by jhines » Wed Sep 21, 2016 2:19 pm
If you added 1 column to the detection design matrix, then only 1 beta estimate will be produced by PRESENCE. That beta value will be labelled with the first "real" parameter name found in the design matrix which has a non-zero value for that column (pA in your case). If that covariate is also in the rows for pB, then the effect for pB is the same as for pA. If the covariate is in all rows of the detection design matrix, then the effect will also apply to rA, rBA and rBa, though you won't see a separate beta for the other "real" parameters.
If you had another column in the design matrix with the covariate in the rows for pB only, then you would get a beta estimate for pB. This model would be different from the one you mentioned in that there would be a different effect of the covariate on detection of species A than species B.