I'm running a selection of MCMCglmm models in R, and have a basic question about the outputs.

Based on what I've been reading, one indication that the model is mixing well is a large effective sample size. I'm finding this is true for most predictor/independent variables in the model -- with eff.samp ranging somewhere in the hundreds or thousands (my data set is based on about n = 800 dyads as data points).

However, for some of the terms I'm finding the effective sample size is extremely small - as in, around 10. I would think this is a sign to be suspicious of the output for these variables in particular, yet these variables often have very small p values (<.001 or even .000).

I've been unable to find a clear explanation of the effective sample size and its relationship to the posterior estimates and pMCMC values. What does a small effective sample size mean, in this type of model?

In case it's relevant, some background on the data: My data are hierarchically nested by dyad ID, and by individual #1 & #2 IDs, and also by geographic location (2 possible locations). I have one response variable and up to four predictor variables in this set of models. The response variable is count data and is zero-inflated, so I am using a zero-inflated poisson distribution.