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Analysis of Variance and F-Tests for Partial Linear Models With Applications to Environmental Health Data

Fish consumption during pregnancy exposes the fetus to both the neurotoxicant methylmercury and nutrients known to be beneficial for brain development. When nutrient status is not measured, maternal methylmercury levels may be a partial biomarker for both toxic and nutrient exposures. It is therefore necessary to employ a flexible model—such as the partial linear model—that will allow for possible nonlinear trends of methylmercury. To enhance interpretations of fitting a partial linear model, we propose analysis of variance (ANOVA) inference tools including ANOVA decomposition and significance tests. The ANOVA decomposition explicitly gives the proportion of variation explained by the model and separates the contributions from the parametric and nonparametric components. Semiparametric F-tests are constructed based on ANOVA decomposition with the normality assumption. The proposed F-tests are applicable to testing significance of the parametric, the nonparametric, and the combination of both. The ANOVA investigation also yields new byproduct estimators, which can be viewed as penalized least squares estimators. Simulation results demonstrate that the performance of the new estimators and ANOVA F-tests is comparable to alternative methods in practical applications. This methodology is applied to reanalyze the Seychelles Child Development Study Main Cohort data to explore nonlinear relationships of prenatal methylmercury exposure through maternal fish consumption with prenatal and postnatal child development.