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On Testing an Unspecified Function Through a Linear Mixed Effects Model with Multiple Variance Components
Author(s) -
Wang Yuanjia,
Chen Huaihou
Publication year - 2012
Publication title -
biometrics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.298
H-Index - 130
eISSN - 1541-0420
pISSN - 0006-341X
DOI - 10.1111/j.1541-0420.2012.01790.x
Subject(s) - mathematics , variance function , nonparametric statistics , statistics , linear model , likelihood ratio test , variance based sensitivity analysis , mixed model , variance (accounting) , one way analysis of variance , analysis of variance , linear regression , accounting , business
Summary We examine a generalized F ‐test of a nonparametric function through penalized splines and a linear mixed effects model representation. With a mixed effects model representation of penalized splines, we imbed the test of an unspecified function into a test of some fixed effects and a variance component in a linear mixed effects model with nuisance variance components under the null. The procedure can be used to test a nonparametric function or varying‐coefficient with clustered data, compare two spline functions, test the significance of an unspecified function in an additive model with multiple components, and test a row or a column effect in a two‐way analysis of variance model. Through a spectral decomposition of the residual sum of squares, we provide a fast algorithm for computing the null distribution of the test, which significantly improves the computational efficiency over bootstrap. The spectral representation reveals a connection between the likelihood ratio test (LRT) in a multiple variance components model and a single component model. We examine our methods through simulations, where we show that the power of the generalized F ‐test may be higher than the LRT, depending on the hypothesis of interest and the true model under the alternative. We apply these methods to compute the genome‐wide critical value and p ‐value of a genetic association test in a genome‐wide association study (GWAS), where the usual bootstrap is computationally intensive (up to 10 8 simulations) and asymptotic approximation may be unreliable and conservative.