# Modeling multinomial problems with unknown sample size in BUGS

I'm trying to estimate the design effect of a series of relatively small sample size surveys ($n\sim 70$) with multiple responses. Design effects roughly correspond to how much larger actual sample variance than would be expected from naive random sampling. The simplest way to parametrize this is for Effective Sample Size to be equal to SampleSize/D, where D is an unknown parameter.

Throwing it naively into BUGS is impossible due to the inability to handle multinomial models with unknown sample size. I tried to do a multinormal approximation with a structured covariance matrix, but I can't figure out/find a closed form for the precision matrix of a multinomial variable.

Does anyone have any ideas about the best way to proceed? This seems like it should be a common problem.

The covariance matrix has diagonal entries $np_i(1-p_i)$ and off-diagonal entries $-np_ip_j$. JAGS and BUGS allow you to invert a matrix numerically (sigma[1:K, 1:K] <- inverse(tau[,]) in WinBUGS), so you don't actually need a closed-form expression for the precision matrix.