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.

https://besjournals.onlinelibrary.wiley.com/doi/epdf/10.1111/j.2041-210x.2012.00261.x

A general and simple method for obtaining R2 from generalized linear mixed-effects models

Information theoretic approaches and model averaging are increasing in popularity, but this approach can be difficult to apply to the realistic, complex models that typify many ecological and evolutionary analyses. This is especially true for those researchers without a formal background in information theory. Here, we highlight a number of practical obstacles to model averaging complex models. Although not meant to be an exhaustive review, we identify several important issues with tentative solutions where they exist (e.g. dealing with collinearity amongst predictors; how to compute model-averaged parameters) and highlight areas for future research where solutions are not clear (e.g. when to use random intercepts or slopes; which information criteria to use when random factors are involved). We also provide a worked example of a mixed model analysis of inbreeding depression in a wild population. By providing an overview of these issues, we hope that this approach will become more accessible to those investigating any process where multiple variables impact an evolutionary or ecological response.

https://www.otago.ac.nz/threatenedbirdgroup/Publications_files/Grueber_et%20al%20JEB_2011_incl%20Corrigendum.pdf

Multimodel inference in ecology and evolution: challenges and solutions

The use of linear mixed effects models (LMMs) is increasingly common in the analysis of biological data. Whilst LMMs offer a flexible approach to modelling a broad range of data types, ecological data are often complex and require complex model structures, and the fitting and interpretation of such models is not always straightforward. The ability to achieve robust biological inference requires that practitioners know how and when to apply these tools. Here, we provide a general overview of current methods for the application of LMMs to biological data, and highlight the typical pitfalls that can be encountered in the statistical modelling process. We tackle several issues regarding methods of model selection, with particular reference to the use of information theory and multi-model inference in ecology. We offer practical solutions and direct the reader to key references that provide further technical detail for those seeking a deeper understanding. This overview should serve as a widely accessible code of best practice for applying LMMs to complex biological problems and model structures, and in doing so improve the robustness of conclusions drawn from studies investigating ecological and evolutionary questions.

A brief introduction to mixed effects modelling and multi-model inference in ecology.

Genetics in conservation management: Revised recommendations for the 50/500 rules, Red List criteria and population viability analyses

Meta-analysis has changed the way researchers conduct literature reviews not only in medical and social sciences but also in biological sciences. Meta-analysis in biological sciences, especially in ecology and evolution (which we refer to as ‘biological’ meta-analysis) faces somewhat different methodological problems from its counterparts in medical and social sciences, where meta-analytic techniques were originally developed. The main reason for such differences is that biological meta-analysis often integrates complex data composed of multiple strata with, for example, different measurements and a variety of species. Here, we review methodological issues and advancements in biological meta-analysis, focusing on three topics: (1) non-independence arising from multiple effect sizes obtained in single studies and from phylogenetic relatedness, (2) detecting and accounting for heterogeneity, and (3) identifying publication bias and measuring its impact. We show how the marriage between mixed-effects (hierarchical/multilevel) models and phylogenetic comparative methods has resolved most of the issues under discussion. Furthermore, we introduce the concept of across-study and within-study meta-analysis, and propose how the use of within-study meta-analysis can improve many empirical studies typical of ecology and evolution.

Methodological issues and advances in biological meta-analysis

Studies evaluating the impact of inbreeding depression on population viability of threatened species tend to focus on the effects of inbreeding at a single life-history stage (e.g., juvenile survival). We examined the effects of inbreeding across the full life-history continuum, from survival up to adulthood, to subsequent reproductive success, and to the recruitment of second-generation offspring, in wild Takahe (Porphyrio hochstetteri) by analyzing pedigree and fitness data collected over 21 breeding seasons. Although the effect size of inbreeding at individual life-history stages was small, inbreeding depression accumulated across multiple life-history stages and ultimately reduced long-term fitness (i.e., successful recruitment of second-generation offspring). The estimated total lethal equivalents (2B) summed across all life-history stages were substantial (16.05, 95% CI 0.08-90.8) and equivalent to an 88% reduction in recruitment of second- generation offspring for closely related pairs (e.g., sib-sib pairings) relative to unrelated pairs (according to the pedigree). A history of small population size in the Takahe could have contributed to partial purging of the genetic load and the low level of inbreeding depression detected at each single life-history stage. Nevertheless, our results indicate that such "purged" populations can still exhibit substantial inbreeding depression, espe- cially when small but negative fitness effects accumulate across the species' life history. Because inbreeding depression can ultimately affect population viability of small, isolated populations, our results illustrate the importance of measuring the effects of inbreeding across the full life-history continuum.

https://www.otago.ac.nz/threatenedbirdgroup/Publications_files/Gruber_etal_ConsBiol_2010.pdf

Inbreeding depression accumulation across life-history stages of the endangered Takahe.

Continuous inbreeding exposes deleterious recessive alleles to selection and can thereby lead to partial purging of the genetic load and reduced inbreeding depression. Purging has been well documented in experimental laboratory populations, but evidence of reduced inbreeding depression due to purging in wild populations is largely lacking. This study examines the inbreeding load associated with juvenile survival at a protected island site of a bottlenecked population of the Stewart Island robin Petroica australis rakiura. Based on a complete pedigree of the island population, we found little evidence that inbreeding coefficients explained any additional variation in juvenile survival, once demographic factors such as the effects of density, timing of fledging and age of mother were taken into account. Lethal equivalents, a standardized measure of the strength of inbreeding depression, were close to zero (B=0.24, 95% CI=−1.92-1.04, n=326) and lower than that documented for an island population of a widespread congener (B=6.71, 95% CI=−0.66-14.08, n=238) and for several other species for which significant inbreeding depression was detected (B=1.30-7.47). There are several reasons as to why studies can fail to detect pedigree-based inbreeding depression in wild populations, but evidence presented here of low lethal equivalents and relatively high survival of both inbred and non-inbred juveniles are consistent with a population that has undergone partial purging of its genetic load during historical population bottlenecks. Although our study does not imply a complete absence of inbreeding depression, it is one of the first studies of a wild population where weak inbreeding depression for juvenile survival appears to be associated with a prolonged bottleneck.

https://www.otago.ac.nz/threatenedbirdgroup/Publications_files/Laws%26Jamieson_AnimCons_2010.pdf

Is lack of evidence of inbreeding depression in a threatened New Zealand robin indicative of reduced genetic load

Inbreeding depression is known to vary greatly between populations and among species. Some of this variation is due to differences in genetic load between populations, while some is due to differences in the environment (e.g. local weather conditions) or demography of the population (e.g. age structure and breeding experience) in which inbreeding is expressed. Although the effects of these factors in isolation are well understood, there is still relatively little known about the interface between inbreeding on one hand, and environment and demography on the other in wild populations. We examined how environmental and demographic factors mediated the effects of inbreeding in a threatened species of bird. The Stewart Island robin, Petroica australis rakiura, has been subjected to a prolonged bottleneck for over 150 years. A complete pedigree of a reintroduced island population, extending back seven seasons to its founding, was available for analysis along with survival data (at the level of the brood) obtained from intensive monitoring over two breeding seasons. We found no strong support that the degree to which a brood was inbred affected its survival at either the hatching, fledging or recruitment stages. The inbreeding coefficient of the mother did have an effect on brood survival when analysed over all three life history stages, but only as a result of an interaction with female age, with broods of one-year-old inbred females suffering greater mortality than those of older inbred females. Although habitat type, temperature, rainfall and year were the best predictors of brood survival for most life history stages, their effects were weak and there were no interactions with inbreeding. Furthermore, there was no strong evidence of inbreeding depression associated with two periods of severe weather. This population is atypical in that inbreeding depression appears to be weak even under severe environmental conditions, and may be indicative that this bottlenecked population has either reduced genetic load or has fixed deleterious alleles.

https://www.otago.ac.nz/threatenedbirdgroup/Publications_files/Laws_etal_JAvianBiol_2010.pdf

Limited inbreeding depression in a bottlenecked population is age but not environment dependent

Species’ declines around the globe have reached such a rapid rate that without the implementation of extreme conservation methods, we are likely to witness the next mass extinction. Translocation is a popular and growing conservation technique, especially to islands. However, despite the widespread use of this method, the long-term genetic outlook of populations established on islands (namely those with small carrying capacity, and isolated from other populations) is often poorly quantified, and there are conflicting data supporting the relative roles of evolutionary processes such as drift, balancing selection, directional selection and genetic purging. Taking examples from avian translocations, this paper reviews recent data on the genetic dynamics of island populations established for conservation purposes. The goal of conservation genetics is to preserve species’ adaptive potential for the long term. In general, population bottlenecks cause losses of genetic diversity, but there is mixed evidence for the importance of other evolutionary processes, such as maintenance of genetic diversity by balancing selection. There is a clear need for long-term studies that test for associations between genetic change and success of populations established via translocation to conservation islands, and for integration of these observations with ecological and climate data. Combining data types into the future will provide a holistic view on the interactions between the role selection plays, how island populations established by translocations may respond, and how this information can improve conservation strategy.

Rebecca J. Laws

Papers

Multimodel inference in ecology and evolution: challenges and solutions

Inbreeding depression accumulation across life-history stages of the endangered Takahe.

Is lack of evidence of inbreeding depression in a threatened New Zealand robin indicative of reduced genetic load

Limited inbreeding depression in a bottlenecked population is age but not environment dependent

Evolutionary genetics of translocated island populations of birds: data and opportunities