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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'.

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  1. No. You hope that the loadings stay the same. If they don't, the latent variables aren't the same thing.
  2. People don't usually worry about strict invariance. You do worry about loadings and intercepts.
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  • $\begingroup$ Thanks. This clears things up for a part. In my situation, or perhaps in any situation: would you always prefer scalar invariance (intercepts) over metric / weak invariance (factor loadings)? $\endgroup$
    – Amonet
    Commented Mar 19, 2018 at 9:26
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    $\begingroup$ It depends on what sorts of comparisons you want to make between groups/over time. If you want to make valid comparisons of latent covariances/variances/correlations/slopes, you need metric invariance; if you want to make valid comparisons of latent means, you need strong (scalar) invariance. $\endgroup$
    – jsakaluk
    Commented Mar 20, 2018 at 22:01

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