Adjusting Bias of Wald Test for Smooth Components and Their Interactions in Generalized Additive Model

Abstract

Generalized additive model (GAM) uses covariates based on smooth functions and can easily predict non-linear relationship between response variables and covariates. However, in spite of its flexibility, it has been known that p-values for its Wald and likelihood ratio tests do not preserve the nominal significance levels. S. N. Wood (2013) found that Wald statistics follow the mixture of weighted chi-square distribution and it has been often utilized for statistical inference. However, its performance was not carefully investigated and I found that it can lead to inflated results in certain scenarios. In my thesis, I extended his method and the proposed method was evaluated with simulation data for various hypothesis tests such as joint test for two or more smooth functions or interactions. With extensive simulations, I confirmed that the proposed method generally performs better than Woods method. Furthermore, the proposed method was applied to the gene-by-smoking interaction association analyses. Four SOX9-associated SNPs were known to be associated with lung functions, and their interaction effect with pack-years was evaluated with Korean cohort data. Interaction test between pack-years and SNPs with linear mixed effects model was not significant but a generalized linear mixed model resulted in significant interaction, which reveals that GAM is useful for covariates with nonlinear relationships with response variables. In conclusion, GAM is useful for modeling non-linear relationship and the proposed method enables valid statistical inferences.