This open access book offers an introduction to mixed generalized linear models with applications to the biological sciences, basically approached from an applications perspective, without neglecting the rigor of the theory. For this reason, the theory that supports each of the studied methods is addressed and later - through examples - its application is illustrated. In addition, some of the assumptions and shortcomings of linear statistical models in general are also discussed. An alternative to analyse non-normal distributed response variables is the use of generalized linear models (GLM) to describe the response data with an exponential family distribution that perfectly fits the real response. Extending this idea to models with random effects allows the use of Generalized Linear Mixed Models (GLMMs). The use of these complex models was not computationally feasible until the recent past, when computational advances and improvements to statistical analysis programs allowed users to easily, quickly, and accurately apply GLMM to data sets. GLMMs have attracted considerable attention in recent years. The word "Generalized" refers to non-normal distributions for the response variable and the word "Mixed" refers to random effects, in addition to the fixed effects typical of analysis of variance (or regression). With the development of modern statistical packages such as Statistical Analysis System (SAS), R, ASReml, among others, a wide variety of statistical analyzes are available to a wider audience. However, to be able to handle and master more sophisticated models requires proper training and great responsibility on the part of the practitioner to understand how these advanced tools work. GMLM is an analysis methodology used in agriculture and biology that can accommodate complex correlation structures and types of response variables.

In the generalized linear model (GLM) (which is not highly general) y = Xβ + ϵ, the response variables are normally distributed, with constant variance across the values of all the predictor variables, and are linear functions of the predictor variables. Transformations of data are used to try to force the data into a normal linear regression model or to find a non-normal-type response variable transformation (discrete, categorical, positive continuous scale, etc.) that is linearly related to the predictor variables; however, this is no longer necessary. Instead of using a normal distribution, a positively skewed distribution with values that are positive real numbers can be selected. Generalized linear models (GLMs) go beyond linear mixed models, taking into account that the response variables are not of continuous scale (not normally distributed), GLMs are heteroscedastic, and there is a linear relationship between the mean of the response variable and the predictor or explanatory variables.

Sugarcane, a highly productive crop, is frequently challenged by different biotic agents, such as the pokkah boeng disease that can cause drastic yield losses of up to 40%. This airborne fungal disease is caused by various Fusarium species integrated into a complex. Integrating novel compounds and biological control agents is of paramount importance to cope with these fungi in sustainable systems. In this study, we aimed to evaluate the effect and compatibility of phosphite (Phi) and Trichoderma sp. in the control of Fusarium sp., in vitro and in planta. Using in vitro tests, we evaluated the effects of Phi (0, 500, 1000, 2000, 4000, and 8000 µg mL⁻¹) and the compatibility of Phi + Trichoderma (isolates Taz 001, 013, and 016) on the pathogen complex. Using in planta tests, we evaluated the compatibility of Phi + Trichoderma (Taz-016) in the control of Fusarium in sugarcane plants under greenhouse conditions. A synergistic effect of Phi + Trichoderma was observed in vitro on the control of the pathogen, especially when combining 1000 µg mL⁻¹ + Trichoderma Taz-016. In the in planta trial, combining 4000 µg mL⁻¹ Phi + Trichoderma Taz-016 showed the best control of Fusarium infection, improving plant height, culm length and leaf dry weight.

The use of agrochemicals has caused environmental problems and toxicity to humans, so natural alternatives for disease control during harvest and postharvest have been evaluated. The aim of this study was to evaluate cinnamon essential oil, neem oil, and black sapote fruit extract for in vitro inhibition of fungi isolated from chayote fruit. The extracts were applied at 300, 350, and 400 ppm in Petri dishes and the mycelial growth of Fusarium oxysporum, Fusarium solani, Goetrichum sp., and Phytophthora capsici was evaluated for 7 days, and the percentage of mycelial growth inhibition per day was calculated. Cinnamon oil showed a fungicidal effect at all concentrations. Neem oil at 400 ppm showed a 42.3% reduction in the growth of F. solani and 27.8% reduction in the growth of F. oxysporum, while at 350 ppm it inhibited the mycelial growth of Phytophthora capsici by 53.3% and of Goetrichum sp. by 20.9%; finally, the black sapote extract at 400 ppm inhibited 21.9–28.6% of the growth of all fungi. The growth of postharvest fungi on chayote fruit could be prevented or reduced by applying the plant extracts evaluated at adequate concentrations.

Generalized linear mixed models (GLMMs) have been recognized as one of the major methodological developments in recent years, which is evidenced by the increased use of such sophisticated statistical tools with broader applicability and flexibility. This family of models can be applied to a wide range of different data types (continuous, categorical (nominal or ordinal), percentages, and counts), and each is appropriate for a specific type of data. This modern methodology allows data to be described through a distribution of the exponential family that best fits the response variable. These complex models were not computationally possible up until recently when advances in statistical software have allowed users to apply GLMMs (Zuur et al. 2009; Stroup 2012; Zuur et al. 2013). Researchers in fields other than statistical science are also interested in modeling the structure of data. For example, in the social sciences there have been applications in the field of education when several tests are applied to students; in longitudinal personality studies when the occurrence of an emotion is repeatedly observed over time over a set of people; and in surveys to investigate the political preference of a population, among others.

In this chapter, we will review generalized linear mixed models (GLMMs) whose response can be either a proportion or a percentage. For proportion and percentage data, we refer to data whose expected value is between 0 and 1 or between 0 and 100. For the remainder of this book, we will refer to this type of data only in terms of proportion, knowing that it is possible to change it to a percentage scale only when multiplying it by 100. Proportions can be classified into two types: discrete and continuous. Discrete proportions arise when the unit of observation consists of N distinct entities, of which individuals have the attribute of interest ^“y. ^”N must be a nonnegative integer and ^“y^” must be a positive integer; here, y ≤ N. Therefore, the observed proportion must be a discrete fraction, which can take values $$ \frac{0}{N},\frac{1}{N},\cdots, \frac{N}{N} $$0 N , 1 N , ⋯ , N N . A binomial distribution is the sum of a series of m independent binary trials (i.e., trials with only two possible outcomes: success or failure), where all trials have the same probability of success. For binary and binomial distributions, the target of inference is the value of the parameter such that $$ 0\le E\left(\frac{y}{N}\right)=\pi \le 1 $$0 ≤ E y N = π ≤ 1 . Continuous proportions (ratios) arise when the researcher measures responses such as the fraction of the area of a leaf infested with a fungus, the proportion of damaged cloth in a square meter, the fraction of a contaminated area, and so on. As with the binomial parameter π, the continuous rates (fractions) take values between 0 and 1, but, unlike the binomial, the continuous proportions do not result from a set of Bernoulli tests. Instead, the beta distribution is most often used when the response variable is in continuous proportions. In the following sections, we will first address issues in modeling when we have binary and binomial data. When the response variable is binomial, we have the option of using a linearization method (pseudo-likelihood (PL)) or the Laplace or quadrature integral approximation (Stroup 2012).