Problem with correlation between intercept and slope

[MaxNachbauer]

Dear forum

I am struggeling with a problem concerning the correlation between random intercept and random slope. I would be very helpful for some advice.

I am running a random-intercept-random-slope model with two levels (students within classes). My DV is math achievement, my IV is SES (socio-economic status of parents).

This is the model (SES is groupmean-centered):
    MATHij = β0j + β1j*(SESij) + rij
    β0j = γ00 + u0j
    β1j = γ10 + u1j

My problem is, that I receive different results for the correlation between random intercept and random slope:
- If I use the covariance between intercepts and slopes, the variance of the intercepts and the variance of the slopes (all provided in the normal output), the correlation is r = 0.224
- If I save the residuals of the intercepts and slopes and correlate them, the correlation is r = 0.380

Does anybody know, what the reason for the different values of the correlation is?

Best

Max

[GeorgeLeckie]

Hi Max,

The correct correlation to report is the one based on the estimated variances and co-variances.

The correlation based on the predicted random effects differs in part due to the fact these are shrinkage estimates.

You should find some coverage of this in various multilevel text books.

As an aside if SES is group-mean centred then I would have thought you would also want to include group mean SES as a second covariate, otherwise you implicitly assume that there is no school-level association between SES and math achievement.

Best wishes

George