According to our second goal, we investigated whether the five CHC factors (Gf, Gv, Gsm, Gc and Gs) and the subtests scores had the same meaning for French and French-speaking Swiss children. As a prerequisite to testing for factorial invariance we considered a baseline CHC model that was estimated for each group separately. As reported in Table 2, all goodness-of-fit indexes indicated that this baseline CHC model fit French data well and that all loadings and cross loadings were significant. Consistently with our previous analyses and with other studies, AIC values indicated that the CHC model was better than the four factors model for French children. This baseline CHC model was tested for French-speaking Swiss
168 children. All goodness-of-fit indexes showed that this baseline CHC model also fit French-speaking Swiss data well; all loadings and cross loadings were significant. The difference in respective AIC values suggested that this CHC model was better than the WISC-IV model with four-factors.
169
Model χ2 df CFI RMSEA SRMR AIC comp ∆χ2 ∆df ∆CFI
Phase 1: Baseline Model fit for each culture
Switzerland 118.72 84 .967 .041 .0499 190.71 - - -
France 257.34 84 .969 .043 .0317 329.34 - - -
Phase 2 : factor invariance across cultures
Model B1: configural invariance 376.22 168 .969 .030 .0317 520.22
Model B2: first- order factor loadings invariant 388.24 179 .969 .029 .0315 570.24 2 vs 1 12.02 11 0.000 Model B3: Model B2 and subtest intercepts invariant 646.60 190 .931 .042 .0334 806.60 3 vs 2 258.36* 11 -0.038 Model B3b: Model B2 and subtest intercepts partial invariant 403.40 187 .967 .029 .0317 569.40 3b vs 2 15.16 8 -0.002 Model B4: Model B3b and second-order factor loadings invariant 406.12 191 .968 .029 .0319 564.12 4 vs 3b 2.72 4 0.001 Model B5: Model B4 and subtest residual variances invariant 441.55 205 .964 .029 .0321 571.55 5 vs 4 34.43* 14 -0.004 Model B5b: Model B4 and subtest residual variances and partial invariant 423.03 202 .967 .028 .0316 559.03 5b vs 4 17.81 11 -0.001 Model B6: Model B5b and disturbances of first order factors invariant 424.57 205 .967 .028 .0321 554.57 6 vs 5b 1.54 3 0.000 Model B7: Model 6 and g variance invariant 431.66 206 .966 .028 .0332 559.66 7 vs 6 7.09* 1 0.001
*p < 0.05
Table 2:Multi-sample goodness-of-fit indices for the CHC model in five factors
170 We then tested configural invariance across groups (Model B1, Table 2). This model provided an adequate fit (RMSEA = .030, SRMR = .03, CFI = .969). In the second step, we constrained the first-order factor loadings to be equal (model B2). The ∆χ2 and the ∆CFI were not significant, indicating that the first-order factor loadings did not differ between the two samples, and thus that the subtests scores measured the same latent variables in both groups.
Next, subtest intercepts were constrained to be equal between the two samples (model B3).
The difference between model B3 and B2 resulted in a significant difference and a degradation of fit (∆CFI = -0.038; ∆χ2 = 258.36, ∆df = 11, p > .05). Inspection of MIs showed that misfit could be attributed to Block Design, Word Reasoning, and Similarities intercepts.
Thus intercepts of Block Design, Word Reasoning, and Similarities were freed. Results indicated that model B3b was not significantly different than model B2, suggesting partial measurement invariance. Next, first-order and second-order loadings were constrained to be equal between French and French-speaking Swiss children (model B4). The ∆χ2 and the ∆CFI were not significant. This result indicated that first-order and second-order loadings did not vary across groups and could be considered invariant. Model B5 tested the invariance of residuals variance (strict factorial invariance). The addition of this constraint resulted in a significant degradation of fit according to the ∆χ2 (34.43, ∆df = 14, p > .05). However, the
∆CFI was not significant, indicating that invariance could still be tenable. Nevertheless, in order to obtain more precise information, modifications indices were inspected. MIs showed that misfit could be attributed to Comprehension, Picture Completion and Arithmetic residual variances. Thus, Comprehension, Picture Completion and Arithmetic residual variances were freed, while all other residuals variances were invariant (Model B5b). Results indicated that model B5b was not significantly worse than model B4, suggesting partial measurement invariance. The next step (model B6) was to constrain the factor disturbances of the first-order factors on the basis of model B5b (factor unique variance). Model 6 allows us to examine whether subtests scores measured the same attributes (except for Comprehension, Picture completion and Arithmetic residual variances). The ∆CFI and the ∆χ2 were not significant (∆χ2 1.54, ∆df =3, p > .05,). The addition of this constraint did not result in a significant degradation of fitThe two groups could be considered invariant for this parameter.
Finally, in the seventh and last level of measurement invariance, g factor variance was constrained to be equal across the two samples. The addition of this constraint resulted in a significant degradation of fit according to ∆χ2 (7.09, ∆df = 1, p > .05). However, the ∆CFI was not significant, indicating that invariance could still be tenable. This finding indicated that the two groups could still be considered invariant for this parameter.
171
Figure 2. The CHC model with five factors(model B7). Standardized loadings. When not equal across groups : France/Switzerland
172 5.7 Discussion
This study is very important, because it is the first one to investigate measurement invariance of the WISC-IV factorial structure between French and French-speaking Swiss samples. This comparison between these two French-speaking samples gives important information about the universality of the meaning of the WISC-IV index scores and about subtests results. In a first step, we tested whether the four indexes (VCI, PRI, WMI, and PSI) had the same meaning for French and Swiss-French children (models A1-A7). In a second step, we investigated whether the five CHC factors (Gf, Gv, Gsm, Gc and Gs) had the same meaning for French and Swiss-French children (models B1-B7). Therefore, the present study allowed us to compare the four factor-scoring models with an alternative CHC model. We considered that this comparison was more appropriate in order to investigate the nature of intelligence structure, and to assess the universality of the underlying latent variables.
Consistently with our previous analyses, the CHC-based model better fitted the French and the Swiss-French data than did the four-factor model (Lecerf et al., 2010; Reverte et al., 2014). This finding provides support for separating PRI into Gf and Gv.
Taken together, results were similar for the measurement invariance of the four-factor model and the CHC based model. Indeed, in both models, configural invariance was satisfied, indicating that the factorial structure of the attributes was equal across groups. We also found that metric invariance was satisfied (first-order factor loadings). The relations between subtest scores and their respective underlying attributes were the same across groups. However, the next level of measurement invariance, intercept invariance was not demonstrated either for the four-factor structure or for the CHC model. This finding could be critical, because it has been suggested that intercept invariance is the last level of invariance needed to compare scores across groups. More precisely, we found that subtests intercepts of Arithmetic, Picture Concepts, Block Design, Word Reasoning, Matrix reasoning and Similarities differed significantly between the two groups in the four-factor structure. Results indicated that the mean of Arithmetic, Picture Concepts, Word reasoning and Matrix Reasoning were higher for the French children while the mean of Block design and Similarities were higher in the Swiss group. Similarly, in the CHC model, we found that subtests intercepts of Block Design, Picture Completion and Arithmetic differed significantly. The lack of support for scalar invariance could be critical, because it indicated that mean difference between French and Swiss French children on these particular scores could not be accounted for by mean difference on the factor that these tests scores were supposed to measure. In other words,
173 intercepts differences could lead to an overestimation or an underestimation of group performance in some latent attribute (VCI, PRI, Gc, etc.). It could be suggested that the two groups differed in the subtest specific ability involve in these tasks. For instance, processing speed could be more recruited in the Block Design test for Swiss French children than for French children. Furthermore, in a previous study, we found differences between French and Swiss-French Children for the subtests Picture Completion and Arithmetic. Thus, it is possible that the resolution of these subtests tap other specific ability.
Because scalar invariance was not demonstrated, partial measurement invariance models were tested. Results indicated that partial second-order factors loadings invariance model was satisfied. This model indicated that distinct but related attributes could be accounted for by one common underlying higher –order factor in both groups. Next, the residual variances invariance model could still be considered tenable. Because the large variation in sample sizes could result in excessive power in the French sample we decided to rely on both the ∆χ2 and the ∆CFI. Partial measurement invariance was tested and showed that residual variances were not equal across groups for Comprehension, Picture Completion and Arithmetic. Finally and according to the ∆CFI, all following measurement invariance models were also satisfied (unique variances of first-order factors, g variance).
The analyses above did not fully support the measurement invariance of the four-factor and the CHC model across groups. It could be suggested that the lack of scalar invariance and residual variances invariance could be in part due to the sample size difference and/or to the range of age of the samples. Firstly, 249 children compose the Swiss sample, while the French sample is composed by 1103. Secondly, the Swiss sample was more homogeneous according to age range than did the French sample (8-12 vs. 6-16 years, respectively). However, it should be noted that when measurement invariance was conducted with a more homogeneous French sample according to age range (8 to 12 years, N = 500), results were similar.
Finally, our results were relatively consistent with previous studies that revealed measurement invariance for gender, culture, clinical vs. normative samples (Chen et al., 2008, 2010, 2012), and supported factor pattern, factor metric and disturbance invariance. For instance, in a study comparing China, Hong Kong, Macau and Taiwan WISC-IV standardization samples, Chen et al. (2010) also observed measurement invariance across samples. Bowden and colleagues (2011) found similar results comparing the U.S. and the Canadian WAIS-IV standardization samples. It should be finally noted that some of the studies conducted on measurement invariance did not test scalar invariance as we did, and did
174 not compare the four-factor model with a CHC five-factor model, as we did (Chen et al., 2010). In addition, some studies investigated measurement with the 10 core subtests, while we used the 15 subtests.
Conclusions
This paper provides a first step of the issue of invariance in measurement between French and Swiss French children. Using the data reported here, it could be concluded that the WISC-IV measures the same attributes for French and Swiss French children. However, while factor loadings appeared to be invariant over groups, it was not possible to conclude that strict measurement invariance was granted, because measurement intercepts invariance was not satisfied. French and Swiss-French children showed significant differences with respect to six intelligence subtests of the WISC-IV. This finding has important applied implications for clinical interpretation of the WISC-IV results by Swiss-French practitioners. Indeed, this finding suggested that Swiss practitioners shloud interpret these six subtests scores with caution. A simple but impractical solution would be to discount the 6 biased subtests.
Concerning residual variances invariance, measurement invariance was satisfied. The present data suggested that the attribute of intelligence as measured with the WISC-IC was similar for French and Swiss-French children. However, some raw scores origins were different. This finding reinforces the importance of the development of local norms. Because the French-speaking Swiss sample size was relatively modest in this first study, measurement invariance with larger samples and with clinical samples should be performed in the future.
175
Chapitre 6
Scores composites CHC pour le WISC-IV: normes francophones.
Objectifs de cette étude : L’objectif de cette étude a une portée plus clinique. En effet, sur la base des résultats des études précédentes, des tables de conversion de notes CHC ont été mises sur pieds. Celles-ci, permettront aux cliniciens d’interpréter les scores du WISC-IV, non seulement selon la méthode standard, mais également sur la base du modèle CHC.
Rappelons qu’actuellement l’interprétation du WISC-IV se fait sur la base de quatre indices qui sont l’indice de Compréhension verbale (ICV), l’Indice de Raisonnement perceptif (IRP), l’Indice de Mémoire de travail (IMT) et l’indice de Vitesse de traitement (IVT). Cependant, même si cette structure est plus contemporaine que dans les versions précédentes du WISC, elle n’est pas encore totalement en accord avec le modèle CHC. Or, dans certaines études (Keith et al., 2006 ; Chen et al., 2009) ainsi que dans les études précédentes qui composent ce travail, il a été démontré qu’une structure basée sur la théorie CHC s’ajuste mieux aux données. Ainsi, en mettant au point ces tables de conversions, le clinicien pourra non seulement interpréter les résultats du WISC-IV sur la base du modèle standard à quatre facteurs mais également sur la base d’un modèle CHC à cinq facteurs. Il aura ainsi plus d’informations sur le fonctionnement cognitif de l’enfant qu’il évaluera.
Lecerf, T., Golay, P., Reverte, I., Senn, D., Favez, N., Rossier, J. (2012). Scores composites CHC pour le WISC-IV : normes francophones. Pratiques Psychologiques, 18, 37-50. doi : 10.1016/j.prps.2012.03.001