Abstract
It is widely believed that regression models for binary responses are problematic if we want to compare estimated coefficients from models for different groups or with different explanatory variables. This concern has two forms. The first arises if the binary model is treated as an estimate of a model for an unobserved continuous response and the second when models are compared between groups that have different distributions of other causes of the binary response. We argue that these concerns are usually misplaced. The first of them is only relevant if the unobserved continuous response is really the subject of substantive interest. If it is, the problem should be addressed through better measurement of this response. The second concern refers to a situation which is unavoidable but unproblematic, in that causal effects and descriptive associations are inherently group dependent and can be compared as long as they are correctly estimated.
Highlights
This paper is about the interpretation of binary response models when making group comparisons
Sikora (2015, p. 273) tells us that in her study: “To avoid problems inherent in comparing logit coefficients or odds ratios between groups ... the key findings are presented as predicted probabilities that are supplemented with tabulated relative risk ratios...”, and Kleykamp (2013, p. 847) warns us before she reveals her results that: “... group comparisons expressed through interactions are problematic in non-linear models because such models cannot distinguish group coefficient differences from group differences in residual variation or unobserved heterogeneity”
We argue that the first version of the group comparison problem only exists if we are genuinely interested in the unobserved continuous variable and that, if we are, the problem should be resolved by more serious efforts to measure this variable
Summary
We describe the interpretation of coefficients in some common regression models, to the extent that is needed to draw on in later sections. We will consider the interpretation of regression coefficients as causal effects This is closest to the spirit in which the question of group comparisons has been discussed in the literature. We begin by considering models with only one explanatory variable, because the issues which are discussed in Section 3 can be described already in this context. For our purposes it is sufficient to note that regression coefficients with causal interpretations can be validly estimated if the observed data are appropriate for this purpose. This is the case if the values of Xi were randomly assigned to the units and certain other assumptions are satisfied. We will assume throughout that the estimators that we mention are to be applied to such data, but methods and assumptions of estimation and inference are not otherwise discussed
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