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After establishing the invariance of the factor structure, differences in the latent means between the gender groups and the educational-level groups were calculated.
The study also investigated the existence of systematic country/cultural differences in the latent factor means for these constructs and mathematics achievements.
Therefore, partial measurement invariance across the elementary school and the middle school sample was established, and differences in the latent means between both groups were subsequently computed.
Differences in the latent mean achievement and the motivational construct were similar to those that have been described in the literature as "paradoxical" and "perplexing".
The empirical examples demonstrate the importance of addressing endogenous explanatory variables in latent regressions for Item Response Theory (IRT) models when estimating causal differences in the latent variable or examining differential item functioning.
Measurement invariance is a fundamental property for an instrument designed to provide group comparisons, as it ensures that mean differences in scores reflect true differences in the latent variable instead of systematic biases in the items.
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This implies that the gender difference in the latent factor means of distress stays the same, in standardized units, over the course of the study.
That is, if a difference of the CES-D scores between two group means is observed, one would want to be sure it is caused by a difference in the latent construct of interest, not by response bias.
Although the differences in the groups' latent means were similar, whether the non-invariant items were considered or excluded, future gender and educational-level comparisons of the scores obtained in the Ktpc should be interpreted cautiously, given that only partial measurement invariance was obtained.
For these groups, we found significant differences in the slope of the latent period-spore production relationship (latent period × combination interaction, Table 4).
The first key question of the present paper was whether these imputation methods would allow unbiased estimation of subpopulation differences and coefficients in the latent regression model.
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