The coefficient of determination R2 and intra-class correlation coefficient from generalized linear mixed-effects models revisited and expanded

The coefficient of determinationR2quantifies the proportion of varianceexplained by a statistical model and is an important summary statisticof biological interest. However, estimatingR2for generalized linear mixedmodels (GLMMs) remains challenging. We have previously introduced a ver-sion ofR2that we calledR2GLMMfor Poisson and binomial GLMMs, but notfor other distributional families. Similarly, we earlier discussed how to estimateintra-class correlation coefficients (ICCs) using Poisson and binomial GLMMs.Inthis paper, we generalize our methodsto allothernon-Gaussian distributions,in particular to negative binomial and gamma distributions that are commonlyused formodellingbiological data. Whileexpanding ourapproach,we highlighttwo useful concepts for biologists, Jensen’s inequality and the delta method,both of which help us in understanding the properties of GLMMs. Jensen’sinequality has important implications for biologically meaningful interpretationof GLMMs, whereas the delta method allows a general derivation of varianceassociated with non-Gaussian distributions. We also discuss some special con-siderations for binomial GLMMs with binary or proportion data. We illustratethe implementation of our extension by worked examples from the field of ecol-ogy and evolution in theRenvironment. However, our method can be usedacross disciplines and regardless of statistical environments

Mating preferences of zebra finches

Kurzfassung zu: "Causes of between-individual differences in mating preferences". Dissertation an der Ludwig-Maximilians-Universität München, Fakultät für Biologie, durchgeführt am Max-Planck-Institut für Ornithologie in Seewiesen, betreut von Dr. Wolfgang Forstmeier und Prof. Bart Kempenaers (2009)

Correlates of male fitness in captive zebra finches - a comparison of methods to disentangle genetic and environmental effects

Bolund E, Schielzeth H, Forstmeier W. Correlates of male fitness in captive zebra finches - a comparison of methods to disentangle genetic and environmental effects. BMC Evolutionary Biology. 2011;11(1): 327.Backgound. It is a common observation in evolutionary studies that larger, more ornamented or earlier breeding individuals have higher fitness, but that body size, ornamentation or breeding time does not change despite of sometimes substantial heritability for these traits. A possible explanation for this is that these traits do not causally affect fitness, but rather happen to be indirectly correlated with fitness via unmeasured non-heritable aspects of condition (e.g. undernourished offspring grow small and have low fitness as adults due to poor health). Whether this explanation applies to a specific case can be examined by decomposing the covariance between trait and fitness into its genetic and environmental components using pedigree-based animal models. We here examine different methods of doing this for a captive zebra finch population where male fitness was measured in communal aviaries in relation to three phenotypic traits (tarsus length, beak colour and song rate). Results. Our case study illustrates how methods that regress fitness over breeding values for phenotypic traits yield biased estimates as well as anti-conservative standard errors. Hence, it is necessary to estimate the genetic and environmental covariances between trait and fitness directly from a bivariate model. This method, however, is very demanding in terms of sample sizes. In our study parameter estimates of selection gradients for tarsus were consistent with the hypothesis of environmentally induced bias (ßA = 0.035 ± 0.25 (SE), ßE = 0.57 ± 0.28 (SE)), yet this differences between genetic and environmental selection gradients falls short of statistical significance. Conclusions. To examine the generality of the idea that phenotypic selection gradients for certain traits (like size) are consistently upwardly biased by environmental covariance a meta-analysis across study systems will be needed

partR2:partitioning R2 in generalized linear mixed models

The coefficient of determination R(2) quantifies the amount of variance explained by regression coefficients in a linear model. It can be seen as the fixed-effects complement to the repeatability R (intra-class correlation) for the variance explained by random effects and thus as a tool for variance decomposition. The R(2) of a model can be further partitioned into the variance explained by a particular predictor or a combination of predictors using semi-partial (part) R(2) and structure coefficients, but this is rarely done due to a lack of software implementing these statistics. Here, we introduce partR2, an R package that quantifies part R(2) for fixed effect predictors based on (generalized) linear mixed-effect model fits. The package iteratively removes predictors of interest from the model and monitors the change in the variance of the linear predictor. The difference to the full model gives a measure of the amount of variance explained uniquely by a particular predictor or a set of predictors. partR2 also estimates structure coefficients as the correlation between a predictor and fitted values, which provide an estimate of the total contribution of a fixed effect to the overall prediction, independent of other predictors. Structure coefficients can be converted to the total variance explained by a predictor, here called ‘inclusive’ R(2), as the square of the structure coefficients times total R(2). Furthermore, the package reports beta weights (standardized regression coefficients). Finally, partR2 implements parametric bootstrapping to quantify confidence intervals for each estimate. We illustrate the use of partR2 with real example datasets for Gaussian and binomial GLMMs and discuss interactions, which pose a specific challenge for partitioning the explained variance among predictors

Sperm velocity in a promiscuous bird across experimental media of different viscosities

Schmoll T, Rudolfsen G, Schielzeth H, Kleven O. Sperm velocity in a promiscuous bird across experimental media of different viscosities. Proceedings of the Royal Society B: Biological Sciences. 2020;287(1931): 20201031.In species with internal fertilization, the female genital tract appears challenging to sperm, possibly resulting from selection on for example ovarian fluid to control sperm behaviour and, ultimately, fertilization. Few studies, however, have examined the effects of swimming media viscosities on sperm performance. We quantified effects of media viscosities on sperm velocity in promiscuous willow warblers Phylloscopus trochilus. We used both a reaction norm and a character-state approach to model phenotypic plasticity of sperm behaviour across three experimental media of different viscosities. Compared with a standard medium (Dulbecco's Modified Eagle Medium, DMEM), media enriched with 1% or 2% w/v methyl cellulose decreased sperm velocity by up to about 50%. Spermatozoa from experimental ejaculates of different males responded similarly to different viscosities, and a lack of covariance between elevations and slopes of individual velocity-by-viscosity reaction norms indicated that spermatozoa from high- and low-velocity ejaculates were slowed down by a similar degree when confronted with high-viscosity environments. Positive cross-environment (1% versus 2% cellulose) covariances of sperm velocity under the character-state approach suggested that sperm performance represents a transitive trait, with rank order of individual ejaculates maintained when expressed against different environmental backgrounds. Importantly, however, a lack of significant covariances in sperm velocity involving a cellulose concentration of 0% indicated that pure DMEM represented a qualitatively different environment, questioning the validity of this widely used standard medium for assaying sperm performance. Enriching sperm environments along ecologically relevant gradients prior to assessing sperm performance will strengthen explanatory power of in vitro studies of sperm behaviour

General methods for evolutionary quantitative genetic inference from generalized mixed models

P.d.V. was supported by a doctoral studentship from the French Ministère de la Recherche et de l’Enseignement Supérieur. H.S. was supported by an Emmy Noether fellowship from the German Research Foundation (SCHI 1188/1-1). S.N. is supported by a Future Fellowship, Australia (FT130100268). M.M. is supported by a University Research Fellowship from the Royal Society (London). The collection of the Soay sheep data is supported by the National Trust for Scotland and QinetQ, with funding from the Natural Environment Research Council, the Royal Society, and the Leverhulme Trust.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 behavioural traits, have inherently non-normal distributions. The generalised linear mixed model (GLMM) framework has become a widely used tool for estimating quantitative genetic parameters for non-normal 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 formulae 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.Publisher PDFPeer reviewe

Morphologie einiger Zweigsänger (Aves: Sylviidae) Nordasiens

Morphologische Daten von Zweigsängern Sylviidae sind nur spärlich publiziert. Basierend auf Beringungstätigkeit in drei Regionen Nordasiens (Ussurien, Baikalsee und Zentralkasachstan) präsentieren wir Flügellängen, Teilfederlängen der achten Handschwinge, Tarsuslängen, Schwanzlängen und Gewichtsmessungen von 27 Zweigsängerarten aus 6 Gattungen. Außerdem zeigen wir flügelmorphologische Profile basierend auf Einzelfedermessungen des Handflügels für 15 Arten aus 5 Gattungen. Von allen untersuchten Arten zeigte der Koreabuschsänger Cettia canturians den rundesten Handflügel, während der Wanderlaubsänger Phylloscopus borealis den spitzesten Handflügel (mit langen Handschwingen 6-8) zeigte. Für einige Arten waren Tests auf Unterschiede in den Messwerten zwischen den Regionen möglich. Wander- und Dunkellaubsänger Phylloscopus fuscatus unterschieden sich nicht zwischen den Fangorten, während am Baikalsee gefangene Gelbbrauenlaubsänger Phylloscopus inornatus kleiner, aber schwerer waren als Individuen aus Ussurien. Auch Klappergrasmücken Sylvia curruca unterschieden sich in Tarsus- und Schwanzlängen, aber nicht in Flügelmorphologie zwischen den Fangregionen Baikalsee und Zentralkasachstan. Die präsentierten Ergebnisse zeigen, wie innerhalb relativ kurzer Fangzeiträume interessante morphologische Daten gesammelt werden können.Data on the morphology of warblers belonging to the family Sylviidae have only scarcely been published. Here we present data on maximum wing chord, length of the 8th primary, tarsus length, tail length and body weight of 27 species (from 6 genera) of Sylviidae. The data are based on ringing activities in three regions of Northern Eurasia (Ussuriland, Lake Baikal and Central Kazakhstan). For 15 species (from 5 genera) we also show wing tip profiles based on measurement of individual feathers of the outer wing. Of all species studied, the Korean Bush Warbler Cettia canturians showed the bluntest wing tip, while the arctic warbler Phylloscopus borealis showed the most pointed wing (with long primaries 6-8). For a number of species we compared morphological measurements between regions. Arctic and Dusky Warblers Phylloscopus fuscatus did not differ in measurements between sites, while Yellow-browed Warblers Phylloscopus inornatus caught at Lake Baikal were smaller but heavier than individuals caught in Ussuriland. Furthermore, Lesser Whitethroats Sylvia curruca differed in tarsus and tail length, but not in wing morphology between Lake Baikal and Central Kazakhstan. The results show how interesting morphological data can be collected within relative short periods of ringing activity