The left-hand side of the Fundamental Theorem of Natural Selection: A reply.

In a recent paper, Grafen (2018) discussed the left-hand side in the equation stating Fisher's (1930, 1958) "Fundamental Theorem of Natural Selection" (FTNS). Fisher's original statement of the FTNS is, in effect, "The rate of increase in fitness of any organism is equal to its genetic variance in fitness at that time" with the rate of increase in fitness understood as the one "due to all changes in gene ratios" (Fisher, 1930, p. 35). For purposes of exposition, Grafen (2018) considered what is today called the analogous discrete-time model, and restated the FTNS on p. 181 as "The increase in population [mean fitness] due to changes in gene frequencies [is equal to the] additive genetic variance in fitness [divided by the] mean fitness". Allowing for the fact that Grafen's statement of the FTNS relates to a discrete-time model, his statement is in effect a discrete-time version of Fisher's. It has however been widely accepted for many years, ever since Price's (1972) deep analysis of the FTNS, that Fisher's wording does not correctly describe the content of the FTNS. The same is therefore true of Grafen's statement. The confusion caused by these misstatements is unfortunate and adds to a continuing misunderstanding of the FTNS, whose source can also be found in Fisher's (1941) own explanation. Our purpose is to review the detailed analysis of the calculations leading to the FTNS to clarify the points at issue.

Analysis of Variance Models with Stochastic Group Weights.

The analysis of variance (ANOVA) is still one of the most widely used statistical methods in the social sciences. This article is about stochastic group weights in ANOVA models - a neglected aspect in the literature. Stochastic group weights are present whenever the experimenter does not determine the exact group sizes before conducting the experiment. We show that classic ANOVA tests based on estimated marginal means can have an inflated type I error rate when stochastic group weights are not taken into account, even in randomized experiments. We propose two new ways to incorporate stochastic group weights in the tests of average effects - one based on the general linear model and one based on multigroup structural equation models (SEMs). We show in simulation studies that our methods have nominal type I error rates in experiments with stochastic group weights while classic approaches show an inflated type I error rate. The SEM approach can additionally deal with heteroscedastic residual variances and latent variables. An easy-to-use software package with graphical user interface is provided.

On the interpretation and relevance of the Fundamental Theorem of Natural Selection.

The attempt to understand the statement, and then to find the interpretation, of Fisher's "Fundamental Theorem of Natural Selection" caused problems for generations of population geneticists. Price's (1972) paper was the first to lead to an understanding of the statement of the theorem. The theorem shows (in the discrete-time case) that the so-called "partial change" in mean fitness of a population between a parental generation and an offspring generation is the parental generation additive genetic variance in fitness divided by the parental generation mean fitness. In the continuous-time case the partial rate of change in mean fitness is equal to the parental generation additive genetic variance in fitness with no division by the mean fitness. This "partial change" has been interpreted by some as the change in mean fitness due to changes in gene frequency, and by others as the change in mean fitness due to natural selection. (Fisher variously used both interpretations.) In this paper we discuss these interpretations of the theorem. We indicate why we are unhappy with both. We also discuss the long-term relevance of the Fundamental Theorem of Natural Selection, again reaching a negative assessment. We introduce and discuss the concept of genic evolutionary potential. We finally review an optimizing theorem that involves changes in gene frequency, the additive genetic variance in fitness and the mean fitness itself, all of which are involved in the Fundamental Theorem of Natural Selection, and which is free of the difficulties in interpretation of the Fundamental Theorem of Natural Selection.

The Impact of Non-additive Effects on the Genetic Correlation Between Populations.

Average effects of alleles can show considerable differences between populations. The magnitude of these differences can be measured by the additive genetic correlation between populations ([Formula: see text]). This [Formula: see text] can be lower than one due to the presence of non-additive genetic effects together with differences in allele frequencies between populations. However, the relationship between the nature of non-additive effects, differences in allele frequencies, and the value of [Formula: see text] remains unclear, and was therefore the focus of this study. We simulated genotype data of two populations that have diverged under drift only, or under drift and selection, and we simulated traits where the genetic model and magnitude of non-additive effects were varied. Results showed that larger differences in allele frequencies and larger non-additive effects resulted in lower values of [Formula: see text] In addition, we found that with epistasis, [Formula: see text] decreases with an increase of the number of interactions per locus. For both dominance and epistasis, we found that, when non-additive effects became extremely large, [Formula: see text] had a lower bound that was determined by the type of inter-allelic interaction, and the difference in allele frequencies between populations. Given that dominance variance is usually small, our results show that it is unlikely that true [Formula: see text] values lower than 0.80 are due to dominance effects alone. With realistic levels of epistasis, [Formula: see text] dropped as low as 0.45. These results may contribute to the understanding of differences in genetic expression of complex traits between populations, and may help in explaining the inefficiency of genomic trait prediction across populations.

Analyzing average and conditional effects with multigroup multilevel structural equation models.

Conventionally, multilevel analysis of covariance (ML-ANCOVA) has been the recommended approach for analyzing treatment effects in quasi-experimental multilevel designs with treatment application at the cluster-level. In this paper, we introduce the generalized ML-ANCOVA with linear effect functions that identifies average and conditional treatment effects in the presence of treatment-covariate interactions. We show how the generalized ML-ANCOVA model can be estimated with multigroup multilevel structural equation models that offer considerable advantages compared to traditional ML-ANCOVA. The proposed model takes into account measurement error in the covariates, sampling error in contextual covariates, treatment-covariate interactions, and stochastic predictors. We illustrate the implementation of ML-ANCOVA with an example from educational effectiveness research where we estimate average and conditional effects of early transition to secondary schooling on reading comprehension.

Clarifying the Relationship between Average Excesses and Average Effects of Allele Substitutions.

Fisher's concepts of average effects and average excesses are at the core of the quantitative genetics theory. Their meaning and relationship have regularly been discussed and clarified. Here we develop a generalized set of one locus two-allele orthogonal contrasts for average excesses and average effects, based on the concept of the effective gene content of alleles. Our developments help understand the average excesses of alleles for the biallelic case. We dissect how average excesses relate to the average effects and to the decomposition of the genetic variance.