# Exploratory factor analysis with factor evolving over the time

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.

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