This research was supported by National Institutes of Health, Institute
of General Medical Sciences Grant No. 12868-04 and National Heart
Institute Contract No. Pll-43-67-1440.
The authors also express appreciation for a grant of free time from the
Triangle Universities Computer Center.
A COMPUTER PROGRAM FOR ANALYSIS OF DATA
BY GENERAL LINEAR MODELS
C. FRANK ST.A..RMER
J A1'12 S E. GRI ZZLE
Department of Biomathematics
Duke University Medical Center
Department of Biostatistics
University of North Carolina
Institute of Statistics ¥imeo Series No. 560
February 1968
v
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TABLE OF CONTENTS
Page
Introduction . . .
1
Examples of Models
2
Examples of Multivariate Hypothesis Construction
u
,
Computations Required to Estimate Parameters and to
Test Hypotheses.
10
Program Outline . .
12
Control Card Design
Data Card Design. . •
Hypothesis Card Design.
Summary of Control Card Design.
Examples of Models . . .
Multiple Regression
Hotelling's T2 . • .
Analysis of Variance:
Analysis of Variance:
13
14
18
19
20
Case I
Case II.
20
24
28
32
Description of Output.
35
Appendix I . .
39
Worked Examples with Examples of Control Cards
and Program Printout
Appendix II.
Program
70
A Computer Program for Analysis of
General Linear Models
I. Introduction
Numerical techniques associated with data analyses that employ linear
models tend to separate into two approaches:
1. Programs which have very simple, minimal amount of input, but
which are severely restricted in the designs that can be
analyzed; or
2. programs which require relatively more tedious input but which are
practically unrestricted as to the type of data that can be analyzed, except for the restrictions of linearity of the parameters in the model and of storage limitation due the computer
employed.
As an example of the first type, a general collection of programs for
evaluating hypotheses might contain a specific program to perform the "t"
tes t, a different program to perform the paired "t" tes t, and others for
performing analyses of variance of various designs.
For those who prefer the more general approach, the theory of the
multivariate general linear model (MGLM) has unified most techniques
associated with the analysis of variance into special cases of the MGLM.
With this generality in mind, we have prepared a general computer
program which is based on the theory of the multivariate linear model.
This single program has the capability of computing the estimates of
parameters and test statistics required for any of the situations envisioned above, as well as a great many more.
2
To introduce the ideas we consider the univariate model
E(Y)
A
8
(nxm) (m x1)
(n x 1)
(1)
where A is a matrix of known coefficients which depends on the design of
the experiment and the model thought to be appropriate, 8 is the parameter vector, and the elements of the vector of observations Yare
assumed to be normally and independently distributed.
Some examples of
application of the univariate general linear model follow.
II. Examples of Models
1. Regression:
(2)
E(y )
n
=
]J
+ an181 + a 8
n2 2
Y1
1
an
a
Y2
1
a
a
Here, Y
21
.
]J
12
8
22
A
1
(3)
8
82
Yn
1
a
n1
a
n2
2. Analysis of variance (completely randomized design, 3 treatments):
3
( 4)
E(y )
n
=
anll\ + an2S2 + an3133 •
Y2
all
a
a
a
2l
a
12
a
22
l3
,
13
23
A
Here, Y
13
anI
Yn
a
a
n2
1
13
2
13
3
n3
where
ail
a
a
i2
i3
={
={
={
1 ify. was subjected to treatment 1,
1
0 otherwise
1 if y. was subjected to treatment 2,
1
0 otherwise; and
1 if y. was subjected to treatment 3,
1
•
0 otherwise.
In evaluating the above models, one could ask, for example, whether
13
1
= 13 2 =
0 in the regression model, or whether 13
1
effect) in the analysis of variance model (ANOVA).
13
2
= 13 3 (no treatment
Both of these hypo-
theses can be obtained from the general hypothesis structure
H :
o
c
13
(sxm) (mxl)
o
(sxl)
(6)
, j,
,. 'j
4
)
.T"'/
where C is an arbitrary matrix chosen to form the hypothesis by yielding
the appropriate linear combinations of the elements of the parameter
vector 6.
For instance, to test 6
1
=
6
2
=
0 in the regression model,
we have
=
C
(:
1
0
:)
:1)
and 6
(7)
62
yielding
C
6
(2 x 3)
(3 xl)
To test 6
1
6
62
C
:y (:: ) (: )-
( :J
3
0
1
0
0
(8)
in the ANOVA model, we have
( -:)
1
-1
1
0
61
and 6
(9)
62
63
yielding
-1
C
6
(2x3)
(3xl)
(10)
o
Suppose the observations (y's) are vectors with p correlated
characteristics instead of single variables, as would be the case for
multiple observations made on the same individual.
A random sample of n
of these vectors could be arranged in a rectangular array to form an
5
nxp matrix, Y, where the first row of Y is the vector of characteristics
observed on the first individual, the second row is the vector observed
on the second individual, etc.
Assuming that the p-dimensional observa-
tion vector has a multivariate normnJ. dj3tribution, and that the n observation vectors are independent, 'we can extend t:lC univariate model
to encompass the p correlated variahles.
E(Y)
(nxp)
A
(nxm)
The model now appears as
B
( 11)
(mxp)
and the hypothesis can be generalized to
H :
o
c
S
U
o
(sxm)
(mxp)
(pxu)
(sxu)
( 12)
where C and U are arbitrary matrices designed to yield the appropriate
hypothesis.
In the construction of the multivariate model represented by
equation (11), the matrix A, called the design matrix, is the same
matrix of known constants that appeared in the univariate model, and
the matrix B is an mxp matrix of p<lrameters in which each of the p
characteristics has m parameters associated with it.
To continue the analogy of multivariate analysis to the univariate
regression and
M~OVA
models, we simply attach one additional column to
the matrix Y for each additional observed characteristic, and attach
one additional column to the parameter matrix
characteristic.
S
for each additional
The multivariate case is simply the model for p cor-
related variables, each of which uses the same design matrix A.
Specific examples of the various types of linear models and hypotheses
will be presented in Section VI to illustrate how several models can be
6
formulated, and how hypotheses can be tested within the framework described.
III. Examples of Multivariate Hypothesis Construction
Our computer program is designed to test hypotheses which can be
structured in the form
o
C
S
u
(s xm) (mxp) (p xu)
H :
o
(s xu)
The matrix C is structured to form linear combinations of the rows
of the
matrix S whereas the matrix U is structured to form linear combi-
nations of the columns of the matrix S.
Interpreting the role of the mat-
rix C in terms of the actual model, the matrix C allows one to form linear
combinations among the m components of the model.
Since the matrix U
operates on the columns of the matrix S, one may use it to form linear
combinations of the p "univariate" models.
Starting with a univariate model, consider the multiple regression
model
E (y.)
1.
. Yl
1
an
a
Y2
1
a
a
2l
12
22
a
13
a
23
a
a
, and S
\.
Yn
Sl
24
, A
where Y
j.l
14
\
1
a
nl
a
n2
a
n3
an)
\::
S4
7
o
To test (31
and (3
C
~
2
= o we choose
(:
1
0
0
0
1
0
:)
and U
(1) •
Then C (3 U = 0 gives
(:
(:)
]J
1
0
0
0
1
0
:)
( 1)
(31
= (
:: )
(32
To test (33
0, so that
(34' form the equivalent hypothesis of (33 - (34
C
whence C (3 U
= (0 0 0 1 -1) and U
(1),
0 is
]J
(31
(0
0
1 -1)
0
(1)
(32
(33 - (34
O.
(33
(34
To understand the role played by the matrix U, consider a bivariate
(p
= 2) regression model identical in structure to the one above except
for the addition of a second variate.
E(y. )
1
lx2
Now
8
and the matrices Y and S have two columns, as follows:
Yll
III
Y21
B11
, and S
Y
Yn2
Ynl
11
2
S12
S22
S13
S23
S14
S24
Suppose the second observed variable is the first variable observed ten
minutes later, and we want to test the equality of the regression lines,
i.e., test whether the response relationship has changed after ten
minutes.
Then
10000
c
Then C S U
=
o
1
0
0
0
o
0
1
0
0
o
0
0
1
0
o
0
001
0 states the desired hypothesis:
1
0
0
0
0
o
1
0
0
0
o
0
100
o o o
1
o o o o
9
o
o
(J
II
\
o
o
o
(314
The extension of the paired t test to its multivariate analog, a
form of the Rotelling T2 test, is obtained by constructing the appropriate
U matrix.
Consider the example of mean blood flow measured before and
after administration of a drug.
E(F
before
F
after
The model then appears as
)
The following matrices then allow evaluation of the paired "t"
test.
Let
C = (1) and U
=( _11
)
then
o.
C (3 U
Similarly, for the T2 test with four observed variables (two blood
pressures and two heart rates) we can write the parameter matrix as
and test the equality of BP andRR, ioe.,
=
~BP
1
by setting
~BP
,
2
~RR
1
~RR '
2
10
C
(1) and U
1
a
-1
a
a
1
a
-1
I
J
then
(1)
C S U
(fl
(fl
BP
BP
1
fl BP
2
fl HR
fl
1
HR )
1
a
-1
a
a
1
a
-1
2
- ~BP , fl HR 121
fl
HR )
2
= (00).
In constructing hypotheses for the analysis of variance, care must
be taken so that the matrix C is constructed according to the restrictions imposed in computing the estimates.
(See examples 3 and 4 in
Section VI).
IV. Computations Required to Estimate Parameters and to Test Hypotheses
The best linear unbiased estimate of the parameter matrix
S
(mxp)
The matrix S*
E
(AI A)-l
AI
(mxn) (nxm)
y
B is
(13)
(m><n) (nxp)
of sum of products due to error is
S*
E
(pxp)
(Y -
AS)
I
(Y - AS),
which simplifies to
S*
E
(pxp)
yly
(pxp)
yl
A
B
(pxn) (nxm) (mxp)
(14)
11
The matrix
S~
of sum of products due to the hypothesis is
S'
C'
[C (A'A)-l c'r l C
S
(pxm) (mxs) (sxm) (mxm) (mxs) (sxm) (mxp)
S*
H
(p xp)
(15)
If we modify the observation matrix Y and the estimated parameter
matrix S by making arbitrary linear combination of the p observations
made on each experimental unit, linear combinations can be represented
by YU and we can define a new SE and SH matrix as
Y'AS]
[Y'Y
U
(16)
(pxp) (pxu)
(p"p)
and
[C (A'A)-l C,]-l
C
S
U
(sxm) (mxm) (mxs) (s xm) (mxp) (pxu)
U'
S'
C'
(uxp) (p xm) (mxs)
(17)
The classical univariate F or multivariate test statistic can be
computed from the matrices (16) and (17).
If the rank(C)
= s, rank(U)
= u, and rank(A) = m, then
F.1.
where i
( 18)
1,2,.",u, and S a n d S
H..
E ..
1.1.
SH and SE'
are the diagonal elements of
1.1.
The three multivariate test statistics in common use are
constructed from SH and SE' as follows:
1. Largest root
e=
Al
(19)
HAl
2. Trace
Tr
-1
Trace (SHSE )
6 A. ,
i
1.
(20)
12
3. Likelihood ratio
. t h e 1..th 1argest root
were
h
A. 1.S
1.
-1
A
n(1+A.)
1.
i
0
(21)
f
0,
A
being the 1arges t root of this equation.
v.
Program Outline
l
(22)
The computer program consists of three primary subroutines:
MODEL and HOTEST.
MANa VA ,
After reading the control cards that define the model,
MANOVA is called to read the data and form A'A, Y'Y and A'Y.
The data
cards to be analyzed contain not only the dependent variables and covariables but also a series of integers that identify the cell in the
experimental design from which the observations were obtained.
Utilizing
this cell information and covariates, the program constructs A'A automatical1y.
In order to allow regression problems, paired "t" tests and
Hote11ing T2 tests to be analyzed using the same program the observed
data are usually considered as being obtained from the same cell or
treatment group, while in the analysis of variance and covariance, observed data are considered as being obtained from many cells.
By a double pivoting method for solving simultaneous equations
and matrix inversion, Sand (A'A)-l are calculated.
Accuracy of the
inversion is then checked by evaluating
A'y - A' AS
A'y - (A'A) (A'A)-l A'Y,
which should be zero except for rounding and truncation errors made in
the computation,
The error matrix is calculated by computing the dif-
ference between the total sum of products matrix (Y'Y) and the sum of
13
products due to the model y'AS.
From this error matrix an error cor-
relation matrix is computed.
MODEL prints the estimated parameter matrix and reconstructs parameters lost through the reparameterization.
HOTEST is then called to evaluate various hypotheses.
model has been specified as a factorial design (NOMU
gram generates U
= (l)and
=
If the
0), the pro-
the appropriate C matrices to test for no
treatment effect in each of the main effects, no interaction if interactions are specified on the model control cards, and no covariate
effectso
Following this, HOTEST reads the cards necessary to test
the user's specific hypotheses.
Using a specific C and U matrix, the error (SE) and hypothesis
(SH) matrices are determinedo
hood ratio evaluated.
F ratios are computed and the likeli-1
The matrix SHSE
is transformed to a symmetric
matrix and its eigen values and eigen vectors found by the Jacobi
method.
The eigen vectors are then transformed back and printed as
discriminant function weights.
From the eigen values Roy's largest
'root criterion (the canonical correlation coefficient) is evaluated
and Hotelling's trace is computed.
Following this, another C and U
matrix are acquired and the procedure is repeated.
Control Card Design
The first control card contains a series of parameters defining
the model; it is punched according to Fortran format (715).
The numbers punched must be right justified in their respective
fields.
14
NME
Number of main effects or factors in the design.
NCOV
NRHM
Number of covariates.
= Number
NIT
Col. 1-5.
Col. 6-10.
of dependent variables.
Col. 11-15.
Number of two-factor interactions (used in factorial ANOVA).
Col. 16-20.
NCON
IPOTT
NOMU
Number of C and U matrices to be read by the program.
=
0 normally.
o
Col. 26-30
Col. 21-25.
0
for inclusion of mean in the model (restricted model
1 for exclusion of mean from the model which has only one
main effect.
Col. 31-35.
The second control card contains the number of levels in each
main effect or factor and is punched according to format (1615).
If
there are more than sixteen main effects, then more than one card is
required to punch the levels.
The third card contains the factors
of each two-factor interaction (for a two-way design with main effects
A
=
1 and B
=
2, to compute the AB interaction
=
two-factor
inter~
action, the numbers 1 and 2 would be punched on the interaction card)
to be included in the model (also punched 1615).
then this card (third) is excluded.
If NIT
=
0 in Card 1,
It is important to remember that
only two-factor interactions can be computed by this program automatically.
However, the higher order interactions can be computed by in-
putting the contrasts as if they were covariables.
Normally, however,
higher order interactions are lumped into the error term.
Data Card Design
The design of the data card containing the observed dependent
15
variables (y's), covariates, and cell identification is completely arbitrary since the user must supply a subroutine to read in the data to be
analyzed.
The cell identification consists of a set of integers, I., one
1
integer for each main effect in the experimental design.
As examples of
typical cell identification, the data for multiple regression and
Hotelling T2 are usually obtained from one cell, hence II
=1
0
= cell ident
For a one-way ANOVA with six levels
II
1 if Y is observed in cellI;
2 if Y is observed in cell 2;
6 if Y is observed in cell 6,
For a two-way design (3 x 2) the cell identifications are
A effect
1,1
1,2
2,1
2,2
3,1
3,2
B effect
2 if Y is observed in cell 1,2;
II
= 3, 1 2 = 1 if y is observed in cell 3,1.
The subroutine to read in the data must have the following basic
construction:
16
Subroutine READIN (Y, INDEX, X, NME, NCOV, NRHM)
Dimension Y(l), INDEX (1), X(l)
Read data
Return
End
The array Y is the array of observed dependent variables.
The array X is the array of observed covariates.
The array INDEX is the array of cell identification integers.
As an example, for a 3x2 design with three dependent variables observed
and two covariates, the subroutine might appear as
Subroutine READIN (Y, INDEX, X, NME, NCOV, NRHM)
Dimension Y(l), INDEX (1), X(l)
Read (1,1) (Y(I),I=1,3), (X(I), I
1,2), (INDEX (I), I
1,2)
1 Format (5 F 10.4, 2 I 5)
Return
End
The read statement could also be written as
Read (1,1) (Y(I), I = 1, NRHM), (X(I), I
I
1,NCOV), (INDEX(I),
1, NME).
17
A data card would be punched as follows:
Variable
Column
Yl
1 - 10
Y2
11- 20
Y3
21 - 30
xl
31 - 40
x
41 - 50
2
Index
Index
55
l
60
2
So a typical observation from cell 2,1 would be punched as
Variable
Value
Yl
5.3
Y2
16.4
11- 20
Y3
18.9
21- 30
xl
2.1
31 - 40
x
3.6
41 - 50
2
Index
l
Index
2
Column
1 - 10
2
55
1
60
Each set of data for analysis must end with a blank set of data.
(If each observation requires M cards to punch, the M blank cards must
be included at the end of the data.)
When the program finds a set of
data coming from cellO (the blank set), it recognizes this as the last
set of data cards.
18
Hypothesis Card Design
Testing a hypothesis of the form
H
o
13
C
o
D
(sxm) (mxp) (pxu)
requires a control card punched with the rank of the matrix C(s) and
the rank of the matrix D(u), respectively, with a 3 I 5 format.
If
D
(pxp)
ted.
I
, a 1 can be punched in column 15 and the D matrix omit(pxp)
Numbers must be right justified.
The matrix C is punched by rows on the following cards (Format
16 F 5.1), and then matrix D is punched by rows on the cards following
matrix C (Format 16 F 5.1).
For instance, if
and D
C
1
0
-1
0
o
o
1
-1
the set of cards defining the hypothesis would appear as follows:
Card
1
2
Parameter
Value
Column
Rank (C)
2
5
Rank (D)
2
10
Matrix C
0.0
1.0
0.0
0.0
0.0
1.0
3
8
13
18
23
28
-
5
10
15
20
25
30
19
3
Parameter
Value
Column
Matrix U
1.0
0.0
-1.0
12 - 15
0.0
18 - 20
0.0
1.0
23
28
33
37
3 - 5
8 - 10
0,0
-1.0
-
25
30
35
40
There are NCON sets of hypothesis cards (control + the matrix C
+
the matrix U) where NCON is specified on the first control card.
Each set of hypothesis cards consists of a control card, the matrix
C, and the matrix U, in that order.
Summary of Contror Card Design
Card
Format
1
7 I 5
Parameter
NME
= Number
of main effects in
design
NCOV
Number of covariates
NRHM
Number of dependent variables
NIT
NCON
IPOTT
Number of two-factor interactions
=
Number of hypotheses to be
tested
o normally
a include mean in analysis
NOMU
1 no mean in analysis
2
(or more)
161 5
Level (I) , I = 1,2, ... NME It levels
in each main effect.
3
(or more)
1 6 I 5
Int
= 1st and 2nd factor in
22
ll
1st interaction
Int
20
Format
Parameter
1nt
= 1st and 2nd factor in
22
21
2nd interaction
1nt
Int} Int 2 .. 1st and 2nd factor in
1 '
:i th
interacti,on
(Card 3 is omitted if NIT
Free format
4
i
= 0)
Data cards
(Under control of READIN subroutine)
Each data card must contain Y ,
i
i
\,
i
Index.,
i
1
..
1,2,. '. . NRHM
= 1,2, ... NCOV
= 1,2, ..• NHE
Blank S9t of data cards
If NeON
~
0, then NCON sets up hypothesis cards in the following order:
3 I 5
1
Rank (C), rank (U)
o in col, 15 1f matrix U is to be read
and 1 in col. 15 if U • I, and the com···
puter generates U
2
1 6 F 5.1
Mat rix C (punched by rows)
3
1 6 F 5.1
Matrix U (punched by rows)
omitted if a 1 is in col. 15 of the
control card
VI, Examples of Models
Multiple Regression
Case:
A series of animals are studied where cardiac output and mean
blood pressure are measured while heart rate and respiration are
varied.
See page 39 for example.
21
Question:
Did cardiac output and mean blood pressure change with
changes in heart rate and respiration rate?
Model:
E(y. )
1
(lX2)
where
cardiac output of the i-th animal
mean blood pressure of the
i~th
animal
heart rate of the i-th animal
respiration rate of the i-th animal
Yll
Y12
1
all
a
Y2l
Y22
1
a
a
y
A
(n x 2)
(n"'3)
Ynl
! ]ll
/,
6
(3 x 2)
Yn2
2l
12
22
, and
\
\
.~.
1
anI
a
n2
]l2
611
612
6 21
6 22 !
Hypotheses to be tested:
1. Does heart rate ...affect cardiac output and mean blood pressure?
20 Does respiration rate affect cardiac output and mean blood
pressure?
To answer question 1, we test the hypothesis that the heart rate regression coefficients are zero:
22
Using the hypothesis structure
c
S
U
(s x3)
(3x2)
(2 x u)
= 0,
let
C
(0
1
0) and
(lx3)
U
(2 x 2)
=(
1
0
0)
1
which yields upon multiplication
(0
1
0)
(0
].11
].12
Sn
S12
S2l
S22
(0
0),
0).
To answer question 2, we test the hypothesis that the respiration rate
regression coefficients are zero:
for which
C
=
(0
0
1) and U =
(~ ~ )
These two tests will not in general be independent, however.
make the tests simultaneously, using
One should
23
giving
Control Cards
Parameter
Value
Column
NME
1
5
NCOV
2
10
NRHM
2
15
NIT
0
20
NCON
3
25
IPOTT
0
30
NOMU
0
35
1
5
Card 1
Card 2
Level (1)
Card 3
Omit
Data Cards
Yl
Y2
a
l
a
1 (all
observations come from
the same cell)
2
Blank Set of Data Cards
Hypothesis Control Card
C matrix card
No U matrix
Rank (C)
1
5
Rank (U)
2
10
Identity matrix
1
15
0.0
1 - 5
1.0
6 - 10
0.0
H - 15
24
Parameter
Hypothesis Control Card
C matrix
Value
Column
Rank (C)
1
5
Rank (U)
2
10
Identi ty matrix
1
15
0.0
1 - 5
0.0
6 - 10
1.0
11 - 15
No U matrix
Hypothesis Control Card
C matrix
Rank (C)
2
5
Rank (U)
2
10
Identi ty matrix
1
15
0.0
1 - 5
1.0
6 - 10
0.0
11- 15
0.0
16 - 20
0.0
21 - 25
La
26 - 30
No U matrix
Hotelling 1 2 (simultaneous paired "t" tests on p pairs of characteristics).
Case:
Mean blood flow, mean blood pressure, cerebra-vascular resistance are measured in a series of experimental subjects both
before and after the administration of epinephrine.
See page 47
for example.
Question:
Did the drug change the blood flow, pressure, and resistance
significantly?
25
Model:
E(y. )
j.l
1.
(lx6)
(lx6)
where
Yil
=
blood flow before
Yi2
blood pressure before
YiJ
resistance before
Yi 4
blood flow after
Yi5
blood pressure after
Yi 6
resistance after
I
. Y
ll
Y12
1
1
Yl6
Y
A
(n x 6)
(nxl)
Ynl
Yn2
=
1
,
and
Yn6
1
(3
(lx6)
Hypothesis to be tested:
Did the means of flow, pressure, and resistance change with administration of the drug?
We form
(blood pressure)
0,
(heart rate)
0,
(resis tance)
0,
26
by setting
C
so that
(1)
1
0
0
0
1
0
0
0
1
-1
0
0
0
-1
0
0
0
-1
= (1) and U =
C
U
S
(l x1) (l x6) (6 x 3)
0
(l x3)
gives
(~1]..12]..13]..14]..15]..16)
= (]..I1 - ]..14' ]..12 - ]..15' ]..13 - ]..16)
(0
0
1
0
0
0
1
0
0
0
1
-1
0
0
0
-1
0
0
0
-1
0) .
Control Cards:
Parameter
Card 1
Value
Column
NME
1
5
NCOV
0
10
NRHM
6
15
NIT
0
20
NCON
1
25
IPOTT
0
30
NOMU
0
35
1
5
Card 2
Level (1)
Card 3
Omit
27
Value
Parameter
Data Cards
Y1
Y2
Y3
Column
1 (all observations come from
cell 111)
Y6
Blank Set of Data Cards
Hypothesis Control Card
Rank (C)
1
5
Rank (D)
3
10
Identity matrix
a
15
C matrix
La
1 - 5
D matrix
La
1 - 5
0.0
6 - 10
0.0
11- 15
0.0
16 - 20
La
21- 25
0.0
26 - 30
0.0
31 - 35
0.0
36 - 40
La
41 - 45
-La
46 - 50
0.0
51 - 55
0.0
56 - 60
0.0
61 - 65
-1.0
66 - 70
0.0
71 -
75
0.0
76 - 80
0,0
1 - 5
-La
6 - 10
28
Analysis of Variance:
Case:
Case 1
A series of twelve animals were studied by dividing them into
six groups according to their diet and sex.
The cell identi-
fication is as follows:
I
Sex { :
1:
II
I:
III
8J
Diet
The cardiac output, heart rate, and initial body weight of
the animals were measured.
The dependent variables were
cardiac output and heart rate.
Since it was felt that the
initial body weight might have a significant influence on
the treatment effects, it was measured and used as a covariate.
Questions:
See page 52 for example.
1. Did diet affect the measured dependent variables?
2. Did sex affect the measured dependent variables?
3. Was initial body weight an influencing factor?
Model:
E(y. )
1
(lx2)
where
a.
1
w.
1
1 if animal received treatment combination i,
{
o
if animal did not receive treatment combination i,
initial body weight,
29
Yn
Y12
Y21
Y22
A
Y
(12 x2)
(12x 7)
Yn1
a a a a a
1 a a a a a
a 1 a a a a
w
1
w
2
w
3
0
1
Yn2
811
1
0
0
0
0
w
4
a a
1
a
0
a
w
5
0
0
1
0
0
0
W
0
0
0
1
0
0
w
7
a
0
a
1
a a
W
0
0
0
0
1
0
w
9
0
0
0
0
1
0
w
10
0
a a
0
a
1
w
0
0
0
0
0
1
w
12
6
s
n
812
8 22
8 32
8
(7 x2)
841
842
8 51
8 52
8 61
8 62
Y1
Y2
Y1
regression coefficient of initial weight on cardiac output,
YZ
regression coefficient of initial weight on heart rate,
8i1
mean of cardiac output for the i-th treatment combination
U= 1,2, ... ,6),
8i2
mean of heart rate for the i-th treatment combination
(i
=
1,2, ... ,6).
30
Hypotheses to be tested:
1. Did cardiac output and heart rate vary with diet?
2. Did cardiac output and heart rate vary with sex?
3. Did initial body weight affect cardiac output or heart rate?
1- ( (3 41 + (311) - (13
+ (3 )
21
51
(13
41
(13
42
(13
2. (13
42
11
(13
3. y
12
1
=
+
(3
+
(3
+
(3
+ 13
11
12
12
) - (13
+
0,
)
0,
) - «(322 + (352 )
0,
) - (13
31
(3
61
+ (362) = 0;
32
) - (13
+ 13
+ 13 61)
0,
+ 13 22 + (332) - «(342 + 13 52 + (362)
O·,
21
+
(3
31
41
51
0,
O.
Y2
For tes t 1,
1
-1
0
1
-1
0
0
0
-1
1
0
-1
0
C = (
1
)
and
u
=
:)
(:
For test 2,
c
(l
1
1
-1
.;.1
-1
0)
and
u=c
a
1
)
For test 3,
C
= (0
0
0
0
0
0
1)
and
U
=( :
0
1
)
31
Control Cards
Parameter
Value
Column
NME
1
5
NCOV
1
10
NRHM
2
15
NIT
a
20
NCON
3
25
IPOTT
a
30
NOMU
1
35
Card 2
Level (1)
6
5
Card 3
Omit
Card 1
Data Cards
Y1' Y , w, I (I
2
= 1,2, .•. ,6), depending on which cell the
observation came from.
Blank Set of Data Cards
Parameter
Hypothesis Control Card
C matrix
Value
Column
Rank (C)
2
5
Rank (U)
2
10
Identity matrix
1
15
1.0
1 - 5
-1.0
6 - 10
0.0
11- 15
1.0
16 - 20
-1.0
21 - 25
0.0
26 - 30
0.0
31 - 35
1.0
36 - 40
0.0
41 - 45
-1.0
46 - 50
La
51 - 55
0.0
56 - 60
-1. a
61 - 65
0.0
66 - 70
32
No U matrix
Parameter
Hypothesis Control Card
Value
Colunm
Rank (C)
1
5
Rank (U)
2
10
Identity matrix
1
15
1.0
1 - 5
1.0
6 - 10
La
11 - 15
-1.0
16 - 20
-1.0
21 - 25
-1.0
26 - 30
0.0
31 - 35
Rank (C)
1
5
Rank (U)
2
10
Identity matrix
1
15
0.0
1 - 5
0.0
6 - 10
0.0
11- 15
0.0
16 - 20
0.0
21 - 25
0.0
26 - 30
1.0
31 - 35
C matrix
No U matrix
Hypothesis Control Card
C matrix
Analysis of Variance.
Case 2
Case:
Same as above.
Question:
Same as above,
See page 61 for example.
where
f1
if animal received Diet I,
Lao therwise;
33
{~ otherwise;
if animal received Diet II,
aiZ
i f animal received Diet III,
a
b
i3
=
(~ otherwise.
{~
i1
i f animal is male,
otherwise;
if animal is female,
{~ otherwise;
biZ
w.
initial body weight.
1
This
~ode1,
using the familiar concepts of treatment "effects" and
3
Z
sex "effects", is overspecified, and the restrictions L S.
0, L ' . = 0
i=l 1
i=l 1
A
must be made.
A
Therefore,
13 3
-(13
1 +
Sz) ,
and
'z
-, 1·
The matrices appear then as
Y11
Y12
1
1
0
1
w
1
1
0
1
W
1
0
1
1
w
1
0
1
1
w
4
1
-1
-1
1
W
1
-1
-1
1
w
6
1
1
0
-1
w
7
1
1
0
-1
W
1
0
1
-1
wg
1
0
1
-1
w10
1
-1
-1
-1
1
-1
-1
-1
, A
Y
(nxZ)
Yn1
. YnZ
1
z
3
s
s
Wl)
w
1Z
34
(3
III
112
(31
(31
~
~
(32
(32
T
T
l
Yl
1
Y
2
Relating the parameter matrix (where superscript represents the example
number) we see that
+
(3 IV
1
T
IV +
II + (3 2
T
ll+
IV
l ,
IV
l ,
IV + (3IV)
II - ((3 1
2
+
IV _
II + (3 2
=
IV
T
l
IV
Tl
'
,
IV + (3IV)
II - ((3 1
2
-
IV
Tl
'
IV
III
Y
Y
To test the hypotheses considered under the earlier model, we require
three pairs of card U matrices:
L C
1
o
0
o
1
0
~)
U
2. C
(0
o o
1
0),
U
3. C
(0
o o
0
1) ,
U
=
(~ ~)
(~ ~)
(~ :)
and
.
35
Control Cards
Parameter
Card 1
Value
Column
NME
2
5
NCOV
1
10
NRHM
2
15
NIT
1
20
25
IPOTT
°0
NOMU
0
35
Card 2
Level ( 1)
3
5
2
10
Card 3
Level (2)
Int (1,1)
Int (1,2)
NCON
Data Cards
Y1'Y2'w
30
1
2
1 (1 = 1,2,3);
1 1
1 (1
2 2
~ A-B
10
= 1,2)
interaction
where 1 , 1 define the cell where the obser2
1
vation was made.
Blank Set of Data Cards
For factorial designs such as the one above, the estimateS of parameters
3
~
3
~
L S. = 0, L 1 = 0, and the
i=l l
i=l i
C and U matrices are automatically generated to test for no treatment,
are computed subject to the restrictions
no interaction, no covariate effect, with U = I.
VII. Description of Output
The output for the program is initiated by a printout of the means,
variances, and cell populations of the dependent variables and covariables.
This printout is incorrect if the data are not sorted so that all
data from each cell appear together.
However, the overall analysis that
follows will still be valid.
Next the matrices (A'A) and(A'A)
-1
are printed.
Following this,
36
the accuracy of inversion is checked by comparing A'Y with (A' A) B.
Next appear the uncorrected sums of squares for the model.
The error correlation matrix appearing next is constructed from
the error matrix by
SE ..
1J
I
where SE
=
SE .. SE ..
.
11
JJ
(Y'y - Y'AS).
The error matrix SE is also printed, as is the estimated parameter
matrix
B.
Each of the following output pages contains the relevant informaEach page contains the C
tion concerning each hypothesis to be tested.
matrix, D matrix, the estimated contrast matrix
= C B D, the error
matrix
A
SE
(pxp)
D' (Y' Y - Y' AB) D,
the sum of products matrix due to the hypothesis
SH
=
D'B'C' [C(A'A)-lC' ]-1 C
S u.
(pxp)
A new correlation matrix is constructed using the modified error
matrix (modified by multiplication by D).
SE
ij
I 'SE .. SE ..
11
1J
The univariate F values are computed for each of the p observed
dependent variables, as follows:
37
S
F.
1
. Hii/rank (C)
S
Eii/(n-rank (A»
To compute the discriminant function weights we maximize the
treatment separation defined by the hypothesis.
Utilizing the deter-
minen tal equation
we solve for the s eigenvalues (s = min(rank (C), rank (U») and the
associated eigenvectors.
tion weights.
These eigenvectors are the discriminant func-
These are normalized by dividing each of the p weights
by its associated variance.
The three multivariate indicators are then printed:
-1
Hotelling's Trace
Trace (SHSE )
Roy
e
It
1
max
+ Itmax
Wilks
For s
= 1 or 2, the exact F ratio is computed to facilitate evaluation
of significance levels.
For s
>
2, the tables developed by Schatzoff
can be used.
Below are the limitations of the program:
1) Number of dependent variables (p) ~ 35, and rank of A matrix < 35.
However, both of these dimensions can be increased easily.
2) No limitation on number of observations.
38
3) The program is in single precision arithmetic and no special
effort has been made to control accuracy.
A revision written
in double precision arithmetic and with special attention to
accuracy is being prepared.
4) The error message at the end is the 360 system end of file
message.
39
APPENDIX I
Example 1
Yl
Mean P
Y2
Cardiac Output
xl
Response Rate
x
2
Heart Rate
1l0.4
1. 76
.07
7.8
102.8
1.55
.07
8.9
101.0
2.73
.07
8.9
108.4
2.73
.07
7.2
100.7
2.56
.07
8.4
100.3
2.8
.07
8. 7
102.0
2.8
.07
7.4
93.7
1.84
.07
8.7
98.9
2.16
.07
8.8
96.6
1.98
.02
7.6
99.4
.59
.02
6.5
96.2
.80
.02
6.7
99.0
.80
.02
6.2
88.4
1.05
.02
7.0
75.3
1. 80
.02
7.3
92.0
1. 80
.02
6.5
82.4
1.77
.02
7.6
77 .1
2.30
.02
8.2
74.0
2.03
.474
7.6
65.7
1. 91
.474
8.3
56.8
1.91
.474
8.2
62.1
1. 91
.474
6.9
61.0
.76
.474
7.4
53.2
2.13
.474
59.4
2.13
.474
G
58.7
1.51
.474
7.5
58.0
2.05
.474
7.6
6.9
- - - - - - - - - - - - - - _ .. -IITE5Tl JOB
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_ _ _ _ _ _ _ _R.E..A.DJ.l.lll_Y.HIo y ( 2l.L llll.LX !.lL._
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_____..
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_
4
INDEX!ll-I
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Before
Mean P
Mean F
After
CVR
Mean P
Mean F
CVR
Obs.
1
202.0
125
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202
129
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2
292
105
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292
108
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3
203
86
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198
89
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4
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188
76
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5
148
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96
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6
249
103
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7
193
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97
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260
117
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IITEST5 JOR
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Y1
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Rate
Female
xl
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Weigh t
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40.9
38.1
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Weight
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130 IB(I,01)-O.O
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131 11(1,1)=1.0
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IP(II-1) 100,101,,100
101 DO 102 1=1,IP
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WRITE (3,3)
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21 WHITP. p, 2) (COM (I,J) ,J=1 ,ft)
11
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15 , , 0 P, "
, P = '. F6 • 4 )
31 FOR"AT(1HO,'LIKELIHOOD RATIO = ',F6.4,· ASY~PTOTIC CHI-SOOARE = ,
*, F10.4,' WITH', 16,' DF',68 P ,F6.4)
,
35 FOR"AT(1HO,31REIACT F FOR LI&E11ROOD RATIO IS,Flv.4,7H WITH (,14,
*18,,14,108) OF, P
,F6.4)
41 FOP"AT(lRO,35X,24HUUIV1RIITE P VALUES WITH,IS,3H , ,15,2 0 8 DEGREE
*5 OF FREEDOfl)
45 FOR"AT(1RO,23HHOFftILIZED DISCRIftINAHT,14I,4E10.8,5{/381,4E20.8»
47 FOF"AT(1HO,'"IXI"O! DISCRlftIllBT FONCTION YIELDS A UNIVARIATE F ('
* ,,14,',',14,') = ',F10.4)
S.OC~9
=
5.M\10
5.0011
=
5.0012
=
5.(\1'\13
5.0('14
5.0(115
S.()r16
48 fORPUT(lHO,561,19BSIGMIFICAMCE LEVELS)
S.C011
K=l
S.C"18
5.01119
S.or;)!)
00 /JOO 1=1,"
00 /JOO ..1=1,1
H (I,..1) =550 (K)
400
5.00/1
K=~+l
IF(N-l) 120,46,120
S.O"22
5.('1('21
c
12C NeOL='
CO"PU~E
CORRELATIOJ ftlTRIX
DO 23 1=1,8
5.0021&
5.0025
S.0026
DO 23 ..1=1,8
IF(J-l)110,110,111
(1.1+1) /2- (1-..1)
S.(\('27
l1C
S.OO2~
GO TO 23
'1' K=(J.J+J)/2-(J-I)
23 B(I,..1)=SE1(~)/FLOAT(IEDF)
DO 24 1= 1, N
DO 24 ..1=1,5
24 A(I,J)=B(I,J)/SOPT(B(I,I)*B(..1,J»
WRITE (3,6)
DO 2S l=l,M
S.Of'?'l
S.i)"ll)
S.(l011
5.0 0 n
5.0"ll
s.o'n~
5.1""'15
'"
,=
=
5.l"I",~r;
6& • v 8;
1,~3Q,rs,IEDF,H,TE,T!T)
S. C·r'\f'l7
e
e
e
e
'?, :
IL COPiPU'tES THE !'lAXI"OK SEPABA1ION F:iMcnON FOR T;,;:; [;'l,:, \'
HRUiH.ES OF A "ULTIVARIATE ANALYSIS OF VARIANCE.
INPUT V':HBLE:r
S5~
ERROR MATRIX OF DI"E~SION N
SSO
SO" OF SQUARES AWD CROSSPRODOCTS FRO~ ftANOVl
LEVRL(I) -1 + DP(I)
NtINE
IORSER OF LIRES II THE ANALYSIS
IEDF
ERROR DEGREES OF FR!lDOft
REQUIRED SOB~OUTI"ES 1) GIUSS
2) ftlTIIV (fUTRIl II1V!RT)
3) CYV (EIGElIYALUE)
') DISCR (C1LCOLl TES DISCBlfUIA NT FU NC'l'ION)
DI"ENSION 5E1(630) ,SSO(630) ,LEVEL(50),ROOT(50),TE(35,35)
DI"ENSIOR TET(35,35) ,E(35,35) ,1(35,35) ,B{35,J5) ,P(50) ,BETA (50)
DI"F.NSION ERROR(35.3S),8(35,35)
4 FOFftAT(lHO,45I,30HDISCRlftINAIT FUICTION WEIGHTS)
5 FOP"AT(10HOLA"BD&
,E16.7,6H
,P6.~,4F20.8,5(/38X,4E20.8»
6 FOF~AT(lH ,551,208 CORRELATION "lT~IX)
,
c
D'
2,
C
C
C
C
C
C
JUT: E RUDII (SE1,
\.]:;1
f(:
--I
',J
25 WFITF (3,1) (A
('/,16
(! ,,1)
~.f)/"l17
K=1
5.COR
DO 69 != 1,N
~.(,1'~9
DO 69 .1=', I
ERFOR(I,J)=SE1 (Kl
s.or/JO
69
S. Ci'PH
S.nl'4;»
5. CMJj
,.1=1,1)
1(=1<+'
[10 70 1=',"
DO 7r J=I,N
7G ERPOP(I,J,=ERROR(J,I)
no 800 1=1,M
DO erc J=1,"
80C E(1,Jl=ERROF(I,J)
CALL KATIHVfF.,N,FOOT,O,DETER")
46 DO 2r 1=1,"
5.1'f\4"
S.<'045
5.0"4"
S.0041
5."048
5.t)('4Q
5.1)"';"
S.oor;1
S.0()1;2
DO 2/"1 J=: 1, N
H (I ,J) =8 (J, I)
TE(I,J'=H(r,Jl+F,RRO~(I,J)
5.0053
5.0(\')4
20 TET(1,J)=ERROR(I.J)
IR~NJC'=TS
5.001)">
DO H; 1= 1, N
5.0 0 56
5.0051
5.0058
5.00<;Q
5.0060
5.0061
5.0062
5.""63
5.0064
K= (I"I+!) /2.
ROOT fl) (H (I,ll/IS)/ (SEl (K) /IEDF)
S.Or.~5
5.0('66
5. {}('61
5.0068
=
16 P(T)=PRBF{FLOAT(IS),FLOAT(IEDF),ROOT(I)'
WInTP (J, 41) IS,1EDP'
WRTTF f3,1) fROOT(I), I=l,N)
WPTTF (3,48)
WRI'1'E (3,7) (P(Il, 1=1,)1'
IF(N-1'121,21,121
121 ITDF=IRAIK*N
C
CO"POTE STfP-DOWN STATISTICS
CAlL GAUSS(TE,N,M,N,A,B'
CALL GAUSS(TET,N,N,N,A,B)
B,.=FLOAT (M) /2. 0
wnKS
5.1)070
S.0(\71
IFfIRANK-N)125,125,126
126 IRANK="
5.0072
5.0073
C
5.0('74
= 1.0
DO 26 I=1,N
BETA(I) = TPT(I,I)/TE(I,I)
26 WILKS = WILKS * BETACI)
S. (\()f;q
CO"PUTE THY. DISCPI .. INANT FUNCTION
125
WRTTF.(3,~,
5.0C75
CALL
5.01'76
5.0071
s.O(nS
Ifl :: IS
S.Otl7CJ
5.(I"A,
5.0081
5.01"82
S.OOR]
5.0084
S.O~R5
5.0086
5.0flR7
S.(\('\R8
DISCR(R,E,N,IS,ROOT,TE,B,A,TRACE,ERRO~
IFf IS-Jl) 127,121,128
128 91=1
121 FIGF.M=~OOT(1,
DO 11 1=1,.1
10= (1*1+1) /2
IF(EIGEH-FOOT(l') 129,13r,130
129 EIGEN=ROOT (1)
130 PER=POOT(I'.10~.O/TRACE
WRITF (3,5) IlnOTCI), PER, (TE{J,I), J=1,N)
DO 44 K= 1, N
44 P(K) =TF(K,I}/SOPT {SE' (K1)/IEI1F)
11 WRITE (3,45)- -(P(K" «::1,N)
--J
-l>-
5.04'99
S.0090
5.0091
5.0092
S.ooe)]
5.009'
C
C
C
C
5.0095
5.0096
5.0091
5.0098
5.0099
5.0100
5.n101
5.0102
5.1)11)3
S.Ole4
5.01:15
5.01<'6
5.0101
5.0108
5.0109
5.0111)
5.0111
5.0112
5.0 113
5.0114
5.0115
5.0116
5.0117
5.0118
5.0119
5.0120
5.0121
5.0122
5.0123
P = EIGEN * IEDF / IS
WRI'I'l" (3,"7) IS,IEDF,F
'1'1 = TB1CP.
TRlcr = TRACE * PLOlr(IED~J
SIG = PRBF(FL01T(ITDP),1000.0,TR1CE/ITDF)
WRITE (3,9) Tl,TR1CE,ITDF,SIG
.
COftPUTE HEC~iS STATISTIC FOR SIGNIFICANCE OF EICH EFFECT
D. 6. HP.CK, CHARTS OF SOBE UPPER PERCENTAGE POIITS OF THE
DISTRIBUTION OF THE LARGEST CHARACTERISTIC ROOT, A8N. 8ATH. STAT.
YOLo 31 (1960) PP. 625 - 6'2.
ISH :: YS
IF (IS-I) 132,132, 131
131 ISH=.
132 H"=(11BS(I5-I)-1)/2.0
IH = (IEDF - • - 1)/2
THFTI = EIGER 1 (1.0 + EIGER)
WRITE (3,8) THETA,ISft,H!,18
ITDF
IS * H
Eft = IEDF + (IS - • - 1) / 2.0
SIG = .9999
CHI = 0.0
IF(VILKS) 55,55,56
56 CBI=-EB*ALOG(iYLKS)
SIG = PRBF(PLOAT(ITDF), 100a.o,CRI/ITDF)
55 VRITE (3,31) WILKS,CHI,ITDF,SIG
IE :: 2 * MB + 2
1H = 2 * 8" + 2
IF(I5-1)51,140,51
140 F=(1.0-VILKS)*(NH+l)/(VILKS*(Hft+l.0»
SIC; '= PRBF (FL01T (IR), PL01T (I E), P)
WRITE (3,35) F,IH,IE,SIG
51 IF (15-2) 21,50,21
50 l" :: (1.0 - SQRT (WILKS» *IEI (5 QR'!' (WIUS). (18 t 1))
1H .. 2 • 18 + 2
IE = 2 • IE
SIG = PBBP (pLOIT (IF), FL01T (IE), F)
WRITF. (3,35) F.IR,JE,SIG
21 RETORI
EID
=
'-l
V1
IOl'! 05/360 BASIC FOITRAM rf (E) CO!lPlLATIO!l(
LB'EL:l0CT67
s.on':' 1
C
e
C
C
C
S. 001' 2
S.oon]
FUNCTIOI PRBP (DA, DB, FR)
FlICT PROBABILITY OF 'IWOOft oceORRElCE OF AN F-RATIO.
01::: WOftER1TOR DP.GREES OF FRFRDOft.
DB::: DEROPIIATOR D!GRFES OF FRErDO".
FR::: F-PATIO TO BE F.Y1LQITED.
PBBF IS RFTORRED IS A DEClftAL-FR1CTIOJ PROBleILITY.
PRRF ::: 1.(\
TP(DI.DB.FR) 2,15,2
CO"PUT~S
S.I'('!' "
21F(FR-l.0)5,6,6
S.lHl"S
S.001\6
S.0("'7
6 l='(ll
S.(H'l('A
GO TO 10
5 I ::: DB
S.0009
s.0010
S.C"'11
S.0012
S.Cl'13
5.001"
5.0"''5
5.0016
DATE: 68.u81
B ::: C8
F ::: 1I'R
B ::: III
F ::: 1.0 / PI'
10 II ::: 2.0 / (9.0 •
I)
DB = 2.0 / (9.0 • B)
Z ::: 18S«((1.0 - DB) • 1'••0.333333 1 •• 0.666667 4- II)
111'(8-4.0)7,8,8
7 Z=7..(1.0+O.08.Z••'/0••3)
1.0 4- Al) / SQBT(BB • P
,....
S.0(,17
8 PRBF=0.S/(1.0+~.(O.19685'+Z.(O.1151q"+Z.(O.00034"tZ.v.019S27»»
S.0019
S.001Q
5.0020
S.0021
S.OC22
S.0(2)
IF (FR-1.0)9,20,zr
9 PRBF=1.0-PR8P
20 IP(PRBP-.l'OOl)21,15,15
21 PBDP=.OOOl
15 BETOI.
!IID
"-J
0\
IBft OS/36C BASIC FORTRAN IV (E)
LE'El: 1OCT67
DI~PNSIO"
8(35, J5) ,E(35,35) ,ROOT(5C) ,VECTOR(35,J5) ,8(35,35)
DI"EISION 1(35,35) ,ERROR (35,JI))
s.erl')
IF(IPAIK-J)2,2,10
C
s.oe05
IF FAIK OP COl .LT. I ~IU< TRIAMGOLARIZE 58 ELSE TRIAMGOLARIZP
2 CALL G10SS(H,I,.,IRANK,A,B)
S.0("n6
5.0('('7
00 12 I=1,IRAIIK
DO 12 J= 1, II
5.Ctlt'\8
SOP!
5.0f'01J
DO 11 It= 1, I(
11 SO"
S. CO 10
= 0.0
= S0'"
... E (.1, It) • A(1, It)
5.0n13
DO 10 .1=l,IFARIt
5.t)C 10
5.<J'-'15
SO"
= 0.0
= S0'"
DO 13 It= 1,1
~.0016
13 SO"
5.0017
14 E (I,.1)
5.001Q
C
5.()01<J
S.I)I']')
... A(I,K) • B(K,.1)
= SOft
GO TO 20
TRIANGULARIZE SF'-l<
10 CALL GAOSS(ERROR,I,I,I,B,A)
CALL ftITINV(B,N,X,O,DETFRft)
DO 15 1= 1,11
DO 15 .1=1,Jf
5.0021
S.~(\~2
= 0.0
5.0('23
SO,.
5.0(\24
5.(""5
5.0 n 26
DO 16 K=1,1
16 SO"
= S0'"
15 A (I,.1)
=
... H(I,K) • B(K,J)
5Uft
no 11 r=1,N
DO 17 .1=1,8
5.l'f""
5.0028
S. or, 2')
S0'" = 0.0
DO 18 P:= 1, •
18 So~ = SO" ... B(K,I) • A(K,.1)
11 EU,.1) = SOP!
5.(\1110
S.r.tlH
s.ef'l2
C
5.0014
S.on15
S.01)36
CO"'POTF TR!CF, EIGPNVF.CTORS AND EIGENVALORS
2(' TPACE = 0.0
L
=
TRANK
S.0039
IF(IPAWK-M)3,3,Q
1& L=H
3 no 21 r=1,L
21 TRICE = TPICF ... F(I,I)
S.OOlQ
CAl.L CVV (L.A,!, ROOT)
5.0 n "l)
5.00'"
DO 22 T=1.1f
DO 22 .1=1,11
5.0n~n
5.00'2
S.On')
S.OOll,.
5.00,.5
5.00'6
5.00117
S~-l
12 B(.1,I) = SOft
DO 10 1= 1, TRANK
5.0f'11
5.0012
5.('If'13
DATE: 6tS.\Jd'
SUBROUTINE DISCR(H,E,N,IRAIK,POOT,VECTOR,B,A,TRACE,EHBOR)
S.ON'1
5.001'\2
S.ON;"
CO~PILATION
SUP! = o.t'
DO 21& K=1,L
24 SU~
SUft ... BCT,J) • A(.1,K)
22 YFCTOPfI,.1) = SO"
=
Rl"TIIPN
fWtl
"'-l
"'-l
IBft
LEYEL : 10CT 61
5.0""2
C
5.00" 1
5 .. 0(\<'lJ
A(l,J)
11' B(1,J)
5.0009
5.001')
s.on11
5.('1'12
5.01'11
5.()On
5.0015
5.0016
5.0011
5.0(\1A
5.01'19
5.0(\10
5.01'71
5.0('22
S.on23
DATE: 68.",d1
= lA(1,J)
= 1(1,J)/1(1,1)
IFfIPAlI~-1.0)9,l~,9
5.00(\6
5. (\(\n7
s.orl" A
BASIC FORTFAH IV eE) CO"PJLATIOR
5U8POOTINY- G.OSS(1~,RROW,MCOL,IRA"[,A,8)
DI~E'SIO" 11(35,35),1(35,35),8(35,35)
GAUSS PERFORftS AM Ell flIIlATION PROCEDURE 011 THE tllTH11 lA (JiFOil ,NCOL)
DO 10 J=1,lICOL
S.(H""1
s. Onn Ii
05/36~
9
DO 1lJ T=2,IRIIK
IL :: I - 1
DO 13 ,1= 1, WROW
S0"
<,.0
=
DO 12 I(=1,lt
12 SUfi
SUft f A(~,I).B(K,J)
13 I(J,J)
AA(I,J) - SO"
DO 14 J= 1 ,HROW
=
14 B (I ,J)
=
= A (l,J)
I
A (1,1)
DO 17 I::2,IFIBK
K=I-1
DO 17 J=l,1I:
11 I(I,Jl=O .. C
15 DO 1~ 1=1,181_11:
All = SORT (le5 (I (I, I»))
DO 16 J=l,lICOL
AA(I,J)
S.OO?I$
16 l(I,J)
5."025
5.01)2"
RETOPIt
= I(I,J)
= l(I,J)
I
AIl
E1U)
---J
ex>
LFYFL:"
f6
1B"05/360 BASIC PO •.
j
s.crr1
SU~POUTINF
S.C!'!) 2
5.C(\~9
5.0010
1(' S (T ••1)
5.00 "
5.0012
:: 2"
= N- 1
1(\ 1=1.11
If. J=1."
=0 • r-
0180
0190
0410
Oq1~
0420
OO 15 1=1.M
croo
15 5(1.1) ::1.C
011110
0450
1)1160
01110
SET 5 :: IOF1ITITI ~ATRIX
101TJI'9 :: C
1I0YTFR
NOITRR ~ 1
DO 1r(l K = '.12
IfP :: 1 + 1
no 1"0 " = JlP.I
IF(lBSIO(K.ft») - TFL) 1CO.l0C.20
IFflBS(D(K.JIf)) - ABS(O(".") - O(K.K») / 2.0) 3v.30.40
BETA=2.0*0 (K. ff) / (D (ft .ft)-D (I. K»)
SINE=BPT1*SORT(1.0/(2.0*(1.C + 8ET1**2 +SQRT(l.v + BETA**2))})
GO TO 45
BETA= (0 (ft. ft) -0 (If .K)) / (2.0 *0 (K. ft) )
C
=
18
5.01'11)
S.C016
5.(\011
S."01~
S.(H"9
20
S.(H'20
30
5.0021
S.0022
5.1)(\~)]
CYV(N.S.D.VALU)
IF (!I- 1• 0 ) q. 21 • 9
9 '!'FL=O.O(\OOl
5. NH'8
S.Cl"lQ
DA:n•• 68.0dl
eE) COMPILA'IION
JACOnl ITERATION
FIGFNVALUFS ARE RETURNED AS DrlGONAL-ELF.~EWIS OF ftATHIX D
EIGFMY~CTORS ARE ~FTURNED AS COLU"NS OF "ATkIX S
Dlf!!WSIOM BI(5(,). [!fI(50). SK(50). 5"(50)
10
12 lf2
DO
DO
s.oon
IV
DI~EMsrON D(15.31)).S(35.35) .VALU(35)
EIGEMVALU~S AND ~IGENYLCTORS
BY THF
C
C
C
5.01'''1
S.OOf'lt
5.('0(\1)
5.rO"6
5.00"7
_r~Ali
ItC
01180
0490
f}500
0510
01)20
OSlO
054Q
0")50
056')
SI"E'SIGN~S08Tll.0/l2.0*~1.0&BET1**2&ABSIBJI'TA<*SQRX'1.0&BETA**2«<
5.t)t'l24
*<.BETA<
5.0(21)
45
S.0026
'11 DO C;O
CSO=SOPT(1.~-SIRE**2)
J= 1, I(
5.('11'27
BK(J)=O(J.K)*CSO-D(J.~*SIWE
S.0028
BfttJ)=D(J.1f)*SINE+O(J,")*CSO
COITI ,ure
S.O(,2~
5('
S.0f'3~
s.oe)q
s.e011)
fI'
55 D(".J)
o (Pt ,K)::('. 0
S.O(\)7
S.OO,q
S.01'40
S.N~41
S.0(\42
S.OColil
D (Jr. ") =0. (\
00 6 (\ .1= 1 • It
SIC (J) =5 (K .J) *CSO-S (",J) *SIIfF
60 S"(J)=S(K.J)*SINF+S(ft.J)*CSO
DO 70 J=l,M
S (K ,J) =SK (J)
S ( .. ,.1) =SJIf(J)
S.OC'lJ4
5.0045
S.0,,"6
S.0047
5.0"48
=
D(J,ft)
D(K.K,=BK(K)*CSO-BK(ft)*SIJE
o (~. ", =B!'I (IC) *SIR E+ Pft (PI) *CSO
S.0016
s.oo 1J1
=
DO «;5 J
1. I
D (J.ll') =BK (J)
o (I': , J) =0 (J , K)
D (J.
=8" (J)
5.0031
5.0032
S.0013
10
COIiTIttn~
10(1 COPlTIIWE
BIGAtJ=T\(1.2)
DO
110 J< = '.N2
<'590
060"
0610
0620
0630
0640
061)0
0660
(\610
061\0
('690
0100
0110
0120
0131)
0140
1\ 7 I)tl
()160
(771)
1)76f)
(\7 Ill')
:; q 11
('Cit"
"Q20
-----J
\J:)
5. o(lia 9
IfP = ~
S.on~o
5.0051
S.00'l2
5.0051
S.0054
5.0055
S.0<'56
S.0051
5.0058
5.0059
5.0060
105
110
115
120
C
5.0061
5.0062
5.0063
5.006_
5.0065
5.0066
5.0067
5.0068
5.0069
130
140
3Q5
2
1009
21
500
+
1
=
DO 110 "
liP,.
IF{ABS(BIGIU) - IB5(D(I,")))105, 110, 110
BlGIU=D(K,")
COllTIlIUE
BIGIL=BIGAU
DO 120 " = 1, 112
IP = " + 1
DO 120 K = lIP,lI
IF{IBS(BIGIL) - IBS(D(~,")))115, 120, 12C
BIGAL=D(J{,ft)
COJrrIROE
FIJD "II OFF DlAGOWIL ELEftEIT ABD TEST
IF(IBS(BIGAL) - TEL) 2· ,2 , UO
IY (10 - liOITER) 305,305,18
KBlTY (3,1(09)
COJrTUUE
PORftlT (18 158 .0 COll'EBGEICE)
DO 500 1-=1,1
'ALU(I)=D(I,I)
BE'l'UBW
no
0830
1)840
0850
0860
OlJ11J
0880
0890
0900
0910
0920
0930
091t0
0950
09&0
0971}
0990
1020
1030
1040
C»
o
Ut~'t~
i\...
l~'"
t
: t)
s.rr"',
(il
CO'PILATICtf
L
E: 6t1.<t(jj
OT"ENSJON TPT(35,2),Y(SC),TNDEr(50),X(SO),TSUD(Sv),KCELLf35,35)
DI"ENSION SE(6J,,),IM(5c),Z(5r),A(35,35),RE~RE(6lv)
M"~
NO. "AI~ FFFPCT5
ICOf =10. COfARIATfS
'RRP.
RO. RIGKT RANO SIDES
NIT
N~. INTP,RACTTONS
LEfFL(I) = LEfFLS OF THF "lIN EFFECTS
TNTfr,2) = T~TERACTIOIS ~ARTrD TN THf ANALYSIS
S-.{'('It''}
=
=
=
c
c
c
c
C
C
S.0(\"6
s.n-'\"I?
S. n ""'8
S.()(\;)9
s.on1()
5.0011
5.('!O12
S.I)011
2
,.
Fnp~~T(12I,P.20.A,121,E2C.8,121,F2C.6)
FOP"AT'tx,1~F12.4<
6 POR'UT (181)
161 FOP·AT(lHr,~H N = ,I5,3X,30H"~AMS AND VARTANCES FOB CELL lr.IJ)
162 FOR"-AT(1RO,RR
"EU,&F2~.R,CJ{/,9X,6F'2r-.R»
163 FORftAT(1P.C,89VAPIANCF,hE20.8,4(/,9X,6F2r. A»
169 FOR"ATl1HI',34RIHANS AND VARIANCES FOP COVARIATES<
90r FOP~ATI1Rl,10H ~FANS AND VARIANCES ARE CORP-LeT ONLY IF TKF DATA AR
*E SOPTPO PY CELL
)
1002 FOP"ATf1Rf,scr258 REDUCED INcrDENCF ftlTPIYl
1004 FORftAT(lHl,57115RRfDUCED ISVFRSE)
1019 POP~ATf1Hr,50~,28HUICORRFCTFDSOft CF PRODUCTS )
1028 FOP~AT"Hl.2hI3RRR~16X18HHACK SOBSTITUTION~2X1udDlrF~R~NCF.)
lC29 FOPl!~T f1Hl,31R"ATRTI IS SINl;OL1R, CFLLSTZFS FOLLOw)
5."014
S.0015
5.M'16
5.CC17
5.0e1"
5.01'19
S.0020
5.01'21
s.ot':n
S.002}
s.OOl'
s.(\t'25
S.0"26
5.0021
5.0('28
S.0"29
5."")1)
S.OO31
S.0032
1030 FOP"AT(5X,2515)
5C WRIT! (3,ft)
IBrp :< 0
IDF • n
00 20 1=1,30
51 (1) =0.0
512 fll =O.f
st (I) =0. C
st21J) =o.C
00 2C .1=1,30
RHPI(T,.1) =C.O
A (1,.1) =o.C
2<' "CFU (1,.1) =0.0
DO 26q .1K=1,465
269 It folK) =0.0
IRE
10. PlAII FP'FCTS
ICOY
MO. COflRTATFS
IRR~ = MO. RIGHT RIMO ftE"BERS
lIT
JO. IJTERICTICNS
C
MIlT = 0
C
C
C
5.00~'
lTPAN IV
DI"~NSrON lrVPL(~~),INT(35,2),C(63C),B(35,35),HH"(j~.3~)
,YYf63~)
5. Co "" '1
5.('''''''''
S.0014
5.0035
5.0('36
5.0037
5.0038
5.00l9
.,ASIC
SOPQOUTINF ~AMOV'(N~F.,NCOY,NPH",NIT,LF.VFl,TNT,C,B,NHft,YY,N,IDF,SF,
*1,J(0l1",REGP F l
DII'!ENSION SI (3')) ,5J2 (35), Sf (~5), Sf2 (35), 'U(3S) , UtU (35)
~."'(':'2
5.01) 3~
();;;/Jt~(j
=
=
=
IFfMIT)lCO,28,100
100 DO 26 I=l,M1T
'"
:< 1M (l ,1)
ft2
.1'"" 11,2)
IN (I) = (LEfEL CPr1) - 1) (LEVEL ("2) - 1)
26 NINT
MINT + IN(I)
WINT
MO. OF REDUCED LEVELS 01 I!TFRACTI0NS
C
=
=
=
*
00
,...-
4'
5,
S.0042
L
~.OO(n
=
5.004"
5.00'S
C
S.on46
S.on47
S.O{HI8
S.01)49
S.OI"""
s.on'>l
S.01"52
5.0053
S.ons..
S.005S
5.(\tl56
5.00!i7
5.0(\58
S.0(\59
5.{\060
5.0(\61
5.(\0f'2
5.C063
5.00611
IF{M~E)101.341.101
11ft :
-
Ileov
tn, FC
:
I'" E
rFfN~~)lC4,ln5,104
105 l"IFC:1
e
5.0082
5.0083
5.0084
5.0085
5.0f\R6
S.Ot:'R7
5.0088
5.0089
5.00<J0
I
VPITF '3,900<
C
S.OOfH
=
101 DO 21 l=l,M"!
21 LTOT = LTOT .. LEVPt{I)
IF (RIT) 122.341,122
122 DO 33 I=1,WIT
LT = ,ltY. .. J
33 LlVEL(tT) = IN (I) .. 1
341 IF (ICOY) 1(13,34,1(13
103 DO 2ro I=l,ICOV
tT = LT & 1
200 LEVEL(LT) = 2
34 11 : LTOT - 1"'E + MINT + NCOY
e
e
S.0"65
S.0066
S.0':'67
5.0068
5.0069
5.0070
5.0071
S.0072
5.0073
5.1)074
5.0075
5.0076
5.0077
S.0078
S.007<J
5.0080
=1
DO 3C I=l,NI'IE
I PT (I , 1) = t + 1
IPT(I.2)
L + LEVFL{I) - 1
){\ L = L .. tFVEL (1) - 1
IPT = START AND STOP LOCATIONS OF liTER ACTIONS 11 REDUCED "ITRIX
28 LTOT
1
1.'1' = R"~
LTOT = TOTAL IUBBFR OF LEVELS .. ftU
H = STZE OF REDUCED ~ATRII
1ft = SIZE OF REDUCED ftATRIX - COVARI1TES
1fC :
('
104 It= 1
Wc=n
IC:1
11[1=-1
DO 7f, I=l,NftEC
16 ISUB (1) = 1
12 CALL RFADIRfY,INOFI.X,lIftE,ReOY,NRHft)
IP(lft~)1C6.13,106
106 DO 173 I=l,W!!
101 J=nO'EI(I)
IP'TWDEX(I)) l1J,173.109
10<J ICY1.L(I,J)=JCE1L(I,J)+'
173 COITIaUE
13 DO 82 II1,a"EC
Ir(I'D)55C.'>3,~"r
550 IF(IMOEX(I)-ISUBfI))11~,111,110
111 IFfIIOEl(I))82,11Q,82
110 DO 160 K=l,RRRft
IP(WC-1) 114,1600,112
112 'lR(~)=(St2(l)-5T(K).SY(K)/.C)/(.C-1.0)
160C XftU'K< I 51'K< I MC
160 COllTTNUB
IDP •
IDF & lIC
WPTTF '3,161<
WRIT~
Rc,~rSOB~K<,
'1:.1,162< ~XPltJ"K<,
IF (ltC-l) 113,1601,113
K",N"F.<
U1,IU~Hft<
ex>
N
5.(
!
-:-C}:?
5.·.·1'(\1
S.('nllfl
5. 1' .ft, (l'l
."
_.~f·F(~. h.J) \9'1'" fKl.~- •• HF'H!1
16(1 I'(~CO~ "4.'64."Q
114 DO l~R K=l.NCOV
IFf Il C-'l166.16r.2.115
11!'> UF (PC) = (5 17 (K) -s t (tc) *SX (K) IKC) I
$."I'\Q';
1602
S.f'''Q7
168
5.(':~q~
S.l'\r'\q
5.(' 1:'"
S.(\l·"l
,r
s.r 2
5.011'1
S.(;1:"7
16'l
S.0121
S.O 12,.
5.0125
5.0126
5.r 127
S.012Q
5. ''29
'
5.013)
s.t) 111
5.(\132
s.on1
5. 0 13"
5.(1 11Ii
5.0136
S.0137
S.0139
s.onCJ
5.0UO
S.1)11&1
5.C1l&2
S.r.1113
5.01114
~".NCOY<
1~5 K.'.WFH~
f.r
•
ST2~'<
IIlC •
r:,
t
(.(\
IF("PF.l99 Q .'4.9QQ
9q9 IP tIllnEX (1))
53 DO 52 Ittt. R"~
53.".
52
,
rS"R~K<
~3
I.nEX~~<
GO TC 111
82 CONTUOE
14 HC ,
flC & 1
IP(NCOV) 869.58.R(,9
S.0116
S.t'111
~o
SI~K<
s.n1~CJ
S.<'118
S.011Q
S.0120
S.(\121
S.0122
~J~U~K<.
166 no 167 K.'.NCOY
SI~I{< • ;'.('
167 SY2~~< • l.o
164
5.011"
5.0111
5.r.112
S.O 113
S.OlU
5.011«;
.3.1~2<
WPTTF
(~C-l. C)
~C
rF(tfC-" "6.'66.116
116 Wpy'!'r.O.H-3) (YAFfn.K=l.ffCOYl
S.rl"4
S.f: 1""5
5.(1-"6
S.Cl':'<I
/
I~"~'< t Sl~~<
CO~Tnm!:
WRIT." ~]. 16Q<
86CJ DO 59 K=l.NCOY
SXT-K< • SI~K( & 11'<
59 SI2IK< , SX2~K< & X~K< • XlK<
SR JK=l
DO Q6 K= 1 • MRR"
SY fK) =SI (K)~T (r)
SY2 (It) =5Y2 (I) +T (It) **2
DO 96 J=1.K
IT (J~ =11 (JIH'Y fJ' *y (I)
96 JK=JK·p
C
~OR" V. 'ECTOR OP RFrOCED 15CIDEJCES CODED AS -1.v.+l
DO 170e" Lll .. !'
HOC Z'U< t
c. r
L : TRDEX (1)
IF (10""-1)770.7 7 1.710
771 Z (1) =1.~
170 IF(Wr.ftU-l)772.32. 77 2
772 Z (1) :1.0
IF (W"-l) 32 .. 32.. 7 73
713 DO 17 .1=2,1"
17
~(J)
L
=1
::
o.e
DO 16 K=l,IIfE
+ IIDEItl)
lYfrWDEI(I)-LEVEL(I)) 18,77'.. 774
L1 :: L
77. LL=~LE'!t(K)-l
LL 1 :: L ~ 1
DO 19 J-LL1.LL
191.(J)
=-1.r-
co
(..,;
~.01'S
.01116
5.011"
5.0,.8
5.01'9
5.0150
5.0151
5.0152
5.0153
5.0151
5.0155
S.Ol56
S.0157
5.0158
S.0159
S.0160
C
C
S.0161
S.0162
S.0163
S.(t16t
5.0165
S.0166
C
5.0167
S.0168
S.0169
5.0170
5.0171
S.0172
S.0173
5.017.
S.0175
S.0176
5.0177
5.0178
5.0179
S.0180
S.0181
S.0182
S.0183
S.018'
S.0185
S.0186
S.0187
S.0188
S.0189
5.0190
C
C
S.0191
5.0192
5.0193
5.0191
GO TO 16
'8 Z (t 1 ) - '.0
16 t ... 1. + Llyn (If) - 1
Clt.CUtlft II'I'ERIC'rl0IS
IF (ITT)775.32,775
775 ftJ=LTCi'-IIIE
DO 29 Ka1.lIT,.
11· Ift(I,1)
12 ,. Ift(I,2)
131 - IPT (n , 1)
132 - IP'J' (I " 2)
III - IP'J' (12,1)
KI2 .. IPT (K2,2)
DO 2' IIl-lll,132
no 29 112=111,KI2
lIoJ ... lIoJ +- 1
29 Z,IIJ) - Z(ftl).Z(1I2)
CILCULITE COYIRII'l!S
l2 IP(ICOY) 776,31,776
776 11I1=IIH1
DO 27 oJalln,l
oJ1 ... 01 - 1ft
Z (01) ., I (011)
27 COftIlOE
UPDA'J'! REDUC!D IICIDEIC! III'lRII liD RIGHT RAID SIDE
31 011.,1
DO S" oJ=l,1
IP (Z (oJ) 777,5.,777
777 DO 90 11=l,IBRft
90 BBft(11,oJ) ... RRft(lfl,oJ) + T(11).Z(~
23 DO 22 1=1,1
22 I (01 ,)() =A (.J.)() of Z «If) .Z (ol)
5' COITIIUI
GO TO 12
11 VRITI 13,6<
VRI?E (3,1002)
DO 102 1=1,_
102 IRITE (3,') (A (1,01) ,oJ= 1, I)
CHECK TO srI IF IIATRII IS SI.GULlB
IP.IBI)180,205,780
780 DO 17. 1=1,1..1
K = tEYELfI)
DO 111 01=1,)(
IP(ICEtL(I,J»781,781,171
781 IBITE(3,1029)
DO 175 L-l,IIIE
11=L!fEL (L)
175 V.I'l1 (3, 1030) L, (Icnt fL,lI) ,"-1,n.
GO TO 81
17. COITIIUE
COIIPUTE REDUOD IIDIDEIC! I'YERSE
205 IDF • IOF - I
011=1
DO 500 1=1,'
DO SOO 01=1,1
OD
~
,0195
...... 01'16
5.n1CJ7
5.0198
5.0199
5.02(\4)
5.0201
5.0202
5.0201
5.020'
5.0205
5.0206
5.0207
5.0208
5.0209
5.0210
5.0211
5.0212
5.0213
5.02"
5.0215
S.0216
5.0211'
5.0218
5.0219
5.0220
5.0221
5.0222
C(Jlt)=l(I,J)
500 JI=JI+'
DO 98 111,_
DO 98 Kt1,IBRIl
98 Bltt,I<'8B8'lI,I<
ClLL ftl~I_'(I,_,8,IBBI,DETBIB)
VIITE (3,333) DftlRII
31] POBlII'f(1RO,1.BDITIIIIIIIT ... ,120.8)
VitI'll (1, 100')
DO 612 1=1,_
612 ••ITr (1,') (I (1,.1) ,.1-1,1)
C08P1'l1 ICCURICY 01' 1_'115101
C
n=1
DO 663 1=1,_
DO 663 .1=1,1
It-I (t,JJ
I (I, JJ =C (ttL)
C (l'L):R
663 lL=ltL+1
DO ' " 1=1,_
DO 66• .1=1,1
66. I (I,JJ =1 (.1,1)
HITI' (3,1028)
DO 170 1=1,.
5U8a (!.
DO 171 .la1,_
. 171 5Uft=SUlfl (I,JJ *8 (1,J)
DIFF=BRft(1,I)-SU"
170
.IITE(3,2)BB~(1,1),SUft.DIFF
COIPUTE ER80l 'fIRB ... fT - B • If
ItJ... 1
DO 1<' 1= 1,IIBII
DO 1n .1=1,:1
11=0.0
DO 1S K=1,.
15 11=11+8(1,1) .RBlI (J,I)
8IGRF(KJ) =11
5E (1'.1) =fl (1.1)-1'
10 KJ=KJ+'
.8ITF(l,1019)
..RtTl (3,1('21)
1021 POB8IT(1BO,5X,5BTO!IL)
DO 10.0 1·','188
1-(1*1+1)/2
10.0 .RITF(3,10") 11(1)
10., FORIIIT(18 ,8I,!12.t)
.RITl' (3, 1022)
1022 FOlIlIT(1BO,108REGBE5SIOJ)
DO 10.2 1=1,'RBII
1= (I.I+I)n
10.2 "RITI(3,10'3) BIGBI CK)
10t3 rORBIT(1R ,81,112.'.
VRYTI(3,1023J
1023 PORBIT(1RO,5I,58BRBOR)
DO 10•• 1=1,IBRII
C
5.0223
5.02211
5.0225
5.0226
5.0227
5.0228
5.0229
5.0230
5.0231
5.0232
S.021J
5.023'
S.0235
5.0236
5.0217
5.0238
5.0239
S.02ltO
S.02"
S.02t2
5.02'3
5.02t'
S.02'S
S.02lt6
S.02lt7
C;)
In
5.02'8
5.02'9
5.0250
5.0251
5.02!i2
5.0253
5.025'
5.0255
5.0256
5.0257
5.0258
5.0259
5.0260
5.0261
5.0262
5.026)
5.026'
5.0265
5.0266
5.0267
5.0268
5.0269
5.0270
5.0271
Ie (1.1....1 )/2
10"
10.5
.RY~P(),10.5)S~(I)
PORftl~(1B
,7I,!12.t)
IfRIT" (3,1106)
1106 PORftAT(1Hl,20I,'PRWOR RATRII')
,.09 PORRITflR ,8E16.7)
DO 1188 l=l,118ft
1= (1). (1-1) /2+1
J. (I). fI .... ')/2
1188 IfRITFfJ,1.C9) (5EfL),L-I,J)
c
CORR~LATIOJ BITRII 0=1
DO 1500 1=1,118"
DO 1St''' J=1,I
1= U.I+I) /2- (I-J)
LRs (1.1....1) /2
LC=(J·J+J)/2
150C REGRE(I) =SE (I) /(SQRT (SE (LR» .SORT(SE(LC))
.RI'1'F(3,1501)
1501 PORftAT(lHO,20I,'COPIELATIOI BATRII')
DO 1502 I=l,IRBft
1= (1). (1-1) /2... '
a.. .
J= (I).
,)/2
1502 If BITE (3, "09) (REGRF(L),L=I,J)
81 RETURJ
EJD
0>
0'>
LEnt: 1OCT67
5.0"'"
5.01'02
5.0003
5.00".
S.Of'OS
S.00f'6
5.001:'7
5.0009
5.0009
5.0010
5.0011
5.0012
5.0013
5.001.
S.001S
5.0016
5.(\1)17
5.0018
5.0019
5.0"20
5.0021
5.0"22
5.0023
5.002.
5.0025
5.0026
5.0027
5.0028
5.0029
5.0010
5.00::'1
5.00:)2
5.003)
5.0030
5.0035
5.0016
5.0037
5.0018
S.0f'39
18ft OS/36C BASIC FORTP1H IV (E) COftPILATION
1
2
3
o
5
11
10
12
20
21
22
SOBROOYIIP SBS~Clft.CO'.C,B,ft,IP,IS,IO.SH.SE.Sil)
DIKE.SIOI SR(630).SE(630),SE1(63(\),lft(l5,35),COI(35,JS).C(63~)
DIREISIOI PSIIC6JO),B(35,l5).OK(l5)
POBI1~(lH ,PE16.7'
FOR"I~(lBr,2~x.'rRROR SUft or PRODUCTS R1TRI. c SE')
POPftl~flHC,20X.'~STIftl~FD COITB1ST = C - 8 - U')
POIIUl' (180)
PORI1~flRO,~X.'SOP. OF PRODUCTS DUE TO NO = SUI)
nITr(l,3)
C1LL 1~BS1(.R,IP,In,SE.SE'.UI)
DO 12 J-l,IO
DO 1fl 1-1,"
SOft-f.. 0
DO 11 l=l,IP
SOP.=SOft+B(K.I).Aft(I.J)
OlfI)--SOft
DO 12 T=1,"
."(I,J)-OlfI)
DO 21 ..lal,IS
SOft-O."
DO 2(\ 1=1,11
SOllaSO....COI(J,J} -III(E,I)
OK fJ) -SOft
DO 22 1=1,15
llfl,I)-OICI)
DO 37 1-1,15
37 'IITE(3,1) (lll(I,J) ,J=l,IO)
51
'IIT!(3,O)
"I'n(3,2)
DO 51 Y=1,10
I- (I) • '1-1) /2+ 1
J- (I). (1+')/2
.IITE(3,1) (SE1(L),l=I,J)
.IIT'!(3,O)
'II~P(),5)
C1LLT~P(CO.,IS,ft.UI)
CILL 11'BSI(COI.II,IS,C,PSIB,~)
CILL SIIYSS(PSII.IS,IS.PSYI,IS,IS,D)
ClLL lTBSI(ll.IS.IO,PSIft,SR,OI)
5.0tt.,
5.0<'02
5.0<"0
5.0005
5.000'
bb.O~l
DO 22 1-1,10
s.oe."
5.0~')
DAT~:
50
DO 50 1=1,1.
I- (1). (1-1)/2+'
J. ( I ) . (1+')/2
.IITE(l,1) (SR (L) .J.:aI,J)
IftOl1
EID
co
....,
DII 05/360 BASIC FORTRA. If (E)
LftBI.: 10C'r67
5.0010
5.(u)11
5.0012
5.0013
5.001'1
5.0015
S.0t'16
5.00'7
s.on18
5.0019
s.on20
GO '1'0 8
., IL= (.1.J+J) /2- (.I-I)
8
50P.:cSO~I! ~. I) -SF. flL)
9
O".}<'SOIl
11
SOft-C-.O
DO 10 If= 1.IP
10 SOIl=501l+1' ~ • .1) -OK (I)
IL=(I-I+I)/2-(I-.I)
11SE1(KL)=SOIl
""OP'
DO
_'.'.1
on
S.OOOl
5.0002
SOBROOTIIE TRIP(COI.IS.II.OK)
DIIIERSIOJ COJ(35.35).UI(3S)
s.00n3
s.oon.
5.0005
5.00"6
5.0007
11='
5.00~A
s.oa09
5.0n10
S.0011
5.0012
S.0013
s.OO"
5.0015
Dl'lE: 68.081
SOBROOTIIB ITB51(lft.IP.Io.sr.SE1.0K)
DIREWSIOI lR(3S.J5).SE(630).SB1(630).01(3~)
DO 11 1-1.10
DO 9 .J-l.IP
SOft-O.O
DO 8 K-l.IP
IF (1-.1)" .6.6
6 KL-(I.I+K)/2-(I-.J)
5.0001
5.0£'02
5.I'(t03
5.0n".
5.001l1j
5.0t""
s.nOf'7
5.0nt\8
5.0t'~9
CORPILATIO'
DO 3 .1-1.15
IF(ll-11) 12.'2,13
12 DO 2 141.11
2 01 (1') -COl (.1,1')
13 DO 1 I-ll.IS
1 COif (.1.1) -COl (1,.1)
IP(K1-ft) 20.20.3
20
DO'
1'=1'1."
• COl (K. J) -01 (I')
3 1'1-11+'
BITO.I
BID
00
00
LP.Y~
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s.eot'~
S.0~'-"2
C
EQUIYlLEICE II~OW,JROV<. SICOLUft.JCOLUIt<, 'A8AI. T.
EQOIY1LEICI 1181.Bll<
.
DOUBLE PBECI5IOJ OPTERR.IRAI.5VAP.T,A.B.PTYOT
III'fIALIZ1TIOI
C
C
C
5.0005
5.0t'06
5.0007
5.0008
S.0~n9
5.0010
5.0011
5.0012
5.0013
S.OO"
5.0015
5.0016
5.0017
5.0018
5.0019
5.0020
C
10
10
15
20
OftER"". DO
DI'J'P.R" • 1.0
DO 2'-' .11'.11
IPIfO'fIJ<• .,
3C DO 550 Itl,.
C
SElRCR POt PIYO'J' nEftDT
C
C
C .0 Iftll'O.DO
IJO lltn.c.O
.5 DO 1r.S .1",'
50 IP IIPIYOTlJ<-l< 6(. 105, 6r
60 DO lCO ((".'
70 IF IIPIYOTII<-l< 80. l n C. 7.r
80 IP IIBSllftIX<-ABStIIJ.I«< RS, ,~c, 10e
85 1801 • .1
Icnt"""
95 lftU'IIJ,I<
100 COUIlln
105 COIITIIUP.
110 IPIYOTIICOLUft<.IPIYOTIICOLUft<Sl
g"
C
5.0021
5.0022
5.0023
S.002t
S.002S
5.0026
5.0027
5.0028
5.0029
5.0010
5.0031
5.0012
S.OOll
S.003'
5.0015
C
C
130
"0
150
160
170
20C
205
210
220
C
C
5.0016
5.0017
IJTEBCRllGE ROIS TO PUT PIYor !LEBEIT O. DIlGOI1L
230
250
260
270
310
320
C
(E)· CO"PILATI01
SUBROUTINE ftATIMYI1,',B,ft,DETEPft<
ftATRIX INYERstOI WITH ACCOftP1IYIMG SOLOTIOI OF LIIEAR hQOATI0NS
D1"E1I5101 IPIYOT(IiC) ,1(35.35) .B(35,31) .IROEI(5u,.U-,plYO'fr-;O)
C
, 5.000)
5.0N"
-TRAI IV
IF IIBOV-ICOLU"< ,.0. 26r, "0
OETERft=-OETEBft
DO 200 LI',I
SVIPIAIIROV,L<
IIIROI,L<IIIICOLUft.L<
ISICOtUft,L<.SIIP
IFlft< 260. 260, 210
DO 250 Lll, ft
SliP • Blt,lBOI(
BIL,IIOI< • 81t.ICotDft<
8~L,ICOLD"< • SliP
IJDEISI,1<.IBOI
IJDEISI,2<'ICOtDR
PIYOrSI<.lSIcotDft,IcotUft<
DITIRBIDETEIR*PIYOTII<
DIYIOI
pryor
101 BY PIYOT EtEftllr
330 lIICOLUft,ICOLUB<tl.0
3..0 DO 350 L",'
5Wl~<
D.
: b8.Ul:ll
DlL"<'2110
DIU0231J
nU'02'"
nn402'"
OALIlC2.5
OAt.0211'
OAL.n247
01L41)2118
nlL4!\2.8
OU'02"CJ
DU"f)250
OAtlln25'
0ltll0252
01LII025]
Olun')ll
nILII"'2-;5
D1L'U~255
0.U~2,)6
OUII0257
OU.t'2S8
DU_/)25Q
DALlI<'260
01t_0261
DAL'0262
DU'026]
oUIIO 2611
01L"0265
OAL'02li6
01t1l~267
OU_0268
DILII0269
nALII0271
0&L.0272
01Lll027]
01t1l027.
DlL.0275
DI1"0276
01t'0277
DIL.0281
01t'0282
DI"0283
0&t.028.
DILII0285
01L'0286
OAL.0287
01L.0288
01L40289
00
ID