Fixed-effect variance and the estimation of repeatabilities and heritabilities: issues and solutions.

Linear mixed-effects models are frequently used for estimating quantitative genetic parameters, including the heritability, as well as the repeatability, of traits. Heritability acts as a filter that determines how efficiently phenotypic selection translates into evolutionary change, whereas repeatability informs us about the individual consistency of phenotypic traits. As quantities of biological interest, it is important that the denominator, the phenotypic variance in both cases, reflects the amount of phenotypic variance in the relevant ecological setting. The current practice of quantifying heritabilities and repeatabilities from mixed-effects models frequently deprives their denominator of variance explained by fixed effects (often leading to upward bias of heritabilities and repeatabilities), and it has been suggested to omit fixed effects when estimating heritabilities in particular. We advocate an alternative option of fitting models incorporating all relevant effects, while including the variance explained by fixed effects into the estimation of the phenotypic variance. The approach is easily implemented and allows optimizing the estimation of phenotypic variance, for example by the exclusion of variance arising from experimental design effects while still including all biologically relevant sources of variation. We address the estimation and interpretation of heritabilities in situations in which potential covariates are themselves heritable traits of the organism. Furthermore, we discuss complications that arise in generalized and nonlinear mixed models with fixed effects. In these cases, the variance parameters on the data scale depend on the location of the intercept and hence on the scaling of the fixed effects. Integration over the biologically relevant range of fixed effects offers a preferred solution in those situations.

Towards robust evolutionary inference with integral projection models.

Integral projection models (IPMs) are extremely flexible tools for ecological and evolutionary inference. IPMs track the distribution of phenotype in populations through time, using functions describing phenotype-dependent development, inheritance, survival and fecundity. For evolutionary inference, two important features of any model are the ability to (i) characterize relationships among traits (including values of the same traits across ages) within individuals, and (ii) characterize similarity between individuals and their descendants. In IPM analyses, the former depends on regressions of observed trait values at each age on values at the previous age (development functions), and the latter on regressions of offspring values at birth on parent values as adults (inheritance functions). We show analytically that development functions, characterized this way, will typically underestimate covariances of trait values across ages, due to compounding of regression to the mean across projection steps. Similarly, we show that inheritance, characterized this way, is inconsistent with a modern understanding of inheritance, and underestimates the degree to which relatives are phenotypically similar. Additionally, we show that the use of a constant biometric inheritance function, particularly with a constant intercept, is incompatible with evolution. Consequently, current implementations of IPMs will predict little or no phenotypic evolution, purely as artefacts of their construction. We present alternative approaches to constructing development and inheritance functions, based on a quantitative genetic approach, and show analytically and through an empirical example on a population of bighorn sheep how they can potentially recover patterns that are critical to evolutionary inference.

Meta-analysis of magnitudes, differences and variation in evolutionary parameters.

Meta-analysis is increasingly used to synthesize major patterns in the large literatures within ecology and evolution. Meta-analytic methods that do not account for the process of observing data, which we may refer to as 'informal meta-analyses', may have undesirable properties. In some cases, informal meta-analyses may produce results that are unbiased, but do not necessarily make the best possible use of available data. In other cases, unbiased statistical noise in individual reports in the literature can potentially be converted into severe systematic biases in informal meta-analyses. I first present a general description of how failure to account for noise in individual inferences should be expected to lead to biases in some kinds of meta-analysis. In particular, informal meta-analyses of quantities that reflect the dispersion of parameters in nature, for example, the mean absolute value of a quantity, are likely to be generally highly misleading. I then re-analyse three previously published informal meta-analyses, where key inferences were of aspects of the dispersion of values in nature, for example, the mean absolute value of selection gradients. Major biological conclusions in each original informal meta-analysis closely match those that could arise as artefacts due to statistical noise. I present alternative mixed-model-based analyses that are specifically tailored to each situation, but where all analyses may be implemented with widely available open-source software. In each example meta-re-analysis, major conclusions change substantially.

Multivariate selection and intersexual genetic constraints in a wild bird population.

When selection differs between the sexes for traits that are genetically correlated between the sexes, there is potential for the effect of selection in one sex to be altered by indirect selection in the other sex, a situation commonly referred to as intralocus sexual conflict (ISC). While potentially common, ISC has rarely been studied in wild populations. Here, we studied ISC over a set of morphological traits (wing length, tarsus length, bill depth and bill length) in a wild population of great tits (Parus major) from Wytham Woods, UK. Specifically, we quantified the microevolutionary impacts of ISC by combining intra- and intersex additive genetic (co)variances and sex-specific selection estimates in a multivariate framework. Large genetic correlations between homologous male and female traits combined with evidence for sex-specific multivariate survival selection suggested that ISC could play an appreciable role in the evolution of this population. Together, multivariate sex-specific selection and additive genetic (co)variance for the traits considered accounted for additive genetic variance in fitness that was uncorrelated between the sexes (cross-sex genetic correlation = -0.003, 95% CI = -0.83, 0.83). Gender load, defined as the reduction in a population's rate of adaptation due to sex-specific effects, was estimated at 50% (95% CI = 13%, 86%). This study provides novel insights into the evolution of sexual dimorphism in wild populations and illustrates how quantitative genetics and selection analyses can be combined in a multivariate framework to quantify the microevolutionary impacts of ISC.

Interplay of robustness and plasticity of life-history traits drives ecotypic differentiation in thermally distinct habitats.

Phenotypic plasticity describes the ability of an individual to alter its phenotype in response to the environment and is potentially adaptive when dealing with environmental variation. However, robustness in the face of a changing environment may often be beneficial for traits that are tightly linked to fitness. We hypothesized that robustness of some traits may depend on specific patterns of plasticity within and among other traits. We used a reaction norm approach to study robustness and phenotypic plasticity of three life-history traits of the collembolan Orchesella cincta in environments with different thermal regimes. We measured adult mass, age at maturity and growth rate of males and females from heath and forest habitats at two temperatures (12 and 22 °C). We found evidence for ecotype-specific robustness of female adult mass to temperature, with a higher level of robustness in the heath ecotype. This robustness is facilitated by plastic adjustments of growth rate and age at maturity. Furthermore, female fecundity is strongly influenced by female adult mass, explaining the importance of realizing a high mass across temperatures for females. These findings indicate that different predicted outcomes of life-history theory can be combined within one species' ontogeny and that models describing life-history strategies should not assume that traits like growth rate are maximized under all conditions. On a methodological note, we report a systematic inflation of variation when standard deviations and correlation coefficients are calculated from family means as opposed to individual data within a family structure.

Indirect genetics effects and evolutionary constraint: an analysis of social dominance in red deer, Cervus elaphus.

By determining access to limited resources, social dominance is often an important determinant of fitness. Thus, if heritable, standard theory predicts mean dominance should evolve. However, dominance is usually inferred from the tendency to win contests, and given one winner and one loser in any dyadic contest, the mean proportion won will always equal 0.5. Here, we argue that the apparent conflict between quantitative genetic theory and common sense is resolved by recognition of indirect genetic effects (IGEs). We estimate selection on, and genetic (co)variance structures for, social dominance, in a wild population of red deer Cervus elaphus, on the Scottish island of Rum. While dominance is heritable and positively correlated with lifetime fitness, contest outcomes depend as much on the genes carried by an opponent as on the genotype of a focal individual. We show how this dependency imposes an absolute evolutionary constraint on the phenotypic mean, thus reconciling theoretical predictions with common sense. More generally, we argue that IGEs likely provide a widespread but poorly recognized source of evolutionary constraint for traits influenced by competition.

Evolution of adaptive diversity and genetic connectivity in Arctic charr (Salvelinus alpinus) in Iceland.

The ecological theory of adaptive radiation predicts that the evolution of phenotypic diversity within species is generated by divergent natural selection arising from different environments and competition between species. Genetic connectivity among populations is likely also to have an important role in both the origin and maintenance of adaptive genetic diversity. Our goal was to evaluate the potential roles of genetic connectivity and natural selection in the maintenance of adaptive phenotypic differences among morphs of Arctic charr, Salvelinus alpinus, in Iceland. At a large spatial scale, we tested the predictive power of geographic structure and phenotypic variation for patterns of neutral genetic variation among populations throughout Iceland. At a smaller scale, we evaluated the genetic differentiation between two morphs in Lake Thingvallavatn relative to historically explicit, coalescent-based null models of the evolutionary history of these lineages. At the large spatial scale, populations are highly differentiated, but weakly structured, both geographically and with respect to patterns of phenotypic variation. At the intralacustrine scale, we observe modest genetic differentiation between two morphs, but this level of differentiation is nonetheless consistent with strong reproductive isolation throughout the Holocene. Rather than a result of the homogenizing effect of gene flow in a system at migration-drift equilibrium, the modest level of genetic differentiation could equally be a result of slow neutral divergence by drift in large populations. We conclude that contemporary and recent patterns of restricted gene flow have been highly conducive to the evolution and maintenance of adaptive genetic variation in Icelandic Arctic charr.

The danger of applying the breeder's equation in observational studies of natural populations.

The breeder's equation, which predicts evolutionary change when a phenotypic covariance exists between a heritable trait and fitness, has provided a key conceptual framework for studies of adaptive microevolution in nature. However, its application requires strong assumptions to be made about the causation of fitness variation. In its univariate form, the breeder's equation assumes that the trait of interest is not correlated with other traits having causal effects on fitness. In its multivariate form, the validity of predicted change rests on the assumption that all such correlated traits have been measured and incorporated into the analysis. Here, we (i) highlight why these assumptions are likely to be seriously violated in studies of natural, rather than artificial, selection and (ii) advocate wider use of the Robertson-Price identity as a more robust, and less assumption-laden, alternative to the breeder's equation for applications in evolutionary ecology.

A framework for power and sensitivity analyses for quantitative genetic studies of natural populations, and case studies in Soay sheep (Ovis aries).

Studies of the quantitative genetics of natural populations have contributed greatly to evolutionary biology in recent years. However, while pedigree data required are often uncertain (i.e. incomplete and partly erroneous) and limited, means to evaluate the effects of such uncertainties have not been developed. We have therefore developed a general framework for power and sensitivity analyses of such studies. We propose that researchers first generate a set of pedigree data that they wish to use in a quantitative genetic study, as well as data regarding errors that occur in that pedigree. This pedigree is then permuted using the data regarding errors to generate hypothetical 'true' and 'assumed' pedigrees that differ so as to mimic pedigree errors that might occur in the study system under consideration. Phenotypic data are then simulated across the true pedigree (according to user-defined genetic and environmental covariance structures), before being analysed with standard quantitative genetic techniques in conjunction with the 'assumed' pedigree data. To illustrate this approach, we conducted power and sensitivity analyses in a well-known study of Soay sheep (Ovis aries). We found that, although the estimation of simple genetic (co)variance structures is fairly robust to pedigree errors, some potentially serious biases were detected under more complex scenarios involving maternal effects. Power analyses also showed that this study system provides high power to detect heritabilities as low as about 0.09. Given this range of results, we suggest that such power and sensitivity analyses could greatly complement empirical studies, and we provide the computer program PEDANTICS to aid in their application.