr2 E--rc
e if! n i2
(a-I) (b-2)
0
- [ (a-I) rr2
a
- [ (a-I) rr2
2
2
(a-l)rr + rr
c
rc
2
rr
+~
E
.
1]
E--e if! nib
(a-I) (b-2)
rc
+ rr2
_l_
a-I if! n i3
.
.
.
a
a
.
.
a
.
··
2
c
(a-I) rr
I-
.po
00
e
e
e
49
Straightforward matrix algebra can then be used to show that the
.expected mean square for rows reduces to
E(MSR)
-
-
(J2 [
~
ab
_1_/ 1 +
L:
L:
if1 jf1,3 n ij \
1
)
2(2b-2)
)
(a_1)2(b_2)2
+
\
(a-l) 2
+
L:
_1_
·..Ll , 3 n 1 J·
Jr
11
,
+
11
\
11
_
1
)
(a_l)2
+ 2 (2b- 2) ) ]
(b_2)2
2
(a-l) (b-2)
)
In the same manner
E(MSC)
=
tr(V ·G )
2 C
E(MSRC) =
tr (V2· GRC2 )
G
RC2
(a-l) (b-l)
(a-l) (b-l)-2
and
where
=
G
RC
can be used to derive the expected mean squares for columns and interaction.
Table 4.1 summarizes the expected mean squares for the case
of two missing cells in the same row.
50
Table 4.1
Expected mean squares for a two-way disproportionate design
with two missing cells in the same row using the method of
unweighted means
Coefficients of Variance Components in E(MS)
2
2
2
000rc
c
e
Source
K1
K2
:;:
1 +
1[1 1
2(2b-2}
j E
ab
+ (a_1)2(b_2)2 ·1
Rows
+
1
b
:;:
2
(a-1) (b-2)
1
. I:
(a-1) 2 )
2
1 -
+ 11 + 2(2b-2} j'E ]
(b-2) 2
3
K3
a~ [ (1
Columns
+
K4
:;:
+
11 +
2 (2a-1)
j. I:1
2
(a-1) (b-1) (b-2)
1 +
a
:;:
2
(a-1) (b-1)
2a- 1 j. I:
(a-1) 2
2
+ 11 - (b_l}2(b_2) j . ; ]
K5
Interaction
~~[ll -
=
(a-l) 2
(b~
.r:
1) (b-2) j
+
(1 -
+
(1 - (b_1)2(b_2) j. I:3 ]
Error
1
1
1 ) . I:
(a_1)2
2
1
where the missing cells are (i j 1) and (i j 2) and
1
1
I:
1
:;:
I:
-1 ,
I:
n ..
ifi 1 jfj 1 j 2 1J
and
I:
2
:;:
1
1
- - + I:
n ..
n ..
ifi 1 1J 1
ifi 1 1J 2
I:
:;:
(a-1) (b-l)
(a-1) (b-1)-2
2
r
0-
,
51
As expected) Table 4.1 shows that as a and b become large
Kl
and
K2
=
=
K3
1
ab
K5
=
K4
1
L:
L:
i
j
(i
f(i l j 1)
l j 2)
1
n ..
~J
.
Also) for the special case of equal observations in the cells (n..
n)
~J
the expected mean squares in Table 4.1 reduce to
2
(a-I) (b-2)
]~~ +
2 + brr2
[ 1+ (a-I)2(b-2) ] rrrc
r
[ 1 + (a-l/(b-l)
]~~ +
[1 +
E(MSR)
=
E(MSC)
=
E(MSRC)
=
1 rr2 +
n e
E(MSE)
=
rr
e
-nl[l+
~
i rc
-:(-a-_" "1),.: 2(~b-_" 1-C-) ] rr;c
2
+ arrc
and
2
The preceding results for the expected mean squares for two missing cells in the same row can be generalized to the case of M missing
cells (denoted by iljl) i j 2) ... iljm) in the same row.
l
The procedure
for obtaining the expected mean squares in the M missing cell case is
exactly the same as that just described for two missing cells.
is:
That
(a) A value is estimated for each of the M missing cells by
minimizing the sum of squares for interaction with respect to each
of the missing cells after having replaced the observations in each
cell by their cell means.
For example) the missing value formula for
52
~
i
~
j
(4.9)
Y..
~J
r(i l j 1) ... (i l jm)
(a-l) (b-M)
The missing value formulas for the other missing cells are exactly the
same except that jl is replaced by the appropriate subscript.
(b) The variance-covariance matrix, V , of the vector of cell means,
M
Xl, is set up by using formulas similar to (4.7) for Var(X . . ),
~l J
Cov(X. . , X. . ) etc.
~lJl
~lJ2
l
This matrix has essentially the same form as
the matrix given in (4.8) for the case of two missing cells.
is substituted into (3.6.1) for the row, column and interaction mean
squares.
Straightforward matrix algebra is then used to obtain the
expected mean squares (see Table 4.2) which are similar to those given
in Table 4.1.
53
Table 4.2
Expected mean squares for a two-way dispropcrtionate design
with M missing cells in the same row using the method of
unweighted means
Coefficients of Variance Components in E(MS)
2
2
2
00cr
rc
c
e
Source
Kl
Rows
1[(1 +
ab
+ (I
=
K2
M(2b-M)
(a-I) 2 (b-M) 2
.r:
)
b
-
1+ (a-I)M(b-M)
1
1) . r:Z
- (a-I) 2
+ (I + M(2b-M) ) . 2:: ]
(b-M) 2
3
K3
Columns
.1..[ (1 +
ab
+ (I +
+
:~[ ( 1 +
+
Error
M(2a-l)
) or:
1
(a- 1) Z(b- 1) (b-,M)
I
1 -
1 +
a
=
M
(a,-I) (b-l)
Za- 1 ) . r:
(a_l)Z
2
(1 - (b-l)~b-M)
K5
Interaction
K4
=
I
02:: ]
3
=
(a-i) 2
(b~ 1)
(b-M)
1
. r:
1
l
1) . r:2
(a-I) 2
(1 - (b-l)~b-M) I· 3]
1:
-
1
r:
Z
r:
jh 1 j Z oo·jm
=
1
and
2
r
0-
(a-I) (b-I)
CM =: (a-I) (b-l)-M
54
(Note that if M = 2 the coefficients in Table 402 reduce to those
given in Table 4.1 and if M = 1 they reduce to those given in
Table 3.1.)
The results obtained above for two or M missing cells in the same
row extend directly to two or M missing cells in the same column.
For
example, if there are M missing cells in the same column then the expected mean square for columns is given by
acic
E(MSC)
+
2
rc
( 1 + (a-M)M(b-l) )
0-
(
1 +
(4.10)
M(2a-M)
). I: + (1 + M(2a-M) ) . I:
(a_M)2(b_l)2
1
(a-M) 2
2
b-_-\)--:2~). I:3 ]
+ ( 1 - -(
I:
I:
l
3
=
I:
I
I:
ifil···im jfj 1
I:
=
,
n ..
~J
I
- - + I:
n ..
jfjl ~l J
jfj 1
I
n.
~2j
I:
2
=
I
I:
ifi l " .. i m
n ..
~J
and
1
I
+ ... + I:
n. .
jfjl ~mJ
Similar results hold for E(MSR) and E(MSRC).
The expected mean squares given in Tables 4.1 and 4.2 can be used
to obtain unbiased estimates of the variance components
2
, and
rc
0-
2,
r
0-
2,
c
0-
2
for the cases of two or M missing cells in the same row
e
0-
by using the formulas given in (3.22) above (Procedure M estimates)
with G
replaced by G
or G
.
RCl
RC2
RCM
In addition, biased estimates of
"-
these variance components may be obtained by using (3.24) (Procedure M
55
estimates) with
1
s
1
n ..
replaced with
~J
1
s
2:
i
2:
j
1
n ..
~J
r(missing
cells)
" estimated variance components, given in
The biases of the Procedure M
(3.25), are examined in Chapter 5 for several designs.
The variances
"
of the Procedure M and M
estimated variance components can again be
derived analytically by using (3.26) and (3.28) with VI replaced by V
2
or V and G
replaced with G
or G
.
M
RCI
RC2
RCM
However, as in the case of
one missing cell (Chapter 3) these variances were not derived
analytically due to the complexity of the matrix algebra involved in
solving (3.26) and (3.28).
Instead, Chapter 5 gives numerical results
obtained by using the computer program VC to estimate the variances of
the Procedure M and
Mestimates
of the variance components for several
designs with two and three missing cells in the same row.
4.3
Two or More Missing Cells in Different Rows and Columns
In this section the estimation of variance components for two-way
disproportionate designs with missing cells in different rows and
columns based upon the method of unweighted means is examined.
Although the procedures for estimating the variance components are
exactly the same as described above for the case of two or M missing
cells in the same row, the matrix algebra involved in the present case
is much more tedious and the results are quite complex.
First, consider a design which has two missing cells which are
in different rows and columns (denote the two cells by (iljl) and
56
For example, the design might have the following n .. pattern:
~J
n
X
n
n
n
12
n
X
2l
n
3l
n
32
13
(4.11)
23
33
The missing value formulas for these two missing cells can be derived
as in Section 4.2 by minimizing the interaction sum of squares with
These formulas are
respect to the missing cells (X. .
~l J
1
given by Burrows [1966J as
Xi .
=
lJ 1
GO]-
(a-I) (b-l)[a 1: y.. +b 1: y.. [a 1:. Y.. +b 1: Y.. • .J..
~lJ
. .J.. ~J 1
. .J.. ~2J
. .J.. ~J 2
JrJl
~r~l
JrJ 2
~r~2
2
2
(a- 1) (b- 1) - 1
G]
(4.12)
and
where
G
=
1:
i
1:
j
l(i j 1) (i j 2)
l
2
By again using (3.12), X. .
~lJl
and X. .
terms of the model parameters.
~2J2
(in (4.12)) can be written in
Unfortunately, this model representa-
tion is quite complicated, namely
57
X. .
~lJ1
=1J.+r.
~1
+c. +fF1 (g . . )+f Z [
J1
~ J
F
1 Z
L:
.J...
Jr- J 1J Z
g.. ]
~lJ
where
f
=
a(a-Z) (b-1)
=
Z
(a-1) (b-1) + 1 )
)
(a-1)b(b-Z)
3
(a-1) b
=
-
=
(a-1) (b-1)
=
-a (b-1)
=
- (a-1) (b-1) + 1
)
Z
+ 1 )
)
and
The model representation for X. .
~ZJZ
i
Z
is exactly the same as (4.13) with
replacing i ) jz replacing j1 and vice-versa.
1
Using (4.13) the variance of X. .
~lJ1
and the covariances of X. .
with the other cell means can be shown to be
~lJ1
58
2
Var(X
i
j )
1 1
=
(J2
+
+ (Jrc [ £2 + (b-2) £; + £2 + (b-2) £;
(J2
r
c
F2
3
1
2
+ (a-2) (£6 + £2) + (a-2) (b-2)
7
+
r1-e [
F2
=
(J
2
+
r
£2
1
n ..
+ £2
~lJ2
(J
2
+
c
I:
2
2
• v
(X
.) +
rc rc i 1 J 1
+ (a-2) (b-2) £2]
2
_1_ +
jFj1j2 n i1j
(J
8
where
I:
2
=
£~ ]
(J
£2
3
n. .
~2J1
2
• v
(X . . )
e e ~lJ1
:
~
59
=
.E
1
.E
ifi 1 i 2 j=j1 j 2
and
n ..
.E
4
~J
=
. ,. 2
=
..,..2
v
+-
v
r
f21 2
+-
0-
F
f31
C
F
0-
=
1
.E
.E
;
n ..
ifi 1i 2 jfj1j2 ~J
for
rc
2
(4.14)
rc
for
=
2
:.zli
F
rc +
n~~
)
and
for
~J2
2
+
:~. )
for
~J
i,h i
1 2
jfj 1 j 2
The above formulas also give the variance of X. .
~2J2
and its co-
variances with the other cell means when (i j 2) and (i j 1) are inter1
2
changed.
A comparison of the equations given in (4.14) with those in
(4. 7) indicates why the results for missing cells in different rows and
60
columns are much more complex than those for missing cells all in the
same row.
Using (4.14) the variance-covariance matrix, V , of the
2
vector of cell means for the case of two missing cells in different
rows and columns can then be written.
For example, for the design
given in (4.11) the variance-covariance matrix of the vector of cell
means, Zl
by
= (XlI'
Y12' Yl3 , Y2l , X , Y , Y3l , Y , Y ), is given
23
32
22
33
e
e
e
v2 =
-2
O"r+c
2
0"
r
2
0"
r
2
0"
c
-
2
+ (1.4)0"
rc
2
+ 040"12
2
+ 060"13
2
+ 040"21
2
+ O"eoce(X11,X21)
2
0"
r+c
2
0"
r
0
2
0"
c
2
+ O"eove(X
2
060"
rc
-
2
0"
c
2
040"32
-
2"
-
020"33
2
+ 060"31
n)
2
+ 0"12
0
0
0"
2
0"
c
0
0
0"
0"2
r
2
0"
c
0
0
2
2
0" + + (1.4)0"
r c
rc
2
0"
r
-
2
0"
c
-
2
+ O"e ov e(X 22 )
2
+ 060"23
-
2
+ 0"13
0
2
040"13
2
c
,
2
0"
r+c
2
-+- CJ21
Sym.
2
c
0
2
+ 040"12
2
O"r+c
2
0"
r
2
+ 040"21
2
O"r+c
+
-
2
040"23
2
060"31
2
0 20"3~
2
+ 060"32
0
0"2
c
0
2
0"23
(D/e) 1 + dia g (
::j)
-
0\
.....
62
where
1
2 [(6) (6)
L:
_1_ + (9) (-6)
L:
1.'-1
-} 2 j-1
- , 2 n ...iJ,
(15)
+ (9)(-6)
L:
L:
L:
r!-
1.'=1,2 J'./.1,2
ij
r
1
n1.'j
L:
ir1,2 j=1,2
~
+ (-3) 2
V (X )
e 2l
= Ve(X l1)
with i
placed with j2
:::;
2
2/
rc + (je n'j
1.
(j
L:
L:
1
ir1,2 jr 1 ,2 n ij
J
•. ,
1 = 1 replaced with i 2 = 2 and jl
= 2,
and
=1
re-
D and C are defined in (4.8),
i r+c =
(j2
r
+
(j2
c
Once V is determined, equation (3,6.1) can again be used to obtain
l
the expected mean squares for rows, columns and interaction.
result for the expected mean square for rows is given by
The
63
E(MSR) = tr(v • G )
2 R
2c
(X. .
X. . )
(X . . )+v (X . . ) _ rc ~(a1~11)} ~2J2 +M ]
rc ~lJ1
rc ~2J2
+ ----------a""";b---------v
(4.15)
+ f .L:
7
where M
= number of missing cells
3
<.~.:
.:.:}
2); v
rc
(X. . )} v
~lJ1
rc
(X. . )}
v (X. . )} v (Xi . )} c (X. . ) X. . )} c (X. . ) X. . ) and
e ~lJ1
e
2J2
rc ~lJ1
~2J2
e ~lJ1
~2J2
were defined in (4.14); and f
F ···} fS}F were defined in (4.13).
~2J2
L:
1
L:
4
Further
simplification of (4.15) to a less complex form does not appear to be possible.
Note here that (4.15) reduces to the formula given in Table 3.1
for the case of 1 missing cell (i j 1) when the proper substitutions are
1
made.
That is} if (i j 1) is the only missing cell then in (4.15)
1
a + b _ 1
1
replace vrc(Xi1j1) with (a-1) (b_1) and ve(Xi1j1) with (a_l)2
as given in (3.15);
Var(X . . ) and Cov(X. '} X. . ) with 0 since X. .
~2J2
missing cell;
~lJ1
~2J2
~2J2
is no longer a
64
with
M
1
I:
i
;
1
with
I:
j
;
n ..
~J
f(i l j 1) (i 2 j 2)
=
1
f
7
F
=
1
a-I
= b-l'
0
I:
_1_
n
l ijl
and
-
and
-
1
~--:-:~---:"""""
(a-I) (b-l)
;
and
,
and
ifi
;
I:
I:
__1_
respectively.
ifi l jfj 1 n ij
The resulting formula for E(MSR) will be that given in Table 3.1.
The results of using (3.6.1) to derive the expected mean squares
for rows, columns and interaction for the case of two missing cells in
different rows and columns are summarized in Table 4.3, where the missing cells are (iljl) and (i j 2); M is the number of missing cells
2
(.!.!.,
two);
so =
I:
1;
i j
n ij
.L (missing)
r cells
I:
v (Xi . ),
e
2J 2
are the same as defined in (4.14) above; f ,
l
(4.13);
"0'
fS' F are defined in
e
Table 403
e
e
Expected mean squares for a two-way disproportionate design with two missing cells in
different rows and columns using the method of unweighted means
Coefficients of Variance Components in E(MS)
Source
0-
Kl
..!.-[SO
ab
Rows
2
rc
=
+ v
v
+ v (X. . )
e ~2J2
1 +
- 2c (X. . , X. . ) / (a_I)
e ~lJl
~2J2
+
?(fl 2:1
o
0-2
c
2
r
0-
=
K2
(X. . )
e ~lJl
I
0- 2
e
rc
(X.
. ) + v
~lJl
rc
(X.
. )
~2J2
ab
b
2c
+f 2' 2: - ( f 3' 2:1 +f 5' 2: 2
2
J
rc
(X.
. , X.
~lJl
. )
~2J2
- M
(a-I)
ab
-
---
+ [f6+f7]02:3+fsOM'2:4)/(a-l))t
j
K3
"!'-b[SO
a
=
+ v
e
K4
(X ~lJl
.. )
+ v
e
(X ~2J2
.. )
v
rc
=
(X.
1 +
Columns
. ) + v
~lJl
2c
+ [£2+£5]'
S +£8 'M' E4) / (b_l)l]
-
(X.
. )
~2J2
ab
- 2c (X . . , X. . )/(b-l)
e ~lJl
,l.2J2
+ } (f3'2:l+f6'2:3-(fl°2:l+f7'2:3
rc
rc
a
(X. . , X. . )
~lJl
(b-l)
ab
~2J2
- M
continued
(j\
\J1
Table 4.3 (continued)
Coefficients of Variance Components in E(MS)
2
(5
rc
2
(5
e
Source
2
c
(5
(52
r
10
K5 =
C2 [. SO + v (X. . ) + v (X . . )
ab
a
e ~lJ1
e ~2J2
1
Interaction
2c (X.
+
Error
e
• , X.
~lJ1
• )
~2J2
(a_I) (b_1)
]
- FI
1
0\
0\
e
e
e
67
FI
=
2
f1'~1
f2'~2
f3'~1
f S ' ~2
F ( (b_1) + (b-1) + (a-1) - (a_1) (b_1)
f 7 . ~3
f8'M'~4
- (a-1)(b-l) -(a-l)(b-l) )
=
and
(a-1) (b_1)
(a-I) (b-1) -2
As in previous cases, the formulas given in Table 4.3 show that as
a and b become large
K1
=
K3
=
KS
=
1
ab
-f
and
K2
=
K4
=
~
~
i
j
1
n ..
1.J
(missing)
cells
1
The extension of the results in Table 4.3 to the case of M missing
cells in different rows and columns is straightforward but as might be
expected the matrix algebra involved is quite complicated.
A brief
summary of the calculations needed to obtain the expected mean squares
in this general M missing cell case will be given here.
First, the
missing value formulas for the M missing cells (denote the missing
values by X. . ,
1. 1J 1
x. . , "', x. . )
1. 2J 2
1.mJ m
are derived as in previous cases
by minimizing the interaction sum of squares.
The resulting formulas
for these missing values are given in general by
68
2: y.. + b 2: y.. - G]
• .1.
~lJ
. .1. ~J1
JrJ1
~r~l
Xi 1 j 1 = -----::[:--:(-a_-=l:7'")"'7.(b:-_-:"l7'")---:-"1':;"":][::-:(:-a--:-l:"7)-=(':'"""b_-=l:7'")-+--::-:M:-----=l-:;'"]- - -
[(a-1) (b-1) + M - 2][a
- [a
where
2:
2: y.. + b
2:
2: Y.. -(M-1)G]
1=i i ... i jfj. ~J
j=j2 j 3... j ifi. ~J
2 3
m
~
m
J
[(a-1)(b-1) - 1J[(a-1) (b-1) + M - 1]
G
=
2:
i
.1
r
(4.16)
2:
j
(missing)
cells
[(a-1)(b-1) + M - 2][a
2:
• .1.
y..
~2J
+ b
2:
. .1.
y.. ~J 2
G]
JrJ 2
~r~2
Xi 2 j 2 = -----=[~(-a_-::1:7'")-:-:(b:---""':"l":"")-----:-l.....,][,,-,(:--a,;;:..,-1="",,)-'('-b--::1""-)-+---"-'M:--::"'-""l"""]- - -
- [a
2:
2:
y..
+ b
2:
2:
y..- (M-1) G]
i=i i .•. i jfj. ~J
j=j1 j 3 •.. j ifi. ~J
1 3
m
~
m
J
[(a-1) (b-1) - 1][ (a-1) (b-1) + M - 1]
In terms of the model parameters (4.16) becomes
69
X.
~1 j 1
=
IJ.
+ r.
~1
f
2
+F-
f
4
+F
f
1
+-
+ c.
F
J1
F
f
6
+F
f
7
+F
fa
+F-
L
jfj j ... j
12m
L
ifi i " ' i
23m
~lJ
3
g .. +F
~lJ
L
g ..
~J
i=i i ... i
23m
L
jfj j ... j
23m
gij
L
L
ifi i ... i
12m
L
ifi i ···i
12m
ifi i " ' i
12m
L
jfj j ... j
12m
(4. 17)
gij
g ..
~J1
2:
gij
L
gij
j=j j ... j
23m
jfj j ... j
12m
where
f
1
= (a-1) [(a-1) (b-1) + M - 2] - b + M - 1 )
f 2 = (a-1) [(a-1) (b-1) + M - 2] + M - 1 )
f
f
3
4
1
(missing)
cells
i=i i ···i
23m
L
g ..
f
f
f
S
+-
L
j =j j ..• j
23m
= (b_1) [(a_1) (b-1) + M - 2] _ a + M - 1 )
= -[ (a-1) (b-1) + M - 2] _ a - b + M - 1 = _ab
)
70
f
S
f6
f
7
= -[ (a-I) (b_l)
+ M - 2J - a + M - 1
= (b-l) [(a-I) (b-l)
= _b(a_l)
,
+ M - 2J + M - 1 }
-[ (a-I) (b-l) + M - 2J - b + M - 1
= _a(b_l)
,
-[ (a-I) (b_l) + M - 2J + M - 1 = -(a-I) (b_l) + 1 ,
S =
f
F
= [(a-I) (b-l) - lJ[(a-l) (b-l)
and g . .
~.e,Jm
+ M - lJ
was defined in (4.13).
The camp lica ted
Similar formulas hold for X. j , X•• ,
X. .
~2 2
~3J3
~mJm
formulas for X. • , X. • , ••• X. • in (4.17) can be visualized more
~lJl
~2J2
~mJm
readily for a particular design by using the following notation:
Suppose we have a 4 x S design with missing cells (1,1), (2,2) and
(3,3).
Then (4.17) can be written as
L:
g ..
jfl, 2, 3
""" 0
f 'g 2l
3
fl' g12
0
ff g3l
f ·g
4 32
f ·g 4l
6
f ·g
7 42
~J
J
(4. IS)
fl' g13
f 'g
2 14
f 'g 2 lS
f 'g
4 23
f ·g
S 24
f S 'g 2S
f 'g
S 34
f ·g
S 3S
f ·g
f 'g
0
f 'g
7 43
S 44
S 4S
(4.19)
where J
4
and J
S
are (4 x 1) and (S x 1) column vectors of one's.
71
Similarly,
+ ",
~
jfl,Z,3
r-
0
f( gZl
f 3 'glZ
0
f 4 'g13
f S ,g14
f S ' glS
f l ,gZ3
f 2 ,gZ4
fZ,gZS
-
, J
f 4 ,g3l
f 3 ,g3Z
f 7 'g4l
f 6 'g4Z
0
f 7 ,g43
f S ' g34
f S ,g3S
f S ,g44
f S ,g4S
S
and
1
= lJ.+r 3 +c 3 +-F [f l
~
g3' + f Z
~
g3'
j=l,Z
J
jfl,Z,3
J
r-
0
f 4 'gZl
f 4 ,glZ
0
f l ,g3l
f l ,g3Z
f 7 ,g4l
f 7 ,g4Z
f 3 , g13
f S ' g14
fS'glS-
f 3 ,gZ3
f S ' g24
fS'gZS
f 2 ,g34
f 2 ,g3S
f S ,g44
f S ,g4S
0
f 6 'g43
The author has found that the notation given in (4, 19) made the calculations of the expected mean squares for a particular design much
easier as opposed to the equations given in (4,17),
72
Having set up (4.17) the variance-covariance matrix,
V , of the
M
vector of cell means can then be set up and used in equation (3.6.1)
to obtain the desired expected mean squares for rows, columns and
interaction.
The results of this calculation are given in Table 4.4,
where the missing cells are (i j 1) ... (imjm)
1
4.3j v
rc
(X .. ), c
~J
rc
j
SO is defined in Table
(X .. ,X ,), v (X .. ) and c (X .. ,X ,) are derived
k 'V
~J
e ~J
e ~J k 'V
directly from equation (4.17) (as was done previously in the case of
two missing cells using (4.13))j
L.
1
=
1
n ..
L
j=j j ... j
L
i=i i ···i
12m
,
~J
12m
fi
L.
2
=
L
i=i i ···i
1
-,
L
n
..
jfj j ... j
L.
3
=
L
iii i " ' i
L
j=j j ... j
1
n ..
L
iii i ···i
L
jfj j ... j
1
j
n ..
12m
L.
4
,
~J
12m
fS' F are defined in (4.17)
f ·L.
f ·L.
2 f( L.1
2 2
3 1
FI - F ( (b-l) +
+
(b-1)
(a-I)
f
7
·L.
3
f
S
-
·M·L.
4
=
(a-1) (b-1)
(a-l)(b_1) -
M
j
f 4· L.1
(a-1) (b-l)
- (a-l)(b-l) - (a_l)(b_l) )
SM
,
~J
12m
12m
f ,
1
~J
12m
12m
and
-
f .L.
f ·L.
5 2
6 3
+
(a_I) (b-1)
(a-1)
Table 4.4
e
e
e
Expected mean squares for a two-way disproportionate design with M missing cells in
different rows and columns using the method of unweighted means
2
rr
e
Source
Kl
Coefficients of Variance Components in E(MS)
2 2 2
rr
rr
rr
rc
c
r
=
imjm
alb
[so + ..4:.
~J-~lJl
iJ'
mm
Rows
=
K2
imjm
ve(X ij )
iJ'
m
L I
m
- 2 . . . . k'>"
~J=~lJl
'V
~J
1 + . {: .
Vrc(Xij)/ab
~J-~lJl
iJ'
mm
c (X .. ,Xk ,)
e
~J
' .1
'V
-
(a-l)
+
':.'"'"
2)."
L
.;-;
.
~J=~lJl
2 '
+ F \El'~1+f2'~2-([f3+f4J'~1+(M-l)f5'~2
ij
mm
L....
k'>"
'V
~J
c
rc
(X .. ,X k ,)
~J
'V
a b ( a- 1)
b
M
as
1
+
(f6+(M_l)f7)·~3+f8·M·~4)/(a-l)).t
continued
-...l
W
Table 4.4 (continued)
Coefficients of Variance Components in E(MS)
Source
0-
2
0-
K3 =
2
0-
c
cr
2
r
K4 =
imjm
10 [so + ..~.
~J-~lJl
imjm
- 2
L
ij=i l j 1
t
L
1 +
v e (X ij )
imjm
Columns
2
rc
e
v
ij=i l j 1
m
k-!, > ij
imjm
c (X .. , Xk-!,)
e ~J
(b_l)
- 2
L
ij=iljl
+
i
+
(f2+(M_l)f5)·~2+f8·M·~4)!(b_l))]
(f3·~1+f6·~3-«fl+f4}~1+(M-l)f7·~3
rc
(X .. )!ab
~J
L
c rc (X ~J
.. ,Xk,v
k-!, > ij
ab(b-l)
imjm
a
+ M
ao
continued
'-l
.po
e
e
e
e
e
e
Table 4.4 (continued)
Coefficients of Variance Components in E(MS)
(J2
Source
(J2
rc
e
(J2
(J2
c
r
KS =
imjm
~[so+.b.
1.J-1. J
v (X .. )
e 1.J
1 1
Interaction
1
imjm
imjm
+ L L
Ce
2
ij=i j 1 kt>ij
1
Error
(Xij,X
kV
]
(a-I) (b-1) - FI
1
-...J
\J1
76
Note that for M equal to two the results in Table 4.4 reduce to those
given in Table 4.3.
As an example of the expected mean squares given in Table 4.4
consider the following 3 x 3 design with missing cells (1}1)} (2}2) and
n
X
n
n
n
X
21
31
n
12
n
13
23
X
32
For this design equation (4.19) becomes
f 'g
1 12
0
J'
3
Xu :::; IJ. + r 1 + c + F
1
f ·g
3 21
f 'g
3 31
where
J'
:::;
[l}l}l]
f
:::;
a[ (a-2) (b_1) + 1]
}
b[ (a_1) (b-2) + 1]
}
3
f
f
1
:::;
3
:::;
4
-ab}
F
0
f ·g
4 32
fl' g13
f ·g
4 23
. J3
(4.20)
0
}
:::;
[(a-1) (b-1) -1][(a-1)(b-1)+2]
and
gij was defined in (4.13) .
Note for this particular design that f } f } f } f and fa given in
S
6
2
7
(4.17) do not exist since there is a missing cell in every row and
column.
Similarly}
77
f
0
flog Zl
f og
4
g
0
3
f 4 gl3
0
lZ
0
f( gZ3
3
0
f 3 g3Z
0
3l
oj
and
f
0
f
0
4
g
Zl
f og
l 3l
4
0
g
f og
3 l3
lZ
f og
3 Z3
0
f og
l 3Z
oj
3
0
Using the results given in Table 404 the coefficients of the
expected mean squares for this 3 x 3 example can then be shown to be
-2ce(Xll'XZZ) - 2ce(XllX33) - 2ce(XZZ,X33)
(a-l)
KZ = 1 +
1:...
ab
[3 + v
rc
(X
11
) + v
(x..) + v
rc --ZZ
rc
(X
33
)
78
-2ce(Xll,X22) - 2ce(Xll,X33) - 2ce(X22,X33)
(b-l)
2· Z
(f 1 + f 4 ) ]
1 (f
)
F
3 (b-l)
+
,
K4 == 1 + 1:....
[3
ab
where
SO
==
Zl
Z
i
==
.J.
r
Z
j
(missing)
cells
1
n ..
1J
,
(a-I) (b-l)
C3 == -,...(a--:.:-l.....
) ...,..(b,....;-:.:-l.)...."":_--::'-3
79
and
1
1
1
1
1
-2 [f f ( - + - ) + f f ( - + - )
l
3
F
n 12
n 2l
1 4 n 13
n 23
+ f f (_1_ +..2-.-) ]
3 4 n
n
3l
Similar formulas can be derived for v
v e (X
33
32
rc
(X
22
), v
rc
(X
33
), v (X ),
e 22
), crc(X ll , X33 ), c rc (X 22 , X ), ce(X ll , X ) and c (X , X )
33
e 22
33
33
by using (4.20).
The results given in Tables 4.3 and 4.4 for the expected mean
squares for designs with missing cells in different rows and columns
can be used along with equation (3.22) to obtain unbiased estimates of
th e
'
var~ance
components
2,
r
~
2 , ~2 and ~2 (Procedure M estimates).
c
rc
e
~
These variance components can also be estimated by again using the
biased estimating equations given in (3.24) (Procedure
Mestimates).
'"
The variances of the Procedure M and M estimated variance components
can then be derived analytically by using (3.26) and (3.28).
How-
ever, as in Chapter 3 and Section 4.2 above, the variances of the
estimated variance components are so complex that it did not appear
worthwhile to derive them analytically; and therefore, they are
examined numerically in Chapter 5 by using the computer program VC for
'"
both Procedure M and M
estimates for several different design patterns.
The results presented for expected mean squares in this section
have produced, as the reader has no doubt noticed, complex formulas
which do not appear to reduce to forms similar to the relatively
simple formulas derived for designs with one missing cell, Table 3.1,
80
or missing cells in the same row (or column), Tables 4.1 and 4.2.
Accordingly, it seems worthwhile to examine these complex formulas
numerically for various designs to determine if there is any possibility of simplifying them.
For example, for certain designs can they
be approximated well by the equations given in (3.23)?
Chapter 5
presents the numerical values of the coefficients Kl, " ' , K5 in
Tables 4.3 and 4.4 for several different designs and then examines the
possibility of simplifying or approximating these coefficients.
In this chapter we have examined the expected mean squares for the
method of unweighted means for designs with missing cells in the same
row and in different rows and columns.
The procedures employed in
these cases can also be used to obtain the expected mean squares for
designs that have a combination of these two cases.
That is, for
designs that have some missing cells in the same row (or column) and
some missing cells in different rows and columns.
For example, a
design with the following n .. pattern:
1J
X
n
n
2l
3l
n
n
n
12
22
32
n
13
X
(4.21)
X
Due to the wide variety of design patterns for these types of designs
no general formula for their expected mean squares has been derived
but it seems obvious that these formulas will be a compromise between
the cases of M cells missing in one row (or column) and M cells missing
in different rows and columns.
Of course, for any particular design
81
pattern} such as that given in (4.2l)} the expected mean squares can
be derived by using the procedures described in detail in the present
chapter.
In Chapter 5 numerical results are obtained for the expected
mean squares and the variances of the resulting estimated variance
components (using Procedures M and
M)
for several designs which have
some missing cells in the same row and some missing cells in different
rows and columns.
82
5.
5.1
COMPUTER STUDY
Introduction
The purpose of this chapter is to describe the results obtained by
using a computer program, VC, which computes and prints out the fo11owing quantities for a given two-way disproportionate design (with
missing cells) and a given parameter set
~2
rc
=1
1.
and ~2
e
2
(~.~., ~r
= 16,
2
~c
= 4,
= 1):
the coefficients K1, K2, K3, K4 and K5 of the expected mean
squares for the method of unweighted means (see Tables 3.1
and 4.1 through 4.4, these coefficients are used in Procedure
2.
the variances of the unbiased estimated variance components
obtained by Procedure M (recall that Procedure M estimates
the variance components by using (3.22))j
3.
the variances of the biased estimated variance components
obtained by Procedure
M(Procedure Mestimates
the variance
components by using (3.24))j
4.
the variances of the unbiased estimated variance components
obtained by a procedure based upon the method of fitting
constants (This procedure computes the mean squares for rows
adjusted for columns, for columns adjusted for rows, and for
interaction adjusted for rows and columns.
The analysis of
variance procedure is then used to estimate the variance
components.
This procedure will be called Procedure A.
has been examined in detail by Bush and Anderson [1963J).
It
83
The above output from the computer program VC gives the information necessary to (a) study the coefficients Kl ... K5 to determine
if they can be approximated by simpler formulas) (b) study the biases
" (c) study
of the estimated variance components obtained by Procedure M)
the magnitudes of the variances and mean square errors of the estimated
Mrelative
variance components obtained by Procedures M and
to each
other and to the corresponding variances obtained by Procedure A and
(d) study for Procedures M,
Mand
A how the variances and mean square
errors of the estimated variance components vary over different designs
(n .. patterns) and over different parameter sets.
q
A brief description of VC will be given first.
5.2
General Description of the Computer Program VC
In general) for a particular design (i.e.) a given n .. pattern)
- -
~J
and a particular parameter set VC computes and prints out the quantities given above in 1 through 4.
The program then checks to see if
another parameter set has been inputed for the design under study and
if so recomputes 1 through 4 for this new set.
If a new parameter set
is not given then the program reads in another design and parameter set
etc.
The main inputs to VC for each design under study besides the n ..
~J
pattern of the design are (a) parameter sets for
(b) the component matrices (see (5.2»
2
~r)
2
~c)
2
~
rc and
2
~e)
of the variance-covariance
matrix of the cell means and (c) matrices needed in the computations
of the variances of the estimated variance components for Procedure A
(see Bush and Anderson [1963J).
IBM System/360 model 75.
VC was written in Fortran IV for the
84
In particular, the procedure used by VC to compute K1 ... K5 for
the method of unweighted means is the following:
= X'GX)
Recall from Chapter 3 that for any mean square (MS
t
computed by unweighted means the formula for computing its expected
value for a particular design is
E(MS)
E(x' GX) = tr(V' G)
(5.1)
where V is the variance-covariance matrix of the cell means,
X,
and
the ki's are coefficients of the variance components which depend on
the dimensions and n .. pattern of the design.
~J
In order to use (5.1)
to compute the individual coefficients k , k , k and k which
1
4
2
3
222
2
multiply ~ , ~ , ~ and ~ respectively it is necessary to partition
e
rc
c
r
the V matrix into component matrices that contain only terms that
.
~nvo
1
ve
2
~e'
2
~rc'
For example, suppose we have a 2 x 2 design;
i.e., the n .. arrangement is given by
- ~J
Then the variance-covariance matrix of the cell means for this design
has the form
85
-2
2
o-r+c + o-ll
0-
0-
2
2
0-
r
2
o-r+c +
r
0-
2
12
-
2
0
c
0
0-
2
c
v
0-
2
0
0-
'-
2
where
2
o-r+c +
0
c
o-r+c =
0-
2
+
r
0-
2
0-
c
2
2
and
c
0- ••
~J
0-
2
21
0-
2
0-
r
= 0-
0-
2
rc
2
r
2
r+c
+
0- 2
22.-
2
+~
n ..
~J
This variance-covariance matrix can be partitioned into
2
0-
r
2
0-
r
V
0-
0-
2
r
2
r
0
0
0-
0
0
0
2
c
0
0-
0-
2
c
2
c
0
0
0-
0-
2
c
0
0
0-
0
0
0-
2
e
nn
2
r
2
r
0-
0-
2
2
0-
r
2
c
0
r
0
2
0
0
0
0
0
0-
0
0
0-
0
0
0
rc
+
+
=
2
rc
0
0-
0-
2
c
2
c
0
0
0-
2
c
2
rc
0
0-
2
rc
0-
0
0-
0
n
0
0
0
0
2
e
12
+ V
V + V + V
rc
e
c
r
+
0-
0
0
n
2
e
0
21
0-
0
0
0
n
2
e
22
Using (5.2) we can write (5.1) as
(5.2)
86
E(MS) =tr(V'G) =tr[(V
e
+V
rc
+V
c
+V).GJ
r
= tr (V . G) + tr (V . G) + tr (V . G) + tr (V . G)
e
rc
c
r
kl~e2 + k ~2
2 rc
+.k ~2 + k ~2
3 c
4 r
tr(V . G)
e
where
(5.3)
etc.
Thus} the matrices Ve } Vrc } Vc and Vr are used as input to VC and then
(5.3) is used to calculate Kl ... K5 for the method of unweighted
means.
Having computed Kl ... K5} VC then computes the variances of the
estimated variance components obtained by Procedure M by using equation
(3.26).
dure
The variances of the estimated variance components for Proce-
Mare
calculated by VC by using (3.28) and the corresponding
variances for Procedure A are obtained by the computational method
described in detail in Bush and Anderson [1963J.
5.3
Designs Investigated
The designs investigated by using the computer program VC consisted of 3 x 3} 4 x 4 and 6 x 6 designs with one or more missing
cells.
The 3 x 3 designs investigated were derived from those studied
by Bush and Anderson [19p3J and Hirotsu [1966J by deleting one} two
or three cells from the full
(!.~.
that these authors studied.
These 3 x 3 designs are given in Table 5.1
along with the base designs
they were derived.
(i.~.)
no missing cells )3 x 3 designs
the full 3 x 3 designs) from which
The 4 x 4 and 6 x 6 designs examined by VC and the
base designs from which they were derived are given in Tables 5.2 and
5.3} respectively.
The notation used to denote the various designs in
87
Table 5.1
Design
The n .. arrangements for the 3 x 3 designs investigated and
~J
a
the base designs from which they were derived
Base Design
16-1
2 2
2 2 2
2 2 2
18
2
2
2
2
2
2
X
X
A.
2
2
2
18
1
2
3
1
2
3
X
B.
1
2
3
17-1
1 1
2 2 2
3 3 3
1
2
3
15-1
1 1
2 2
3 X
18
2
3
1
3
1
2
X
C.
1
2
3
17-1
2 3
2 3 1
3 1 2
1
2
3
16-1
2 3
3 1
1 X
15-1
2 X
3 1
1 2
1
2
3
12-2
2 X
X
1
1 2
1
2
3
D.
1
1
4
18
1
4
1
4
1
1
E.
1
1
1
18
1
2
3
1
3
5
17-1
1 4
1 4 1
4 1 1
14-2
2 2
2 X 2
2 2 2
13-2
1 4
1 X 1
4 1 1
X
X
17-1
1 1
X
1 2 3
1 3 5
1
1
1
13-1
1 1
2 3
3 X
12- 3
2 2
2 X 2
2 2 X
X
15-2
1 1
2 X 2
3 3 3
1
2
3
13-2
1 1
X
2
3 X
14-2
2 3
2 X 1
3 1 2
1
2
3
13-2
2 3
1
X
1 X
12-3
1 4
1 X 1
4 1 X
1
1
4
14-1
1 X
4 1
1 1
1
1
4
L5-2
1 1
X
1 X 3
1 3 5
1
1
1
11-2
1 1
X
3
3 X
10-3
1 1
X
1 X 3
1 3 X
X
X
12-3
1 1
2 X 2
3 3 X
X
12-3
2 3
2 X 1
3 1 X
X
9-3
2 X
1
X
1 2
X
1
2
X
10-2
1 X
1
X
1 1
aAn X in a cell of the design means there are no observations in
this cell.
88
Table 5.2
Design
A.
B.
The n .. arrangements for the 4 x 4 designs investigated and
1J
a
the base designs from which they were derived
Base Design
2
2
2
2
1
1
1
1
32
2 2
2 2
2 2
2 2
32
1 1
2 2
2 3
2 3
2
2
2
2
1
2
3
6
30-1
222
222
222
222
28- 2-1
X 2 X 2
222 2
2 2 2 2
2 2 2 2
24-4-1
XX 2 2
XX 2 2
2 2 2 2
222 2
24-4-2
X 2 2 2
2 X 2 2
2 2 X 2
2 2 2 X
X
1
1
1
31-1
111
222
233
236
30-2
X 1 X 1
1 2 2 2
123 3
1 2 3 6
1
1
1
1
24-2
111
222
233
X 3 X
29-2
X 1 1 1
1 X 2 2
1 2 3 3
123 6
1
1
1
1
23-2
111
222
2 X 3
2 3 X
X
X
1
1
29-3
X 1 1
2 2 2
2 3 3
236
1
1
1
1
20-3
111
222
2 3 X
2 XX
27-4
XX 1 1
XX 2 2
1 2 3 3
123 6
X
1
1
1
20-4
1 1 1
X 2 2
2 X 3
2 3 X
1
1
1
1
17-4
111
222
2 XX
2 XX
X
2
2
2
28-2-2
X 2 2 2
2 X 2 2
222 2
222 2
X
X
2
2
26-3
X 2 2
2 2 2
222
2 2 2
a An X in a cell of the design means there are no observations in
this cell.
89
Table 5.3
The n .. arrangements for the 6 x 6 designs investigated and
1)
a
the base designs from which they were derived
Design
Base Design
48
A.
1
1
1
1
1
1
2
2
2
2
1
1
1
1
1
1
1
1
46-2
1
1
1
1
1
1
1
1
X
1
2
2
2
2
1
1
X
1
1
2
2
2
2
2
2
2
2
1
1
1
1
1
1
1
1
2
2
2
2
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
44-3
1
1
2
2
2
2
1
1
2
2
2
2
1
1
2
2
2
2
1
1
1
1
1
1
2
2
2
2
1
X
1
1
1
1
1
1
58
B.
1
1
1
1
1
1
X
1
1
1
1
1
1
X
1
1
1
1
X
2
2
2
2
2
2
X
1
1
1
1
1
1
1
1
1
1
56-2
1
1
1
1
X
1
1
1
1
1
2
2
3
2
2
3
2
1
1
1
1
1
1
1
X
3
1
1
1
2
3
2
2
2
2
3
1
1
1
1
1
1
1
1
1
1
1
2
2
1
1
2
6
2
2
1
1
1
3
1
1
3
1
2
2
1
1
2
6
1
1
1
2
2
2
2
2
X
2
1
1
1
3
X
3
1
3
1
2
X
1
1
2
X
1
1
2
2
1
X
1
2
2
1
1
2
2
2
X
1
1
1
1
1
1
1
1
1
1
1
1
1
1
2
2
2
2
1
1
X
X
X
X
1
1
1
1
2
2
2
2
3
1
1
1
3
1
3
1
3
1
2
2
2
X
1
X
47-4
46-4
1
1
X
1
2
2
50-3
1
1
3
1
1
1
1
1
1
40-4
42-4
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
2
2
1
1
1
1
1
1
2
3
2
2
3
2
3
1
1
1
3
1
2
1
1
1
X
X
X
X
continued
90
Table 5.3 (continued)
Design
Base Design
76
c.
2
2
2
2
1
3
1
3
3
1
3
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
3
3
3
3
1
1
6
6
6
6
72-2
1
1
1 1
1 1
1 1
1 1
68-3
X 2 1
2 X 1
1
1
3
1
3
3
1
1
3
3
1
3
1
3
3
1
6
6
6
6
2
2
1
3
3
I
2
2
1
3
3
1
1
3
3
1
3
3
2
2
1
3
3
2
2
1
3
3
1
X
2
2
1
3
3
1
2
2
1
3
3
1
62-4
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
3
3
3
3
1
1
6
X 6
6
X
52-4
X I
I X I
I
I
I X 6
I
I 6 X
1
1
I
1
1
1
1
1
1
1
1
3
3
1
1
3
3
1
I
I
X X
X X
aAn X in a cell of the design means there are no observations in
this cell.
91
Tables 5.1 through 5.3 is as follows:
within the 3 x 3) 4 x 4 and
6 x 6 designs, the designs that come from the same base design are
labeled A or B or etc.
Thus, for the 3 x 3 designs there are the 5
base design patterns A, B, C, D and E studied by Bush and Anderson
[1963J.
Then within designs that come from the same base design the
designs are first given a number which represents the total number of
observations in the design, and then a number representing the number
of empty cells
~n
the design.
For example, for the 3 x 3 - A designs
given in Table 5.1 the first design investigated is labeled 16-1.
This means that there are a total of 16 observations and one missing
cell in the design.
For some of the 4 x 4 - A designs (Table 5.2) a
third index number had to be added to distinguish between designs with
the same number of observations and missing cells.
In these cases the
third number is nothing more than an index which labels the designs in
order of appearance.
For each of the 3 x 3 and 4 x 4 designs given in Tables 5.1 and
5.2 the following 13 sets of parameter values were used by VC for
computing the variances of the estimated variance components for the
three estimation procedures under study:
2
r
2
c
2
rc
~
~
~
1
16
4
16
1
4
16
1
1
4
16
1
1
1
1/4
1/4
1/4
1/4
1
1
1
~
2
e
1
1
1
1
1
1
1
~
2
r
16
16
1
4
16
4
2
c
~
4
16
1
1
1
4
~
2
rc
1
1
4
4
4
4
~
2
e
1
1
1
1
1
1
For each of the 6 x 6 designs given in Table 5.3 the parameters were
92
allowed to take the following 10 sets of values:
(J2
r
(J 2
c
(J 2
rc
(J 2
e
1
16
4
1
16
4
16
1
4
16
1
1
4
1
1
4
16
1
4
4
1/4
1/4
1/4
1
1
1
1
4
4
4
1
1
1
1
1
1
1
1
1
1
2
Note that the above parameter combinations include the cases of (J ~
rc
222
2
or < (J and (J «(J) ~ or < (J
Note also that all three estimation
e
r
c
rc
procedures have the properties that (a) the variance of the estimated
row variance component, Var (;2), remains invariant under changes in
r
remains invariant under changes in (J and
(Jc' (b) Var (~2)
c
r
2
2
(c) Var
(~2
rc
2
) remains invariant under changes in both (J2 and (J.
r
C
Therefore, the variances of the estimated variance components computed
by VC for the above parameter combinations imply additional combinations of interest.
(J 2
r
=
2
16, (J
c
=
2
1, (J
rc
For example, the results for the parameter sets
= 4,
(J2
e
=
2
1 and (J
r
= 4,
imply the results for the parameter set (J2
r
(J2
e
= 1.
(J2
c
= 4,
2
16, (J
c
(J2
rc
=4
2
and (J = 1
e
= 4,
(J2
rc
= 4,
All three estimation procedures also have the property that
2
.
"2
2 2 2
for a fixed (J «(J), an increasing (J
increases Var (~) (Var «(J )) .
r
c
rc
r
c
5.4
Results
In this section the results of running VC for the designs and
parameter sets given in the previous section will be examined in
detail.
These results are summarized in Appendix Tables 9.1 through
93
9.10 (which give mean square errors for all three estimation procedures
under study), Table 5.5 (expected mean squares for Procedure M) and
Tables 5.6, 5.7 and 5.8 (biases of the estimated variance components
'"
for Procedure M).
Note in the following discussion that for Procedures
A and M the mean square error of an estimated variance component is the
same as its variance (which was computed by VC) since both of these
methods are unbiased estimating procedures.
On the other hand, for
'" the mean square error of an estimated variance component
Procedure M
'" is a biased estimation proceis not the same as its variance since M
dure.
For Procedure
Mthe
biases of the estimated variance components
were computed on a desk calculator by using (3.25).
The meaq square
errors of the estimated variance components were then computed using
the square of these biases added to the variances computed by VC.
5.4.1
Mean Square Errors of the Estimated Variance Components
The results given in Tables 9.1 through 9.10 for the mean square
errors of the estimated variance components for the three estimation
procedures under study indicate the following general conclusions
(here
"2 ,,2
"'2
, ~ and ~
are used to denote estimates of the variance
r
c
rc
~
components, the " does not imply an estimate for a particular estimating procedure):
1.
"
For all the designs and parameter values studied Procedure M
compares favorably with Procedure M.
and
-,-~
IJ
; i.,~
Mhave
In fact, Procedures M
mean square errors which are relatively close
together especially for (a);2 and for (b) ~2 (;2) in the
rc
r
c
2
2
2
case of ~ (~) larger than ~
Moreover, both Procedures M
r
c
rc
Ii;
and
M compare
')1/ r~
1''/:
.'JU;)'
C:~f'J
quite favorably with Procedure A:'
94
2.
For the case of an equal number of observationsin each cell
(non-missing) of the design
4 x 4 - A)
the
(~.~.
three estimation procedures give mean square
"2 and relatively close for
rc
~
errors which are identical for
"2
~
r
,,2
and
designs 3 x 3 - A and
The one noticeable exception to this conclusion is
~.
c
when there are a considerable percentage of missing cells
(3 x 3 - A design 12-3) and ~2 ~ ~2
r
Mis
Procedure
3.
rc
or
2
.
e
~
In this case,
better for estimating ~2 and ~2.
r
c
For designs with an unequal number of observations in the
cells, if ~2 (~2) is much larger than ~2 then Procedures M
r
c
rc
"
and M
have mean square errors for ~,,2
r
"2
c
(~)
which are usually
smaller than the mean square errors for Procedure A (here
"much larger" may vary somewhat depending upon the de,sign but
usually if
~
~2r (~2)
~ 4~2rc this result will hold).
c
2
2
is greater than ~ then Procedures M and
rc
e
M give smaller
mean square errors for ~"2 than does Procedure A.
rc
when
(J"
2
rc
is less than or equal to
~
2
e
Also, if
However,
then Procedure A will
give smaller mean square errors for ~"2
particularly i f
rc'
there are not many missing cells in the design
3 x 3 - B designs 17-1 and 15-1).
less than or equal to
~
2
rc
(~.~.
For the case of ~2 (~2)
r
c
no one method gives in general a
,,2
smaller mean square error for ~
r
,,2
(~)
c
"
although M is quite
often the best procedure; instead, the design pattern determines which method is better.
general that when
~
2
e
for
~
2 (~2) and
r
c
However, it does appear in
~
2
rc
are equal but larger than
" will give the smallest mean square errors
that Procedure M
,,2
~
r
,,2
(~
c
).
For the case of a relatively high percentage of
95
(~o~.
missing cells in a design
3 x 3 designs with three
A
missing cells) Procedure M gives mean square errors for
and
.. 2
~
r
which are quite often much smaller than those given by
c
Procedures A and M.
~
where
A2
~
2
r
2
(~)
c
<
= cr2
rc
This result is especially noticeable
2
e
Table 504 summarizes these
or cr .
results for designs with an unequal number of observations
in the cells.
4.
....2
~2
(cr) for Procedure A decrease as
r
c
~
The mean square errors of
the number of observations in each row (column) of the
design (i.e. n.
- -
gether.
~
= ~j
n.
0'
~J
i
= 1,2,···,a)
become closer to.. 2
In fact, the mean square errors of cr
r
.... 2
(~)
c
may
actually be reduced by decreasing the number of observations
in the design so that the number of observations in each row
This property of Procedure A is
(column) are nearly equal.
especially noticeable when cr2 (~2) is much larger than cr 2
r
c
rc
For example, consider the 3 x 3 - E designs 17-1 and 10-3
for the parameter combination cr
cr
2
e
=
1.
2
r
=
16
'
cr
2
c
2
16, cr
rc
=
1 and
In this case the mean square error using Procedure A
.. 2
for cr is 338 for the 17-1 design and 302 for the 10-3 design
r
(Table 9.5).
Thus, here is a case where one design (17-1)
has nearly double the number of observations of another
design (10-3) and yet the mean square error of
is larger than for 10-3.
.. 2
~
r
for 17-1
The reason for this apparent con-
tradiction is that in 17-1 the number of observations in the
three rows of the design (n., i
~
=
1,2,3) are 2, 6 and 9
respectively while the corresponding totals for the 10-3
96
Table 5.4
General results using mean square errors to determine the
best estimating procedure
Parameter Combinations
0-
2
0-
r
2
0-
c
2
rc
Best Estimating Procedure
0-
2
2
0-
r
M or M
2
"
M or M
> 0-rc
0-
2
2
< 0-e
re
.5
0-
2
re
Varies with
design a
< 0-e2
Varies with
design a
> 0-2e
~ 0-
=
0-
2
0-
2
2
rc
2
0-
rc
"
M or M
A
e
> 0-e
=
2
2
rc
...
2
> 0-rc
.5
0-
c
A
M
2
A
> 0-e
M
and 0- if 0- (0- ) ~ 0or
rc
r
ere
and there is a high percentage of missing cells in the design (~.~.
aprocedure
Mis
best for estimating
e
3 x 3 - E design 10-3).
0-
2
2
2
2
2
97
design are 2, 4 and 4.
Thus, the n. 's for the 10-3 .design
~
are much closer together than for the 17-1 design and this
results in a smaller mean square error for
.
~2
~r us~ng
Procedure A1 ,2.
This property of Procedure A indicates that this estimating procedure is not very satisfactory for estimating
variance components for designs where the number of observations in the rows (columns) have wide imbalance, especially
when ~2 (~2) is much larger than ~2
r
c
rc
Instead, Procedure M or
A
M is recommended in these cases since these procedures do not
appear to be as sensitive to the design pattern as is Procedure A.
That is, the mean square errors of
Procedures M and
Musually
A2
c
For the 3 x 3 - E design
example given above the mean square errors of
Mare
using
(~)
r
increase as the number of observa-
tions in a design are decreased.
Procedures M and
~2
~
A2
~
r
.
us~ng
both 279 for design 17-1 and 298 and
296 respectively for design 10-3.
5.
For the 6 x 6 designs investigated (Tables 9.8, 9.9 and 9.10)
all three estimating procedures have mean square errors for
the estimated variance components which are relatively close
lSear1e [1967] gives examples for a one-way classification where
increasing the total number of observations increases the variances of
the estimated between-group variance components using Procedure A. As
in the present case, this result is due to imbalance in the number of
observations in each group of his design.
2Gay1or [1960] shows that the optimal designs for estimating
variance components in two-way classifications are as close to
balanced designs a~ is possible under certain restrictions.
98
together.
In fact, Procedures M and
Mhave
In addition, Procedures M and
Mare
nearly identical.
mean square errors for
;2 and ~2 which are approximately equal for a given parameter
r
c
combination for all the 6 x 6 designs studied.
for the parameter combination ~2
r
~2
~
mean square errors of
r
For example,
= 16 ~2 = ~2 = ~2 = 1
'
c
rc
e
the
~
for M and M for all the 6 x 6 - A
and B designs are 107 while the corresponding values for the
6 x 6 - C designs are 106 or 107.
~2
~
mean square errors of
r
and
On the other hand, the
.
for Procedure A vary somewhat
~2
~
c
from design to design for a given parameter combination due to
the imbalances in the number of observations in the rows and
columns of the 6 x 6 designs.
was discussed in 4 above.
This property of Procedure A
The mean square errors of
~2
~
rc
for
the 6 x 6 designs for all three estimating procedures depend
mainly on the number of observations in a designj and therefore, the 6 x 6 - C designs generally have the smallest mean
square errors for
6.
~2
v
rc
~
For Procedures M and M, as the number of missing cells in a
design increase the mean square errors of ~2 and ~2 also
r
c
increase.
~2
~
However, this result does not always hold for
In estimating this variance component by Procedures M
rc
~ ~t
. appears t h at ~f
an d M
L
of
~2
~
rc
~
2 <
e
~
2 t h en t h e mean square error
rc
also increases with the number of missing cells.
the other hand, if
On
~2e ~ ~2rc , then the total number of observa-
tions in the design appear to be a much more important factor
in determining the magnitude of the mean square error than
99
does the number of missing cells.
For example} consider the
If ~2
rc
4 x 4 - B designs 20-3 and 27-4.
= .25 and ~2e = 1
then Table 9.7 shows that the mean square errors of ~"2 for
rc
"
Procedures M and Mare
.52 and .51 respectively for design
20-3 and .33 and .34 for design 27-4.
2
rc
~
2
~e
=
1, the mean square errors are both 1.21 for design
20-3 and 1.05 for design 27-4.
~
2
e
Similarly for
However, when
1, this pattern is reversed.
~
2
rc
4 and
In this case the mean
square errors of ~"2 for 20-3 are 7.74 and 7.73 respectively
rc
for M and
Mwhile
for 27-4 the corresponding values are 8.40
and 8.41.
7.
For Procedures M and
if ~2 (~2) is much larger than ~2
r
c
rc
M,
then the design pattern has relatively little effect on the
mean square errors of
"2
~
r
,,2
(~).
c
That is, several designs can
have a wide range of total number of observations and number
of missing cells and still the mean square errors of the
estimated variance components for rows (columns) will not
change appreciably.
For example, for the ten 4 x 4 - B
designs studied by VC and for the parameter combination
2
= .25 and ~e2 = 1 the range of the mean
rc
,,2
"
for Procedures M and M
is only 176
square errors for ~
c
r
to 180 3 (Table 9.7).
~
2
= 16
r
'
2
16,
c =
~
~
«i)
3It is interesting to note here how close the mean square errors
176 to 180 are to
2~4/(a_l)
r
(= 171) which would be the variance of
;2 (;2) if this estimated variance component were distributed as
r
c
~2.X2(a_l)/(a_l).
r
100
8.
For designs with more cells missing in the same row than are
missing in the same column
(~.~.
24-2) the mean square error of
4 x 4 - B designs 30-2 and
"2
~
r
is greater than the mean
square error of ~2 for all three estimating procedures.
c
Of
course, the reverse will be true for designs with more cells
missing in the same column than are missing in the same row.
9.
As would be expected the 3 x 3 designs had mean square errors
for all three estimating procedures which were much more
sensitive to the number of missing cells than were the 4 x 4
and 6 x 6 designs studied.
This is due to the fact that it
is not just the number of missing cells but the number of
missing cells relative to the total number of cells in the
design that determines the effect on the mean square errors
of the estimated variance components.
In fact, for the 6 x 6
designs studied where the percentage of missing cells was
quite small and the number of observations large there was
relatively little effect on the mean square errors as the
number of missing cells was varied.
To summarize, for the majority of the designs and parameter
combinations that VC computed Procedures M and "M appear to be as good
as or better than Procedure A for estimating variance components with
respect to the mean square errors of the estimates.
In fact, Proce-
" are a 1ways b e tt er th an Proce d ure A f or
d ures M an d M
est~mat~ng ~
when
~2r (~2)
is much larger than ~2.
c
rc
2 ( ~2)
r
c
The only case where Procedure A
is consistently better than Procedures M and
when
..
Mis
for estimating ~2
rc
~2rc 5 ~2e and there are a small proportion of missing cells in a
Wl
A
design.
Procedure M usually has mean square errors that are relatively
close to those of Procedure M and in some cases (notably for the case
of estimating ~2 (~2) when (a) there are a large proportion of missing
r
c
cells in a design and (b) ~2 < ~2 ) significantly less than those of M.
r
rc
Furthermore, Procedure
M and A.
Mis
In fact, Procedure
computationally much easier than Procedures
Mcan
easily be carried out on a desk
calculator while Procedures M and A although they can be done on a
desk calculator quite often require the use of an electronic computer.
Accordingly, of the three estimation procedures studied for estimating
variance components Procedure
Mis
recommended.
However, if one is
interested only in unbiased estimates of the variance components then
Procedure M is recommended over Procedure A because (a)- its mean square
errors are usually as small as or smaller than those of Procedure A
and (b) it is computationally easier than Procedure A.
5.4.2
Expected Mean Squares for the Method of Unweighted Means
As discussed in Chapter 4, the coefficients Kl, K2, K3, K4 and
K5 of the expected mean squares for the method of unweighted means
have rather complicated analytical formulas especially for designs
with missing cells in different rows and columns.
Therefore, VC was
set up to calculate and print out these coefficients so that they could
be examined numerically to determine if they could be approximated by
simpler formulas.
In particular, for the 3 x 3, 4 x 4 and 6 x 6
designs with missing cells studied it was of interest to determine how
well Kl ... K5 could be approximated by the coefficients for designs
with no missing cells given in (3.7)j namely, K2
Kl
= K3 = K5 =
(harmonic mean)-l.
= K4 = 1.0 and
Recall also from Tables 3.1, and
102
4.1 through 4.4 that as the number of rows and columns of a design
become large that Kl ... K5 converge to the coefficients given in
(3.7).
In addition} examination of the analytical formulas for K5} K3
and Kl (Tables 4.3 and 4.4) indicated that perhaps these coefficients
A
A
A
could be approximated by K (see equation (3.23») K·K4 and K·K2
respectively.
Kl ... K5}
Thus} Table 5.5 presents the numerical values for
K}
(harmonic mean)-l) K'K4 and K·K2 for all the designs
given in Tables 5.l} 5.2 and 5.3.
Examination of Table 5.5 shows the following:
A
1.
For all of the designs studied K5 is quite close to K.
How-
ever} (harmonic mean)-l does not seem to be a good approximation to K5.
A
K is
cl~ser
In fact) for all the cases given in Table 5.5
in magnitude to K5 than is (harmonic mean)
-1
It
is interesting to note that if in the expected mean square
for interaction for the method of unweighted means} K is used
to estimate the variance component for interaction (recall
Procedure
of rr
2.
2
rc
Mapproximates
K5 with K) the bias in this estimate
is negligible (see Table 5.6).
For most of the designs studied Kl and K3 are relatively close
to K·K2 and K.K4} respectively.
These approximations improve
as the number of observations in the cells of a design become
(!.~.
more nearly equal and for the equal sampling designs
A
n,.
~J
= nj designs 3 x 3 - A and 4 x 4 - A) Kl equals K·K2 and
K3 equals K.K4 as shown previously in Chapters 3 and 4.
If
K.K2} K'K4 and K are used to approximate Kl} K3 and K5 and
the resulting expected mean squares are then used to estimate
103
Table 5.5
Coefficients of the variance components in the expected mean
squares for the method of unweighted means for several twoway disproportionate designs with missing cells
Analysis of Variance Table for the Method of Unweighted Means
Coefficients of Variance Components in E(MS)
2
e
Source
cr
Rows
Columns
Interaction
Error
K1
K3
K5
1
Design
Coefficients
2
crrc
cr
K2
K4
1
a
2
c
2
cr
r
b
a
Approximations to
Coefficients
....
....
K2
K4
K1
K3
K5
K·K2
K·K4
"b
K
HM- 1 c
16-1
14-2
12-3
1. 25
1. 60
2.00
1. 25
1. 60
2.00
.625
.800
1.00
.625
.800
1.00
.500
.500
.500
.625
.800
1.00
.625
.800
1.00
.500
.500
.500
.556
.611
.667
17-1
15-1
15-2
13-2
12-3
1.25
1. 25
1. 60
1. 60
2.00
1. 25
1. 25
1. 60
1. 60
2.00
.811
.729
.988
.969
1. 22
.617
.867
.821
1. 18
1. 22
.545
.656
.553
702
.608
.704
.806
.838
1.07
1. 22
.704
.806
.838
1.07
1. 22
.563
.645
.524
.667
.611
.611
.685
.667
.741
. 741
17-1
16-1
15-1
14-2
13-2
12-2
12- 3
9-3
1. 25
1. 25
1. 25
1. 60
1. 60
1. 60
2.00
2.00
1. 25
1. 25
1. 25
1. 60
1. 60
1. 60
2.00
2.00
.690
. 784
.815
.981
.961
1. 13
1. 22
1. 50
.690
. 784
.815
.981
.961
1. 13
1. 22
1. 50
.577
.619
.633
.588
.688
.666
.608
. 750
.704
.791
.806
.952
1. 07
1.10
1. 22
1. 50
.704
.791
.806
.952
1. 07
1.10
1. 22
1. 50
.563
.625
.646
.595
.667
.690
.611
.750
.611
.667
.685
.685
.741
.759
. 741
.833
3 x 3
A. -
B.
C.
continued
104
Table 5.5 (continued)
Design
Coefficients
Approximations to
Coefficients
,.
,.
K5
K.K2
K·K4
"b
K
lIM-I c
.891
1.03
1. 19
1. 48
1. 50
.729
.792
.800
.850
.750
.899
1.02
1. 26
1. 43
1. 50
.899
1.02
1. 26
1. 43
1. 50
.719
.813
.786
.893
.750
.750
.833
.833
.917
.833
.865
.954
1.11
1. 37
1. 56
.865
.954
1.11
1. 37
1. 56
.633
. 781
.660
.821
.774
.839
.964
1.11
1. 30
1. 56
.839
.964
1.11
1. 30
1. 56
.671
.771
.695
.810
.778
.707
.796
.763
.852
.852
1.11
1. 22
1. 23
1. 44
1. 67
1. 49
.556
.666
.617
.722
.833
. 747
.556
.611
.617
.722
.833
.747
.500
.500
.500
.500
.500
.500
.556
.666
.617
.722
.833
.747
.556
.611
.617
.722
.833
. 747
.500
.500
.500
.500
.500
.500
.531
.563
.563
.594
.625
.625
1.11
1. 33
1. 23
1. 33
1. 23
1.44
1. 44
1. 67
1. 49
1. 67
1.11
1. 22
1. 23
1. 22
1. 23
1. 44
1. 44
1. 67
1. 49
1. 67
.748
.940
.840
.949
.882
.935
1. 09
1.02
1.08
1. 31
. 748
. 741
.840
.861
.882
.935
1.09
1.02
1.08
1. 31
.630
.598
.637
.719
. 736
.551
.763
.566
.725
.800
.715
.823
.806
.950
.893
.850
1.09
.997
1.08
1. 32
.715
.755
.806
.871
.893
.850
1.09
.997
1.08
1. 32
.644
.619
.655
.714
.726
.590
.756
.597
.722
.792
.667
.667
.698
.750
.760
.667
.802
.698
.792
.844
1.08
1. 20
1. 17
1. 20
1.08
1. 12
1. 17
1. 20
.899
1.00
.971
1.06
.886
.933
.985
1.02
.823
.833
.845
.877
.890
1.00
.987
1.05
.890
.933
.987
1.05
.824
.833
.844
.875
.833
.847
.861
.889
K2
K4
1. 25
1. 25
1. 60
1. 60
2.00
1. 25
1. 25
1. 60
1. 60
2.00
.891
1.03
1. 19
1. 48
1. 50
1. 25
1. 25
1. 60
1. 60
2.00
1. 25
1. 25
1. 60
1. 60
2.00
30-1
28- 2-1
28- 2- 2
26-3
24-4-1
24-4-2
1.11
1. 33
1. 23
1. 44
1. 67
1. 49
31-1
30-2
29-2
24-2
23-2
29-3
20-3
27-4
20-4
17-4
6 x 6
~
46-2
44-3
42-4
40-4
3 x 3
D.
17-1
14-1
13-2
10-2
12-3
a
K1
K3
E.
17-1
13-1
15-2
11-2
10-3
4 x 4
A.
B.
continued
105
Table 5.5 (continued)
'a
Coefficients
Design
K2
K4
K1
56-2
50-3
46-4
47-4
1.08
1. 20
1. 17
1. 20
1.08
1. 12
1. 17
1. 20
.827
.996
.945
1.01
72-2
68-3
62-4
52-4
1.08
1. 20
1. 17
1. 20
1.08
1. 12
1. 17
1. 20
.781
.847
.834
.945
Approximations to
Coefficients
K5
"
K·K2
"
K·K4
"b
K
.840
.861
.931
.938
.763
.780
.814
.806
.826
.940
.951
.968
.826
.877
.951
.968
.765
.783
.813
.807
.778
.801
.833
.829
.763
. 785
.853
.895
.715
.701
.716
.771
.773
.842
.841
.925
.773
. 786
.841
.925
.716
.702
.719
.771
.731
.727
.750
.796
K3
lIM- 1
6 x 6
B.
C.
a These coefficients are used in Procedure M.
b
A
K
=
1
s
.L
r
c HM- 1
~
j
1
~J
where s is the number of occupied cells in
the design.
.E
.E
_1_] where M is the number of missing
i
j
nij
n ..
(missing)
cells
=..!-.
[M
ab
+
.L
r
(misSing)
cells
cells in the design .
c
106
the variance components for rows and columns by the
~alysis
of variance procedure the biases in these estimates are
relatively small.
That is, if we approximate the expected
mean squares for rows, columns and interaction by
A
E(MSR)
A
E(MSC)
=
A
(J
2
+ b
rc
(J
2
r
2
+ K4
e
(J
2
+ a
rc
(J
2
c
(J
(l<. K4)
(J
" 2
K
(J
E(MSRC) =
and estimate
2
+ K2
e
(K. K2)
+
e
(J2
r
and
(J2
c
(J
2
rc
from these approximate expected mean
squares; then, the biases in the resulting estimates, namely
(Kl - K2· K5)
b
Bias (~2)
c
=
(K3 - K4·K5)
a
(J
(J
2
e
and
2
e
are quite small. For example, for the 4 x 4 - B designs
studied these biases are the following when
Design
4x4-B
31-1
30-2
29-2
24-2
23-2
29-3
20-3
27-4
20-4
17-4
Bias in Estimated Row
Component When Kl
(J
2
= 1:
e
Bias in Estimated Column
Component When K3
Approximated by K.K2
"
Approximated by K·K4
.012
.036
.014
-.002
-.006
.035
-.002
.019
.000
-.007
.012
.003
.014
-.004
-.006
.035
-.002
.019
.000
- .007
"
"
Thus, it would appear that although K'K2
and K'K4
are far
from perfect approximations to Kl and K3 they are quite
107
satisfactory in many cases when the analytical formulas for
K1 and K3 are relatively complicated to compute.
3.
For the designs studied neither
Knor
(harmonic mean)-l are
good approximations to K1 and K3 even though these quantities
converge to one another as the number of rows and columns of a
design become large.
Even the 6 x 6 designs studied had re1a-
tive1y large differences between K1 and K3 versus "K and (harmonic mean)
-1
.
In addition, as the number of missing cells of a
design increases so does the magnitude of the differences between these quantities.
4.
For the designs studied K2 and K4 are not relatively close to
1.0, although as expected these coefficients become closer and
closer to 1.0 as the number of rows and columns of the designs
increases.
Only for the 6 x 6 designs with two missing cells (K2
= K4 = 1. 08) is the magnitude of these coefficients relatively
close to 1.0.
However, it should be pointed out here that K2 and
K4 are independent of the n .. pattern of a design.
~J
Therefore, once
the values of these coefficients are computed for a particular
design these values apply to all designs with the same dimensions
(number of rows and columns) and number and pattern of missing
cells.
For example, the values of K2 and K4 for the 3 x 3 - B
design 15-2 also apply to the 3 x 3 - C designs 14-2, 13-2 and
12-2.
Thus, the results given in Table 5.5 give the values of K2
and K4 for mos t cases of interes t for 3 x 3 and 4 x 4 des igns wi th mis sing
cells.
dure
to
The biases in the estimated variance components for Proce-
Mwhich
K are
sets K2 and K4 equal to 1.0 and K1, K3 and K5 equal
studied in Section 5.4.3.
W8
In summary, in terms of the relative magnitude of the coefficients
and the resulting biases introduced in estimated variance components it
appears for the cases studied that
A
Kis
a good approximation for K5 and
A
K·K2 and K·K4 are reasonable approximations for K1 and K3 respectively.
K2 and K4 cannot be approximated well by 1.0 for the 3 x 3 and 4 x 4
designs but for the 6 x 6 designs this approximation is fairly good
if the design does not have a large number of missing cells.
Of
course, in practice how well K1 ... K5 are approximated by simpler
formulas is a function of how the resulting expected mean squares are
to be used.
In tenns of using expected mean squares to estimate
variance components, Section 5.4.1 indicated that Procedure
M(which
A
approximates K2 and K4 with 1.0 and K1, K3 and K5 with K) is usually
as good as or' better (in terms of mean square errors of the estimates)
than Procedure M which uses the exact analytical formulas K1 ... K5.
The effect of using the approximations studied in this section for
K1 ... K5 to set up approximate tests of hypothesis deserves some
study.
In addition, it
i~
important to remember here that designs with
one missing cell or missing cells in only one row (or column) have
exact analytical fonnu1as (Chapters 3 and 4) which are relatively easy
to compute; and therefore, in these cases the approximations studied
here will not save much computational time.
5.4.3
A
Biases of the Estimated Variance Components for Procedure M
The biases of the estimated variance components obtained by using
Procedure
Mwere
calculated for all the designs in Tables 5.1, 5.2 and
5.3 by using equation (3.25) and the results given in Table 5.5.
109
These biases are given in Tables 5.6, 5.7 and 5.8 for the parameter
combinations ~2
rc
respectively.
22222
= 1; ~rc = 1, ~e = 1 and ~rc 4, ~e
e
~
.25,
=
1
The most striking result in these tables is that the
biases of the estimated variance component for interaction are quite
small for all cases studied.
Thus, when estimating
~
2
it appears
rc
....
that very little bias is introduced by using Procedure M as opposed
to using an unbiased estimating procedure.
column variance components
(f or a
·
b ~ases
.
g~ven
2 ,
rc
~
~
.. 2
(~
r
.. 2
~
and
c
For the estimated row and
) the tables show that the
2 comb'~nat~on
. ) .~ncrease as ()
h num b er
ate
e
0
f
missing cells in a design increases and (b) the number of rows and
(i.~.
columns of the designs decreases
the 3 x 3 designs have larger
biases than do the 4 x 4 designs while the biases for the 6 x 6 designs
studied are relatively small even for the case of ~2
rc
= 4 and ~2e = 1).
In general it is difficult to determine from Tables 5.6-5.8 the re1ative importance of the biases for
upon the true values of ~2 and ~2.
r
c
.. 2
~
r
.. 2
and
~
since this will depend
c
For example, if ~2
r
= 16
and ~2
rc
=
1 then Table 5. 7 shows that the biases in ~2 for all the designs
r
studied are less than 4 per cent of 16 (= ~2).
r
~2 = 1 and ~2
r
= 4,
rc
the biases in
.. 2
~
r
~
2
e
=
On the other hand, if
1 then Table 5.8 shows that in several cases
are greater than 1 (=
2
~).
r
One method that is
commonly used to study the relative importance of the biases of an
estimation procedure is to calculate the contribution of these biases
to the corresponding mean square errors of the estimators.
This in
effect indicates how large the bias squared is relative to the mean
square error of the estimators.
Accordingly, the percentage contribu-
tion of the biases to the mean square errors of
.. 2
~
r
and
.. 2
~
c
for
1,
110
Procedure
Mwere
computed for all the designs and parameter comb ina-
tions given in Section 5.2.
The result of these calculations is that
the maximum percentage contribution of the biases to the mean square
errors is 10 per cent and in most cases the contribution is much less
than 10 per cent.
For example, Table 5.9 gives the results for the
3 x 3 - B designs studied.
This table shows that in most cases the
percentage contribution is less than 1 per cent.
Thus, it would appear for Procedure
Mthat
the biases of the
estimated variance components are relatively unimportant when
compared with the corresponding mean square errors.
It is also worth noting here that the biases of ~2 and
r
Tables 5.6-5.8 are always positive.
rrc2
in
This indicates that the Proce-
A
dure M estimated row and column variance components usually have a
positive bias.
111
Table 5.6
Biases
a
of estimated variance components obtained by
Procedure
Design
Rows
Mwhen
~2rc = .25 and ~2e = 1
ColLunns Interaction
Design
3 x 3
4 x 4
A.
A.
16-1
14-2
12-3
.063
.150
.250
.063
.150
.250
.000
.000
.000
17-1
15-1
15-2
13-2
12-3
.110
.045
.195
.139
.288
.045
.091
.139
.203
.288
-.018
.011
.029
.035
-.003
17-1
16-1
15-1
14-2
13-2
12-2
12-3
9-3
.059
.076
.082
.181
.141
.205
.287
.333
.059
.076
.082
.181
.141
.205
.287
.333
.014
-.006
-.013
-.007
.021
-.024
-.003
.000
17-1
14-1
13-2
10-2
12-3
.075
.100
.180
.260
.333
.075
.100
.180
.260
.333
.010
-.021
.014
-.043
.000
D.
.021
.042
.044
.083
.125
.092
.000
.000
.000
.000
.000
.000
31-1
30-2
29-2
24-2
23-2
29-3
20-3
27-4
20-4
17-4
6 x 6
.036
.106
.065
.078
.051
.124
.109
.155
.119
.169
.036
.050
.065
.049
.051
.124
.109
.155
.119
.169
- .014
-.021
- .018
.005
.010
- .039
.007
-.031
.003
.008
46-2
44-3
42-4
40-4
.016
.036
.028
.039
.014
.022
.030
.032
-.001
.000
.001
.002
56-2
50-3
46-4
47-4
.014
.044
.029
.042
.016
.019
.027
.030
-.002
-.003
.001
-.001
72-2
68-3
62-4
52-4
.014
.033
.027
.037
.011
.019
.030
.029
-.001
-.001
-.003
.000
B.
A.
B.
E.
17-1
13-1
15-2
11-2
10-3
.098
.079
.199
. 232
.344
.098
.079
.199
.232
.344
-.038
.010
-.035
.011
-.004
Co 1urnns Interaction
.021
.061
.044
.083
.125
.092
30-1
28-2-1
28-2-2
26-3
24-4-1
24-4-2
B.
C.
Rows
C.
a The biases were computed by using equation (3.25). Note that
"
f or ~A2 an d ~A2 d epen d on 1y upon the values of v~2 and v~2 and
the b1ases
r
c
2
2
rc
e
are independent of the size of ~r and ~c .
112
Table 5.7
Biases
a
of estimated variance components obtained by
Procedure
Design
Rows
Mwhen
Columns
~2
rc
=
1 and
Interaction
b
~2 = 1
e
Design
Rows
Columns
.042
.123
.087
.166
.251
.184
.042
.083
.087
.166
.251
.184
.057
.168
.108
.140
.094
.206
.192
.281
.211
.295
.057
.091
.108
.091
.094
.206
.192
.281
.211
.295
46-2
44-3
42-4
40-4
.026
.061
.049
.064
.024
.037
.052
.057
56-2
50-3
46,-4
47-4
.024
.069
.050
.067
.026
.034
.048
.055
72-2
68-3
62-4
52-4
.024
.058
.048
.062
.021
.034
.051
.054
3 x 3
4 x 4
A.
A.
16-1
14-2
12-3
.125
.300
.500
.125
.300
.500
17-1
15-1
15-2
13-2
12-3
.172
.108
. 345
.289
.538
.107
.154
.289
.353
.538
17-1
16-1
15-1
14-2
13-2
12-2
12-3
9-3
.121
.138
.144
.331
.291
.355
.537
.583
.121
.138
.144
.331
.291
.355
.537
.583
17-1
14-1
13-2
10-2
12-3
.137
.163
.330
.410
.583
.137
.163
.330
.410
. 583
17-1
13-1
15-2
11-2
10 ... 3
.161
.141
.349
.382
.594
.161
.141
.349
.382
.594
30-1
28- 2-1
28-2-2
26-3
24,-4-1
24-4-2
B.
C.
B.
31-1
30-2
29-2
24-2
23-2
29-3
20-3
27-4
20-4
17-4
6 x 6
A.
D.
B.
E.
C
a The biases were computed by using equation (3.25).
Note that the
biases for ~2 and ;2 depend only upon the values of ~2 and ~2 and are
r
c
2
2
rc
e
independent of the size of ~ and ~
r
c
b
2 and
The biases for ~"2 depend only upon the value of ~e'
thererc
fore, the biases for ~"2 in the present case are the same as those given
rc
in Table 5.6.
113
Table 5.8
Biases
a
of estimated variance components obtained by
Mwhen
Procedure
Design
Rows
Columns
~2
rc
= 4 and ~2e = 1
Interaction
b
Design
Rows
Columns
.124
.370
.259
.496
.753
.552
.124
.248
.259
.496
.753
.552
.140
.416
.281
.388
.267
.536
.522
. 784
.579
. 798
.140
.256
.281
.256
.267
.536
.522
. 784
.579
. 798
46-2
44-3
42-4
40-4
.066
.161
.134
.164
.064
.097
.137
.157
56-2
50-3
46-4
47-4
.064
.169
.135
.167
.066
.094
.133
.155
72-2
68-3
62-4
52-4
.064
.158
.133
.162
.061
.094
.136
.154
3 x 3
4 x 4
A.
A.
16-1
14-2
12-3
.375
.900
1. 50
.375
.900
1. 50
17-1
15-1
15-2
13-2
12-3
.422
.358
.945
.889
1. 54
.357
.404
.889
.953
1. 54
17-1
16-1
15-1
14-2
13-2
12-2
12-3
9-3
.371
.388
.394
.931
.891
.955
1.54
1.58
.371
.388
.394
.931
.891
.955
1. 54
1. 58
17-1
14-1
13-2
10-2
12-3
.387
.413
.930
1.01
1. 58
.387
.413
.930
1. 01
1. 58
17-1
13-1
15-2
11-2
10-3
.411
.391
.949
.982
1. 59
.411
.391
.949
.982
1. 59
B.
-
30-1
28-2-1
28-2-2
26-3
24-4-1
24-4-2
B.
31-1
30-2
29-2
24-2
23-2
29-3
20-3
27-4
20-4
17-4
6 x 6
C.
A.
D.
B.
E.
C.
a The biases were computed by using equation (3.25).
Note that
the biases for ~2 and ~2 depend only upon the values of ~2 and ~2 and
r
c
2
2
rc
e
are independent of the size of ~ and ~ .
r
c
2
"2
bThe biases for ~
depends only upon the value of ~e) and thererc
fore) the biases for
given in Table 5.6.
"2
~
rc
in the present case are the same as those
114
Table 5.9
cr
2
rc
a
Percentage contribution of the bias of row.... and column
variance components estimated by Procedure M to the mean
square error, 3 x 3 - B designs
cr
2
e
Design
cr
2
r
3x3-B
17-1
15-1
15-2
.25
1
4
1
1
1
Rows
cr
2
c
%
%
< 1
< 1
1
2
Columns
1
< 1
< 1
< 1
13-2
< 1
2
12-3
17-1
15-1
15-2
13-2
12-3
17-1
15-1
15-2
13-2
12-3
17-1
15-1
15-2
13-2
12-3
17-1
15-1
15-2
13-2
12-3
17-1
15-1
15-2
13-2
12-3
17-1
15-1
15-2
13-2
12-3
3
< 1
< 1
< 1
< 1
< 1
< 1
< 1
3
2
6
< 1
< 1
< 1
< 1
1
1
<
< 1
< 1
< 1
< 1
2
1
6
5
10
< 1
< 1
2
2
4
3
< 1
< 1
< 1
< 1
< 1
< 1
< 1
2
3
6
< 1
< 1
< 1
< 1
1
1
<
< 1
< 1
< 1
< 1
1
1
5
6
10
< 1
< 1
2
2
4
16
1
4
16
1
4
16
1
4
16
1
4
continued
115
Table 5.9 (continued)
2
rc
(J"
4
a
2
e
(J"
Design
1
17-1
15-1
15-2
13-2
12-3
(J"
2
r
16
Rows
<
<
<
<
1
1
1
1
< 1
The percentage contribution was computed by
[bias (estimator)]2
[bias (estimator)]2 + Var (estimator)
x (100) .
116
6.
UNWEIGHTED MEANS FOR DESIGNS WITH TWO LEVELS OF NESTING
6.1
Introduction
In this chapter the method of unweighted means will be examined
for estimating variance components from a two-way classification with
s .. classes within each (ij) cell of the design and an unequal number
~J
of observations within each of the s .. classes.
The mathematical
~J
model for this design can be written as
(6.1)
=
i
1,2,""
a; j
= 1,2, .. ·,
b;
k = 1,2, ... ,s .. ; -t = 1,2,· .. ,n" k
~J
~J
is the observed value for the -tth observation in the sk
where Y
ijk-t
. t h e ~.th rowan d J.th co 1 umn. I n t h e mo d e 1 , ~ ~s
. a constant;
class ~n
r., c., rc .. , sk(") and e. 'k' are normally independently distributed
~
J
~J
~J
. b1 es
ran dom var~a
respectively.
!.~.,
~
= n ~J
..
n. 'k
~J
~
~
j
k
.
~
2
~ ,
r
2
~ ,
c
2 ,
~
rc
~
2 an d
s
2
e
~,
As in Chapter 3 totals and means are denoted by dropping
subscripts,
k
~J
'
. h means zero and
w~t
var~ances
n
.
~J
J
=n
ijk
= n~...
~ n ..
,
~
,
i
1
~
~J
y
ijk-t
J
= Yijk
n.
~
~J
-
,
1
n ..
-t
n .. = n. ,
= Y
j
and
~
j
1
n
'
117
The model given in (6.1) will be referred to as a model with two
levels of nesting.
An example of a design with two levels of nesting
is the following:
Columns
1
sl(l1)
1
3
2
s2(11)
sl(12)
s2(12)
B [:;] EJ 5!J
sl (13)
s2 (13)
c;] G;]
s3 (12)
B
sl (21)
Rows
2
s2 (22)
sl (22)
I I I
n 2U
n 221
sl (31)
sl(32)
!
sl (23)
s2 (23)
~ c:;] ~
s2(32)
sl(33)
s2 (33)
5;] G;J ~ 5;] 5;]
3
s3(33)
B
Here a
= 3} b = 3} sll = s13 = s22 = s23 = s32
s21 = s31 = 1.
.
3 and
118
In order to apply the method of unweighted means to a design with
model (6.1) it is first necessary as described in Chapter 3 to compute
the mean for each of the (ij) cells of the design.
Now in Chapter 3
the method of unweighted means was used for designs which had model
(3.1)}
l.~.}
for designs with only one level of nesting within a cell.
In this case the estimation of the cell means is straightforward
because the arithmetic mean of the (ij) cell is known to be the best
linear unbiased estimate of the cell mean.
In the present case (two
levels of nesting within a (ij) cell) the best linear unbiased estimate
y. 'k}
of the (ij) cell mean is to weight the class means}
~J
by the
reciprocals of their variances} i.e.
s ..
s ..
~J
(cell mean) ..
~J
r,
k=l
~J
Yijk
l:
2
cr
2 +_e_)
(cr
s
n
ijk
Of course} the relative values of cr
2
e
k=l
and cr
2
s
1
(6.2)
2
cr
+_e_)
s
n
ijk
(i
are in general unknown}
hence the best linear unbiased estimate of the cell mean cannot be
obtained for model (6.1).
However if cr
2
s
= O} then from (6.2) the best
linear unbiased estimate of the cell mean is
SO.
~J
1
y ij --n ..
~J
l:
s ..
n
l:
L
k=l
t=1
~J
1
n
=-ijk Yijk
n ..
~J
k=l
ijk
Yijkt
(6.3.1)
the simple arithmetic mean of all observations in the (ij) cell which
weights each class mean within the (ij) cell by the number of observations in the class means (n. Ok)'
~J
On the other hand if cr
2
e
= O} then the
119
best linear unbiased estimate of the cell mean is
s
-*
Yij
1
s. ,
L
~J
s ..
ij
q
-
y.~J'k
k=l
1
s, .
~J
n
ijk
L
L
k=l
t=l
Yijkt
n
ijk
(6.3.2)
which gives equal weight to each class mean within the (ij) cell.
Accordingly, Section 6.2 examines the expected mean squares for
designs with model (6.1) using the method of unweighted means with cell
means computed by the limiting cases given in (6.3.1) and (6.3.2).
Based on these two sets of expected mean squares the variance components are estimated by again using the analysis of variance procedure.
In Section 6.3 the variances of the estimated variance components are
studied.
Throughout this chapter it is assumed that the designs under
study have n. 'k >
~J
6.2
a
for all i, j and k.
Expected Mean Squares and Estimated Variance Components
In order to estimate the variance components in model (6.1) by the
analysis of variance procedure based upon the method of unweighted means
using either
y."
~J
or y~. as an estimate of the (ij) cell mean, essen~J
tial1y the same procedure is followed as was described in Chapter 3.
In particular, the (ij) cell means are computed by (6.3.1) or (6.3.2)
and then the formulas given in (3.3) and (3.4) are used to compute the
mean squares for rows, columns and interaction.
Then (3.6.1) is used
to calculate the expected values of these computed mean squares.
The
mean squares for classes nested within the (ij) cells can be computed
by the following weighted formulas:
120
MSS
w
(using
yij)
= MSS
w
= L: L: L: L:
i j k .(,
(Y ijk
-y .. ) 2/ L:L:
~J
s ..
.,
~ J
~J
-
ab
(6.4.1)
and
MSS
[~'~')
w
-*
= MSS*
(using yij)
w
L: L: L: L:
i j k .(,
the (class mean-cell mean)
2
(Y ijk
-*
Yo.) 2/ L:L:
~J
s ..
..
~ J
~J
-
ab
terms are weighted by the number of
observations in the class means (n. 'k) in (6.4.1)J.
~J
The following un-
weighted formulas may also be used to compute the mean squares for
1
classes:
(using
MSS
w
Y. ,) = MSS
~J
L:L:L:
w
i j k
(y,
'k
~J
2/ L:
yij)
~
j
~
s
ij
-
ab
(6.4.2)
and
MSS
w
(using y-*ij)
= MSS*
L:L:L:
w
i j k
[i.e.) the (class mean-cell mean)
(6.4.2)J.
2
(y ijk
-* 2/ L:L:
y,,)
~J
.,
~ J
so,
~J
ab
terms are given equal weight in
The expected values of the mean squares in (6.4.1) and
(6.4.2) can be computed directly using the definitions in (6.1).
The
mean square for subclasses nested within the sk(ij) classes is given by
MSE
Once the expected values of the mean squares are computed the estimates
of the variance components in (6.1) are obtained by equating these
expected mean squares to the computed mean squares.
The resulting
estimates of the variance components are unbiased.
1
Crump [1954J discusses for various parameter combinations the use
of weighted versus unweighted mean squares for a design with two levels
of nesting.
121
As in Chapter 3 the first step in using (3.6.1) to calculate the
expected mean squares for rows} columns and interaction using the
method of unweighted means for a design with model (6.1) is to determine the variance-covariance matrix of the (ij) cell means y .. or
~J
y-*...
~J
Now in terms of the model parameters (6.3.1) becomes
eijk-f,
n ..
~J
and (6.3.2) is
-*
y ij =
I-L
(6.5)
e ij k-f,
s .. (n. 'k)
sk(ij)
+ L: L:
s ..
k -f,
~J
+ r.~ + c. + rc .. + L:
~J
J
k
~J
~J
Thus} the variances and covariances of the cell means yij and Ym-f, are
given by
L:
0"2 + 0"
Cov (Y ) Ym-f,)
ij
r
2
c
2
2
+ 0"
+ 0"
rc
s
n
k
2
ijk
2
2
0"
+~
n ..
n ..
if
~J
~J
2
j = -f,
if
= 0"
r
i =m
i =m
j 1= -f,
0"
2
if
c
}
i I=m
j = -f,
}
and
= 0
if
i I=m
j 1= -f,
Using the notation given in (3.5)} the variance-covariance matrix} V}
-
of the vector of cell means yl = (Y ) Y12}
11
~
...
}
Yab) can then be
122
written as
er
2
k
{ D/c } a + diag (
V
n
L:
s
2
ijk
er
2
+~ )
(6.6)
n ..
2
n ..
~J
~J
where
D
and
=
C = er
2
c
I
b
•
-* and -*
Similarly, the variances and covariances of Y..
Ym-t are
~J
2
er
-* = er + er + er + _s_ + er2 L:
Cov (y-*ij' Ym-t)
r
rc
c
s ..
e k
~J
2
1
2
2
2
if
(sij) n ijk
j = -t,
if
= erc2
i =m
if
i =m
j
I
i
1m
-t,
j = -t
and
=0
if
ilm
j
Thus, the variance-covariance matrix, V*
1.
1
I
-t.
of the vector of cell means
*
V* = { D/c }
2
er
(
_s_
a + diag
s ..
~J
(6. 7)
123
Using (6.6)} (6.7)} (3.6.1) and the formula for G given in (3.3)
R
it can then be shown for the case of all n. k > 0 that the expected
~j
mean squares for rows using the method of unweighted means are given
by
2
= a tr(D·P)
+ a(a-l) tr(C·Q) +
(a~ ~ ~ ~
i
+ <alb
If-)
~ ~
i
j
ij
j
k
n ijk)
- - 0-
n
2
2
s
ij
o-~
2
= b 0-r2
+
i rc
1
+ <ab ~ ~ ~
+
(-1..
ab
2
k
j
i
2
o-s
n ijk)
n ..
~J
~ ~
_1_)
i j
ij
n
2
0-e
and
(6.8)
y .. )
/ using -*
E(MSR
~J
+
(-1..
ab
1
+ (ab
~ ~ _1_)
i
j
s. .
~J
0-
2
s
~ ~ ~
i
j k
In the same manner the expected mean squares for columns and interaction may be derived by using
124
* = tr(V* .GC)
E(MSC )
tr(V' G )
E(MSRC) =
where G
C
RC
and G
,
and
are defined in (3.3).
RC
The expected mean squares for the sk(ij) classes within the (ij)
cells can be derived directly from (6.4.1) and (6.4.2).
The results
of these calculations for the weighted mean squares for classes are
2
ijk
- .E.E.E
n ,
(n
i j k
iJ
.E .E Sij - ab
i j
n
E(MSS )
w
= cr
E(MSS*) = cr
w
+[
2
e
2
e
+[
n ..
(n - .E .E 2.J. )
s. ,
i j
1.J
.E .E s. , - ab
1.J
i j
l]
]
cr
cr
2
s
and
2
s
The corresponding expected values for the unweighted mean squares for
classes are
E(MSS)
w
~[
n ijk ) _1_
.E .E .E (1 i j k
n ij
n ijk ]
.E .E s,. - ab
1.J
i j
i
e
2
(ab -.E
i
.E
j
.E .E
i j
~
s"
1.J
Sij n ijk
n~j
- ab
)] cr2
s
and
125
.[ L:i---"L:~:..:..L:-=-(_l
;,
.
=
k
J
L: L:
s.. n" k
~l-=).....,.-...;;;"l~J
1.)
1.J
si)' - ab
j
i
Tables 6.1 and 6.2 give a summary of the expected mean squares for
designs with two levels of nesting using the method of unweighted
means with cell means Y
and Y~j' respectively.
ij
Examination of
Tables 6.1 and 6.2 shows that if the number of observations in each
(ij) cell of the design are equal (i.e. n"
-
-
1.) k
= m and s .. = s)
1.)
then the
expected mean squares in both tables reduce to
E(MSR)
E(MSC)
1
sm
= - cr
2 +.! 2 + cr2 + b cr2
cr
e
rc
r
s s
2
2
1 2 +.! 2
cr
cr + crrc + a cr
sm e
c
s s
1
sm
2 +.! 2
2
cr + cr
rc
e
s s
E(MSRC) = - cr
2
e
+
2
m cr
s
2
e
E(MSS) = m
w
+
cr
E(MSS ) = cr
w
cr
E(MSE)
= cr
2
s
2
e
As mentioned previously, the results given in Tables 6.1 and 6.2 can be
used to estimate the variance components in (6.1) by setting the
expected mean squares equal to the computed mean squares.
The results
126
Table 6.1
Expected mean squares for a two-way disproportionate design
with two levels of nesting within each cell using the method
of unweighted means with cell means y..
~J
Coefficients of Variance Components in E(MS)
2
2
2
000e
s
rc
Source
K1
K2
=
1 (Z L: _1_)
ab . . n ..
~ J
~J
Rows
1
Columns
ab
Interaction
(L: L: _1_ )
1
ab (L: Z
i j
2
ijk )
2
n ..
n
1
b
~J
n
2
ijk
)
1
n2
1
~ )
ab (L: L: L:
2
k
j
i
n ..
1
ab
i j n ij
(L: L: L:
2
a
i j k n ..
~J
1
)
n ..
-~J
~J
K3
=
2
Classes
L: n ijk )
(n - L:L:
a
..
i j k n ~J
L: L: s .. - ab
i j ~J
1
(using MSS )
w
s
n ijk )
Classes
(using MSS )
w
Error
1
L: L: L: (1 - n
-n
ij
ii k
iik
L: L: s .. - ab
i j
~J
(ab - L:L:L:
i j k
1 -
L: L: s ..
i j
~J
2
ijnijk)
2
n ..
-
~J
ab
1
a
2
r
0-
=
1
(L: L: L:
ab
i j k
1
2
c
0-
The mean square for classes nested within the (ij) cells may be
computed in two different ways as shown in (6.4.1) and (6.4.2).
127
Table 6.2
Expected mean squares for a two-way disproportionate design
with two levels of nesting within each cell using the method
of unweighted means with cell means y~.
~J
Coefficients of Variance Components in E(MS)
2
Source
CT
K1*
Rows
Columns
CT
e
=
CT
2
rc
1
1
L:_
)
ab (L:
s
..
i j q
1
1
1
)
(L: L: L:
ab
2
i j k (s .. )(n. 'k)
1 (L: L: _1_ )
ab . . s ..
1
1
1
)
ab (L: L: L:
2
i j k (s, .)(n. 'k)
1 (L: L: _1_ )
ab . . s ..
1
~
~J
~J
~
~J
J
J
CT
2
c
2
CT
r
=
1
1
)
ab (L: L:L:
2
i j k (sij) (n
ijk )
~J
Interaction
2
s
*
K2
b
a
~J
~J
K3* =
n, ,
Classes
(n - L:L:~)
. , s.,
~ J
~J
a
1
L:L: s ..
(us ing MSS*)
w
1
Classes
(using MSS: )
w
Error
~J
i j
L: L: L: (1 - - )
i j k
L:L: s, ,
i j ~J
Sij
-
-
ab
1
-n
ijk
ab
1
1
a The mean squares for classes nested within the (ij) cells may be
computed in two different ways as shown in (6.4.1) and (6.4.2),
128
2
of this calculation from Table 6.1 are
"2
r
cr
=
MSR - MSRC
b
=
1
bY' (GR - GRC ) 1..
1
crc
"2
MSC - MSRC
a
"2 =
rc
2
MSRC _ K2 (MSS ) +
- K1) (MSE)
w
K3
K3
cr
"2
s
cr
t
(6.9)
(K2
'G
K2
RC 1.. - K3 (MSSw) + K3 - K1) (MSE)
=
1..
=
MSS - MSE
w
K3
cr"2
-a 1..' (GC - GRC ) 1..
MSE
e
Similarly, the results from Table 6.2 are
,*
(GR - GRC ) :i.*
1
,*
(G C - GRC ) :i.*
=
"bY
(~Z) *
c
=
'81..
r
(irc )*
(;2)*
s
and
1
(cr"Z ) *
"Z
cr
e
:i.
,* G
*
RC Z
=
MSS* - MSE
w
K3*
=
MSE .
KZ*
KZ*
+ (K3* - K1*) MSE
K3* (MSS*)
w
ZIn (6.9) and (6.10) MSS
(6.10)
and MSS* respectively are used to estiw"Z
w"Z
mate the variance components cr
and cr. These variance components
rc
s
can also be estimated in a straightforward manner from Tables 6.1 and
6.2 using the unweighted mean squares MSS
w
and MSS*
w
129
6.3
Variances of the Estimated Variance Components
Examination of (6.9) and (6.10) shows that the formulas for the
estimated row and column variance components for a design with two
levels of nesting have exactly the same form as the corresponding
estimates given in (3.8) for the two-way classification with one level
of nesting.
"2
~
r
and
.. Z
~
c
Therefore, the results of Hirotsu for the variances of
given in (3.9) can be used (after replacing the variance-
covariance matrix of the cell means, V, given in (3.5) by V given in
(6.6) or V* given in (6.7»
to obtain the variances of the estimated
row and column variance components for a design with two levels of
nesting.
Thus, the following variances can be shown to hold for the
estimated row and column variance components in (6.9) and (6.10):
+
zis
a (a-l)
4
+ _ _s=----_ [ (H3S _ HZS) + (H5S + HZS - H3S - H4S) ]
aZ(b_l)Z
(a_l)Z
~
~4
+
2
e 2 [ (H3 - H2) +
a (b-l)
(MS + H2 - H3 - H4) ]
2
(a-l)
n
( L:
k
2
ijk)
2
n ..
1.)
n
+
1
(6.11)
L:L:
L:
L:
(a_l)2 i j mfi tfj
(L:
k
2
ijk
2
n ..
1.)
)
(~)]
mt
}
130
where
HI
1
n ..
= E E
i j
H2
:: E E
=E
,
.(,
2
ijk ) ( E
2
k
n ..
~J
n
2
H4S :: E E E ( E ~)( E
2
k n ..
k
i j rn
~J
=E
E E E ( E
ijrn.(,
~J
n2 . 2
ijk )
2
n ..
~J
k
2
ijk )
2
n i .(,
H4
2
n
rnjk )
2
J
n
rnj
H5 :: E E E E
n
2
ijk ) ( E
2
k
n ..
n
H5S
,
n i .(,
~J
H3S :: E E E ( E
k
i j t
k
-1
1
n ..
n
and
EE ( E
H2S = E E ( E
k
i j
~J
E E
i j
HIS
i j
1
2
n ..
i j
H3
,
~J
n2
ijk )
2
n ..
~J
=E
1
n ..
E E
i j rn
1
n
rnj
~J
ijrn.(,
1
n ..
~J
-n 1.(,
rn
2
n
rn.(,k )
2
n
rnt
with a and b interchanged,
~
2
r
replaced by
H4 interchanged and H3S and H4S interchanged.
,
131
2
2
b
(J4 +.1£..
== b2 l a-1 r
a-1
r
2(J2
+
. S
a(a-1)
b (J4
ir irc + (a-I) (b-1)
rc
2(J2
e
a(a-1)
i
(ir +....E£)Hl*
b-l
(J2
«(J2 +....E£ ) HlS*
r
b-l
4
(J
+
+
(H3S* _ H2S*) + (HSS
2
a (b_1)2·
S
[
*
*
*
*
+ H2S - H3S - H4S ) ]
(a_l)2
4
(J e
+
a 2 (b_1)2
[ (H3* _ H2*) + ~::"'----=':=----='==---=:':"':-'.L
(HS* + H2* - H3 * - H4*) ]
2
(a-1)
(6.12)
1
~~ ~
(a_l)2 i j mfi
+
where
H1* ==
~ ~
(
k
i j
H2* ==
H3* ==
~ ~
(
k
~ ~ ~
(
~ ~ ~
i j t
,
1
)2
2
(sij) n ijk
~
i j
i j t
H3S * ==
1
)
2
n
(sij) ijk
~
~ (_1_)(~
~j Sij
k
~
k
1
s ..
~J
HlS* ==
i j
,
1
) ( ~
2
n
k
(sij) ijk
1
-Sit
~ ~
H2S * ==
~ ~
i j
J
1
,
s ..
~J
1
2
S ..
,
~J
1
)
2
n
(s it) Hk
,
132
r.r.r.
H4*
(
i j m
r.
k
(s 0 0)
1.J
1
1
H4S* = r.r.r.
so.
i j m 1.J
H5*
r. r. r. r.
=
k
i j m -t
H5S* = r. r. r. r.
i j m -t
"2 * ]
Var[ (0")
C
s
r.
(
1
2
1
s ..
1.J
) (
n
1
r.
k
ijk
,
)
2
(s 0) n Ok
mJ
mJ
,
mj
1
2
(sij) n ijk
r.
) (
k
1
)
2
(sm-t) nm.(,k
1
sm-t
"2 * ] with a and b interchanged, 0" 2 replaced
Var[ 0")
r
r
2 H3* and H4* interchanged and H3S* and H4S*
by 0" ,
c
interchanged.
Unfortunately, the variances of the estimated row by column
interaction variance components in (6.9) and (6.10) cannot be obtained
from (3.9) due to ,the additional term in their estimating equations
involving MSS
w
and MSS*.
w
In fact, due to the unequal sampling within
the (ij) cells of the design it can be shown that MSS
(MSS*) is not
w
independent of MSRC (MSRC*); and in addition, that SSS
w
w
(SSS*) over
its variance is not distributed as a chi-squared variate.
w
Thus, the
variances of the estimated interaction variance components in (6.9)
and (6.10) are quite complicated and they were not derived.
The analytic formulas for the variances of the estimated row and
column variance components given in (6.11) and (6.12) can be used to
o
f or wh a t parame t er com bOt
exam1.ne
1.na 1.ons
0
(
1..e.
- •
0"2 ,
r
0" 2 ,
c
0" 2 ,
rc
0"2
s
an d
0"2)
e
"2
"2
the
, .. variances of 0" r and 0"c are smaller (or larger) than the variances
133
of
( v~2)* or (~2)*.
v
r
Tha t '~s, f or wh'~c h parame t er va 1 ues ~s
. y.. b e tt er
c
~J
(or worse) than y~. in terms of minimizing the variances of the
~J
estimated row and column variance components obtained by using the
analysis of variance procedure based upon the method of unweighted
means?
"'2 *
Accordingly, the numerical values of the variances of cr"'2 and (cr)
r
r
were computed from (6.11) and (6.12) for 40 parameter combinations
using four designs with two levels of nesting.
following s .. and n"
~J
~J
k
The designs had the
arrangements:
Design
I
II
tl~~ t1
i1 EJGJ QGJ
00 00 00
s 1 (11) s2(11)
[]
[2]
[]
00
00
00
00 00 00 0 G 0
00
00
00
00 00 00 0 0 [2]
Here a
=b =
n. '1 = 1 and n .. 2 = 2 for
n
1, n
all (ij) cells.
n
Here a
~J
=b =
3; s ..
~J
~J
=
2;
ij1
ij3
3; s ..
ij2
~J
=
3;
2 and
3 for all (ij) cells.
134
IV
III
0~ ~~ ~~
0
B
00
B
00
B
G
0
G
~~ ~0
00
88
G G 0
B
88 00 00 00
0 0 0
B
~8 ~8 ~0
Here a = b = 3 ,. s .. = 3',
n
ijl
1, n
B
80
G
~J
1 and
ij2
0~
Here a = b = 3', s .. = 3 ,.
~J
n
B
n
4 for all (ij) cells.
ij3 =
n
ijl
ij3
1, n
1 and
ij2 =
= 10 for all (ij) cells.
"2 * were then
The ratios of the variance of cr"2 to the variance of (cr)
r
r
computed for all designs and parameter combinations and these ratios
are given in Table 6.3.
The results in Table 6.3 indicate the following conclusions:
(a) The variances of cr. . 2 and
"2 * are quite close to each other for
(cr)
most of the cases examined.
The only noticeable exception to this
r
r
cone 1 usion is wh en cr 2 is large compared to cr 2, cr 2 and cr 2 (e.g.
s e r e
r --
cr
2
e
= 1'
1 and cr
2
r
= 2).
In this case, the ratios are
,,2 * are somewhat
relatively large implying that the variances of (cr)
r
"2
smaller than the variances of cr •
r
(i._e. cr2e
= 1 and cr
s
=~)
8
and cr
(b) For the case of cr
2
an d cr 2 no t more t h an
rc
r
2
s
.
«cr
tw~ce
2
e
cr2 the
e
135
Table 6.3
Ratios of the variances of the estimated row variance
components obtained by using the analysis of variance
procedure based upon the method of unweighted means with
cell means y.. and y-*..
~J
~J
Design
Parameters
cr
2
e
cr
2
s
2
cr
rc
I
2
cr
r
Var (~2)
r
Var
«;.2)*
r
II
/
Var ( crZ)
r
Var (0-2) -k
r
III
/
Var (ci)
r
(~2)
i(
Var
r
IV
/
/
Var (crZ)
r
2
Var (0- ) *
r
1
1
1
1
1
1/8
1/2
1
2
8
1
1
1
1
1
2
2
2
2
2
.99
1.00
1.00
1.02
1.08
.99
1.00
1.01
1. 02
1.09
.98
1.00
1.02
1. 07
1. 30
.97
1. 01
1. 07
1. 17
1. 75
1
1
1
1
1
1/8
1/2
1
2
8
2
2
2
2
2
2
2
2
2
2
.99
1.00
1.00
1.02
1. 07
.99
1.00
1.00
1.02
1.09
.98
1.00
1.02
1.06
1. 27
.97
1.01
1.06
1. 16
1. 67
1
1
1
1
1
1/8
1/2
1
2
8
1
1
1
1
1
8
8
8
8
8
1.00
1.00
1.00
1. 01
1.03
1.00
1.00
1.00
1. 01
1. 03
.99
1.00
1. 01
1.02
1.10
.99
1.00
1.02
1.05
1. 23
1
1
1
1
1
1/8
1/2
1
2
8
2
2
2
2
2
8
8
8
8
8
1.00
1.00
1.00
1. 01
1.03
1.00
1.00
1.00
1. 01
1.03
1.00
1.00
1.01
1.02
1. 10
.99
1.00
1.02
1.05
1. 23
1
1
1
1
1
1/8
1/2
1
2
8
8
8
8
8
8
8
8
8
8
8
1.00
1.00
1.00
1.00
1.02
1.00
1.00
1.00
1.00
1.03
1.00
1.00
1. 01
1.02
1.08
.99
1.00
1.02
1.04
1. 19
1
1
1
1
1
1/8
1/2
1
2
8
1
1
1
1
1
16
16
16
16
16
1.00
1.00
1.00
1.00
1.02
1.00
1.00
1.00
1.00
1.02
1.00
1.00
1.00
1.01
1.05
1.00
1.00
1.01
1.02
1. 12
continued
136
Table 6.3 (continued)
Design
Parameters
CT
2
e
CT
2
s
2
CT
rc
II
I
CT
2
r
Var
Var
(,;.2) /
r
"2 ) *
(CT
r
Var (~2)
r
Var (i)*
r
III
/
IV
Var (;2)
r
Var (;2)
r
Var
Var
/
(;2) *
r
/
(;2) *
r
1
1
1
1
1
1/8
1/2
1
2
8
2
2
2
2
2
16
16
16
16
16
1.00
1.00
1.00
1.00
1.02
1.00
1. 00
1.00
1.00
1.02
1.00
1.00
1.00
1.01
1.05
1.00
1.00
1.01
1.02
1. 12
1
1
1
1
1
1/8
1/2
1
2
8
8
8
8
8
8
16
16
16
16
16
1.00
1.00
1.00
1.00
1.01
1.00
1.00
1.00
1.00
1.01
1.00
1.00
1.00
1.01
1.05
1.00
1.00
1.01
1.02
1.11
137
"2 is smaller than that of (rr)
"2 * .
variance of rrr
On the other hand) when
r
2
2
2
2
2
2
and rr are not more than 8 times
rr »rr (e.g. rr = 1) rr = 8) and rr
see
s
rc
r
,,2
rr2 the variance of (;2)* is noticeably smaller than that of rr . (c) Of
e
r
r
the four designs studied) Design IV (the design with the most imbalance
in the n. 'k1s) has ratios which are the most sensitive to varying
~J
parameter values.
In fact) when rr
2
s
=
2
8 the ratios for Design IV are
relatively large ranging from 1.75 (rrrc
rr
2
= 1)
2
rrr
=
2'
2) to 1.11 (rrrc
= 8)
16) .
r
Thus) the ratios in Table 6.3 suggest the following procedure for
estimating the row variance component in model (6.1) based upon the
method of unweighted means if the relative magnitude of the variance
2
If rr «
s
2
while if rr »
s
components are known:
y, .
estimated by
-*
by Yij'
~J
2
then the (ij) cell mean should be
e
2
rr then this mean should be estimated
e
rr
2
2
However) if (a) rr e is approximately equal to rr
s
(~.~.
2
rr s
2
2
2
2
2
,,2
~ rr ~ 2rr ) or (b) rr
and/or rr »rr then the variances of rr and
e
s
rc
r
s
r
2
(;2)* are approximately equal and it makes very little difference
r
whether
Y.,
~J
or y~, is used to estimate the (ij) cell mean.
~J
By
symmetry the same procedure is recommended when estimating the column
variance component.
On the other hand) if no information is available concerning the
relative magnitudes of the variance components then y~. is recommended
~J
over y .. f or
~J
-* (~.~.
y ij
,.
est~mat~ng
when
one and since
rr~ «rr~)
y~,
~J
relatively large.
.
. t e c ho~ce
'
rr2 (rr2 )s~nce
th
e '~nappropr~a
r
c
0
f
still results in relative efficiencies near
provides better estimates of rr
2
2
2
(rr ) when rr is
r
c
s
138
7.
SUMMARY AND SUGGESTIONS FOR FUTURE RESEARCH
7.1
Summary
The method of unweighted means which gives a relatively simple
analysis for crossed classifications with unequal numbers of observations in the cells has been limited to designs where every cell has at
least one observation.
The principle objective of this dissertation
was to examine the method of unweighted means with regard to estimating
the variance components for a two-way classification with unequal
numbers of observations in the cells and with one or more cells having
no observations.
To do this two procedures based upon the method of
unweighted means were developed to estimate the variance components for
a design with missing cells.
These two procedures were then compared
with a procedure based on the method of fitting constants which has
been studied extensively by many authors and found to produce relatively good estimates of the variance components (Bush and Anderson
[1963J).
The estimation procedure for all three methods of estimation
is the analysis of variance procedure which involves calculating mean
squares, computing the expectations of these mean squares under the
variance component model and then equating calculated mean squares to
expected mean squares and solving the set of equations for the variance
components.
Specifically, the mean square errors of the estimated
variance components for each of the three estimation procedures were
compared numerically for several parameter combinations and two-way
designs with missing cells.
The first of the two estimating procedures based upon the method
of unweighted means, Procedure M, estimates the variance components
139
(~, ~2 and ~2 ) by first estimating observations for the missing
r
c
rc
cells in the design.
These observations are estimated in a manner
such that the row by column interaction sum of squares is minimized
after having replaced the observations in each cell by their cell
means.
Having estimated values for the missing cells, Procedure M
computes the mean squares by the method of unweighted means
a~d
then
equates these mean squares to their expectations and solves these
equations for the variance components.
The second procedure, Procedure
~
M, is the same as Procedure M except that approximate expected mean
squares are used which are essentially those that would be obtained by
using the method of unweighted means for a design where there are no
missing cells.
Procedure M gives unbiased estimates for the variance
A
components while Procedure M gives biased estimates.
A
The estimates of the variance components for Procedures M and M
are given in Chapters 3 and 4 for designs with different patterns of
missing cells.
The expectations of the mean squares for these proce-
dures were derived analytically by expressing the mean squares in
~.~.
matrix notation;
the mean square for rows for the method of un-
weighted means is given by MSR
= ~rGR~
where
means and G is a square symmetric matrix.
R
~r
is a vector of cell
Then, the results of
Lancaster [1954] and Searle [1956] were used which state that if the
quadratic form Z
= ~IG~
is a function of variables
mu1tinorma1 distribution with means
then
E(Z)
=
tr(V·G) +
~IG~
~
~,
having a
and variance-covariance matrix V,
140
The analytical results in Chapters 3 and 4 shew that the formulas for
the expected mean squares for the method of unweighted means used in
(1.~.)
Procedure M
the expected mean squares which account for the
fact that the design has missing cells) are relatively simple for
designs with one missing cell (Table 3.1) or missing cells in the same
row. or column (Tables 4.1 and 4.2).
However, for designs with missing
cells in different rows and columns these expected mean squares are
quite complex (Tables 4.3 and 4.4).
squares used in Procedure
M(1.~.,
The approximate expected mean
the expected mean squares which
ignore the fact that the design has missing cells) also have formulas
which are relatively simple (equation (3.23)).
The variances of the estimated variance components for Procedures
M and
Mcan
be derived analytically by expressing the estimated
variance components in matrix notation
variance component for Procedure
Mis
(~.~.)
the estimated row
~2
given by ~
r
=
y'(G
-
R
and again using the results of Lancaster and Searle which state for
2 =
~'G~
as defined above that
Var (2)
=
2 tr(v.G)2 + 4~'GVG~
(7.1)
However) only the variance of the estimated row by column interaction
component
h2
) for a design with one missing cell was derived
rc
(~
analytically due to the enormous amount of matrix algebra involved in
solving (7.1) and the resulting complexity of the variance formula
for ~"2 (equation (3.27)).
rc
Instead, it was decided that for comparison
purposes it was better to write a computer program to calculate the
variances of the estimated variance components obtained by Procedures M
141
" rather than perform tedious matrix algebra which would lead to
and M
complex analytical formulas (of questionable value) for these variances.
In addition, the computer program was set up to (a) compute
the corresponding variances for the procedure based on the method of
fitting constants (denoted by Procedure A) and (b) compute the
coefficients of the expected mean squares for the method of unweighted
means used in Procedure M.
The computer program, written for the
IBM/360 model 75, used (7.1) to compute the variances for Procedures M
and
Mand
a computational method given by Bush [1962] to compute the
variances for Procedure A.
The results of running the computer program for twenty-six 3 x 3
designs, sixteen 4 x 4 designs, twelve 6 x 6 designs and 13 sets of
parameter values are given in Chapter 5.
A summary of these results
indicated that for the majority of the designs and parameter combina"
tions studied Procedures M and M
appear to be as good as or better than
Procedure A for estimating variance components with respect to the
mean square errors of the estimates.
In fact, Procedures M and Mare
always better than Procedure A for estimating ~2 (~2) when ~2 (~2) is
r
much larger than
~2.
rc
ere
The only case where Procedure A is consistently
better than Procedures M and
Mis
for estimating ~2 when ~2 5 ~2 and
rc
rc
e
there is a small proportion of missing cells in the design.
square errors of Procedure
Procedure M.
.
proport~on
0
Mare
The mean
usually relatively close to those of
However in some cases, notably when there are a large
f"
'
m~ss~ng ce 1 1
s '~n a d
es~gn
an d
2.~s d
'
d by
om~nate
r
~
~
2 ,
rc
" is significantly better than Procedure M for estimating
Procedure M
and
2
~.
c
A
Based on these results, and the fact that Procedure M is
~
2
r
142
A
computationally much easier than Procedures M and A (Procedure M
can easily be carried out on a desk calculator) it is recommended that
of the three procedures investigated Procedure
Mbe
used to estimate
the variance components for a two-way classification with unequal
numbers and missing cells.
However, if only unbiased estimates of the
variance components are desired then Procedure M is recommended over
Procedure A because its mean square errors are usually as small as
or smaller than those of Procedure A and it is computationally easier
than Procedure A.
A
In addition to the comparison of Procedures M, M and A, the
results from the computer runs also indicated that (a) the complicated
analytical formulas for the expected mean squares for the method of
unweighted means used in Procedure M can be approximated satisfactorily
in many cases by simpler formulas and (b) the biases in the estimated
variance components for Procedure
Mare
relatively unimportant when
compared with the corresponding mean square errors of the estimates.
In Chapter 6 the method of unweighted means was examined for
estimating the variance components for a two-way classification with
two levels of nesting within each of the cells of the design (classes
within cells and observations within classes).
For this type of
design it was assumed that every cell had at least one observation.
The estimating procedure was again based upon equating computed mean
squares to their expected values; but due to the two levels of nesting
two different cell means were considered for use in the method of
unweighted means, namely (a) the mean of class means weighted by the
numbers of observations per class
j
1.~.,
the simple arithmetic mean of
143
the observations in a cell (denoted by
y..
)
q
and (b) the unweighted
class means within cells (denoted by Y~j).
mean of
Accordingly,
based upon the method of unweighted means two sets of estimates of
the variance components were obtained using
of the cell means.
y..
and y~. as estimates
q
~J
The resulting variances of the estimated row and
column variance components were obtained analytically using the
results of Hirotsu [1966J.
A numerical comparison was then carried
out using these analytic variance formulas for
"2
"2
and ~ to determine
r
c
~
for various parameter combinations which of the two cell means leads
to the estimated variance components with the smallest variances.
The
numerical comparison evaluated the variances for four different 3 x 3
designs and 40 parameter combinations.
The results showed that if
~2r (~2)
and/or ~2 dominate the between classes variance component,
c
rc
(a)
~2,
or if (b) ~2 and ~2 are approximately equal then the variances of
s
e
s
the estimated row (column) variance components are nearly equal for
both cell means.
t h an
.
On the
oth~r
hand, if
~2r (~2)
c
and
~2rc
are not more
O d by ~2, 2 t h en ()
a ~of ~ 2.~s d
om~nate
y .. .
g~ves
s s e
~J
tw~ce ~
.
var~ances
which are a little smaller than those given by Y~. and (b) if ~2
~J
s
2 Yij
-* gives variances which are smaller than those given
dominates ~e'
by
Y...
~J
Based on these results, it is suggested if no prior knowledge of
the relative magnitudes of the variance components is known, that y-*..
~J
(the unweighted cell mean) be used in estimating ~2 and ~2 for the two
r
c
levels of nesting design.
the inappropriate choice of
The reason for this recommendation is that
Y: j
(!.~. when ~; is dominated by ~~) still
results in variances for &2 and &2 that are approximately equal to
r
c
144
those given by -Yij while if
2
~s
-* provides better
is relatively large Y
ij
estimates of ~2 and ~2 than does
r
c
7.2
1.
y...
~J
Suggested Future Research
Extend the two-way model to three-way and higher order cross
classifications with missing cells to see if similar results are
obtained.
2.
Study the level of significance and power of approximate tests of
hypotheses of the variance components for designs with missing
cells based upon the expected mean squares used in Procedures M
and
M)
the properties of these tests should be compared to
previous work on the properties of tests for designs with no
missing cells.
3.
Study the possibility of approximating the variances of the
variance component estimators by simpler formulas which do not
require the use of a computer.
4.
Investigate the use of other estimation procedures for the
variance components from unbalanced data which do not necessarily
produce unbiased estimates but are easy to compute and have other
attractive properties such as relatively small mean square errors.
5.
Compare Procedures M and
besides Procedure A,
~.~.
Mwith
other estimation procedures
Henderson's Method 1, KochIs symmetric
sums) maximum likelihood) etc.
6.
" to estimate
Examine the possibility of using Procedures M and M
the variance components for a two-way classification with missing
cells and two levels of nesting within each of the occupied cells.
145
8.
LIST OF REFERENCES
Anderson, R. L. 1961. Designs for estimating variance components.
Institute of Statistics, Mimeo Series No. 310. North Carolina
State University at Raleigh, North Carolina.
Anderson, R. L. and Bancroft, T. A. 1952. Statistical Theory in
Research. McGraw-Hill Book Co., Inc., New York, New York.
Blischke, W. R. 1968. Variance of moment estimators of variance
components in the unbalanced R-way classification. Biometrics
24:527-539.
Burrows, P. M. 1966. Estimation of multiple missing values in
standard experimental designs. Rhod. Zambo Mal. J. Agric. Res.
4:45-47.
Bush, N. 1962. Estimating variance components in a multi-way
classification. Unpublished Ph.D. thesis, Department of Experimental Statistics, North Carolina State University at Raleigh,
North Carolina. University Microfilms, Ann Arbor, Michigan.
Bush, N. and Anderson, R. L. 1963. A comparison of three different
procedures for estimating variance components. Technometrics 5:
421-440.
Crump, P. P. 1954. Optimal designs to estimate the parameters of a
variance component model. Unpublished Ph.D. thesis, Department
of Experimental Statistics, North Carolina State University at
Raleigh, North Carolina.
Eisenhart, C. 1947. The assumptions underlying the analysis of
variance. Biometrics 3:1-21.
Gaylor, D. W. 1960. The construction and evaluation of some designs
for the estimation of parameters in random models. Unpublished
Ph.D. thesis, Department of Experimental Statistics, North
Carolina State University at Raleigh, North Carolina. Institute
of Statistics, Mimeo Series No. 256.
Gaylor, D. W., Lucas, H. L., and Anderson, R. L. 1970" Calculation
of expected mean squares by the abbreviated Doolittle and
square root methods. Biometrics 26:641-655.
Gosslee, D. G. 1956. Level of significance and power of the unweighted means' test. Unpublished Ph.D. thesis, Department of
Experimental Statistics, North Carolina State University at
Raleigh, North Carolina.
Gosslee, D. G. and Lucas, H. L. 1~~5. Analysis of variance of disproportionate data when interaction is present. Biometrics 21:
115-133.
146
Hartley, H. O. 1967. Expectations, variances, and covariances of
ANOVA mean squares by 'synthesis'. Biometrics 23:105-114.
Hartley, H. O. and Rao, J. N. K. 1967. Maximum-likelihood estimation
for the mixed analysis of variance model. Biometrics 54:93-108.
Harville, D. A. 1969. Variances of variance-component estimators for
the unbalanced 2-way cross classification with application to
balanced incomplete block designs. Annals Math. Stat. 40:408416.
Henderson, C. R.
components.
1953. Estimation of variance and covariance
Biometrics 9:226-252.
Hirotsu, C. 1966. Estimating variance components in a two-way layout
with unequal numbers of observations. Rep. Stat. App1. Res. JUSE
13:29-34.
Hirotsu, C. 1968. An approximate test for the case of random effect
model in a two-way layout with unequal cell frequencies. Rep.
Stat. Appl. Res. JUSE 15:13-26.
Koch, G. G. 1967. A general approach to the estimation of variance
components. Technometrics 9:93-118.
Koch, G. G. 1968. Some further remarks concerning 'a general
approach to the estimation of variance components'. Technometrics
10:551-558.
Lancaster, H. O. 1954. Traces and cumu1ants of quadratic forms in
normal variables. J. Roy. Stat. Soc., Supp1. 16:247-254.
Low, L. Y. 1964. Sampling variances of estimates of components of
variance from a non-orthogonal two-way classification. Biometrika
51:491-494.
Mahamunu1u, D. M. 1963. Sampling variances of the estimates of
variance components in the unbalanced 3-way nested classification.
Annals Math. Stat. 34:521-527.
Searle, S. R. 1956. Matrix methods in components of variance and
covariance analysis. Annals Math. Stat. 27:737-748.
Searle, S. R. 1958. Sampling variances of estimates of components of
variance. Annals Math. Stat. 29:167-178.
Searle, S. R. 1961. Variance components in the unbalanced 2-way
nested classification. Annals Math. Stat. 32:1161-1166.
Searle, S. R. 1967. Computer simulation of variance components
estimates. Paper No. BU-233-M, Mimeo Series, Biometrics Unit,
Cornell University, Ithaca, New York.
147
Searle, S. R. 1968. Another look at Henderson's methods of
estimating variance components (with discussion). Biometrics
24:749-787.
Searle, S. R. 1970. Large sample variances of maximum likelihood
estimators of variance components using unbalanced data.
Biometrics 26:505-524.
Webster, J. T.
597-604.
1968.
An approximate F-statistic.
Technometrics 10:
Yates, F. 1933. The analysis of replicated experiments when the
field results are incomplete. Empire J. Exp. Agr. 1:129-142.
Yates, F. 1934. The analysis of multiple classifications with
unequal numbers in the different classes. J. Amer. Stat. Assoc.
29:51-66.
148
9.
APPENDIX.
MEAN SQUARE ERRORS OF ESTIMATES OF VARIANCE COMPONENTS
The mean square errors of the estimates of the variance components
for each design and parameter combination given in Section 5.3 are
shown in Tables 9.1 through 9.10 for the estimation procedures A, M and
A
M.
Note that since Procedures A and M are unbiased procedures, their
mean square errors are the same as their variances.
149
Table 9.1
Mean square errors of estimates of variance components for
3 x 3 - A designs
Parameters
2 2 2
2
cr cr cr
cr
r c rc e
1
1 .25
1
Rows
Procedure
...
a
A
M
M
Columns
Procedure
...
A
M
M
Interaction
Procedure
...
M
M
A
b
18
16-1
14-2
12-3
1. 79 1. 79 1.77
2.09 2.13 2.05
2.75 2.75 2.44
1. 79 1. 79 1.77
2.09 2.13 2.05
2. 75 2. 75 2.44
.34
.44 .44 .44
.63 .63 .63
1. 21 1.21 1.21
264.
276. 266. 266.
284. 269. 269.
273. 273. 272.
c
19.2 18. 7 18.6
20.3 19.5 19.4
20.8 20.8 20.5
19.2 18.7 18.6
20.3 19.5 19.4
20.8 20.8 20.5
Design
16
1 .25
1
18
16-1
14-2
12-3
4
4 .25
1
16-1
14-2
12-3
16 16 .25
1
16-1
14-2
12-3
1
18
16-1
14-2
12-3
2.38
2.84 2.92 2.83
3.69 3.92 3.62
6.00 6.00 4.75
18
16-1
14-2
12-3
20.4
22.0 21. 7 21. 6
24.1 23. 7 23.4
27.0 27.0 25.8
1
18
16-1
14-2
12-3
272.
286. 277. 277.
297. 283. 283.
291. 291. 290.
1
18
16-1
14-2
12-3
20.4
22.0 21. 7 21. 6
24.1 23.7 23.4
27.0 27.0 25.8
18
16-1
14-2
12-3
272.
286. 277. 277.
297. 283. 283.
291. 291. 290.
1
4
16
16
1
1
1
4
16 16
1
1
1
1
1
1
1
276. 266. 266.
284. 269. 269.
273. 273. 272.
2.38
2.84 2.92 2.83
3.69 3.92 3.62
6.00 6.00 4. 75
1.18
1. 56 1. 56 1. 56
2.32 2.32 2.32
4.58 4.58 4.58
continued
150
Table 9.1 (continued)
Parameters
2
2 2 2
cr cr crrc cr
e
c
r
1
1
4
1
4
1
4
1
16
4
1
4
4
4
1
1
Columns
Procedure
Rows
Procedure
Design
a
A
M
'"
M
18
16-1
14-2
12-3
7.38
10.0 10.8 10.1
15.7 17.7 15.0
34.0 34.0 22.8
16-1
14-2
12-3
36.4 37.0 36.3
45.1 47.1 44.4
67.0 67.0 55.8
18
16-1
14-2
12-3
307.
329. 322. 321.
354. 345. 342.
379. 379. 368.
18
16-1
14-2
12-3
A
M
'"
M
10.0 10.8 10.1
15.7 17.7 15.0
34.0 34.0 22.8
Interaction
Procedure
A
M
'"
M
10.2
13.6 13.6 13.6
20 . 3 20. 3 20. 3
40.6 40.6 40.6
31.4
36.4 37.0 36.3
45.1 47.1 44.4
67.0 67.0 55. 8
aprocedure A",is based upon the method of fitting constants, and
Procedures M and M are based upon the method of unweighted means.
Procedures A and M give unbiased estimates for the variance components
while Procedure Mgives bia,sed estimates.
b
The mean square errors for the base design (i.e. design 18) are
given only for those cases .computed by Bush [1962]-for Procedure A.
c , .
"'2
Procedures A, M and M have the properties that M. S. E. (cr) remains
r
invariant over any change in cr2 ; M.S.E. (~2) remains invariant over any
c
c
change in cr2 ; and M.S.E.(~2 ) remains invariant over any change in cr2
rc
r
2
r
and cr. Thus, if a cell of the table is left blank it implies that the
c
results for this cell have already been given.
151
Table 9.2
Mean square errors of estimates of variance components for
3 x 3 - B designs
Parameters
2 (]"2 (]"2 (]"2
(]"
r c rc e
1
16
4
1 .25
1 .25
4 .25
16 16 .25
1
4
1
1
1
1
Columns
Procedure
Rows
Procedure
,.,
M
M
A
M
M
b
18
17-1
15-1
15-2
13-2
12-3
2.01
1. 93
2.28
2.3.5
3.16
1. 99
1.92
2.35
2.37
3.12
1. 97
1. 88
2.28
2.24
2.72
1. 76
2.00
2.09
2.49
3.18
1. 79
2.04
2.15
2.5.1
3.08
1. 76
2.01
2.06
2.47
2. 70
18
17-1
15-1
15-2
13-2
12-3
292.
328.
278.
316.
272.
284.
268.
267.
271.
271.
275.
268.
267.
271.
271.
275.
1
17-1
15-1
15-2
13-2
12-3
22.5
19.6
22.4
20.1
22.1
19.2
19.0
20.1
20.1
21. 7
19.2
19.0
20.0
20.0
21. 2
18.6
20.7
19.8
22.3
22.6
18.6
19.4
19.6
20.8
21.5
18.6
19.4
19.5
20.6
21. 2
1
17-1
15-1
15-2
13-2
12-3
268.
296.
278.
307.
295.
266.
269.
269.
273.
275.
266.
269.
269.
273.
275.
1
18
17-1
15-1
15-2
13-2
12-3
2.63
3.09
3.15
3.95
4.28
6.79
3.17
3.08
4.23
4.28
6.59
3.10
2.98
3.93
3.86
5.16
2.59
2.95 2.91
3.27 3.24
3.973.96
4.54 4.55
7.04 6.55
2.82
3.15
3.63
4.19
5.15
18
17-1
15-1
15-2
13-2
12-3
22.5
25.5
22.8
26.5
24.5
28.9
22.3
22.0
24.4
24.4
28.0
22.2
21.9
24.1
24.0
26.6
1
1
1
,.,
,.,
a
A
Design
Interaction
Procedure
A
M
M
.39
.43 .51 .52
.59 .69 .69
.66 • 74 .73
1.01 1.05 1.04
1. 61 1.60 1. 60
c
1. 31
1.59
1. 88
2.44
2.98
5.32
1.68
1.97
2.51
3.02
5.31
1. 69
1. 97
2.50
3.01
5.31
continued
152
Table 9.2 (continued)
Parameters
2
2 2 2
cr cr cr
cr
r c rc e
16
16
1
4
16 16
1
4
16
1
1
1
1
1
1
4
4
4
1
Collnnns
Procedure
Rows
Procedure
Design
a
A
M
"
M
b
18
17-1
15-1
15-2
13-2
12-3
302.
339.
289.
330.
286.
304.
279.
278.
285.
285.
294.
279.
278.
285.
285.
292.
A
A
M
M
18
17-1
15-1
15-2
13-2
12.. 3
20.9
21.8
24.1
24.3
27.1
30.0
21. 6
22.5
23.8
25.1
28.0
21. 6
22.4
23.5
24. 7
26.6
1
18
17-1
15-1
15-2
13-2
12-3
274.
279.
307.
292.
323.
317.
277.
279.
283.
287.
294.
277.
279.
283.
287.
292.
1
18
17-1
15-1
15-2
13-2
12-3
8.15
10.3
11.7
16.4
18.7
36.8
11.2
11. 1
18.3
18.5
35.5
10.6
10.3
15.6
15.5
23.7
11.4
11.9
18.5
19.2
38.7
10.8
11.4
17.8
19.1
35.4
10.0
10.7
15.0
16.2
23.7
1
17-1
15-1
15-2
13-2
12-3
40.6
39.4
48.7
48.7
71.8
37.9
37.5
48.1
48.2
68.9
37.2
36.7
45.4
45.2
57.1
18
17-1
15-1
15-2
13-2
12-3
340.
385.
338.
391.
350.
398.
324.
323.
347.
347.
383.
324.
323.
345.
344.
371.
1
1
Interaction
Procedure
A
M
11. 3
14.3
15.4
21. 5
22.3
42.7
13.9
14.6
20.8
22.2
42.6
"
M
13.9
14.6
20.8
22.1
42.6
continued
153
Table 9.2 (continued)
Parameters
(J'2 (J'2 (J'2 (J'2
r c rc e
4
4
4
1
Rows
Procedure
Design
Columns
Procedure
,.
,.
A
M
18
17-1
15-1
15-2
13-2
12-3
M
A
M
M
34.1
38.3
41. 2
49.0
52.8
75.6
37.0
38. 1
47.3
49.3
68.9
36.2
37. 4
44.5
46.4
57.2
Interaction
Procedure
x
A
M
M
aprocedure A",is based upon the method of fitting constants, and
Procedures M and M are based upon the method of unweighted means.
Procedures A and",M give unbiased estimates for the variance components
while Procedure M gives biased estimates.
b
The mean square errors for the base design (i.e. design 18) are
given only for those cases computed by Bush [1962]-for Procedure A.
cprocedures A, M and
Mhave
the properties that M.S.E. (~2) remains
r
invariant over any change in (J'2; M.S.E. (~2) remains invariant over any
c
c
change in (J'2; and M.S.E. (~2 ) remains invariant over any change in (J'2
r
rc
r
and (J'2 • Thus, if a cell of the table is left blank it implies that the
c
results for this cell have already been given.
154
Table 9.3
Mean square errors of estimates of variance components for
3 x 3 - C designs
Parameters
2 2
cr2
cr2 cr cr
r c rc e
1
16
4
1 .25
1 .25
4 .25
16 16 .25
Columns
Procedure
Rows
Procedure
,..
,..
Interaction
Procedure
,..
a
A
M
M
A
M
M
A
M
M
1
b
18
17-1
16-1
15-1
14-2
13-2
12-2
12-3
9-3
1.83
1. 93
1. 96
2.27
2.35
2.45
3.16
3.67
1.86
1.95
1. 99
2.33
2.36
2.54
3.12
3.67
1. 83
1.93
1. 97
2.25
2.23
2.44
2.72
3.11
1. 83
1. 93
1. 96
2.27
2.35
2.45
3.16
3.67
1. 86
1. 95
1. 99
2033
2.36
2.54
3.12
3.67
1. 83
1. 93
1. 97
2.25
2.23
2.44
2.72
3.11
.42
.49
.57
.60
.80
1.02
.98
1. 60
2.38
.54
.61
.64
.80
1.04
1.03
1. 60
2.38
.54
.61
.65
.80
1.03
1.04
1. 60
2.38
1
18
17-1
16-1
15-1
14-2
13-2
12-2
12-3
9-3
265.
271.
282.
286.
292.
271.
306.
284.
279.
267.
268.
268.
271.
271.
273.
275.
279.
267.
268.
268.
271.
271.
273.
275.
279.
17-1
16-1
15-1
14-2
13-2
12-2
12-3
9-3
19.0
19.9
20.1
21. 1
20.0
22.2
22.1
22.7
18.9
19.2
19.2
20.1
20.1
20.6
21. 6
22.7
18.8
19.1
19.2
20.0
19.9
20.5
21. 2
22.1
19.0
19.9
20.1
21.1
20.0
22.2
22.1
22.7
18.9
19.2
19.2
20.1
20.1
2006
21. 6
2207
18.8
19.1
19.2
20.0
19.9
20.5
21. 2
22.1
271.
282.
286.
292.
271.
306.
284.
279.
267.
268.
268.
271.
271.
273.
275.
279.
267.
268.
268.
271.
271.
273.
275.
279.
1
1
Design
17-1
16-1
15-1
14-2
13-2
12-2
12-3
9-3
c
continued
155
Table 9.3 (continued)
Parameters
2 2 2
2
CT
CT
r CTc CTrc e
1
4
16
16
1
1
1
4
1
1
1
1
Design
Rows
Procedure
,.
a
M
A
M
1
18
17-1
16-1
15-1
14-2
13-2
12-2
12-3
9-3
2.57
3.01
3.13
3.17
4.07
4.28
4.26
6.78
7.42
3.01
3.14
3.18
4.21
4.27
4.50
6.59
7.42
2.91
3.05
3.09
3.89
3.85
4.15
5.16
5.72
1
18
17-1
16-1
15-1
14-2
13-2
12-2
12-3
9-3
20.9
22.1
23.0
23.3
25.3
24.4
26.5
28.9
29.4
21. 9
22.2
22.3
24.4
24.4
25.0
28.0
29.4
21.8
22.1
22.2
24.1
24.0
24.6
26.6
27.7
1
18
17-1
16-1
15-1
14-2
13-2
12-2
12-3
9-3
274.
282.
293.
297.
306.
285.
320.
304.
297.
277.
279.
279.
285.
285.
287.
294.
297.
277.
278.
279.
285.
284.
286.
292.
295.
1
18
17-1
16-1
15-1
14-2
13-2
12-2
12-3
9-3
Columns
Procedure
,.
A
M
2.57
3.01
3.13
3.17
4.07
4.28
4.26
6.78
7.42
3.01
3.14
3.18
4.21
4.27
4.50
6.59
7.42
2.91
3.05
3.09
3.89
3.85
4. 15
5.16
5. 72
20.9
22.1
23.0
23.3
25.3
24.4
26.5
28.9
29.4
21. 9
22.2
22.3
24.4
24.4
25.0
28.0
29.4
21. 8
22.1
22.2
24.1
24.0
24.6
26.6
27.7
M
Interaction
Procedure
,.
A
M
M
1. 35
1. 70
1.81
1. 86
2.62
2.99
2.91
5.31
6. 50
1. 74
1. 85
1. 90
2.62
3.01
2.96
5.31
6. 50
1. 74
1. 86
1.91
2.62
3.00
2.98
5.31
6. 50
continued
156
Table 9.3 (continued)
Parameters
2 2 2
2
CT
CT
CT
CT
r c rc e
16 16
1
4
16
1
1
1
1
4
4
4
Design
Rows
Procedure
,.
a
M
A
M
1
18
17-1
16-1
15-1
·14-2
13-2
12-2
12-3
9-3
1
18
17-1
16-1
15-1
14-2
13-2
12-2
12-3
9-3
8.36
11. 2
11.3
11.3
17.3
18.7
17.3
36.8
37.4
10.9
11.2
11.3
18.3
18.5
19.0
35.5
37.4
10.2
10.5
10.6
15.6
15.5
16.1
23.7
24.9
17-1
16-1
15-1
14-2
13-2
12-2
12-3
9-3
38.2
39.1
39.5
48.4
48.6
49.4
71.8
71.4
37.3
37.8
37.9
48.1
48.2
49.0
68.9
71.4
36.6
37.1
37.2
45.4
45.2
46.1
57.2
58.9
18
17-1
16-1
15-1
14-2
13-2
12-2
12-3
9-3
313.
329.
341.
345.
369.
349.
382.
398.
387.
323.
324.
324.
347.
347.
349.
383.
387.
322.
323.
324.
3.45.
344.
347.
371.
375.
1
1
Columns
Procedure
,.
A
M
M
274.
282.
293.
297.
306.
285.
320.
304.
297.
277.
279.
279.
285.
285.
287.
294.
297.
277.
278.
279.
285.
284.
286.
292.
295.
11. 2
11. 3
11. 3
17.3
18.7
17.3
36.8
37.4
10.9
11. 2
11. 3
18.3
18.5
19.0
35.5
37.4
10.2
10.5
10.6
15.6
15.5
16.1
23.7
24.9
Interaction
Procedure
,.
A
11.1
14.5
14.6
14.7
21. 2
22.3
22.2
42.6
45.5
M
14.1
14.3
14.4
21.2
22.1
22.0
42.6
45.5
M
14.1
14.3
14.4
21.2
22.1
22.0
42.6
45.5
continued
157
Table 9.3 (continued)
Parameters
2 2 0-2
2
000r c rc e
4
4
4
1
Design
Rows
Procedure
...
a
M
A
M
18
17-1
16-1
15-1
14-2
13-2
12-2
12-3
9-3
Columns
Procedure
...
A
M
M
33.3
38.2
39.1
39.5
48.4
48.6
49.4
71. 8
71.4
37.3
37.8
37.9
48.1
48.2
49.0
68.9
71.4
36.6
37.1
37.2
45.4
45.2
46.1
57. 2
58.9
Interaction
Procedure
,.
A
M
M
aprocedure A is based upon the method of fitting constants, and
Procedures M and Mare based upon the method of unweighted means.
Procedures A and M give unbiased estimates for the variance components
while Procedure Mgives biased estimates.
b
The mean square errors for the base design (i.e. design 18) are
given only for those cases computed by Bush [1962]-for Procedure A.
c , .
Procedures A, M and M have the properties that M. S. E.
.
.
~nvar~an
t over any c h
ange'~n
(rr
2
0-;
c
M•S•E
• ("'2)
0c
,.2
r
(0-)
remains
. .~nvar~an
. t over any
rema~ns
change in 0-2 ; and M.S.E. 2 ) remains invariant over any change in 0-2
r
rc
r
and 0-2 • Thus, if a cell of the table is left blank it implies that the
c
results for this cell have already been given.
158
Table 9.4
Mean square errors of estimates of variance components for
3 x 3 - D designs
Parameters
2 (J'2 (J'2 (J'2
(J'
r c rc e
1
16
4
1 .25
1 .25
4 .25
16 16 .25
1
4
1
1
1
1
Design
Aa
M
b
18
17-1
14-1
13-2
10-2
12-3
2.03
2.16
2.59
2.82
3. 67
2.07
2.23
2.64
3.03
3. 65
1
18
17-1
14-1
13-2
10-2
12-3
266.
277.
290.
279.
315.
283.
1
17-1
14-1
13-2
10-2
12-3
19.8
20.8
20.9
23.5
22.8
1
17-1
14-1
13-2
10-2
12-3
1
1
1
Columns
Procedure
Rows
Procedure
A
M
"
M
2.04
2.19
2.49
2.89
3. 10
2.03
2.16
2.59
2.82
3.67
2.07
2.23
2.64
3.03
3.65
2.03
2.19
2.49
2.89
3.10
269.
271.
274.
277.
279.
269.
271.
273.
277.
278.
c
19.5
19.9
20.8
21.8
22.7
19.4
19.9
20.6
21. 7
22.1
19.8
20.8
20.9
23.5
22.9
19.5
19.9
20.8
21.8
22.7
19.4
19.9
20.6
21. 7
22.1
277.
290.
279.
315.
283.
269.
271.
274.
277.
279.
269.
271.
273.
277.
278.
2.73
3.27
3.45
4.61
4.78
7.52
3.29
3.49
4.66
5.17
7.40
3.19
3.39
4.22
4.75
5.71
~
18
17-1
14-1
13-2
10-2
12-3
2.73
3.27
3.45
4.61
4.78
7.52
3.29
3.49
4.66
5.17
7.40
3.19
3.39
4.22
4.75
5.71
18
17-1
14-1
13-2
10-2
12-3
21. 3
23.0
24.1
25.4
27.9
29.9
22.6
23.1
25.2
26.3
29.4
22.5
22.9
24.8
25.9
27.7
Interaction
Procedure
A
A
M
M
.57
.71 .77 .77
.88 .95 .96
1. 31 1. 32 1. 31
1. 61 1.71 1. 76
2.19 2.19 2.19
1. 60
2.06
2.29
3.44
3.81
6.31
2.13
2.37
3.46
3.93
6.31
2.12
2.38
3.45
3.98
6.31
continued
159
Table 9.4 (continued)
Parameters
2
2 2 2
0"
0"
0"
0"
r c rc e
16
16
1
4
16 16
1
4
16
1
1
1
1
1
1
4
4
4
1
Columns
Procedure
Rows
Procedure
Design
a
A
M
""
M
18
17-1
14-1
13-2
10-2
12-3
276.
288.
301.
293.
329.
303.
280.
281.
288.
291.
297.
280.
281.
287.
290.
295.
A
M
"
M
18
17-1
14-1
13-2
10-2
12-3
21. 3
23.0
24.1
25.4
27.9
29.9
22.6
23.1
25.2
26.3
29.4
22.5
22.9
24.8
25.9
27.7
1
18
17-1
14-1
13-2
10-2
12-3
276.
288.
301.
293.
329.
303.
280.
281.
288.
291.
297.
280.
281.
287.
290.
295.
1
18
17-1
14-1
13-2
10-2
12-3
8.69
11. 7
11. 9
19.1
18.3
38.2
11.5
11.9
19.3
20.3
37.4
10.7
11.1
16.3
17.3
24.9
11. 7
11.9
19.1
18.3
38.2
11.5
11. 9
19.3
20.3
37.4
10.7
11.1
16.3
17.3
24.9
1
17-1
14-1
13-2
10-2
12-3
39.2
40.6
49.8
51. 6
73.1
38.3
39.0
49.5
51. 1
71.4
37.5
38.2
46.4
48. 0
58.9
18
17-1
14-1
13-2
10-2
12-3
315.
335.
350.
357.
393.
396.
325.
327.
350.
354.
387.
325.
326.
347.
351.
375.
1
1
Interaction
Procedure
"
A
M
M
11. 7
15.3
15.7
23.3
24.2
45.3
15.0
15.5
23.3
24.0
45.3
15.0
15.5
23.3
24.1
45.3
continued
160
Table 9.4 (continued)
Parameters
2 2
cr2
cr
r c rc e
cicr
4
4
4
1
Design
Rows
Procedure
a
"
M
M
A
18
17-1
14-1
13-2
10-2
12-3
Columns
Procedure
A
M
"
M
33.9
39.2
40.6
49.8
51. 6
73.1
38.3
39.0
49.5
51.1
71.4
37.5
38.2
46.4
48.0
58.9
Interaction
Procedure
"
M
A
M
~Procedure A"is based upon the method of fitting constants, and
Procedures M and M are based upon the method of unweighted means.
Procedures A and M give unbiased estimates for the variance components
while Procedure Mgives biased estimates.
b
The mean square errors for the base design (i.e. design 18) are
given only for those cases computed by Bush [1962]-for Procedure A.
c Procedures A, M and M
" have the
.
propert~es
that M.S.E. (cr....2) remains
r
2
2
invariant over any change in cr ; M.S.E.(rr ) remains invariant over any
c
c
. cr2 ; an d M•S• E
..
.
h·
2
c h ange ~n
• (,,2
cr ) rema~ns
~nvar~ant over any c ange ~n cr
2
r
rc
r
and cr. Thus, if a cell of the table is left blank it implies that the
c
results for this cell have already been given.
161
Table 9.5
Mean square errors of estimates of variance components for
3 x 3 - E designs
Parameters
2 (J2 (J2 (J2
(J
r c rc e
1
16
4
1 .25
1 .25
4 .25
16 16 .25
1
4
1
1
1
1
Columns
Procedure
Rows
Procedure
Design
,.
,.
a
A
M
M
A
M
M
b
18
17-1
13-1
15-2
11-2
10-3
2.07
2.11
2.48
2.72
3.78
2.04
2.14
2.49
2.87
3.76
2.02
2.11
2.39
2.72
3.19
2.07
2.11
2.48
2.72
3.78
2.04
2.14
2.49
2.87
3.76
2.02
2.11
2.39
2.72
3.19
18
17-1
13-1
15-2
11-2
10-3
295.
325.
275.
315.
297.
283.
269.
270.
273.
276.
279.
269.
270.
272.
275.
279.
1
17-1
13-1
15-2
11-2
10.,.3
22.5
19.8
22.7
22.3
23.1
19.4
19.7
20.5
21.4
22.9
19.4
19.6
20.4
21.3
22.3
22.5
19.8
22.7
22.3
23.1
19.4
19. 7
20.5
21.4
22.9
19.4
19.6
20.4
21.3
22.3
1
17-1
13-1
15-2
11-2
10-3
325.
275.
315.
297.
283.
269.
270.
273.
276.
279.
269.
270.
272.
275.
279.
1
18
17-1
13-1
15-2
11-2
10-3
2.91
3.39
3.44
4.63
4.61
7.67
3.25
3.38
4.44
4.96
7.56
3.17
3.27
4.09
4.54
5.82
2.91
3.39
3.44
4.63
4.61
7.67
3.25
3.38
4.44
4.96
7.56
3.17
3.27
4.09
4.54
5.82
1
18
17-1
13-1
15-2
11-2
10-3
23.4
26.1
23.2
27.8
26.5
30.0
22.5
22.8
24.9
25.9
29.7
22.4
22.7
24.5
25.4
28.0
1
1
Interaction
Procedure
,..
A
M
M
.45
.49 .64 .65
.92 .96 .96
.90 .95 .96
1.48 1.49 1. 48
2.42 2.41 2.41
c
1.48
1. 78
2.33
2.83
3.65
6.63
1. 90
2.37
2.88
3.66
6.61
1. 91
2.36
2.89
3.65
6.62
continued
162
Table 9.5 (continued)
Parameters
2 2 2 cr2
cr cr cr
r c rc e
16
16
1
4
16 16
1
4
16
1
1
1
1
1
1
4
4
4
1
Columns
Procedure
Rows
Procedure
Design
a
A
M
'"
M
18
17-1
13-1
15-2
11-2
10-3
306.
338.
286.
332.
310.
302.
279.
280.
287.
290.
298.
279.
280.
286.
289.
296.
A
M
"
M
1
18
17-1
13-1
15-2
11-2
10-3
23.4
26.1
23.2
2.7.8
26.5
30.0
22.5
22.8
24.9
25.9
29. 7
22.4
22.7
24.5
25.4
28.0
1
18
17-1
13-1
15-2
11-2
10-3
306.
338.
286.
332.
310.
302.
279.
280.
287.
290.
298.
279.
280.
286.
289.
296.
12.4
12.5
20.2
17.8
38.4
11.4
11.7
18.8
19.9
37.8
10.7
10.9
16.0
16.9
25.1
10... 3
9.89
12.4
12.5
20.2
17.8
38.4
11.4
11. 7
18.8
19.9
37.8
10.7
10.9
16.0
16.9
25.1
1
17-1
13-1
15-2
11-2
10-3
44.4
40.4
55.4
49.1
73.2
38.1
38.6
48.9
50.5
71.9
37.4
37.9
46.0
47.4
59.2
1
18
17-1
13-1
15-2
11-2
10-3
353.
393.
336.
408.
370.
395.
325.
326.
349.
353.
388.
324.
325.
346.
350.
376.
1
18
17-1
13-1
15-2
11-2
Interaction
Procedure
'"
A
M
M
12.4
15.9 14.4 14.4
15. 6 15. 5 15. 5
22.121.921.9
23.6 23.6 23.6
46.0 45.9 45.9
continued
163
Table 9.5 (continued)
Parameters
2
2 2 2
er
er er er
r c rc e
4
4
4
1
Columns
Procedure
Rows
Procedure
Design
a
A
M
18
17-1
13-1
15-2
11-2
10-3
'"
M
A
M
38.3
44.4
40.4
55.4
49.1
73.2
38.1
38. 6
48.9
50.5
71.9
'"
M
Interaction
Procedure
A
M
'"
M
37.4
37.9
46.0
47.4
59.2
aprocedure A is based upon the method of fitting constants, and
Procedures M and Mare based upon the method of unweighted means.
Procedures A and M give unbiased estimates for the variance components
while Procedure Mgives biased estimates.
b
The mean square errors for the base design (i.e. design 18) are
given only for those cases computed by Bush [1962]-for Procedure A.
the properties that M.S.E.(~2) remains
r
2
invariant over any change in er ; M.S.E.(~2) remains invariant over any
c
c
. er2 ; an d M•S•E
..
.
• ("'2
er ) rema~ns
~nvar~ant
over any ch
ange·~n er2
c h ange ~n
2
r
rc
r
and er. Thus, if a cell of the table is left blank it implies that the
cprocedures A, M and
Mhave
c
results for this cell have already been given.
164
Table 9.6
Mean square errors of estimates of variance components for
4 x 4 - A designs
Parameters
2
2 2 2
Q'"
Q'"
Q'"
Q'"
r c rc e
1
16
4
1 .25
1 .25
4 .25
1
1
1
Design
a
A
M
"
M
A
M
"
M
30-1
28-2-1
28-2-2
26-3
24-4-1
24-4-2
.99
1.07
1.03
1.11
1. 21
1.13
.98
1.07
1.03
1.11
1.19
1.12
.98
1.06
1.02
1.10
1.19
1.12
.99
1.03
1.03
1.11
1. 21
1.13
.98
1.02
1.03
1.11
1.19
1.12
.98
1.02
1.02
1.10
1.19
1.12
30-1
28-2-1
28-2-2
26-3
24-4-1
24-4-2
178.
189.
180.
189.
198.
177.
175.
176.
176.
177.
177.
177.
175.
176.
176.
177.
177.
177.
30-1
28-2-1
28-2-2
26-3
24-4-1
24-4-2
12.0
12.8
12.2
12.9
13.6
12.3
11.8
12.1
12.0
12.2
12.4
12.2
11. 8
12.1
12.0
12.2
12.4
12.2
12.0
12.1
12.2
12.9
13.6
12.3
11.8
11. 9
12.0
12.2
12.4
12.2
11.8
11. 9
12.0
12.2
12.4
12.2
178.
179.
180.
189.
198.
177.
175.
176.
176.
177.
177.
177.
175.
176.
176.
177.
177.
177.
1. 38
1.47
1.48
1. 63
1. 84
1. 75
1. 38
1.48
1. 49
1. 70
1. 94
1. 75
1. 38
1.47
1.48
1. 68
1. 90
1.71
30-1
28-2-1
28~2-2
16 16 .25
1
4
1
1
1
1
1
1
1
ColLunns
Procedure
Rows
Procedure
26-3
24-4-1
24-4-2
30-1
28-2-1
28-2-2
26-3
24-4-1
24-4-2
1. 38
1. 51
1. 48
1. 63
1. 84
1. 75
1. 38
1. 59
1.49
1. 70
1. 94
1. 75
1. 38
1.58
1. 48
1. 68
1. 90
1.71
30-1
28-2-1
28-2-2
26-3
24-4-1
24-4-2
13.2
14.1
13.5
14.4
15.3
14.0
13.1
13.6
13.3
13.9
14.4
14.0
13.0
13.6
13.3
13.8
14.4
13.9
Interaction
Procedure
"
A
M
M
.17
.17
.20
.20
.23
.27
.27
.17
.20
.20
.23
.27
.27
.20
.20
.23
.27
.27
.60
.68
.68
.79
.94
.94
.60
.68
.68
.79
.94
.94
.60
.68
.68
• 79
.94
.94
b
continued
165
Table 9.6 (continued)
Parameters
2 2 ()2 ()2
()
()
r c rc e
16
16
1
4
16 16
1
4
16
1
1
1
1
1
1
4
4
4
1
1
1
1
1
1
Rows
Procedure
Design
a
A
M
'"
M
30-1
28-2-1
28-2-2
26-3
24-4-1
24-4-2
182.
194.
185.
194.
204.
183.
180.
182.
181.
183.
184.
183.
180.
182.
181.
183.
184.
183.
Columns
Procedure
A
M
'"
M
30-1
28-2-1
28-2-2
26-3
24-4-1
24-4-2
13.2
13.5
13.5
14.4
15.3
14.0
13.1
13.3
13.3
13.9
14.4
14.0
13.0
13.3
13.3
13.8
14.4
13.9
30-1
28-2-1
28-2-2
26-3
24-4-1
24-4-2
182.
184.
185.
194.
204.
183.
180.
181.
181.
183.
184.
183.
180.
181.
181.
183.
184.
183.
3. 70
4.22
4.22
4.97
6. 04
5.95
3.78
4.31
4.39
5.62
7. 10
5.91
3.73
4.20
4.27
5.41
6. 77
5.59
30-1
28-2-1
28-2-2
26-3
24-4-1
24-4-2
3. 70
4.26
4.22
4.97
6.04
5.95
3. 78
5.00
4.39
5.62
7.10
5.91
3. 73
4.86
4.27
5.41
6. 77
5.59
30-1
28-2-1
28-2-2
26-3
24-4-1
24-4-2
18.8
20.5
19.9
21. 7
24.0
22.7
18.8
21.0
19.9
22.1
24.6
22.6
18.7
20.8
19.9
21.9
24.3
22.3
30-1
28-2-1
28-2-2
26-3
24-4-1
24-4-2
201.
215.
206.
218.
231.
210.
199.
205.
202.
208.
215.
210.
199.
205.
202.
208.
215.
209.
Interaction
Procedure
'"
A
M
M
5. 10
5.82
5.82
6. 79
8.14
8. 14
5. 10
5.82
5.82
6. 79
8.14
8. 15
5. 10
5.82
5.82
6. 79
8.14
8. 15
continued
166
Table 9.6 (continued)
Parameters
(J2 (J2 (J2 (J2
r c rc e
4
4
4
1
Design
Rows
Procedure
A
a
A
M
M
30-1
28-2-1
28-2-2
26-3
24-4-1
24-4-2
Columns
Procedure
A
M
M
A
18.8
19.8
19.9
21. 7
24.0
22. 7
18.8
19.8
19.9
22.1
24.6
22.6
Interaction
Procedure
A
A
M
M
18. 7
19.7
19.9
21. 9
24.3
22.3
a
Procedure A is based upon the method of fitting constants, and
Procedure M and Mare based upon the method of unweighted means.
Procedures A and M give unbiased estimates for the variance components
while Procedure Mgives biased estimates.
b "
2
Procedures A, M and M have the properties that M. S. E. (<Y)
remains
r
invariant over any change in (J2; M.S.E. (~2) remains invariant over any
2
A2
c
c
2
change in (J ; and M. S. E. «(J ) remains invariant over any change in (J
r
rc
r
and (J2. Thus, if a cell of the table is left blank it implies that the
c
results for this cell have already been given.
167
Table 9.7
Mean square errors of estimates of variance components for
4 x 4 - B designs
Parameters
2 (J"2 (J"2 (J"2
(J"
r c rc e
1
16
4
1 .25
1 .25
4 .25
16 16 .25
1
1
1
1
Rows
Procedure
Columns
Procedure
Design
a
A
M
"
M
A
M
M
31-1
30-2
29-2
24-2
23-2
29-3
20-3
27-4
20-4
17-4
1.11
1. 20
1.18
1. 18
1.14
1. 23
1. 26
1. 35
1. 28
1.41
1.07
1. 20
1.13
1. 20
1. 15
1. 21
1. 29
1. 29
1. 29
1.4.5
1.07
1. 20
1.12
1. 20
1.14
1. 22
1.27
1. 28
1. 27
1.43
1.11
1.13
1.18
1.13
1.14
1. 23
1. 26
1.35
1. 28
1. 41.
1.07
1.08
1.13
1.14
1.15
1.21
1. 29
1.29
1. 29
1.45
1.07
1.08
1.12
1.13
1.14
1. 22
1027
1. 28
1. 27
1.43
31-1
30-2
29-2
24-2
23-2
29-3
20-3
27-4
20-4
17-4
201.
216.
209.
1960
184.
219.
191.
232.
183.
195.
176.
178.
177.
178.
1770
178.
179.
179.
179.
180.
176.
178.
177.
178.
177.
178.
179.
1780
179.
180.
31-1
30-2
29-2
24-2
23-2
29-3
20-3
27-4
20-4
17-4
13.5
14.5
14.1
13.4
12. 7
14. 7
13.3
15. 7
12.9
13.9
12.1
12.5
12.3
12.5
12.3
12.5
12.7
12.7
12.7
13.2
12.1
12.5
12.3
12.5
12.3
12.5
12.7
12.7
12.7
13.1
13.5
13.5
14.1
12.6
12.7
14. 7
1303
1507
12.9
13.9
12.1
12.1
12.3
12.3
12.3
12.5
12.7
12. 7
120 7
13.2
12.1
12.1
12.3
12.3
12.3
12.5
12. 7
12. 7
12. 7
13.1
201.
2000
209.
1830
184.
219.
191.
232.
183.
1950
176.
176.
177.
178.
177.
1'78.
179.
179.
179.
180.
176.
176.
177.
177.
177.
178.
179.
178.
179.
180.
31-1
30-2
29-2
24-2
23-2
29-3
20-3
27-4
20-4
17-4
"
Interaction
Procedure
"
A
M
M
.19
.20
.22
.32
.35
.20
.46
.24
.46
.68
.26
.27
.29
.38
.41
.27
.52
.33
.52
.70
.26
.27
.30
.38
.40
.27
.51
.34
.52
.70
b
continued
168
Table 9.7 (continued)
Parameters
2
2 2 2
0000r c rc e
1
4
16
16
1
1
1
4
1
1
1
1
1
1
1
1
Columns
Procedure
Rows
Procedure
Design
a
A
31-1
30-2
29-2
24-2
23-2
29-3
20-3
27-4
20-4
17-4
1. 60
1. 74
1. 75
1. 74
1. 68
1.83
1. 89
2.08
2.05
2.17
1.49
1. 76
1. 62
1. 76
1. 64
1.84
1. 93
2.07
1. 96
2.28
1.48
1. 76
1. 60
1. 74
1. 63
1.82
1. 90
2.04
1. 91
2023
31-1
30-2
29-2
24-2
23-2
29-3
20-3
27-4
20-4
17-4
15.0
16.2
15.8
15.1
14.2
16.6
15.1
17.9
15.0
15.9
13.4
14.0
13.7
14.1
13.8
14.2
14.5
14.8
14.5
15.3
13.3
14.0
13.7
14.0
13.7
14.2
14.4
14. 7
14.5
15.2
31-1
30-2
29-2
24-2
23-Z
Z9-3
20-3
27-4
20-4
17-4
207.
222.
215.
ZOZ.
189.
225.
198.
240.
190.
Z02.
181.
183.
182.
183.
18Z.
184.
185.
186.
185.
187.
181.
183.
182.
183.
182.
184.
185.
185.
185.
187.
31-1
30-Z
29-2
24-2
23-Z
29-3
20-3
27-4
20-4
17-4
M
'"
M
'"
M
A
M
1. 60
1. 67
1. 75
1. 68
1. 68
1.83
1. 89
2.08
2.05
2.17
1.49
1.55
1. 62
1. 62
1. 64
1.84
1. 93
2.07
1. 96
2.28
1.48
1.54
1. 60
1. 60
1. 63
1. 82
1. 90
2.04
1. 91
2.23
15.0
15.2
15.8
14.2
14.2
16.6
15.1
17.9
15.0
15.9
13.4
13.5
13.7
13. 7
13.8
14.2
14.5
14.8
14.5
15.3
13.3
13.5
13.7
13.7
13.7
14.2
14.4
14.7
14.5
15.2
Interaction
Procedure
'"M
A
M
.67
.73
.77
.90
.93
.80
1.16
.97
1. 27
1.53
.73
. 79
.84
.96
.99
.86
1. 21
1.05
1. 33
1. 56
.73
• 79
.84
.96
.99
.86
1. 21
1.05
1. 33
1. 55
continued
169
Table 9.7 (continued)
Parameters
2 2 2
2
cr cr cr
cr
r c rc e
16 16
1
4
16
1
1
1
1
4
4
4
1
1
1
1
Rows
Procedure
Design
a
A
M
M
31-1
30-2
29-2
24-2
23-2
29-3
20-3
27-4
20-4
17-4
31-1
30-2
29-2
24-2
23-2
29-3
20-3
27-4
20-4
17-4
4.53
5.04
5.26
5.29
5.05
5.65
5.93
6.96
7.13
7.20
3.96
5.32
4.61
5.32
4.66
5.88
6.06
7.35
6.33
7.78
3.89
5.20
4.49
5.16
4.52
5.70
5.82
7.04
5.98
7.40
31-1
30-2
29-2
24-2
23-2
29-3
20-3
27-4
20-4
17-4
22.1
24.0
23.9
22.9
21. 7
25.2
23.6
28.3
25.2
25.8
19.2
21. 6
20.4
21.6
20.5
22.6
22.9
25.0
23.4
25.8
19.1
21.5
20.3
21.5
20.4
22.4
22.7
24.7
23.0
25.3
31-1
30-2
29-2
24-2
23-2
29-3
20-3
27-4
20-4
17-4
230.
248.
241.
227.
213.
253.
224.
272.
220.
232.
200.
207.
204.
207.
204.
209.
210.
216.
212.
218.
200.
206.
203.
206.
204.
209.
210.
216.
211.
218.
Columns
Procedure
A
A
M
M
207.
206.
215.
189.
189.
225.
198.
240.
190.
202.
181.
181.
182.
182.
182.
184.
185.
186.
185.
187.
18l.
181.
182.
182.
182.
184.
185.
185.
185.
187.
4.53
5.07
5.26
5.09
5.05
5.65
5.93
6.96
7.13
7.20
3.96
4.44
4.61
4.56
4066
5.88
6.06
7.35
6.33
7.78
3.89
4.33
4.49
4.44
4.52
5.70
5.82
7.04
5.98
7.40
Interaction
Procedure
...
A
M
M
5.95
6.64
6.88
6. 78
6.82
7.49
7.98
9.16
9.38
9.51
5.42
6.10
6.21
6.48
6.54
6.96
7.74
8.40
9.06
9.48
5.42
6.10
6.22
6.47
6.53
6.96
7.73
8.41
9.06
9.47
continued
170
Table 9.7 (continued)
Parameters
2 2 2 0"2
0" 0" 0"
r c rc e
4
4
4
1
Columns
Procedure
Rows
Procedure
Design
a
A
M
31-1
30-2
29-2
24-2
23-2
29-3
20-3
27-4
20-4
17-4
"-
M
,.
A
M
M
22.1
23.0
23.9
21. 8
21. 7
25.2
23.6
28.3
25.2
25.8
19.2
20.1
20.4
20.3
20.5
22.6
22.9
25.0
23.4
25.8
19.1
20.0
20.3
20.2
20.4
22.4
22.7
24.7
23.0
25.3
Interaction
Procedure
,.
A
M
M
aprocedure A,.is based upon the method of fitting constants, and
Procedures M and M are based upon the method of unweighted means.
Procedures A and M give unbiased estimates for the variance components
while Procedure
gives biased estimates.
M
b
,.2
A
Procedures A, M and M have the properties that M. S. E. (0") remains
r
.
.
~nvar~an
t over any c h
2 M•S• E
ange·~n 0";
• ("2)
0"
c
c
. .~nvar~ant
.
over any
rema~ns
change in 0"2; and M.S.E. (~2 ) remains invariant over any change in 0"2
2
r
rc
r
and 0". Thus, if a cell of the table is left blank it implies that the
c
results for this cell have already been given.
171
Table 9.8
Mean square errors of estimates of variance components for
6 x 6 - A designs
Parameters
(J2 (J2 (J2 (J2
r c rc e
1
1 .25
1
16
1 .25
1
4
4 .25
1
1
1
1
1
16
1
1
1
4
4
1
1
16 16
1
1
4
1
1
1
Rows
Procedure
a
A
M
'"
M
A
M
'"
M
Interaction
Procedure
"
A
M
M
46-2
44-3
42-4
40-4
.58
.60
.59
.61
.57
.60
.59
.61
.57
.60
.59
.61
.58
.59
.60
.60
.57
.58
.59
.60
.57
.58
.59
.60
.19
.21
.23
.29
.22
.24
.26
.32
.22
.24
.26
.32
46-2
44-3
42-4
40-4
108.
112.
106.
111.
105.
105.
105.
105.
105.
105.
105.
105.
46-2
44-3
42-4
40-4
7.23
7.47
7.15
7.44
7.04
7.12
7.10
7.15
7.04
7.12
7.09
7.15
7.26
7.26
7.29
7.24
7.04
7.07
7.10
7.13
7.04
7.07
7.10
7.13
46-2
44-3
42-4
40-4
.73
.76
.76
• 78
.72
.77
.75
.77
.72
.76
.74
.77
.73
.75
.77
.77
.71
.73
.75
.77
.71
.73
.75
.76
.38
.41
.44
.50
.41
.44
.47
.53
.41
.44
.47
.5'3
46-2
44-3
42-4
40-4
110.
114.
108.
113.
107.
107.
107.
107.
107.
107.
107.
107.
46-2
44-3
42-4
40-4
7.74
8.00
7.70
7.99
7.51
7.65
7.60
7.68
7.51
7.64
7.60
7.67
7.77
7.78
7.84
7.79
7.50
7.55
7.61
7.65
7.50
7.55
7.61
7.65
111.
110.
111.
110.
107.
107.
107.
107.
107.
107.
107.
107.
1.55
1. 61
1. 68
1. 70
1. 46
1.52
1.59
1. 66
1. 46
1.52
1.59
1. 65
2.19
2.31
2.43
2.50
2.14
2.25
2.38
2.46
2.14
2.25
2.38
2.46
Design
46-2
44-3
42-4
40-4
46-2
44-3
42-4
40-4
1.56
1. 64
1. 69
1. 70
1.47
1. 67
1.59
1. 67
1.46
1. 68
1.59
1. 67
Columns
Procedure
b
continued
172
Table 9.8 (continued)
Parameters
2 ()2 ()2 ()2
()
r c rc e
4
4
4
1
16
4
4
1
Rows
Procedure
Design
a
A
M
"
M
46-2
44-3
42-4
40-4
9.99
10.4
10 • 2
10.4
9.56
9.99
9. 84
10.0
9.55
10.0
9. 84
10.0
46-2
44-3
42-4
40-4
118.
122.
117.
121.
114.
115.
115.
115.
114.
115.
115.
115.
Cohnnns
Procedure
"
A
M
M
10.0
10. 1
10.3
10.2
9.55
9. 69
9.85
9.99
Interaction
Procedure
'"
A
M
M
9.54
9. 69
9.85
9.98
aprocedure A",is based upon the method of fitting constants, and
Procedures M and M are based upon the method of unweighted means.
Procedures A and M give unbiased estimates for the variance components
while Procedure
gives biased estimates.
M
b "
Procedures A, M and M have the properties that M. S. E.
«(),,2r
remains
invariant over any change in ()2 j M.S.E.(~2) remains invariant over any
c
c
change in ()2 j and M.S.E. (~2 ) remains invariant over any change in ()2
r
rc
r
and ()2. Thus, if a cell of the table is left blank it implies that the
c
results for this cell have already been given.
173
Table 9.9
Mean square errors of estimates of variance components for
6 x 6 - B designs
Parameters
2 2
clcr2 cr cr
r c rc e
1
1 .25
1
16
1 .25
1
4
4 .25
1
1
1
1
1
16
1
1
1
4
4
1
1
16 16
1
1
4
1
1
1
Columns
Procedure
a
A
M
~
A
M
M
Interaction
Procedure
,..
A
M
M
56-2
50-3
46-4
47-4
.59
.63
.60
.64
.56
.60
.59
.60
.56
.60
.59
.60
.58
.57
.58
.58
.57
.57
.58
.59
.56
.57
.58
.59
.11
.14
.17
.16
.15
.17
.21
.20
.21
.20
56-2
50-3
46-4
47-4
114.
123.
113.
122.
104.
105.
105.
105.
105.
105.
105.
105.
b
56-2
50-3
46-4
47-4
7.54
8.11
7.52
8.13
7.00
7.12
7.08
7.12
7.00
7.12
7.08
7.12
7.42
7.16
7.18
7.21
7.01
7.03
7.07
7.08
7.01
7.03
7.07
7.08
56-2
50-3
46-4
47-4
.76
.80
.78
.82
.70
.77
. 74
.76
. 70
.76
.74
.76
.75
.75
.77
.77
.71
.72
.74
• 75
.71
.72
.74
• 75
.30
.33
.38
.37
.33
.36
.41
.40
.33
.37
.41
.40
56-2
50-3
46-4
47-4
116.
125.
U5.
125.
107.
107.
107.
107.
107.
107.
107.
107.
56-2
50-3
46-4
47-4
8.12
8.66
8.09
8.71
7.47
7.65
7.59
7.65
7.47
7.64
7.58
7.64
7.98
7.74
7.78
7.82
7.48
7.51
7.58
7.61
7.48
7.51
7.58
7.60
114.
110.
110.
110.
107.
107.
107.
107.
107.
107.
107.
107.
1. 66
1. 70
1.77
1.82
1.45
1.50
1.58
1. 63
1. 44
1.50
1.58
1. 63
2.18
2.28
2.42
2.42
2.03
2.15
2.31
2.29
'2.03
2.15
2.31
2.29
Rows
Procedure
Design
56-2
50-3
46-4
47-4
56-2
50-3
46-4
47-4
1.71
1. 68
1. 74
1. 76
1. 45
1. 68
1.58
1. 66
1. 44
1. 68
1.58
1. 66
A
.15
.17
continued
174
Table 9.9 (continued)
Parameters
2
2 2 2
cr cr crrc cr
e
r c
4
4
4
1
16
4
4
1
Columns
Procedure
Rows
Procedure
Design
a
A
A
A
M
M
56-2
50-3
46-4
47-4
10. 7 9. 51 9. 50
11.1 10.0 10.0
10.79.829.82
11.3 9.98 9.98
56-2
50-3
46-4
47-4
125.
133.
124.
134.
114.
115.
115.
115.
Interaction
Procedure
A
M
M
10.5
10 • 3
10.4
10.6
9.52
9. 64
9.81
9.93
9.51
9. 64
9.81
9.92
A
M
A
M
114.
115.
115.
115.
aprocedure A is based upon the method of fitting constants, and
Procedures M and Mare based upon the method of unweighted means.
Procedures A and M give unbiased estimates for the variance components
while Procedure
gives biased estimates.
M
b
Procedures A, M and M have the properties that M.S.E.(~2) remains
r
invariant over any change in cr2 ; M.S.E.(~2) remains invariant over any
A
c
2
"'2
c
change in cr; and M.S.E.(cr ) remains invariant over any change in cr2
r
r
rc
2
and cr • Thus, if a cell of the table is left blank it implies that the
c
results for this cell have already been given.
175
Table 9.10
Mean square errors of estimates of variance components for
6 x 6 - C designs
Parameters
2 2 2
2
CT
CT
CT
CT
r c rc e
1
1 .25
1
16
1 .25
1
4
4 .25
1
1
1
1
1
16
1
1
1
4
4
1
1
16 16
1
1
4
1
1
1
Rows
Procedure
,.
a
M
M
A
Columns
Procedure
,.
A
M
M
72-2
68-3
62-4
52-4
.59
.59
.57
.60
.56
.57
.57
.59
.55
.57
.57
.59
.62
.59
.60
.59
72-2
68-3
62-4
52-4
118.
115.
108.
114.
105.
105.
105.
105.
105.
105.
105.
105.
72-2
68-3
62-4
52-4
7. 75
7.61
7.20
7.56
6.98
7.03
7.02
7.09
6.98
7.03
7.02
7.09
72-2
68-3
62-4
52-4
.80
.80
.79
.70
• 74
.72
.75
.69
.73
.72
.75
72-2
68-3
62-4
52-4
120.
118.
111.
116.
106.
107.
107.
107.
106.
107.
107.
107.
72-2
68-3
62-4
52-4
8.46
8.30
7.87
8.18
7.44
7.55
7.52
7.61
7.44
7.55
7.52
7.61
Design
.77
72-2
68-3
62-4
52-4
72-2
68-3
62-4
52-4
2.00
2.03
1. 96
1.83
1.44
1. 63
1.55
1. 64
1.43
1. 62
1.55
1.64
.55
.56
.57
.58
.55
.56
.57
.58
8.28
7.74
7.67
7.49
6.97
6.98
7.03
7.06
6.97
6.98
7.03
7.06
.83
.79
.79
.78
.69
.70
.73
.74
.69
.70
.72
.74
8.97
8.41
8.31
8.10
7.43
7.47
7.53
7.58
7.43
7.47
7.53
7.58
129.
120.
118.
115.
106.
107.
107.
107.
106.
107.
107.
107.
1.97
1. 90
1.87
1.83
1.43
1.48
1.56
1. 62
1.42
1.48
1. 56
1. 61
Interaction
Procedure
,.
A
M
M
.07
.08
.09
.13
.12
.12
.12
.12
.13
.13
.16
.16
.26
.27
.29
.33
.29
.30
.32
.36
.29
.30
.32
.36
2.44
2.35
2.40
2.41
1.97
2.04
2.16
2.23
1.97
2.04
2.16
2.23
b
continued
176
Table 9.10 (continued)
Parameters
(J"2 (J"2 (J"2 (J"2
r c rc e
4
4
4
1
16
4
4
1
Rows
Procedure
Design
a
A
M
M
72-2
68-3
62-4
52-4
11.6
11.5
10.9
10.9
9.48
9.89
9.75
9.94
9.47
9.88
9.75
9.94
72-2
68-3
62-4
52-4
132.
129.
122.
126.
114.
115.
115.
115.
114.
115.
115.
115.
Columns
Procedure
A
A
Interaction
Procedure
"
M
M
A
A
M
M
12. 1 9.47 9.46
11.4 9.59 9.59
11.29.769.76
10.8 9.90 9.89
a
Procedure A"is based upon the method of fitting constants, and
Procedures M and Mare based upon the method of unweighted means.
Procedures A and M give unbiased estimates for the variance components
while Procedure
gives biased estimates.
M
bprocedures A, M and
.
.
t
~nvar~an
Mhave
the properties that
2 M•S•E• ("2)
over any ch
ange·~n (J";
(J"
c
c
M.S.E.(~2) remains
r
. .~nvar~ant
.
over any
rema~ns
change in (J"2; and M.S.E. (~2 ) remains invariant over any change in (J"2
2
r
rc
r
and (J". Thus, if a cell of the table is left blank it implies that the
c
results for this cell have already been given.
,