Basic Features of the Analysis of Germination Data with Generalized Linear Mixed Models

Abstract

Germination data are discrete and binomial. Although analysis of variance (ANOVA) has long been used for the statistical analysis of these data, generalized linear mixed models (GzLMMs) provide a more consistent theoretical framework. GzLMMs are suitable for final germination percentages (FGP) as well as longitudinal studies of germination time-courses. Germination indices (i.e., single-value parameters summarizing the results of a germination assay by combining the level and rapidity of germination) and other data with a Gaussian error distribution can be analyzed too. There are, however, different kinds of GzLMMs: Conditional (i.e., random effects are modeled as deviations from the general intercept with a specific covariance structure), marginal (i.e., random effects are modeled solely as a variance/covariance structure of the error terms), and quasi-marginal (some random effects are modeled as deviations from the intercept and some are modeled as a covariance structure of the error terms) models can be applied to the same data. It is shown that: (a) For germination data, conditional, marginal, and quasi-marginal GzLMMs tend to converge to a similar inference; (b) conditional models are the first choice for FGP; (c) marginal or quasi-marginal models are more suited for longitudinal studies, although conditional models lead to a congruent inference; (d) in general, common random factors are better dealt with as random intercepts, whereas serial correlation is easier to model in terms of the covariance structure of the error terms; (e) germination indices are not binomial and can be easier to analyze with a marginal model; (f) in boundary conditions (when some means approach 0% or 100%), conditional models with an integral approximation of true likelihood are more appropriate; in non-boundary conditions, (g) germination data can be fitted with default pseudo-likelihood estimation techniques, on the basis of the SAS-based code templates provided here; (h) GzLMMs are remarkably good for the analysis of germination data except if some means are 0% or 100%. In this case, alternative statistical approaches may be used, such as survival analysis or linear mixed models (LMMs) with transformed data, unless an ad hoc data adjustment in estimates of limit means is considered, either experimentally or computationally. This review is intended as a basic tutorial for the application of GzLMMs, and is, therefore, of interest primarily to researchers in the agricultural sciences.

Germination and the Initial Seedling Growth of Lettuce, Celeriac and Wheat Cultivars after Micronutrient and a Biological Application Pre-Sowing Seed Treatment

“…The Pearson’s correlation between germination and other examined parameters was calculated using simple correlation coefficients (r). Arcsine transformation was applied to seed germination data and percentages of infected seed before being subjected to the analysis of variance [ 66 ]. Cluster and principal component analysis (PCA) was performed based on comparing the impact of treatments.…”

Why Seed Physiology Is Important for Genebanking

“…Different link functions may be used to relate the response variable to the predictor [83], but the most common is probably the logit link function, that is, binary logistic regression analysis, for example, [84]. A further extension of this method is to fit a generalized linear mixed model (GLMM) to take into account fixed (factor-related) and random effects (e.g., if seeds are sown across multiple test units such as Petri dishes or rolled paper towels) [82,85]. It is rarely appropriate to use analysis of variance (ANOVA), because the nature of germination data is likely to violate the assumptions of the analysis; that is, that the errors after fitting the model follow the same, normal distribution across treatment groups [82].…”

Multiple Comparison Procedures for the Differences of Proportion Parameters in Over-Reported Multiple-Sample Binomial Data

“…Another common single step adjustment/correction method is the Dunn method (Dunn of 1961 [26]) which controls the FWER by dividing α by m, the number of comparisons made, with m in Equation (7). Hence, treating one group as the control group and comparing the remaining of the groups with the one control group.…”

“…Hence, treating one group as the control group and comparing the remaining of the groups with the one control group. The rest of this procedure is easy to implement because it is the same as the previously discussed (single step) Bonferroni method in Equation (8) of Section 6.1, but with using m from Equation (7).…”

“…On the hand, there are literatures which discussed misclassified binomial data, for one-sample and two-sample binomial data, with one-type or both-type of misclassifications, Bayesian or frequentist (i.e., likelihood-based) methods, but none of them are with the multiplicity adjustments for (pairwise) MCPs, either; for example, Rahardja in 2019 [6] reviewed such literature. Additionally, there are past papers which studied various ways to analyze binomial data, but they did not include misclassifications nor multiplicity adjustments for (pairwise) MCPs; for example, among many papers, Gianinetti in 2020 [7], Giles and Fiori in 2019 [8], Hodge and Vieland in 2017 [9]. Stamey, et al in 2004 [10] proposed multiple comparison method for comparing Poisson rate parameters where counts are subject to misclassification.…”