# The all-zero treatment problem with conditional binomial data and GLMMs

There are numerous advantages in using generalized linear mixed models (GLMMs) for the analysis of non-normal data, especially discrete data. Yet there are also multiple challenges with using GLMMs that are not of relevance, or of less relevance, when fitting linear mixed models to data. It is usually easy to fit a GLMM to data when the response variable has a binomial distribution conditional on the random effects. This is of high relevance in plant pathology, where disease incidence (number of diseased plants, y, out of n assessed plants in each experimental unit) is typically analyzed. With a GLMM fitted to such data, one can estimate the probability of a plant being diseased, π, through the inverse-link function for each treatment, of direct interest in plant epidemiological research. A not uncommon situation occurs when all of the plants are disease-free for all blocks for one (or more) of the treatments. This all-zero treatment creates a quasi-complete separation problem, making it very challenging to fit a GLMM to the data. Essentially, the model is ill-conditioned with such a dataset. Work to date shows that pseudo-likelihood estimation will not converge with the all-zero treatment problem. Likelihood approximation methods (e.g., Gauss-Hermite quadrature) typically do converge, but parameter estimates for one or more of the treatment effects (e.g., τᵢ in a model with an intercept) are unreliable, even for treatments without all zeroes. Standard errors (SEs) are typically greatly inflated, as are the treatment-effect parameter estimates (τᵢ). However, estimated expected values for some of the treatments, and for some treatment differences, may be reliable. The problematic treatment-effect parameters may not always correspond to the treatment with all zeroes, and it is not always easy to identify difficulties. Parameter estimates and their estimated SEs for a given dataset can depend on the optimization method used with integral approximation (e.g., quasi-Newton vs. Newton-Raphson vs. conjugate gradient optimization). The presentation will discuss implications of some of the ad-hoc ways of adjusting the data to allow for successful model fitting with integral approximation, and seek advice on moving forward when datasets have this all-zero property for some treatments. Bayesian analysis with informative priors for treatments with all zeroes is one potential approach to deal with this challenge.