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

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Abstract
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.