PSYCHOMETRIKA--VOL. ~, NO. 2
JUNE,1970
THE RELATION BETWEEN SAMPLE AND POPULATION
CHARACTERISTIC VECTORS*
NORMAN
UNIVERSITY
CLIFF
OF SOUTHERN
CALIFORNIA
Dataare reportedwhichshowthe statistical relation betweenthe sample
and populationcharacteristic vectors of correlation matriceswith squared
multiple correlations as communality
estimates. Samplingfluctuations were
foundto relate onlyto differencesin the squareroots of characteristic roots
andto samplesize. Aprinciple for determining
the number
of factors to rotate
andinterpret after rotation is suggested.
A number of statistical
estimation procedures involve the computation
of the characteristic vectors of a covariance or correlation matrix, but very
little
is known concerning the sampling behavior of such characteristic
vectors. Characteristic vectors are of particular importance in the currently
used methods of estimation in factor analysis, in which case R~, the sample
squared multiple correlation of variable j with all the remaining variables in
the matrix, is often substituted for unity as the diagonal element of any row j
of the correlation matrix R. This paper reports a Monte Carlo study of the
sampling characteristics of the characteristic vectors of such reduced correlation matrices.
Method
The procedure used follows a methodology essentially the same as that
used by JSreskog [1963], Browne[1968], Hamburger[1963], Cliff and Pennell
[1967], and Pennell [1968]. In it, a number of sample correlation matrices
are generated from a given correlation matrix according to a procedure
developed by Pennell and Young [1967] following a scheme set forth by
Browne [1968] and summarized by Cliff and Pennell [1967]. The method
assumes multivariate normality for the variables correlated.
Five population matrices were used in the present study. Four contained
12 variables, and the fifth, 20. Since the purpose of the study focussed on
commonfactor analysis, all five of the matrices had the property that their
rank, in the population, reduced exactly to four if the correct "communali* This study was supported by the National Science Foundation, Grant GB4230.
The author wishes to express his appreciation for the use of WesternData Processing
Center and the Health Sciences ComputingFacility, UCLA.He also thanks Dr. Roger
Pennellfor extremelyvaluableassistance in a numberof phasesof the study.
PSYCHOMETRIKA
ties," which are not the population squared multiple correlations, were inserted as diagonal entries. Twoof the matrices were designed to have common
factor loadings which were "univocal," i.e., they had only one non-zero
loading. This also implies that sections of the correlation matrix consisted
of zero elements. The remaining two matrices consisted of variables with
"complex" loading patterns, i.e., few zero loadings and few or no zero correlations. The two membersof each pair differed in the pattern of their characteristic roots; one univocal and one complex matrix had nearly equal nonzero characteristic roots, while the other two matrices had non-zero roots
that were of different sizes, the smallest being quite small.
It was hypothesized that the behavior of characteristic vectors of sample
reduced correlation matrices would depend not on the characteristic
roots
and vectors of either the correlation matrices themselves or upon those of the
correlation matrices with population R~ as diagonal estimates, but rather
upon those of the "expected value" reduced correlation matrices. The expected values of R~ , the sample squared multiple, is given by the formula
(1)
E(/~) = R~ + .(n -- 1) (1 2
(N 1)
where E(/~) is the expected value of R~ . N is the sample size, and n is the
number of variables. This formula is adapted from Kendall and Stuart [1961,
v. 2, p. 341] and is accurate as an approximation within 1/2N. The expected
value of a sample Pearson correlation coefficient is the population parameter
p within the same degree of precision, 1/2N [Kendall, & Stuart, 1963, v. 1,
p. 390]. The approximations improve as R~ departs from 0.50 and as p departs from zero, respectively.
The sample sizes assumed in the present study were 100 and 600. Consequently, the expected value of the correlation matrix with/~ in the diagonal
will be closely approximated by the population correlation matrix itself with
the diagonal replaced by (1). Such matrices were used here as "expected
value" (EV) correlation matrices. The characteristic
roots and vectors
these expected value matrices were computed. The characteristic
roots are
given in Table 1. These roots and vectors will be referred to as EV roots and
vectors.
An identification problem exists with respect to considering the sampling
characteristics of characteristic vectors. Population characteristic vectors
maybe identified by their roots, so long as the latter are distinct, but this
cannot be done in samples since the sample roots will differ from sample to
sample as well as from the population values. What does a sample characteristic vector estimate? Howis it matched with any one population parameter rather than another? Discussing the whole set of characteristic vectors
of a matrix is not fruitful because, on the one hand, they may be permuted,
provided the roots are correspondingly permuted, and thus their order of
Table.1
CharacCeris~icRoo~s of Expected Value CorrelationMatrices
Population Matrix
Complex,
Complex,
Univocal,
Univocal,
different,
similar,
different,
similar,
12ovariable
IO0
600
12-varlable
i00
600
12-variable
i0o
600
12-variable
I00
600
Complex,
different,
20-variable
I00
1
2.842 2.786
2.020 1.952
2.805 2.796
1.422 1.359
2.896
2
1.o80 I.O01
1.128 1.057
1.040
.961
1.223 1.147
1.215
¯ 640 .561
.893
.825
.761
.672
1.074 .997
.838
.171 .096
.623
.557
.342
.243
¯ 973 .892
.386
.o15 - .067
.136
.070
.036 - .027
- .071 - .152
.152 .072
6
- .008 - .067
.o18 - .064
.036 - .027
.071 - .152
.116 .072
7
- .008 -..071
.o18 - .064
- .018 - .063
- .078 - .153
.116 .072
8
- .126 - .108
.001 - .066
- .018 - .063
- .078 - .153
.112 .072
9
- .040 - .120
- .055 - .129
- .049 - .138
- .085 - .155
.1±2 .056
I0
- .066 - .128
- .081 - .152
- .049 - .138
¯ 085 - .155
.112 .036
ii
- .IIi - .176
- .162 - .226
- .076 - .155
- .090 - .161
.lO5 .005
12
- .116 - .178
- .168 - .232
- .076 - .155
.090 - .161
.o73- .003
aColumn contains roots 13-20 for N.= i00; matrix not studied at N = 600..
~.
166
PSYCHOMETRIKA
listing is not fixed. Onthe other hand, any one set of characteristic vectors.
is always an orthogonal transformation of any other set of the same order.
That is, any sample set is a transformation of any pop~ation set of the same.
order and there is no way to guarantee circumstances under which the transformation will be an identity transformation.
The approach to the identification
problem taken here was to use the
rank in magnitude of characteristic roots to identify the vectors, both population and sample. The sampfing problem was then posed as follows: when
sample and population normalized characteristic
vectors are ordered by
magnitude of characteristic root, what size can one expect for the elements
of the transformation that carries the sample vectors into the pop~ation? .
If two symmetric matrices are of the same order, then matrices composed of their normalized characteristic vectors are orthogonal transformation of each other because the characteristic vectors of each set are orthogonal.
The EVand sample characteristic vector matrices, V and l? are such matrices;
their columns are n characteristic vectors, so
(2)
VV’ = tF’
= ~
Then
(3)
lTiT’V = V
So ~V is a transformation that changes 17 into V. Its elements are also the
scalar products of each vector of V with each of those of 17 (from the definition
of the scalar product). Also, because a scalar product, ~’~ x,.y., , between
vectors X and Y is
(4)
c~ = IX[ IV[ cos ~,
where IX] and ]Y] are the respective lengths of X and Y [Birkhoff & MacLane,
1953, p. 158], the elements of IT’V are the cosines of the angles between their
constituent vectors.
The procedure used here was to generate a number of sample values of
reduced R, compute their characteristic vector matrices l?, multiply 17’V,
and tabulate the distribution of each entry of T -- ]7’V. If t.~M , the scalar
product (cosine) between sample vector m and population vector M,
typically small for m ~ M, and close to unity for m = M, then the sample
vector resembles the corresponding EVfactor closely. If the tin, are not near
zero and tmity, this impfies that the sample vectors are hodgepodgecombinations of the EVones.
These t..M are also the "Tucker phi’s" [Tucker, 1951] between sample
and population principal factors since the phi is
NORMAN CLIFF
167
and
and correspondingly for ]~u .
The overall index of a vector’s (and factor’s) sampling stability was
the root mean square over samples of the sample tram ¯ The mean is of little
interest since the sign of a characteristic vector is arbitrary. These tmMwere
compared to various characteristics
of the respective EV matrices in an
attempt to discover the parameters which influence the size of t~.
It is also of interest to find someoverall index of the degree to which a
given population vector, and the corresponding factor, are "recoverable" by
rotation of the sample principal factors. Equation (3) repeats the wellknownfact that a given population vector is always completely recoverable
from the complete set of sample factors, but this is certainly not the case if
only the r sample vectors corresponding the r largest roots are rotated, as is
the typical case in applications of factor analysis. The sum of the squared
cosines between a population vector and the r largest sample vectors is a
good index of its recoverableness. It is the phi that could be obtained, on the
average, between sample and pop~ation vectors by appropriate rotation
of the sample vectors corresponding to the largest roots.
One hundred sample correlation matrices were derived by the Permell
and Young [1967] procedure for each population matrix and sample size
combination. Sample and expected value characteristic
vectors were arranged by order of size of characteristic
root. Then each sample characteristic vector matrix was multiplied by the transpose of the appropriate E¥
characteristic matrix, yielding the cosines of the angles between sample and
population vectors. The root mean squares over samples of the entries in
this cosine matrix were computed as an index of the sampling stability of the
characteristic vectors.
Results
Twoof the nine matrices of these root mean square cosines are given
in Table 2. It is immediately apparent that the angle of rotation between
sample and EV vectors depends on the difference in size of the corresponding
EV characteristic
roots. This seemed reasonable on intuitive grounds. It
remained to attempt to specify the nature of the relationship more exactly.
Closer examination led to the hypothesis that it was the difference in
the square roots of the characteristic roots rather than the roots themselves
that were important. This hypothesis was tested for the positive EVroots,
using Fig. 1 which plots root mean square t.,M , the cosine of the angle between sample vector m (m= 1, 2,- ¯ -, n) and EVvector M(M = 1, 2,...,
against the reciprocals of the differences in square roots of EVcharacteristic
roots, [~2 _ ~1. The plots appear to be linear in the lower ranges. The
Table 2
Mean ~qua~ed Cosines ~ between Smuple
and Population
Factors
Unt~ocal~ ~ame-s~zed Factors
N = 100
~opulat~on
Fac~o~
.~ample Facto~
~
~
~
~
5
6
7
8
9
I0
11
i~
6~ 098 ~ 098 o~ 007 oo~ oo~ o0~ 0o~ oo~ 0o~
2
z3~552 z3~ z~oz9 0o7 ~07 o0~ 005 005 0o7 00~
zoo ~65 539 z28 o2~ 008 005 009 o07 006 006 005
08k z2k z~5 566 02o oz3 oo7 ozo 008 008 006 008
oo~ 006 oo7 oo7 156 111 117 132 lOl 116 lO3 136
oo~ 009 o13 009 122 122 1~ o9~ lO8 119 113 1~o
7
006 OlO 006 o13 116 13o 118 lOO 132 12o 131 118
006 oo8 0o6 Oll lO9 127 lo2 139 095 12~ 1~5 130
?
005 ook 008 oz2 ~z8 z~k ~ ~z6 ~3 z29 zz? ~o8.
10
oo5 oo7 OlO 0o7 lOl lOl 12o 127 1~6 118 132
11
003 007 ozo o~o lO3 1~6 lk9 132 1~8 13o zz8 101
0o3 Oll o10 o12 099 1~3 lOl 131 131 1~ 117 119
Oo~le~ Dif£e~en~-slzed Fac~oPs
N = 600
~opulation
sampleFactor
Facto~
5
6
7
8
9
Z0
ZZ
Z~
1
99~ 00~ OOl o0o ooo ooo oo0 ooo oo0 o00 oo0 o00
2
003 966OZ9 002 0O2 001 ooi 0oi 0oi 001 ooi ooi
OOZo~9 9k9 o08 003 004 003 003 002 002 002 002
oo~ oo2 o09 833 o1~2. 034 025 OZ7 o~5 009 006 007
ooo ooi 003 025 z9o ~o2 25z ~57 o78 ol¢.~.o~9
o~
6
o00 0oi 003 o31 232 233 179 122 083 051 030 032
7
ooo oo1 oo~o38 236 199 z55 z28 079 062 047 o51
0oo ooi 002 023 iP_~lO~lll161 179 I~6 069 077
9
000 001 003 015 080 093 114155 i~6 164 114 113
~o
0oo ooi oo2.Oll043 069 o87 11~9 258 175 128 075
ooo 0oi ooi 005 o25 027 o35 o57 080 158 299 3~3
ooo oo1 002 006 o~1 032 039 049 o77 187 275 310
*Dec~mls
NORMAN CLIFF
N = 600
12 va~. CB ¯
I~ vat. CD ¯
.2.
0.0
~
REC~PROCAL OF ROOT DIFFERENCE
@
N = I00
.8.
12 vat. CB ¯
12 vat.
20 vat. CD~
.\
.6.
0
0.0
RECIPROCAL OF ROOT DIFFERENCE
169
PSYCHOMETRIKA
Recovery o] Population Factors ]rom Sample Data
Table 3 presents data on the degree of resemblance (phi-square) between
a sample factor and the corresponding population factor, the factors being
identified by order of root-size. The entries on each line correspond to the
diagonal entries on Table 2, but data is presented for all nine matrices studied.
With a few exceptions, these entries are large for all the real factors, and small
for the error factors. Those entries for real factors that are not high (less
than .85) correspond to instances where there was one or more population
roots, whether real or error, which were nearly equal in size to the given one.
For N = 100, there will be a high degree of resemblance (phi-square) greater
than .81) between sample and population factors only in those few cases
where the root is really different from all the others. There will be a fair
degree of resemblance on the average (phi-square greater than .50) in
majority of cases, but difficulty in recognizing all the factors is to be expected
in a fair proportion of samples. It mayalso be noted that there is very little
resemblance between sample and population "error" factors.
A low resemblance between corresponding factors will not be serious
if the r largest sample factors are simply a transformation of the r largest
population factors. Data on the question of whether or not this will be the
case is presented in Table 4. Its entries are "multiple phi-squared," the
maximumcosine-squared possible between a population factor and a linear
combination of the four largest sample factors. Thus it is a measure of the
degree to which population factor is within the space of the sample factors.
Here, the results are -¢ery encouraging for all the N = 600 data, but less so
for N = 100. As long as the real roots are all substantially greater than the
error roots in the population, the factors are recoverable, but bearing in
mind that these figures are averages it appears that when this is not the
case one or even two population factors may not be recoverable, even when a
Procrustes rotation is used. The situation is especially bad in the case of the
20-variable matrix where the presence of 16 error dimensions seems to provide that many more ways in which the solution may wobble.
Table 4’s data concerns the "recoverableness" of population principal
factors, and one can wonder about the "composition" of the sample factors,
i.e., the degree to which a given sample factor includes real rather than error
variance. The symmetry of the matrices in Table 2 shows that the proportion
of a sample factors’ variance that is from real rather than error population
factors will be virtually identical to the "recoverableness" of the corresponding population factor. Therefore, when N was 600, even the smallest
sample factor is almost completely derived from real variance, but when
N = 100 the fourth (and sometimes even the third) sample factor contains
good proportion of error when the corresponding root is small.
One may conceptualize the results from the point of view of factor
Table 3
Mean Square Cosines (Phi-Squared) between Population
Factor and Corresponding Sample Fector
Fsctor
Matrix
~=lOO
"Error"
"Real"
x
~
3
~
5
6
7
8
9
lO
11
12
cS
.878 .694 .638 .651
.172
.188
.170.
.121
.135
.152 .227 .204
~CD-12
.962 .820 .710 .h3~8
o177
,.166
,132
,093
,103
.138 .21a. ,15o
us
.641 .552 .539 .566
.156
o122
,118
,139
,IA3
o118 ,118 ,ll7
UD
,955 °732 .659 ,551
,231
,155
,341
,317
,138
,150 ,194 ,177
CD-20*
,915 ,662 ,541 ,307
.16i
,143
.132
.105
,058
,050 ,056 ,051
°057
,074
,059
,063
,060
,049 ,050 ,055
N=600
CS
,983 ,895 ,850 .916
.213
.251
.158
.221
,263
,311 ,400 ,471
CD
¯ 933 ,966 °949 ,833
,190
,233
,155
,161
,146
,175 ,299 ,310
US
,81~ ,72o ,685 °732
.115
.I01
,152
,147
.138
.113 ,143 ,122
UD
,991 ,934 .921 ,916
.160
.425
.120
.256
.222
.189 .239 .265
*Factors 13-20 on second line
ma~le~
"Recoverableness"of Population Factors:
Average Eultiple-Phi-Squaredwlth
Four Largest Sample Factors
Factors
Matrix
~:zoo
"Real"
CB
~972 .938 .906 ,865
~
6
7
8
9
~o ~1 ~
.o52 .o~9 .058 .0~1 .033 .03~.017.020
CD-12
.984 .939 .869 .537
.121
,091
,090
.I0~
.101
~060 .052 .051
US
.951 .9~ .932 .819
.02~
.036
.035
.031
.029
.029 .930 .036
I 2
3 ~
.092 .~I0 .011 .009 .074 .0?8.060.063
o~-2o, .95o .81~ .?18 .~95 .090 .09~ .085 .084 .068 .076.082.070
¯ 064 .066 .071 .061 .066 o0~8.046.03~
UD
.982
.925
.891
.695
N=600
CS
CD-I~
US
.995 .990 .964 .979
.997 .990 .977
.B~
~993 ,992 ,990 .989
.99~ ,988 .981 .9~9
.oo7 oolo .oo9 .ol1
.029
.035
.043
.0~6
.oo5 .oo4 .003 .003
.019
.014
.007
.009
.oo~ .oo~ .oo~ ,oo~ .oo~ .oo~ .oo~ .oo~
.002 .002 .o17 .o16 .o~3 .oz~.o1~.OlO
*Factors 13-20 on second line
NORMAN CLIFF
175
analysis as follows. In factor analysis, only the r largest roots and the corresponding vectors are retained for further study, the remainder being discarded. The population factor matrix F, is defined as
(10)
in which H, is a super matrix:
I, being an r by r identity matrix and 0 an n - r zero matrix. In samples,
we have the corresponding equation
or
(13)
Someidea of the irdiuences affecting the difference between F, and/~, can
be gathered by expressing (13) in supermatrices
T ....
0
Carrying out the multiplication,
1,I
LT.....
(17)
Letus consider
thetwopartsof therightsideof (17)separately.
Thesecond
statesineffect
thatinsofar
as T, ....contains
non-zero
elements,
theerror
vectors
of theEV matrixwillbe involved
in thesamplefactormatrix.
This
section
of T is theoneinvol~ing
cosines
between
"real"and"error"
vectors.
Thee~dence
presented
earlier
argues
thatthesecoefficients
willbenearzero
if thedifferences
in thecorresponding
rootsarelarge.
Thesedifferences
will
be1~rgeif thefirstr EV rootsarealllargeandtheremaining
n --r arenear
~,ero
ornegative.
Membersof the two setscan approach
eachotherin sizefromeither
direction.
One or moreof theerrorrootscanbe moderate
sizedif allthe
~(/~)arenotaccv_rate
communality
estimates.
Consideration
of conditions
under which one or more of the r largest are relatively small suggests that
~ ~
PSYCHOMETRIKA
this will be the case if the rotated factors do not exhibit good orthogonal
simple structure, or if all factors are not equally well defined, but those are
only preliminary suggestions..
It may be noted that neither A~-r nor -~-r enter into (17) directly: the
absolute size of their elements is not important. What is important is that all
of the elements of An-r be as different as possible from those of Ar.
The first part of the expression may not be as great an influence on
sampling fluctuations as the second, especially if the possibility of rotation
is introduced. The transformation which gives a least squares fit of
toFr = VrAris
(18)
H =
(see Mosier, 1939), provided .~r and T~ are non-singular. The transformation
is oblique in general. If the rightmost part of (17) is zero, then the fit
exact, although the transformation remains oblique. In either case, the degree
of obliquity depends on how different the ~ and )~M(m, M <_ r) are, both
within and between sets since if they are all equal their matrices are scalars
and would cancel out each other, leaving T~, . This will be an orthogonal
transformation if Tn .... = 0.
These considerations support the practice of looking for a break in the
size of the characteristic roots and rotating those factors corresponding to
roots larger than the break-point. The reasoning here is that the sample
characteristic roots will follow the EVrather closely in size (C]. Cliff and
Hamburger, 1967); therefore, a sample break will correspond to an EV break,
and the elements of T, .... will be small if r is taken as indicated. In that
case the r retained factors will be relatively uncontaminatedby the remaining
ones. This is no guarantee that r is the numberof factors in the population,
but the indication is that even if there are one or two more population factors
any additional sample factors retained will include a substantial proportion
of error.
The present results mayexplain a minor aspect of Pennell’s [1968] data.
He studied the influence of loading size and sample size upon the sampling
errors of loadings and found that, except for two instances, the sampling
-~
variances of loadings, as determined empirically, were proportional to N
and (1 - 1~)~. The present results suggest an explanation for his two disparate instances. The hypothesis is that in those two instances the EVroots
for factors beyond r were of moderate size, were close in magnitude to smaller
real roots, and the variables studied had appreciable loadings on the correo
sponding vectors. This could appreciably increase the instability of the loading
since it would have the effect of randomly adding or subtracting from the
loading, or looked at another way, give the test vector the appearance of
"wobbling" in the commonfactor space. The latter occurs not simply because the vector wobbles, but because the definitions of the commonfactor
NORMAN
CLIFF
177
space changes from sample to sample. Suppose, for example, there was a
good-sized difference betweenthe fourth and fifth roots, but the fifth, sixth,
seventh, etc. were of nearly equal size. One should resist the temptation to
keep five factors, even if he has theoretical grounds for doing so, because it
will contain a substantial proportion of variance from the sixth, seventh...
etc., population factors, which are presumably error.
It is tempting, although undoubtedly premature, to take Formula (9)
literally,
and decide on the number of factors by computing the complete
table of differences i~ successive roots, the corresponding cosines, and the
resulting expected proportions of variance from smaller factors. If further
studies were to confirm the present ones, it ~vould be worthwhile to work out
such decision procedures in more detail. They would be based not on the
"significance" of a factor but on the proportion of real variance ia it, although the nature of the currently available significance tests would indicate
that the decisions would be similar.
The results presented here are for the principal-factors-with-squaredmultiples procedure for getting factor loadings. This procedure, while it is
currently very widely practiced, does not have a very well-integrated theoretical basis. Our results are designed to be of interest to users of this method.
Currently, a related but muchbetter grounded procedure for getting factor
loadings is gaining acceptance. This is the Rao [1955] procedure Harris
[1962, 1967] has discussed. In it, rather than subtracting S~ = 1 -- R~ from
the diagonal of the correlation matrix, it is pre- and postmultiplied by S
Thus S-1RS-1 2.
is factored rather than R -- S
JSreskog [1963] has done a Monte Carlo study of S-IRS -~ but he
ports results only for individual loadings rather than for the complete vectors.
He did find, however, what he termed an "ill-conditioned
case," one in
which the roots were very nearly equal. This gave loadings with very high
sampling errors, indicating that the same I~inds of effects will occur, but the
parametric question of their magnitude is left open. It would be of considerable interest to study the s~mpling behavior of the characteristic vectors of
S-~RS-1. Since the multiplication by S-~ tends to have ~he effect of equalizing the error roots, the sampling behavior of the characteristic vectors maybe
improved slightly for a given correlation n~trix. Browne’s[1968] results are
somewhat in support of this view. Although he did not explicitly study
S-~RS-~, he did study a closely related matrix, and fotmd its sampling
behavior to be slightly better than that for R - S~. Again, parametric irfformarion of the present kind is not available from his study.
REFERENCES
Birkhoff, G. & MacLane, S. A brief survey of modern algebra. Rev. ed. New York: Macmillan, 1953.
Browne, M. W. A comparison of factor analytic techniques.
Psychometrika, 1968, 33,
267-334.
178
PSYCHOMETRIKA
Cliff, N. &Hamburger, C. D. The study of sampling errors in factor analysis by means
of artificial experiments.PsychologicalBulletin, 1967, 58, 430-445.
Cliff, N. &Pennell, R. The influence of communality,factor strength, and loading size on
the sampling characteristics of factor loading. Psychometrika, 1967, 32, 309-326.
Harris, C. W. SomeRao-Guttmanrelatio~hips. Psychometrika, 1962, 27, 247-263.
Harris, C. W.Onfactors and factor scores. Psychometrika,1967, 32, 363-379.
JSreskog, K. G. Statistical estimation in factor analysis. Stockholm:Almquistand Wiksell,
1963.
Kendall, M. G. & Stuart, A. The advanced theory of statistics. Vol. 2. London: Charles
Griffin, 1961.
Kendall, M. G. &Stuart, A. The advancedtheory of statistics. (2nd Ed.) Vol. 1. London:
Charles Griffin, 1963.
Pennell, R. J. & Young,F. W. An IBM7094 program for generating random factor matrices. BehavioralScience, 1967, 12, 165-166.
Pennell, R. J. The effect of communalityand N on the sampling distributions of factor
loadings. Psychometrika,1968, 33, 423-440.
Tucker, L. R. A methodfor synthesis of factor studies. Personnel Research Section Report
No. 984, 1951, Department of the Army(mimeo.).
Manuscript received 12/30/67
Revised manuscriptreceived ~ /~5 /69