An empirical comparison of inference using order-restricted and linear logit models for a binary response

In many applications with a binary response and an ordinal or quantitative predictor, it is natural to expect the response probability to change monotonically. Two possible models are a linear model with some link, such as the linear logit model, and a more general order-restricted model that assumes monotonicity alone. The order-restricted approach is more complex to apply, and we investigate whether it may be worth the extra effort. Specifically, suppose the order restriction truly holds but a simpler linear model does not. For testing the hypothesis of independence, is there the potential of a substantive power gain by performing an order-restricted test? For estimating a set of binomial parameters, how large must the sample size be before the consistency of the order-restricted estimates and inconsistency of the model-based estimates makes a substantive difference to mean square errors? We conducted a limited simulation study comparing estimators and likelihood-ratio tests for the linear logit model and for the order-restricted model. Results suggest that order-restricted inference is preferable for moderate to large sample sizes when the true probabilities take only a couple of levels, such as in a dose-response experiment when all doses provide a uniform improvement over placebo. If the true probabilities are strictly monotone but deviate somewhat from the linear logit model, the logit-based inference is usually more powerful unless the sample size is extremely large. When the true probabilities may have slight departures from monotonicity, the order-restricted estimates often perform better, particularly for moderate to large samples.