Linear mixed‐effects models are powerful tools for analysing complex datasets with repeated or clustered observations, a common data structure in ecology and evolution. Mixed‐effects models involve complex fitting procedures and make several assumptions, in particular about the distribution of residual and random effects. Violations of these assumptions are common in real datasets, yet it is not always clear how much these violations matter to accurate and unbiased estimation. Here we address the consequences of violations in distributional assumptions and the impact of missing random effect components on model estimates. In particular, we evaluate the effects of skewed, bimodal and heteroscedastic random effect and residual variances, of missing random effect terms and of correlated fixed effect predictors. We focus on bias and prediction error on estimates of fixed and random effects. Model estimates were usually robust to violations of assumptions, with the exception of slight upward biases in estimates of random effect variance if the generating distribution was bimodal but was modelled by Gaussian error distributions. Further, estimates for (random effect) components that violated distributional assumptions became less precise but remained unbiased. However, this particular problem did not affect other parameters of the model. The same pattern was found for strongly correlated fixed effects, which led to imprecise, but unbiased estimates, with uncertainty estimates reflecting imprecision. Unmodelled sources of random effect variance had predictable effects on variance component estimates. The pattern is best viewed as a cascade of hierarchical grouping factors. Variances trickle down the hierarchy such that missing higher‐level random effect variances pool at lower levels and missing lower‐level and crossed random effect variances manifest as residual variance. Overall, our results show remarkable robustness of mixed‐effects models that should allow researchers to use mixed‐effects models even if the distributional assumptions are objectively violated. However, this does not free researchers from careful evaluation of the model. Estimates that are based on data that show clear violations of key assumptions should be treated with caution because individual datasets might give highly imprecise estimates, even if they will be unbiased on average across datasets.

Biological responses to climate change have been widely documented across taxa and regions, but it remains unclear whether species are maintaining a good match between phenotype and environment, i.e. whether observed trait changes are adaptive. Here we reviewed 10,090 abstracts and extracted data from 71 studies reported in 58 relevant publications, to assess quantitatively whether phenotypic trait changes associated with climate change are adaptive in animals. A meta-analysis focussing on birds, the taxon best represented in our dataset, suggests that global warming has not systematically affected morphological traits, but has advanced phenological traits. We demonstrate that these advances are adaptive for some species, but imperfect as evidenced by the observed consistent selection for earlier timing. Application of a theoretical model indicates that the evolutionary load imposed by incomplete adaptive responses to ongoing climate change may already be threatening the persistence of species.

One contribution of 13 to a theme issue 'The role of plasticity in phenotypic adaptation to rapid environmental change'.Phenotypic plasticity is a major mechanism of response to global change. However, current plastic responses will only remain adaptive under future conditions if informative environmental cues are still available. We briefly summarize current knowledge of the evolutionary origin and mechanistic underpinnings of environmental cues for phenotypic plasticity, before highlighting the potentially complex effects of global change on cue availability and reliability. We then illustrate some of these aspects with a case study, comparing plasticity of blue tit breeding phenology in two contrasted habitats: evergreen and deciduous forests. Using long-term datasets, we investigate the climatic factors linked to the breeding phenology of the birds and their main food source. Blue tits occupying different habitats differ extensively in the cues affecting laying date plasticity, as well as in the reliability of these cues as predictors of the putative driver of selective pressure, the date of caterpillar peak. The temporal trend for earlier laying date, detected only in the evergreen populations, is explained by increased temperature during their cue windows. Our results highlight the importance of integrating ecological mechanisms shaping variation in plasticity if we are to understand how global change will affect plasticity and its consequences for population biology.This article is part of the theme issue 'The role of plasticity in phenotypic adaptation to rapid environmental change'.

Induced defences, such as the predator avoidance morphologies in amphibians, result from spatial or temporal variability in predation risk. One important component of this variability should be the difference in hunting strategies between predators. However, little is known about how specific and effective induced defences are to different types of predators. We analysed the impact of both pursuing (fish, Gasterosteus aculeatus) and sit‐and‐wait (dragonfly, Aeshna cyanea) predators on tadpole (Rana dalmatina) morphology and performance (viz locomotive performance and growth rate). We also investigated the potential benefits of the predator‐induced phenotype in the presence of fish predators. Both predators induced deeper tail fins in tadpoles exposed to threat of predation, and stickleback presence also induced longer tails and deeper tail muscles. Morphological and behavioural differences resulted in better escape ability of stickleback‐induced tadpoles, leading to improved survival in the face of stickleback predation. These results clearly indicate that specific morphological responses to different types of predators have evolved in R. dalmatina. The specific morphologies suggest low correlations between the traits involved in the defence. Independence of traits allows prey species to fine‐tune their response according to current predation risk, so that the benefit of the defence can be maximal.

Although there are many examples of contemporary directional selection, evidence for responses to selection that match predictions are often missing in quantitative genetic studies of wild populations. This is despite the presence of genetic variation and selection pressures – theoretical prerequisites for the response to selection. This conundrum can be explained by statistical issues with accurate parameter estimation, and by biological mechanisms that interfere with the response to selection. These biological mechanisms can accelerate or constrain this response. These mechanisms are generally studied independently but might act simultaneously. We therefore integrated these mechanisms to explore their potential combined effect. This has implications for explaining the apparent evolutionary stasis of wild populations and the conservation of wildlife.

This chapter asks: How can evolutionary potential be measured? The question is deceptively simple: whilst evolutionary potential is typically defined on a per-trait basis, it has become clear that the complex genetic architecture of quantitative traits requires other ways to quantify evolutionary potential and constraints. This chapter reviews knowledge about multivariate evolutionary potential in the wild and the extent to which genetic covariances, as summarized in the G-matrix, impact evolutionary trajectories of natural populations both in terms of rate and direction. In terms of constraints, genetic covariances among traits can slow down the rate of adaptation, and influence the direction of the response to selection. However, the constraints posed by genetic covariances are insurmountable only if G-matrices are stable. The chapter thus reviews firstly theoretical predictions about the stability of G in relation to selection, migration and drift, and secondly methods available to test differentiation among matrices. To date, a majority of studies imply conservatism of G-matrices; however, a couple of recent studies have revealed that differentiation of G-matrices among wild populations can also be very fast, especially during colonisation of new habitats. Furthermore, as an increasing number of methods have been proposed for comparing G-matrices, we assessed how these methods perform under different hypothetical scenarios. The chapter shows that limited statistical power could often lead to erroneous conclusion of matrix conservatism, suggesting caution is needed in interpreting the results of matrix comparisons. The chapter concludes by identifying areas in need of further research.

The genetic variance-covariance matrix (G-matrix) summarizes the genetic architecture of multiple traits. It has a central role in the understanding of phenotypic divergence and the quantification of the evolutionary potential of populations. Laboratory experiments have shown that G-matrices can vary rapidly under divergent selective pressures. However, due to the demanding nature of G-matrix estimation and comparison in wild populations, the extent of its spatial variability remains largely unknown. In this study, we investigate spatial variation in Gmatrices for morphological and life-history traits using long-term data sets from one continental and three island populations of Blue tit (Cyanistes caeruleus), which have experienced contrasting population history and selective environment. We found no evidence for differences in G-matrices among populations. Interestingly, the phenotypic variance-covariance matrices (P) were divergent across populations, suggesting that using P as a substitute for G may be inadequate. These analyses also provide the first evidence in wild populations for additive genetic variation in the incubation period (i.e. the period between last egg laid and hatching) in all four populations. Altogether, our results suggest that G-matrices may be stable across populations inhabiting contrasted environments therefore challenging the results of previous simulation studies and laboratory experiments.

Observed phenotypic responses to selection in the wild often differ from predictions based on measurements of selection and genetic variance. An overlooked hypothesis to explain this paradox of stasis is that a skewed phenotypic distribution affects natural selection and evolution. We show through mathematical modeling that, when a trait selected for an optimum phenotype has a skewed distribution, directional selection is detected even at evolutionary equilibrium, where it causes no change in the mean phenotype. When environmental effects are skewed, Lande and Arnold's (1983) directional gradient is in the direction opposite to the skew. In contrast, skewed breeding values can displace the mean phenotype from the optimum, causing directional selection in the direction of the skew. These effects can be partitioned out using alternative selection estimates based on average derivatives of individual relative fitness, or additive genetic covariances between relative fitness and trait (Robertson-Price identity). We assess the validity of these predictions using simulations of selection estimation under moderate sample sizes. Ecologically relevant traits may commonly have skewed distributions, as we here exemplify with avian laying date - repeatedly described as more evolutionarily stable than expected - so this skewness should be accounted for when investigating evolutionary dynamics in the wild.