A Compound Logistic Regression Model for Binary Responses

Logistic regression is the most commonly used method for constructing predictive models for binary responses. One significant drawback to this approach, however, is that the asymptotes of the logistic response function are fixed at 0 and 1, and there are many applications for which this constraint is inappropriate. More flexible models have been proposed for this application, most proceeding by supplementing the logistic response function with additional parameters. In this article we extend these models to allow correlated responses and the inclusion of covariates. This is achieved through the \emph{compound logistic regression model}, for which the mean response is a function of several logistic regression functions. This permits a greater variety of models, while retaining the advantages of logistic regression.

💡 Research Summary

The paper introduces a “compound logistic regression” framework that extends the classic logistic model for binary outcomes by allowing the asymptotic probabilities (lower and upper limits) to differ from the fixed values of 0 and 1. The authors begin by highlighting the limitation of the standard logistic function, which forces the response curve to approach 0 and 1 at the extremes of a predictor, a restriction that is often unrealistic in biomedical applications such as correlates of protection (COP) for vaccines.

To overcome this, they start from the generalized logistic form proposed by Richards (1959) and simplify it by fixing two shape parameters, yielding a four‑parameter model: lower asymptote (a_L), upper asymptote (a_U), a threshold location (z_{thr}), and a slope parameter (\alpha). In its simplest one‑dimensional version the model is

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