Background

Epidemiological studies of rare events, which are common in the medical literature, often involve modelling sparse data sets.

Assessing the fit of these models may be complicated by the large numbers of observed zeros in the data set.

Methods

Poisson models, fitted as generalized linear models, were used to investigate the referral patterns of patients suffering from end-stage renal failure in south west Wales.

The usual method for assessing the goodness of fit is to compare the deviance with a X2 distribution with appropriate degrees of freedom.

However, this test may be invalid when the data set is sparse, as the deviance values may be unusually low compared to the degrees of freedom.

This would suggest that there is a problem with underdispersion when, in fact, the large numbers of zeros in the data set make the comparison with the X2 distribution unreliable.

A simulation approach is advocated as an alternative method of assessing model fit in these situations.

Results

Three models are considered in detail here.

The first modelled the total referrals in each of the 245 wards in the study area and included two explanatory variables.

These observations were not unusually sparse and both the X2 goodness of fit test and the simulation methodology outlined here suggested that the model did not fit.

The second model included the population'at risk'as an offset and the model improved considerably. (...)