Correlations shift to zero with increased predictors

I’m working with a hierarchical model from an experiment with multiple variables manipulated orthogonally within each of many human participants, with multiple observations in each condition specified by the full crossing of the manipulated variables. In lme4 syntax, the model would be akin to Y ~ A*B*C*D + (A*B*C*D|Participant). I have some decent prior expectation for particular correlations (across participants) amongst some of the variables’ effects on the outcome, but opt for a lkj_corr_cholesky(1) prior for simplicity. When I run the model with just one or two of the variables (ex. Y ~ A+B + (A+B|Participant)), I get posterior distributions on the correlations that mesh with my expectations (ex. participants’ deviations in the effect of A correlate with participants’ deviations in the effect of B), but when I run the model with all the variables and their interactions (and hence a much larger correlation matrix), the posterior distributions on all the correlations shift towards zero considerably. Any idea what might be going on?

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The variance of the marginal correlations shrinks as K increases under an LKJ prior with fixed shape parameter.

Ah, and I presume it wouldn’t be sensible to “avoid” this by providing no prior on the correlation matrix?

“No” prior is equivalent to the LKJ prior with shape 1, in which case it will still have smaller marginal variances as K increases.

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