Up to this point, the response variables have all been continuous measurements such as weights, heights, lengths, temperatures and growth rates. A great deal of the data collected by scientists, medical statisticians and economists, however, is in the form of counts (whole numbers or integers). The number of individuals who died, the number of firms going bankrupt, the number of days of frost, the number of red blood cells on a microscope slide, and the number of craters in a sector of lunar landscape are all potentially interesting variables for study. With count data, the number 0 often appears as a value of the response variable (consider, for example, what a 0 would mean in the context of the examples just listed). In this chapter we deal with data on frequencies, where we count how many times something happened, but we have no way of knowing how often it did not happen (e.g. lightning strikes, bankruptcies, deaths, births). This is in contrast to count data on proportions, where we know the number doing a particular thing, but also the number not doing that thing (e.g. the proportion dying, sex ratios at birth, proportions of different groups responding to a questionnaire).
Straightforward linear regression methods (assuming constant variance, normal errors) are not appropriate for count data for four main reasons:
- The linear model might lead to the prediction of negative counts.
- The variance of the response variable is likely to increase with the mean. ...