ENSURING POSITIVENESS OF THE SCALED DIFFERENCE CHISQUARE TEST STATISTIC ALBERT SATORRA UNIVERSITAT POMPEU FABRA UNIVERSITY OF CALIFORNIA


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1 PSYCHOMETRIKA VOL. 75, NO. 2, JUNE 200 DOI: 0.007/S Y ENSURING POSITIVENESS OF THE SCALED DIFFERENCE CHISQUARE TEST STATISTIC ALBERT SATORRA UNIVERSITAT POMPEU FABRA PETER M. BENTLER UNIVERSITY OF CALIFORNIA A scale ifference test statistic T that can be compute from stanar software of structural equation moels (SEM) by han calculations was propose in Satorra an Bentler (Psychometrika 66:507 54, 200). The statistic T is asymptotically equivalent to the scale ifference test statistic T introuce in Satorra (Innovations in Multivariate Statistical Analysis: A Festschrift for Heinz Neuecker, pp , 2000), which requires more involve computations beyon stanar output of SEM software. The test statistic T has been wiely use in practice, but in some applications it is negative ue to negativity of its associate scaling correction. Using the implicit function theorem, this note evelops an improve scaling correction leaing to a new scale ifference statistic T that avois negative chisquare values. Key wors: momentstructures, goonessoffit test, chisquare ifference test statistic, chisquare istribution, nonnormality.. Introuction Moment structure analysis is wiely use in behavioral, social, an economic stuies to analyze structural relations between variables, some of which may be latent; see, e.g., Bollen an Curran (2006), Grace (2006), Yuan an Bentler (2007), an references therein. In such analyses, it frequently happens that two neste moels M 0 an M, are compare using estimation methos that are nonoptimal (asymptotically) given the istribution of the ata; e.g., maximum likelihoo (ML) estimation is use when the ata are not multivariate normal. In those circumstances, the usual chisquare ifference test T = T 0 T, base on the separate moels gooness of fit test statistics, is not χ 2 istribute. A correction to T by a scaling factor was propose by Satorra (2000) an Satorra an Bentler (200) to improve the chisquare approximation. The latter is the focus of this paper. Satorra an Bentler s (200) correction provie a simple proceure to obtain an approximate scale chisquare statistic using han calculations on regular output of structural equation moeling (SEM) analysis; it has the rawback, however, that a positive value for the scaling correction is not assure. The present paper evelops a simple proceure by which a researcher can compute the exact scaling correction of ifference test statistic base only on output from stanar SEM programs. Research supporte by grants SEJ an PR from the Spanish Ministry of Science an Technology an by USPHS grants DA0007 an DA0070. Requests for reprints shoul be sent to Albert Satorra, Department of Economics an Business, Universitat Pompeu Fabra, Ramon Trias Fargas 2527, Barcelona, Spain The Psychometric Society 243
2 244 PSYCHOMETRIKA 2. Setup an Notation Throughout we ahere to the notation an results of Satorra an Bentler (200). Let σ an s be pimensional vectors of population an sample moments, respectively, where s tens in probability to σ as sample size n +.Let n(s σ)be asymptotically normally istribute with a finite asymptotic variance matrix Γ (p p). Consier the moel M 0 : σ = σ(θ) for the moment vector σ, where σ( ) is a twicecontinuously ifferentiable vectorvalue function of θ, a qimensional parameter vector. Consier a iscrepancy function F = F(s,σ) in the sense of Browne (984), an the estimator ˆθ base on F or on an (asymptotically equivalent) weighte least squares (WLS) analysis with weight matrix V = 2 2 F(s,σ)/ σ σ evaluate at σ = s. Let M 0 : σ = σ (δ), a(δ) = 0, an M : σ = σ (δ) be two neste moels for σ. Here, δ is a (q + m)imensional vector of parameters, an σ (.) an a(.) (an mvalue function) are twicecontinuously ifferentiable vectorvalue functions of δ, a compact subset of R q+m.our interest is in the test of a null hypothesis H 0 : a(δ) = 0 against the alternative H : a(δ) 0. For the evelopments that follow, we require the Jacobian matrices Π(p (q + m)) := σ (δ)/ δ an A(m (q + m)) := a(δ)/ δ, which we assume to be regular at the true value of δ, sayδ 0. We also assume that A is of full row rank. By using the implicit function theorem, associate to M 0 (more specifically, to the restrictions a(δ) = 0), there exists (locally in a neighborhoo of δ 0 ) a onetoone function δ = δ(θ) efine in an open an compact subset SofR q, an a θ 0 in the interior of S such that δ(θ 0 ) = δ 0 an σ(δ(θ)) satisfies the moel M 0.LetH = δ(θ)/ θ ((q + m) q) be the corresponing Jacobian matrix evaluate at θ 0. Hence, by the chain rule of ifferentiation, Δ = σ/ θ = ( σ (δ)/ δ )( δ(θ)/ θ ) = ΠH. Since a(δ(θ)) = 0, it hols that with A evaluate at δ 0, AH = 0 with r(a) + r(h) = q + m, an r(.) enoting the rank of a matrix. Thus, H is an orthogonal complement of A.Let P ((q + m) (q + m)) := Π VΠ. Associate to M, the less restricte moel σ = σ (δ), the goonessoffit test statistic is T = nf (s, σ), where σ is the fitte moment vector in moel M with associate egrees of freeom r = r 0 m an scaling factor c given by c := r tr{u Γ }= r tr{vγ} r tr { P Π VΓVΠ }, () where U := V VΠP Π V. (2) We refer to Satorra an Bentler (200) for further etails. 3. Scaling Correction for the Difference Test When both moels M 0 an M are fitte, for example, by ML, then we can test the restriction a(δ) = 0, assuming M hols, using the chisquare ifference test statistic T := T 0 T. Uner the null hypothesis, we woul like T to have a χ 2 istribution with egrees of freeom m = r 0 r. This is the restricte test of M 0 within M. For general istributions of the ata, the asymptotic chisquare approximation may not hol. To improve on the chisquare approximation, Satorra (2000) gave explicit formulae that exten the scaling corrections propose by Satorra an Bentler (994) to the case of ifference, Wal, an score types of test statistics. General expressions for those corrections were also put forwar in Satorra (989, p. 46). Specifically, for the test statistic T we are consiering, Satorra (2000, p. 24) propose the following scale test statistic: T := T /ĉ, where c := m tr{u Γ } (3)
3 ALBERT SATORRA AND PETER M. BENTLER 245 with U = VΠP A ( AP A ) AP Π V. (4) Here, ĉ enotes c after substituting consistent estimates of V an Γ, an evaluating the Jacobians A an Π at the estimate ˆδ 0 when fitting M 0 (or M ). Since tr{u Γ } can be expresse as the trace of the prouct of two positive efinite matrices, tr{u Γ } > 0, an thus c > 0; the same for ĉ > 0. Consequently, T is ensure to be a nonnegative number. A practical problem with the statistic T is that it requires computations that are outsie the stanar output of current SEM programs. Furthermore, ifference tests are usually han compute from ifferent moeling runs. Satorra an Bentler (200) propose a proceure to combine the estimates of the scaling corrections c 0 an c associate to the chisquare goonessoffit test for the two fitte moels M 0 an M in orer to compute a consistent estimate of the scaling correction c for the ifference test statistic. A moifie (easy to compute) scale test statistic T with the same asymptotic istribution as T was propose. Both statistics were shown to be asymptotically equivalent uner a sequence of local alternatives (so they have the same asymptotic local power). Their proceure to compute T is as follows (see Satorra an Bentler, 200, p. 5).. Obtain the unscale an scale goonessoffit tests when fitting M 0 an M, respectively; that is, T 0 an T 0 when fitting M 0, an T an T when fitting M ; 2. Compute the scaling corrections ĉ 0 = T 0 / T 0, ĉ = T / T, an the unscale chisquare ifference T = T 0 T an its egrees of freeom m = r 0 r ; 3. Compute the scale ifference test statistic as T := T / c with c = (r 0 ĉ 0 r ĉ )/m. Here, r 0 an r are the respective egrees of freeom of the moels M 0 an M. The basis for computing the scaling correction for the ifference test statistic is the following alternative expression for U of (4) (see Satorra an Bentler, 200, p. 50) U = U 0 U, (5) where U is given in (2) an U 0 := V VΠH(H Π VΠH) H Π V. Since (5) implies mc = tr{u Γ }=tr { (U 0 U )Γ } = r 0 c 0 r c, (6) it follows that c = (r 0 c 0 r c )/m. This is the theoretical basis for Satorra an Bentler s (200) proposal as given in steps 3 above. 4. Problem with the Current Scale Difference Test ForanarbitrarymatrixV>0, (5) an (6) are exact equalities when the matrices U, U 0 an U are evaluate at a common value δ (as, e.g., the fitte value uner moel M 0 or uner M ); however, they are just asymptotic equalities when U, U 0 an U are evaluate at ifferent estimates that converge to the same true value uner the null hypothesis. Uner Satorra an Bentler s (200) proposal, c evaluates U 0 an U at the estimates ˆδ 0 an ˆδ, respectively. Since ˆδ will in general not satisfy the null moel M 0 (i.e., it will not be of the form δ = δ(θ) for the function implie by the implicit function theorem), when it eviates highly from M 0,the estimate ifference c = (r 0 ĉ 0 r ĉ )/m may turn out to be negative. This may happen in small samples, or when M 0 is highly incorrect; a result can be an improper value for T. Satorra an Bentler (200, p. 5) warne on the possibility that an improper value of T coul arise.
4 246 PSYCHOMETRIKA In orer to be sure to avoi a negative value for c, an hence T, currently one woul nee to resort to computing T using (3). Unfortunately, this is impractical or impossible for most applie researchers who only have access to stanar SEM software. Fortunately, as we show next, the exact value of T can also be obtaine from the stanar output of SEM software, using a new han computation. 5. A New Scale Test Statistic T Denote by M 0 the fit of moel M to a moel setup with starting values taken as the final estimates obtaine from moel M 0, an with number of iterations set to 0. Consier ĉ (0) := T (0) / T (0), where T (0) an T (0) are the stanar unscale an scale test statistic of this aitional run. Note that the estimate ĉ (0) uses moel M but the matrices Π an A are now evaluate at ˆδ 0 := δ(ˆθ), where ˆθ is the estimate uner M 0. Since now all the matrices involve in (5) are evaluate at ˆδ 0, the equality (6) hols exactly, an not only asymptotically, as when U 0 is evaluate at ˆδ 0 an U at ˆδ. The scaling correction that is now compute is ĉ (0) := ( r 0 ĉ 0 r ĉ (0) ) /m, (7) which, in view of (6), is the scaling correction of (3) when U is evaluate at the estimate ˆδ 0 when fitting M 0, an thus it is a positive number. The new scale ifference statistic is thus efine as T (0) := (T 0 T )/ĉ (0). (8) Clearly, T (0) = T ; that is, T (0) coincies numerically with the scale statistic (3) propose in Satorra (2000). 6. An Illustration We use these ata just for illustrative purposes, an because they provie an example where the stanar scaling correction fails to be positive. We use a latent variable moel iscusse for ata by Bentler, Satorra, an Yuan (2009). The Bonett Woowar Ranall (2002) test shows that these ata have significant excess kurtosis inicative of nonnormality at a onetail 0.05 level, so test statistics erive from ML estimation may not be appropriate an we o the scaling corrections. The moel consiere is a structure means moel, with the mean cigarette sales inirectly affecting the mean rates of the various cancers. The specifie moel is V j = λ j F + E j, j = 2,...,5, F = βv + D, V = μ + E, where the V j s enote observe variables; F, D, an E j are the common, resiual common, an unique factors, respectively; λ j enotes a factor loaing parameter, β is the effect of cigarette smoking on the cancer factor, an μ is the mean parameter for rates of smoking. The units of measurement for the factor were tie to V 2, with λ 2 =. The following values for the ML an Satorra Bentler (994) (SB) scale chisquare statistics are obtaine T = , T = , r = 9, ĉ =.6434, along with the egrees of freeom r an the scaling correction ĉ. The moel oes not fit, though for the sake of the illustration we are aiming for, this is not of concern to us.
5 ALBERT SATORRA AND PETER M. BENTLER 247 Restricte Moel M 0 The same moel is now fitte with the ae restriction that the error variances of the kiney an leukemia cancers, E 4 an E 5, are equal. This moel gives the following statistics T 0 = , T 0 = , r 0 = 0, ĉ 0 = Difference Test Our main interest lies in testing the ifference between M 0 an M, which we o with the chisquare ifference test. The ML ifference statistic is T = = , which, with egree of freeom (m), rejects the null hypothesis that the error variances for E4 an E5 are equal. Since the ata are not normal, we compute the SB (200) T scale ifference statistic. This requires computing the scaling factor c = (r 0 ĉ 0 r ĉ )/m given by [ 0(.4322) 9(.6434) ] / = = The scaling factor c is negative, so the SB ifference test cannot be carrie out; or if carrie out, it results in an improper negative chisquare value. New Scale Difference Test As escribe above, to compute the scale statistic T, we implement (7) an (8). The output that is missing in the prior runs is T (0) the value of the SB statistic obtaine at the final parameter estimates for moel M 0 when moel M is evaluate. This can be obtaine by creating a moel setup M 0 that contains the parameterization of M with start values taken from the output of moel M 0. Moel M 0 is run with zero iterations, so that the parameter values o not change before output incluing test statistics is prouce. The new result gives T (0) = , T (0) = , r = 9, ĉ (0) =.469, where as expecte, T (0) = T 0 as reporte above (i.e., the ML statistics are ientical), an the value ĉ 0 is han compute. As a result, we can compute ĉ (0) = ( r 0 ĉ 0 r ĉ (0) ) [ ] /m = (0)(.4322) (9)(.469) =.0, which, in contrast to the SB (200) c computations, is positive. Finally, we can compute the propose new SB correcte chisquare statistic as T = T (0) = (T 0 T )/ĉ (0) = ( )/.0 = 29.79, which can be referre to a χ 2 variate for evaluation. 7. Discussion The implicit function theorem was use to provie a theoretical basis for the evelopment of a practical version of the computationally more ifficult scale ifference statistic propose by For this particular example, the Appenix of Satorra an Bentler (2008) illustrates this proceure with EQS (Bentler, 2008). In the same reference, there is a secon illustration with a larger egree of freeom.
6 248 PSYCHOMETRIKA Satorra (2000). 2 The propose metho is only marginally more ifficult to compute than that of Satorra an Bentler (200) an solves the problem of an uninterpretable negative χ 2 ifference test that applie researchers have complaine about for some time. Like the metho it is replacing, the propose proceure applies to a general moeling setting. The vector of parameters σ to be moele may contain various types of moments: means, prouctmoments, frequencies (proportions), an so forth. Thus, this scale ifference test applies to methos such as factor analysis, simultaneous equations for continuous variables, loglinear multinomial parametric moels, etc. It can easily be seen that the proceure applies also in the case where the matrix Γ is singular, when the ata is compose of various samples, as in multisample analysis, an to other estimation methos. It applies also to the case where the estimate of Γ reflects the fact that we have intraclass correlation among observations, as in complex samples. Hence, this new statistic shoul be useful in a variety of applie moeling contexts. Simulation work will be neee to unerstan its virtues an limitations, relative to other alternatives, in such contexts. References Bentler, P.M. (2008). EQS 6 structural equations program manual. Encino: Multivariate Software. Bentler, P.M., Satorra, A., & Yuan, K.H. (2009). Smoking an cancers: caserobust analysis of a classic ata set. Structural Equation Moeling, 6, Bollen, K.A., & Curran, P.J. (2006). Latent curve moels: a structural equation perspective. New York: Wiley. Bonett, D.G., Woowar, J.A., & Ranall, R.L. (2002). Estimating pvalues for Maria s coefficients of multivariate skewness an kurtosis. Computational Statistics, 7, Browne, M.W. (984). Asymptotically istributionfree methos for the analysis of covariance structures. British Journal of Mathematical an Statistical Psychology, 37, Grace, J.B. (2006). Structural equation moeling an natural systems. New York: Cambrige University Press. Satorra, A. (989). Alternative test criteria in covariance structure analysis: a unifie approach. Psychometrika, 54, 3 5. Satorra, A. (2000). Scale an ajuste restricte tests in multisample analysis of moment structures. In D.D.H. Heijmans & D.S.G. Pollock, A. Satorra (Es.), Innovations in multivariate statistical analysis: a festschrift for Heinz Neuecker (pp ). Dorrecht: Kluwer Acaemic. Satorra, A., & Bentler, P.M. (994). Corrections to test statistics an stanar errors in covariance structure analysis. In A. von Eye & C.C. Clogg (Es.) Latent variables analysis: applications for evelopmental research (pp ). Thousan Oaks: Sage. Satorra, A., & Bentler, P.M. (200). A scale ifference chisquare test statistic for moment structure analysis. Psychometrika, 66, Satorra, A., & Bentler, P.M. (2008). Ensuring positiveness of the scale ifference chisquare test statistic. Department of Statistics, UCLA Department of Statistics Preprint. Yuan, K.H., & Bentler, P.M. (2007). Structural equation moeling. In C.R. Rao & S. Sinharay (Es.) Hanbook of statistics 26: Psychometrics (pp ). Amsteram: NorthHollan. Manuscript Receive: 4 JUN 2008 Final Version Receive: 25 MAY 2009 Publishe Online Date: 20 JUN Satorra (2000) provies also Monte Carlo evience on a specific moel context an various sample sizes of the superiority of the scaling correction over other alternatives such as the ajuste (mean an variance correcte) statistic.
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