Sampling Bias and Logistic Models

Abstract

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.

Geostatistical Inference Under Preferential Sampling

“…We hold the view that, in most geostatistical applications, spatial correlation reflects, at least in part, smooth spatial variation in relevant, but unobserved, explanatory variables rather than being an inherent property of the phenomenon being studied; an example to the contrary would be the spatial distribution of the prevalence of an infectious disease during an epidemic where, even for a uniformly distributed population in a completely homogeneous environment, the process of transmission from infectious to susceptible individuals would induce spatial correlation in the prevalence surface. This in turn leads us to emphasize that our paper is not a plea for uniform sampling, but rather for ensuring that any model for a set of data should respect whatever sampling design has been used to generate the data; for a thorough discussion that also uses point process models of sampling designs, albeit in a very different setting, see McCullagh (2008).…”

On the Singularities of the Information Matrix and Multipath Change-Point Problems

“…When sampling is biased and the marginal distribution of Z is available, the joint distribution should be used. This issue has been recently discussed by McCullagh [49] and Bergeron, Asgharian, and Wolfson [6]. If, however, the marginal distribution of Z is not available, the conditional approach seems to be appropriate, as discussed by Mandel and Ritov [48], for example.…”

Conditional Logistic Regression With Longitudinal Follow-up and Individual-Level Random Coefficients: A Stable and Efficient Two-Step Estimation Method

“…Perhaps this could be relaxed by generalizing the multiperiod multinomial probit model (Geweke, Keane, and Runkle 1997). Another avenue of interest is to account more explicitly for the spatial sampling process in the inference, as recently suggested by McCullagh (2008). Indeed, he showed that, in cases in which the configuration of covariates is random, the conditional distribution of the response on the sampling units may be different than what is inferred from the marginal model (2.4).…”