Unrepeatable repeatabilities: a common mistake
TL;DR: The correct calculation of repeatability is outlined, a common mistake is pointed out, how the incorrectly calculated value relates to repeatable values is shown, and a method for checking published values and calculating approximate repeatability values from the F ratio is provided.
Abstract: -Repeatability is a useful tool for the population geneticist or genetical ecologist, but several papers have carried errors in its calculation We outline the correct calculation of repeatability, point out the common mistake, show how the incorrectly calculated value relates to repeatability, and provide a method for checking published values and calculating approximate repeatability values from the F ratio (mean squares among groups/ mean squares within groups) Received 6 February 1986, accepted 25 August 1986 REPEATABILITY is a measure used in quantitative genetics to describe the proportion of variance in a character that occurs among rather than within individuals Repeatability, r, is given by: r = (VG + VEg)/ VP, (1) where VG is the genotypic variance, VEg the general environmental variance, and Vp the phenotypic variance (Falconer 1960, 1981) In addition to its use in assessing the reliability of multiple measurements on the same individual, repeatability may be used to set an upper limit to the value of heritability (Falconer 1960, 1981) and to separate, for instance, the effects of "self" and "mate" on a character such as clutch size (van Noordwijk et al 1980) Repeatability is therefore a useful statistic for population geneticists and genetical ecologists Recently, we have noticed an increasing number of published papers and unpublished manuscripts in which repeatability was miscalculated Our purpose is fivefold: (1) to outline the correct method of calculating repeatability; (2) to point out a common mistake in calculating repeatability; (3) to show how much this mistake affects values of repeatability; (4) to provide a quick way of checking published estimates, and to calculate an approximate value of repeatability from published F ratios and degrees of freedom; and (5) to make recommendations for authors, referees, editors, and readers to prevent the promulgation and propagation of incorrect repeatability values in the literature CALCULATION OF REPEATABILITY Repeatability is the intraclass correlation coefficient (Sokal and Rohlf 1981), which is based on variance components derived from a one-way analysis of variance (ANOVA) The intraclass correlation coefficient is given by some statistical packages; otherwise it can be calculated from an ANOVA ANOVA is described in most statistics textbooks (eg Sokal and Rohlf 1981; Kirk 1968 gives a detailed treatment of more complex designs of ANOVA), so we will not repeat it here, but give the general form of the results from such an analysis in Table 1 Repeatability, r, is given by r = sA / (S + SA)' (2) where S2A is the among-groups variance component and s2 is the within-group variance component These variance components are calculated from the mean squares in the analysis of variance as:
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TL;DR: Two types of repeatability (ordinary repeatability and extrapolated repeatability) are compared in relation to narrow‐sense heritability and two methods for calculating standard errors, confidence intervals and statistical significance are addressed.
Abstract: Repeatability (more precisely the common measure of repeatability, the intra-class correlation coefficient, ICC) is an important index for quantifying the accuracy of measurements and the constancy of phenotypes. It is the proportion of phenotypic variation that can be attributed to between-subject (or between-group) variation. As a consequence, the non-repeatable fraction of phenotypic variation is the sum of measurement error and phenotypic flexibility. There are several ways to estimate repeatability for Gaussian data, but there are no formal agreements on how repeatability should be calculated for non-Gaussian data (e.g. binary, proportion and count data). In addition to point estimates, appropriate uncertainty estimates (standard errors and confidence intervals) and statistical significance for repeatability estimates are required regardless of the types of data. We review the methods for calculating repeatability and the associated statistics for Gaussian and non-Gaussian data. For Gaussian data, we present three common approaches for estimating repeatability: correlation-based, analysis of variance (ANOVA)-based and linear mixed-effects model (LMM)-based methods, while for non-Gaussian data, we focus on generalised linear mixed-effects models (GLMM) that allow the estimation of repeatability on the original and on the underlying latent scale. We also address a number of methods for calculating standard errors, confidence intervals and statistical significance; the most accurate and recommended methods are parametric bootstrapping, randomisation tests and Bayesian approaches. We advocate the use of LMM- and GLMM-based approaches mainly because of the ease with which confounding variables can be controlled for. Furthermore, we compare two types of repeatability (ordinary repeatability and extrapolated repeatability) in relation to narrow-sense heritability. This review serves as a collection of guidelines and recommendations for biologists to calculate repeatability and heritability from both Gaussian and non-Gaussian data.
2,104 citations
Cites background or methods from "Unrepeatable repeatabilities: a com..."
...(2) ANOVA-based repeatabilities The most commonly used method to calculate repeatabilities in behavioural and evolutionary biology is the ANOVAbased method popularised by Lessells & Boag (1987)....
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...In order to test if the repeatability is significantly greater than zero, the p value from the ANOVA tests is appropriate (Donner, 1986; Lessells & Boag, 1987)....
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...It expresses the proportion of the total variation that is reproducible among repeated measurements of the same subject or group (Lessells & Boag, 1987; Shrout & Fleiss, 1979)....
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...An ANOVA-based repeatability can be calculated as (Donner, 1986; Lessells & Boag, 1987): RA = MSA − MSWMSA + (n0 − 1)·MSW , (4) n0 = 1k − 1 · ⎛ ⎜⎜⎜⎝N − k∑ i=1 n2i N ⎞ ⎟⎟⎟⎠ (5) where RA is the ANOVA-based repeatability estimate, MSA is the mean between-individual sum of squares and MSW is the mean…...
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TL;DR: Meta-analysis is used to ask whether different types of behaviours were more repeatable than others, and if repeatability estimates depended on taxa, sex, age, field versus laboratory, the number of measures and the interval between measures.
1,671 citations
Cites background or methods or result from "Unrepeatable repeatabilities: a com..."
...The majority of repeatability estimates (708 of 759) considered in this meta-analysis were calculated as suggested by Lessells & Boag (1987) ....
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...…per individual (mean effect size ¼ 0.47, 95% confidence limits ¼ 0.43, 0.52; hereafter reported as 0.43 # 0.47 # 0.52) were higher than those that did correct for different numbers of observations per individual (0.35 # 0.37 # 0.38, Qb ¼ 23.0, N ¼ 759, P < 0.001) (Lessells & Boag 1987)....
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...In an important paper, Lessells & Boag (1987) pointed out that...
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...In an important paper, Lessells & Boag (1987) pointed out that MSa (the mean square among individuals) depends on n0, the coefficient representing the number of observations per individual....
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...The majority of repeatability estimates (708 of 759) considered in this meta-analysis were calculated as suggested by Lessells & Boag (1987)....
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TL;DR: This paper generalizes the methods called for Poisson and binomial GLMMs to all other non-Gaussian distributions, in particular to negative binomial and gamma distributions that are commonly used for modelling biological data and can be used across disciplines and regardless of statistical environments.
Abstract: The coefficient of determination R2 quantifies the proportion of variance explained by a statistical model and is an important summary statistic of biological interest. However, estimating R2 for g...
1,389 citations
TL;DR: An overview of how mixed-effect models can be used to partition variation in, and correlations among, phenotypic attributes into between- and within-individual variance components is provided.
Abstract: Growing interest in proximate and ultimate causes and consequences of between- and within-individual variation in labile components of the phenotype - such as behaviour or physiology - characterizes current research in evolutionary ecology. The study of individual variation requires tools for quantification and decomposition of phenotypic variation into between- and within-individual components. This is essential as variance components differ in their ecological and evolutionary implications. We provide an overview of how mixed-effect models can be used to partition variation in, and correlations among, phenotypic attributes into between- and within-individual variance components. Optimal sampling schemes to accurately estimate (with sufficient power) a wide range of repeatabilities and key (co)variance components, such as between- and within-individual correlations, are detailed. Mixed-effect models enable the usage of unambiguous terminology for patterns of biological variation that currently lack a formal statistical definition (e.g. 'animal personality' or 'behavioural syndromes'), and facilitate cross-fertilisation between disciplines such as behavioural ecology, ecological physiology and quantitative genetics.
854 citations
Cites methods from "Unrepeatable repeatabilities: a com..."
...With regard to univariate analyses, repeatability (Glossary) is often estimated using analyses of variance (e.g. Lessells & Boag 1987)....
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TL;DR: This article examined the behavior and population genetics of the invasive Argentine ant (Linepithema humile) in its native and introduced ranges, and provided a mechanism to explain its success as an invader.
Abstract: Despite the severe ecological and economic damage caused by introduced species, factors that allow invaders to become successful often remain elusive. Of invasive taxa, ants are among the most widespread and harmful. Highly invasive ants are often unicolonial, forming supercolonies in which workers and queens mix freely among physically separate nests. By reducing costs associated with territoriality, unicolonial species can attain high worker densities, allowing them to achieve interspecific dominance. Here we examine the behavior and population genetics of the invasive Argentine ant (Linepithema humile) in its native and introduced ranges, and we provide a mechanism to explain its success as an invader. Using microsatellite markers, we show that a population bottleneck has reduced the genetic diversity of introduced populations. This loss is associated with reduced intraspecific aggression among spatially separate nests, and leads to the formation of interspecifically dominant supercolonies. In contrast, native populations are more genetically variable and exhibit pronounced intraspecific aggression. Although reductions in genetic diversity are generally considered detrimental, these findings provide an example of how a genetic bottleneck can lead to widespread ecological success. In addition, these results provide insights into the origin and evolution of unicoloniality, which is often considered a challenge to kin selection theory.
823 citations
References
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01 Jan 1981
TL;DR: The genetic constitution of a population: Hardy-Weinberg equilibrium and changes in gene frequency: migration mutation, changes of variance, and heritability are studied.
Abstract: Part 1 Genetic constitution of a population: Hardy-Weinberg equilibrium. Part 2 Changes in gene frequency: migration mutation. Part 3 Small populations - changes in gene frequency under simplified conditions. Part 4 Small populations - less simplified conditions. Part 5 Small populations - pedigreed populations and close inbreeding. Part 6 Continuous variation. Part 7 Values and means. Part 8 Variance. Part 9 Resemblance between relatives. Part 10 Heritability. Part 11 Selection - the response and its prediction. Part 12 Selection - the results of experiments. Part 13 Selection - information from relatives. Part 14 Inbreeding and crossbreeding - changes of mean value. Part 15 Inbreeding and crossbreeding - changes of variance. Part 16 Inbreeding and crossbreeding - applications. Part 17 Scale. Part 18 Threshold characters. Part 19 Correlated characters. Part 20 Metric characters under natural selection.
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TL;DR: This chapter discusses research strategies and the Control of Nuisance Variables, as well as randomly Randomized Factorial Design with Three or More Treatments and Randomized Block Factorial design, and Confounded Factorial Designs: Designs with Group-Interaction Confounding.
Abstract: Chapter 1. Research Strategies and the Control of Nuisance Variables Chapter 2. Experimental Designs: an Overview Chapter 3. Fundamental Assumptions in Analysis of Variance Chapter 4. Completely Randomized Design Chapter 5. Multiple Comparison Tests Chapter 6. Trend Analysis Chapter 7. General Linear Model Approach to ANOVA Chapter 8. Randomized Block Designs Chapter 9. Completely Randomized Factorial Design with Two Treatments Chapter 10. Completely Randomized Factorial Design with Three or More Treatments and Randomized Block Factorial Design Chapter 11. Hierarchical Designs Chapter 12. Split-Plot Factorial Design: Design with Group-Treatment Confounding Chapter 13. Analysis of Covariance Chapter 14. Latin Square and Related Designs Chapter 15. Confounded Factorial Designs: Designs with Group-Interaction Confounding Chapter 16. Fractional Factorial Designs: Designs with Treatment-Interaction Confounding
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201 citations