PSYCHOLOGIC~L
RESEARCH
FINAL REPORT
by
George E. Nicholson, Jr., Thomas E. Jeffrey, l~Tilliam G. Howe
Institute of Statistics, University of North Carolina
Chapel Hill, North Carolina
Mimeograph Series No. 100
April 15, 1954
Limited Distribution
•
ii
TABLE OF CONTENTS
SECTIONS
PAGE
INTRODUCTION
iii
1.
PREDICTION IN FUTURE SM1PLES
•. • . . . . . • . .
2.
THE ADDITION OF TESTS TO A KNOWN FACTOR STRUCTURE
3.
ESTD·1ATION OF RELIABILITY OF MENTAL TESTS PARALLEL
4.
PRINCIPAL C011PONENT FACTORIZATION
• • • • • • • .
1
~
FO~IS
...
36
47
iii
Introduction
The ma.terial in this report arose from considering certain
methodological problems which arise in the field of test construction
and personnel selection.
During the pa,st few years various members of
the Institute of Sta.tistics research program in statisticB have done
work on certain aspects of some of these problems a.nd the existence of
this work, coupled with the esta,blishment of the Psychometric La.boratory
at Chapel Hill in 1952, seemed to ma.ke a. survey of what had been done am
an attempt to relate this with problems of interest to research workers
in the field of menta.! testing desirable.
It was the original intention of this project to formulate outstanding problems in the field of statistical problems of mental testing,
survey the theory for what had been accomplished and relate this theory
to the problems by means of expository extension and example.
In this
wa.y it wa,s hoped that statisticians might be made a.wa.re of problems in a
manner which would command their attention, workers in the testing field
would be made familiar with new methods and computational techniques would
be developed for the utiliza.tion of techniques well lmown but not extensively applied beca.use of laborious figure work.
This report is principally the work of Professors G. E.
Jr., T.E. Jeffrey and Mr. William G. Howe.
Nicholso~
Some of the topics were dis-
cussed with Drs. Dan Teichroew a.nd Masil B. Danford.
Dr. Ledyard Tucker
of the Educational Testing Service wa,s generous in discussing severa.l
.
points and, together with other members of the Educational Testing Servic'e ,
supplied data for examples.
Dr. Hubert Brogden and Mr. Ha.rry Harman of
the Adjutant General's Office a.lso discussed the progress of the report
a.nd supplied da,ta from the Personnel Research Branch files.
Fina.lly, ccn-
-eiderB.ble ref"erence was made to unpublished thesis results of G. E.
son, Jr., D. Teichroew and M. B. Danford.
Nic~o 1-
1.
A
Prediction in Future Samples
problem which arises in the field of psychology as well as
other fields is the following.
Supoose a test battery consisting of E aptitude tests (predictors) is applied to a group of students who are being considered for
admission to a particular school.
The battery is intended to furnish
the admissions officer with information to assist him in predicting
the success of the applicants in their first year.
after the students
have comoleted the first year a criterion measure (average academic
grade, say) is obtained for each student.
validated, i.e., the multiole
correl~tion
score and criterion is computed.
The test battery is then
coefficient between test
A prediction equation is obtained by
using the regression coefficients calculated on a leAst squares basis
as weights to be applied to the test scores of succeeding students.
This gives rise to an estimate of the future criterion score of the
student and in this sense is useful for predicting the probable success
or failure of members of succeeding classes.
when the criterion scores of the second group of students are
available, these may be compared with the estimates in order to determine how well the oredicting equation is performing.
The determination
of the ugoodness tt of the predicting equation is usually made on the
basis of the multiple correlation coefficient obtained' in the first
(validation) semple.
However, the object of the procedure is to pre-
dict the criterion scores in the second sample and the correlation between the predicted scores and the observed scores in the second sample
and the correlation between the predicted scores and the observed scores
in the second sample can be rerarded as the actual measure of the "good-
2
ness" of the prediction equation.
It has been observed by several
writers that generally the second multiple correlation coefficient is
sm aller than the first and this effect has been called the "shrinkage II
of the multiple correlation coefficient.
To fix ideas and establish a notation, let us consider a set of
variates y, xl' ..• , xp where y is the dependent or criterion variate
and
1
1
, ... ,
p are the independent or predictor variates.
X
Let
y~(l)
denote the B-th observation on y and x' a the B-th observation on
~!'"'
•
xi (i=1,2, .• ,p;
(1~1~21
~=1,2, ..
"Nl) in a sample 81 ,
Let ya (2), xi a (2)
••• ,Pj,a=1,2, ... ,N2) be defined for an independent sample 8 2, We
assume that xi are fixed constants,
In particular let
xl~(l)=Xla(2)=1
and Xj~(l), Xja (2) (j=2, ... ,p) be deviations from sample means in 8(1)
and 8(2) respectively.
The observations yare random variables such
that
(1)
independently distributed with a common unknown variance cr2 •
(2)
The expected value of y
known constants.
+
a X +
t'p py
= ~lXl
This implies that Y.
eY where eY is a normal
Y
==
+ , •• + PpX
n_
"'(
v~riato
+ e
p where Pi are un-
Y
zero mean, and such that
E(e. e.) = cr2 if i=j and zero otherwise.
J.
J
La t A.. = 8 xi x. ; A == j- a..
J.J
Y
Y JY
~J-
7;
7=
C = j- c. .
~J-
A-I
3
and
where the b. are defined by the p normal equations
J.
X.
J.a
=0
Denote by A(l) and A(2) respectively the two matrices of normal
equations which define the two sets of reGression coefficients with
which we are concerned, i.e.,
A(l)
= -I-a..
(1) 7=' ,[Sx. (1) x. (1) 7
J.J
J.Cl
J Cl -
and similarly for A(2).
for the little
A(2)/N 2 •
XIS
Also A(l)/Nl is the sample covariance matrix
in sample one and a sliailar interpretation holds for
Now consider the following 2 x 2 table.
Sample 1
Sample 2
Y(l)
Y(2)
In the above table Y(l) represents the least squares predicting equation calculated from sample land Y(2) represents the least squares predicting equation derived from sample 2.
~2 is the result of correlat-
ing the estimated criterion variates in S2 on the basis of the prediction
equation derived from Sl'
It is clear that in this notation R2 is simply the square of
ll
4
the multiple correlation coefficient claculated from the l-st sample
2
and Rl2 is the square of the correlation between the observed criterion
values in the 2-nd sample and those estimated from the predictors in the
2-nd sample and the regression coeffici3nts calculated from the l-st
sample.
The
which we consider arises from the observation of
~roblem
2
research workers that generally Rl2
is lower than R2 .
11
If sample I and sample 2 are independently drawn from the same
population, then ail and R~2 are independent estimates of the population
2
parameter p • Under the assumption already made the distribution of
these sample estimates is well known.
Accordingly questions related
to the behaviour of the best linear prediction to be expected in any
sample may be answered by using this distribution.
For example, the chance that R~2 will be less than ail is 1/2
since both follow the same distribution.
The chance that Ril-
R~2
> d
may be answered by working directly with the known distribution.
The distribution of R~. depends on N., the number of observations
u.
1in the sample, p, the number of predictors, and p2, the parameter. Previous considerations of this problem have resulted in proposals to correct the observedR~. for bias to obtain a better estimate of
J.J.
first order approximation to an unbiased estimate of
2
N.-l
L-1
i-P
P ;;;; 1 - -1.N
2
- R..
1.1- -
7
p2 is
p2.
A
5
More exact expressions are available for obtaining unbiased estimates
of p2 but the problem of inferring how well a predicting equation will
perform in a new sample is not answered by this kind of analysis.
The set of regression coefficients estimated from the first
sample are known to be unbiased estimates of the population regression
coefficients.
For example, let ~
=~
~l' ~2' •.• , ~p
of population regression coefficients and let b
l
_7
be the vector
= L-bll , b 2l , ..• bp1 _7
be the vector of regression coefficients estimated on the basis of
sample 1.
Then E(b ) ~~.
l
expected that max (b i - ~i)
If the first sample is large then it may be
=0
is very small and it may be considered
1
that this problem reduces itself to considering the sampling fluctuations
of the simple correlation coefficient since for a fixed set of
~
the mul-
tiple correlation coefficient simply reduces to the correlation between
the criterion and a fixed linear function of normally distributed variabIes which is a single normal variable.
Then the problem becomes that
of studying the distribution of the ordinary correlation coefficient.
The question, however, remains of why there should be any shrinkage
since it would be expected that greater values of the second coefficient
would occur as frequently as smaller ones.
to run as follows.
The intuitive answer seems
If the first sample is large and has ample degrees
of freedom for estimating B then it is reasonable to assume that the
regression weights are close to the population weights.
Since b is tho
unique set of coefficients which maximizes the multiple correlation co-
6
2
efficient in the sample, the observed Ril is an over estimate of P •
In a second sample, however, of approximately the same size the b
used being cloSG to the
~
an unbiased estimate of p
upward.
of the population will tend to make
2
and hence less than Ril which is biased
Accordingly, the phenomenon of shrinkage should be observed
whenever two fairly large samples are used and when the degrees of
freedom available for estimation of ~ are large.
Furthermore, in
this case the phenomenon ought to be adequately explained by appealing
to the distribution of the simple correlation coefficient.
The prob-
lem, however, will depend on the number of degrees of freedom available
for estimating
~
in sample 1 and the size of the second sample relative
to the first.
Lot us consider the following formulation of the problem.
The
best prediction in a particular sample is obtained when the least squares
prediction equation derived from that sample is used.
A measure of the
adequacy of this prediction equation is given by Ril.
The actual mea-
sure of prediction efficiency is, however, Ri2 since it is the second
sample in which the prediction is required and not the first.
Now among
the various interpretations which can be made of the correlation coefficient, one is that the square of the correlation coefficient is the proportion of variance of one of the variables which is explained by a
linear least squares regression on the other.
which we adopt.
It is this interpretatim
We propose that the accuracy of a predicting equation in
a sample be measured by this moans.
If then the y
a
in a sample of N.
~
7
observations arc to be estimated by an estimation procedure yea) which
gives rise to N.~ estimates Y(t (k) then we shall define the correlation
between y and Y(k) to be
:z
s(y _ Y (k) )2
a
a
1 __a
-..,~
( Y - -)2
Y
a
s
a
It should be noted that the relation
I-s(y - y)(y (k) - Y(k» 72
-_,;;,(t a
(t
s(y(t - Y(t (k»2
=1
-
_
(t
S( Y - -y) 2
(t
a
is not true unless Y (k) is the estimate based on the least squares
a
regression calculated from the same sample.
We, therefore, select
as our definition
S(y (2) _ Y (1»2
(1)
(t
a
a
a
a
= 1-------S(y (2) _ Y(2»2
The failure of Ya (1) to be equal to y(t (2) is measured by
s(ya (2) - Ya (1»2 = B(ya (2) - Y(t (2»2 + S(Ya (2) _ Ya (1»2
which may be partitioned into the two components shown.
The first term
represents the failure of tho observations to follow the best least
squares predicting equation and the second part represents the failure
of the predicting equation derived from Sample 1 to be the same as that
calculated from Sample 2.
It, therefore, seems natural to use the ratio
8
the efficiency of the predictor Yex(l) in Sample 2.
E
=
measure the efficiency of the predicting equation.
E
<1
Let
It is clear that
and assumes its maximum value for Yex(l) = Y (2).
ex
If Yex(l) is taken as y(2) we see that E
< 1 - R2
Also from (1)
showing that for a
large R2 using a simple predicting equation like the mean of the y's
has very low efficiency.
It may be emphasized here that in terms of our assumptions
the problem of finding a set of constants b , ••• , bp which will
l
maximize
is the same as the problem of finding constants a ,
l
is a minimum, i.e., a
i
= bi
and
... , a p
such that
9
=r2
However this relationship is not true for an arbitrarily determined
set of constants.
The definiffiion of multiple correlation coefficient
must then be chosen, made specific and we have chosen what is sometimes
referred to as the correlation index.
SL-Ya(2) - Ya(1)_7
2
= S(Ya(2) - Ya(2»2+~~L-bi(2) - b i (1)_7
J~
;-b.(2) - b.(l) 7a .. (2)
-
J
J
~J
-
= Ql +Q 2 •
It follows from all the assumptions that ()1. is distributed like x~
If we let b.(2) - b.(l)
and is independent of Q2'
~
~
2-P
= w. it follows that
~
tho set w. has a normal multivariate distribution with zero moans and
~
-7.
covariance matrix Z = I-C(l) + C(2)
-
This covariance matrix Z and
the matrix of the form Q = A(2) are both positive definite. Therefore
2
Q may (by a familiar transformation) be transformed into the form
2
where the real positive k. are the roots of the determinantal equation
~
IA(2)
C(l) - (k-l) I = 0
and the zi are normally distributed random variables with zero means
and unit variance.
The final result then is
10
S(y (2) _ Y (2»2
We now consider the reciprocal of the ratio
a
a
-~---..;.;.....--:~
=E
S(y (2) _ Y (1»2
a
a
which is
and
Special Cases.
Assume that b.(l)
~
= 6., i.e., the first sample is so large that
.~
the estimates of the 8. arc equal to the B..
~
tributed exactly as
X
.~
In this case Q is dis-
2
2
p and
where F N P is Snedecor's F with P and N2-P dogrees of freedom.
p, 2Assume that A(2) =A(l)/k and consider the determinantal equatim
!A( 2)C(1)-().-1) I=:o •
11
Under this assumption all of the r oats are equal, A. = k+l and
J.
Q
2
2
= (l+k)Xp
'
Then
1
Pr["'f!f<x7=PrrF
l!l -
-
-
<
p, 2-P N
(x-l)(N 2-P)
p(l+k)
-
7.
In order to obtain an upper confidence limit for P at the e level of
significance, choose x such that it satisfies the equation
(x-I)(N2 -P)
F
e
=
Pr
L-Fp, N2-P -<
p(l+k)
that is
where
F
e-
7=
1 - e
The confidence interval is
1<
1:E
<:x:
with confidence coefficient 1 - e •
Since E is tho relative efficiency of the prediction equation Y(l)
derived from S(l) the interpretation of the confidence interval gives an
estimate of how well Y(l) will predict in 3(2).
Tho practical signifi-
cance of this result is obvious for large samples A(2)/N should be
2
approximately equal to A(I)/N1 •
For k
= N2/NI
12
The expected value of ~ is
E
1
E(-E) = E .L~l
+
(1
N2
+ r.-)
I'll
Fp, N
.
2P
p
N2-p-2 -
7
_ Nl (N2-2) +- N2P
Nl (N2-P-2)
which for large N is approximately
2
1+
E...
1-
E-
1
Nl
E(-) =-...;:;,.E
N
2
We noto from this that if p is fixed and N1 and N2 are both large then
E(~)
E
=1
vThich indicates that the regression equation calculated from
a very largo sample will predict equally well in another very large
sample provided tho number of predictors is rclativGly small.
N
2
= rNl and p = SNI where 0
E(2:)
E
<:
s < I and r > O.
= (l+s)r
(r-s)
and
1
E(1)
E
r-s
= r(l+s}
Then
Let
13
For this expression to have its maximum value s must be a minimum.
Hence we have the conclusion above that the ratio of the number of
predictors to the sample size N should.be small. However if r is
1
very large for small s we have an efficient arrangement. For
N
A(2) =
Nf A(l)
then the following table gives the expected relative
1
efficiency of the predictor Y(l) in 52.
pIN1
N2/N1
.1
.01
.05
.1
.89
.47
.2
.94
.71
.45
.5
.97
.86
.73
1
.98
.90
.82
.33
2
.99
.93
.86
.50
.~
10
.99
.95
.90
.63
.45
00
.99
.95
.91
.66
.50 .
.5
1
Another possible measure of prediction is
1
-
E'
N S(Ya(2) - Ya (1»2
-..;;..--.--,;,~--:::
=-1
N2 S(Y8(1) -
Y~(1»2
This is the ratio of the sample variance of the residuals about the
predictor from S(l) applied in 8(2) to the predictor of 5(1) applied
in 5(1).
By the samo arguments as before
1- may
E'
be expressed as
14
P
l:
. 1
J.=
where X2
N1-P
2
"-.Z.
J. J.
is independent of the numera tor and the quanti ties in
the numerator have already been defined.
If we assume A(l)
A{2'
=~
by the previous argument we have
-E'
1
and therefore
1
E (if) =
N1
'N2 •
from which it follows that if k
N2+kp
N - p-2
l
N
=-E
N1
It is of interest to note this expression is independent of N •
2
In order to present an empirical example based on the analysis
describod above two sets of data were used.
The first set of data was
obtained through the courtesy of Dr. Ledyard Tucker and Marjorie Olsen
of the Educational Testing Service, Princeton, N.J., and is from Law
15
Schools
ad~ussions
tests as described below.
The notation
S(y (l)-y(l»(Y (2)-Y(2»
a
R:l.
S(Yr? (2 )-Y (1) )2
13
S(Y~ (2)-YC 2)')2
a
-=Y(=l=))=;2=(=Y=(=2)=-1=(=2)=);=2
2V=/;::S(=Y=(=l=)
S
a
a
is usod.
Law Schools Admissions Test:
Sample
School
1. Buffalo
2. Columbia
3. Cornell
4. NYU
5. Penn.
Records for 1,000 frcshman students.
I
II
Total
44
64
47
35
97
44
64
47
35
97
128
94
70
194
88
School
6.
7.
8.
9.
Sample
I
II
Rutgers
55 55
So. Calif. 62 62
Virginia
27 27
Yale
69 69
Total
~ 506
Total
110
124
54
138
1000
Raw scores for six of tho ton sw)tests of the Law School Admissian Test and the total scores over all ton subtosts wore available.
Tho criterion was the first year average grades converted to a common
scale for all schools.
Sinco Penn. hAd the greatest number of studonts, this sample
was used.
SDmple I was used for determining the multiple correlation.
The rogression weights wore determined and used in Sample II, to predict the criterion.
Sample II was divided into two parts.
The first
fifty constituted the first group for which the criterion was prodicted
and then the entire 97 were used and the criterion predicted.
This vIas done for three predictors and then for six pr:3dictors.
Tho subtests used as predictors are identifiod as follows:
16
Part 2 - Sont0ncc Completion
Part 3 - Paragraphs
Part 5 - Reading Comprehension
Part 8 - Figure Classification
Part 7 - Debates
Part 10- Reading Comprehension
Parts 2, 3, and 10 were usod as tho throe prodictors while all six
woro used in tho second case.
Rosults:
-
Confidence
Interval
0/0
1-1.27
1-1.40
1-1.17
1-1.26
1-1.46
1-1.65
1-1.29
1-1.39
95
99
95
99
95
99
95
99
-
97 50 3 .2436 .1256 .1139
1.17
1.0967
97 97 3 .2436 .1229
.0697
1.23
1.0638
97 50 6 .3345 .2004 .1939
1.21
1.13
97 97 6 .3345 .1652
1.38
1.13
.0771
In the above calculations tho estimate of P is taken to be
instead of
1
E=
2
1 - R12
2~
1 - R22
Tho following tables givu the calculations upon which this
example is bnsed.
17
Law Sohoo1 Admission Test Data
Penn.
Two samples of 97 each
Table of Intercorrelotions for Sample I
Part 2
Part 3
.7387
Part 5
.7162
.6484
Part 6
.2759
.3995
.3286
Part 7
.5495
.5192
.5542
.2772
Part 10
.8230
.7570
.7329
.3321
.5932
Criterion
.4576
.4319
.4651
.4017
.4357
.4672
Raw Score Regression Weights:
(1)
For three predictors (Part scores 2, 3, and 10)
Predicting criterion in
b
.1097
;:
2
b
3
b
10 =
b
( 2)
O
Snmplc II
= .0961
=
.1126
Multiple Rll = .4919
b2
==
.0878
b
;:
.0005
b
5
= .1055
;:
.1642
;:
.1204
b
6
b
7
b
;:
10
.0320
b
.9254
O
::::
==
.3544
N == 97; Ri2 = .2497
6.4245
For six predictors (Part scores 2, 3,
3
N == 50; Ri2
5, 6, 7 and 10)
Predicting criterion
in Samplo II
Multiple R1l= .5784
N
= 50:
N = 97:
R'12
=. 4La 7
RI12 = .4064
18
A second set of
theory dovoloped.
d~ta
was used in order to illustrate the
These data were supplied by the Adjutant General's
Office and consisted of test scores of a group of 651
~1
who attended
the 3rd Armored Division Clerks Training School Course at Ft. Knox,
Kentucky.
These data were divided into six sub groups, each made up of
complete classes in terms of starting data.
These six sub-groups were:
Sub-Group No.
No. of E. M.
1
113
130
104
97
101
106
2
3
4
5
6
Sub-group 3 (104 cases) was used as one population for determining the regression weights.
Sub-groups 3 and 4 (201 cases) and sub-
groups 3, 4, and 5 (302 cases) were used as the other two.
those three populations the regression
wei~hts
For each of
were obtained using 3, 6,
and 8 predictors.
The Predictors:
The predictors are 7 of the 10 aptitudo scores derived from the
army classification battery.
Three of these 10 scores (Army Radio Code
Aptitude Score, Electrical Information Score, and Radio Information
Score) could not be uRed since thero were so many unsatisfactory or
missing scores.
Civilian Education was used as a predictor and at
first it was intended to use the Adjutant General's Office Aptitude
Area IV Score.
scores used.
This score is the sum of threo of the other soven
It was later decided against using tho Area IV Score and
only eight prodictors wore used in all.
19
Prodictor
No.
Name
Civilian Education
Reading and Vocabulary Test - RV
Arithmetic Roasoning Test - AR
P2ttern Analysis Test - PA
Mechanical Aptitude Test - MA
Army Clerical Speed Test - ACS
Shop Mechanics Test - SM
Automotive Information Test - AI
1
2
3
4
5
6
7
8
The Effect on the Multiple CorrelAtion Due to
IncreasinG the Number of Predictors
for ThroG Populations of Different Sizo
Popu11tion No.
Throe
(Nos. 1, 4, 5)
Number of PrJdictors
Six
Eight
(Nos. 1, 3-7 incl.)
(Nos. 1-8 incl.)
I
(N
=>
104)
.5047
.5679
.5897
II
(N
= 201)
.4997
.5861
.6141
III
(N
= 302)
.5246
.6378
.6512
PredictinG the Criterion:
Sinco three populations wore used to determine the regression
weights for each of throe grouDs of prodictors, thero wore nine prediction oquotions.
These arc given on the
att~ched
work shoots.
For each
predictor grouD the criterion was predicted for each of the populations
shown below:
Po-pulation siZG used in
determining illultiple
rogression weights
Size of population for which
criterion is -predicted
A
B
c
I
(N
= 104)
53
106
214
II
(N
201)
101
219
450
III
(N
= 302)
130
219
349
20
Hogr"ssion weights wer;; dGtermined for three diffGNnt sized
groups.
Group I:
Sample 3 (104 men)
Group II:
Sample 3 plus sample 4 (201 men)
Group nIl
Sample 3 plus sample 4 plus sample
Tables of
Intercorre1~tions
Group I (N
1
2
3
4
5
6
7
8
Critorion
=:
104)
2
3
4
5
6
7
8
485
454
156
232
150
166
-065
470
388
091
205
167
167
025
413
441
378
314
443
178
464
542
309
434
297
229
170
591
609
267
121
000
268
548
255
034
= 201)
1
2
3
4
5
6
7
8
403
353
099
210
165
126
-007
423
430
240
337
237
298
171
459
477
476
407
487
362
505
551
315
449
365
283
227
633
636
307
180
056
284
587
333
170
2
3
4
5
6
7
8
449
237
357
276
313
186
451
508
485
415
494
401
583
556
375
450
379
348
267
623
639
338
178
101
341
611
347
212
Group III (N
1
1
2
3
4
5
6
for the throe groups:
1
Group II (N
1
2
3
4
5
6
7
8
Criterion
5 (302 men)
447
359
128
248
224
166
7
8
048
Critvrion 422
= 302)
21
Results
N1
N2
p
2
R11
Rl~
2
R12
53
106
214
53
106
214
53
106
214
3
3
3
1
104
104
104
10).j.
104
104
104
104
104
6
6
8
8
8
.2547
.2547
.2547
.3225
.3225
.3225
.3477
.3477
.3477
.2515
.2580
.2140
.3841
.4014
.3257
.3463
.4385
.3641
.2314
.1859
.2134
.3319
.3566
.3189
.3004
.3975
.3535
2
201
201
201
201
201
201
201
201
201
101
219
450
101
219
450
101
219
450
3
3
3
6
6
6
8
8
8
.2497
.2497
.2497
.3435
.3435
.3435
.3771
.3771
.3771
.2974
.2128
.2485
.5074
.3078
.3721
,'4703
.3626
.3969
302
302
302
302
302
302
302
302
302
130
219
349
130
219
349
130
219
349
3
3
3
6
6
6
8
8
8
,2752
.2752
.2752
.4068
.4068
.4068
.4241
.4241
.4241
.2438
.2373
.2413
.3738
.3437
.3516
.3839
.3827
.3798
Group
6
22
Those data giv3 rise to certain
unm~pectGd
results_
In
particular it will bo noted that in 10 out of the 27 cases computed
2
,2 1S
. graa t or th an R11R12
will be
gro~ter
It has already been pointod out that R~2
than Ri2 and that in general Ril is greater than
~2-
These data were a source of diffic111ty to analyze and n1lmerous errors
and
omissions dictated rather restrictively the samples used.
As a
consequence it is not at all clear that the samples are random samples
from the sruno population and an extensive analysis would have to be
undertakon to locate the cause of the unusual behavior of these samples.
It was felt that the expenditure of time would not be warranted since
the investigators had no direct connection with the original data and
as a consequence ara in no position to perform such an analysis adequately.
23
2.
The Addition of Tests to a Known Factor Structure
The specific problem deals with relating the Army Classification
Battery, consisting of ten tests, to the factor structure obtained by
D:c • .ndkins in the "Factor
l.. nalysis
of Reasoning Tests."
Personnel
Research Section Report No. 878 gives the analysis for this battery
of sixty-six tests.
These sixty-six tests plus eleven additional
tests not included in the analysis, were administered to 200 subjects.
The eleventh measure was a criterion score.
This was added to the
ten tests of the Army Classification Battery.
The seventy-seven
tests are listed by name in Table 1.
Dwyer's* method was used to obtain the projections of the added
tests on the orthogonal factor axes of the reasoning battery.
This
method depends on the fundamental factor theorem that the orthogonal
factor matrix post multiplied by its transpose will reproduce the
correlation matrix.
Written as an equation, FF' = R.
The development of Dwyer's method:
consider r tests whose
table of intercorrelations, when factored, results in an orthogonal
factor matrix of k dimensions.
The problem:
to find the projections
of test t, an aduitional test, for this factor matrix when the intercorrelations of test t with the other r tests are known.
tions of test t on the k dimensions are defined as:
The following equations must be satisfied since FF'
The proJec-
atl , a t2 , ••• , a tk •
= R.
*Dwyer, Paul S. "The Determination of the Factor Loadings of
a Given Test from the Known Factor Loadings of Other Tests." Psychometrika, Vol. 2, No.3, September 1937, pp. 173-178.
24
atla ll + a t2a l2 +
a tla
2l
+ a
• 0 0
+
atia li +
• 0 0
+ atka
lk ::: r lt
+ Sti a 2i + .•• + a tks 2k ::: r 2t
0
•
::: r
+ ••• + atia
+ . •• + atka
a +
t2 22
·atla. . •+ .a . a. . . . . . . . . . . . . . .
jl
t2 j2
'0'
Ji
jt
jk
·.. . .....
Applying the theory of least squares we get the following normal
equations:
••• +
( 1)
· . ... ... .. .. .. . .. ..........
The solution of these k normal equations gives the desired
projections, a tl , a t2 , '0" a ti , .•• , a tk • The communality for test
22
2
t is then: h2
::: a 2
+ a
+ ••• + a
+ •• + a • This method is
t
tl
t2
ti
tk
applicable to any orthogonal system but is especially adapted to the
principal component method of factoring where L a'
j
Ji
a
Jk
::: 0 for i
f
k.
25
Rewriting the normal equations (1) in matrix form:
a2
jl
La
a
E a2
j j2
I:
ajla ji
ajla jk
a
r.
a j2aji ••• E aJ2a jk
j
a
..• ...• .• • ·..
.......•
I:
j
r.
j
j
a
Jl j2
r. ajla ji
j
a
Jl j2
j
j
2
La a
• •• E a ji
j2
ji
j
j
I:
j
...
L. aJia k
j
J
o
.. • .• ..• ..·..... • .
Ea a
r. a 2
1:: ajla jk
j2 jk ·.. ~ ajia jk ... j jk
(2)
L
j
t2
L a r
j2 jt
j
.•
.....
a
ti
ajlr jt
L ajir
Jt
j
.....
a
J
J
tl
T
A
r.
tk
j
ajkr jt
-
P
Let the left hand matrix be denoted as A, the middle matrix
as T, the right hand matrix as P.
order 16 x 16.
Matrix A in this problem is of
It is a symmetric matrix.
unknowns and its order 1s 16 x 11.
16 x 11.
Matrix T is the matrix of
The order of matrix P is also
All elements in matrices A and P can be determined.
The orthogonal factor matrix from the reasoning battery is
reproduced here in Table 2.
This factor matrix when pre-multiplied
by its transpose gives matrix A.
Matrix A appears in Table 3.
The correlations between each of the eleven added tests and
the original sixty-six tests appear 1n Table 4.
This matrix of corre-
lations when pre-multiplied by the transpose of the orthogonal factor
26
matrix gives the product matrix P.
This product matrix is given in
Table 5.
If both sides of the matrix equation (2) are pre-multiplied by
the inverse of matrix h, we get:
but
A-lA
=I
(the identity matrix)
therefore
The inverse of matrix A is given in Table T.
h
-1
when the rows of
are multiplied by the columns of P, we get the projections of the
ad~ed
tests on the sixteen dimensions of the reasoning battery.
Table
7 gives these projections and the communalities for each of the eleven
added tests.
Table 8 gives the transformation matrix obtained by Dr. Adkins
when the reasoning battery was rotated to simple structure.
shows the projections of these
ad~ed v~riables
Table 9
upon the oblique reference
axes obtained by Dr. Adkins.
The projections of the eleven added tests, shown in Table 7, are
con6iBt~ntly
low for several of the tests.
This seems to indicate that
the factorial composition of these tests with low communality cannot
be adequately described in terms of the factors isolated in the reasoning study.
Without a complete refactorization of all seventy-seven
tests it is not possible to say whether or not some additional factors
would be generated by the addition of these eleven variables.
27
Table 1
REASONING STUDY TEST BATTERY
Test
No.
12.
3.
4.
5.
6.
7.
8.
9·
10.
11.
12.
13.
14.
15.
16.
17.
18.
19·
20.
21.
22.
~3.
24.
25.
26.
27.
28.
29.
30.
31.
32.
33.
34.
35.
36.
37.
38.
39.
40.
41.
42.
43.
44.
Test Name
Absurdities
Arithmetic
Block Count ing
Camouflaged Outlines
Cards
Circles
Conclusions
Decoding
Designs
False Premises
Figure Analogies
Figure Classification I
Figure CLassification IIA
Figure Classification lIB
Figure Series
Figures
First and Last Letters
Forms
Geometrical Puzzles
Identical Forms
Incomplete Outlines
Letter Series
Logical Puzzles
Map Planning
Matrices VI
Mechanical Information
Mechanical Movements
Mixed Series
Mutilated Pictures
Mutilated words
Nim
Number Series
Numerical Operations I
Numerical Operations II
Numerical Puzzles
Overlapping Circles
Painted Blocks
Paper Folding
Picture ~nalogies
Picture hrrangement
Picture Classification
Picture-Group Naming
Planning a Circuit
Practical Situations
Test
No.
45
46.
47.
48.
49.
50.
5152.
53.
54.
55.
56.
57.
58.
59.
60.
6162.
63.
64.
65.
66.
*
67.
68.
69.
70.
Test Name
Progressive Matrices B
Progressive Matrices C
Progressive Matrices D
Progressive Matrices E
Reading
Reading II
Reasons
Sentence Order
Series
Street Gestalt Completion
Suffixes
Surface Development
Things Round
Topics
Verbal Analogies
Verbal Classification I
Verbal Classification II
Vocabulary
Word-Group Naming
Word Selection
\olord Squares
Education
FOLLOWING TEST TO BE ADDED
Reading and Vocabulary Test
(RV-l)
Arithmetic Reasoning Test
(AR-l)
Pattern Analysis Test
(PA-l)
Mechanical Aptitude Test
(Mh-5)
71.
72.
73.
74.
75.
76.
77.
Army Clerical Speed Test
(ACS-l)
Army Radio Code Aptitude Test
(,~RC-l)
Shop Mechanics Test
(SM-l)
Automotive Information Test
(AI-l)
Electrical Information Test
(EI-l)
Radio Information Test
(RI-l)
Criterion Rating
The analysis of the reasoning battery included tests 1-66.
67-77 inclusive are the tests which are to be added.
Tests
28
Table 2
CENTROID FACTOR MATRIX F
Test
No.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
r
II
III
IV
V
56
65
58
54
56
58
37
71
54
38
70
55
60
62
67
53
44
48
60
52
80
69
65
58
78
33
40
80
42
32
32
63
57
-40
03
26
18
37
18
-13
09
16
06
20
16
31
31
16
33
-20
-03
24
-10
-10
-01
-19
14
16
16
23
-02
-04
-18
11
15
-26
-12
-14
31
14
18
27
-07
05
18
19
-14
-10
02
24
-15
-03
-12
09
-10
-12
14
-13
-11
05
14
-08
-06
08
06
-05
33
20
-03
12
19
-08
-21
-24
-07
-17
-09
04
-19
-08
13
-20
-19
-12
-10
-20
17
18
-15
10
11
39
-12
03
-13
20
20
-14
-10
-05
37
16
04
-12
09
11
-22
-06
07
13
-06
08
12
04
-11
-05
20
-10
-24
06
-07
12
14
02
-12
-06
11
-28
-13
09
25
08
-11
-07
-18
VI VII
-12
-07
-10
-03
10
-15
12
-11
04
11
17
-02
-25
-09
-10
11
23
-07
-05
-12
-06
..16
-02
-04
-01
09
13
03
10
16
-14
-10
-14
Dectmals have been omitted.
VIII
08 -12
-07
09
06
04
05
06
-02
-10
17
-02
-06
-03
09
-02
11
-07
-17
05
03
-08
-02
14
12
03
-13
-20
-06
03
-28
-15
-04
-15
IX
X
08 -07
20
-12
-07 -03
16 -15
-06
09
02 -07
-07 -16
17 -02
-06 -15
-17 -03
04 -11
-06 -12
20 -10
09 -21
06
04
16
03
12 -12
10 19
12 -05
-14 -22
-06 -10
-04
07
-10 16
18 -03
06 18
05 -05
18 03
-06 -12
-16 -11
-07
05
09 17
-18
07
-24 -05
-08
13
03
-20
03
03
-02
-07
-18
-08
-06
13
14
11
-19
14
16
-07
19
05
07
-03
10
-03
13
11
06
08
04
-10
-04
-16
O
XI
15
-06
02
05
-12
04
-14
05
02
16
11
-18
-10
-06
-05
-17
05
13
-10
08
09
-07
05
-08
-08
26
12
-13
06
-07
03
-11
13
XII XIII XIV
-06
05
-12
-14
-06
14
02
16
12
-17
-11
-07
-11
-07
14
10
-11
-18
-15
-16
-02
14
06
14
01
13
-12
08
07
08
05
07
17
-07
-11
06
-05
10
02
-05
-09
-06
07
-06
-13
14
10
12
11
14
-08
05
12
08
-03
04
13
07
-06
-10
21
-10
06
-07
03
10
-04
11
03
-15
13
09
04
-04
-09
17
08
02
-13
-09
14
11
-04
19
05
-07
06
-03
-07
-06
-09
13
04
12
-07
-14
-11
05
-13
XV XVI
-07 -04
11 -08
12
-17
10 16
08
-07
13 -13
-08 -05
08 -09
-06
09
02
-07
-02
07
06
-13
-11
-10
-06
12
-05
10
17
16
02
02
13
05
07
-05
-12
07
-09
05
-10
16
-16
-02
-05
05
08
07
-10
-05
-09
-05
09
01
..10
10
08
-09
-04
14
-14
10
-12
14
29
Table 2 (continued)
CENTROID FACTOR MATRIX F0
Test
No.
I
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
69
34
52
55
62
55
58
40
53
44
47
54
57
54
65
68
63
31
61
64
36
42
56
38
31
76
77
78
74
73
70
77
56
II
III
IV
-12
-13
25
-11
35
07
-12
-11
-33
29
-02
26
20
31
23
-37
-26
-39
-42
20
-21
-29
39
-38
-36
-04
-14
-09
-34
-34
-05
-04
-20
12
-03
-22
05
08 -05
-05 -04
08
-08
16
15
13
10
-26
-17
-19
-18
12
-16
-17
17
-05
-20
24
-06
27
16
19
-31
-10
-16
-21
02
-24
08
-17
11
14
03
21
13
-11
-19
-18
-15
19
11
-27
16
-08
16
-05
12
-16
-30
-10
13
12
13
-09
10
02 -05
-06 -27
V
VI VII VIII
-16 -19 -13
06 04
-25
12 -14
07
-28 -13 -14
-04 07 -11
07 -08 -07
26 -08 08
-08 -17 13
09 17 -08
-12 09 -10
05 16 07
19 13 14
16 16 07
27 04 11
11
08 -04
10 -08 -10
04 -13 06
-18 10 22
09 -19 19
16 07
05
32 27 -14
-16 29 -16
-05 11 -12
-17 06 12
-23 24 31
08 02 05
04
10 -22
12 -08 07
06 04 -13
-06 -04 -03
-08 08 14
07 -14 12
09 -03 -20
Decimals have been omitted.
-18
-04
09
-13
-12
-08
-27
-13
09
-13
10
-08
-14
-07
-04
10
13
26
08
-02
09
15
-09
26
08
04
-03
04
02
10
-10
06
15
IX
X
-09 -10
03 -06
04 -02
13 17
-05 -10
-04 -24
14 -06
15 13
13 -05
-14 08
16 06
10 12
10 17
11 19
14 12
12 -07
03 -11
-06 17
06 -16
-08 -18
-04 -08
-12 -11
-06 06
-15 -10
14 14
09 -16
-15 09
06
-24
12 02
06 -03
09 -16
-15 -08
-05 -09
XI XII XIII XIV
11 19
04 -04
12 -02
06 -11
08 -06
-10 -16
05
-05
-06 -13
-20 -11
08 02
17 14
06 12
-03 09
-06 -02
10 -08
07 08
14 12
02
-05
04
05
10 04
06
-14
-17 -12
-07 11
03 11
-05
03
-15
-12
04
02
06
07
04 -04
-07 -13
02 -10
07 -16
06
02
19
09
-06
-14
-05
-13
02
-18
-12
-07
09
-03
04
07
09
12
06
-06
11
03
-02
12
-06
-14
-22
03
-10
-13
03
03
-04
-11
-07
XV
XVI
-08 02
11 -02
-12 -19
08 04
04
07
-08
-13
04 -06
-08 08
02 16
12 -06
-16 -03
-03 -10
-20 05
-08 07
05 08
-05 -07
19 -07
06 -06
05 -08
03 -08
-08 07
-06 -07
03 -03
-12 12
-08 -17
05 19
13 -07
14 -05
-06 -08
07 04
-08 10
14 11
-13 -03
02
-06
-14
-06
-06
03
-03
03
08
-09
-15
08
08
-04
11
-03
11
-05
-05
-06
-22
17
09
-14
-10
06
09
12
-03
12
06
04
02
Table 3
F'o F0
o
=
A
I'C'\
I
I 22.1345
II
II
III
IV
2030 -3910 -(379
2030 3.4334
1443
1498
V
VI
VII
3881
-4665
0321
1264
0137
-0644
1938 -0263 -0387
VIII
-2508
-3910
1443 1.8234
IV
-0379
1498
1~3A
V
3881
1264
VI
-4665
0137
VII
0321
-0644
-0316 -1405
VIII
-0738
-2508
-0387
IX
-0270
-1340 -0791
X
-1767
2087
XI
0170
-0369
XIII
XIV
0170
1378
2657
1910
2318
1910
2087 -0369
-0452
1239
1302
0775
0844
-0316 -0487 -0791
1832
-0040
-0510 -0857
-0291
0378
XII
1378
-0452 -0040 -0565
0415
-0620
0298
XIII
2657
1239 -0510 -0109
0382
-0273
-0338
XIV
1910
1302
-0857
0857 -0824
-0585
0355
XV
2318
0775
-0291
0002
0950 -0258
0227 -0063
XVI
1910
0844
0378
0385
-0186 -0108
-0156 -0430
-2005
0014
-1405
-0208 -0289
-0231
-0263
(168 1.3281 -011.6
0538
-0947 -0240
0357
-0914
0475
-0387
(014
-0098
-0699
-0620
0346 -0340
0298
-0338
-0300
-0116 1.0881
-0405
1026
0202
0467
0467
0538
-0405
.8715
-c208
-0947
1026
0461
.9658 -1001
-0289
-0240
0202
0467
-1001
0357 -0098
0346
1832 -0231
-2005
0168
-1340
XI
1068 -0914
-0699
1068 -0565
XVI
-0109
0857
0002
0385
0382
-0824
0950
-0186
-0273 -0585
0355
-0258 -0108
0227
-0156
-0506 -0063
-0430
0752
-0510
-0367
.9013
-0545
0325
0348
-0160 -0374
0752
-0545
.8581
0225
-0311
-0388 -0401
-0340 -0510
0325
0225
.6779
0345
0124
0210
0348 -0311
0345
.7124
0798
-0549
-0300 -0160 -0388
0124
0798
-0506 -0274
-0401
0210
-0083
0478
-0195
-0524
-0367
xv
XII
X
-0738 -0270 -1767
III
1.4582
IX
-0083 -0195
0478
-0524
0020 -0262
-0065
-0088
.6186
-0146 -0082
0359
-0549
-0146
.6416 -0023
0229
0020
-0065
-0082
-0023
.5638
-0382
-0262
-0088
0359
0229
-0382
.5935
----------------------------------------------------------------------Decimals have been omitted except for diagonal elements.
e
e
Table 4
CORRELATIONS BETWEEn ADDED TEST3 AND REhSONING BATTERY TESTS
Test
No.
ro'f
.1<"\
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19 .
20 '
21
22
23
24
25
26
27
28
29
30
31
32
33
67
51
53
25
28
25
34
30
50
31
31
44
40
40
39
41
19
37
30
36
30
65
47
50
36
47
17
26
60
20
29
28
47
46
68 69 70 71
47
71
4c
3c
4E
47
25
6c
38
33
45
45
49
46
48
33
32
32
44
35
61
58
57
46
59
29
35
64
22
27
30
57
51
23
43
56
50
57
39
20
51
49
30
49
35
52
47
53
48
24
30
53
41
56
40
33
51
61
25
28
53
38
39
41
43
33
43
36
16
36
42
12
38
26
40
33
36
32
26
32
47
35
43
40
43
42
43
41
37
42
26
23
24
28
24
24
43
32 38
72 73 74
36 29 39 30
31 22 38 26
37 20 34 28
16 16 33 25
28 32 39 35
29 22 37 22
26 16 17 -03
40 32 44 36
24 23 39 37
27 18 09 04
31 36 35 28
33 25 24 15
18 27 40 33
30 24 32 24
35 36 34 21
20 24 23 20
24 24 34 16
30 23 24 09
25 26 33 18
40 22 34 20
49 31 47 24
44 30 47 34
30 27 40 26
29 23 41 23
42 28 45 25
01 21 42 71
14 19 33 40
45 31 41 28
35 17 15 19
29 14 27 24
17 20 28 15
43 19 35 23
58 34 35 17
75
32
32
27
35
39
18
13
33
37
25
36
25
35
37
76
07
12
01
05
06
-04
02
07
04
14
12
05
07
09
01
23 04
22 02
25 01
32 05
21 07
42 06
32 19
24 20
27 01
30 12
29 08
32 05
32 06
21 04
19 07
13 08
28 13
23 16
22
77
26
19
23
25
20
14
07
27
21
17
32
16
25
24
21
13
21
21
11
28
29
25
21
13
29
27
18
26
26
15
13
15
23
Test
No.
3
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
51
15
32
40
31
46
46
27
47
23
31
26
37
30
35
67
58
27
60
42
28
38
23
37
16
61
63
64
72
58
56
50
52
68
1
32
37
52
49
44
43
30
42
30
38
36
44
40
51
52
46
15
46
45
18
32
42
26
20
64
56
59
62
54
60
55
50
19
42
26
60
35
45
16
27
47
28
46
45
47
56
32
35
04
27
51
23
21
68
10
07
42
47
52
37
35
46
48
27
70
39
24
29
45
41
29
32
27
37
36
29
27
25
21
36
45
42
15
41
32
25
27
42
16
12
42
49
47
4l
4
47
49
32
71
55
21
37
27
20
37
37
21
32
24
24
27
32
29
39
39
26
21
41
34
29
25
29
32
28
39
42
41
40
44
35
38
46
17
27
24
29
25
23
17
27
27
24
19
18
21
27
37
36
13
33
31
15
24
29
21
07
34
37
40
34
32
33
38
34
73
3
23
33
32
33
31
31
25
35
30
35
30
30
27
30
50
36
10
42
36
19
28
38
16
08
45
47
47
~i
48
45
39
74
2
. 14
17
27
29
07
09
11
22
34
30
20
18
17
18
32
28
01
20
22
12
14
41
04
-03
27
31
31
29
25
35
24
15
75
27
26
18
33
34
30
21
11
22
28
20
20
25
20
30
31
24
09
21
30
16
20
26
76
17
09
01
06
10
16
03
02
12
01
00
01
07
05
06
09
10
04
00
03
15
11
-02
08
02
01
34
34
39
05
06
00
1
37
41
25
10
12
19
12
12
~8
05
77
22
09
18
18
18
15
10
03
22
14
31
21
2)
12
26
27
25
12
23
24
07
15
25
11
13
31
32
35
30
25
27
27
34
Decimals have been omitted.
e
e
Table 5
PRODUCT MATRIX P
(\I
r(\
Dimension
No.
I
= F0
R
a
R = Correlation Matrix of Adued Tests vs Reasoning Battery Tests
a
hdCLed Tests
67
68
~9
70
71
t
72
73
74
75:
76
77
15·7255 17.1918 15.2227 13.5197 12.4217 10.0895 13.2731 8.7890 10.4592 2.6978 8.1468
II
-8709
1008
1.4501
1406
-4515
-0629
0335
5520
3600
-1842
-0319
III
-7056
-5322
1086
-0692
0218
-2186
-3326
-2140
-2962
-1161
-2270
IV
-080:;1
-1002
2073
3964
-3510
0625
3091
6848
2448
-0793
0925
V
3290
0982
4098
-0313
3148
1011
0684
-1686
0362
-0348
1555
VI
-4581
-4827
-2244
-3228
-2891
-2259
-3278
-1064
-1633
-0048
-0926
VII
-0814
-1032
-0379
-1406
-0284
-0848
-1644
-2747
-1556
-0801
-0647
VIII
0437
-1751
-2276
-0038
-2113
0280
0464
0081
-0407
-0752
0621
IX
0139
1269
-1483
-0045
-0534
-0657
0221
-0406
-1113
0514
-0784
X
-3103
-2228
-0371
-0851
-1622
-1979
-0939
0570
-1129
-1731
0035
XI
0059
0364
-0206
0946
0')09
0906
1018
2385
0848
0074
1810
XII
1163
1544
0918
0359
1547
0926
1124
1670
-1085
-0333
0522
XIII
1724
2222
3013
1249
1653
1260
1746
0784
1086
0292
0753
XIV
0877
1445
1632
1779
-0567
1300
0449
1240
1548
-0568
0267
XV
1486
2592
1359
1240
0~30
0368
1234
1500
1285
1063
0537
XVI
0754
0937
2084
1677
0231
1298
1488
1572
1284
0567
1681
Decimals have been omitted except for those values greater than unity.
e
e
Taole 6
..:"\
IC"\
A- 1
Factor
No.
I
I
0467 -0017
II -0017
III
IV
II
III
0086
v
0012 -0125
VI
0186
3140 -0124 -0249 -0201 -0181
0086 -0124
6099 -0999
0012 -0249 -0999
V -0125
IV
03 69
0380
7335 -0355 -0105
IX
x
0004 -0023
0005
0077
0163
0851
0424 -0856
0218
0095
0554
0430 -1355
2071 -0196
VII
VIII
1095 -0191
092~
-0201
0369 -0355
7976
0018 -0554
0186 -0181
0380 -0105
0018
9571
0004
0163
0095
VIII -0023
0851
0554 -C191
0924 -0964 -0666 1.1207
0424
0430
0306 -0353 -0693
VI
VII
IX
X
0005
0077 -0856 -1355
XI
0013
XII
-0059
XIII
0218
XIV -0126 -0631
XV -0175 -0361
0051
0460 -0362
2071 -1511
1280
xv
XVI
0164 -0649 -0631 -0361 -0370
0504
0456
0941
0354 -0330
0150 -1058 -0070 -0369
1192 -1166
0299
0999
0850
0370
0099
0481 -0519
0638 -1015 -0355
0078
1022
0391
0166
0854
0043
0622
0478 -0338 -0425
0330
0611
0094
0356
0683
0677 -0875
1007
1022 -0338 -1123 1.6115 -0843 -0196
0026 -0040
0673
0279
0463 -096+ -0353
0189
1413 -1179
1413 1.1497
0189 -0490 -1179
0481
XIV
1280 -0443 -0347
0463 1.1846 -0666 -0693 -0490
0999
XIII
0013 -0059 -0160 -0126 -0175 -0146
0461 -1511
0306 -0362
XII
0478 1.2561 -1123
0724
0520
0295
0504 -0443
0724 -0519
0391 -0425
0520 -0843 1.4616 -1796
1202
0243
0150 -0347
0295
0638
0166
0330
0683 -0169 -1796 1.6835
0504
0342 -0852
1192
0850 -1015
0854
0611
0677
0026
1202
0504 1.6567
0094 -0338
0354 -0070 -1166
0372 -0355
0043
0094 -0875 -0040
0243
0342
0099
0622
0356
0279 -0852 -0338
0164 -0196
-0160 -0649
1095 -0554
0051
XI
0456
0941 -1058
XVI -0146 -0370 -0330 -0369
0299
0078
1007
0673
0094 1.8266
1133
1133 1.7324
Decimals have been omitted except for those values greater than unity.
e
e
Table 7
ADDED SECTION OF ORTHOGONAL FhCTOR l-liATRIX
-:t
tI"\
Test
I
67
706
774
688
615
566
456
601
400
474
122
370
68
69
70
71
72
73
74
75
76
7?
e
II
III
IV
V
-281 -219 009 065
-007 -123 -033 -174
354 172 095 074
-004 069 259 -186
-170 172 -235 083
000 035 -024
-03b
-021 -085 235 -121
123 -091 453 -253
068 -078 153 -104
-054 -018 -052 -070
-034 -037 055 038
VI
-142
-115
104
-041
000
-011
-065
•081
040
043
078
VII VIII
-157
-159
-004
-135
-076
-103
-175
-245
-147
-089
-070
041
-115
-085
051
-215
075
094
065
-005
-082
085
IX
-012
140
-083
038
-071
-062
053
-002
-093
034
-074
X
XI
XII XIII
XIV
XV
XVI
-108 -099 -001 -005 -041 -005 -L61
-056 -061 057 010 -035 167 -090
-010 024 016 124 -002 -086 046
031 071 -024 -044 050 008 066
-067 -058 046 039 -138 -142 -140
-122 115 049 -006 065 -094 062
048 046 059 014 -144 -003 062
165 205 248 -092 -011 132 129
-035 068 -206 -026 026 046 040
-161 006 -087 -002 -134 178 061
090 262 013 -047 -064 -050 195
h
2
703220
764370
693515
524064
568294
278851
517914
696847
354845
130269
291322
e
35
Table 8
TRANSFORMATION MATRIX FOR THE REASONING BATTERY
Oblique Reference Axes
B'
C'
D'
I
22
21
19 18
II -03
06
13 -26
28
III -19 -35 -19
IV
42 -17
07 -30
V 31
22
32
23
VI -04
34 -17 -23
VII -01
02
36 14
VIII -02 -24
17 23
IX -54
11
36 28
06 29
X -22
31
XI -17 12
20
37
XII
09 09 09 -60
XIII -60 25
26
05
XIV
00 05
15 -23
XV -05
08 04 9 029
XVI
24 -13 -25
11
A'
E'
F'
02
19
25
-17
05
09
18
03
-33 -06
-12 -28
22
31
50
-51
17 -20
14
41
06
-05
-40 -21
04 30
06 -38
04
37
-29 -21
G'
J'
H'
I'
K'
14
16 -13 21
10
17 -06 -06 -02 -19
02
04 17 16 -07
01 -15
20 -21 -02
14
16 09 -35 -12
41
-06 11
09 -37
22 -30 -21
-16
23
16 -12
16 -45
18
-08 -19 -01 06 -08
00
08
-39 -19 -12
06 -10
10 36 19
41 -48
20
-13 -29
42
60
10 18
-52
28 -39 -14 -21
-57
46
08
27 03
-25
-19 -31
75 -20 18
L'
N'
M'
17
17 22
25 -06 -14
28
07 41
09 -16 32
-27 10 37
00 42
27
31 -27 -08
-12
29 -06
32 43 -12
00 15
-31
04
-36 -11
-06 -18 -11
08
17 -25
18
05 -36
66
25
-37
-17
15 -34
p'
0'
14
-09
18
-11
18
07
-10
32
-08
-11
22
40
47
00
-02
20
-38
21
14
20
61
24
-29
16
35
-25
-07 -08
-22 -10
-48 -16
Decimal points have been omitted.
Table 9
OBLIQUE FACTOR MATRIX
Reference Axes
F'
G'
H'
I'
07
03 -05
27
10
13
05 -05
05
08
11
17
17 -05
14
12
11
07
09 -08
14
08 17 11
03 -02
02
14
17 -05
09 -14
22
07 03
05 -02 17
08 -06 09 -14 -05
07
12
02
12
05
15 -04
00
02 -15 -06 -01
03
14
07 11 19 -12
05
17
06
11
13
-01
15
05
-11
-16
11
09
01
-18
-24
-01
-04
-15
00
03
13
Test
A'
B'
67
26
06
18
17
68
69
70
71
72
73
74
75
76
77
C'
D'
l!;'
08
11
07
19
12
15
01
12
09
10
Decimal points have been omitted.
-09
06
17
J'
K'
L'
M'
18 11 -05
21
04
10 31
45
11
08
29 ..01
10
15
07
17
04 09 -03
32
04
04 05
07
18 12 -01
16
18 -01 -11
12
18 03 13
03
18
13 12 -03
06
10 -15
-03
0'
p'
06
02
03 -02
04
23
10
02
14
23
00
08
11
02
11
00
10 -10
06 -12
09
05
10
16
13
31
03
19
33
60
19
-02
30
N'
36
3.
Estimation of Rel1ability of Mental Tests from Para.llel Forms.
The relia.bili ty coefficient of a. test ma.y be defined to be the
correla.tion coefficient between two pa.rallel forma of a test.
In the
development of this fundamental concept in the theory of mental tests
the notion of the existence of a.n indefinitely large number of parallel
forms of a. particular test 1s basic.
If we assume that a. test score is of the form
where g ia a subscript denoting the test and i the sUbscript denoting
the individua,l, then the popula,tion pa.rameter to be estimated is
where h is a subscript denoting a. test which is a. parallel form of
test g.
Suppose we select a. random sample of K testa from the group of
parallel forma and apply these tests to ea.ch of N individuals who are
chosen in some specified wa,y.
results in N scores.
follows.
Each a.pplication of a. para.llel form
Consequently the matrix of observations is a.s
37
Tests
Individua.ls
1
2
3
K
1
X
ll
X
12
X
13
XlK
2
X
21
X22
X?3
X
2Jf
3
~l X32 ~3
X
3K
N
This Bet of Bcores ma.y be regarded a.B ariBing from a. K variate
probability distribution where E(X
il
)
= E(X i2 ) = ... = E(X iK ) = ~,
2
(E denotes "expected" va.lue) the varia,nce aX
P gh = P for all g a,nd h.
= C12
for a,ll g and
g
These are necessary conditions for a set of
tests to be rega,rded a.s para.llel forms.
The parameter P is the intra.-cla.ss correla,tion coefficient or,
in terms of our previous discussion, the reliability coefficient.
In order to obtain an estima,te of P we use the likelihood ra,tio
criterion and obta,in a. confidence interva.l procedure.
We transform the origins,l scores by means of an orthogona.l
transforma.tion so tha.t
end
38
where
K
r.
j=l
c
gj
K
r.
= 0
j=l
c
2
= 1
gj
and
K
E c jC
= 0
j=l g hj
for g
1h
•
As a result of this transformB.tion we ha.ve a. transformed
observa~
tion matrix
where
E(Y
il
) =
JK
~, E(Y
ij
2
0y
)
0
::;
=0
j 11
2
cry
::;
2
cr £1 + (K-l)
1
2
£1 -
j
e..7 = O22
This implies
(J
0
2
1
and Pgh = O.
We IDB·Y observe a.t this point tha.t the expression
not les8 than zero.
p_7 : ;
2Lr 1 + (K-l)
e.7 ~
oi is necessarily
-1
0, hence p~ K-l
which shows that the va.lue of the reliability coefficient is directly
affected by the number of parallel forme used to estimate it.
We sha.ll
return to this point B.nd others logica.lly connected with it later.
39
The frequency function for the sample of Y' s me.y now be written
a.s
1
--l----;;;....--~'
e
'2 N(K-l). N(K-l)
(27f)
(12
It follows that
and
where
Y
1
=
N Y
E - il
1=1
j
N
are chi-squa.re random variables and independent.
Our purpose is to estime,te P and to test hypotheses covering it.
Therefore we consider the ra.tio
~ = 1 + (K - l)p
1 - P
40
which is independent of unlmown parameters B.nd is in one toone correspondence with p.
u/V
and the re.tio
:B
rlt.!
.L·2'
~
The likelihood ratio sta.tist1c
is eetima.ted a.s
ha.S the well-known :Beta. frequency function
1
N(K-l}-,
2
1
;"
'-'-
Although in most pra.ctica.l problems there is no interest in testing
hypotheses concerning p the test is derived here to round out the discuseion.
To test the hypothesis H : ;"
O
= ;"0
a.ga.inst the a.1terna.tive
we require a rule for rejection of the hypothesis
U
V
~
c 2 where c
1
EO
if
U
V
~
c
l
or if
B.nd c 2 a.re constants determined by
Probe.bil1ty
f
¥~ 01 or ¥~ c2 for HO true _7 =
€
e.nd
7
d Probability.Lr V
U S: 01 or V
U ~ c given tha.t H is true_ ;"=;,, = O.
d;"
2
o
41
c
Letting
r
l
c
= 71
and
r2 = 72 we ma,y determine
N-l
1
(1)
Br(N-ll N(K-l)
2
'.I.
'
2
-
(X)2 -
7
a best test by solving
1
dX
=1
-
E
and
(2)
where
=
€
is the proba,bllity of me,king the error of rejecting H when
O
it is in fa.ct true.
The two equa.tiona in "1 a,nd "2 determine the rule
for rejection of the hypothesis H at the € level of signifioanoe.
O
Values of 7
1
B.nd "2 a.re tabula.ted for given values of N and K.
For example if H : P = 0 substitute and get A = 1 then do not
O
O
reject H if
O
In genera.l the rule for non-rejection is
In order to estimate P which is usually what is w.nted the
statement above is converted as
42
L.!!<r..<.!:!L
''1
2 V-
- V '1 1
which is a true ste,tement with probe,bility 1 -
€
•
If N> 10 the equal tails of the F distribution may be used to
determine '1
1
and '1 2 '
slightly greater.
We determine '1
1
In this case the probability level is not
€
but
The difference is not, however, of practica,l importance.
and '1
2
8S
and
where '1
1
end '1
2
ere tabled for conventional values of € .
Since ma,ny
F tables give only critical values for the upper tail of the distribution it should be recalled that
FN-l,N(K-l), €
As an example if K
and
= 2,
N
= 121,
€
=
1
F
N(K-l) ,N-l,l-€
= 10 0/0,
we ha,ve to determine
From the ordinary F te,ble 12
example, K
= 3,
N
= 121,
€
= 1.352
= 10 0/0,
B,nd. 1
1
1
= 1.352
= .7396.
If, for
then we have to determine
If N(K-l) 1s large, then FN-l,N(K-l) is approximB,tely distributed like
2
xN-1/N-1.
In t his OB.se we have
and.
Looking in the F ta,b1e for Pr £F240 120
,
1..
= 1.254
1
1
We defined
Now
and 1
1
~ ~
1
_7 =
= .7958.
.05 we ha,ve
44
Y =
11
Xi1 + Xi2 + ••• + XiI{
= VK Xi
VK
end
where
_
X
1
=
~
Xij
a.nd
l..
j=l
K
= ~
~
Xij
1=1 j=l NK
X••
eo
N
U=K E
1=1
2
N
X~-NKX
•• = K
(Ii - X •• )
E
1=1
We ha.ve
V
=
N
K
E
E
i=l g=2
~
g
Rece.ll1ng tha.t the X' B and Y' B a.re rela.ted by an orthogona.l tra.neforlIl8.tion we know tha.t
N
E
K
E
2
X
1=1 h=l
ih
=
N
E
K
E
1=1 h=l
y
2
ih
hence
N
K
-
V = E E (X 1h - X1)
1=1 h=l
2
A compa,ct sUIDma,ry of the ca,lculations described a,bove is as
follows.
Displa.y the ca,lculations in an ena,lysis of varia,nce ta,ble
considering the me.trix of raw scores as one way cla.Bsifica.tion with
rega.rd to iOOividua.ls • Then
u=
Sum of Squares between Ind1vidua.ls
v = Sum of Squares within Individua.ls.
In such a, table we ha.ve
Source of Va.riation
d.f.
SS
Between Ind.ividua,ls
N - 1
U
Within Individua.ls
where a
2
I
=E
N
1:
N{K - 1)
Sr
N-l
~
E .M.S.
= sI
N~K-l)
= sE
Ka
2
I
2
+ a
E
2
°E
and E.M.S. means expected va.lues of the
i=l
square.
With this Bet up in mind
which is eatima.ted a.s
M.S.
46
2
2
If we ha.ve a. good estim8.te of the ratio of 0I to 0E and it is
equal to
0:
(S8.y), then
A. - 1
Now i f we wish to estima,te the value of r
I
for some other number of
replica.tiona, a8,y K', then
r± =
If K
= pK',
¥it
+ (k-k' )r
I
then
which is the sO-C8,11ed Spearmen-Brown prophecy formula, for estimB,ting
the reUe,bility of a. test if it is increased in length p times.
47
4.
Principal Component Factorization
The purpose of factor analysis:
~<.
Thurstone" states,
"The factorial methods were developed primarily
for the purpose of identifying the principal dimensions or categories of
mentality; but the methods are gener81, so that they have been found useful for other psychological problems and in other sciences as well."
The factorial method is generally employed by the investigator to test
an hypothesis when he is unable to devise some other critical test.
Factorial methods m8Y also be used to explore a range of phenomenon to
determine the underlying constructs.
For example:
one might wish to
investigate the basic abilities or skills involved in mechanical aptitude.
To do this the investigator would design a series of tests which would
differentiate between persons with mechffi1ical ability and those without
this ability.
He would no doubt include tests of manual dexterity,
spatial visualization, coordination, etc.
Tests of verbal ability would
probably be considered not applicable and would not be included in the
test battery.
which has a
This test battery would be given to a group of subjects
conside~able
range in mechanical skill.
From the scores
obt8ined on these tests, the investigator 1iould attempt to tease out by
factorial methods the underlying abilities involved.
His gamble from
the beginning of the experiment being that the number of fundamental
abilities involved is fewer than the number of tests represented in the
test battery.
~<.
"1. 1. Thurstone, IIMultiple Factor Analysis, II The University of
Chicago Press, 1947, Chicago, Illinois, p. 55.
48
Factor Analysis
In the
~d
sens~
Multiple Regression:
that one is attempting to predict the dependent
variable from two or more independent variables, there is no direct
relation between factor analysis and multiple regression technique.
In the factorial problem we start with the test scores or other measures
and attempt to determine the factors.
Here the factors could be con-
sidered to be the dependent variables and the test scores the independent variables.
Generally there is no ldistinction made between
dependent and independent variables in factor analysis.
A close
parallel exists when one knows the factor scores of an individual
and attempts to predict his performance on a test of known factorial
composition.
Here the factor scores would represent the independent
variables while the predicted test score would be the dependent variable.
The fundamental equation of factor analysis is written
s .. = c.lx·l . + c x 2 . + c. x . + •.• + c. x i
J~
J
~
j2 ~
J3 3~
Jq q
where s .. is the standard score of individual i in test j, the xTs
J~
are stand3rd scores of individual i in each of the uncorrelated reference abilities, and the
CIS
Are the weights assigned to the standard
scores in the reference abilities for the determination of the observed
standard score Sji.
The equation is written to represent q terms in
the right-hand member and it is hoped that this number of reference
abilities will be relatively small compared with the number of tests,
n, in the whole battery.
The
CIS
are
c~lled
the test coefficients or factor loadings.
determine these we st8rt with a score matrix S or order n x N, where
To
49
the elements of S are Sji' the standard score of individual i on test
j.
N refers to the number of individuals and n denotes the number of
tests.
The complete correlation matrix is defined as the symmetric
matrix of intercorrelations between the tests with unity in the
diagonals.
This matrix is denoted R and is of order n x n.
I
In matrix
notation:
R
I
=2
N
SSt with elements
I
rJ"k = -
N
~
N i=l
sJ"iski
Most factorial problems are not concerned with explaining the
complete variance of the tests.
That
is shared with other tests is more
p~rt
co~nonly
of the test variance which
of interest.
The variance
of the test is considered to be divided into three parts, viz., the
cornmon variance, the specific variance, and the error variance.
last two are known as the unique variance of the test.
variance of a test is termed the communality.
The
The common
When the communalities
are placed in the diagonal cells, the correlation matrix is known as
the reduced correlation matrix and denoted as R.
We wish to find a matrix F, called the orthogonal factor matrix,
such that when post multiplied by its transpose will reproduce the
correlation matrix R.
This factor matrix will have order n x r where r
is the number of factors.
It is hoped that r will be materially smaller
than n, the number of tests.
To obtain the factor matrix F a number of methods have been
50
employed.
One of these is the Method of Principal Components set forth
by Rotelling.
~-
This method extracts the factors in such a way as to
maximize the amount of test variance
This method while having desirable
comput~tional
labor.
~ccounted
for by each factor.
char~cteristics
requires considerable
For desk calculators it represents
amount of work if the test battery is of any size.
A
staggering
Consequently other
factor methods were devised which would tend to approxtmate some of the
desirable fentures of the principnl component solution but reduced the
computational work.
The complete centroid method developed by Thurstone
is one exnmple of such a compromise.
yield a unique factor matrix.
The centroid method does not
It requires the reflection of test
vectors during the extraction of factors and therefore it does not
readily lend itself to automptic machine calculation.
The principal
component method is better suited for machine calculation.
Since the
fatigue of the operator is not a factor hero, the amount of computational
labor is not so important as
1~
the case with desk
Rotelling's method is iterative.
vector U of n elements.
c~lculators.
It starts with a trial column
This trial vector is usually taken proportional
to the column sums of the reduced correlation matrix.
factor is so determ:i_ned as to
m~ke
The proportionality
the largest entry in U equal to unity.
The following matrix multiplication is then carried out
RxU=V
-II-,
Hsrold Rotelling, IfAnnlysis of a Complex of Statistic:,l Variables
into Principal Components," Journal of Educational Psychology, XXIV
(September and October, 1933).
51
This column vector V is again divided by its largest entry to give a
second estimate of the vector U.
This procedure is repeated until two
successive U vectors are obtained which are identical to the required
number of decimal places.
~t
this point the test projections on the
first factor axis are obtained by multiplying the last column vector
1
V by l/(U V )2, where U and V are the last values of U and V obtained
ff
f
f
when convergence has been achieved.
One of the available modifications of the method is that of
t
powering the matrix (i.e., At or R).
The advantages of this procedure
together with the mathematics underlying Rotelling's iterative method
may readily be seen in the following discussion.
The n characteristic vectors V ,
l
V
ll
V =
l
..............
..
, Vn
of A, such that
V
ln
V
n
V
nl
are
=
V
nn
determined such that
or to put it a little more simply V'V
= I,
i
=j
i
f J
the unit matrix.
It will
be assumed that the Vi are so ordered that the associated characteristic
roots are
~l ~ ~2 ••• ~ ~n·
As shown in matrix theory AV i
= ~iVi.
Now any column vector Z of order n x 1 may be expressed in the
form
S2
(2)
Thus
2
annn
A. V
..
.
• ,. +
Therefore, for the j-th element of AtZ
f.( t)
• •• +
J
where
D.. v ..
1. J1
.A.t
nJ n
C
: : : c.. •
1J
Then
t+l
CljA. l +
f .( t+l)
J___
+ c .A.t+l
nJ n
:2
f j( t)
t
cljAi + • .• + c nJ.A.tn
:;:
C2 · A. 2
A. +.....:...2(
)
1 clj ~
t
C
A.
t
A.n
A. 2+· • . + ~j.(.c)
clj 1
t
t
C .A.
c 2 ' A. 2
J
+ n (...!!)
+
1 + .....9.(r:-)
c lj A.
cij 1
l
53
flnd
f .(t+l)
Lim
t ->
Hence, the
r~tio
00
J
f/t)
for flll j.
= '"1
of every element of the (t+l)st iterRte to the t-th
t
iterate approaches "'1' as t increAses and A Z approaches VI for
t
t
V
n
••• +
Herein the speed of convergence of an
arbitr~ry
vector depends on the
closeness of the roots and the choice of the initial vector.
The
further separated the roots nre and the closer Z is to the actual vector
VI' the quicker the convergence.
that the speed depends on the
It is also Gvident from the above
"'2
rntio~,
the closer this quotient is to
1
unity the slower the convergence.
If At is used, the convergence
"'2 t
"'2
depends on (~) which is certninly much less than
r-.
1
CAution concerning powering should be added;
mulation of rounding errors as the matrix is
A word of
1
this involves the accur~ised
to high powers,
thus making for some loss of efficiency.
Another interesting point is the fact
th~t
the diagonal elements
of At as t is increased tend to be proportional to the squares of the
Vjlfs, that is to the squares of the elements of the first
vector.
The proof is as follows:
ch~racteristic
54
In mntrix
not~tion
z
~
(2) may be written
Va where
a
a
"~ [~:]
1
,
(r]
Then since VfZ
...
a
:::0
VIVa = a from (1)
a
where Z.
1
= Va (")
1
(n)
:::0
•
Now the ith elements of ~tZ. arc precisely the diagonal elements of
1
At end theroforo
+
V~ At
and
•.• +
ln n
55
(t)
(t)
where aii
Dividing
l1
t
and a jj
are the i-th and j-th diagonal elements of A •
(t) by a~~)
JJ'
ii
2
(t)
a ..
JJ
=
2
t
+ V. ().. /Al)
VilAi + ...
m
t
n
+ V~ At
In n
and
t ->
00
(t)
a..
J.J.
""]tJ
a
2
~
jj
V'l
J.
T
V
jl
This was proved by Sir Cyril Burt in a pnper published in the British
Journ~l
of Educational Psychology.
Much of the material used in this
section together with other methods of finding characteristic roots
and vectors, is to be found in Mr
Seymour Geisser's unpublished
master's thesis, University of North Carolina,
2.
1952.
Description of the computing procedure:
In adapting this iter3tive procedure for solution using the IBM
calculating punch, the capabilities of the machine have been limited to
those present on the
st~nd~rd
602 A machine.
avnilable on these machines at added cost.
Additional features are
If this procedure is to be
56
used on mnchines possessing ?dditional counter nnd storage capacity,
sane modificntion in the method described here may result in greater
efficiency.
The computing method outlined in this pnper is a modification
of a method reported by King and
A deck of
c~rds
is preppred contnining the
reduced correlation matrix.
c~rds
Y.
Priestley~
Figure 1 shows a
for a fifteen variable problem.
v~lues
s~mple
given in the
layout of matrix
This matrix will h?ve sixteen
rows since a sum row is added to provide a summational check for the
multipliC'stion of R x U = V.
In this fifteen variable problem forty
mntrix cnrds will be required.
This deck of matrix cards will be used
repeatedly from iteration to iteration until the final location of the
first factor is determined.
Each matrix card is identified by n code
number which appecrs in columns
1-4.
matrix card by row and column.
Matrix card no. 1
This code number identifies the
h~s
the code 0101.
It contains clements from the first pair of rows and elements from the
first three columns.
0602.
M~trix
card no. 27 would have the code number
Each matrix card contains six elements of the reduced correlation
matrix as shown in Figure 1.
These data e.r e punched into columns 5-40
inclusive.
A second deck of forty cards will be required for this sample
problem.
This deck will contain the elements of the vector U
of these cards which are
,e
c~lled
multiplier cards
Each
carries three
~~, H, KinC, Jr, and William Priestley, Jr., r~ass Spectrometer
Calculations on the IBM 60?-A Calculating Punch," IBM Technical Newsletter No.3, December 1951, pp, 5-17.
57
elements of the U vector.
columns 1-4.
These multiplier cards are coded also in
The first multiplier card has the code number 0101 and
u2 ' and u . The fifth multiplier card has the
3
code number 0105 and contains elements u13 ' u14' and u15: The five
multiplier cards bearing the code 0101, 0102, 0103, 0104, 0105 contain
contains elements
~,
elements of the U vector 1-3, 4-6, 7-9, 10-12, and 13-15 respectively.
These multipliers are used in turn to multiply the elements appearing
on matrix cards 1-5.
Since the storage capacity of the 602 A is not
adequate to store these multipliers, it will be necessary to read the
values into the machine when multiplying each successive pair of rows.
The values from these first five multiplier cards are reproduced seven
times so that forty multiplier cards in all are used.
The code numbers
in columns 1-2 are changed for each reproduction of the multiplier cards
so that the first two columns of the card carry code numbers from 01 to
08 in our example.
~fuen
both the matrix cards and the mUltiplier cards are so coded,
it is possible to position the correct multiplier card in front of each
matrix card by running the cards through the sorter.
With the multiplier
cards placed face down in the hopper of the sorter and the matrix cards
placed face down on top of the multiplier cards, we sort the cards on
columns
4, 3, 2, and 1 in that ordor. The cards are picked up from the
stackers from right to left placing the next pack picked up on top of
the preceding pack.
The cards will then be in the proper order.
this example it is not necessary to sort on all four columns.
columns 4 and 2 arc needed.
For
Only
However, when a large problem is being
solved, the code numbers may require two digits and for this reason the
space has been provided.
58
Read Cycle:
The planning chart for this problem is given in Figure 2.
wiring of the control panel appears in FiGure 3.
The
The first three elemQnts
of vector U which appear in multiplier card 0101 are read into storage
unita:6L, 7L, and 7R.
These multipliers can be either plus or minus.
The sign of the value is indicated in columns 41, 47, and 53 for each
A 5 punch is used to indicate a
of the three elements respectively.
negative value
~Thile
a 0 punch is used to show that the value is positive.
Each value is carried to four decimal places.
1.0000.
No value can exceed
The code number of the multiplior card is read into counter 3.
This code number will be used later to indicate which pair of elements
of vector V is punched on the trailer card.
In order to read the code
number in only once, the value is read in through a pilot selector.
The eight multiplier cards bearing the code --01 have an 11, sometimes
known as an X punch in column 67.
This X is sensed by the control
brushes and is used to pick up pilot selector six which is impulsed to
read and then drop out.
The position of control brushes varies from
one machine to tho next so that the card column corresponding to the
position of one of tho twenty control brushes in another machine may
be different from tho ones used here.
for all nlultiplier cards,
Column 76 has an 11, or X, punch
This punch is used to imoulse the skip out
hub from the read brushes and also to pick up pilot selector
the control brushes.
5 from
Pilot selector 5 allows the read hub to be
impulsed from the skip out hub for all multiplier cards.
It is necessary
to impulse read through the pilot selector since for matrix cards we
want to go through the program steps but still skip out the matrix
card without punching.
If the skip out hub were directly wired to
S9
the read hub both types of cards would skip out without programming.
Pilot selector
S is
wired to read drop out.
All multiplier cards
have an 11 or X punch in column 78.
By means of the control brushes
pilot selector 1 is picked up.
transferred pilot selector one
'~en
reads in the multiplier values into storage units 6L, 7L, and 7R.
selector 1 is wired to road and drop out.
Pilot
It is necessary to use a
pilot selector here since we are reading in values from two separate
detail cards.
If we were to read in the values from the multiplier
card directly from the read oycle hubs and then read the matrix card
values in from the read cycle hubs, the storage units 6L, 7L, and 7R
would be set to zero when the matrix cards wore read in.
After the
values are read in from the first multiplier card this card skips out
to the stacker and the first matrix card is read.
The first matrix card carries the first six values from the reduced correlation li1atrix.
indicated in
Fi~ure
2
These are re<:'.d into the storage units as
These values can also be either plus or minus.
The signs nre indicated in columns S, 11, 17, 23, 29, and 3S,
Again
as for the multiplier cards a 5 punch indicates
is
th~t
the
v~lue
negative while as 0 punch shows that tho vclue is positive.
values ar0 punched to three decimal places.
These
Since the column sums
will be used to compute a summational check space must be allowed for
two whole numbers in addition.
allowed for the
nt~crical
Therefore, five columns have been
value of each entry.
have an 11 or X punch in column 71.
brushes and
im~ulses
The matrix cards all
This is read by tho reading
the skip out hub, however the machine will go
through tho proerams and the card does not skip out until program
60
step 7 is reached.
Program Step 1 (For Sign Control)
Since both the multiplior and the multiplicand can be either
plus or minus, three pilot selectors must be used to determine the two
products developed at anyone time.
'Ie viill multiply hil and r 21 by ul
The machine will multiply several multiplicands by
at tho same time.
the
s~me
multiplier as rapidly as it will multiply one multiplicand
and one multiplier.
The officiency of the machine is increased by de-
veloping as many products simultnneously as is possible.
only two can bo developed at one time.
In this case
The standard machine has only
six counters with a total of thirty positions.
Each product for the
summational check could have as many as cloven places for a large problem.
One additional counter spt.lce will be needed to determine the sign
of the nroduct.
Therefore t~o twelve position counters will be needed.
Counters 1 and 2 arc coupled to givG one twelve position counter.
Counters
4, 5,
and 6 arc coupled to give a fourteen position counter.
Counter 3 is being used for code designation.
space on the standArd
6o~
A machine
The left hand position of storage
unit 6L is read out and wired to permit a
No
2.
This exhausts the counter
5 to
energize pilot selector
If a zero appears in the. left hand position, pilot selector No. 2
will not be enercized.
The remaining five positions of storage unit
6L arc road out and read into storage unit lR.
The left hand position
of storage unit 2L is read out and wired so that a 5 energizes pilot
selector No. 3 while a zero will not.
The left hand position of storage
unit 3R is rend out and wired so that a 5 will energize pilot selector
61
No.4 while a zero will not.
selector No.2 is used to
Pilot selector No.3 coupled with pilot
contr~l
the sign for the odd numbered rows
of the reduced correlation matrix while pilot selector No.4 coupled
with pilot selector No.2 is used to
numbered rows.
contr~l
the sign for the even
For each successive multiplication the sign control
for the ·previous multiplication must be dropped out.
impulsing the digit selector from the program exit.
This is done by
The digit
emits a timed impulse for 9, 8, 7, 6, 5, 4, 3, 2, and 1.
emitted for zero
~ut
select~r
No impulse is
The 9 hub of the digit selector is wired to drop
hubs of pilot selectors f, 3, and 4.
In this way the pilot se-
lectors a.re reset before the 5 impulse is emitted from the storage
units.
Pilot selectors are double pole double throw relays.
Each
selector has two SWitching elements which work together mechanically
but are electrically independent.
ally a pilot selector.
The diagram below shC'ws schematic-
The two sections (a.) and (b) constitute the
switching features of one pilot selector.
If the pilot selector is
not energized the C or common hub of both section (8.) and (b) are
connected to the N or n0rmal hub.
When the selector is energized
the a.rmature SWings to the dotted position shown in the diagram and
both
secti~ns
are now connected to the T or transferred hub.
N
(a)
~ /-f.
C
d
T
62
N
(b)
This makos it ?ossible in this problem for us to control the sign of
tho two products developed simultDneously with only three selectors.
Figure
4 shows
the connections through the pilot selectors so as to
add or subtract into the counters
P. S. No. 3
P. S. No.2
4
P. S. No.
N
N
From program
exit
Plus
Minus
Plus
++
+-+
++
-+
Counters
Counters
(1,2)
(4,5,6)
The dcvelODmont of one product will be followed through the pilot
selectors.
Pilot selectors No.2, No.3, and No
4 aro returned to
normal position at the beginning of the sign control
progr~m.
If u
l
is
63
negative a
5 punch in the loft hand position of the storage unit 6L
causes pilot selector No. 2 to transfor to the position shown by the
2
dotted line. h
ll must be positive so pilot selector No. 3 is not encrgizcd. The impulse from the program exit chub is connected to the common
hub of P. S, No. 2 and exits from the T hub and is then wired to the C
hub of P. S, No. 3 coming out in this case of the N hub of P. S. No. 3
and then to counters 1, 2 where it is wired to subtract.
Program Step 2:
The multiplier
~
is read out of storage unit lR while the mul-
tiplicands hil and r 2l are read out of storage units 2L and 3R respectively.
The machine is imoulsed to multiply.
The product Ulhil is de-
veloped and added or subtracted into counters 1 and ? depending upon
the setting of pilot selectors No. 2 and No.3.
The product u r
is
l 21
developed and added or subtracted into counters 4, 5, and 6 depending
upon the setting of pilot selectors No. 2 and No.4.
Program Step 3:
This is the second sign control program.
No.3, and IJo.
4 are
reset.
Pilot selectors No.2,
The loft hand position of storage unit 7L
is read out to pilot selector No.2 while tho remaining part of storage
unit 7L is read out and read into storace unit lR.
The left hand
position of storage unit 2R is read out to pilot selector No. 3 while
the left hand position of storage unit
No.4.
4L
is read out to pilot selector
64
Program Step 4:
The multiplier u2 is read out of storage unit lR while the
multiplicands r 12 and h~2 are read out of storage units 2R and 4L respectively. The machine is impulsed to luultiply. The product u2r
lt
is developed and added or subtracted into counters 1 and 2 depending
upon the setting of pilot selectors No. 2 and No.3.
The product
u2h~2 is developed and added or subtracted into counters 4, 5, and 6
depending upon the setting of pilot selectors No. 2 ond No.4.
Program Step
5:
This is the third sign control pror·ram.
No.3 and No.4 aro resot.
Pilot selectors
N~.
2,
The left hand position of storage unit 7R
is read out to pilot selector No. 2 while the remaining part of storage
unit 7R is read out to storage unit lR.
The loft hand position of
storage unit 3L is read out to pilot solector No. 3 while the left hand
position of storage unit
4R is read out to nilot selector No. 4
Program Step 6:
The mnltiplier u
multiplicands r
respectively.
is read out of storage unit 1R while the
3
and r?3 arc read out of stor~gG units 3L and 4R
13
Tho machine is impulsed to multiply.
The product
u r
is developed and added or subtracted into counters 1 and ? de3 13
pending upon the setting of pilot selectors No. 2 and No.3. The product
u r
is developed 2nd added or subtracted into counters 4, 5, and 6,
3 23
depending upon the sottin~ of pilot selectors No. 2 and No.4.
65
Program Stop 7:
The machine is impulsed to read the next card.
The first matrix
card skips out and the second multiplier card is read into the machine.
The romaining four multiplier cards with their corresponding matrix
cards pass throueh the machine in the same manner.
matrix card, No.5, has been computed, the values v
When the last
and v have been
2
accumulated in counters 1 Dnd ?, and counters 4, 5, and 6. Blank
l
trailer cards wore plAced After every fifth ;;lAtrix cl1rd
card has an 11 or X punch in column 79.
This trailer
This X punch is sensed by the
reading brushes and causes the machine to skip the first sever program
stops.
Program Step 8:
The code number is read out of counter 3 and the counter is reset to zero.
This numb0r is read into storcgo unit 6R.
Storage unit
6R is Dupulscd to read out 9nd to punch into columns 1-4 of the trailer
card.
Tho code numbJr for tho first trailer card is 0101.
Program Stop 9:
Tho product v
and 6R.
is read out of counters 1 and 2 and read into 6L
l
Count3rs 1 and ? are reset to zero. The ninth hub from the
right of stora?e units 6L and 6R combined is I'dred through a balance
test hub to detor",:ine the sign of the Droduct.
If the sign is negative
an 11 of X punch is wired to punch in tho column containing the first
digit.
Storago units 6L and 6R arc read out and impulsed to punch into
columns 61-68.
66
Program stop 10:
is rend out of counters 4, 5, and 6 and read into
2
storage units 7L and 7R. Counters 4, 5, and 6 are reset to zero. The
The product v
ninth hub from the right of storage units 7L and 7R combined is rend
through the balance test hub to determine the sign.
An 11 or X punch
is punched over the left hand digit of the result to indicate a negative
vnlue.
storage units 7L and 7R arc read out and impulsed to Gunch into
columns 69-76.
Program Step 11:
The machine is impulsed to read tho next card.
This completes the
comput~tion
of the first two elements of
vector V,
Tho remaining clements arc computed in a similar manner.
The actunl
runnin~
time for a single iterntion depends on the number
of matrix cards involved.
It requires approximately eight seconds to
develop one set of products such as ulh 2 and u r • For the example
ll
l 2l
given in Figure 1 thero arc forty matrix cards. The time required would
be 320 seconds
~lus
apDroximatoly two seconds for tho punching of each
tr;;iler card or n:1 additional 16 soconds making a total time of 336
seconds.
This \o1onld be 5 minutes and 36 seconds.
After the eight trailer cards have been ounched, those can be
separated out by sorting on column 79.
into tho 11 nocket.
Tho
S11m
The trailer cards will 811 fall
These cards arc listed nnd sum"1ed on tho tabulator.
must check rather closely with the cor:mutGd SUl11mntional chock.
67
The lnrgest entry is used as a divisor fer all entries in the V column
vector.
This constitutes the second approximation of the U vector.
A
now set of multiplier cards is made up and tho multiplication of R x U
is carried out again.
Disadvantages of the Method:
Because of inadequate storage capocity, a considerable number of
cards must be used to repeatedly road in the multiplier values.
sample
proble~
forty multiplior cards were required.
iable problem 160 multiplier cards would be needed.
cards would have to be used for each iteration.
large number of iterations would be required.
In the
In a thirty varThis number of
In many instances a
The number of cards used
might well becomo extromely large.
Another disadvantage is tho extremely slow convergence rate of
the iterative solution when two or more roots of the reduced correlation matrix are approximately equal.
In tho next section of this re-
port a discussion of an altornative solution is given which tends to
speed up the rate of convergence.
The same bosic plan for using the
602 A can be utilized for this alternative solution.
3.
A.
An iterative method for the simultnneous extraction of two or more roots:
History of the Method.
Because of the disndvantages of Rotolling's iterative mothod as
outlined in Section I, a groat deal of tinill and effort have gone into
investigations of new tochniquos w'ich were
stantially tho comnutrtional time involved.
dos~gned
to reduce sub-
Mr. R. R. Morris of Eastman
68
Kodak Company became interested in the problGffi about five years ago
and attempted to find a new method which would satisfy this intent.
To describe his approach it is necessary to indicate the type of data
with which he was working.
The illanufacturc and control of color film is a very complicated
procoss; it is nocessary to take obsorvations on a largo numbGr (say
40 or 50) of hi2hly correlated variables.
For theoretical and uractical
reasons it is thought that sev::ral linear functions of these vnriables
will suffice for control purposos and also give some indication of the
underlyin~
chemical
re~ctions
which
occur in tho film.
foundotion of this approach lios in the fact
are known to exist in the film end theory
th~t
The theoretical
at least three dyes
)ostul~tes th~t
these three
dyes have an additive effect (i.e., Y ~ ox + bX + cx ). Under this
l
2
3
hyuothesis the model becomos precisely that of principel component theory.
In colorimetry the most accurate quantitativG data is obtained
by a spectrophotometric curve of tho color patch to be measured.
This
consists in ·.·.loasuring the transmittance (or refLctance) of thJ "laterial
across the visible spectrum fram 400-700 millimicrons at 10 millLmicron
intervals.
The procedure then involvos taking a large number of patches,
corresponding to N or sample sizo, and
com~uting
the 31 x 31 covsriance
matrix taking as the 31 vnriables tho wavelengths at 10
from
400-700~.
~
intervals
Covariance matrices are used here since the variables
arJ all in the same units; thus there is no need to use correlation
matrices.
Still there is no essential diffcrence in tho method between
the two approaches.
o~teinQd,
If in the nnalysis illorc than three vectors were
this moy indicntc tho non-additivity of tho process or the
formation of some new dyes.
This finding is then reported to the
physicists and chemists who explore it further.
Perhaps the most effective use of principal components is in the
quality control field where an almost impossible number of control charts
would otherwise be necessary.
Controls are kept on the values of the
three or four linear functions themselves employing Rotelling's methods
as outlined in Techniques of Statistical Analysis.
Moreover, investi-
gation and indeed determination of new films are facilitated by this
technique which, as shown above, saves a great deal of labor and still
gives results sufficiently accurate for most applications.
Rotelling's method was therefore employed on the covariance
matrices of these variables.
In the majority of cases it was found
that three or four vectors were sufficient to remove 95-99
%
of the
trace of the covariance matrix and that, although convergence of the
first vector was fairly rapid, the remaining ones converged comparatively slowly.
Because of the great amount of computation necessary, Morris then
•
devised a method of obtaining simultaneously the vectors corresponding
to the k largest roots of a non-negative symmetric matrix A of order
n x n •
B.
Description of the Method.
Take an arbitrary k x n semi-orthogonal matrix Xo ~i.e.,
XOKb
=
I(k), X consists of the first k rows of an orthogonal matrix of order
o
n x n_7 Form XOA = YO (k x n).
The rows of YO will be "closer" to the
70
first k characteristic vectors of A than those of Xo in the sense that
the rows of YO' considered as vectors, are nearer to the space generated
by the k characteristic vectors.
ing the matrix.
This follows from the section on power-
In fact, if A is actually of rank k, the rows of YO will
lie in the space of the k characteristic vectors.
However, the rows of
Yo are no longer orthonormal so that if YO is left unmodified and repeatedly pre-multiplied by A, all k rows will converge to the first
characteristic vector of A.
This can be altered to give convergence
to the first k vectors of A by making the rows of YO mutually orthogonal
and of unit length.
I(k).
Thus, YO is modified to obtain Xl such that XIXi
=
YO must be modified so that the rows of YO and Xl generate the
same space and this requirement is equivalent to Xl
= TO(k
x k)Y where
O
TO is non-singular.
Then the iterative scheme is
=
and at each step,
IT.1. I ~
0
The iteration is terminated when T becomes a diagonal matrix.
i
The problem remaining is to define T. This can be done in an
71
infinity of ways, for if any matrix M is found that will orthonormalize
the rows of Y, then NM where N is any k x k orthogonal matrix will also
orthonormalize the rows for
MYY'M' = I(k)
so
NMYY'M'N' = NN' = I(k) •
Now there are two criteria for defining T:
(1) Convenience in computation or programming,
(2)
Speed of convergence.
Method I.
The most convenient method is the Gram-Schmidt or Square root
process which is as follows:
Consider
YY'(k x k)
I{k)
K{k x k)
L(k x k)
where K and L are upper and lower triangular matrices (t ..
~J
1
>j
for upper triangular matrix and t
= a for
K'K
= YY'
XX'
= LYY'L' = LK'KL' = (LK,)2 = I(k)I{k) = I(k).
and K'L
= I(k).
ij
= a for
i < j for lower).
Then L may be taken as T, for if X
= LY
This is the standard
72
squaro root method as outlined in P. S. Dwyer's papers and others:
it
takes the first row of X to be the first row of Y normod to unit length;
the second row of X is a linear combination of the first two rows of Y
such
th~t
the first two rows of X
~re
orthogonal and of unit length, etc.
Method II.
The fastest convergence as shown by empirical evidence seoms to
be obtained by defining T as the appropriately normalized k x k matrix
of characteristic vectors of YYf(k x k).
Lot V be the matrix having as its rows the characteristic vectors
of yyl, each normed to unit length ond let./\ be the dingonal matrix
whose clements nrc the corresponding roots.
Then V(YYt) == A V and
I
1
W' == I(k).
Take
T.: A - ~
V.
Thus if X ==
I
XXI == ./\. - 1 V(YYI )VI.!\ -
A
-1
VY,
I I I
'2 =A- '2 J\. wlj\. - '2
== I(k)
•
This method, though involving loss iteration, will be much more
difficult to program than Method I on computers such as the IBM 602A,
604 and CPC.
This results from a lack of storage capacity on these
machines together with the difficulty in extracting the roots and vectors
of a k x k matrix, evon one as small as
4 x 4.
!'-iorris uses Hethod I routinely employing the CPe.
C.
Discussion of the Method.
The chief advantages of this method arc first, clements of A
arc not altered by computation of residuals, thus helping control Dound-
73
ing errorj and second, speed of convergence is much increased, particularly if the roots are close together.
The last statement applies
especially to Method II.
However, it seems that using Method II, the off diagonal elements of YY' become so small with respect to the diagonal elements,
that it is rather difficult to extract the characteristic vectors of
YY', at least on a desk machine.
This can be avoided by, at this
point, assuming that the vector associated with the largest root is
sufficiently stable, as it should be at this stage, and then using
Method I to get the other orthogonal rows of X.
The initial starting matrix X might be a little less arbi-
o
trary.
If all the correlations are positive, it would be advisable
to use l/~ as the elements of the first row of XO.
Supposing the
column sumBof A to be employed as the first row of XO' there would be
some difficulty in finding the other rows of X such that they are
o
orthogonal to the first row.
Therefore this method is not recommended.
On the other hand, the choice of the lower k - I rows seems to
be indeterminate.
Determining the sequence of signs in the other rows
by using the centroid method of factoring would appear of little
value because of the great discrepancy between characteristic vectors
and centroid factors folloWing the first.
At Kodak, Fisher and Yates' orthogonal polynomials are used as
X when the correlations are all positive.
o
A mathematical description of Method I, employing a 2 x n matrix
as XO' gives an insight into the advantages and disadvantages of this
74
procedure.
Using the notation of the section on powering the matrix,
xb _7
consider the first row of X ~or the first column of
o
as the
transpose of a linear combination of the n true characteristic vectors,
+ a V )1, and in the same manner take the second row
n n
+ b V ) I.
Then
n n
••• + a A V
n n n = Yl
If the first and second rows of X had been postmultiplied by A, the re-
o
sult would be the transpose of the above, and the calculations would
come out the same.
Now, proceeding as in Method I,
2 2
a A +
l l
2
.. '. :I- B.2A
nn
2
2
albl;'l + ••• + a nbn An
(1)
(2)
(2)
0)
=
2
2
alblAl + ••• + a nbnAn
b2A,2 + •.• + b2~2
1 1
n n
v (1)
K
(2)
"nIT
=
I
0
2
\;1 0)--ftF-
75
1
o
\f[i)
L
=
(2)
Consider the second row of L opera.ting on Y1 8,nd Y , or
2
_1
J
G3) -m=
a b
A 2
b1
8'2b 2 A2 2
n ( n)
- + -.--"- (-) + ... + n~
a.
2, Al
8
1
1
8'1
1
2
2
8,
A 2
8
A 2
2
2
n
n)
(-)
+ ... + 2
1 + 2
Al
8
8
1
1
1
r
Y + Y
2
1
(r
1
(3)
1
=
_m=J2
~) _mj-"2 { (~_{ b;:2 )A2V ...+(bn-b;:n )
76
Therefore, to a. certain order of a,pproxima,tion V1 has been removed;
then on i tera.tion the result will lie in a. spa.ce orthogona,l to V1 .
From the a,bove discussion for Method I, it ia evident that the
number of iterations to obta.in V to the desired e.ccura.cy is not rel
duced by this method, since the first row is left unchanged by the
transforma.tion 8.nd iterated.
It ia a.lso a,ppa.rent that the convergence
of the second vector is still a. function of the third root and will not
in genera.l be changed (i.e., same number of iterations) from that of
the usua.l method;
however, residual ma.trices need not be computed and
this is where the saving arises.
The results me,y be genera,lized to the ca.se where X conta.ins
o
more than two rows.
For example, if X contains three rows, then the
o
speed of convergence to the second vector should be considerably
quickened, while for the third the nUI'lber of iterations would rema,in
a,pproxime,tely the same, but the two residual me.trices need not be computed.
Thus it seems a.dvantageous when using Method I to set k = 4
a,t least and proceed
8S
outlined.
On the other hand, if there is a, relia.ble estime.te of the rank
of the ma.trix ba.sed on theory or experience, Method II would give the
qUickest convergence, but a.s mentioned before, if k is ra.ther la.rge
the work involved would still be a.pprecia,ble.
to using
In this case, it amounts
principal components on a. k x k ra.ther than an n x n ma.trix.
Since in psychological de.ta. k will uBua.lly be unknown, Method I
is recommended for this rea.son and for the grea.ter convenience in programming.
77
There a,re as yet no pUblished accounts of this method, Morris
ha,ving given it only in discussion form a,t an IBM seminar et Endicott.
D.
Sample Problems.
A sma,ll fioti tious example constructed by Morris will llluetre.te
the possible se.ving.
Method II w.e employed on the following ma,trix:
.050108
4.350108
3.150108
A.
1
A.
2
4
.850108
.950108
.050108
3.150108
= 5.0004321
= 5.0000000
~=
A.
.950108
- .349892
4.350108
,
=2
·1 000
o=
, the multiple extra,otion pro-
Sta.rt1ng With X
010 0
cedure took 23 itera.t10ns while Rotelling' a itera.tive method would take
better than 20,000 itera.tions to get five pla,oe a.ccure.cy on the first
vector.
Though obviously e, situation such a.s this will not usua.lly oc-
cur in pre,ctice, this ma,y give some indice,tion of relative efficiency.
In order to giva e, more pre,ctioe.l example a. sma.!l problem has
been undertaken here.
It consists of twelve body measurementsj
de,te, is fallible e,nd he,s been fectored previously by the centroid
the
78
method in which four factors were extracted.
The dia.gonal elements of
the reduced correle.tion ma.trix were ta.ken a.s the communa.l1ties from
the previous study.
2 x 12 we.s set up;
As a. starting point en orthogona.l ma.trix of order
the elements of the first row being e.ll 1/\ff2 and
in the second row, the first half
1/Vi2
end the second ha.lf -1/Vf2 .
To compare Hotelling's method with the multiple extraction
procedure, the first vector was determined by Rotelling's method.
The
starting vector wa.s selected proportione.l to the column sums of the
reduced correla.tion matrix.
After twelve itere.tions two successive
u I s were the same to six decima,ls.
Using the multiple extra.ction
method the first vector was also found accura.te to six decima,ls s.fter
twelve trials while the second axis had not yet become stable.
P:
number of a.dditionsl itera.tions will be necessary to ga.in convergence.
This is due to the rela.tive closeness of the second a,nd third roots;
the situa.tion would no doubt be improved by using a. 3 x 12 matrix a,s
Xi rather tha.n a 2 x 12.
4.
Dia.gona.l Estima.tes for the Reduced Correla.t10n Ma.trix.
To date no solution to this problem ha.s been obtained, though
some idee,s have been a.dve.nced end ere a.t present being inveetige.ted.
Guttman in a. recent pa.per discussed the different estima,tes which could
be used.
He recommends tha,t the mUltiple correla,tion between ee.ch
varia.ble B.nd the other n-l varia.bleB be used a.B the estima.te.
The
computa.tional la,bor involved in a. large study would ma.ke this method
seem impra.ctica.l.
W. G. Howe he.s proposed that the "best" est1ma.tes of these communalities a.re those obta.ined by fe.ctorizing the unreduced correla.tion
79
matrix (with one's in the d1agonals,R ) ,preferably us1ng chara.cter1sl
tic vectors.
Then, after these vectors, V(k x n), have been so nor-
malized that VVI = ').., the d1a,gona.l elements of V'V are selected and
inserted in the dia.gona.ls of R .
l
The new ma.trix R 1s i tera,ted until
a new set of characteristic vectors, say Vi' is obta,ined.
However,
the computa,tion here will be much less, since V itself may be used as
the init1al a,pprox1ma tion and should converge rapidly to Vi.
procedure is then continued until the diagonal elements become
to wit ViVi
= Vi+1Vi+l
or Vi
This
sta~le;
= Vi + l •
The scheme a,s outlined a,bove has a ba.d drawback in that the rank
of R (actua.lly the rank of the popula,tion reduced correla,t1on ma,trix)
is unlmown.
In the above it has been assumed the,t V·V ::: R, or tha.t k
vectors remove enough of the varia.bility .
But since this rank is un-
known, it is very difficult to tell s,t wha,t sta.ge to stop fs,ctoring.
Investigation is being conducted along these lines.
However, a,t present the highest entry in each colmnn 1s the
estimate of the communa.lity that has been used here.
2
hh
r 13 r'14 1r1S r'16 1r17 r'18 r 19 1r-1 ,10 1r1 ,ll 1r1 ,12 r"1,13 1r1 ,14 1"l,lS
- Mat ix Ca d #1- - Matr ix Car d #2- ~ }!atr 1x Car k1 1f3- ~z.rat 1x Car ~ #4- ~I~tr ix Cal ~ #Sr 21 h 222 r 23 1'24 ~2S
26 1t"27 • 1'28 r 29 1t"2,lO ~2,ll 1t"2,12 1t"2,13 1t"2,14 r 2 ,lS
2
r31 r 32 h 33 1'34 1r3S r 3b 1'37 1'3~ 11')9 1r3,10 1r3,ll 1r3,12 1'3,13 1t"3,14 r 3 ,lS
~ }~tJ 1x Ca d 116- - }latl 1x Car ~ JJ7_ f-I·:atr 1x Cal k1 /18- ~}~at ~x Car ~ 119- ~ }:atl Iix Ca ~ nor 12
r41
r42
r43
r
r
r
S1
S2
S3
-!·.at ix Ca d
r
r
o1
n
r
02
1'72
- ratl ix Ca
1"
Pl
1'82
r
r
n
n
- ;''atl ix Ca
1'10,1 1'10,2
1'11,1 r 11 ,2
- :'atl ix Ca
r 12 ,1 1'12.2
1'13,1 1'13,2
I-
r-7at ix Ca
r 1L ,l 1'14,2
~t4
1r
IrS4
~55
lin- -t:at
1'6,
'"ou
4s
2
1r46
!r5o
r'47
r'4e
1r49
Ir
1'58
IrS9 I-5,10 IrS,ll Ir.5,12 l-5,13 ll'5,14 !oS,lS
k1 413- ~ }:at ~x Ca ~ #lis- ~ }latr Iix Ca ~ #1~
S7
p.x Car d JJl~ I-
Ir !)
o
h~b6
1r
I~tr 1x
67
2
1'68
Ca
1r69
10 4 ,10 1r4 ,ll
106
..
..
I"
~I
I!
v
2
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