Inference for Multiple Isotonic Regression

The isotonic regression for two or more independent variables is a classic problem in data analysis. The classical solution involves enumeration of upper sets, which is computationally prohibitive unless the sample size is small. Here it is shown that the solution may be obtained through a single projection onto a convex polyhedral cone. The cone formulation allows an exact test of the null hypothesis of constant versus isotonic regression function. Categorical covariates can be included so that the model is of several hyper-surfaces; then the null hypothesis for the test is that the regression function depends only on the categorical covariate. Simulations show that when the constraints hold, the test tends to have higher power than the standard parametric test that does not use the constraints, even when the parametric assumptions are true. For other types of inference, the speed of the algorithm allows for bootstrap confidence intervals and related inference for moderate-sized samples. The methods are illustrated with data from an observational study concerning diet and lifestyle factors as predictors of blood plasma micronutrients in healthy adults.