An Assessment of Mixed-Modeling Approaches for Characterizing Profiles of Time-Varying Response and Predictor Variables

Abstract

A general statistical modeling approach was tested for characterizing the relationship between pathogen inoculum density (or other biological response variables) and environmental variables when the data are collected as temporal profiles of observations within multiple locations or years. The approach, based on the use of linear mixed models, simultaneously accounts for serial correlations of the observations within each time profile, the random effects of location–year (or other grouping factors), and the cross-correlation of the environmental variables, and is appropriate when the environmental effects on the response variable or its transformation (Y) are distributed over several times (e.g., days). Stability and precision of parameter estimates for environmental effects over multiple time lags were achieved through the use of polynomial constraints within a likelihood-based full mixed-model fit; from the parameter estimates, marginal effects of environmental variables and weights for individual time lags were determined. The mixed model was directly expanded, through the incorporation of smoothing functions, to potentially account for possible longer-term trends in the temporal profiles unrelated to the environmental variables being considered. The new approach described here (with or without a smoothing function) generalizes a previously used—and computationally less demanding—two-stage (composite) approach. In the previous approach, constrained parameter estimates and associated weights were first determined without consideration of serial correlation, cross-correlation of environmental variables, and the random effects of location–year; then, a mixed-model fit was accomplished using the fixed time-lag weights derived in the first step. Using data for inoculum density of Gibberella zeae on wheat spikes from 27 location–years, similar results were achieved with the full mixed model and the two-stage approaches, in terms of both the calculated parameters and predictions of Y. With the use of smoothing functions, the precision of the predictions was improved but the general conclusions regarding environmental effects on Y were not affected. Thus, in the particular example data set, previously derived conclusions regarding environmental effects on inoculum density were robust in terms of the statistical methodology used in analysis; most researchers will find the two-stage approach much easier to implement for the analysis of multiple profiles of time-varying observations.