The context: I am doing a confirmatory factor analysis (CFA) in a longitudinal setting (100+ individuals, 4 time points). The individuals filled out a questionnaire at 4 time points (some exceptions as there are missing data). The questionnaire items are used to measure a Latent Variable (LV), e.g. behaviour / intelligence / motivation. The study I am doing the CFA for involves a treatment or incentive, that is expected (or hoped) to have an effect on this LV. In the example below I will use 'motivation in school' as the LV.
Question 1: If the LV is expected to change over time, should I also be expecting changing factor loadings? E.g. a (linear) increase with time as the 'incentive' to get students in school more motivated.
Question 2: If question 1 is the case, or if factor loadings are fluctuating quite a lot, how should one determine that the measurement model and / or the items in the measurement model are adequate? Differently put: how would I know if I am measuring the same construct over time? (Or at least have some confidence about it).
I am really puzzled about this, because I am probably mixing things up in my head with measurement invariance. For the longitudinal CFA I already did, I 'established' strong / scalar invariance, i.e. the factor loadings are set equal over time and the item intercepts also. The test for strict invariance 'failed'.