Constrained Statistical Inference: A Hybrid of Statistical Theory, Projective Geometry and Applied Optimization Techniques

In many data applications, in addition to determining whether a given risk factor affects an outcome, researchers are often interested in whether the factor has an increasing or decreasing effect. For instance, a clinical trial may test which dose provides the minimum effect; a toxicology study may wish to determine the effect of increasing exposure to a harmful contaminant on human health; and an economist may wish to determine an individual's optimal preferences subject to a budget constraint. In such situations, constrained statistical inference is typically used for analysis, as estimation and hypothesis testing incorporate the parameter orderings, or restrictions, in the methodology. Such methods unite statistical theory with elements of projective geometry and optimization algorithms. In many different models, authors have demonstrated constrained techniques lead to more efficient estimates and improved power over unconstrained methods, albeit at the expense of additional computation. In this paper, we review significant advancements made in the field of constrained inference, ranging from early work on isotonic regression for several normal means to recent advances of constraints in Bayesian techniques and mixed models. To illustrate the methods, a new analysis of an environmental study on the health effects in a population of newborns is provided.