4 - A general formulation

Summary

• Logistic regression can be used to estimate odds ratios using data from a case-control sample as though the data had arisen prospectively. This allows regression adjustment for background and confounding variables and makes possible the estimation of odds ratios for continuous exposures using case-control data.

• The logistic regression of case-control data gives the correct estimates of log odds ratios, and their standard errors are as given by the inverse of the information matrix.

• The logistic regression model is in a special class of regression models for estimating exposure-outcome associations that may be used to analyse case-control study data as though they had arisen prospectively. Another regression model of this type is the proportional odds model. For other models, including the additive risk model, case-control data alone cannot provide estimates of the appropriate parameters.

• Absolute risks cannot be estimated from case-control data without additional information on the proportions of cases and controls in the underlying population.

Preliminaries

The previous chapters have introduced the key features of case-control studies but their content has been restricted largely to the study of single binary exposure variables. We now give a more general development. The broad features used for interpretation are as before:

1. • a study population of interest, from which the case-control sample is taken;

2. • a sampling model constituting the model under which the case-control data arise and which includes a representation of the data collection process;

3. • an inverse model representing the population dependence of the response on the explanatory variables; this model is the target for interpretation.