Thanks, Ben!
Looks like I need to read-up on what rstanarm is doing behind the scenes. Your post was very educational. However, I worry I am still missing something. I tried running the model with QR = TRUE and all other arguments left at their default, removing the priors and also switching from standardized back to centred predictors (either at -1/1 or mean centred). This corresponds to the following model:
mod_nopriors = stan_glmer(switched ~ cond + order + choices + relprefc1 + ici + bore + fat + (cond|ID),
However, this produces even more extreme estimates:
Estimates:
Median MAD_SD
(Intercept) -6.6 2.6
cond 1.2 0.9
order 0.2 0.6
choices 5.2 3.0
relprefc1 -0.4 0.3
ici -4.7 2.4
bore 0.8 0.5
fat 0.5 0.4
Error terms:
Groups Name Std.Dev. Corr
ID (Intercept) 3.8
cond 3.8 0.14
Num. levels: ID 96
Once back-transformed, the intercept (representing performance at the mean of all predictors) is now .0013. If I re-introduce the prior on my intercept (prior_intercept = normal(-1, 1)), but exclude the prior on my coefficients (I think the help file says you should not use informative priors if QR = TRUE) then it looks a little better:
Estimates:
Median MAD_SD
(Intercept) -2.7 0.4
cond 0.6 0.3
order 0.1 0.3
choices 2.7 1.2
relprefc1 -0.2 0.1
ici -2.1 0.7
bore 0.3 0.2
fat 0.2 0.1
Error terms:
Groups Name Std.Dev. Corr
ID (Intercept) 0.82
cond 0.80 0.14
Num. levels: ID 96
Not bad. Still, the intercept seems lower than I would have expected.
Nonetheless, maybe I am asking the wrong question. What I am trying to accomplish here is to use my model to (a) draw judgments as to the strength of association between each predictor and the dependent measure, and, (b) produce counterfactual plots for each variable that I can include in my article to demonstrate the nature of these associations. For (a) I figure I already have this sorted, using the posterior of each coefficient. In this sense, the absolute value of the intercept is less important. For (b), from skimming various texts on regression, it seems the thing to do is to present a plot depicting the effect of each variable at the mean of all other variables – hence the centred predictors. For example, looking at the effect of cond, I would use:
plot_dat = posterior_linpred(mod, re.form=NA, transform=TRUE, newdata=expand.grid(cond=c(-1,1), order=0, choices=0, relprefc1=0, ici=0, bore=0, fat=0))
This produces estimates of .04 and .11 for cond == -1 and cond == 1, respectively, which I would interpret as representing the mean expected performance in each condition for a “typical” participant (due to re.form=NA). However, the estimates for each condition are very low, and in fact the basic proportion for each condition, ignoring all other predictors, would be .08 and .22 – which do not even fall within the 95% HDI of the relevant estimates. Is there a better way to summarize the results graphically?
Cheers!
Jon