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Procedures for ConstrainedConfirmatory FactorTransformationA. Ralph HakstianPublished online: 10 Jun 2010.

To cite this article: A. Ralph Hakstian (1975) Procedures for Constrained ConfirmatoryFactor Transformation, Multivariate Behavioral Research, 10:2, 245-253, DOI:10.1207/s15327906mbr1002_9

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PROCEDURES FOR @-COXSTRAIXED COXFIRMATORY FACTOR TRXXSFOR3fA.TIOK

A, RALPE HAKSTI-%X University of BrLtish Coiumbia

Ontlined is a model fo r transformation of one factor matrix t o congm- e3ce wizh a second or target mntrix in which the correiations arllclrlg the trscsformed factors a r e constrained, to certain pre-specli'ied ~ ~ I i l e s . Proce- d u r e ~ a r e developed for :n:plernenting the model. and these procedures are ilt:~straied v:ith esarn$e fac tor soIurio::s. App::c:rt:on of the p:.ocvlures 1s

briefly discussed.

Confirmatory factor analysis, or the problem of tr.al:sfis~r~~ing a factor pattern to f i t a s closely as possible :a "target" or hypo- thesis matrix, 11;s been extensil-ely st:rdied. FOP the or'tilogofial factor model, the ge2:eral orthogcnrrl Proerirstes proceciure, n ~ o s t thoroughly outlined by Schijuema~n Cli966), is 15-ideiy used, and extensions of the basic orthogonal leas-c sqnares fitting formzlat.ion have recently appeared (Brawne, 197%; PIaks';ian, 1973). The oblique eoafirrasatory application has see11 progressive refinement siliee Moaier's ( 1939) approximate procedure, with B~owne's (1969) exact reference structure solution, and, most rece~ltly, the exact primary pattern fitting fullctiol~s of Erowne and Ti~istof (1969) and Gruraeus (1970). For more detxil on confirmatory fae- tor a~ralysis techniques, the render is referred to IiIakstian ( 19'72).

Althongh, in factor analysis generally and the co::firmator.- factor analysis problem specifically, we traditionally dlstinguisl: between orthogonal an~l oblique t r~~nsforn~a t ion , anotl~er. dichotomy might. exist behireen a-collstraived and ~$-? ; r ;c~ i : s f r~ i~ ie~ i l confirma- tory transformation, with the basis for the 6isri:ction being the existence or non-existence of constraints on the off-cliagonnl ele- ments of the matrix @, of primary-factor ixterco~relations. Sea; in this Iighe, the obiigua confirmatory procedures noted above are @-unconstra.n"ned, in that any d e g e e of obliquity short of factor collapse is permitted. The orttiogo~i.al fitting proced?,;,res, conyrrse- iy, are @-constrainedl in that the factor corl-elations 2iz.e specified (to be zero) in advance.

It may be, ho\~-ever, that the inyestigator xi-s-ishes to coxstraic the primary-factor correlations, not renifol.rn!q- to zero, but rather to various non-zero 1-alues. These vzlses may represezlt I > ( > r i ~ ~ c l ~ 011

the degree of factor obliquity desired, ol* :key may represent Integral part of the structural hypothesis beir:g assessed. 111 any

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ease, current procrv,stean procedrures cFa not irsclrrde this nsn- orthogonal2 albeit +-constrained, confirmatory applieatio:~, We thus are concerned, in the rest of this paper, with establishing the @-const?*ai.ri ed eonfirmatory .m o d d acd procedures f o r its imple- mentation. The orthogonal ca~f imafo rg appZicatian is, of course, a special case of this more general model.

%Ye begin by writing the general factor transformatiorr equa- tion :

where P, of order p ~~ariables s q factors, is the primary-factor pattern matrix, A, p x g, is an initial, orthoiona~ factor. loading matrix, and G, q x q, is a ~??akrix of unit-Ieng51 colurr,,ns of direc- tion c s s i~~es mspping the factors represerrted by A into tlzose repre- sented by P. The matrix G, q x q , containing the primar-y-factor in tereorreiatisns, is n function of G, as

Eqvrazisn [2] reflects tkre well-known constraint, in most. factor analytic applications, to unit-length factors. Fcrther col~strafning G so that @ = 1 defines or~hogonai transformatio~z.

If we write the canonical decomposition of @ as

it can be seen ths t an infinite number af matrices, 6, exist satisfy ing Equation [21, since G may be twitten

246 MULTEVARIATE BEHAVIORAL RESEARCH

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where V and L, both g x q, are as in Ecirration [ 3 ] , and T, q x q, is any ~rth~ano~srznl CTT = TI" = 1) matrix. Since (G') -1 in Equation El] may thus be written TL-'V', by letting L-lT' be written t', we can rewrite Equatioxl [I] as

I t should be ciesr that for an orthogonal salixtiorz, @ = I - S.

We have, in Equation e51, the basis for the @-constrained con- firmatory model. If we let B, p x q, be the target or hypothesis matrix, and t', a factoring of the specified factor correlation matrix (as above), our model is

where the uaskno~x~~r matrix is T, and t- must be presem as a post- multiplier to restrict the factor corre1atr"ons to those specified, Our eriteriou is maximal column-wise cozgrzie?rce bet~veen ATI.' and B, expressed, overall, as the scalar p~ocluct fuzction

where t.r stands for trace, and we have the constraints

SO that G , as specified in Equation [4J1 \%-ill satisfy Equation [2]. P t ill be noted that the criterion in Equation [7 ] is maximat

pairsvise factor congme.rrce, as opposed to the residrzal least squares criterion often employed in the cie~elopmenk of confirmatory tech-

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niqrres, These eri!er.ia Iezlcl to identical so lu t io~~s in the orthogo?ral special case of Cquntisn [GI-that is, tf-it11 Z- -- I. Cliff's (3966) orthogonal confirmatory criterion is ss in Eqzatisz [?I (with kJ set to I), and kke soI:,~tio3 is procedrrrall!; identical to Schiinemann's (5966), the Istter based on tke residua!: least sqrzares criterion.

Thfs identity does cot extend. kioxs-erer, to the more general model embodied ir: Equation [ G I , where the scakir product Sune-

ti02 i3 Eq~zatim~ [7 ] does not, in this apy,l:'eation, csrrespoud to the residual !east sq:.lnres dietar3ce fu:~ctinn.' We can, ~-io~etP~',eiess, formafate the p ~ c s c ~ t p~obIem in terms &at ~eestahlish the e~rxi- valenee berv-een the scalar- procluct and distawe criteria. If we re- write thc model in Equation [6] ;IS

where E. p x q, is an "error" matrix, subject, of course, to zhe csastraiats in Equation IS], sri'e can aker this formulation through poslmultiplicatio~a of Equatfm [9] by U- I, yielding

where B" = BY-', and E* = KG-:, For the model represented by Equation [IbSa], the sca:a~ product fu::ction,

and the weighted distance Snnctio:~~

I. The mthr :s indebted $0 an anonymas re~<ew.er for pointing ont the So:lowi?;.,g tlterr,at:ve formri!atior: of the +-conetrained cont'lrmatory problem.

248 MULTIVAR!A?E BE~LAv!QrS?tL RESEARCH

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A. Ralph H a k s t ~ a ~

El21 g(T) " t?,[E"'EKf = t i LE'EYIJ = tf[I;;LT- ;(t-<) - :E1j = t r [ E ~ E ' f - >>ric l i t i i~un~,

Iead to identical solution, but a soIr,t;on sligl-.tij* ciiffcre~~t from that implied by Equation [TI. The weighted distsnce fr~nction in I3q::a- tion [I21 is that employed by Brov;.;ne and Krl'stof (1969) in their obliclrre primary-pattern procedure, 2nd rep~esents E~II dternatlve criterion for the @-eonstrained esnf i~na tory application to ",at for w-,F-l?ieh solution is dm-eloped 4n the next sectio~.

Maxiw~iai,ng tktt Criterion Frbnct ;OH

Returning to the model in E q ~ ~ t i o n [GI c.nd the fu:~ctioa to be maximized in Eq~~sltiolr ['i], \ye TT-rite the fu:?ction to be cliffen.- ectiated, incorporatinrg the esl:st~ai~:.ts espressed i ! ~ Eqnat;nml [8j, as

el31 f* (T) = tr[U'T'A'lff - 57 [AQTT - 913,

where A, q x q, is a nlatrix of (unknown) Eagrnnge mu1tipkiel.s. Taking the partial derivative of f* (T) in Equation f13f, ~ 9 t h re- speet to (the elements s f ) T (see SehiSnemann, 1965) and setti~ag chis matrix t o the null matrix, n-e hzye

If we let the symmetric mztrix .'I -t A' = Q , Eq:iatim [ i d ] may be rewritten

AIECr = TQ,

[8] j , we have T'A'BU' = Q,

Zf we write the product of the three k~ieri-n rnbtrices, -l'BGf, as S, q x q, we have

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T'S - $, so that, since Q is syarnetric, the rsnlmaivrr orthonarrnal tmnsfor- rnatio:~, TI must render the product T'S symmetric. SOW^, it is ~-elH k~lov-n that a non-symmetric matrix such as S ccn be represented by its "'basic structure" (Horst, 1963) as

where, in this ease, the left and right orthonormal matrices, X and Y, respectively, are q x q, and the middle cliagoca! matrix, D, is also q x g, with ail diagonal elements pcssiti\-e if 5 is of full rank (9-hi& it generally ~ - 2 l be). The orthonarmaI matrices X and 97 are obtnined by the eanorriea1 decomposi&ions

!Xu 5% = YD2B; and SS" X5@s2X'.

Taking for the matrix T, the orthmormaI: prodnet XY' thus satis- flea the symmetry ~equirement in E q ~ a t i s 3 [16Q. TT~~II T deter- mined, P-the primary pattern maxima~Iy congruent to B--is given by Equation [ 5 j 9 so that (GI-' - TU, and, as ern be seen from Ecjuaiions f2] and [3J, the exact, pre-specified values in @ are retained.

If, instead of ihe formulation for S as >YBr', 311 Eqratisns [16) and f 173, we let 5- - AfBU-l, we haw, upon sole-ing fo r X an~d T, by $:quation [18], z;.nd subsequeu tly for T = X Y f , the solu- titan SOP T when the problem is stated 4n ternls of the weighted distance f'i~rncthn deseIoped in Equatiom [LO] C~z.sugh.1 [f2], A slightly different P matrix, f rsrri that follotving from the earlier developmect resnlts from application sf Eyuat;ion [ S ] , but the pre-specified entries o f o are retained: of course, ~s before.

The following steps summarize the preceding development (in terms of Equaticx~s [6 J and [9] 1 :

6 1 ) Obtain the eano!~ical deesrnpoaitiorn of the pre-specified

250 MULTBVARiATE EEHRVIORAL RESEARCH

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factor intercorrelation matrix, (P, as G = VL2V', and establish the "constraining matrix," U, as U = L-IT'. (If orthogor~al co~zfi~?na- tory rotation is desired, @ = I - V.) It should be noted here that the formulation of a--equal to G'G in Equation 121 and to Vt2Y', in Equation 131 and just previously-is as a gramiuiz matrix, (and, in fact, a positis~e-definite matrix, to ensure against coliapse of the factor space), If the prespecified Q matrix is non-gramim, the procedures developed cannot be applied, but, in practice, this fact is not likely to present a problcm fo r the majority of @ ma- trices that ~vould be specified. 3Iatrices containing very large cor- relations (-80, say, or larger) and, particularly, a pattern of cor- relatiorls with some extremely Isrge and others near-zero or nega- tive wo~ald be those most fikeEy to be non-gradiarr. The user st.iI: knows., of course, from the decompositiorl of his pre-spwified CI

mr*trix, whether it is no11-gramian, since this conc!ition resr:lts in one or mare negative elements of L2,

(2) Construct the matrix $r by S = A'BUP, where A and B are, respectively, the urtrotated and Brget matrices.

(3) Obtain canonical decompositions of 8'5 and 55': S'S .=

YD2YY'; SS' = XD2X'. (4) Construct the matrix X'SY, If any of the cjiagonal ere-

nments of this diagonal matrix is ncgntiz .~, the corresponding co1- umn of S rnrrst be reflected, yielding SL, to ensure a maximum for P(T) .

( 5 ) T- X*YF. (6) I' - ATE.

ILLVSTRATIVE EUIIPLEP AND CONCLCSIOYS

Tke procedures described iil the previous seerlor1 .ityere applied to Molzinger anck Marman's Xi?[p P~~ichcilngica! S7cr?.iables example (Harman, 1967). For tinis set of nine variables, three multiple- group factors were given, and the Gram-Schmidt orthonormalized factors were used in the present analyses. The author orthsgonaily rotated these factors to csnonical (or principal-axes) f o m , a pro- cedure which although actuzllg rrnnecessary, ~enderecl the starting factor pattern similar to nnrotated matrices encountered in practice.

Results of application of the preceding technique are present- ed in Table I, 111 this table, the principal-axes starting matrix is

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A Ratph klakstian

designated A, and the target matrix. B. The a-awcronsE~ained pri- mary pattern-presented for cornpaz*atire purpcsses-was obtairretl by using the approximate least squares procedure first derelopetl by &!osier (19391. The :D-rortstraiwed algcrithm was employetl three times, yieldiag '%-Coristrainecl Pattern 1,'' which is, as ;I

glance st; the anccs~lipunying factor corre:azisas ~ - i I 1 reveal, an orthcsgonal Procreretes soIntion (41 - I = U\, and tv-o additional (a-constrzi~ed pattea-ns. in which the factor corre!aticms were re- stricted PO the ~ a i i i ~ s g i ~ e n below the corresponding pattes:~ matrix,

Tabie 1 Unrotated and Target Matrices, and a-k'acsnstrained and Constsefrred Confirmatory Pr;mnrg Pct terns fo r the Nine PsychologicaI Variables

- (Decimal pcrints omitted)

- - - -- - ". Unrotated i arget cp-Urieorrstraineti

- Kstrix 4h) Mstrlx (B) Pa t t e m Variable k 11 11i S I1 111 I 11 11r-

111

Factor Correlatiuns 100 00 I00 00 00 ZOO

Fae: o r Cnrse:azions 100 25 180

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It can be seen, from Table 1. that srmccessi.r.ely better simple structures resulted from allo~s-ing the factor correlations to in- crease from the strict orthogonalit;v embodied in "@-Constrainecl Pattern I." The actual correlation values specified in "@-Con- strained Pattern 3'bere close to those obfained in the oblique multiple-group solution presented ill Harman (1967).

The procedures developed in this paper may be useful both in confirmatory at:d exploratory factor analytic applications. Using the notion of factor intercorrelation restriction, the investigator may more thoroughly evaluate specified oblique structural hypo- theses, I n exploratory factor analysis, so!utions obtained by "blind" transformation methods may be "polished"-towards an idealized structure sraggestect by the "blind" so';ution--at -the same time .that unwanted or excessive obliqrritg of axes is a~oided.

Browne, M, W. On oblique Prwrustes rotation, P.wgchonwtrikn, 1969, $1, 425- 132.

Bro~~me, N. W. Odhogonal rotation to a partially specified target. British JoztrnaP of Mathematical & Statistical P~yckotogy, 1972, 95, 115-120.

Browne, &f. W., L Krishf, W. On the oblique rotation of a factor rnatrlx t o a specified pattern. Pspchoraefrika, 1969,34, 237-248.

Cliff, X. Orthogonal rotation to congruence. Fsychont.etrika, 1966, 31, 33-42. Gruvaeus, G. T. A general approach to Procrustes paitern mtatioc. Pq1-

chov~etqika, 1976,35, 493-505. Kskstian, A. R. Optimizing the reso!ntion between sslient and non-sallie~t fac-

tor pattern coefficients. British J n ~ ~ m a t of hlatk~mrttical 6: Stcltisf~ccl Psychology, 1972,.25,229-265.

Hakstian, A, R. Procedures for weighting factors and variables In orthogonal confirmatory factor anaiysis. Illltlfi?*rc~-icrfe Reha.c,iorcb Resea~cC, 1973, 8, 379-390.

H a m a n , I-I. H. Modem facfor n?uzlysis. (2nd ed.) Chicago: University of Chicago Press, 1967.

Horst, P. M ~ t m k algebra f o ~ so&cil sc;enris:s. h'ex- York: Holt, Rinehzrt and Winston. 1963.

Mosipr, C. I. De~er~nicing n simple structure n-her. loadings for cer:tSir: tests are knon-n. Psyrizonzetr~ko. 1989, $, 149-162.

Schbnemann? P. H. On the formal ciifferentiation of traces and de;erminants. Research ;IIemorannEztm Xo. 27, The Psychometric Laboratory, University of Xorth Czrolina, 1965.

SckBnemann, P. H. -4 generalized clo!il!ioc of the orthogonal Procn~stes pro& lem, Psychon~efrika, 1966, S1,1-IO.

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