Sampling bias and logistic models

Summary. In a regression model, the joint distribution for each finite sample of units is determined by a function px(y) depending only on the list of covariate values x=(x(u1),…,x(un)) on the sampled units. No random sampling of units is involved. In biological work, random sampling is frequently unavoidable, in which case the joint distribution p(y,x) depends on the sampling scheme. Regression models can be used for the study of dependence provided that the conditional distribution p(y|x) for random samples agrees with px(y) as determined by the regression model for a fixed sample having a non‐random configuration x. The paper develops a model that avoids the concept of a fixed population of units, thereby forcing the sampling plan to be incorporated into the sampling distribution. For a quota sample having a predetermined covariate configuration x, the sampling distribution agrees with the standard logistic regression model with correlated components. For most natural sampling plans such as sequential or simple random sampling, the conditional distribution p(y|x) is not the same as the regression distribution unless px(y) has independent components. In this sense, most natural sampling schemes involving binary random‐effects models are biased. The implications of this formulation for subject‐specific and population‐averaged procedures are explored.