Interpret var (bcons.1) & var (bcons2.1)

[Zhen2018]

Hi, I fitted a negative binomial model in MLwiN , and wondered if anyone could help me interpret the random part of the output please. It shows var (bcons.1) =1.00 (SE: 0.00) , var (bcons2.1) = - 0.04 (SE: 0.00). I would be very grateful if you could advise me how to interpret var (bcons.1) and var (vcons2.1). Thanks!

[ChrisCharlton]

I passed this question on to Professor Kelvyn Jones and he gave the following answer:

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.

[Zhen2018]

Many thanks to you and Professor Kelvyn Jones for answering my questions. That's very helpful!