Summary
1. Linear regression models are an important statistical tool in evolutionary and ecological studies. Unfortunately, these models often yield some uninterpretable estimates and hypothesis ...tests, especially when models contain interactions or polynomial terms. Furthermore, the standard errors for treatment groups, although often of interest for including in a publication, are not directly available in a standard linear model.
2. Centring and standardization of input variables are simple means to improve the interpretability of regression coefficients. Further, refitting the model with a slightly modified model structure allows extracting the appropriate standard errors for treatment groups directly from the model.
3. Centring will make main effects biologically interpretable even when involved in interactions and thus avoids the potential misinterpretation of main effects. This also applies to the estimation of linear effects in the presence of polynomials. Categorical input variables can also be centred and this sometimes assists interpretation.
4. Standardization (z‐transformation) of input variables results in the estimation of standardized slopes or standardized partial regression coefficients. Standardized slopes are comparable in magnitude within models as well as between studies. They have some advantages over partial correlation coefficients and are often the more interesting standardized effect size.
5. The thoughtful removal of intercepts or main effects allows extracting treatment means or treatment slopes and their appropriate standard errors directly from a linear model. This provides a simple alternative to the more complicated calculation of standard errors from contrasts and main effects.
6. The simple methods presented here put the focus on parameter estimation (point estimates as well as confidence intervals) rather than on significance thresholds. They allow fitting complex, but meaningful models that can be concisely presented and interpreted. The presented methods can also be applied to generalised linear models (GLM) and linear mixed models.
Repeatability (more precisely the common measure of repeatability, the intra‐class correlation coefficient, ICC) is an important index for quantifying the accuracy of measurements and the constancy ...of phenotypes. It is the proportion of phenotypic variation that can be attributed to between‐subject (or between‐group) variation. As a consequence, the non‐repeatable fraction of phenotypic variation is the sum of measurement error and phenotypic flexibility. There are several ways to estimate repeatability for Gaussian data, but there are no formal agreements on how repeatability should be calculated for non‐Gaussian data (e.g. binary, proportion and count data). In addition to point estimates, appropriate uncertainty estimates (standard errors and confidence intervals) and statistical significance for repeatability estimates are required regardless of the types of data. We review the methods for calculating repeatability and the associated statistics for Gaussian and non‐Gaussian data. For Gaussian data, we present three common approaches for estimating repeatability: correlation‐based, analysis of variance (ANOVA)‐based and linear mixed‐effects model (LMM)‐based methods, while for non‐Gaussian data, we focus on generalised linear mixed‐effects models (GLMM) that allow the estimation of repeatability on the original and on the underlying latent scale. We also address a number of methods for calculating standard errors, confidence intervals and statistical significance; the most accurate and recommended methods are parametric bootstrapping, randomisation tests and Bayesian approaches. We advocate the use of LMM‐ and GLMM‐based approaches mainly because of the ease with which confounding variables can be controlled for. Furthermore, we compare two types of repeatability (ordinary repeatability and extrapolated repeatability) in relation to narrow‐sense heritability. This review serves as a collection of guidelines and recommendations for biologists to calculate repeatability and heritability from both Gaussian and non‐Gaussian data.
Fitting generalised linear models (GLMs) with more than one predictor has become the standard method of analysis in evolutionary and behavioural research. Often, GLMs are used for exploratory data ...analysis, where one starts with a complex full model including interaction terms and then simplifies by removing non-significant terms. While this approach can be useful, it is problematic if significant effects are interpreted as if they arose from a single a priori hypothesis test. This is because model selection involves cryptic multiple hypothesis testing, a fact that has only rarely been acknowledged or quantified. We show that the probability of finding at least one ‘significant' effect is high, even if all null hypotheses are true (e.g. 40% when starting with four predictors and their two-way interactions). This probability is close to theoretical expectations when the sample size (N) is large relative to the number of predictors including interactions (k). In contrast, type I error rates strongly exceed even those expectations when model simplification is applied to models that are over-fitted before simplification (low N/k ratio). The increase in false-positive results arises primarily from an overestimation of effect sizes among significant predictors, leading to upward-biased effect sizes that often cannot be reproduced in follow-up studies (‘the winner's curse'). Despite having their own problems, full model tests and P value adjustments can be used as a guide to how frequently type I errors arise by sampling variation alone. We favour the presentation of full models, since they best reflect the range of predictors investigated and ensure a balanced representation also of non-significant results.
The coefficient of determination R2 quantifies the proportion of variance explained by a statistical model and is an important summary statistic of biological interest. However, estimating R2 for ...generalized linear mixed models (GLMMs) remains challenging. We have previously introduced a version of R2 that we called for Poisson and binomial GLMMs, but not for other distributional families. Similarly, we earlier discussed how to estimate intra-class correlation coefficients (ICCs) using Poisson and binomial GLMMs. In this paper, we generalize our methods to all other non-Gaussian distributions, in particular to negative binomial and gamma distributions that are commonly used for modelling biological data. While expanding our approach, we highlight two useful concepts for biologists, Jensen's inequality and the delta method, both of which help us in understanding the properties of GLMMs. Jensen's inequality has important implications for biologically meaningful interpretation of GLMMs, whereas the delta method allows a general derivation of variance associated with non-Gaussian distributions. We also discuss some special considerations for binomial GLMMs with binary or proportion data. We illustrate the implementation of our extension by worked examples from the field of ecology and evolution in the R environment. However, our method can be used across disciplines and regardless of statistical environments.
Summary
Intra‐class correlations (ICC) and repeatabilities (R) are fundamental statistics for quantifying the reproducibility of measurements and for understanding the structure of biological ...variation. Linear mixed effects models offer a versatile framework for estimating ICC and R. However, while point estimation and significance testing by likelihood ratio tests is straightforward, the quantification of uncertainty is not as easily achieved.
A further complication arises when the analysis is conducted on data with non‐Gaussian distributions because the separation of the mean and the variance is less clear‐cut for non‐Gaussian than for Gaussian models. Nonetheless, there are solutions to approximate repeatability for the most widely used families of generalized linear mixed models (GLMMs).
Here, we introduce the R package rptR for the estimation of ICC and R for Gaussian, binomial and Poisson‐distributed data. Uncertainty in estimators is quantified by parametric bootstrapping and significance testing is implemented by likelihood ratio tests and through permutation of residuals. The package allows control for fixed effects and thus the estimation of adjusted repeatabilities (that remove fixed effect variance from the estimate) and enhanced agreement repeatabilities (that add fixed effect variance to the denominator). Furthermore, repeatability can be estimated from random‐slope models. The package features convenient summary and plotting functions.
Besides repeatabilities, the package also allows the quantification of coefficients of determination R2 as well as of raw variance components. We present an example analysis to demonstrate the core features and discuss some of the limitations of rptR.
Summary
The use of both linear and generalized linear mixed‐effects models (LMMs and GLMMs) has become popular not only in social and medical sciences, but also in biological sciences, especially in ...the field of ecology and evolution. Information criteria, such as Akaike Information Criterion (AIC), are usually presented as model comparison tools for mixed‐effects models.
The presentation of ‘variance explained’ (R2) as a relevant summarizing statistic of mixed‐effects models, however, is rare, even though R2 is routinely reported for linear models (LMs) and also generalized linear models (GLMs). R2 has the extremely useful property of providing an absolute value for the goodness‐of‐fit of a model, which cannot be given by the information criteria. As a summary statistic that describes the amount of variance explained, R2 can also be a quantity of biological interest.
One reason for the under‐appreciation of R2 for mixed‐effects models lies in the fact that R2 can be defined in a number of ways. Furthermore, most definitions of R2 for mixed‐effects have theoretical problems (e.g. decreased or negative R2 values in larger models) and/or their use is hindered by practical difficulties (e.g. implementation).
Here, we make a case for the importance of reporting R2 for mixed‐effects models. We first provide the common definitions of R2 for LMs and GLMs and discuss the key problems associated with calculating R2 for mixed‐effects models. We then recommend a general and simple method for calculating two types of R2 (marginal and conditional R2) for both LMMs and GLMMs, which are less susceptible to common problems.
This method is illustrated by examples and can be widely employed by researchers in any fields of research, regardless of software packages used for fitting mixed‐effects models. The proposed method has the potential to facilitate the presentation of R2 for a wide range of circumstances.
Human‐altered environmental conditions affect many species at the global scale. An extreme form of anthropogenic alteration is the existence and rapid increase of urban areas. A key question, then, ...is how species cope with urbanization. It has been suggested that rural and urban conspecifics show differences in behaviour and personality. However, (i) a generalization of this phenomenon has never been made; and (ii) it is still unclear whether differences in personality traits between rural and urban conspecifics are the result of phenotypic plasticity or of intrinsic differences. In a literature review, we show that behavioural differences between rural and urban conspecifics are common and taxonomically widespread among animals, suggesting a significant ecological impact of urbanization on animal behaviour. In order to gain insight into the mechanisms leading to behavioural differences in urban individuals, we hand‐raised and kept European blackbirds (Turdus merula) from a rural and a nearby urban area under common‐garden conditions. Using these birds, we investigated individual variation in two behavioural responses to the presence of novel objects: approach to an object in a familiar area (here defined as neophilia), and avoidance of an object in a familiar foraging context (defined as neophobia). Neophilic and neophobic behaviours were mildly correlated and repeatable even across a time period of one year, indicating stable individual behavioural strategies. Blackbirds from the urban population were more neophobic and seasonally less neophilic than blackbirds from the nearby rural area. These intrinsic differences in personality traits are likely the result of microevolutionary changes, although we cannot fully exclude early developmental influences.
The green-brown polymorphism of grasshoppers and bush-crickets represents one of the most penetrant polymorphisms in any group of organisms. This poses the question of why the polymorphism is shared ...across species and how it is maintained. There is mixed evidence for whether and in which species it is environmentally or genetically determined in Orthoptera. We report breeding experiments with the steppe grasshopper Chorthippus dorsatus, a polymorphic species for the presence and distribution of green body parts. Morph ratios did not differ between sexes, and we find no evidence that the rearing environment (crowding and habitat complexity) affected the polymorphism. However, we find strong evidence for genetic determination for the presence/absence of green and its distribution. Results are most parsimoniously explained by three autosomal loci with two alleles each and simple dominance effects: one locus influencing the ability to show green color, with a dominant allele for green; a locus with a recessive allele suppressing green on the dorsal side; and a locus with a recessive allele suppressing green on the lateral side. Our results contribute to the emerging contrast between the simple genetic inheritance of green-brown polymorphisms in the subfamily Gomphocerinae and environmental determination in other subfamilies of grasshoppers. In three out of four species of Gomphocerinae studied so far, the results suggest one or a few loci with a dominance of alleles allowing the occurrence of green. This supports the idea that brown individuals differ from green individuals by homozygosity for loss-of-function alleles preventing green pigment production or deposition.
Context-dependent biological variation presents a unique challenge to the reproducibility of results in experimental animal research, because organisms' responses to experimental treatments can vary ...with both genotype and environmental conditions. In March 2019, experts in animal biology, experimental design and statistics convened in Blonay, Switzerland, to discuss strategies addressing this challenge. In contrast to the current gold standard of rigorous standardization in experimental animal research, we recommend the use of systematic heterogenization of study samples and conditions by actively incorporating biological variation into study design through diversifying study samples and conditions. Here we provide the scientific rationale for this approach in the hope that researchers, regulators, funders and editors can embrace this paradigm shift. We also present a road map towards better practices in view of improving the reproducibility of animal research.
Methods for inference and interpretation of evolutionary quantitative genetic parameters, and for prediction of the response to selection, are best developed for traits with normal distributions. ...Many traits of evolutionary interest, including many life history and behavioral traits, have inherently nonnormal distributions. The generalized linear mixed model (GLMM) framework has become a widely used tool for estimating quantitative genetic parameters for nonnormal traits. However, whereas GLMMs provide inference on a statistically convenient latent scale, it is often desirable to express quantitative genetic parameters on the scale upon which traits are measured. The parameters of fitted GLMMs, despite being on a latent scale, fully determine all quantities of potential interest on the scale on which traits are expressed. We provide expressions for deriving each of such quantities, including population means, phenotypic (co)variances, variance components including additive genetic (co)variances, and parameters such as heritability. We demonstrate that fixed effects have a strong impact on those parameters and show how to deal with this by averaging or integrating over fixed effects. The expressions require integration of quantities determined by the link function, over distributions of latent values. In general cases, the required integrals must be solved numerically, but efficient methods are available and we provide an implementation in an R package, QGglmm. We show that known formulas for quantities such as heritability of traits with binomial and Poisson distributions are special cases of our expressions. Additionally, we show how fitted GLMM can be incorporated into existing methods for predicting evolutionary trajectories. We demonstrate the accuracy of the resulting method for evolutionary prediction by simulation and apply our approach to data from a wild pedigreed vertebrate population.
PDF
