Variability and uncertainty of variance components in fixed-effects and mixed-effects ground-motion models

It has become common in the development of ground-motion models (GMMs), especially using mixed-effects regression with crossed random effects, to calculate standard deviations, referred to as variance components, from sample statistics of the residuals (i.e. random effects and within-group errors) rather than using the variance components reported by a mixed-effects regression program, calculated using the same algorithms used in the mixed-effects regression, or estimated using Bayesian inference. This practice leads to underestimating the standard deviations because it does not account for the uncertainty (i.e. standard errors) associated with fitting the random effects and within-group errors during the regression analysis. In this study, we used a series of ground-motion models for Fourier amplitude spectra developed using mixed-effects regression of an Next Generation Attenuation (NGA)-West2 database to show that residual-based standard deviations can be significantly smaller than regression-based standard deviations. These differences are exacerbated for those variance components with the fewest number of observations or when the residuals are partitioned (e.g. by earthquake magnitude). Using residual-based variance components not only results in smaller standard deviations but can lead to biased inferences when attempting to compare the efficacy of a model based solely on a comparison of the values of the variance components or their related mean square errors. It can also lead to biased fixed-effects coefficients if the variance components are derived from the residuals of a mixed-effects regression rather than estimated during a regression. We show that variance components can be decomposed into various random-effect grouping factors and partitioned into subsets of predictor variables, such as magnitude, from the total residuals of a non-Bayesian mixed-effects regression program using Bayesian inference. This process does not require the development of an entire GMM using Bayesian mixed-effects regression.