TY - JOUR

T1 - Estimation of variance components, heritability and the ridge penalty in high-dimensional generalized linear models

AU - Veerman, Jurre R.

AU - Leday, Gwenaël G. R.

AU - van de Wiel, Mark A.

PY - 2019/1/1

Y1 - 2019/1/1

N2 - For high-dimensional linear regression models, we review and compare several estimators of variances τ2 and σ2 of the random slopes and errors, respectively. These variances relate directly to ridge regression penalty λ and heritability index h2, often used in genetics. Several estimators of these, either based on cross-validation (CV) or maximum marginal likelihood (MML), are also discussed. The comparisons include several cases of the high-dimensional covariate matrix such as multi-collinear covariates and data-derived ones. Moreover, we study robustness against model misspecifications such as sparse instead of dense effects and non-Gaussian errors. An example on weight gain data with genomic covariates confirms the good performance of MML compared to CV. Several extensions are presented. First, to the high-dimensional linear mixed effects model, with REML as an alternative to MML. Second, to the conjugate Bayesian setting, shown to be a good alternative. Third, and most prominently, to generalized linear models for which we derive a computationally efficient MML estimator by re-writing the marginal likelihood as an n-dimensional integral. For Poisson and Binomial ridge regression, we demonstrate the superior accuracy of the resulting MML estimator of λ as compared to CV. Software is provided to enable reproduction of all results.

AB - For high-dimensional linear regression models, we review and compare several estimators of variances τ2 and σ2 of the random slopes and errors, respectively. These variances relate directly to ridge regression penalty λ and heritability index h2, often used in genetics. Several estimators of these, either based on cross-validation (CV) or maximum marginal likelihood (MML), are also discussed. The comparisons include several cases of the high-dimensional covariate matrix such as multi-collinear covariates and data-derived ones. Moreover, we study robustness against model misspecifications such as sparse instead of dense effects and non-Gaussian errors. An example on weight gain data with genomic covariates confirms the good performance of MML compared to CV. Several extensions are presented. First, to the high-dimensional linear mixed effects model, with REML as an alternative to MML. Second, to the conjugate Bayesian setting, shown to be a good alternative. Third, and most prominently, to generalized linear models for which we derive a computationally efficient MML estimator by re-writing the marginal likelihood as an n-dimensional integral. For Poisson and Binomial ridge regression, we demonstrate the superior accuracy of the resulting MML estimator of λ as compared to CV. Software is provided to enable reproduction of all results.

UR - https://www.scopus.com/inward/record.uri?partnerID=HzOxMe3b&scp=85070841242&origin=inward

U2 - 10.1080/03610918.2019.1646760

DO - 10.1080/03610918.2019.1646760

M3 - Article

JO - Communications in Statistics: Simulation and Computation

JF - Communications in Statistics: Simulation and Computation

SN - 0361-0918

ER -