Testing for No Effect When Estimating a Smooth Function by Nonparametric Regression: A Randomization Approach

Abstract When a linear regression function is estimated by ordinary least squares, the null hypothesis of no relationship between the response and the design variable can be tested by the normal theory F test. This article describes a new test for use in the case in which a smooth regression function is estimated by a nonparametric procedure such as kernel estimation or local regression. The test is constructed as an approximation to an exact permutation test. The test statistic is the ratio of sums of squares that are defined by analogy to the analysis of variance. The null permutation distribution of the test statistic is approximated by matching its exact mean and variance to the moments of a gamma distribution. A simulation study shows that the approximation is excellent for several regression procedures and for normal, heavy-tailed, and skewed error distributions. Simulation results are also used to investigate the power of the test as a function of the smoothing parameter. The test is applied to data on the relationship between lymphocyte concentrations and immunological status in men with human immunodeficiency virus infection.