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Journal ArticleDOI

Effect size, confidence interval and statistical significance: a practical guide for biologists.

01 Nov 2007-Biological Reviews (Wiley)-Vol. 82, Iss: 4, pp 591-605
TL;DR: This article extensively discusses two dimensionless (and thus standardised) classes of effect size statistics: d statistics (standardised mean difference) and r statistics (correlation coefficient), because these can be calculated from almost all study designs and also because their calculations are essential for meta‐analysis.
Abstract: Null hypothesis significance testing (NHST) is the dominant statistical approach in biology, although it has many, frequently unappreciated, problems. Most importantly, NHST does not provide us with two crucial pieces of information: (1) the magnitude of an effect of interest, and (2) the precision of the estimate of the magnitude of that effect. All biologists should be ultimately interested in biological importance, which may be assessed using the magnitude of an effect, but not its statistical significance. Therefore, we advocate presentation of measures of the magnitude of effects (i.e. effect size statistics) and their confidence intervals (CIs) in all biological journals. Combined use of an effect size and its CIs enables one to assess the relationships within data more effectively than the use of p values, regardless of statistical significance. In addition, routine presentation of effect sizes will encourage researchers to view their results in the context of previous research and facilitate the incorporation of results into future meta-analysis, which has been increasingly used as the standard method of quantitative review in biology. In this article, we extensively discuss two dimensionless (and thus standardised) classes of effect size statistics: d statistics (standardised mean difference) and r statistics (correlation coefficient), because these can be calculated from almost all study designs and also because their calculations are essential for meta-analysis. However, our focus on these standardised effect size statistics does not mean unstandardised effect size statistics (e.g. mean difference and regression coefficient) are less important. We provide potential solutions for four main technical problems researchers may encounter when calculating effect size and CIs: (1) when covariates exist, (2) when bias in estimating effect size is possible, (3) when data have non-normal error structure and/or variances, and (4) when data are non-independent. Although interpretations of effect sizes are often difficult, we provide some pointers to help researchers. This paper serves both as a beginner’s instruction manual and a stimulus for changing statistical practice for the better in the biological sciences.

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Citations
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Journal ArticleDOI
TL;DR: In this article, the authors make a case for the importance of reporting variance explained (R2) as a relevant summarizing statistic of mixed-effects models, which is rare, even though R2 is routinely reported for linear models and also generalized linear models (GLM).
Abstract: Summary The use of both linear and generalized linear mixed-effects models (LMMs and GLMMs) has become popular not only in social and medical sciences, but also in biological sciences, especially in the field of ecology and evolution. Information criteria, such as Akaike Information Criterion (AIC), are usually presented as model comparison tools for mixed-effects models. The presentation of ‘variance explained’ (R2) as a relevant summarizing statistic of mixed-effects models, however, is rare, even though R2 is routinely reported for linear models (LMs) and also generalized linear models (GLMs). R2 has the extremely useful property of providing an absolute value for the goodness-of-fit of a model, which cannot be given by the information criteria. As a summary statistic that describes the amount of variance explained, R2 can also be a quantity of biological interest. One reason for the under-appreciation of R2 for mixed-effects models lies in the fact that R2 can be defined in a number of ways. Furthermore, most definitions of R2 for mixed-effects have theoretical problems (e.g. decreased or negative R2 values in larger models) and/or their use is hindered by practical difficulties (e.g. implementation). Here, we make a case for the importance of reporting R2 for mixed-effects models. We first provide the common definitions of R2 for LMs and GLMs and discuss the key problems associated with calculating R2 for mixed-effects models. We then recommend a general and simple method for calculating two types of R2 (marginal and conditional R2) for both LMMs and GLMMs, which are less susceptible to common problems. This method is illustrated by examples and can be widely employed by researchers in any fields of research, regardless of software packages used for fitting mixed-effects models. The proposed method has the potential to facilitate the presentation of R2 for a wide range of circumstances.

7,749 citations


Cites methods from "Effect size, confidence interval an..."

  • ...…a model (Orelien & Edwards 2008), and (iii) information criteria are not comparable across different datasets under any circumstances, because they are highly dataset specific (in other words, they are not standardized effect statistics which can be used for meta-analysis; Nakagawa & Cuthill 2007)....

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Journal ArticleDOI
TL;DR: The first new effect size index is described is a residual-based index that quantifies the amount of variance explained in both the mediator and the outcome and the second new effectsize index quantifying the indirect effect as the proportion of the maximum possible indirect effect that could have been obtained, given the scales of the variables involved.
Abstract: The statistical analysis of mediation effects has become an indispensable tool for helping scientists investigate processes thought to be causal. Yet, in spite of many recent advances in the estimation and testing of mediation effects, little attention has been given to methods for communicating effect size and the practical importance of those effect sizes. Our goals in this article are to (a) outline some general desiderata for effect size measures, (b) describe current methods of expressing effect size and practical importance for mediation, (c) use the desiderata to evaluate these methods, and (d) develop new methods to communicate effect size in the context of mediation analysis. The first new effect size index we describe is a residual-based index that quantifies the amount of variance explained in both the mediator and the outcome. The second new effect size index quantifies the indirect effect as the proportion of the maximum possible indirect effect that could have been obtained, given the scales of the variables involved. We supplement our discussion by offering easy-to-use R tools for the numerical and visual communication of effect size for mediation effects.

2,359 citations

Journal ArticleDOI
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 from "Effect size, confidence interval an..."

  • ...Notably, 95% confidence intervals can function as an indicator of statistical significance as well as an indicator of uncertainty and the presentation of p values should not replace the more informative confidence intervals (Nakagawa & Cuthill, 2007)....

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Journal ArticleDOI
TL;DR: In this article, the authors focus on parameter estimation (point estimates as well as confidence intervals) rather than on significance thresholds for linear regression models and propose a simple alternative to the more complicated calculation of standard errors from contrasts and main effects.
Abstract: Summary 1. Linear regression models are an important statistical tool in evolutionary and ecological studies. Unfortunately, these models often yield some uninterpretable estimates and hypothesis tests, especially when models contain interactions or polynomial terms. Furthermore, the standard errors for treatment groups, although often of interest for including in a publication, are not directly available in a standard linear model. 2. Centring and standardization of input variables are simple means to improve the interpretability of regression coefficients. Further, refitting the model with a slightly modified model structure allows extracting the appropriate standard errors for treatment groups directly from the model. 3. Centring will make main effects biologically interpretable even when involved in interactions and thus avoids the potential misinterpretation of main effects. This also applies to the estimation of linear effects in the presence of polynomials. Categorical input variables can also be centred and this sometimes assists interpretation. 4. Standardization (z-transformation) of input variables results in the estimation of standardized slopes or standardized partial regression coefficients. Standardized slopes are comparable in magnitude within models as well as between studies. They have some advantages over partial correlation coefficients and are often the more interesting standardized effect size. 5. The thoughtful removal of intercepts or main effects allows extracting treatment means or treatment slopes and their appropriate standard errors directly from a linear model. This provides a simple alternative to the more complicated calculation of standard errors from contrasts and main effects. 6. The simple methods presented here put the focus on parameter estimation (point estimates as well as confidence intervals) rather than on significance thresholds. They allow fitting complex, but meaningful models that can be concisely presented and interpreted. The presented methods can also be applied to generalised linear models (GLM) and linear mixed models.

2,065 citations

Journal ArticleDOI
TL;DR: Two lower bounds on sample size in SEM are developed, the first as a function of the ratio of indicator variables to latent variables, and the second as afunction of minimum effect, power and significance.

1,100 citations

References
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Book
01 Dec 1969
TL;DR: The concepts of power analysis are discussed in this paper, where Chi-square Tests for Goodness of Fit and Contingency Tables, t-Test for Means, and Sign Test are used.
Abstract: Contents: Prefaces. The Concepts of Power Analysis. The t-Test for Means. The Significance of a Product Moment rs (subscript s). Differences Between Correlation Coefficients. The Test That a Proportion is .50 and the Sign Test. Differences Between Proportions. Chi-Square Tests for Goodness of Fit and Contingency Tables. The Analysis of Variance and Covariance. Multiple Regression and Correlation Analysis. Set Correlation and Multivariate Methods. Some Issues in Power Analysis. Computational Procedures.

115,069 citations


"Effect size, confidence interval an..." refers background in this paper

  • ...When any three of these four parameters are fixed, the remaining one can be determined (Cohen, 1988; Nakagawa & Foster, 2004)....

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  • ...Cohen (1988) has proposed ‘conventional’ values as benchmarks for what are considered to be ‘small’, ‘medium’, and ‘large’ effects (r ¼ 0.1, 0.3, 0.5 and d ¼ 0.2, 0.5, 0.8, respectively)....

    [...]

Journal ArticleDOI

49,129 citations


"Effect size, confidence interval an..." refers background in this paper

  • ...When any three of these four parameters are fixed, the remaining one can be determined (Cohen, 1988; Nakagawa & Foster, 2004)....

    [...]

  • ...Cohen (1988) has proposed ‘conventional’ values as benchmarks for what are considered to be ‘small’, ‘medium’, and ‘large’ effects (r ¼ 0.1, 0.3, 0.5 and d ¼ 0.2, 0.5, 0.8, respectively)....

    [...]

Book
19 Jun 2013
TL;DR: The second edition of this book is unique in that it focuses on methods for making formal statistical inference from all the models in an a priori set (Multi-Model Inference).
Abstract: Introduction * Information and Likelihood Theory: A Basis for Model Selection and Inference * Basic Use of the Information-Theoretic Approach * Formal Inference From More Than One Model: Multi-Model Inference (MMI) * Monte Carlo Insights and Extended Examples * Statistical Theory and Numerical Results * Summary

36,993 citations


"Effect size, confidence interval an..." refers methods in this paper

  • ...…& Omland, 2004; Stephens et al., 2005; note that the IT approach often results in more than one ‘important’ model, in which parameters, or effect sizes, can be calculated as weighted means according to a weight given to each remaining model; for detailed procedures, see Burnham & Anderson, 2002)....

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Book
01 Jan 1983
TL;DR: In this paper, a generalization of the analysis of variance is given for these models using log- likelihoods, illustrated by examples relating to four distributions; the Normal, Binomial (probit analysis, etc.), Poisson (contingency tables), and gamma (variance components).
Abstract: The technique of iterative weighted linear regression can be used to obtain maximum likelihood estimates of the parameters with observations distributed according to some exponential family and systematic effects that can be made linear by a suitable transformation. A generalization of the analysis of variance is given for these models using log- likelihoods. These generalized linear models are illustrated by examples relating to four distributions; the Normal, Binomial (probit analysis, etc.), Poisson (contingency tables) and gamma (variance components).

23,215 citations

BookDOI
01 Dec 2010
TL;DR: A guide to using S environments to perform statistical analyses providing both an introduction to the use of S and a course in modern statistical methods.
Abstract: A guide to using S environments to perform statistical analyses providing both an introduction to the use of S and a course in modern statistical methods The emphasis is on presenting practical problems and full analyses of real data sets

18,346 citations