# Mixed effects count modeling

I am trying to predict the probability that someone's vehicle fails its emission inspection a discrete number of times using a Poisson count model (actually a negative binomial hurdle model, but the same logic applies), $$P(y_i | X_i) = \dfrac{\exp(-e^{X\beta})(e^{X\beta})^{y_i} }{y_i!}$$ where the probability of vehicle $i$ failing emissions $y$ times is a function of different predictors. This is a straightforward problem.

What is not straightforward is that households frequently own multiple vehicles, and I need to accommodate the within-household effects. In some ways I can think of this this as an unbalanced panel.

HH  |  VEH  | FAILURES |  X
A       1        0       x_a1
A       2        0       x_a2
B       1        2       x_b1
C       1        1       x_c1
C       1        0       x_c2
.......


It looks like a panel, but $y_t$ has no practical relationship to $y_{t-1}$, nor does the $X$ for different vehicles. Most panel methods I know involve taking the differences of one variable or another. Is there some other way to incorporate correlated observations that would help me with this problem?

Update: There are two packages for R that claim to do this; MCMCglmm and glmmADMB. The latter appears to have a simpler call syntax, as the former requires conversational ability in Bayesian inference. Unfortunately, I cannot get the example scripts for glmmADMB to run on my system (OSX 10.7, R 2.15.2), because they crash with a matrix type error.

• Maybe a mixed model with a random effect for household? I've never tried to do this with this with a Poisson model, but perhaps you can patch it in somehow. – Jonathan Christensen Dec 5 '12 at 17:45
• I think you are exactly right about the mixed models, though I know very little about them. What sort of loss do we suffer in degrees of freedom? – gregmacfarlane Dec 5 '12 at 21:17
• The error on glmmADMB is fixed as of r227 in the subversion repository. – gregmacfarlane Dec 21 '12 at 0:04