On the use of the Price equation

This paper distinguishes two categories of questions that the Price equation can help us answer. The two different types of questions require two different disciplines that are related, but nonetheless move in opposite directions. These disciplines are probability theory on the one hand and statistical inference on the other. In the literature on the Price equation this distinction is not made. As a result of this, questions that require a probability model are regularly approached with statistical tools. In this paper, we examine the possibilities of the Price equation for answering questions of either type. By spending extra attention on mathematical formalities, we avoid the two disciplines to get mixed up. After that, we look at some examples, both from kin selection and from group selection, that show how the inappropriate use of statistical terminology can put us on the wrong track. Statements that are 'derived' with the help of the Price equation are, therefore, in many cases not the answers they seem to be. Going through the derivations in reverse can, however, be helpful as a guide how to build proper (probabilistic) models that do give answers.     

Natural selection. IV. The Price equation

The Price equation partitions total evolutionary change into two components. The first component provides an abstract expression of natural selection. The second component subsumes all other evolutionary processes, including changes during transmission. The natural selection component is often used in applications. Those applications attract widespread interest for their simplicity of expression and ease of interpretation. Those same applications attract widespread criticism by dropping the second component of evolutionary change and by leaving unspecified the detailed assumptions needed for a complete study of dynamics. Controversies over approximation and dynamics have nothing to do with the Price equation itself, which is simply a mathematical equivalence relation for total evolutionary change expressed in an alternative form. Disagreements about approach have to do with the tension between the relative valuation of abstract versus concrete analyses. The Price equation's greatest value has been on the abstract side, particularly the invariance relations that illuminate the understanding of natural selection. Those abstract insights lay the foundation for applications in terms of kin selection, information theory interpretations of natural selection and partitions of causes by path analysis. I discuss recent critiques of the Price equation by Nowak and van Veelen.     

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.     

Residual-based diagnostics for structural equation models

Summary .  Classical diagnostics for structural equation models are based on aggregate forms of the data and are ill suited for checking distributional or linearity assumptions. We extend recently developed goodness-of-fit tests for correlated data based on subject-specific residuals to structural equation models with latent variables. The proposed tests lend themselves to graphical displays and are designed to detect misspecified distributional or linearity assumptions. To complement graphical displays, test statistics are defined; the null distributions of the test statistics are approximated using computationally efficient simulation techniques. The properties of the proposed tests are examined via simulation studies. We illustrate the methods using data from a study of in utero lead exposure.     

THE COMPONENTS OF KIN COMPETITION

It is well known that competition among kin alters the rate and often the direction of evolution in subdivided populations. Yet much remains unclear about the ecological and demographic causes of kin competition, or what role life cycle plays in promoting or ameliorating its effects. Using the multilevel Price equation, I derive a general equation for evolution in structured populations under an arbitrary intensity of kin competition. This equation partitions the effects of selection and demography, and recovers numerous previous models as special cases. I quantify the degree of kin competition, α, which explicitly depends on life cycle. I show how life cycle and demographic assumptions can be incorporated into kin selection models via α, revealing life cycles that are more or less permissive of altruism. As an example, I give closed‐form results for Hamilton's rule in a three‐stage life cycle. Although results are sensitive to life cycle in general, I identify three demographic conditions that give life cycle invariant results. Under the infinite island model, α is a function of the scale of density regulation and dispersal rate, effectively disentangling these two phenomena. Population viscosity per se does not impede kin selection.     

The distribution of allelic effects under mutation and selection

The Price (1970, 1972) equation is applied to the problem of describing the changes in the moments of allelic effects caused by selection, mutation and recombination at loci governing a quantitative genetic character. For comparable assumptions the resulting equations are the same as those obtained by different means by Barton & Turelli (1987; Turelli & Barton, 1989). The Price equation provides a natural framework within which to examine certain kinds of non-additive allelic effects, recombination and assortative mating. The use of the Price equation is illustrated by finding the equilibrium genetic variance under multiplicative dominance and epistasis and under assortative mating at an additive locus. The limitations of the use of recursion equations for the moments of allelic effects are also discussed.     

Integrating physiological, ecological and evolutionary change: a Price equation approach

We use a general quantitative framework – the Price equation – to partition phenotypic responses to environmental change into separate physiological, evolutionary and ecological components. We demonstrate how these responses, which potentially occur over different timescales and are usually studied in isolation, can be combined in an additive way; and we discuss the main advantages of doing this. We illustrate our approach using two worked examples, concerning the emergence of toxin resistance within microbial communities, and the estimation of carbon uptake by marine phytoplankton in high-CO₂ environments. We find that this approach allows us to exclude particular mechanistic hypotheses with regard to community-level transformations, and to identify specific instances where appropriate data are lacking. Thus Price's equation provides not only a powerful conceptual aid, but also a means for testing hypotheses and for directing empirical research programmes.     

A novel application of the Price equation reveals that landscape diversity promotes the response of bees to regionally rare plant species

Bees are ecosystem service providers that are globally threatened by losses of plant diversity. However, effects of multi‐species floral displays on bees in agro‐ecosystems with variable landscape context remain poorly understood, hindering pollinator conservation tactics. We addressed this knowledge gap through a novel application of the modified Price equation to evaluate responses of bees to diverse floral communities on 36 farms in Washington, USA, over 3 years. We found that floral richness, not floral identity, was the best predictor of floral visits by bees. However, the benefits of regionally rare floral species (i.e. plants found at relatively few sites) were only fully realised when farms were embedded in diverse landscapes. Our analysis used the modified Price equation to demonstrate that plant diversity, rather than specific plant species, promotes pollinator visitation, and that diverse landscapes promote the response of pollinators to regionally rare plant species.     

A first formal link between the price equation and an optimization program

The Darwin unification project is pursued. A meta-model encompassing an important class of population genetic models is formed by adding an abstract model of the number of successful gametes to the Price equation under uncertainty. A class of optimization programs are defined to represent the "individual-as-maximizing-agent analogy" in a general way. It is then shown that for each population genetic model there is a corresponding optimization program with which formal links can be established. These links provide a secure logical foundation for the commonplace biological principle that natural selection leads organisms to act as if maximizing their "fitness", provides a definition of "fitness", and clarifies the limitations of that principle. The situations covered do not include frequency dependence or social behaviour, but the approach is capable of extension.     

The Price equation framework to study disease within-host evolution

The evolution of infectious diseases is known to affect epidemiological dynamics, but, for some viruses and bacteria, this evolution also takes place inside a host during the course of an infection. I develop an original approach to study intrahost evolutionary dynamics of quantitative disease traits. This approach can be expressed mathematically using the ‘Price equation’ framework recently developed in evolutionary epidemiology. This framework combines population genetics and within-host population dynamics models to identify trade-offs that affect disease intrahost evolution and to predict short-term evolutionary dynamics of life-history traits. I show that this can be applied to study the evolution of viruses competing for host cells or to study the coevolution between parasites and the immune system of the host. This framework can also easily incorporate experimental data. Studying intrahost evolutionary dynamics provides insight at the within-host level, because it allows us to better understand the course of chronic infections, and at the epidemiological level, because it helps to study multi-scale evolutionary processes. This framework can be used to address important biological issues, from immune escape to disease evolutionary response to treatments.     

Measures of success in a class of evolutionary models with fixed population size and structure

We investigate a class of evolutionary models, encompassing many established models of well-mixed and spatially structured populations. Models in this class have fixed population size and structure. Evolution proceeds as a Markov chain, with birth and death probabilities dependent on the current population state. Starting from basic assumptions, we show how the asymptotic (long-term) behavior of the evolutionary process can be characterized by probability distributions over the set of possible states. We then define and compare three quantities characterizing evolutionary success: fixation probability, expected frequency, and expected change due to selection. We show that these quantities yield the same conditions for success in the limit of low mutation rate, but may disagree when mutation is present. As part of our analysis, we derive versions of the Price equation and the replicator equation that describe the asymptotic behavior of the entire evolutionary process, rather than the change from a single state. We illustrate our results using the frequency-dependent Moran process and the birth–death process on graphs as examples. Our broader aim is to spearhead a new approach to evolutionary theory, in which general principles of evolution are proven as mathematical theorems from axioms.     

A rule is not a rule if it changes from case to case (a reply to Marshall's comment)

This is a reply to “Queller's rule ok: Comment on van Veelen ‘when inclusive fitness is right and when it can be wrong’ ” by James Marshall in the Journal of Theoretical Biology, in this issue.In order to circumvent the disagreement about the Price equation and focus on the issue of the predictive power of inclusive fitness for group selection models, I derive Queller's and Marshall's rule without the Price equation. Both rules however need a translation step in order to be able to link them to the group selection model in van Veelen (2009). Queller's rule applies to games with 2 players and 2 strategies, and is general. Marshall's rule on the other hand applies only to a small subset of 3-player games. His rule is correct, but for other, similarly small subsets we would get other rules. This implies that if we want a rule that applies to all symmetric games with 3 players and 2 strategies, it will have to use a vector of dimension 2 that represents population structure. More in general: for group selection models with groups of size n, a correct and general prediction will need to use a vector of dimension n−1 that represents population structure.     

Determination of molecular parameters by fitting sedimentation data to finite-element solutions of the Lamm equation.

A method for fitting experimental sedimentation velocity data to finite-element solutions of various models based on the Lamm equation is presented. The method provides initial parameter estimates and guides the user in choosing an appropriate model for the analysis by preprocessing the data with the G(s) method by van Holde and Weischet. For a mixture of multiple solutes in a sample, the method returns the concentrations, the sedimentation (s) and diffusion coefficients (D), and thus the molecular weights (MW) for all solutes, provided the partial specific volumes (v) are known. For nonideal samples displaying concentration-dependent solution behavior, concentration dependency parameters for s(sigma) and D(delta) can be determined. The finite-element solution of the Lamm equation used for this study provides a numerical solution to the differential equation, and does not require empirically adjusted correction terms or any assumptions such as infinitely long cells. Consequently, experimental data from samples that neither clear the meniscus nor exhibit clearly defined plateau absorbances, as well as data from approach-to-equilibrium experiments, can be analyzed with this method with enhanced accuracy when compared to other available methods. The nonlinear least-squares fitting process was accomplished by the use of an adapted version of the "Doesn't Use Derivatives" nonlinear least-squares fitting routine. The effectiveness of the approach is illustrated with experimental data obtained from protein and DNA samples. Where applicable, results are compared to methods utilizing analytical solutions of approximated Lamm equations.     

Prediction of adsorption from multicomponent solutions by activated carbon using single-solute parameters. Part II--Proposed equation

Prediction of multicomponent adsorption is still one of the most challenging problems in the adsorption field. Many models have been proposed and employed to obtain multicomponent isotherms from single-component equilibrium data. However, most of these models were based on either unrealistic assumptions or on empirical equations with no apparent definition. The purpose of this investigation was to develop a multicomponent adsorption model based on a thermodynamically consistent equation, and to validate that model using experimental data. Three barbiturates--phenobarbital, mephobarbital, and primidone--were combined to form a ternary system. The adsorption of these barbiturates from simulated intestinal fluid (without pancreatin) by activated carbon was studied using the rotating bottle method. The concentrations, both before and after the attainment of equilibrium, were determined with a high-performance liquid chromatography system employing a reversed-phase column. The proposed equation and the competitive Langmuir-like equation were both fit to the data. A very good correlation was obtained between the experimental data and the calculated data using the proposed equation. The results obtained from the original competitive Langmuir-like model were less satisfactory. These results suggest that the proposed equation can successfully predict the trisolute isotherms of the barbituric acid derivatives employed in this study.     

A semiparametric Bayesian approach for structural equation models

In the development of structural equation models (SEMs), observed variables are usually assumed to be normally distributed. However, this assumption is likely to be violated in many practical researches. As the non‐normality of observed variables in an SEM can be obtained from either non‐normal latent variables or non‐normal residuals or both, semiparametric modeling with unknown distribution of latent variables or unknown distribution of residuals is needed. In this article, we find that an SEM becomes nonidentifiable when both the latent variable distribution and the residual distribution are unknown. Hence, it is impossible to estimate reliably both the latent variable distribution and the residual distribution without parametric assumptions on one or the other. We also find that the residuals in the measurement equation are more sensitive to the normality assumption than the latent variables, and the negative impact on the estimation of parameters and distributions due to the non‐normality of residuals is more serious. Therefore, when there is no prior knowledge about parametric distributions for either the latent variables or the residuals, we recommend making parametric assumption on latent variables, and modeling residuals nonparametrically. We propose a semiparametric Bayesian approach using the truncated Dirichlet process with a stick breaking prior to tackle the non‐normality of residuals in the measurement equation. Simulation studies and a real data analysis demonstrate our findings, and reveal the empirical performance of the proposed methodology. A free WinBUGS code to perform the analysis is available in Supporting Information.     

An approximation diffusion equation for a long narrow channel with varying cross-sectional area

The diffusion equation for long narrow channels which lie parallel to a rectangular coordinate and have varying cross-sectional areas may be approximated by an equation which involves only one space variable and the average concentration at each value of this space variable. This equation is derived and is discussed along with its assumptions.