title: Assessing variation in life-history tactics within a population using mixture regression models: a practical guide for evolutionary ecologists.
abstract: Mixed models are now well-established methods in ecology and evolution because they allow accounting for and quantifying within- and between-individual variation. However, the required normal distribution of the random effects can often be violated by the presence of clusters among subjects, which leads to multi-modal distributions. In such cases, using what is known as mixture regression models might offer a more appropriate approach. These models are widely used in psychology, sociology, and medicine to describe the diversity of trajectories occurring within a population over time (e.g. psychological development, growth). In ecology and evolution, however, these models are seldom used even though understanding changes in individual trajectories is an active area of research in life-history studies. Our aim is to demonstrate the value of using mixture models to describe variation in individual life-history tactics within a population, and hence to promote the use of these models by ecologists and evolutionary ecologists. We first ran a set of simulations to determine whether and when a mixture model allows teasing apart latent clustering, and to contrast the precision and accuracy of estimates obtained from mixture models versus mixed models under a wide range of ecological contexts. We then used empirical data from long-term studies of large mammals to illustrate the potential of using mixture models for assessing within-population variation in life-history tactics. Mixture models performed well in most cases, except for variables following a Bernoulli distribution and when sample size was small. The four selection criteria we evaluated [Akaike information criterion (AIC), Bayesian information criterion (BIC), and two bootstrap methods] performed similarly well, selecting the right number of clusters in most ecological situations. We then showed that the normality of random effects implicitly assumed by evolutionary ecologists when using mixed models was often violated in life-history data. Mixed models were quite robust to this violation in the sense that fixed effects were unbiased at the population level. However, fixed effects at the cluster level and random effects were better estimated using mixture models. Our empirical analyses demonstrated that using mixture models facilitates the identification of the diversity of growth and reproductive tactics occurring within a population. Therefore, using this modelling framework allows testing for the presence of clusters and, when clusters occur, provides reliable estimates of fixed and random effects for each cluster of the population. In the presence or expectation of clusters, using mixture models offers a suitable extension of mixed models, particularly when evolutionary ecologists aim at identifying how ecological and evolutionary processes change within a population. Mixture regression models therefore provide a valuable addition to the statistical toolbox of evolutionary ecologists. As these models are complex and have their own limitations, we provide recommendations to guide future users.
authors.affiliation: Faculty of Biosciences, Fisheries and Economics, Department of Arctic and Marine Biology, UiT The Arctic University of Norway, 9037 Tromsø, Norway.
authors.forename: Sandra
authors.initials: S
authors.lastname: Hamel
authors.affiliation: Faculty of Biosciences, Fisheries and Economics, Department of Arctic and Marine Biology, UiT The Arctic University of Norway, 9037 Tromsø, Norway.
authors.forename: Nigel G
authors.initials: NG
authors.lastname: Yoccoz
authors.affiliation: CNRS, UMR 5558 'Biométrie et Biologie Evolutive', Université de Lyon, Université Lyon 1, F-69622, Villeurbanne, France.
authors.forename: Jean-Michel
authors.initials: JM
authors.lastname: Gaillard
mesh_terms: D000818:Animals; D001499:Bayes Theorem; D005075:Biological Evolution; D003198:Computer Simulation; D004463:Ecology / Q000379:methods*; D015233:Models, Statistical; D012044:Regression Analysis
keywords: AIC; BIC; bootstrap; finite mixture models; growth; individual trajectory; multi-modal distribution; reproduction