The value of generalized linear mixed models for data analysis in the plant sciences

Modern data analysis typically involves the fitting of a statistical model to data, which includes estimating the model parameters and their precision (standard errors) and testing hypotheses based on the parameter estimates. Linear mixed models (LMMs) fitted through likelihood methods have been the foundation for data analysis for well over a quarter of a century. These models allow the researcher to simultaneously consider fixed (e.g., treatment) and random (e.g., block and location) effects on the response variables and account for the correlation of observations, when it is assumed that the response variable has a normal distribution. Analysis of variance (ANOVA), which was developed about a century ago, can be considered a special case of the use of an LMM. A wide diversity of experimental and treatment designs, as well as correlations of the response variable, can be handled using these types of models. Many response variables are not normally distributed, of course, such as discrete variables that may or may not be expressed as a percentage (e.g., counts of insects or diseased plants) and continuous variables with asymmetrical distributions (e.g., survival time). As expansions of LMMs, generalized linear mixed models (GLMMs) can be used to analyze the data arising from several non-normal statistical distributions, including the discrete binomial, Poisson, and negative binomial, as well as the continuous gamma and beta. A GLMM allows the data analyst to better match the model to the data rather than to force the data to match a specific model. The increase in computer memory and processing speed, together with the development of user-friendly software and the progress in statistical theory and methodology, has made it practical for non-statisticians to use GLMMs since the late 2000s. The switch from LMMs to GLMMs is deceptive, however, as there are several major issues that must be thought about or judged when using a GLMM, which are mostly resolved for routine analyses with LMMs. These include the consideration of conditional versus marginal distributions and means, overdispersion (for discrete data), the model-fitting method e.g., maximum likelihood (integral approximation), restricted pseudo-likelihood, and quasi-likelihood, and the choice of link function to relate the mean to the fixed and random effects. The issues are explained conceptually with different model formulations and subsequently with an example involving the percentage of diseased plants in a field study with wheat, as well as with simulated data, starting with a LMM and transitioning to a GLMM. A brief synopsis of the published GLMM-based analyses in the plant agricultural literature is presented to give readers a sense of the range of applications of this approach to data analysis.