Словари и журналы / Психологические журналы / p41British Journal of Mathematical and Statistical Psycholog

covariance matrix by a robust covariance matrix in the normal theory based likelihood function, Yuan and Bentler (1998a) investigated the rescaled likelihood ratio statistic for SEM.

Now let t = tr(QG)/( p ± q) be the average of the non-zero eigenvalues of QG; then a rescaled statistic can also be constructed as

where tˆ is a consistent estimate of t with the unknown parameters in Q being replaced by their consistent estimates and G by Sy. Empirical results in Curran, West, and Finch (1996), Hu, Bentler, and Kano (1992), and Yuan and Bentler (1998b) indicate that the test statistic TR performs quite well under a variety of distribution conditions when F = FML and the model Žtting is through a simultaneous estimation process. More empirical studies are needed to verify the performance of TR in (22).

Suppose we use the same method for estimating g as for u, that is, a simultaneous estimation process; then U = (sÇ¢WsÇ)± 1sÇ¢W. If W = G± 1, then QG is a projection matrix of rank p ± q, t = 1 and the correction factor is unnecessary. This is just the known result about ADF. If W = W1 in the simultaneous estimation process, then TR becomes the statistic studied by Satorra and Bentler (1988, 1994).

5. Applications

The methods developed are applied to two data sets in this section. The Žrst data set is from Holzinger and Swineford (1939) and consists of 24 cognitive test scores of 145 seventhand eighth-grade students. Jo¨ reskog (1969) studied a correlation structure of nine variables from this data set. We will also use these nine variables in our study. Our second data set is from the Health Outcomes of Women survey, an ongoing longitudinal study conducted by Marshall (1999). The survey consists of three groups, and our analysis is based on a data set of nine variables with 183 cases from the white group.

Example 1

The nine variables of Holzinger and Swineford’s (1939) are: (1) visual perception, (2) cubes, (3) lozenges, (4) paragraph comprehension, (5) sentence completion, (6) word meaning, (7) addition, (8) counting dots, (9) straight-curved capitals. In the original report of Holzinger and Swineford (1939), variables 1, 2 and 3 were designed to measure spatial ability, variables 4, 5 and 6 were designed to measure verbal ability, and variables 7, 8 and 9 were speed tests. Let x be the vector of the nine observed variables; then the conŽrmatory factor model

represents the original design of Holzinger and Swineford. We assume that the measurement errors are uncorrelated, with W being a diagonal matrix.

52 Ke-Hai Yuan and Wai Chan

Previous analyses generally assumed that Holzinger and Swineford’s (1939) data set followed a multivariate normal distribution. However, Mardia’s multivariate skewness and kurtosis (Mardia, 1970, 1974) for these nine variables are M1 = 0.77 and M2 = 3.04, indicating that the data set may come from a distribution with heavier tails than those of a normal distribution when referring these statistics to x2165 and N(0, 1), respectively. We Žt model (23) in the following ways: (a) a simultaneous estimation of all the unknown parameters in the model bythe MLand ADFmethods; (b) Žtting the three submodels each with three indicators separately for factor loading and error variance estimates, and then estimating the factor correlations in the combined model byŽxing the factor loadings and error variances at these estimates. Thus in (b) we have four models to Žt. Our interest is in contrasting the parameter estimates in the different estimation approaches. The multivariate kurtosis for variables 1, 2 and 3 is M21 = 1.38, for variables 4, 5 and 6 is M22 = 1.46, and for variables 7, 8 and 9 is M23 = 2.99; and none of the multivariate skewness values is signiŽcant. It is obvious that the kurtosis M23 is statisticallysigniŽcant. So we mayestimate the Žrst two submodels by normal theory based MLand the third by the ADF method in order to obtain more efŽcientparameter estimates. For the purpose ofcontrasting different estimation processes we Žt the three-indicator submodels and the Žnal combined model in

(b) bythe following approaches: (I) allthe four models are Žtted byML; (II) the Žrst 2 threeindicator models are estimated by ML, the third three-indicator model by ADF, and the combined model by ML; (III) all 3 three-indicator models are estimated by ADF and the combined one by ML; (IV) all the three-indicator submodels are estimated by MLand the combined one by ADF; (V) the Žrst 2 three-indicator models are estimated by ML, the third three-indicator modelbyADF, and the combinedmodelbyADF; (VI) allthe models are estimated by ADF.

Parameter estimates and the corresponding test statistics are reported in Table 1. Because all the parameter estimates are signiŽcant, we omit the standard errors to save space. Notice that with three-indicators the one-factor model is a saturated model, and all the estimation approaches (I–VI) in (b) lead to the same set ofestimates for factor loadings and error variances. Estimation approaches I–III also lead to the same set of estimates of factorcorrelations, and sodo the estimation approaches IV–VI. As can be seen from Table 1, there exist various discrepancies between the parameter estimates by the different approaches. The largest discrepancy occurs with the estimates for w99 between the simultaneous estimation approach (a) and the model segregation approach (b). If model (23) is correct, all the estimates are consistent for w99. Then we have to regard the discrepancy as the result of small sample size. Actually, all the parameter estimates are statistically signiŽcant. How good the model (23) is at describing the relationship among the nine variablesis re•ected bythe test statistics in Table 1. Here the test statistic reported for the simultaneous estimation process by MLis the Satorra and Bentler (1988) rescaled statistic; the statistic for the simultaneous estimation process by ADF is that proposed by Browne (1984); and the statistics for conditions I–VI are calculated according to (22). Because different P matrices are used in (12), the TR statistics for different conditions may not be equal even though the parameter estimates are equal. Here the differences among conditions I–IIIand those among conditions IV–VIare quite small, but this maynot always be the case. The smallest statistic is Satorra and Bentler’s TSB = 49.37, while the largest one is the rescaled statistic TR = 72.94 corresponding to condition IV. All of the statistics are signiŽcant at the 0.01 level, implying that the overall model (23) is not adequate in explainingthe association among the nine variables. This mayalsoexplain whythere exists a substantial difference among the different estimates of w99.

Even though there is no substantial difference in the conclusions regarding model

(23) by the different procedures, there is a difference in the speed of convergence. For example, with default initial values in the EQS set-up (Bentler, 1995) the simultaneous estimation process with ML needs 82 iterations to converge. However, for the same estimation methods and default initial values the 3 three-indicator submodels need respectively 7, 6 and 9 iterations to converge, while the combined model for the estimation of factor correlation needs only 2 iterations. So convergences with the segregated models are much easier.

This example illustrates that one can Žt a model in parts according to the procedures developed in this paper. Even though various discrepancies exist among the different approaches, the conclusion regarding model (23) and its parameters is basically the same. That is, all the parameters are necessary in the parameterization in (23) but the model is not adequate in explaining the relationship among the variables.

Example 2

Because model (23) is statisticallynot adequate in explaining the relationship among the

54 Ke-Hai Yuan and Wai Chan

Because there are double loadings for x8 and x9 in (24), it can be shown that the model based on (x7, x8, x9) is not identiŽable if l81 and l91 are included in the model; and the estimates for l73, l83 and l93 are not consistent if l81 and l91 are ignored. However, it can be easilyshown that the factor loadings l11, l21, l31, l42, l52, l62 and the error variances for the Žrst six variables can stillbe consistentlyestimated byŽtting the two three-indictor submodels. In addition to the simultaneous estimation by the MLand ADF approaches, the following four model segregation approaches are also investigated: (I) both the two three-indicator submodels and the combined model are estimated by ML; (II) the two three-indicator models are estimated by ADF, the combined model by ML; (III) both the three-indicator submodels are estimated by ML, and the combined one by ADF;

Table 2. Parameter estimates and test statistics in Example 2

(IV) both the two three-indicator submodels and the combined model are estimated by ADF.

Parameter estimates and the corresponding test statistics by different methods are reported in Table 2. As in Example 1, all these estimates are statisticallysigniŽcant. Even though the differences among the different approaches for estimates of w99 become smaller, there is a substantial difference between estimates of w77. With regard to test statistics, the smallest one is given by the Satorra and Bentler statistic, while the largest one is the rescaled statistic TR for conditions III and IV. None of them is statistically signiŽcant at the 0.5 level, implying that model (24) is reasonable in explaining the relationship among the nine observed variables.

Even though model (24) Žts the data better, various discrepancies still exist among the estimates by different procedures. Similarly, the various test statistics are still in agreement regarding the model’s adequacy. This example also points out the fact that only identiŽable submodels can be used in the Žrst stage to obtain gˆ .

Example 3

The Health Outcomes of Women survey concentrates on the longitudinal effect of partner’s abuse on the health of women from low-income families. The four variables measured at time 1 in x = (x1, x2, x3, x4)¢ are psychological abuse, partner’s threats, partner’s violence and partner’s sexual aggression, together constituting a partner’s violence factor. There are Žve dependent variables in y = ( y1, . . . , y5)¢. Variables y1 and y2 are measures of self-esteem and stress, together forming an overall stress factor at time 1. The time 2 measurements y3 to y5 are measures of physical health, quality of health and health perception, constituting a health condition factor. In LISREL( Jo¨ reskog & So¨ rbom, 1993) notation, the substantive theory, that family violence at time 1 leads to stress on women at time 1 and that these together predict health condition in the subsequent wave, can be expressed as

and «, d and z are vectors of uncorrelated errors. Our interest here is in evaluating the model structure implied by (25). The problem with this data set is that the MLE of the error variance ( vd2 ) of x2 is negative (a Heywood case). As discussed in the earlier sections, we Žrst estimated the one-factor model x = Lx y + d; however, the error variance estimate for x2 based on this submodel is still negative. Further experiment shows that the submodel implied by x1, x2 and x4, which is just a three-indicator, onefactor model, leads to a positive estimate for vd2. By treating this as a Žxed parameter in the ML estimation process, the Heywood case is avoided. Table 3 reports the parameter estimates by different estimation methods, together with the corresponding

test statistics. The notation ML-MLis for the model segregation approach in which both the three-indicator model and the combined model are estimated by the normal theory based maximum likelihood. Notice that we only Žxed vd2 in estimating the combined model; the other parameters of the three-indicator model are re-estimated. Of course, we can also use the ADFmethod in the simultaneous estimation approach, which also avoids the Heywood case, as can be seen from Table 3. All the test statistics are statistically signiŽcant when referring to the chi-square distribution with 24 degrees of freedom. As in the previous examples, there are some differences in the parameter estimates.

This example demonstrates that when an improper solution exists one can use the method developed in this paper to obtain a set of reasonable solutions.

Because of our limited approach to problematic data sets, the three examples based on the two data sets are used for the purpose of showing how to apply the proposed procedures in practice. As a matter of fact, the traditional methods work quite well with Examples 1 and 2, and the ADF method overcomes the Heywood case in Example 3. However, there are problematic data where traditional methods maynot solve the issues as discussed in the previous sections, and the procedure developed may apply. Actually, due to issues such as non-convergence or there being only a single indicator,

problematic data sets are seldom reported. We hope the procedure developed can help future researchers in putting more of these data sets into the literature.

Finally, we observe that one can use standard SEM software (e.g., AMOS, EQS, LISREL) to obtain parameter estimates gˆ and uˆ , but one has to use the procedures in Section 3 to obtain correct standard errors of uˆ and the procedure in Section 4 for an overall model evaluation.

6. A simulation example

The three examples in the previous section indicate that there may be various discrepancies among the parameter estimates and test statistics obtained by different procedures. These discrepancies are generallycaused bytwo sources of errors: sampling error, which occurs when the overall model is correctly speciŽed; and systematic error plus sampling error, which occurs when the overall model is misspeciŽed. For all the three examples in the previous section the submodels for estimating g are correct because they are saturated. However, if the overall model is misspeciŽed the estimates based on the submodel and those based on the overall model will not converge to the same set of values. Even though the overallmodel in Example 2 is not signiŽcant, we still do not know whether model (24) is correctly speciŽed. These considerations lead us to present the following simulation example. Let l = (0.60, 0.70, 0.80)¢,

f ,N(0, F) and e ,N(0, W) be independent; then x = Lf + e follows N(0, S). With the normal data all the three estimating functions in (9) are optimal. Thus V12 = 0 and the covariance matrix of uˆ (gˆ ) is given by (5). As in Example 1, factor loadings and error variances are estimated by 3 three-indicator submodels. To save space, in addition to reporting the parameter estimates and test statistics corresponding to the simultaneous estimation procedure by MLand ADF, we only report the results of estimating (I) all the three-indicator submodels and the combined model by the normal theory based maximum likelihood (ML-ML), (II) and all the three-indicator submodels and the combined model by the ADF procedure (ADF-ADF). These two procedures correspond to conditions (I) and (VI) in Example 1.

The results of three replications of the above simulation conditions with sample size 200 are given in Tables 4–6. There is little discrepancy between parameter estimates based on either of the simultaneous estimation processes and those based on the corresponding model segregation approach in any of these replications. Actually, because the overall model is correctly speciŽed, the discrepancy among the different parameter estimates by the simultaneous and segregation approaches is only caused by sampling errors, which is re•ected in Tables 4–6. On the other hand, the discrepancies in Tables 1–3 may be caused by systematic errors in addition to sampling errors. For normal data, parameter estimates in the simultaneous procedures are more efŽcient, as re•ected by the slightly larger standard errors from the model segregation approaches. There are also a few exceptions due to the Žnite-sample effect.

Furthermore, there is little difference among the different test statistics. In replication 1, the test statistic TR for condition I is only slightly greater than the corresponding

58 Ke-Hai Yuan and Wai Chan

Table 4. Comparison of simultaneous estimation with segregated estimation, replication 1

TML, and TR for condition II is slightly greater than TADF. With replication 2, the TR for condition I is slightly smaller than TML; the TR corresponding to condition II is also slightly smaller than TADF. With replication 3, the TR is slightly greater than TML in condition I; while TR for condition II is a little greater than TADF. In contrast to the test statistics in Tables 1–3, the statistics in Tables 4–6 have much smaller discrepancies. This implies that, when the overall model is correctly speciŽed, the rescaled statistics are more like those based on simultaneous estimation approaches. Even though gˆ or

covariance structure of the factor model. The three replications were obtained by setting the initial SEED= 1234567 and the normal variables in z1 and z2 were generated in sequence by the function NORMAL(SEED). We hope these details will help readers who wish to verify the results in Tables 4–6.

7. Discussion

We propose a model segregation procedure for dealing with various issues in covariance structure analysis. Properties of parameter estimates can be characterized by means of estimating equations with nuisance parameters. Overall model evaluation can be performed using a rescaled statistic. Examples illustrate several applications of this procedure in practical data analysis.

As illustrated in the examples in Section 5, substantial discrepancies mayexist among model parameter estimates by different approaches. If the overall model is correctly speciŽed, then any identiŽable submodel will lead to consistent parameter estimates. Discrepancy among the parameter estimates is only due to small sample size; this is illustrated in Section 6. However, when the overall model is misspeciŽed, the parameter estimates from the overall model and those from a submodel may not converge to the same quantity. So, with a large sample size, substantial differences between parameter estimates by different approaches may imply a misspeciŽcation in the overall model.

The potential of the proposed procedure to solve some challenging problems in SEM

60 Ke-Hai Yuan and Wai Chan

Table 6. Comparison of simultaneous estimation with segregated estimation, replication 3

does not implythat allthe issues discussed can be solvedbythis procedure in practice. For example, when a variable contains an improper error variance estimate and no submodel containing this variable is identiŽable, the procedure developed here is not applicable. In addition to such a limitation, there are also many unknown aspects with the proposed procedure. For example, how the statistic TR behaves in various conditions, especiallyits Type I and Type II errors, may merit a large-scale Monte Carlo study. More experience of this procedure with practical data is also needed for its better application.

AŽnal remark on applying the model segregation approach to overcome Heywood cases is needed. As discussed in van Driel (1978), there are several causes of negative estimates of error variances. When theyoccur with a large enough sample size, it is quite likely that the overall model is misspeciŽed. In such a case, Žxing these error variances at some positive value gˆ may avoid Heywood cases, but it may also excuse a researcher from Žnding a better model. Our recommendation in this regard is that, when parameter estimates under a segregation approach and a simultaneous approach are signiŽcantly different or the corresponding test statistics lead to quite different conclusions regard-