## The Poisson distribution with offset

Sometimes the underlying scale for count data varies with each observation. For example, observations may be made over varying time periods (e.g. number of epileptic seizures measured over different numbers of days for each patient). Alternatively, the underlying scale may relate to some other factor such as the size of a geographical region over which counts of subjects with a specific disease are taken. To take account of such a varying scale, the scale for each observation needs to be utilised in forming the distribution density. The scale variable is often referred to as the offset. The parameter of interest is then the number of counts per unit scale of the offset variable. If we denote the offset variable by t (even though it is not always time) and the observed number of counts by z, the distribution of y = z/t has density function f (y, t) = (¡t)yte-xt/(yt)!

Note that when t = 1, the density function for the Poisson distribution without an offset variable above is obtained, confirming it as a special case of this distribution.