Natural selection and sexual selection are important evolutionary processes that can shape the phenotypic distributions of natural populations and, consequently, a primary goal of evolutionary biologists is to quantify the strength and mode of selection on traits. Studying selection yields valuable information about which traits are important for a fitness-related function (e.g., changes in beak shape to improve feeding in Darwin’s finches) or how evolutionary processes caused a population to diverge into two distinct phenotypes (e.g., carnivore versus herbivore) rather than maintaining an intermediate phenotype (e.g., omnivore). A classic example was based on phenotypic traits associated with higher survival in house sparrows following a storm. Readers who are new to selection analyses are encouraged to first review the Major Books and Book Chapters section of this work for a general overview of the terminology before delving into the many advances achieved by measuring selection, and specifically the selection gradient, in natural populations. While methods to quantify the mode and total strength of selection on a trait were available since the early 1800s, another approach was needed to characterize when a trait was being directly targeted by selection. Readers are strongly recommended to have obtained a general introduction to linear algebra and regressions in order to better understand how to quantify selection on traits and how the different computational methods differ. One or more unfamiliar technical terms may be presented, and readers are encouraged to refer to the corresponding paper for a more thorough explanation of the term. Linear selection (“directional selection”) changes the mean trait value in a population through a linear function between fitness and the trait value whereas nonlinear selection involves nonlinear fitness functions that can: (1) increase trait variance when values at the tail ends of the distribution are linked to higher fitness (“disruptive selection”), (2) decrease trait variance when intermediate values relate to higher fitness (“stabilizing selection”), or (3) increase/decrease the covariance between two traits (“correlational selection”). The selection differential is a coefficient often used to estimate selection but does not partition direct versus indirect selection, whereby direct selection results in phenotypic changes that are directly related to fitness whereas indirect selection can cause a phenotypic change that is not directly related to fitness because that trait is correlated with another trait that is under direct selection. The selection gradient was then introduced to quantify the direct relationship between a phenotypic trait and fitness and originally defined as the partial derivative of (relative) fitness with respect to the mean value for a phenotypic trait. Selection gradients are also defined as the partial regression coefficients from a multiple least-squares regression where relative fitness is the response variable and phenotypic traits are the independent variables, and can be estimated with non-parametric techniques, such as path analyses or logistic regressions. In all of these contexts, the importance of selection gradients in evolutionary biology has been undeniable. Statistical analyses are, however, only as useful as the statistical models from which they were calculated and there have been substantial efforts to study the factors that could affect estimations of selection gradients. Nonlinear selection is expected to be common in nature yet attempts to quantify it has been challenging. Various methods have been applied to better capture nonlinear selection and visualize fitness surfaces and landscapes, but a consensus has yet to be reached. Moreover, estimates of selection may be affected by environmental variation, missing or skewed data, type of standardization, sampling bias, publication bias, and user error. Despite these challenges, the future of quantifying selection gradients remains a promising and fruitful endeavor. Evolutionary biology is becoming increasingly more open access to increase accessibility, enhance reproducibility, and enable future syntheses and other advancements in quantifying selection. Estimating selection gradients has become far more accessible with the advent of new computational methods that can handle a broader range of data sets and the availability of example data sets, making the present moment a particularly exciting time to measure multivariate selection on quantitative traits. In the following work, shorthand notation is provided to classify the types of selection analyzed (selection = linear, nonlinear, or linear and nonlinear) and outcome of the quadratic regression coefficients in the gamma matrix that are used to quantify nonlinear selection. Specifically, past work documented that numerous publications may have underestimated the strength of stabilizing/disruptive selection by a factor of two so it is useful to identify whether the quadratic regression coefficients for the squared traits (gamma_ii) were “doubled” (correctly multiplied by a factor of two), “not doubled,” “reviewed” (summarized from other studies), “not applicable” (not needed because nonlinear selection was not measured or another method was used to estimate the quadratic regression coefficients), “not reported” (nonlinear selection was calculated but the gamma_ii values were not displayed in the publication), or had “doubling unconfirmed” (nonlinear selection was estimated but it was unclear whether the gamma_ii values were correctly doubled).
Major Books and Book Chapters
Selection on quantitative traits has been featured in numerous books and book chapters. Walsh and Lynch 2018 is currently the most comprehensive and well-written overview of univariate and multivariate selection, but those looking for a more introductory explanation of the topic may want to refer to Bell 2008. The first synthesis of phenotypic selection estimates and other important discussions of natural selection are published in Endler 1986. Chenoweth, et al. 2012 covers various methods to quantify selection coefficients and visualize fitness surfaces, and is complemented by Calsbeek, et al. 2012, which identifies many of the ecological factors that can affect estimates of selection. Jones, et al. 2012 provides a good overview on the contrast between estimating sexual selection and natural selection. Lynch and Arnold 1988 addresses age-specific selection and methods to incorporate individual growth trajectories into estimates of selection and partition the variance in fitness down to age-invariant components using path analysis. Together, these works are an excellent resource to learn the history, mathematics, challenges, and future directions of quantifying selection gradients and evolutionary parameters in general.
Bell, Graham. 2008. Selection: The mechanism of evolution. 2d ed. Oxford: Oxford Univ. Press.
A general introduction to selection, from genetics to speciation, with a brief overview of selection gradients that spans a few pages. The chapter “Natural Selection in Open Populations” is particularly useful for a review of phenotypic selection and examples of selection experiments (selection = linear and nonlinear, gamma_ii = reviewed).
Calsbeek, Ryan, Thomas P. Gosden, Shawn R. Kuchta, and Erik I. Svensson. 2012. Fluctuating selection and dynamic adaptive landscapes. In The adaptive landscape in evolutionary biology. Edited by E. I. Svensson and R. Calsbeek, 89–109. Oxford: Oxford Univ. Press.
A good introduction to the ecological factors that can affect how selection operates on phenotype and shapes the adaptive landscape. The chapter extends general patterns of variation described in Siepielski, et al. 2009 (under Spatial and Temporal Variation) to also cover frequency-dependence, density-dependence, competition, predation, sexual selection, phenotypic plasticity, and abiotic factors (selection = linear and nonlinear, gamma_ii = reviewed).
Chenoweth, Stephen F., John Hunt, and Howard D. Rundle. 2012. Analyzing and comparing the geometry of individual fitness surfaces. In The adaptive landscape in evolutionary biology. Edited by E. I. Svensson and R. Calsbeek, 126–149. Oxford: Oxford Univ. Press.
A comprehensive review of the quantitative methods used to conduct estimations of linear and nonlinear selection coefficients, canonical analysis of nonlinear selection (“A form” versus “B form”), significance testing, and evaluations of the factors that could affect selection coefficients (e.g., multicollinearity, unmeasured traits, nonparametric methods, mixed effects models). There is also extensive discussion on connecting phenotypic selection with genetic selection to identify broader patterns in adaptive evolution (selection = linear and nonlinear, gamma_ii = reviewed).
Endler, John A. 1986. Natural selection in the wild. Princeton, NJ: Princeton Univ. Press.
A seminal book on the conceptual and mathematical background to estimate natural selection, with one of the first published syntheses on phenotypic selection estimates. The book includes a discussion on factors that could lead to false detection, lack of detection, or misleading results when estimating selection. Comparisons and history of mathematical techniques to estimate selection differentials and gradients are included. Selection gradients are referred to as “selection coefficients” in this work (selection = linear and nonlinear, gamma_ii = reviewed).
Jones, Adam G., Nicholas L. Ratterman, and Kimberly A. Paczolt. 2012. The adaptive landscape in sexual selection research. In The adaptive landscape in evolutionary biology. Edited by E. I. Svensson and R. Calsbeek, 110–122. Oxford: Oxford Univ. Press.
A comparison between the Bateman gradient calculated for sexual selection and the linear selection gradient calculated for natural selection. Bateman gradients quantify the linear relationship between reproductive success (fitness variable) and mating success (trait variable) and due to the trait being a composite of environmentally dependent factors, the Bateman gradient cannot replace the selection gradient in the breeder’s equation. Selection gradients are reviewed in a general sense (selection = linear and nonlinear, gamma_ii = reviewed).
Lynch, Michael, and Stevan J. Arnold. 1988. The measurement of selection on size and growth. In Size-structured populations. Edited by B. Ebenman and L. Persson, 47–59. Berlin: Springer-Verlag.
Addresses how selection may vary across ontogeny, phenotypic variance-covariance can be reconstructed for life history stages, and variance in fitness can be partitioned across age-variant factors by using path analysis. Conditional linear selection gradients = [P(i)^-1]*S(i), where P is the phenotypic (co)variance matrix and S is the linear selection differential estimated from the individuals alive during a “census” (i). Similarly, the condition stabilizing selection gradient = [P(i)^-1]C(i)* [P(i)^-1] (selection = linear and nonlinear; gamma_ii = reviewed).
Walsh, Bruce, and Michael Lynch. 2018. Evolution and selection of quantitative traits. Oxford: Oxford Univ. Press.
A definitive guide on evolutionary quantitative genetics. This work is an impressively thorough and comprehensive overview of the many factors that drive changes in traits. Chapter 29 (“Individual Fitness and the Measurement of Univariate Selection”) and chapter 30 (“Measuring Multivariate Selection”) are particularly useful in reviewing the past, present, and future of estimating selection coefficients. The appendices also include useful information about common quantitative methods to analyze quantitative traits (selection = linear and nonlinear, gamma_ii = reviewed).
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