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The lowest level random part of NBD does not really admit to much interpretation; it really allows for more overdispersion than a Poisson, and when it is over dispersed the standard errors of the fixed part take account of this overdispersion and the partitioning of the variance with higher levels can change.

In a Poisson model the level 1 variance if fitted as a linear function of the mean (derived from the fixed part), so that the estimated parameter will be constrained to 1; that is the level -1 variance will be equal to the means (that is what bcons.1 is doing in MLwiN); the variance is not a freely estimated parameter.

In the NBD model, the level -1 variance is a quadratic function of the mean; there is the linear bit which is constrained (associated with bcons1) and the quadratic bit (associated with bcons2.1) that is now estimated. Somewhat surprisingly here this is a negative value (indicating underdispersion) but is numerically not very large. (-0.04). I suggest you fit a strict Poisson (in effect constrain the quadratic term to 0.00) and see if it makes a big difference to the parameters of real interest. You can get underdispersion when analyses a correlated repeated measures data but I have not seen much on it.