Study 2 moved beyond the analysis of factor structure to an analysis of the dimensional structure, testing for four achievement goals as the first-order latent variables and four factors of two competence dimensions as the second-order latent variables. The hypothetical dimensional structure model was compared to other alternative second-order models.
Method
The Participants and Procedure were the same with Study 1.
Results
Dimensional structures of achievement goals: The second-order factor structure
The support of first-order achievement goal structure as a good fit and far better fit than other alternatives does not necessarily guarantee 2 × 2 nature of achievement goal framework. To examine the dimensional nature of the achievement goal model, I further tested the AGQ-C by conducting a second-order factor analysis. The hypothetical model specified that the first-order latent variables were the four achievement goals and the second-order latent variables were four of two competence dimensions (Fig. 4-1).
The results showed that 12 items loaded on their respective first-order latent goals that in turn loaded on the designated second-order factors as I expected. The path coefficients from the second-order factors to the first-order goals ranged from small (.29) to large (.98); all reached the .05 significant levels. As expected, factors within each dimension were found to correlate with each other. The correlation between the approach and avoidance factor was significant though rather small (φ= .10, p < .05); the correlation between the mastery and performance factor was comparatively lager (φ= .35, p < .05). Factors across dimensions (e.g., the approach factor and the mastery factor) were not correlated. The results from the analysis strongly supported the hypothetical second-order model because each fit statistic met the criteria for a good fitting model: χ2(48, N=3137) = 294.15 (p = .000); RMSEA = .040; CFI = .99; GFI= .97. The dimensional
nature of a 2 × 2 achievement goal model was confirmed.
χ2(48, N =3137) = 294.15 (p = .000), RMSEA = .040, CFI = .99, IFI =.99, GFI= .97.
Figure 4-1 The second-order measurement model of achievement goals-dimensional structure.
Estimates are standardized.
Note. All coefficients are significant (p< .01). Error variables are not represented in order to simplify the presentation. V1 to V12 represent the individual items of the scale.
Model comparison
The hypothetical model was further compared to two alternatives. The first was a second-order mastery-performance model where the first-order goals respectively load on only two second-order latent factors: mastery and performance. The second was a second-order approach-avoidance model where the first-order goals respectively load on two other second-order latent factors: approach and avoidance.
As displayed in Table 4-1, the results from these analyses indicated that both alternative models provided good fits to the data; however, the hypothetical model displayed a far better fit than any of the alternative models. In sum, the dimensional nature of the 2 × 2 achievement goal
framework, assuming that the valence dimension was crossed with the definition dimension, was confirmed in the Taiwanese student sample.
Table 4-1 Fit indices of dimensional achievement goal model and other alternative models, all with second-order factor structure (N = 3137)
Overall fit indices
Variable χ2/df CFI IFI RMSEA AIC
Hypothetical second-order model 6.13 .99 .99 .040 354.15 Mastery–performance second-order model 10.41 .98 .98 .055 568.04 Approach–avoidance second-order model 6.73 .99 .99 .043 387.93 Log-likelihood ratio test (model comparison)
df χ2 p
Second-order achievement goal model vs.
Mastery–performance second-order model 1 215.89 < .001 Approach–avoidance second-order model 1 35.78 < .001
Note. CFI = comparative fit index; IFI = incremental fit index;
RMSEA = root-mean-square error of approximation;
AIC = Akaike information criterion.
Testing path coefficient invariance
In a closer look at the results of the hypothetical second-order model, it was found that in each pair of paths from a second-order latent factor to the respective first-order goals the path coefficient disparity within the pair was very large. For example, the path coefficient of the avoidance factor to mastery-avoidance goals (lambda = 0.29, in Figure 4-1) was obviously less than the path coefficient of the avoidance factor to performance-avoidance goals (lambda = 0.98).
Accordingly, further examinations of path coefficient invariance by setting the paths from a second-order factor to its two respective goals as equal (in Table 4-2) were conducted. To test formally the statistical significance of the difference between the two path coefficients, four alternative models were posited. This model comparison approach was appropriate when 4 factors of the valence and definition dimensions were scaled to have a variance of 1 so that their effects are in relation to a standardized metric.
In the constrained model 1, the approach-to-mastery-approach path was constrained to be equal to the approach-to-performance-approach path. If two paths really do differ significantly, then the hypothetical second-order model (two paths are freely estimated) would fit the data significantly better than the constrained model 1. Because the constrained model 1 is nested under the hypothetical second-order model, the chi-square test statistic for the constrained model 1 cannot be any better than that of the hypothetical model. However, if the fit of the constrained model 1 approaches that of the hypothetical model, then two paths do not differ in their contribution to the approach factor. The rest of the path invariance tests followed the same procedure. In the constrained model 2, the avoidance-to-mastery-avoidance was constrained to be equal to the avoidance-to-performance-avoidance path. In the constrained model 3, the mastery-to-mastery-approach path was constrained to be equal to the mastery-to-mastery-avoidance path. In the constrained model 4, the performance-to-performance-approach path was constrained to be equal to the performance-to-performance-avoidance path.
The results (in Table 4-2) showed that the difference in chi-squares for the two models (Δχ2
(Δdf=1) = 537.82, p < .001) was statistically significant. It revealed that the fit of the hypothetical model was better than that of the constrained mode 1. Two path coefficients (approach-to-mastery-approach path and approach-to-performance-approach path) freely estimated by the hypothetical model were not equal (.61 < .93). In the constrained model 2, the coefficients of avoidance-to-mastery-avoidance path and avoidance-to-performance- avoidance path did differ (.29 < .98, Δχ2 (Δdf=1) = 1868.82, p < .001). In the constrained model 3, the coefficients of mastery-to-mastery-approach path and mastery-to-mastery- avoidance path were not equal (.36 < .96, Δχ2 (Δdf=1) = 780.15, p < .001). Finally, in the constrained model 4, the coefficient of performance-to-performance-approach path was not equal to that of performance-to-performance-avoidance path (.17 < .79, Δχ2 (Δdf=1) = 1688.82, p < .001).
It revealed that, in the Taiwanese student sample, each pair of goals has nonequivalent contributions to the correspondent factors. For the valence dimension, the approach factor was mainly derived from the variance of mastery-approach goals (instead of from that of performance-approach goals) while the avoidance factor was mainly derived from the variance of performance-avoidance goals (instead of from that of mastery-avoidance goals). For the
definition dimension, the mastery factor was principally derived from the variance of mastery-avoidance goals (instead of from that of mastery-approach goals); and the performance factor was mostly derived from the performance-approach goals (instead of from that of performance-avoidance goals).
Table 4-2 Path coefficient invariance analyses of the constrained models nested under hypothetical second-order achievement goal model
χ2 (df)
Set path (approach to mastery-approach goals) = path (approach to performance-approach goals)
831.97 (49) p = .000
537.82 (1) p < 0.001 Constrained model 2
Set path (avoidance to mastery-avoidance goals) = path (avoidance to performance-avoidance goals)
2162.23 (49) p = .000
1868.08 (1) p<0.001 Constrained model 3
Set path(mastery to mastery-approach goals) = path(mastery to mastery-avoidance goals)
Set path(performance to performance-approach goals) = path(mastery to performance-avoidance goals)
1982.71 (49) p = .000
1688.56 (1) p<0.001