I am studying neighbourhood effects on antisocial behaviour using multilevel modelling. The dependent variable, antisocial behaviour is a count data which varies between 0 and 10 which is a skewed data. I am wondering if I can still use multilevel regression. I was suggested to use multilevel Poisson regression. But I was not sure which method would be better for my data and also practical to use.
Cound variables are skewed by nature and Poisson regression goes into the right direction. But, there is one problem:
Poisson distributions have only one parameter lambda that controls mean and variance, as lambda = mu = var. Typically, the distribution of real data has more variance than allowed by Poisson distribution, which is called overdispersion. If not accounted for, hypothesis testing and certainty estimates (confidence limits) become unreliable.
Within the framework of Generalized Linear Multilevel Models there exist two approaches for dealing with overdispersion:
I describe both approaches in chapter 7.4.3 of my book:
https://www.researchgate.net/project/Book-New-statistics-for-the-design-researcher
Cound variables are skewed by nature and Poisson regression goes into the right direction. But, there is one problem:
Poisson distributions have only one parameter lambda that controls mean and variance, as lambda = mu = var. Typically, the distribution of real data has more variance than allowed by Poisson distribution, which is called overdispersion. If not accounted for, hypothesis testing and certainty estimates (confidence limits) become unreliable.
Within the framework of Generalized Linear Multilevel Models there exist two approaches for dealing with overdispersion:
I describe both approaches in chapter 7.4.3 of my book:
https://www.researchgate.net/project/Book-New-statistics-for-the-design-researcher
It is possible to model count data with a multilevel Poisson model and moreover it is possible to use a multilevel model to model explicitly non - independence (contra to what I think @Dongshan Zhu writes). The key characteristic of the Poisson model is that the residual variance at level 1 (after taking account of any predictors and any multilevel structure) is equal to the mean (derived by the fixed part estimates). It may be however that this residual variance is in reality greater than the mean and then people use Negative Binomial Distribution models (and lots of other variants). An example would be counts of the number of days in hospital where they average is say around 3-4 days but some people are in for 3 weeks, so that there is a long positive tail to the distribution of the outcome.
This has a chapter on multilevel modelling using Poisson and NBD models and the final chapter covers modelling additional spatial dependence. Both chapters also cover the very useful concept of the offset whereby you can model observed counts in relation to expected counts .
Book Developing multilevel models for analysing contextuality, he...
And here are some accompanying slides
http://www.bristol.ac.uk/cmm/software/support/workshops/materials/drd.html
This post is interesting on the need for explicitly modelling data with a large number of zeros
https://statisticalhorizons.com/zero-inflated-models
Finally I should add that it is not the observed count that is logged which would be the case if you did it yourself prior to model but the predicted or estimated value. In practice at the first iteration of the generalized linear model the observed count is indeed logged ( a small value or start is usually added to get away from an observed count of zero which is undefined when log transformed ) and the parameters of the model are estimated , subsequently these estimates are then used to derive the expected count and it is these expected values in the log form that are modeled in later iterations. Usually and eventually after several iterations the model estimates will converge. The log link of the generalized exponentiates the function of the linear predictors; it does not log transform the outcome variable. For the importance of this distinction see
https://stats.stackexchange.com/questions/47840/linear-model-with-log-transformed-response-vs-generalized-linear-model-with-log/48679
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