PSYCHOMETRIKA--VOL.39, NO. 1MARCH,1974

AN I N D E X OF F A C T O R I A L S I M P L I C I T Y *HENRY F. KAISER
UNIVERSITY OF CALIFORNIA, BERKELEY
AND
UNITED STATES COAST GUARD ACADEMY

An index of factorial simplicity, employing the quartimax transformational criteria of Carroll, Wrigley and Neuhaus, and Saunders, isdeveloped. This index is both for each row separately and for a factor patternmatrix as a whole. The index varies between zero and one. The problem ofcalibrating the index is discussed.
After a factor analysis has been completed, it is of interest to assess how
good the solution is, in the sense of how simple--and thus how interpretable-the final factor pattern matrix is. The ideal solution, most investigators
would agree, is one t h a t is unifactorial, i.e., a solution for which each rowof the pattern matrix has one, and only one, non-zero loading. In this paper
we propose an index, for each row of the factor matrix separately and forthe matrix as a whole, which measures the tendency towards unifactoriality
for a given row and the tendency toward unifactoriality for the entire factor
pattern matrix. For this purpose, we turn to the quartimax criteria for
analytic transformation of Carroll [1953], Wrigley and Neuhaus [1954],and Saunders [1953], and measure the value of their criteria for the data at
hand relative to the optimum value of their criteria. Interestingly, it turns
out t h a t each of their three rationales leads to the same index of factorial
simplicity when we scale the index to lie between zero and one. (It should
be pointed out that, while we develop our index from the quartimax viewpoint, our results are applicable to any factor pattern matrix.)
First we consider Carroll's [1953] criterion. For row j of the patternmatrix V, he proposes t h a t

(1)

= 5:

should be a minimum. Clearly the minimum value t h a t C~ can reach is zero,and this occurs when only one of the q (the number of factors) elements inrow i is non-zero. The worst possible value of Ci, i.e., when C~ is a maximum,* The research reported in this paper was supported in part by the Officeof Computing
Activities, National Science Foundation.
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PSYCHOMETRIKA

occurs when all q of the vi, in row j are equal in absolute value. I t is seent h a t "this is given b y(2)

max C; = q ( q -1)(~E] v i 2 / q ) 2,a= (q -- 1)(~-'~. v,.2)2/q.

(3)

Let us take as our squared index of factorial simplicity, for row j of V,
the quantity(4)(IFS(j))2=C,

1

max C;

2~, ~-, vi.2vi, 2= 1 -"~'(q -- 1)(~E] v i 2 ) / q(5)

8

We define our I F S ( J ) in this way so t h a t it can v a r y from zero to one, yielding
a value of zero when all vi. are equal in absolute v a l u e - - t h e worst possible
situation--and reaching a value of one only when all vi, but one are z e r o - - t h ebest possible situation. (We discuss why we take the square root of (5) for
the actual ( I F S ( J ) ) below under "the problem of calibration.") Noting
that(6)

E Y_,v,.% 2= (Z~,.2) 2 - Z~;.',
e#tssand after a little algebra, we findq E v,.' - ( E v;.3)2(7)$(IFS(J)) 2 =$(q _- 221)(Z] v,. )aFrom Carroll's viewpoint, his overall criterion for the entire patternmatrix is(s)c = E c; = E E E v,.% 2.
iis#t

If we define our overall squared index of factorial simplicity as(9)(IFS)2=

Cmax C '1

we readily find t h a t

Y~ [q ~(10)(IFS) 2 =iv;° 4 -- ( ~[] v;.2) 2]s~E [(q is1)(2Ev;.)]22swhich again can vary from zero--worst possible--to one--best possible.
The latter occurs only when the entire matrix is unifactorial.

HENRYF.

KAISER33

F r o m Wrigley and N e u h a u s ' [1954] point of view a row of V is ideal
when the variance of the squared loadings is a maximum, i.e., when(11)W; = [q ~ vi, 4 -- ( ~v,,2)2]/q 2sis a maximum. The m a x i m u m value t h a t Wi can attain is(12)max W; = [ q ( ~ ] v;.2) 2 -( ~ ] v,,2)2]/q 2.

s$F r o m this viewpoint let us take as our squared index of factorial simplicity(13)(IFS(J))~ _W,
m a x Wi 'q ~]vi. 4-(~]vi,2) 28(14)a(q -- 1)(~-v;.22),8

the same index we proposed when looking at the problem from Carroll'sviewpoint. Overall--for the entire m a t r i x - - w e define(15)(IFS)2 -~ imax Wii

E[q(16)--'Ev;." - (E* ~ ...... ,'[(q -,;. ) ]Jtas before.
Finally, Saunders' [1953] version of the quartimax criterion proposes
that, for row j,Z(17)Si* -"-Vis22$

should be a maximum. The m a x i m u m value of Si* occurs, as might be expected, when one and only one element in row j is non-zero, and is(18)

max Si* = 1.

When all elements in row j are equal in absolute v a l u e - - t h e worst possible
s i t u a t i o n - - S i * is a minimum, and is(19)

rain Si* = 1/q.

Let us redefine Si* as(20)

S~ = a -'k bSi*34PSYCHOMETRIKA

choosing a and b so t h a t S; varies from zero to one (rather than Si*'s varying
from 1/q to 1). For this purpose we have the simultaneous equations in aand b(21)a + b (max Si*) = a "4- b(1) = 1,and(22)

a + b (min S i * ) = a -4- b ( 1 / q ) = O,yielding(23)

a = --1/(q-1),and(24)b = q/(q--1),

from which

Zq~ ~"1) + (q - 1) Z: (~i.)/)is(25)

S~ =(- q -t

Elementary algebra yields(26)

S, = ( I F S ( J ) ) 2 =q ~ v ; , 4 -- ( y ~ v i 2 ) 2(q - 1)(~-v;,22)$aas before. For the entire pattern matrix we can again derive the overallsquared I F S as(27)(IFS)2__Zi

E,~2 [(q --(Z°1)( ] E -v;. ) 2

Thus, all roads lead to Rome: regardless of what version of the quartimax
criterion we use, if we set out to define an index which varies from zero toone we find (7), (14), or (26) for a given row J, or (10), (16), or (27) for theentire factor pattern matrix.
T h e P r o b l e m o] Calibration

Although we know t h a t an I F S can vary from a minimum of zero to a
maximum of one, we still would like to know the meaning of values between
these extremes. I t would seem t h a t only extensive numerical experience
with real data will give us a solid feeling for, say, how big is big, etc. Ho~=ever, the following considerations may be helpful for this problem.
Above we uniformly defined squared indices of factorial simplicity.
T h e reason for this is t h a t throughout we have dealt with fourth-powerfunctions of loadings, and it would seem more consistent with other statisticalHENRY F. KAISER

35

measures, e.g., the correlation ratio, to deal with squared functions. Additionally, numerical appraisal indicates that, were we not to take squareroots above, our indices would be too dramatically depressed for only slightdepartures from unifactoriality.
Further help in the calibration of our index may be gained from considering the special case of a row of a factor matrix with q elements, c ofwhich are non-zero and equal in absolute value, and (q - c) are zero (c iswhat is usually called the complexity of a variable represented by the row).
Elementary analysis for this special case yields
q--cc ( q - 1)=(28)

(IFS(J))2andlim(IFS(J)) 2 = lim q - c(29)

.

.

.

.

.

c(q--1)_ 1c

In Table 1 are given values of I F S ( J ) , as given by the square rootsof (28) and (29), for various values of q and c. This table might possibly
give the wrong impression about what I F S s are observed with real data.
Most real world problems have an average complexity c of, so to speak,
about one and one-half, and thus yield an I F S in the .70s or .80s.
Subjective reflection, based upon Table 1 and primarily observing
I F S s for a substantial number of factor analyses from the real world, suggests
the following evaluation of levels of our index of factorial simplicity:in the .90s, marvelous
in the .80s, meritorious
in the .70s, middlingin the .60s, mediocrein the .50s, miserablebelow .50, unacceptable
Table I
Indices of Factorial Simplicity as a Function of q, the NLmber of Factors,
and c, the Complexity of a Variable's Loadings.ooooou"

32.0000
1 1.0000
2.0000.3333
.5000
.5774
1.0000 1.0000
34.0000
.2500
.4082

.2000 .2582 .2928 .3162 .3333 .3464 .35681 .4472.3162 .3536 .3780 .3953 .4082 .4183 .4264] .5000.4472 .4714 .4880 .5000 .5092 .5164 .52221 .5774.6124 .6325.6455.6547. 6 6 1 4 .6667
.6708
.6742
.7071
1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
5678q = ntmlber of factors

9i0Ii

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PSYCHOMETRIKA

Referring to T a b l e 1, we see t h a t an I F S will be at .50, the borderlineof acceptability, when, for complexity two, there are three factors, for complexity three, there are nine factors, and for complexity four, there is an
infinity of factors. For complexity c of five or greater, an acceptable I F S
is n o t attainable.
REFERENCES
Carroll, J. B. An analytical solution for approximating simple structure in factor analysis.
Psychometrika, 1953, 18, 23-38.
Saunders, D. R. An analytic method for rotation to orthogonal simple structure. American
Psychologist, 1953, 8, 428 (Abstract).
Wrigley, C. & Neuhaus, J. O. The quartimax method: An analytical approach to orthogonal
simple structure. British Journal of Statistical Psychology, 1954, 7, 81-91.
Manuscript received 2/5/73
Revised manuscript received 9/5/73