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Drawing an analogy, my task can be phrased like this: Evaluate a person's IQ based on several tests and examine how IQ changes over time.

I do $K$ different intelligence tests for each person several times (Day 1, Day 2,...). I need to find the factor responsible for IQ and estimate how it changes over time.

If I just needed to find IQ scores, I could do an exploratory factor analysis. Find $M$ important experiments from $K$ and calculate the loadings $L_1,L_2...L_M$.

For example, I may find these loading for the day 1 and use for other days, then I will be able to estimate the IQ dynamic. But in this case i will loose a lot of information from Day 2, and 3.

I try to formalise the problem: if i have some factor $F$ and Variables $V_1,V_2,..V_K$

$V_1 = L_1*F + N(\mu_1,\sigma_1)$

$....$

$V_K = L_K*F + N(\mu_K,\sigma_K)$

this will lead me to a usual factor analysis.

But my factors evolve during experiments

$F = w*DayN+N(\mu_d,\sigma_d)$

Could you help me to estimate $L_{1-K}$ and $w$?

Additional problem: the number of days vary between observers from 3 to 6.

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  • $\begingroup$ If the nature of IQ is changing over time, how can you examine how it changes over time? It's not the same thing. $\endgroup$
    – Jeremy Miles
    Jan 20, 2021 at 19:17

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A multilevel exploratory factor analysis is the solution. You can check an illustration here. This will also account for the varying numbers of days between participants.

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