# Consideration for the use of linear quantile mixed models for describing highly variable data in agriculture

Quantile regression (QR) has become well established in recent decades for characterizing relationships between a response variable (y) and covariables. Instead of modeling conditional expected values, in QR one models the conditional quantiles, with median regression being a special case. QR is usually considered desirable when the effects of the covariate(s) on the quantile depend on the quantile; that is, when the relationship between the center of the distribution and a covariate is different from the relationship at more extreme quantiles. The usual estimation methods are distribution free and are usually considered robust for the conditional median. The data in agricultural studies are often clustered, with correlated observations within clusters. Several proposals have been made to account for the correlations in QR, with quantile mixed modeling (QMM) receiving the most attention since about 2007. QMM generalizes parametric random-coefficient mixed modeling that is frequently used for clustered normal data. The key step is to consider the asymmetric Laplace (AL) distribution as the working model for the distribution of y conditional on the random effects. By fixing the skewness parameter τ of the AL (0 < τ < 1), the location parameter of the AL is the conditional quantile, μ(τ). Then, estimation becomes a maximum likelihood problem. But there are many challenges with this approach related to computation (optimization), inference, and prediction (including the BLUPs), with questions remaining regarding bias and precision of parameter estimates. Bayesian estimation is a valuable alternative to frequentist MLE. I will present an exploration of the use of linear QMM to describe data for wheat yield and quality in relation to the severity of symptoms of the disease Fusarium head blight. Both frequentist (lqmm package in R) and Bayesian approaches (PROC MCMC in SAS) will be presented. The difficulties in taking a QMM approach will be discussed.