Two-term Edgeworth expansions of the distributions of fit indexes under fixed alternatives in covariance structure models

Transcription

1 Economic Review (Otaru University of Commerce), Vo.59, No.4, 4-48, March, 009 Two-term Edgeworth expansions of the distributions of fit indexes under fixed alternatives in covariance structure models Haruhiko Ogasawara Abstract. Asymptotic expansions of the distributions of thirteen fit indexes used in covariance structure analysis in practice are obtained. The fit indexes include the usual log likelihood ratio statistic for a posited model and the functions of this statistic and the corresponding statistic of the so-called baseline model of uncorrelated observed variables. The results are derived by the two-term Edgeworth expansion under fixed alternatives for possibly nonnormally distributed data. A numerical example using a misspecified factor analysis model is shown to see the behavior of the asymptotic results in finite samples. Keywords: Fixed alternatives, nonnormal distributions, Edgeworth expansion, structural equation modeling, RMSEA Purpose In covariance structure analysis, various indexes are used to assess the goodness-of-fit of a posited model. Among these fit indexes, the classic log likelihood ratio chi-square statistic is a basic one with the assumption of multivariate normality for observed variables under the null hypothesis of a true model. Let F be the discrepancy function for Wishart maximum likelihood estimation, i.e. F = F( S, Σ) = ln Σ ln S + tr( Σ S ) p, where Σ= Σ( θ ) is the covariance matrix for p observed variables described by a q parameter vector θ and S is the p p unbiased sample covariance matrix. Then, the vector of the Wishart maximum likelihood Partially supported by Grant-in-Aid for Scientific Research from the Japanese Ministry of Education, Culture, Sports, Science and Technology No Author s address: Department of Information and Management Science, Otaru University of Commerce, 3-5-, Midori, Otaru Japan. hogasa@res.otaru-uc.ac.jp 4

2 estimators ˆθ is obtained by minimizing F over the parameter space. Let ˆF denote F when Σ is given by Σˆ = Σ( θ ˆ). Then, it is well known that under normality with a true model nf ˆ is asymptotically chi-square distributed with the degrees of freedom (df) being υ pp ( + )/ q, where n+=n is the sample size. When the model is slightly misspecified or under a local alternative in normal samples, nf ˆ has asymptotically the noncentral chi-square distribution. Under normality this can be used for testing models. In the local alternative the population covariance matrix is assumed to be a function of n, which is a technical one to have the noncentral distribution and is not a realistic assumption. In practice, it is known that when the sample size is large, nf ˆ tends to become large, which indicates that fixed alternative hypotheses are more realistic. In practice, data are usually nonnormally distributed. Ogasawara (007a) derived the asymptotic distributions of various fit indexes including the chi-square statistic by the single-term Edgeworth expansion under nonnormality and fixed alternatives. In this paper the results are extended to the two-term Edgeworth expansion with simulations for confirmation with a finite sample size. Fit indexes As in Ogasawara (007a), thirteen fit indexes including ˆF are dealt with in this article. The members of the first group ([] to [8] given below) are the fit indexes as functions of ˆF while the members of the second group ([9] to [5]) use the so-called baseline model of uncorrelated observed variables as well as ˆF. They are defined as [] ˆ ln ˆ ln tr( ˆ F = Σ S + Σ S ) p ; [] F ˆ ln Diag( ) ln B = S S ; Fˆ p p( p+ ) [3] GFI = = Fˆ + p Fˆ ; [4] AGFI = ( GFI) ; + p ˆ [5] Abs.GFI exp F υ ˆ = n ; [6] RMSEA Max F υ n, 0 = ; ˆ ˆ { F ( υ / n)} p [7] Γ = = { Fˆ ( υ/ n)} + p { Fˆ ( υ/ n)} + p ; υ 4

3 ˆ p( p+ ) ˆ [8] Γ = ( Γ ) ; υ I Fˆ k Fˆ k kfˆ + k k with B 0 = = Fˆ ˆ B k0 FB k0 [9] NFI, k 0 = 0, k =, k = 0 ; [0] IFI( Δ ), k0 = υ / n, k =, k = 0 ; [], k 0 = 0, k = υ / υ B, k = 0 ; [], k = υ / n 0 B, k = υ / υ B, k = 0 ; [3] FI (RNI), k = υ / n 0 B, k =, k = ( υb υ)/ n; ˆF is the value of ˆF for the baseline model of where ˆF is as before; B uncorrelated observed variables with df = υb p( p + )/ p = p( p )/; Diag() denotes a diagonal matrix whose diagonal elements are those of the argument matrix; and for details of the fit indexes see Ogasawara (007a). 3 Asymptotic expansion of the fit indexes Let I = I( Fˆ, Fˆ B) be the function of ˆF and/or ˆF B. We assume that the following Taylor expansion with the differentiability of I with respect to sample variances and covariances up to the third order is available: < 3> < > I I = I0 + s= σ ( s σ T T) + I s= σ ( s σ T T) s' s' < 3> + I s= σ ( s σ T T) + Op( n ), 6 s' I = I( F, F ), F0 = ln Σ0 ln ΣT + tr( Σ0 Σ T) p, Σ0 = Σ( θ 0), F B0 = ln Diag( ΣT) ln Σ T, ΣT = E( S ), s= v( S), σt = v( ΣT), v( ) is where 0 0 B0 the vectorizing operator taking the nonduplicated elements of a symmetric < p matrix, s > = s s s (p times) is the p-fold Kronecker product of a vector, and θ 0 is given by fitting the (misspecified) model to the true covariance matrix Σ T. w= n( I I ). From (), under a fixed alternative assumption with Let 0 possibly nonnormal data, when the cumulants exist they are given by < > () 43

4 κ ( w) = E( w) = n α + O( n ), / 3/ ( w) = E[{ w E( w)} ] = + n Δ + O( n ), 3 / 3/ 3( w) = E[{ w E( w)} ] = n α3+ O( n ), 4 4 w = w w w = n 4+ O n κ α α κ κ ( ) E[{ E( )} ] 3{ κ ( )} α ( ). Using the asymptotic cumulants of (), it is known that the following distribution function by the two-term Edgeworth expansion is valid for nonnormal data under regularity conditions: w α α / 3 Pr z / =Φ( z) n + ( z ) / 3/ φ( z) α α 6α z α4 αα 3 z 3z α3( z 0z + 5 z) n ( Δ α+ α) φ( z) 3 α 4 6 α 7α + on ( ), where φ( z) = (/ π)exp( z / ) and Φ ( z) = φ( t) dt. The asymptotic cumulants are given by the moments of the observable variables up to the eighth order with the partial derivatives of the fit indexes with respect to the sample variances and covariances of the observable variables up to the third order (see Ogasawara, 007a, b). The remaining asymptotic cumulant other than those available in Ogasawara (007a) is α 4, which is the most complicated one in (). This can be given by using the formula of Ogasawara (006): I0 I0 I0 I0 α = 4 σ σ σ σ 4 3 κabcdefgh + κacκbdefgh + κaceκbdfgh a b c d e f g h Tab Tcd Tef Tgh κ κ + κ κ + κ κ κ + κ κ κ aceg bdfh abeg cdfh ac be dfgh ac eg bdfh κ κ κ + κ κ κ κ κ ac beg dfh bc de fg ha abcd 44 z M( ef, gh) ()

5 + 0 I0 I0 I0 I0 M ( ab, cd) M ( ef, gh, jk) σ σ σ σ σ j k Tab Tcd Tef Tgh Tjk 3 3 I0 I0 I0 I j k l m σtab σtcd σtef σtgh 3 σtab σtcd σtef σ Tgh 5 I0 I 0 M( ab, cd) M( ef, gh) M( jk, lm) σtjk σtlm (4αα 3 + 6αΔ α + 6 αα ), where the partial derivatives denote those evaluated at the population values; a b is p a b ; M(ab, cd) and M(ef, gh, jk) are ne{( sab σ Tab)( sab σ Tcd )} and n E{( s σ )( s σ )( s σ )} up to O(), respectively with ef Tef gh Tgh jk Tjk S = { s ab } and Σ T = { σ Tab} ; and k Σ is the sum of k similar terms. 4 Numerical illustration A numerical example using a misspecified one-factor model as in Ogasawara (007a) is presented, where the true covariance matrix is given by the two-factor model: ' ΣT = ΛTΛT' + ΨT, ΛT =, Ψ = diag(.55,.55,.55,.55,.55,.55,.55). T Normal and nonnormal observations with sample size N=300 were randomly generated by = Γ is a lower-triangular matrix such that x ΓTf, where T = ' T T T Σ Γ Γ and f is a random vector with independent elements. For nonnormal observations the elements of f have standardized chi-square distributions with df=3. The simulation was performed with,000,000 replications. Table shows asymptotic and simulated cumulants except α. The simulated Δ α is defined by n (SD n α ), where SD is the usual standard deviation for a fit index given by,000,000 estimates. Table shows the root mean square errors of the approximate distribution functions of the 45

6 / / standardized estimators ( n ( I I0)/ α ) given by the single- and two-term Edgeworth expansions and Hall s (99) variable transformation. The true distribution functions were given by the simulations. It is known that Hall s method is asymptotically equivalent to the single-term Edgeworth expansion. We see that on average the two-term Edgeworth expansion has reduced the errors. References Hall, P. (99). On the removal of skewness by transformation. Journal of the Royal Statistical Society, B, 54, -8. Ogasawara, H. (006). Asymptotic expansion of the sample correlation coefficient under nonnormality. Computational Statistics and Data Analysis, 50, Ogasawara, H. (007a). Higher-order approximations to the distributions of fit indexes under fixed alternatives in structural equation models. Psychometrika, 7, Ogasawara, H. (007b). Higher-order estimation error in structural equation modeling. Economic Review (Otaru University of Commerce), 47(4), otaru-uc.ac.jp/~hogasa/ 46

7 Table. Theoretical and simulated cumulants for the misspecified one-factor model (N=300) Normal χ (df=3) Normal χ (df=3) Fit index Th. Sim. Th. Sim. Th. Sim. Th. Sim. Δ α (higher-order added variance) α (bias) F F B GFI AGFI Abs.GFI RMSEA NFI IFI FI (RNI) α 3 (skewness) α 4 (kurtosis) F F B GFI AGFI Abs.GFI RMSEA NFI IFI FI (RNI) Note. Th.=Theoretical or asymptotic values. Sim.=Simulated values. 47

8 Table. 5 0 root mean square errors of the approximate distribution functions of the standardized estimators for the misspecified one-factor model (N=300) N* E E Hall N* E E Hall Normally distributed data Chi-square distributed data with df=3 F F B GFI (AGFI) Abs.GFI RMSEA ( ) NFI ( ) IFI (FI, RNI) Note. N*=Normal approximation, E=Single-term Edgeworth expansion, E=Two-term Edgeworth expansion, Hall=Hall s method by variable transformation. 48