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Abstract:
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In this talk, we introduce methods to perform quantile functional regression, which given a sample of many observations per subject regresses the distribution modeled as a functional object on covariates. We introduce custom basis functions called "quantlets" to represent the quantile functions that are orthogonal and empirically defined, so adaptive to the features of the given data set. After fitting the quantile functional regression, we are able to perform global tests for which covariates have any effect on the distribution, and local tests to identify at which quantiles the difference lies while adjusting for multiple testing and to assess whether the covariate affects certain major aspects of the distribution, including location, scale, skewness, Gaussian-ness, or tails. If the differences lie in these commonly used summaries, our method can still detect them, but our method can also detect effects on other aspects of the distribution that might be missed if one restricted attention to those summaries. We illustrate this method on biomedical imaging data for which we relate the distribution of pixel intensities to various demographic and clinical characteristics.
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