A Comprehensive Trial Of The Scree And Kg Criteria For Determining The Number Of Factors.

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A Comprehensive Trial Of The Scree And Kg Criteria For Determining The Number Of Factors Raymond B. Cattell & S. Vogelmann Published online: 10 Jun 2010.

To cite this article: Raymond B. Cattell & S. Vogelmann (1977) A Comprehensive Trial Of The Scree And Kg Criteria For Determining The Number Of Factors, Multivariate Behavioral Research, 12:3, 289-325, DOI: 10.1207/s15327906mbr1203_2 To link to this article: http://dx.doi.org/10.1207/s15327906mbr1203_2

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The b w m l of k f ~ l t i v a r b t Behaz~iorat e Research. 1977,12,289-325

A COMPREHENSIVE TRIAL O F THE SCREE AND KG CRITERIA FOR DETERE\IIR'ESG TEE NUMBER O F FACTORS RAYMOND B. CATTELL and S. VOGELMAKN Unirersity of I I a ~ a i i

ABSTRACT


In most well designed researches larger substantive factors shox a break in size from those of a trk-ial lower degree of variance, n-hich are due tn e m r and other sources. The only short test for the ?:rrn?ber of sxlck factors that has been repeatedly supported by plasmode results is the scree. The present experiment extends this empirical testing of the scree by :ak:ng 15 plasrnodes, which vary in (1) number of variabies, (2) of factors, 1 3 ) ifepree of obliquity. from orthogonal cases on. ( 4 ) presence of sin~plestr~zcture, ( 6 ) presence of error, extend~ngto factoring random numbers, and ( 6 ) size of comrnunality. The results support the scree and s h o ~ it superior tc the Kaiser-Guttnnan unity root criterion. Apart from the scree itself, a brief examination is made of the reliability of its use in the handa of trained and untrained judges. k. second evaluation of the i ~ t z i n s i cvalidity of t h e scree is made by examining internal consisteneg of item and parcel factoring, of differect sampling of variables and of change of population. The resu!r agarr, supports its capacity to locate the correct number of factors within narrow limits.

THEPOSTTION OF THE SCREE A X 5 KG A n i o s ~~ ~ E T R O FOR DS S)ECIDIN@ THE N'L~MBER OF FACTORS A factor analytic research can be ruined at each o f more than half a dozen different stages (Cattell, 1058; Gorsneh, 1974: Vaughan, 1994), but the two points at which a 11-ro11g decision produces misinformation most dangerous for subsequent thenry are an arbitrary, inadequate rotation or \%-hena ~ v r o n gestimate is made of the number of factors. Our concezn here is atvith the latter. Since the amount w r i t t e n on the number of factors problem is enormotls, no review n-ill be attempted beyond referral to standard texts (Bnrt> 1940; Cnitell, 1952; Child, 1970; Ii'l-uchter, 1967; Gorsuch, 1974: Harman, 1964; LawIis & Chatfield, 1974; Laviey &- blaxt~efi,1963: IlIuiaik, 19'52; Rao, 1955;Thurstone, 1941). O:rr position is in accord w ~ t h Gorsuchk essential distinction between psychometric afid statistical bases of decision, and our area of operation is in f l ~ eformer, Our argument is that sampling error simply alters the size arid shape sf factors m d not their number, except in r e s t ~ i c t i o ~uf~ s sample size so extreme as to give certain population factom 110 ~epresentc%ticsnin any incfividuals in the populatjkm. The psychoJULY. 1977

289

Raymond B. Catteiii and S. brogeirnann

metric basis therefore treats the sample as if i t were the population, The ratiocaie in regard to the other so:rree of error-error of measurement-is etiserissed in the next section. A l t h o ~ g hwe intend no review of the whoie field, it is appropriate to remind the reader of the tests for number of factors with which the scree stacds implicitly In comparison. Several methods hinge on starting with some estimate of the comm~znalities and abstracting factors until the r e s i h a l s go zero or negative, as in Burt's (194Ck):Guttman's Ilon7erIbo~nd(19.541 and the Minres method described in ITarrnan iI1967), xdiiclm does this without estimating co~munality.Fro21 way back we have Coombys (1941) sign test and Sokr-:l's (1959) distrjbutiorr of residuals. Contrasted Downloaded by [ECU Libraries] at 13:57 29 June 2014

with psychometric bases are t h e statistical upproaches of Knrt

C 1940). Eartiett (1937), and, for IzigI~esirefinement, the maximum Iikelihoocf rneckod of Lawley (1963) and Rao (1955) as made more prae ticable by Jbreskog (1963,2969). Flany of these methods require rnucl.1 plax~ningand eakcrrlakjor.:, even with Jii~eskag's (1969) Invent;on of t.he shorter maximum lilkelihocid aethod, and most factorists today seem to be using two very short nrethods-the scree (Cattell, 1966 ( a ) ) and the KG (Kaiser-Guttma~)root of unity test (Guttanan, $954; Kaiser, 1910). HOT&-ever, where the number of factors is r ~ u e i a lto some theory and indicstioz~ss r e rot clear, they shorrk! be suppkemented wri th Ionger p r o ~ e d u ~ e s(IV'lrhximi:.rm . likelihood method of Law Hey, 5963 and dijreskog, 1963, 1969). Both rest tk-lelr decision on the plot of sizes of suecessi\-ely extracted roots when the eoxponents program is used and unities are put in t h e diagonal. Bnt they do so in d;ffer.ent ways, the KG looking a t the absolute value and the scree at k3e tangent, etc., c.f the siope. The scree is perhaps properly seen as the most developed of a small family of rnetl~sdsbased sa the slope, of ~x~h4cZ;z the best known are those of Tucker (see Humphregs, Tucker $r. Dachler, 1990, or. Tlrnrstone, P947), Horn (1965) and Linn i 1968). From years of experie~ceusIng the joint verdict of the scree and the KG, Cattell has expressed (1973) some criticisms of the latter and encorrl;tered some problems In the former which are in\-estigated here. Historicdly it is to be ~oteci,how-eyer, that the scree has received most support of any short method. from the use of piasmodes. A plasmode is defined as a numerical example made up to fi: a rnstkzertlaticltI model-in this case the factor modei-and irac!udes aIso concrete, mechanical: models vrhich are 290

MUbT lVARIATE BEHAVIORAL RESEARCH

Raymond 8.Cattell arld S Vogelrna~n


bsunct to fit the model. Thus one can construct "hack stage" a definite number of factors, generate a correlation matrix therefrom, and test a test for the number of factors by asking someone to f a c t o ~the given corl-elation matrix as nsuaY, using the tesr. PreLimi~iaryresults with a mathematiczl plasmode (Tucker, Kcopman & Linm, 1969), with the ball problem (Cattell & Dickrrra~~, 1962), the coffee cups problem (Cattell 23 S:lli;van, 1962), and several w r i e d mathematically constructed pIasrnocies iCattell & C ; O ~ ' S L I C ~ L1963: , Cattell & Jaspars, 196'7), have give:^ strong support to the scree, justifying ike stilI more estended esnrr,ir,atio~~ here.

A problem x-hich psyc!lomet~*ictests of number of factors mxst f:;ce, and which is consjdered more f~r!ly in the origir~alwriting on the scree (Cattell, 1966 (a) ) concerns measurement error, not the excluded srtrnpiing error, -4s IIorn (196.51, Humphreq-s et al. f 1969, 1910), Linn (1968) and Cattell and Finlibeiner (In preparation) among others have shown, random nolma! deviates wiil generate correlations which, in smali sampies (say 30 t o lor?), will produce factors with appreciable (say .3 t o .5) loadings. It must be said that it is in the nature of t h e scree test to include any factor of the same general magnitude as the s:~bstanti~-e ftx-

tors, and if one, or even two, error factors reach that magnitude (with small sampies and low reliability rneasuremects) it n-ill include them irr the designated number of factors to extamact.But this error is lumped together in a commorl error factor in any good subsequent sirnpie structure rotation, and can be recogglized, if not by the simple structure pattern aIoue, by the fact that such a factor will not match with any factor in atlother experiment on the same variables. Although theoretically the curve of successively plotted roots could drop smoothly down to the titi, component (71 being the number of variables) and does so when one, for example, factors random norrnd deriate scores, it js a facr of esperiwce that alnlost dl real data gives x break in the curve, Such a break appears still more clearly after fixing commrrnalities arid rotating to simple structure. One then finds a d e f i ~ ~ i tnumber e of larger snb-


Raymond B. Cattell and S. Vogelrnann

stantive factors separated by an appreciable order of magnitude from a probably much more numerous series of quite trividlg small common factors. But checking fact*or number by rot 8t'B O is only suppHemenlary to the present approach. What is certain is that extensive application 0f the scree test over a wide range of experimen tar data-Prom ability, personality and motivation traits, from state increment measures, learning measures, soeiaf and cultural psychoI'ogy, etc,---has regularly given plots as illustrated in Figrere I , showing 8 more or less sharp break bet~veen the downward curve af successive latent rmt beigenvalue) sizes and the completely straight scree which marks the onset of trivial factors. And where physical plasmodes of karo~vn amber sf factors are plotted, as in the Diskman ball problem (CatteI! & Dickrnan, f 962), Thurstorme's boxes (Thurstone, 1941), @aanyseggs (Coan, 1959), and the Sullivan cups of coffee (Cattell $ Srrllivam~,1962), the onset of the scree is cl.dite sharp and appears where the number of factors (as known from Lhe ulterior evidence) ends,

size

Empirical Insrance Piartnode of f O Rcrors, Bur 2 Error Factors Ad&$;

Size

h' moderace

of

.2

.I

a

2

3

4

5

C-

7

N ~ a b 6 r scf t3e Sdccessive P a c t o r s

Figure 5, Desigri 14: Or~hogonal,18 variables, 6 facbrs, simple struct~~re, small c o ~ r n u n a l i tpresence ~~, of e ~ o factor. r

91%

MULTIVARIATE BEHAVIORAL RESEARCH


Pmbers of t h e Successive Pac-tczs

F l p e 3, Design 15: Orthogonal, 12 variables, 2 factors, simple? structure, small eommunality, na e m s factor. 2. See NABS document No. 03061 for 6 pages of supplementary material. Order from ASIS;NAPS co! Microfiche Publications, P.O.Box 3513, Grend Central Stadon, New Pork, Nl' 10017. Eemit in advance for each NAPS aecessim number. Institutions and organizations may use purchase orders when ordering, however, there is a billing charge for this service. Make cheeks payable to "Microfiche Publications." Photocopies are $5.00. Microfiche are $3.00 each, Outside the United States anc': Canada, postage is $3.00 for a photocopy or $1.00 for a fiche.


Rsymond 13, Cattell and S. VogeEmenn

were warned that a scree might not be possible a t ail if there w7ere eases with as many fact0r.s as variables. All subjects mere shown three actual scree plots (not incltaded heye) to illust~*ate the above points. The resrrlts are shown in Table 2. The true number has been Iocakd infallibly by aI1 judges In plasmadea 1, 7, g9,14 md 15 and on 3 as an average of judges to the nearest digit. St is missed by one or less in piasmodes 2, 4, 5, 6, 10, 11, 12 and 13. Only in one case-pIasmode 8 with 20 obliqrxe simphe strarcture factors put into 20 variables--has it failed by more than one factor. Here 12.42 is the average estimate, and the range of judgments is from 6 to 20. Five of the 12 judges here, however, correctly diagnosed 20, Tt must be pointed out In this odd case that in w-riting a matrix of twenty factors for twenty variables there is considerable chance of accidentally writing out one factor that is a linear combination sf several others. We checked orr this by demonstrating that the matrix was invertable and therefore not such a singular product. However, the first was n e w singular, so we made a second, which also proved near singular. This problem did not arise in the seconef 20 factor plasmode, No. 9, made from actual scores as random normal deviates, and it is noteworthy that aTI 12 judges declared no scree was possible and judged it 20 factor, The anomalous screes in pIasrnade 8 map therefore express tendencies of the matrix to approximate to lower rank. Ko firm conclusion can be Crawn as to the source of the 'boff-by-one9kccres. In two cases the existence of an error common factor has resulted in a partial or full digit overestimate, which is consistent with oilr earlier experience that a Iarge enough common error factor will be irnclrrded by the scree as a factor worthy of extraction; but irr plasmode 14 a11 judges gave 5 factors despite the extra error factor, For the rest, an overestimate by one occurred in ( 2 ) an orthogonal case, 20 a(', 10 F, large d2" and (61, an oblique, 40 V, 8 F, large h%case. An underestimate by one occurred in (4) a a-anctom deviates example, 20 V, 20 F, small M ; (5) orihogonaf, 20 V, I1 F, large h2, and with (13) oblique, 20 V, 12 F', large h2. Thirs no lead to an association of ur-rde~ or over estimates with ostlzogonaiiky, size sf communality, etc., is offered, unless i t be of error in either direction with higher eommurrality. It remains to compare the scree with the KG and since we &re interested both in general accuracy about the true value md 314

MUtTlVARihTE BEHAViORAL RESEARCH


Raymond 5,Cattell and S. Vogeimanr?

JULY, 1977

Raymond B, Cattelr and S, Vogelmann

any systematic bias, Table 3 gives mean deviation both with and without sign.


EVALEATION OF RELATIVEACCPJRACY QF SCREEAND KG TESTS The mean results far Table 8 are ealcukted both for all pIasmodes and without the "'trick" cases, (41, (8) and (91, with the ~~nusual feature of having as many factors as variables, on the assumption that the latter (withsrrt), better represents ;swhat could typically be enee~~ntered in factor andytic research, On this evidence [ ( I ) and (411 the KG has a definite biass,which the scree does not, to trmderestirnate the number of h e t o r s . In tezms of aecuracy, the scree can be ex~aluatedby the mean of all judges, thus carnpa~fnga single outcome for each plasmode against the single KG vdue. But accepting the counter criticism of the KG exponents, that the weakness of the scree Is the variation among judges, we have calc~rIatedit also with variation over judges. Note that this variation is not that in Table 2, &out the mean of judges, but is a b u t the true 2valt6e. By either evaluation the scree turns out to be more accurcite than the KG. For the mean judge it ia better by a Teble 8 Accuracy-of Scree andKG Teste --- - -True Dev. Va!ae - -- KG - -Scree - KG-

----

--

Plasmode - --

8 8 9 9

8 11 f 0.2 20

TO

S

I0

-8

8

7

0

-I

F

8

8

SO 20 10

8 6 8

8 i2.4

i. 2

8 20

3 4

10 -g. E 20

5

G 9

In 11 12 18 24 3%

2

+

E

2 12 5 9 E 2

7 2 2 8 4 5

0

-2

0.

-I4 -12

-a

0 0 -4

-1 3 -28 .86 I .08

2 -3.27 3.67

-

-

Dev. Scree - -O I

-1 -II

20 Ii 25 2.E 11 5

AEI Plasmodes Mean Bias: Mean Error f i) Mean Judge Mean Error ((2 All Judges Without (81, (81,& ( 9 ) -.80 Mean Bias 1.20 Mean Error ( 1 ) Mean Error (2) ----- - -.- - -- ---E = random loading common error factor.

316

-

p p

.21 .35 .65 -

.2

0 0 1 0 -9.6 0

I .b .5 -2

0 6 (1)

(2)

@b

(4) (6)

(6)

-

MUbTIVARkATE BEHAVIORAL RESEARCH


Raymond B. Cattell and S. V~gelmarrn

factor sf 4 and for any judge, even omitting the trick cases, by a factclr of more than 2. CatteII (1973) has criticized the KG on the thearetical ground that since rotation evens up the variance of the set of factors taken, the last unrotated factor taken according to the alpha criterion is not reaHy the last meeting that requirement, The present e m p i r i d evidence supports this criticism that KG has bias. The defects of the KG in genera1 experience, however, are not only that i t underestimates with compsratiuely few variablessay up to 40, but that with many, say 100 to 300 as in questionnai~e item factoring-it grossEy over-estimates. 'It can be noted that with the three 40 variable plasmodes here [Xos. ( I ) , (61, and i'i)]it underestimates only by .33, whereas beIo~f-that number it does so by roughly 4. It underestimates in Fig. 8 by 8 or 9; and in the roughly 200 item study in Fig. 6, i t overestimates by 22.

Evidence on the scree has so far been from plasmodes. A critic could raise the doubt that such building of a known number of factors into a correlation matrix, even with error, still may not entirely reproduce the situation in natural data. We planned, therefore, to include studies on r e d data. Howeyer, since this closes the door to independent knowledge of the actual number of factors, a different methodology and criterion must. now be introduced. The criterion now becomes internal consistency as both number of factors indicated, among stladies which logica!ly should yield the same number of factors. Three experimental approaches recommend themselves, as foliows: I ) To factor the very same data twice, once using items as variables, once using the decidedly smaller number of variables offered by positing homogeneous pareels of several items each. 'She rationale and methodology of this approach has been set outby Burdsa2 and Catteff f 19'74) and Cattell f 19'74) ; the demonstration that the number of commo~l factors from the two should be the same (with readily obtainable conditions and avoidance of certain limits) has been giren by Cattell (1993). 2) To factor tw.0 (odd-even) samples from a comprehensive population, or, as a greater challenge, from two poprrlations. We feel that though two samples from the same person population is theoretically satisfactory it is a relatively trivial unchauenging case, in that almost any test of factor number might, due to the high overail similarity, JULY, 1977

3x9


Raymond B. Cattell and S. Vogelrnann

give similar results. Accordingly, we raised the discrimination power by taking two pop~lationsin ~vhichthe form of the popfilchticlns might be expected t o be different b:lt the number to be the same. Indeed we took the same split persona2ity scales given to two groups differing in wlture and language. To give background to the crossculitnral example it shauId be said that the 16 PF has been given ta sampIes on same twenty foreign countries (Cattell, Eber & Tatsuoka, 1990, p. 250) and translated into now more tlzan ten languages, and the results of the various investigations agree that the nature sf the rotated factors and their number is essentisIly the same. The translation proeedrzre has typically been to rrse a check by back translation. Of course, this is only a note on the background and the present example stands on its own feet. As a trial by the first approach, Fig. 6 sho.c%rsthe scree for the general personality pool of questionnaire items (Cattell, 1973) for normal behavior. (The ecree omits one of its several conditions l a e r ~ a r n e l r ythe angle of the scree ks the rest-but fits the others.)

Number al factors

Figure 6..Same Data Factored by Parcels and by Items. Data consists of 16 PF pool of items (187) on uadergraduate group of 250 QBardsaI $E Vanghan, 1974).

31%

MULTWARIATE BEHPtVBQFBAL RESEARCH

Raymond B. Cattell and S. Vogeimant!

The number of fact.ors indicated Iies in the range normaily found, -20 to: -23.


That the parcels should yield sfightly rnor+22 against 20 irm items -is in accord wit11 the theory of snmrnation of weak common factors in parcel scores (Cattell, 197'4; Burdsal S: Cattell, 197'4). But the etatistica1 theory is that items and parcels should give the same number (Cattell, 1978) and within a small percent difference they do, by the scree. By the ICG, on the other hand, the discrepancy is extreme, indicating roughly a dozen by parcels anit over 40 by item plots.

The results of the second method are shown i n Figure 9. Atthough Sells, Demaree and Will (1970) enriched the pool u p to 300 items with Guilford scale items, the 16 PF arid the Guilford have beet1 shown to occupy essentially the same space (Cattell & Gibbons, 1968). The researches in Fig, 6 and Fig. 7 can therefore be considered to take equivalent marker samples from the total ::.or~~:lat personality behavior sphere of questionnaire items, The agreement, despite Sells' population being military rather thazl student, is close (22 versus 20 and 22).

The third method of testing a test for the number of factors is shown in Figure 8. IIere we hare taken cultures more divergent than from two European countries by comparing Japan and the U.S.A. The problem of exact translation of any one item is here avoided by taking scores for 32 short scales, in ~vhichitem rariarions are likely to csncel. IIere we found agreement at f 6 factcrrs. The change in angle at the 16th factor is slight, b r ~ thas been checked by being brought out more cIearlg in plots enlarging the vertical s a l e . The KG indicates 7 in the Japanese an2 9 i~ the American.

SUM blXRY

(1) Psychologists seem accustomed, if not addicted, to shor-f, easily applicable tests sf number of factors in factor experiments. Since a mistake on number of factors c2n be serious for rest:lting theory, it is urged that supplementary tests be used in crucial situations and that the short tests-mainly the scree and the KaiserGrrttman unit root criterion-be examined as to validity. JULY, I997

319

t

Seven other eigcnvalues


.mge up t n 18.2

necessary

i

Successive principd asis factors

Figure 7. Different Large Sample of Items from Same Large Pod of Items. The 800 items here are from essentially the same pool as that from which the data in Pig. 6 were taken, and are factored on large ( N = 8080) population sample (Sells, Demaree, and Will, 1970). The original research took out only 11 factors, resulting in the mixture of primary and secondary shown elsewhere &@attell,L873) to result from undedactoring, The Kaiser-Guttman indicates over 66 f rectors.

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(2) Tests are statistical and psychometric, the former kind requiring more laborp sls in the maximum likelihood method. As

trsr factor

--9

.

Rg-cre 8. Trandsted I t e m Scales Factored in U.S, and Japa:lese Papdatlon Samples. 32 scales (Forms A md B of the 16 P.F.) given to 117' American (x poizits) and 300 Japanese (* points) undergraduates. This figure may need enlargement of vertical scale to see the break clearly at 15 to 16. JULY. 1977

321

Raymond B. Cat4eBI and %. Vogelmann

f a r as number of factors is concerned, provided one is w o r k i ~ g with an N > 100 m d preferably N > 200, a s a factor analysis eomrnonIp needs, the inference from sample to pop:rIation is not important. A psychometric approach fixing the number in the sample is then appropriate. The scree and KG do this and provide a basis of procedure in which the commr:nalitr"es are then decided by the number of factors. (3) Four methods of evaluating empirically a number of factors test are suggested: (i) by plasmodes built with a known number, (ii) by agreement of results for items and parcels, (iii) by agreement of results from different item samples, and (iv) by Downloaded by [ECU Libraries] at 13:57 29 June 2014

agreement of results from different poptrlatiorr samples.

(4) The first is the only absolutely positive method and it is tried here after constructing 15 pIasmcldes mimicking typical ( a ) odl-thogonality-obliquitjr, pspchoiogfcal datz and s ~ ~ i e across ci (b) number of 1-ariables-8 throrrgh 40, (c) number of factors-2 through 20, (ci) presence and absence of simple structure, (e) mean size of eommwnality, and &f)preser,ce or absence of error, the Iatter being i n t r ~ d e deither as a formIess, low variance emor factor or by random deviates. ( 5 ) Both the scree and the KG kit the target several times, but the scree is definitely superior: (a) in having negligible bias, whereas the KG underestimateci appreciabry in one range and uve~esi;imat~ect egstematlcallg in another and (b) in having greater acccracy, i.e,, smaller deviations around the true values. Since an argument against the scree has been that: judges differ, their variability has been examria& and shown to be small. Even with judge variability the scree is superior in accuracy to the KG. (6) The remaining three metl~odscan be tried with real data, though material suitable le rare. However, a substantiai example of same data factored by items and parcels gives close numbers (20 anci 22) by the scree, a s also does factoring of different items sub-pools from the same domain (22 versus 20 a%d 22) and the same translated items on two ceal turally different popuIations (15 and 1 6 ) using scales as \-zriablex. ( 9 ) On these examples, aiso, the scree is more accurate and consietent,, the KG giving too few factors with fewer ~ariabhes and decidecliy too many with malap. The practice of S:rilding the KG into fac:ur eomprzting programs, because of its extreme cheapness, and proceeding with. no break for the investigator to apply other checks for rirrmlber of factors (Tearing the novice even un-


Raymond B. Cattell and S Vogelmann

aware of the basis of decision) is quite indefensible, Almost eq~ially undesirable is the use of the scree by individraals who have not apgliect it before, and have no explicit training in its use across a series of examples. But, granted some experienced skill, the scree appears to be highly dependable. Even with briefly trained judges it has no systematic bias and misses (in this sample of 15 studies) the true number of factors by plus or minus one, on an arerage, or by about half a factor when deIiberately "tricky" examples are not inclrtded. It is suggested that in crucia1 cases it be supplemented by s statistical test, in the form of Sijreskog's maximum Xlkellhood, which, incidentally, has bee^ foutld by us in not formally reported work to agree well w-ith the scree on rnicfdle and larger size samples. Though it could be said in a purely Iogieat sense that s statistical sample dependent test, such as JBreskog's is il.x*eleant to the present psychometric approach, yet, in practice, with certain assumptions, it is informative to cross-cheek their verdicts.

1, Crawford, C. B. Analytic methods of rotation in the deteminatinr. of the number of factors. Unpublished doetord dissertation, M c G i l U~iversity, Toronto. 1966, 2, Johnson, P. Empiricd statistical sampling distributions of some indices of the number of factors to consider in eqloratory factor analysis, M A . Thesis. Simon Fraser University, British Columbia, 1970.

Albert, A. A, The nicimum rank of a correlation matrix. Proceedings Xafuriat A m ; s m i c Science, 1944, SO, 144-148. Bartiett, M. S. The statistical conception of mental factors, B14'tkh Journal of Psychology, 1937, &8, 97-104. Burdsal, C . k., and Cattefl, R. E. The radial parcel doubIe factoring method of item factor analysis. M~~ltivariate Behve'orak Research, 1974, 44, 140164. Burdsal, C . .4.,and Vaughan, D. M. A contrast of the personality structure of college students found in the questionnaire medium by items compared to parcels. Journal of Genetic Psgchology, 19'74,125, 219-224, B w t , C. L. Factors of the mind. Eondon: University of London Press, 1940. Cattell, R. B. Factor amlgsis, New Pork: Harper, 1952, Greenwood, 1970. Cattell, R, B. Extr-acting the correct number of factors in factor analysis. 1958, f 8,791-836. Educational and PsychoEogical ICfeasz~~enaent, Catcell, R. B. Theory of situational instrument second order and refractioll h c t o r s En personality struckre research. Ps~tchologicalBzclktin, 1961, 58, 160-174. Catten, R. B. The scree test for the number of factors, Multivariate Behaviorat Research, 1966, I , 245-276. (a f JULY, 1973

323

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CatteII, It. B. Mnndbook of ~VFultivariateExperimental Psgchotogy. Chicago: Rand fiEc,h;aIIy,1966, Chapter 6. (b) Cnttell. R. B. Fe?-8anaEfy and m o d by gue~tionnaire.Sas Frit7cisco: JosseyBass. 1973, Catteil, B. B. Eadia! parcel factoring-vs-item factoring in defining personality structwe ir, q~estionnaires.Australaa?z Jaurnai of Psyckoloyy, 1974. 26, 103-219. Cattell, R. B. The scientific use of factor ana.lys&. New Yark: Ple~lum (inn press, 1977). Ccttdl. R. B., & Digman, J. M. A theory of the structure of perturbations in obsemer ratings and questionnaire data in personality research. Bekaz~loralScierccu, 1964: 9 , 341-368. Cattel. R. B.; & D i e b a n , K. A d y n m i c model of physical i~~flpxences demonst~.atingthe necessity of ob:ique simple struct~rre,Fsgckokoyical BuE!etin,


1962, 5 9 , 385-400.

Cattell, ReB., Eber, KGW., & Tatsuoka, M. Ffaazdbook l o r the 16 PP. Iliinois: Chsmpaign I.P.A.T. 1970. Cattell, R, B., & Finkbeiner, C. The srgnificance of width of hyperplsne. ( rn preparation). Cattell, K. B., B Gibbocs, B, D. Persoilality f a e t o ~strwture of the combined GuiIford and Gattell personality questionnairesL J o u r m l of PersomBtg a ? ~ dSoctal Ps~jcXology,1968, 3, IN-120. Cattell, R. B.. & Gorsuch, B. E. The uniqueness and sigr-ificance of simple strucSure denonstrated by cortrasting organic "natural st~.r~cture'bnntl "random stmcture" data. Psychometrika, 1963, 28, 55-67. Cattell. B. B.. & Jaspars, J. A general ptasmde (KO. 30-10-5-2) for factor asalytic exercises and research. ikIultiwa?%ateBehavioral Resea~chMonogWphs. 1967, 67-3. 1-212. Catteil, R. B., & Sullivanl W. The scientific nature of factors: A demonstration by cups of coffee. Behavioral S&e?tce, 1962, 7 , 184-193. GI.-iEd, D. The esserctiak of factor amlusis. Londo~i:EfoEt, Binehart & Winston, 1970. Coarm, it. W. A comparisocrl: of oblique and orthogond factor solutions on eggs. Journal of Ezperime.ntaE Educat.ion, 1959, L?, 151-166. Goornbs. C. 11, A critericrr, for significant common factor variance. Pa1~chowtetrika, 1941, 6, 267-232, F r i l c h r , B. Introdaetioaz to factor uwalysia. Nadras: Affiliated East-West Private, Ltd., 1967. GorsucE: R. L. Factor analysis. Philadelphia: S~unders,3014, G ~ t t m a L. ~ , Some necesssiy conditions for C O ~ ~ factor P S andysis. P s ~ c h o metrika. 1954.18, 149-161. I I a ~ w ~ a El. n , El. Modem f a c t a ~analusis. @?&ago: University of t%cago Press, 1961. How, J. L. A rationale for the numbe+ of factors in factor analysis. Psychomctrika, 1965, s:,, 179-185. EIumphreys, t.E., & Ilger,, D, E. Note ox the criterion for the n~ernber of common factors, Educational and Psgchologdcal Measurement, 3969, 23, 57:-573. Eun~phreys,2. G., Tncker, L. R., & Dachler, P. Evduatiing the Importance of factors in acg given order of factoring. Multivariate Behavioral Research, 19'70, 5, 209. 36reskep, K. G. Statistieat estimation in f ~ c t o roinal.gsk. Uppsala: A1mqest & TGkselk. 1963. JiEreskog, K. G. A general approach to confirmatory maxinmnt likelihood factor sr,d?,ysCs. Psgekiantetdca, 1969, 34, 183-292. Kaiser, H. F. A second generation Little Jiffy. Paychowetrika, 1990,S%,401. Laaiey, D. K., 6, Maxurell, A. E. Factor a ~ ~ a l y sa&s a statisticat method, London: Buttemoath, 1963. 324

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Lawlis, G . F., 6: Chatfield, D. Multivariate upproaches for the behaviors/ sciemcee. Lubbock,Texas: Texas T e c h ~ ~ i cPress, d 1974. Linn, R, E. A Monte Carlo approach to the number of factors problem. Peychometm'koe, 1968,S#,37-71, Muiaik, A. A. Ths foundations of factor analysis. New Yo&: h1cGraw RiII, 19'12. Rao, C. B. Estimation and testa af significance in factor at~alysis.Pa~gcharnsthka, 1955,RO,93-LIZ. Sells, S. B., Demaree, R, B., t Will, D, P. Dimemions of personality: I. Conjoint factor structure of Guilford and Cattell trait markers. .Flult~variats Behavioral Resemch, 197Q,.$, 391-422. Sokd, R. R. A comparison of fire tests for completeness of f a c t o ~extraction. ITransactions Kansas Academy o f Science, 3959,62,141-152, Thwstone, L. L. Mzcltiple fcrctor atuzl2/sis. Chicago: University of Chicago Press, 1944. Tucker, L. B., Koopman, R. F., & Linn: R. L. Evaluation of factor analytic research procedures by means of simulated correlation matrices. PsychamsMka, 1969,94,421. Vaughan, D. On the techicaf standards of ten represectative factor analyses purporting to decide personality theories. Journal of Behavioral Scimee, 1974, $, 1-10.

JULY, 1979

The Scree Test and the Number of Factors: a Dynamic Graphics Approach.

Cattell's Scree Test In Relation To Bartlett's Chi-Square Test And Other Observations On The Number Of Factors Problem.

Erratum: Determining the Magnetic Properties of 1 kg Mass Standards.

Factors Influencing Four Rules For Determining The Number Of Components To Retain.

Application of a framework for determining number of drugs.

Determining standard criteria for tubeless PCNL.

External validation of the number of risk factors score in a palliative care outpatient clinic at a comprehensive cancer center.

Effects of a lifestyle modification programme to reduce the number of risk factors for metabolic syndrome: a randomised controlled trial.

Determining the number of intraparotid lymph nodes: postmortem examination.

Determining the optimal number of lymph nodes harvested during esophagectomy.

The number of mechanically ventilated ICU patients meeting communication criteria.

Update on the parallel analysis criterion for determining the number of principal components.

Expanded criteria for carcinoid liver debulking: Maintaining survival and increasing the number of eligible patients.

The same number of optimized parameters scheme for determining intermolecular interaction energies.

Determining the optimal number of independent components for reproducible transcriptomic data analysis.

A Needs Assessment of the Number of Comprehensive Addiction Care Physicians Required in a Canadian Setting.

Balloon angioplasty for native coarctation of the aorta in children and adults: factors determining the outcome.

Predictive models and prognostic factors for upper tract urothelial carcinoma: a comprehensive review of the literature.

The role of number of tasks in determining the relevance of information.

[Intramedullary tumors--surgical factors determining the outcome].

Support for a comprehensive background check requirement and expanded denial criteria for firearm transfers: findings from the firearms licensee survey.

Factors determining the patients' care intensity for surgeons and surgical nurses: a conjoint analysis.

Sherloc: a comprehensive refinement of the ACMG-AMP variant classification criteria.

Method for determining the extent and copy number of overlapping and nonoverlapping segments of integrated viral genomes.