Jeff,
Yes to both - "Tpref" is the variable used for groups and the model does have different values of p for each group. I removed Tpref from the dataframe in covariate.predictions since, as you said, the indices are different for each group (thanks - that makes sense now). The predicted values are now different for each combination of the individual covariate and the Tpref group, which is what I expected. The predicted real parameter estimates match the subset of combinations I checked by "hand" (Excel).
Thanks for setting me straight!
Joe