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Is it appropriate to use a negative binomial regression with fixed effects when the dependent variable appears somewhat normally distributed within each group, but the overall distribution of the dependent variable appears more like a negative binomial distribution?

I'm using multilevel modeling to test hypotheses about "within group" relationships, without any need to make assumptions about variation between them. For that reason, I'm opting for a "fixed effects" model. Theory would lead us to use a multilevel count model, and overall analysis of the dependent variable supports the claim. The skew of the variable is 231.49 and its kurtosis is 77899.

Here's the log-transformed distribution of the dependent variable:

log-transformed distribution of the dependent variable

However, when we look at the within-group variation, the distributions have skewness closer to 0. Here is a histogram of the per-group skewness for a subset of groups:

histogram of per-group skewness

Given this, when using a fixed effects model to estimate the within-group variation, should I try to fit a fixed effects count model or a fixed effects linear model?

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In my humble opinion, it doesn't matter what the larger population looks like as you state you are concerned with in-group analysis.

Assuming that each group is correctly and consistently tagged, one could treat those as segments and create a unique model for each group. This is commonly done.

This is similar to what CART does. It divides up the population first by a tree and then has regression models at the ends of the branches.

Remember Box wrote: "The most that can be expected from any model is that it can supply a useful approximation to reality: All models are wrong; some models are useful".

As for your exact question, I recommend trying both and see which one performs better*. It is not uncommon to find that a model performs better under a distribution that it doesn't "belong".

As for better, you must use heuristics that matter to the real world. So avoid R squared.

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Rather than the distributions themselves, the assumption about the shape of the error term is what should guide you to the appropriate link function, whether it is the identity link or some other link.

Above you took the log of the dependent variable to show that it appears approximately normal. What does the distribution of the untransformed dependent variable look like?

Because we cannot look at the true error, the next step would be to create model with your predictor under the linear/identity link and plot the residuals. The outcome variable should be in its original metric rather than log-transformed. Are they skewed such that you would consider them Poisson distributed or some other distribution? If so, then the identity is inappropriate and you need to switch to a log link.

The last piece you can use to aid in your decision making would be model fit. If you fit a model first with the identity link and second with the log link, what are the values of the deviance or AIC?

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