SELECTION AMONG FACTORIALLY CLASSIFIED VARI.ABLES
BY
R. M. Patel) C. C. Cockerham) and J. O. Rawlings
Institute of Stutistics
M~eogr~ph Series No. 317
J anu~ry ) 1962
ERRATA SHEEr
Page
9
Line
4th from bottom
Incorrect
Correct
= (1 1) • • •
= (1,
· . • 3.~ ( ~-~
r
,c
·.. (E xi) (E 1/ 2)
· . . ~ (~-~)
r
2
2
• • • (E xi) (E 1/~)
1) • • •
10
7th from top
13
6th from top
15
11th from top
b'E b • • •
- pp-
18
9th from top
• • • and Lbb • ••
18
lOth and 11th lines--equation should be numbered (4.0.12)
27
5th from top
2
~
2
1
[b'~ b]2 • • •
-
A
p-
A
•• • and ~p • • •
m
••• (- ln II
m
•. ·
• . . (- ln
k~
• • • ( kendall, •••
R
•••
k~
27
8th from top
32
6th column,
0.014
23rd and 24th nos. (0.983)
43
Last reference should include (University Microfilms, Ann Arbor)
• •• (Kendell, • • •
0.983
(0.014 )
iv
TABLE OF CONTENTS
Page
..•
.... •
·..·...
..... ·.........
CHAPrER 1 - INTRODUCTION
CHAPrER 2 - REVIEW OF LITERATURE • . . . · . • • • · . . .
CHAPrER 3 - APPLICATION TO A DIALLEL EXPERIMENT. . . .
·...
CHAPrER 4 - COMPARISON OF INDEXES. • • • · . .
·....
CHAPrER 5 - EMPIRICAL SAMPLING STUDIES • • · . . . . . . . • • • •
LIST OF TABLES • • • • •
•
•
Introduction. • • • • • • • • • • • • • • •
Methods • • • • • • • •. • • • • • • •
• • • • • •
• • • • • •
Results from Empirical Sampling • • • • • • • • • • • • • •
'.2.1 Estimates of Index Weights • • • • • • • • • •
5.2.2 Comparison of Indexes • • • • • • • • • • • • • • •
5.2.2.1 Unrestricted Estimated Indexes •••
5.2.2.2 Restricted Estimated Indexes • • • •
.....
·.. • • ·.......
CHAPrER 7 - SUMMARY AND CONCLUSIONS. . . . • • · . · . . . · . . .
LIST OF REFERENCES • • • • • • • • • • . . · .
• • • •
•
CHAPrER 6 - DISCUSSION • •
•
•
v
1
2
8
15
26
26
26
28
28
30
30
35
37
41
43
v
LIST OF TABLES
Page
1.
2.
Analysis of variance for a dial1e1 experiment with no sells
and no reciprocals • • • • • • • • • • • • • • • • • • • ••
9
Total correlations between the optimum index and the estimated
indexes for various parametric combinations. • • • • • •
22
2a.
2b.
2c.
2d.
(n = 7, r = 2) • • • • • • • • •
(n = 7, r = 11) • • • • • • • • •
(n = 19, r = 2) • • • • • • • • •
(n = 19, r = 11) • • • • • • •
·· .. ..· ... ... ... ... ... ... ... ...
•
•
•
•
0
•
•
3. The optimum weights, b, and their unbiased estimates with the
true and estimated standard deviations for a
4.
= 0.1
• • • • •
22
23
24
25
29
Estimated means and standard deviations of correlations between
the estimated indexes and the optimum index for various
parametric combinations • • • • • • • • • • • • • • •
31
·.
4a.
4b.
4c.
4d.
(n
(n
(n
(n
= 7, r = 2) • ·
= 7, r = 11) •
= 19, r = 2) ·
= 19, r = 11).
·• •• ·• ·· • .. ·• ·• ·• •• ·• ·• ·• ·· •• •• ·· ••
• · • • • • • • • · · · · · • · • •
• · · • •
• · · • • • · • ·
·
··
·
31
32
33
34
CHAPrER 1
INTRODUCTION
The selection index problem is of the multivariate type and may be
stated simply.
The potential of an individual unit in a population is
a known function of many non-observable variates.
some inherent error, are the variates observed.
These variates, plus
The problem is to
construct a function of the observables which in some manner best estimates or dis criminates among the potentials of individual units.
Non-linear functions among both observables and non-observables
have sometimes been alluded to in the past, but essentially all of our
knowledge to date is for linear functions.
A solution for the best
linear function was given as long ago as 1936.
It involved population
parameters, however, which must be estimated.
Estimated indexes, while
appealing, are not necessarily best in any sense, and their "goodness "
is the subject of most of the current work on selection indexes.
The present study is concerned with a special class of selection
indexes.
In randomized multifactor experiments, the variables may be
defined such that they are uncorrelated.
This removes much of the
intractability encountered with more general selection indexes, and
allows the development of some analytical results pertaining to estimated indexes.
studies.
These results are augmented by empirical sampling
2
CHAPI'ER 2
REVIEW OF LITEEATURE
8mith' s (1936) ;development of the optimum index is sketched in the
following.
~'
Let
= (gl'
~,···,gk) denote the set of variates determining the
value of an individual unit in the population sampled.
a'
= (al ,
a , ••• ,~) denote the known constant relative values of
2
the corresponding g's such that G = !'~ denotes the value of
an individual unit.
normal distribution.
N
-b'jJ.)
-p
- ppb
, (b'I:
a'jJ.
- -g
(2.0.1)
a'I: b
- pg-
In a sample of n variables, I
and G , the index and true values
j
j
are related by the linear regression equations
= a'jJ. +
- -g
Cov IG
2
CT
I
(2.0.2)
3
If we denote the mean' of a selected set of Its from the sample as I
and the sample mean as
s
I, then the expected difference between the
corresponding G's is
- - Cov IG
= (Is-I)
2
O"I
If the I's in the selected set are in the upper q fraction of the
sample,
I s -I
may be approximated
closely for large samples by
.
(I s -I) = to"I ,
co
co
where E(t) = -1q
J
=
q
xf(x)dx ,
X
J
t(x)dx ,
1
f(x) = - e
21t
x2
-"2
X
o
o
Fcir sizes of sam:Ples-iess than 50, ,the expected value of t may be found
from available tables.
In any case, since'E(t) is a function only of
q, or selection intensity, and sam:Ple size, the gain from selection is
expected to be
A
= E(G -Gil) = CovIG E(t) •
O"I
s..
Smith found the constants,
:£',
which maximized the expected gain, A,
to be
bt
-
= atI:.
-
,E-l
pg pp
or since the e's were assumed to be uncorrelated with the gt s ,
and
(2.0.4)
4
(2.0.6)
For this optimum or best index,
Cov
T_
~
~,
G = a' 1: b = b' 1: b - c;2
-gg-
-pp--~'
and
L\ = C;~E(t)
(2.0.8)
•
Since the optimum index involves population parameters, Smith suggested
that L:.gg and L:ee be replaced by sample. estimates and that selection be
based on
I
.....
b
.....,
' (S
)-1
=bn=aS
£
gg gg +S ee £n.
Hazel (1943) independently
develop~d
similar results for more complex
situations in animal populations.
Bartlett (1939), Nanda (1949) and Cochran (1950) were well aware
of the problems involved in the use of a sample index.
Bartlett and
Nanda attempted to obtain variances of the estimated coefficients.
Bartlett, for a two
~ariate
index, approximated the variance of the
estimated expected gain,
...
E( t )~~
= E( t
)
[....:£' SpJ?.....Ji
•
Nanda extended the approximations to k variates.
(2.0.10)
The degrees of these
approximations are not given and their results are not very informative
except to indicate that as more variates are utilized the sample must
be larger to provide useful estimates.
Cochran (1950) suggested that the probability of selecting the
desired G's might be a useful criterion for measuring the usefulness
5
of I",.
Williams (1961) investigated this by making use of the proba-
b
. .
bi1ity that one of the m largest sample GIS was associated with the
largest sample £'~, which he denoted as
P[l, m, n] •
He found that the same set of weights,
£',
gain also maximized this probability.
He thought this criterion to be
which maximized the expected
more useful than expected gain in comparing sample indexes with others
because the probability criterion· tends to reflect more than just expected gain.
In particular, Williams compared the estimated index, I""
b
with the optimum index, ~, and the base index, I
a
= !I~, in an effort
to find a basis for choosing between I'"
and I a , since I is never
b
o
U
known.
The comparisons were too complex to draw any general conc1u-
sions, but they were of such a nature as to suggest that for many sizes
of samples and parametric situations the base index would be superior
to the estimated index.
This is always true when the estimated index
is based on. a sample size less than k+5, where k is the number of variates.
Williams found the estimated weights,
-b',
to
be biased, which if
...
uncorrected for bias, could lead to negative expected gain.
Lower
bounds to the variances of the estimated weights indicated that the
variances increased as the base weights deviated more from the optimum
weights.
Where the base weights are very different from the optimum
weights, however, is the situation when an estimated index is most
needed.
The optimum index is also the one that has maximum correlation with
the true values, G's.
The correlation between index values and GIS is
6
=
PIG
and the same set o'f weights,
Cov IG
0"
E. t,
r:r
GI
(2.0.11)
'
maximizes this correlation that maxi-
mizes the expected gain (2.0.4), since O"G is constant.
For this opti-
mum index the correlation is
The covariance between any index,
Cov(:u..,
-b
I_ ) =
-b
It
for example, and
~
is
btL:ppb, and substituting (2.0.6)
-
Cov( l"b, ~) =
E.'" t L:gg! = Cov
rsG •
Thus,
P~"G = P~ G PI£~ •
(2.0.12 )
The expected gain from an index relative to the gain from the optimum
index is also the correlation
In comparing two indexes, I a and
has several' ~equivalences
!S,
the ratio of their expected gains
~ = Prs~
~a
PIa~
(2.0.14)
The difference'between their expected gains relative to the expected
gain from the optimum index may be used as a comparison,
7
or just the difference between their expected gains may be used.
The
comparison of probabilities of correct selection is more closely related
to
(2.0.16)
than the other comparisons, in that the probability of correctly selecting the G's is monotonically related to PIG.
This illustrates for
any given difference, Prs~ - P
J1> = (~ - !:J.a)/'\' that the difference
Ia
in discri~natory power: among the G's is further dependent on PJ1>G.
The method of comparing indexes varies with the simplicity of
various formulations and with the tastes of authors, but most often
expected gains are the measure of comparison.
8
CHAPrER 3
APPLICATION TO A DIALLEL EXPERIMENT
In multifactor experiments one may wish to choose the best
combinations of factors.
If the factors are random variables, then the
selection of the best combinations may be fitted into the selection index framework.
In particular, a diallel experiment without selfings or
reciprocals (Griffing, 1956) is to be considered.
A random sample of n
parents or lines are mated in all possible combinations to produce
n(n-l)/2 progenies which are measured in r complete replications. This
----th
is the common single cross experiment. An observation on the ij
progeny or single cross in the tth replication may be represented by
the linear model
(i<j =1,2"".,n)
(t
= 1,2, ••• ,r)
where
I..l.
= the mean,
rt
= the effect of replication
t~
= average effect of i th line (general combining ability),
Sij
=
t,
interaction effect due to the specific combination of
parents i and j (specific combining ability),
eijt
= the error effect.
When all effects except the mean are random and the variances of each
type of effects are homogeneous, the appropriate expectations of the
pertinent mean squares are as given in Table 1.
9
Table 1.
Analysis of variance for a diallel experiment with no selfs
and no reciprocals
M.S.
d.f.
Source
~
;:: n-l
Interaction effects
~
==
Error
~;::
Average effects
E.M.S.
~
2
;:: cr2
+ rcr 2
+r(n-2)cr
s
t
~
;::cr2 + rcr2
s
A
~
A
n(n-3)
2
~
(r-l)(n+l)(n-2)
2
A
.lvJ.
~
;:: cr2
The aim is to choose among the true values expressed as a deviation from the sample mean.
where a dot indicates a mean over the subscript it replaces.
venience the notation n
i
;:: n-i will be used.
For con-
To fit the G' s into the
index notation,
~ (si
;:: [siJ' - -~
n
•
n
+ s . ) + -.J
~
S
]
••
l
+ ~~ (61. + S.j - 2s.) + t i + t j - 2t.]
;:: [Sij] + [Tij ]
;: (1 1) [:1J ]
.
::::a 'g .
-
-J. j
ij
•
The least squares estimates of Sij and Tij based on a fixed model except for error are
10
= ([Y
- -ll. (Y - + Y
) + -n Y' ], [ll.
- (Y
+ Y - 2Y
) ])
_ ij. ~
i ••
•j •
~. • •
~
i. •
•j •
• ••
= (Sij,Tij ) + ([e i -j ..
•
~(e.
~
~..
+e j )+.!l. e
••
~
•••
],[ll.(e.
~
+e j -2e
J. • • •
)])
o.e
0
= (~' + ~') in the index notation.
The variances are
~
I1.
0'2
s
~ ~-l\
0
n (
-1
0
r
=
1::pg =1::gg =
0
222
(0'S + ~a:t)
Ii
g (":3-~)
0
n
r
o
=
1::pp =1::gg +1::ee =1::gg +
2
2 0'
o
n r
2
o
The expected mean squares, M's, are those in Table 1.
n
":3
r
It may' be noted
that this is a fairly restricted case of selection indexes in that it
is only a two variate case and
trices.
1::g~'
ee and 1::pp are all diagonal ma-
1::
The case is important in its own right, however.
11
The optimum weight~ for choosing among the G' s in experiments of
this size are
b' = a'I::
-
I::- l
1 -
gg pp
These weights must conform to the following bounds,
The weights for the base index are
a'
::i:
(1,1) ,
, and the base index is
I aij
= -a'~i'J = Yi J.,
- Y• •• ,
which is the usual basis for choosing the best entries.
By reference
to (3.0.4) the optimum index coincides with the base index when
1.
0-2
~.
-. = - = 0
2•
o-~ = (M; - ~)/r~ = 0
r
r
or
•
In the latter case the indexes are the same except for a scalar multiplier; i.e.,
:£' = C!',
where c is a scalar.
Neither of these condi-
tiona -are probably ever found in pras:tice so that it would always be
best to use the optimum index.
12
For the estimated index the weights are obtained by substituting
mean squares f'or their expected values
-".b
= (1
"
"
~
-~,
1
-
"
~) •
-:;-j'
(3.0.5)
~
Since the mean squares are independently distributed as {iMi/di'
(d is the degrees of' f'reea.om), the expectations of' the estimated
~ = di
weights are easily obtained, using E
and E
l/X~ = (d.:2)
.'
,
J.
Eb' [1 - ~~\)' 1- ~t~-2)]
=
r,b
= L1 -
(~_?) , b2-(~_2)
From Table 1, ~ = nn.,/2 and ~ = ~.
values of' b and n.
the bls
2(1-b2 ->]
2(1-b1 )
•
The bias can be large· f'or small
" and -2/3 f'or ~2
" as
The maximum bias is -2/7 f'or ~1
----> 0 and n = 6, the smallest admissible number that is mean-
ingf'ul f'or the purpose.
The biases are small, however, if' either n is
large or the b' s are near one.
In this connection, the magnitudes of'
theb's are partly dependent on and increase with ~he number of' replications, as can be readily seen in (3.0.4),.
Unbiased estimates can be
constructed.
~. =
""::'
-
[1 -
"
(d2 -2 )l>J.
"
,
~~
= [b.1(~-2)/d2
1 -
(~-~)I\ ]
~~
+
2/~, ~2(~-2)/~ + 2/~]-
(3.0.7)
13
The variances of theb's are also easily obtained.
v.~
bl
~
~.
= V(l - -- F )
M.2
~ 12
= V(l
- ~)
= V[l
- (l-b1 ) F12 ]
= (l-b1 )2
V,F12
where V,F12 is given by Cramer, 1946, and may be obtained as
and
Similarly,
The covariance is evaluated as
where
and the correlation is
From the relationships given in (3.1.2) the variances and covariance of
the unbiased estimates are derived as
2("-l+ ~- 2)
dl(~- 4)
The variances and covariance of the unbiased estimates are less than
those of the biased estimates and considerably less for small n.
In
either case the variances andcovariances decrease as the bls and n
increase_
Recall that increasing r increases the b's and that r-l is
a factor in "-l-
15
CHAPrER 4
COMPARISON OF INDEXES
In comparing indexes reference populations must be kept clear.
The optimum. or best index is d,efined and depends on the parameters n,
222
r, IJ', IJ's and IJ't.
Thus, it is defined for each size of experiment.
While it is possible to estimate the
vari~ce
parameters from one
size of experiment and construct an estimated index to be used for
another size of experiment, the estimation experiment will always be
considered to be of the same size as the experiments to which the indexes are to be applied and over which expectations are taken.
From (2.0.8) the expected gain for the optimum. index was expressed
as
b'Z b E(t)
-
p:p-
= IJ'L E(t) •
~.
The other indexes will be compared to the optimum. index as a ratio of
their expected gains to the expected gain for the optimum. index.
For
example,
( 4.0.1)
Recall that a'
=l'
is a unit vector.
:Note that if
is a scalar, then the correlation is unity.
£' = ca'
-,
where c
The. correlation may be
evaluated for any particular parametric situation.
For the index using the biased weights, and conditional on a-single
set of
...
£,
16
(4.0.2)
and similarly for the unbiased weights,
•
For comparison with
Pr
~'
the expectations
a
Eo...
·.L~I
bb
= P:r~I '
bb
over the distribution of the estimated weights are required.
plicit expressions could be found for these expectations.
No ex-
However,
unconditional or total correlations can be evaluated by taking expectations in the following
manne~:
(4.0.4)
.,..
,..
If we let the vector E(£' - £') =~' denote the bias in the estimated
weights, and which are given in (3.0.6), then
A
Eb.'Z b'" =b'Z b +b'Z B.
-
p:p-
-P:P-
PlF
-
(4.0.5 )
E
.
where
i,.
b
£,zP:P-b = -b'ZP:P-b
-
+ 2b'Z
-
B+ -B'z'PP'":'B. +~'
P:P-
Z ~
·b pp-.co
. b
~
,
is the vector of square roots of the va:riances of the estimated
'
weights'given in (3. o. 8, 3.0.9).
Thus
17
b'E b + b'~!"
=
-
[-b' Ep;p-b.]t (-b' Ep;p-b.'
p;p-
.=E~
-
" "',,
+ 2b' E. B + B' E B +0".' E. 0" ]
- p;p- p;p-£. p;p-£
t
(4.0.6)
For the lIDbiased weights the terms involving the bias,
~
~,
in
(4.0.5)
are zero.
b'E b
RI I =
.~ b
b
-p;p-
"::'[---xr-----'iWi---------rxb' E b]2 [b' E b + O"J,. E
--pp-
-~
p;p-
-
0",..]2
p;p-~
6
The variances of
£,
are given in (3.0.12).
Instead of the correlation between an estimated index and the optimum index for the population of experiments, one may consider the
correlation for each experiment from which the index was estimated.
This correlation for the index with the biased weights is
,
where
~
",pp
E
=
n
l
"
~
. -r
(4.0.8)
0
'"
0
-n .~r
2
is the estimated variance covariance matrix.
The other corresponding
sample correlations are
(4.0.9 )
18
=
" "
£'>;
(4.0.10)
b'r:. b~]i
[-b'r:".pp-b.]2 [.~"
-pp-
Again, unable to evaluate the expected value of these correlations
(4.0.8, 4.0.9, 4.0.10), total or unconditional correlations were resorted to.
"
Er:
PI'
Since the estimated variance covariance matrix is unbiased,
=r:
PI'
" a
Eblr:
pp-
.!.
.!.
=
b]2 [Ea'£ a]2
[E -bIZ.pp- .PP(4.0.11)
The total correlations for the estimated indexes cannot be evaluated
in terms of biases and variances as they were in (4.0.6) and (4.0.7),
" or b'
~
" are correlated.
sinceb'
and z.t,b
=
b'i; b
- pp;b]* [b'r: b +cr" r: cr ]i.
[ -b'r:pp-'
- PP- -sb
" PI>-"
sb .
'
where
(4.0.13 )
l/ThiS solution was suggested by Dr. D. S. Robson, Professor,
Cornell University, N. Y., in personal communications.
19
A
R
~
Eb'.E b
-pp:
-.
IS~~ [E -b'~-PIrb]* [E -SPE.I r£Ji
-
t;
(4.0.14 )
b'.E b + bt.E d
-
=
PIr
-
PIr
where
and
1
(1-b2 ) [
2(d + ~- 2) ~J
Idles
].
.
(4.0.15)
A
Note that d' is very close to ~t in (4.0.5).
Some of the total corre-
lations may be ranked for any parametric situation.
In comparing
RI£~ (4.0.7) with RIsb~ (4.0.12) it may be noted that each term in
~iS larger than the corresponding term in ~t" so that
b
sb
and thus
The comparison of RIAL and R I_ (4.0.14) is more involved. Writing
I ~-b
sb-b
the two correlations in more si~~le notations for corresponding terms,
the argument of inequalities follows:
20
Each of X, Y and Z is positive and y+ is slightly larger than Y since
.
each term in ~'~ is larger than ~'*
sb
sb
X3 + 2~Z + ~y
o
5 x3 + ~y+ + 2~Z + 2XZY+ + z 2x2 + z2(y+)2
5 x2(y+ - Y) + 2XZY+ + z2x2 + Z2(y+)
2
•
Thus, Rr .L S Rr ~I· The comparison of RTl'L with Rr""L is very
s<O-b
sb b
-b~b
b-b
messy and the larger was not definitely established by inequalities,
but munerical results to be presented do satisfactorily establish that
(4.0.16 )
The four total correlations and
for various parametric situations.
Pla~
are tabulated in Table 2
The variance parameters are coded
as follows,
(4.0.17)
so that a represents the fraction of the total variance that is nonerror and r represents the fraction of the non-error variance that is
due to average effects of the parents.
No results are given in Table 2 for a > 0.6 since all of the
correlations are then almost the same.
The base index is perfectly
correlated with the optimum index when r
=0
but the weights are very
21
different, Le. <,~I_:!?I) = ~~ is large.
An increase in
r reduces
but an increase in a increases PIa~ because it reduces ~~.
PIa~
An increase
in r also increases PI I because of the reduction in ~~ ~ The total
ab
correlations are affected in the same way as PI ~ by changes in a and
a
r but they, also increase as r or n increases.
With a
~
0.2 and r
~
11 there is not a great deal of difference
among any of the correlations (see Tables 2b and 2d).
r
~
With n
= 19
and
11, Table 2d, the total correlations are very similar and are
generally slightly higher than the correlation for the base index.
Even when r
= 2,
Table 2c, the total correlations are not very dif-
ferent but they are larger than
For small
v~ues
PIa~
of nand r, Table 2a,
the total correlations.
differ very much, and
when
r is much larger than zero.
PIa~
is generally larger than
It is here, too, that the total correlations
R~:rs
is even negative.
0.2
0.4
0.6
0.0
1.000
0.186
0.548
0.623
0.819
0.4
0.915
0.330
0.671
0.729
0.865
0.6
0.862
0.425
0·720
0.769
0.875
1.0
0.775
0.584
0.788
0.824
0.888
0.0
1.000
0.655
0.868
0.905
0.953
0.4
0.966
0.795
0·915
0.934
0.963
0.6
0.940
0.835
0.926
0.941
0.965
1.0
0.876
0.870
0.930
0.941
0.958
0.0
1.000
0.912
0.969
0.979
0.988
0.4
0.986
0.954
0.980
0.984
0·990
0.6
0.978
0·959
0.980
0.984
0.989
1.0
0.934
0.953
0·972
0.976
0.982
23
Table 2b.
P~Ia
R~I£
R
0.0
1.000
0.707
0.4
0.962
0.6
R
~ISb
R~I
0.876
0.918
0.961
0.834
0·925
0.947
0.972
0.938
0.866
0.936
0.953
0.972
1.0
0.868
0.898
0.944
0.957
0.969
0.0
1.000
0·933
0.971
0.982
0.990
0.4
0.987
0.963
0.982
0.988
0.993
0.6
0.975
0.969
0.983
0.988
0.992
1.0
0.929
0.964
0.979
0.983
0.987
0.0
1.000
0.991
0.996
0.997
0.999
0.4
0.997
0·995
0.997
0.998
0·999
0.6
0.993
0.995
0.997
0.998
0.998
1.0
0.970
0.989
0·993
0.994 .
0.995
0.0
1.000
0.998
0.999
0.999
1.000
0.4
1.000
0.999
0·999
1.000
1.000
0.6
1.000
0.999
0.999
1.000
1.000
1.0
0.995
0.995
0.997
0.997
1.000
r
0.1
0.2
0.4
0.6
(n = 7, r = 11)
~I~
b
s~
24
Table
0.1
0.2
0.4
0.6
2c. (n
= 19, r = 2)
P~Ia
R~rs'
R~I
R~I s'S
R
0.0
1.000
0.678
0.746
0.755
0.780
0.4
0.738
0.885
0·907
0.910
0.925
0.6
0.670
0.924
0.936
0.938
0.945
1.0
0.601
0.956
0.961
0.962
0.965
0.0
1.000
0.922
0.930
0.933
0.943
0.4
0.842
0.969
0.972
0.973
0.976
0.6
0.792
0.977
0.979
0.980
0.981
1.0
0.727
0.984
0.985
0.985
0.986
0.0
1.000
0.987
0.989
0.990
0.991
0.4
0.941
0.994
0.994
0·995
0.995
0.6
0.915
0.995
0.995
0·995
0.995
1.0
0.859
0.995
0·995
0.995
0.995
0.0
1.000
0.997
0.998
0.998
0.998
0.4
0.981
0·999
0.999
0.999
0.999
0.6
0.970
0.999
0·999
0·999
0.999
1.0
0.928
0.998
0.998
0.998
0.998
B
~I~
sb
25
Table 2d.
0.1
0.2
0.4
0.6
en = 19,
r
= 11)
P~Ia
R~I£
R
~I*
b
R~Is'b
R~Is~
0.0
1.000
0.988
0.991
0.991
0.992
0.4
0.935
0.996
0.996
0.996
0.996
0.6
0.907
0.996
0·997
0·997
0.997
1.0
0.849
0.996
0.997
0.997
0.997
0.0
1.000
0.998
0.998
0.998
0.998
0.4
0.979
0.999
0.999
0.999
0.999
0.6
0.966
0.999
0.999
0.999
0·999
1.0
0·922
0.998
0.999
0.999
0.999
0.0
1.000
1.000
1.000
1.000
1.000
0.4
0.996
1.000
1.000
1.000
1.000
0.6
0.993
1.000
1.000
1.000
1.000
1.0
0.968
0.999
0.999
0.999
1.000
0.0
1.000
1.000
1.000
1.000
1.000
0.4
0.999
1.000
1.000
1.000
1.000
0.6
0.998
1.000
1.000
1.000
1.000
1.0
0.985
1.000
1.000
1.000
1.000
26
CHAP.rER 5
EMPIRICAL SAMPLING STUDIES
5.0
Introduction
The means and the variances of the correlations between various
selection indexes and the optimum selection index are required for
their evaluation.
In the previous chapter, total correlations have
been evaluated by taking the expectations of the covariances and variances separately but no explicit solutions could be found for the
expectation of the correlations per see
Therefore, empirical sampling
was resorted to in order to obtain information on the :ma.gnitude of the"
means and variances of the correlations.
In addition, empirical sam-
pling allowed evaluation of certain restricted selection indexes.
5.1
...
Methods
6
The estimated weights, -:-...,.
b or b, are functions
of independent mean
squares in the diallel experiment (,3.0.5, ,3.0.7) and the correlations
bet,:,een indexes involve only the index weights and certa,in population
parameters.
Therefore, it is sufficient, in empirical evaluation of
the correlations, simply to generate the independent mean squares.
Under the assumption of normality of the original observations, the
mean squares,
Mi ," are
distributed as x2M /d with d degrees of freedom.
i
i i
Hence, the mean squares of diallel experiments can be generated by
generating triplets of chi-square random variables with the appropriate
degrees of freedom.
27
Two methods of generation of the independent chi-square variables
were enu>loyed.
For chi-square variables with d < 100, the fact that
i
-2'ln Pk is distributed as >t with two degrees of freedom, where Pk
,
m
is a uniform variate, and thus that (- 1n II pk/m) is distributed as
(1)
,
>t/di
k~
with d '= 2m degrees of freedom,was utilized.
i
(2) For chi-square variables with di > 100, the Wilson-Hilferty
appr.oximation (kendall, M. G. and Stuart, A., 1958), which is very good
even for small degrees of freedom, was utilized.
>t
This approximation to
with d. degrees of freedom is
~
where z is a standardized normal deviate.
One hundred experiments were generated independently for each of
the four sizes of the diallel, Le., for all combinations of:n=7, 19
and r=2, 11.
For each experiment, the biased and unbiased weights,
A.
£
'"'" respectively, and the correlations, P~rs and P~:tS' were comand~,
puted for different parametric situations defined by all
combinations of a = 0.1, 0.2, 0.4, 0.6, 0.8 and 0.9 and 7
poss~ble
= 0.0, 0,2,
0.4, 0.6, 0.8 and 1.0.
In addition, the correlations of certain restricted estimated
indexes with' the optimum index were computed.
A1though there are
several possible restrictions, the particular ones employed in this
study restricted the use of the estimated indexes to those cases where
the estimated coefficients,
b'
A
or
£1,
satisfied the bounds on the
28
optimum weights, 0 ~ b l ~ b 2 ~ 1. The base index ,!f~ was used as the
alternative index in those cases where the bounds were not satisfied.
The three restricted estimated indexes considered were:
1.
2.
3-
!br
l
I ....
t)r
l
I"
£r
2
=
=
=
{£'E
."
if
o ~ ~l
,!f~
otherwise
~'E
if
,!'~
otherwise
r
{h
,!';e
" ~1
~ ~2
o ~ ~l ~ ~2 ~
A
A
o ~ ~l ~ ~2 ~
if
1
1
otherwise
Empirical sampling results are presented only for part of the
parametric combinations of
0:
and 'Y. investigated, 'Y = 0.0, 0.4" 0.6,
1.0 and 0: = 0.1, 0.2, 0.4, 0.6.
results for
0:
The indexes gave essentially the same
> 0.6.
5.2
Results from Empirical Sampling
5.2.1 Estimates of Index Weights.
E,
A
deviations of the unbiased weights,
and standard deviations in Table .3 for
The sample means and standaxd
.
are compared with the known means
0: =
0.1.
These results are Pre-
sented primarily to illustrate the degree of precision obtained from
the sample of 100 experiments.
variables was
utilize~
Since the same set of randOm chi-square
for all parametric combinations of
0:. and 'Y
given size of experiment, it is expected that the deviations of the
in a
29
e
Table 3.
The optimum weights, £, and their unbiased estimates with the
true and estimated standard dev1atio~for a = 0.1
b1
B1
A
er~
er,-,>
~1
(n
= 7,
b2
~
b2
.1
1)2
A
erA
1)
erA
2
= 2)
r
0.1 0.0 0.182
0.228
0.463
0.436
0.182
0.149
0.896
0.806
0.4 0.118
0.167
0.499
0.469
0.366
0.341
0.694
0.625
0.6 0.082
0.133
0.520
0.488
0.430
0.408
0.624
0.561
1.0 0.000
0.056
0.566
0.531
0.526
0.507
0.519
0.467
(n
= 7,
= 11)
r
0.1 0.0 0.550
0·550
0.207
0.211
0.550
0.538
0.454
0.417
0.4 0.423
0.423
0.266
0.271
0.761
0.754
0.242
0.222
0.6 0.328
0.328
0.310
0.315
0.806
0.801
0.196
0.180
1.0 0.000 -0.001
0.460
0.469
0.859
0.856
0.142
0.130
(n
= 19,
r
= 2)
0.1 0.0 0.182
0.184
0.130
0.118
0.182
0.257
0.323
0.258
0.4 0.118
0.120
0.141
0.128
0.622
0.657
0.150
0.119
0.6 0.082
0.084
0.147
0.133
0.702
0.730
0.118
0.094
1.0 0.000
0.003
0.159
0.145
0.791
0.810
0.083
0.066
(n
= 19,
r
= 11)
0.1 0.0 0.550
0.5553
0.054
0.057
0.550
0.534
0.171
0.190
0.4 0.423
0.427
0.070
0.073
0.900
0.897
0.038
0.042
0.6 0.328
0.332
0.081
0.085
0.928
0.926
0.027
0.030
1.0 0.000
0.006
0.121
0.127
0.954
0·952
0.018
0.019
30
estimates from the true· values would show the same trend for all parametric combinations of a and
the true weights,
£,
r.
The differences,however, decrease as
increase.
Even for small experiments (n
= 7,
r
= 2)
the estimated mean and
standard deviations are reasonably close to the expected values indieating the adequacy of the sample size.
5.2.2
Comparison of Indexes.
The means and standard deviations
of the correlations, P~Is' P~II'.' P1,!£r and P~Io/.llo are given in
~
1)
1
. ~rl
Tables 4a, 4b, 4c and 4d for the four sizes of experiments. The correlation between the base index and the optimum. index, P~Ia (4.0.1),
is also given for each case.
The means of the correlation, P1,I*
.
are given only for the small experiment (Table 4a).
deviations for PI. I
t>*br
'
~r2
The standard
were not computed but they will be less than the
2
standard deviations for P~I*
•
~rl
5.2.2.1
Unrestricted Estimated Indexes.
P1,!£and P~ I*
The differences between
reflect the average improvement in the estimated index
b
_
realized by correcting the estimated weights for bias.
metric combinations of a and
r
In all para-
the correction for bias resulted in an
improvement in terms of both higher mean correlations and reduced
standard deviations, the greatest improvement occurring in the parametric situation when the biased index, Is, was the poorest.
The
estimated indexes were poorest, as would be expected, in the smallest
experiment (Table 4a).
In this case, both of the unrestricted estimated
32
Table 4b.
r
0.1
0.2
0.4
0.6
P~Ia
0.0
1.000
0.4
0.962
0.6
0.938
1.0
0.868
0.0
1.000
0.4
0.987
0.6
0.975
1.0
0.929
0.0
1.000
0.4
0.997
0.6
0·993
1.0
0.970
0.0
1.000
0.4
0·999
0.6
0.999
1.0
0.986
(n
= 7, r ::; 11)
P~rs
P~I~
P
~rsr1
P~I~
0.822 a/
(0.391)=
0.885
(0.274)
0.907
(0.239 )
0.932
(0.217)
0.923
(0.215 )
0.944
(0.182 )
0.948
(0.176)
0.954
(0.159)
0.981
(0.066)
0.980
(0.024)
0.977
(0.024)
0·923
(0.061)
Br1
0.987
(0.036)
0.986
(0.016)
0·979
(0.024)
0·919
(0.056)
0.956
.( 0.153)
0·971
(0.125 )
0.973
(0.107)
0.979
(0.056)
0.978
(0.094)
0.985
(0.062)
0.987
(0.050)
0.985.
(0.042)
0.997
(0.009)
0·992
(0.014)
.0.991
(0.015 )
.0.961.
(0.033 )
0.997
(0.007)
.0·995
(0.008)
.0.994.
(0.009)
0.959
(0.031)
0·993
(0.035 )
0.997
(0.014)
.0.996
(0.016)
_0·991.
(0.0;0)
0.997
(0.014)
0.998
(0.008)
.0.998
. (0.010)
. 0.994.
(0.022)
0·999
(0.005 )
.0.998.
(0.012)
.0.998.
(0.004)
.0.014.
(0.983 )
0.999
(0.004)
0.998
(0.008)
.0·999.
(0.002)
.0.983.
(0.013 )
0.999
(0.006)
0.999
(0.002 )
.0·999(0.003 )
0.996
(0.016)
1.000
(0.002)
.1.000
(0.001)
1.000.
(0.002 )
0·997
(0.011)
1.000
(0.000)
1.000
(0.002 )
0.999
(0.003 )
.0.993
(0.007)
1.000
(0.000)
.1.000,
(0.001)
.1.000.
(0.002)
.0.992.
(0.006)
b
~.IStandard deviations in parentheses.
33
Table
PI I
b a
0.1
0.2
0.4
0.6
0.0
1.000
0.4
0.738
0.6
0.670
1.0
0.601
0.0
1.000
0.4
0.842
0.6
0.792
1.0
0.727
0.0
1.000
0.4
0.941
0.6
0.915
1.0
0.859
0.0
1.000
0.4
0.981
0.6
0.970
1.0
0.928
!ojStandard
4c.
(n
= 19, r = 2)
P~I~
P~r.sr1
P~I ...
0.770 a/
(0.398)0.932
(0.122)
0.954 _
(0.073 )
0.971
(0.041)
0.807
(0.134)
0.944
(0.097)
0.960
(0.060)
0.974
(0.035 )
0.950
(0.110)
0.921
(0.100)
0.889
(0.142)
0.786
(0.190)
0.946
(0.105 )
0.925
(0.100)
0.889
(0.142 )
0.785
(0.189)
0.958
(0.101)
0.982
(0.030)
0.985
(0.024)
0.989
(0.017)
0.966
(0.073 )
0.984
(0.027)
0.986_
(0.021)
0.989
(0.015 )
0.983
(0.048)
0.982
(0.029)
0.969
(0.063 )
_0.857
(0.132)
0.980
(0.047)
0.983
(0.029)
-0·970
(0.020)
_0.856
(0.166)
0.996
(0.007)
.0·997(0.005)
0.996
(0.006)
0.996
(0.006)
0·997
(0.005)
0·997
(0.005 )
0.997
. (0.005)
0.996
(0.005)
0.998
(0.003 )
0·997
(0.005 )
(0.006)
0.926
(0.070)
0.998
(0.004)
0·997
(0.005 )
0.997
(0.005 )
.0.926
(0.069)
0.999
(0.001)
.0.999
(0.001)
0.999
(0.001)
0.998
(0.003)
0.999
(0.001)
0.999
(0.001)
0·999
(0.001)
0.998
(0.002)
1.000
(0.000)
0.999.
(0.001)
-0·999.
(0.001)
_0.962 _
(0.036 )
1.000
(0.000)
.0.999 .
(0.001)
0.999.
(0.001)
_0.962_
(0.044)
P~~
b
deviations in parentheses.
~0.996
Sr1
34
Table 4d.
'Y
0.1
0.2
0.4
0.6
P~Ia
0.0
1.000
0.4
0.935
0.6
0.907
1.0
0.849
0.0
1.000
0.4
0·979
0.6
0.966
1.0
0.922
0.0
,1.000
0.4
0.996
0.6
0.993
1.0
0.968
0.0
1.000
0.4
0.999
0.6
0.998
1.0
0.985
!,/Standard
(n = 19, r = 11)
P~I~
b
P
0.991 /
(0.018~
0·997
(0.004)
0.997
(0.004)
0.996
(0.005 )
0.994
(0.011)
0.997
(0.004)
0.997
(0.004)
0·997
(0.005 )
0.999
(0.002)
0.997
(0.004)
0.997
(0.004)
0.923
(0.077)
0.998
(0.003 )
0·997
(0.004)
0·997
(0.004)
0·923
(0.074)
0.998
(0.003 )
0.999
(0.001)
0.999
(0.001)
0.998
(0.002)
0·999
(0.002)
0.999
(0.001)
0.999
(0.001)
0.999
(0.002)
1.000
(0.000)
.0·999
(0.001)
.0·999
(0.001)
.0.960
(0.039)
1.000
(0.000)
. 0·999
(0.001)
0·999
(0.000 )
0.960
(0.038 )
. 1.000
(0.000)
1.000
(0.000)
1.000
(0.000)
0.999
(0.001)
1.000
(0.000)
1.000.
(0.000)
1.000
(0.000)
0.999
(0.000)
1.000
(0.000)
1.000.
(0.000)
1.000.
(0.000)
,0.984
(0.016)
1.000
(0.000)
.1.000.
(0.000)
,1.000.
(0.000)
0.984.
(0.016)
1.000
(0.000)
1.000
(0.000)
1.000
(0.000)
1.000
(0.000)
1.000
(0.000)
1.000
(0.000 )
1.000
(0.000)
1.000
(0.000)
1.000
(0.000)
1.000
(0.000)
1.000
(0.000)
0.993
(0.007)
1.000
(0.000)
1.000.
(0.000)
.1.000,
(0.000)
0.993
(0.007)
P~rs
deviations in parentheses.
~rsrl
-
P~I,",
"Sr
1
35
indexes were inferior to the base index in terms of both lower mean
correlations and very high standard deviations of these correlations.
Increasing the nUIriber of replications (Table 4b) considerably iIIl.J?roved
the estimated biased and unbiased indexes, but resulted in a corresponding improvement in the base index.
On the other hand, an increase
in nUIriber of parents, n, improved the unrestricted estimated indexes
and concomitantly decreased the efficiency of the base index.
As a
resUlt, with large nUIriber of parents, n, and small nUIriber of replications, r, (Table 4c), the unrestricted estimated indexes were superior
to 1'he base index except for the cases where '1
P~I~ increased with increasing
'1,
t~e
= 0.0.
a. With respect to
Both
~~I£
and
c~ing values of
mean correlations tended to reach a maximum. at intermediate
values of '1, particularly for large values of n and r.
5.2.2.2
P~l{)r'
and
Restricted Estimated Indexes.
P~I~
The
differences between
, while consistently favoring the correction for bias,
were-minor with ~rfew exceptions where the decrease in the standard
deviation was large.
All of the restricted estimated indexes were con-
siderably better, in terms of both higher mean correlations and reduced
standard deviations, than the unrestricted estimated indexes in those
situations where unrestricted indexes were no better than the base index.
unrestricted indexes were reasonably good for large experiments
or large values of a and '1'
became less iIIl.J?ortant
an~
The improvement due to the restriction
in fact the restricted indexes became slightly
inferior to the unrestricted indexes for a few cases where '1 was large.
The restricted indexes were consistently better than the base index
except for
(at
r = 0.0,
where the base index was optimum.
This difference
r = 0.0), however, was very small.
The restriction on unbiased weights,
"'"
a ~ ~l
A
~ £2 ~ 1, indicated
little advantage over the restriction on biased weights,
a~
A
A
bl ~ ~2 ~ 1 in the small experiment (Table 4a).
For this reason
this restriction has not been considered in the larger experiments.
;7
CHAPl'ER 6
DISCUSSION
It has been observed that when .y
= 0.0, :e, = c! so that P:s,I = 1
irrespective of the values of other parameters.
a
Also, it is clear from
(;.0.4) that :e, ----> ! as the number of replications is increased regardless of the size of n.
Thus the base index,
!'~,
approaches the
optimum index, or P:s,I ----> 1, as the number of replications is ina
creased. On the other hand, increasing the number of parents, n, for
to approach a = 1 while the value of b is un2
2
l
affected. Thus, when r O,b l < b2 and increasing n will cause :e, to
deviate further from c! resulting in a decrease in the correlation of
a given r, causes b
r
the base index with the optimum index.
The variances of the estimates of the optimum weights are decreasing functions of the degrees of freedom and of the optimum weights.
For small experiments the optimum weights and degrees of freedom are
small and the estimates of the b's have large variances.
Consequently,
the correlation P:s,rs is generally small and many times negative.
Hence, the estimated index
:rs is
considerably inferior to the base
index, I a , for small experiments.
'"
Referring to the bias in the estimates, :e"
(;.0.6) the bias is
largest under the· same conditions as when the variance of the estimate
is largest.
Correction for bias reduces the variance of the estimates,
the amount of reduction being greatest in experiments with small n.
Thus, correction for bias is most important for small experiments, and
becomes less important for larger experiments.
In spiteof the
38
improvement due to correction for bias, I A is still inferior to the
1)
base index, la' in small experiments. Therefore, when one has to
choose in small experiments among the base index, the estimated index
and the estimated index corrected for bias, the base index should be
the choice.
This supports the finding of Williams (1961).
Increasing the size of the experiment, by changing the number of
parents, n, is more effective, than changing r, in increasing the efficiency of the estimated indexes compared to the efficiency of the base
index.
This is due to the fact that the base index loses its effi-
ciency on increasing the number of parents except for r
= 0.0, whereas
the estimated indexes gain in efficiency on increasing n.
Hence the
estimated indexes would be preferred to the base index when the number
of parents is large.
An increase in number of replications improves the base index as
well as the estimated indexes, and leads to little difference among any
of them except for low values of a where the base index is better.
In those experiments where the estimates of the optimum weights
are highly variable the estimates of the £ are often nonsensical in
terms of the known limits on b.
Therefore it appeared practical to
exclude from consideration the nonsensical estimates by imposing the
--
A
*
"-
restriction 0 ~ bl ~ b2 ~ 1 or 0 ~ b.l ~ £.2 ~ 1. (The upper limit of 1
is not really required since £ and £ are always less than unity.) The
"
'
outstanding improvement, in terms of mean correlations and reduced
standard deviations, in the restricted indexes over the unrestricted
ones, is the result of elimination of the unrealistic estimates
39
(negative estimates and the estimates of bl being greater than the
estimates of b2 ). Whenever the restriction was not satisfied a constant index I was used. This had the effect of greatly reducing the
a
variances in the cases where the optimum weights were poorly estimated
since a majority of the time the estimated index was replaced with the
base index.
The choice of the base index as the alternative is also the
explanation for the failure of the restricted estimated indexes to be
superior to the unrestricted estimated indexes when r
= 1.0.
Since I
a
is most inefficient when r = 1.0, applying the restriction often resulted in lower individual correlations.
On
the other hand, when
r = 0.0, the choice of the base index as the alternative was precisely
the right one for the most efficiency.
The importance of the restricted indexes in this study is that
they illustrate a possible method of obtaining indexes which will be
consistently good over a Wide range of par-ametric conditions.
The
particular restricted indexes used in this study can undoubtedly be improved upon by imposing different restrictions.
For example, altering
"" =5 0 :5 '6'2
'" the index I +w.here
the restriction in such a way that for 1)1
b
+'
b
= (0,1) is used. This should help considerably when r is large
because I + is the optimum index if r
= 1.0•. Conversely, the type of
b
alternative might be conditioned by some prior knOWledge of the likely
size of ex and
r.
For example if the average effects are known to be
of importance, perhaps the best procedure would be to use I + in all
b
cases where the bounds are not satisfied.
In any case, some sort of
restrictions offer a very useful alternative.
· 40
The results obtained from the empirica.l sampling indicate that the
tota.l correlations obtained in (4.0.6) and (4.0.7) are the lower bounds
of the expected va.lues of P~Ib and P~IS'
While they are lower bounds
they appear close enough to the expected va.lues to be useful in ranking
the various indexes.
41
CHAPTER
7
SUMMARY AND CONCLUSIONS
The statistical aspects of selecting nonobservable linear
functions of multivariate normal variables with the aid of a selection
index, say ~ = E, rR where R is the vector of observables, have been
considered in this dissertation.
A special class of selection indexes
has been examined which is appropriate for multifactor experiments.
The properties of estimates of the optimum weights, E" were studied for
one of the randomized multifactor experiments, a diallel experiment.
'" were found.
The biases and variances of the estimated weights, E"
The
biases were negative and were functions of the optimum weights and the
number of lines.
The optimum weights were also functions of the number
of lines and replications, as well as parametric components of variance.
Correction for bias reduced the variance of the estimates.
Unable to
. find explicitly the expected values of the correlations between the
estimated indexes and the optimum index,
P~r.s
and
P\I~'
total correla-
b
tions were studied.
These total correlations were compared for various
parametric situations and were also compared with the correlation between the "base index, I
and the optimum index,
a
= -a R where -a is a vectorI of known weights,
I
I
~.
Empirical sampling was utilized to obtain information about expected values and the variances of' the correlations,
P~r.s
and
P~I",.
£
In addition, restricted indexes were considered.· That is, when the:
estimated b's did not satisfy the following restrictions,
42
the base index was used instead of the estimated index.
'0
o.
The conse-
quence of this procedure was compared with the other alternatives.
It is concluded from the analytical and empirical sampling results
that the biased estimated index is unsatisfactory in small experiments.
Correction for bias improves the efficiency in terms of both higher
mean correlations and reduced standard deviations of the estimated
index considerably but not enough to dictate its use over the base
index in
s~l
experiments.
The efficiency of all the indexes is increased by increasing the
number of replications.
For experiments with many lines ana very few replications, estimated indexes are more efficient than the base index.
Restricted
indexes appear to be the most useful, and especially for small experiments.
'"
o ~ £'1
The base index as an alternative for the cases where
'" ~ 1 is not satisfied is the best alternative for small
~ £'2
values of
r,
the fraction of the non-error variance that is due to
average effects of the parents.
other alternatives were suggested
which might further improve the efficiency of restricted indexes.
43
LIST OF REFERENCES
..
Bartlett, M. s. 1939. The standard errors of discriminant function
coefficients. Jour. Roy. Stat. Soc. §:169-l73.
Brim, C. A., Johnson, H. W. and Cockerham, C. Clark. 1959. Multiple
selection criteria in soybeans. Jour. Amer. Soc. Agron. 51:42-46.
Cochran, W. G. 1950. Proceedings of the Second Berkeley Symposium on
Mathematical Statistics and Probability: 449-470.
Cramer, H. 1946. Mathematical Methods of Statistics.
University Press, Princeton: 242.
Princeton
Falconer, D. S. 1960. Introduction to Quantitative Genetics.
Press Co., New York: 324-328.
Fisher, R. A. 1954.
Inc., New York:
Ronald
Statistical Methods for Research Workers.
287-289.
Fisher, R. A. and Yates, F.
New York.
1949.
Stat.istical Tables.
Hafner,
Hafner, Inc.,
Griffing, B. 1956. A generalised treatment of the use of diallel
crosses in quantitative inheritance. Heredity 10:31-50.
Hazel, L. N. 1943. The genetic basis for constructing selection
indexes. Genetics 28:476-490.
Kem,pthorne, O. 1957. An Introduction to Genetic Statistics.
Wiley and Sons, Inc., New York: 510-516.
John
Kendall, M. G. and Stuart, A. 1958. The Advanced Theory of Statistics.
Hafner Publishing Company, New York, Vol. 1: :372-373.
Lush, J. L. 1947. Family merit and individual merit as basis for
selection. Amer. Naturalist 81:241-261 and 362-379.
Nanda, D. N. 1949. The standard errors of discriminant function
coefficients in plant breeding experiments. Jour. Roy. Stat. Soc.
(B) 11:282-290.
Smith, H. F. 1936. A discriminant function for plant selection.
Eugenics 1:240-250.
Ann.
Williams, J. S. 1961. An evaluation of the worth of some selection
indices. Unpub~ished Ph.D. Thesis, North Carolina State College,
Raleigh.
INSTITUTE OF STATISTICS
NORTH CAROLINA STATE COLLEGE
e
;.
'.
(Mimeo Series available for distribution at cost)
273. Schutzenberger, M. P. On the definition of a certain class of automata. January, 1961.
274. Roy, S. N. and J. N. Shrizastaza. Inference on treatment effects and design of experiments in relation to such inferences.
January, 1961.
275. Ray-Chaudhuri, D. K. An algorithm for a minimum cover of an abstract complex. February, 1961.
276. Lehman, E. H., Jr. and R. L. Anderson. Estimation of the scale parameter in the Weibull distribution using samples censored by time and by number of failures. March, 1961.
277. Hotelling, Harold. The behavior of some standard statistical tests under non-standard conditions. February, 1961.
278. Foata, Dominique. On the construction of Bose-Chaudhuri matrices with help of Abelian group characters. February,
1961.
279. Eicker, FriedheIm. Central limit theorem for sums over sets of random variables. February, 1961.
280. Bland, R. P. A minimum average risk solution for the problem of choosing the largest mean. March, 1961.
281. Williams, J. S., S. N. Roy and C. C. Cockerham. An evaluation of the worth of some selected indices. May, 1961.
282. Roy, S. N. and R. Gnanadesikan. Equality of two dispersion matrices against alternatives of intermediate specificity.
April, 1961.
283. Schutzenberger, M. P. On the recurrence of patterns. April, 1961.
284. Bose, R. C. and I. M. Chakravarti. A coding problem arising in the transmission of numerical data. April, 1961.
285. Patel, M. S. Investigations on factorial designs. May, 1961.
286. Bishir, J. W. Two problems in the theory of stochastic branching processes. May, 1961.
287. Konsler, T. R. A quantitative analysis of the growth and regrowth of a forage crop. May, i961.
288. Zaki, R. M. and R. L. Anderson. Applications of linear programming techniques to some problems of production planning over time. May, 1961.
289. Schutzenberger, M. P. A remark on finite transducers. June, 1961.
290. Schutzenberger, M. P. On the equation all+n = b"+mc"+p in a free group. June, 1961.
291. Schutzenberger, M. P. On a special class of recurrent events. June, 1961.
292. Bhattacharya, P. K. Some properties of the least square estimator in regression analysis when the 'independent' variables
are stochastic. June, 1961.
293. Murthy, V. K. On the general renewal process. June, 1961.
294. Ray·Chaudhuri, D. K. Application of geometry of quadrics of constructing PBIB designs. June, 1961.
295. Bose, R. C. Ternary error correcting codes and fractionally replicated designs. May, 1961.
296. Koop, J. C. Contributions to the general theory of sampling finite populations without replacement and with unequal
probabilities. September, 1961.
.
Foradori,
G.
T.
Some
non-response
sampling
theory
for
two
stage
designs.
Ph.D.
Thesis. November, 1961.
297.
298. Mallios, W. S. Some aspects of linear regression systems. Ph.D. Thesis. November, 1961.
299. Taeuber, R. C. On sampling with replacement: an axiomatic approach. Ph.D. Thesis. November, 1961.
300. Gross, A. J. On the construction of burst error correcting codes. August, 1961.
3.01. Srivastava, J. N. Contribution to the construction and analysis of designs. August, 1961.
302. Hoeffding, Wassily. The strong laws of large numbers for u-statistics. August, 1961.
303. Roy, S. N. Some recent results in normal multivariate confidence bounds. August, 1961.
304. Roy, S. N. Some remarks on normal multivariate analysis of variance. August, 1961.
~O5. Smith, W. L. A necessary and sufficient condition for the convergence of the renewal density. August, 1961.
306. Smith, W. L. A note on characteristic functions which vanish identically in an interval. September, 1961.
307. Fukushima, Kozo. A comparison of sequential tests for the Poisson parameter. September, 1961.
308. Hall, W. J. Some sequential analogs of Stein's two-stage test. September, 1961.
309. Bhattacharya, P. K. Use of concomitant measurements in the design and analysis of experiments. November, 1961.
310. Anderson, R. L. Designs for estimating variance components. November, .1961.
311. Guzman, Miguel Angel. Study and application of a non·linear model for the nutritional evaluation of proteins.
Thesis-January, 1962.
Ph.D.
312. Koop, John C. An upper limit to the difference in bias between two ratio estimates. January. 1962.
313. Prairie, Richard R. and R. L. Anderson. Optimal designs to estimate variance components and to reduce product variability for nested classifications. January, 1962.
314. Eicker, FriedheIm. Lectures on time series. January, 1962.
315. Potthoff, Richard F. Use of the Wilcoxon statistic for a generalized Behrens-Fisher problem. February, 1962.
316. Potthoff, Richard F. Comparing the medians of the symmetrical populations. February, 1962.
317. Patel, R. M., C. C. Cockerham and J. O. Rawlings. Selection among factorially classified variables. February, 1962.