Estimating the incidence rate ratio in cross-sectional studies using a simple alternative to logistic regression

Logistic regression is often used for the analysis of cross-sectional studies, and prevalence odds and odds ratios are obtained. Other methods have been proposed for estimating prevalence ratios. An alternative regression method is also available for estimating rate ratios. Its application to cross-sectional studies is discussed.

When dealing with chronic conditions, it is possible to model binomial data using the complementary log-log link function log(-log(1-pi)), where pi is the prevalence, an option available on many statistical software packages. In effect, these are models for the disease incidence rate lambda, which is assumed to be constant over the underlying follow-up period t. This approach is based on the well-known relationship 1-pi-exp(-lambda t). The cumulative effect of age on prevalence (effectively "time of follow up") can be accounted for in the model, by specifying it as an offset.

The regression coefficients associated with the covariates included in the model estimate rate ratios, rather than odds or prevalence ratios. The method is applied to the analysis of the prevalence of respiratory symptoms in 4395 children aged 7-9 years who are residents of Huddersfield (northern England), surveyed in the framework of the SAVIAH (Small Area Variations of Air Quality and Health) study.

By considering saturated models including only sex as a covariate, direct comparison of crude and fitted parameters (odds, prevalence, and rate ratios) shows that, for short follow-up periods, the complementary log-log model is a valid alternative to logistic regression. More complex models including other covariates are also discussed.