Effect size

In statistics, an effect size is a quantitative measure of the magnitude and often direction of a phenomenon. It can refer to the value of a statistic calculated from a sample of data, the value of one parameter for a hypothetical population, or the equation that operationalizes how statistics or parameters lead to the effect size value. Examples of effect sizes include the correlation between two variables, the regression coefficient in a regression, the mean difference, the standardised mean differences, and the risk of a particular event (such as a heart attack). Effect sizes are a complementary tool for statistical hypothesis testing, and play an important role in statistical power analyses to assess the sample size required for new experiments. Effect size calculations are fundamental to meta-analysis, which aims to provide the combined effect size based on data from multiple studies. The group of data-analysis methods concerning effect sizes is referred to as estimation statistics. Effect size is an essential component in the evaluation of the strength of a statistical claim, and it is the first item (magnitude) in the MAGIC criteria. The standard error of the effect size is of critical importance, as it indicates how much uncertainty is included in the observed measurement. A standard deviation that is too large will make the measurement nearly meaningless.…

In statistics, an effect size is a quantitative measure of the magnitude and often direction of a phenomenon. It can refer to the value of a statistic calculated from a sample of data, the value of one parameter for a hypothetical population, or the equation that operationalizes how statistics or parameters lead to the effect size value. Examples of effect sizes include the correlation between two variables, the regression coefficient in a regression, the mean difference, the standardised mean differences, and the risk of a particular event (such as a heart attack). Effect sizes are a complementary tool for statistical hypothesis testing, and play an important role in statistical power analyses to assess the sample size required for new experiments. Effect size calculations are fundamental to meta-analysis, which aims to provide the combined effect size based on data from multiple studies. The group of data-analysis methods concerning effect sizes is referred to as estimation statistics. Effect size is an essential component in the evaluation of the strength of a statistical claim, and it is the first item (magnitude) in the MAGIC criteria. The standard error of the effect size is of critical importance, as it indicates how much uncertainty is included in the observed measurement. A standard deviation that is too large will make the measurement nearly meaningless. In meta-analysis, which aims to summarize multiple effect sizes into a single estimate, the uncertainty in studies' effect sizes is used to weight the contribution of each study, so larger studies are considered more important than smaller ones. The uncertainty in the effect size is calculated differently for each type of effect size, but generally only requires knowing the study's sample size (N), or the number of observations (n) in each group. Reporting effect sizes or estimates thereof (effect estimate [EE], estimate of effect) is considered good practice when presenting empirical research findings in many fields. The reporting of effect sizes facilitates the interpretation of the importance of a research result, in contrast to its statistical significance. Effect sizes are particularly significant in social science and medical research, with the latter emphasizing the importance of the magnitude of the average treatment effect. Effect sizes may be measured in relative or absolute terms. In relative effect sizes, two groups are directly compared with each other, as in odds ratios and relative risks. A larger absolute value always indicates a stronger effect for absolute effect sizes. Many types of measurements can be expressed as either absolute or relative, and these can be used together because they convey different information. A prominent task force in the psychology research community made the following recommendation: ==Overview== ===Population and sample effect sizes=== As in statistical estimation, the true effect size is distinguished from the observed effect size. For example, to measure the risk of disease in a population (the population effect size) one can measure the risk within a sample of that population (the sample effect size). Conventions for describing true and observed effect sizes follow standard statistical practices—one common approach is to use Greek letters like ρ [rho] to denote population parameters and Latin letters like r to denote the corresponding statistic. Alternatively, a "hat" can be placed over the population parameter to denote the statistic, e.g. with \hat\rho being the estimate of the parameter \rho . As in any statistical setting, effect sizes are estimated with sampling error, and may be biased unless the effect size estimator that is used is appropriate for the manner in which the data were sampled and the manner in which the measurements were made. An example of this is publication bias, which occurs when scientists report results only when the estimated effect sizes are large or are statistically significant. As a result, if many researchers carry out studies with low statistical power, the reported effect sizes will tend to be larger than the true (population) effects, if any. Another example where effect sizes may be distorted is in a multiple-trial experiment, where the effect size calculation is based on the averaged or aggregated response across the trials. Smaller studies sometimes show different, often larger, effect sizes than larger studies. This phenomenon is known as the small-study effect, which may signal publication bias. ===Relationship to test statistics=== Sample-based effect sizes are distinguished from test statistics used in hypothesis testing, in that they estimate the strength (magnitude) of, for example, an apparent relationship, rather than assigning a significance level reflecting whether the magnitude of the relationship observed could be due to chance. The effect size does not directly determine the significance level, or vice versa. Given a sufficiently large sample size, a non-null statistical comparison will always show a statistically significant result unless the population effect size is exactly zero (and even there it will show statistical significance at the rate of the Type I error used). For example, a sample Pearson correlation coefficient of 0.01 is statistically significant if the sample size is 1000. Reporting only the significant p-value from this analysis could be misleading if a correlation of 0.01 is too small to be of interest in a particular application. ===Standardized and unstandardized effect sizes=== The term effect size can refer to a standardized measure of effect (such as r, Cohen's d, or the odds ratio), or to an unstandardized measure (e.g., the difference between group means or the unstandardized regression coefficients). Standardized effect size measures are typically used when: the metrics of variables being studied do not have intrinsic meaning (e.g., a score on a personality test on an arbitrary scale), results from multiple studies are being combined, some or all of the studies use different scales, or it is desired to convey the size of an effect relative to the variability in the population. In meta-analyses, standardized effect sizes are used as a common measure that can be calculated for different studies and then combined into an overall summary. Unstandardized effects are also known as simple effects. ==Interpretation== The interpretation of an effect size of being small, medium, or large depends on its substantive context and its operational definition. Jacob Cohen suggested interpretation guidelines that are near ubiquitous across many fields. However, Cohen also cautioned: "The terms 'small,' 'medium,' and 'large' are relative, not only to each other, but to the area of behavioral science or even more particularly to the specific content and research method being employed in any given investigation... In the face of this relativity, there is a certain risk inherent in offering conventional operational definitions for these terms for use in power analysis in as diverse a field of inquiry as behavioral science. This risk is nevertheless accepted in the belief that more is to be gained than lost by supplying a common conventional frame of reference which is recommended for use only when no better basis for estimating the ES index is available." (p. 25) Sawilowsky recommended that the rules of thumb for effect sizes should be revised, and expanded the descriptions to include very small, very large, and huge. Funder and Ozer suggested that effect sizes should be interpreted based on benchmarks and consequences of findings, resulting in adjustment of guideline recommendations. noted for a medium effect size, "you'll choose the same n regardless of the accuracy or reliability of your instrument, or the narrowness or diversity of your subjects. Clearly, important considerations are being ignored here. Researchers should interpret the substantive significance of their results by grounding them in a meaningful context or by quantifying their contribution to knowledge, and Cohen's effect size descriptions can be helpful as a starting point." Similarly, a U.S. Dept of Education sponsored report argued that the widespread indiscriminate use of Cohen's interpretation guidelines can be inappropriate and misleading. They instead suggested that norms should be based on distributions of effect sizes from comparable studies. Thus a small effect (in absolute numbers) could be considered large if the effect is larger than similar studies in the field. See Abelson's paradox and Sawilowsky's paradox for related points. The table below contains descriptors for various magnitudes of d, r, f and omega, as initially suggested by Jacob Cohen, and later expanded by Sawilowsky, and by Funder & Ozer. ==Types== About 50 to 100 different measures of effect size are known. Many effect sizes of different types can be converted to other types, as many estimate the separation of two distributions, so are mathematically related. For example, a correlation coefficient can be converted to a Cohen's d and vice versa. \eta ^2 = \frac{SS_\text{Treatment}}{SS_\text{Total}} . \omega^2 = \frac{\text{SS}_\text{treatment}-df_\text{treatment} \cdot \text{MS}_\text{error}}{\text{SS}_\text{total} + \text{MS}_\text{error}} . This form of the formula is limited to between-subjects analysis with equal sample sizes in all cells. Since it is less biased (although not unbiased), ω 2 is preferable to η 2 ; however, it can be more inconvenient to calculate for complex analyses. A generalized form of the estimator has been published for between-subjects and within-subjects analysis, repeated measure, mixed design, and randomized block design experiments. In addition, methods to calculate partial ω 2 for individual factors and combined factors in designs with up to three independent variables have been published. f^2 = {R^2 \over 1 - R^2}. Likewise, f 2 can be defined as: f^2 = {\eta^2 \over 1 - \eta^2} or f^2 = {\omega^2 \over 1 - \omega^2} for models described by those effect size measures. The f^{2} effect size measure for sequential multiple regression and also common for PLS modeling is defined as: f^2 = {R^2_{AB} - R^2_A \over 1 - R^2_{AB}} where R 2 A is the variance accounted for by a set of one or more independent variables A, and R 2 AB is the combined variance accounted for by A and another set of one or more independent variables of interest B. By convention, f 2 effect sizes of 0.1^2 , 0.25^2 , and 0.4^2 are termed small, medium, and large, respectively. Cohen's \hat{f} can also be found for factorial analysis of variance (ANOVA) working backwards, using: \hat{f}_\text{effect} = {\sqrt{(F_\text{effect} df_\text{effect}/N)}}. In a balanced design (equivalent sample sizes across groups) of ANOVA, the corresponding population parameter of f^2 is {SS(\mu_1,\mu_2,\dots,\mu_K)}\over{K \times \sigma^2}, wherein μ j denotes the population mean within the j th group of the total K groups, and σ the equivalent population standard deviations within each groups. SS is the sum of squares in ANOVA. q = \frac 1 2 \log \frac{ 1 + r_1 }{ 1 - r_1 } - \frac 1 2 \log \frac{1 + r_2}{1 - r_2} where r 1 and r 2 are the regressions being compared. The expected value of q is zero and its variance is \operatorname{var}(q) = \frac 1 {N_1 - 3} + \frac 1 {N_2 -3} where N 1 and N 2 are the number of data points in the first and second regression respectively. \theta = \frac{\mu_1 - \mu_2} \sigma, where μ 1 is the mean for one population, μ 2 is the mean for the other population, and σ is a standard deviation based on either or both populations. In the practical setting the population values are typically not known and must be estimated from sample statistics. The several versions of effect sizes based on means differ with respect to which statistics are used. This form for the effect size resembles the computation for a t-test statistic, with the critical difference that the t-test statistic includes a factor of \sqrt{n} . This means that for a given effect size, the significance level increases with the sample size. Unlike the t-test statistic, the effect size aims to estimate a population parameter and is not affected by the sample size. SMD values of 0.2 to 0.5 are considered small, 0.5 to 0.8 are considered medium, and greater than 0.8 are considered large. d = \frac{\bar{x}_1 - \bar{x}_2} s. Jacob Cohen defined s, the pooled standard deviation, as (for two independent samples): s = \sqrt{\frac{(n_1-1)s^2_1 + (n_2-1)s^2_2}{n_1+n_2 - 2}} where the variance for one of the groups is defined as s_1^2 = \frac 1 {n_1-1} \sum_{i=1}^{n_1} (x_{1,i} - \bar{x}_1)^2, and similarly for the other group. Other authors choose a slightly different computation of the standard deviation when referring to "Cohen's d" where the denominator is without "-2" s = \sqrt{\frac{(n_1-1)s^2_1 + (n_2-1)s^2_2}{n_1+n_2}} This definition of "Cohen's d" is termed the maximum likelihood estimator by Hedges and Olkin, and it is related to Hedges' g by a scaling factor (see below). With two paired samples, an approach is to look at the distribution of the difference scores. In that case, s is the standard deviation of this distribution of difference scores (of note, the standard deviation of difference scores is dependent on the correlation between paired samples). This creates the following relationship between the t-statistic to test for a difference in the means of the two paired groups and Cohen's d (computed with difference scores): t = \frac{\bar{X}_1 - \bar{X}_2}{\text{SE}_{diff}} = \frac{\bar{X}_1 - \bar{X}_2}{\frac{\text{SD}_{diff}}{\sqrt N}} = \frac{\sqrt{N} (\bar{X}_1 - \bar{X}_2)}{SD_{diff}} and d' = \frac{\bar{X}_1 - \bar{X}_2}{\text{SD}_{diff}} = \frac t {\sqrt N} However, for paired samples, Cohen states that d' does not provide the correct estimate to obtain the power of the test for d, and that before looking the values up in the tables provided for d, it should be corrected for r as in the following formula: \frac{d'} {\sqrt{1 - r}}. where r is the correlation between paired measurements. Given the same sample size, the higher r, the higher the power for a test of paired difference. Since d' depends on r, as a measure of effect size it is difficult to interpret; therefore, in the context of paired analyses, since it is possible to compute d' or d (estimated with a pooled standard deviation or that of a group or time-point), it is necessary to explicitly indicate which one is being reported. As a measure of effect size, d (estimated with a pooled standard deviation or that of a group or time-point) is more appropriate, for instance in meta-analysis. Cohen's d is frequently used in estimating sample sizes for statistical testing. A lower Cohen's d indicates the necessity of larger sample sizes, and vice versa, as can subsequently be determined together with the additional parameters of desired significance level and statistical power. \Delta = \frac{\bar{x}_1 - \bar{x}_2}{s_2} The second group may be regarded as a control group, and Glass argued that if several treatments were compared to the control group it would be better to use just the standard deviation computed from the control group, so that effect sizes would not differ under equal means and different variances. Under a correct assumption of equal population variances a pooled estimate for σ is more precise. g = \frac{\bar{x}_1 - \bar{x}_2}{s^*} where the pooled standard deviation s^* is computed as: s^* = \sqrt{\frac{(n_1-1)s_1^2 + (n_2-1)s_2^2}{n_1+n_2-2}}. However, as an estimator for the population effect size θ it is biased. Nevertheless, this bias can be approximately corrected through multiplication by a factor g^* = J(n_1+n_2-2) \,\, g \, \approx \, \left(1-\frac{3}{4(n_1+n_2)-9}\right) \,\, g Hedges and Olkin refer to this less-biased estimator g^* as d, but it is not the same as Cohen's d. The exact form for the correction factor J() involves the gamma function J(a) = \frac{\Gamma(a/2)}{\sqrt{a/2 \,}\,\Gamma((a-1)/2)}. There are also multilevel variants of Hedges' g, e.g., for use in cluster randomised controlled trials (CRTs). CRTs involve randomising clusters, such as schools or classrooms, to different conditions and are frequently used in education research. ====Ψ, root-mean-square standardized effect==== A similar effect size estimator for multiple comparisons (e.g., ANOVA) is the Ψ root-mean-square standardized effect: \Psi = \sqrt{ \frac{1}{k-1} \cdot \sum_{j=1}^k \left(\frac{\mu_j-\mu}{\sigma}\right)^2} where k is the number of groups in the comparisons. This essentially presents the omnibus difference of the entire model adjusted by the root mean square, analogous to d or g. In addition, a generalization for multi-factorial designs has been provided. Provided that the data is Gaussian distributed a scaled Hedges' g, \sqrt{n_1 n_2/(n_1+n_2)}\,g , follows a noncentral t-distribution with the noncentrality parameter \sqrt{n_1 n_2/(n_1+n_2)}\theta and degrees of freedom. Likewise, the scaled Glass' Δ is distributed with degrees of freedom. From the distribution it is possible to compute the expectation and variance of the effect sizes. In some cases large sample approximations for the variance are used. One suggestion for the variance of Hedges' unbiased estimator is \hat{\sigma}^2(g^*) = \frac{n_1+n_2}{n_1 n_2} + \frac{(g^*)^2}{2(n_1 + n_2)}. \beta = \frac{\mu_1 - \mu_2}{\sqrt{\sigma_1^2 + \sigma_2^2 - 2\sigma_{12} }}. If the two groups are independent, \beta = \frac{\mu_1 - \mu_2}{\sqrt{\sigma_1^2 + \sigma_2^2 }}. If the two independent groups have equal variances \sigma^2 , \beta = \frac{\mu_1 - \mu_2}{\sqrt{2}\sigma}. E_{HBG}=\frac{2}{\pi}\arctan\left( \frac{J(\bar X_2-\bar X_1)} {k\,S_{ref}\sqrt{1+\lambda\left(\frac{s_D}{S_{ref}}\right)^2}} \right) where S_{ref}=\sqrt{\frac{s_1^2+s_2^2}{2}}. In the published calibration of the method, the recommended parameter values were \lambda = 2.8759 and k = 0.5069 . Here \bar X_1 and \bar X_2 denote the means at two assessment occasions, s_1 and s_2 denote the corresponding standard deviations, s_D is the standard deviation of the individual gain scores, and J is a small-sample correction factor. In the practical approximation reported for the method, J \approx 1-\frac{3}{4n-5} . In published calibrations, absolute values of E_{HBG} have been interpreted using the following empirical bands. ===Categorical family: Effect sizes for associations among categorical variables=== Commonly used measures of association for the chi-squared test are the Phi coefficient and Cramér's V (sometimes referred to as Cramér's phi and denoted as φ c ). Phi is related to the point-biserial correlation coefficient and Cohen's d and estimates the extent of the relationship between two variables (2 × 2). Cramér's V may be used with variables having more than two levels. Phi can be computed by finding the square root of the chi-squared statistic divided by the sample size. Similarly, Cramér's V is computed by taking the square root of the chi-squared statistic divided by the sample size and the length of the minimum dimension (k is the smaller of the number of rows r or columns c). φ c is the intercorrelation of the two discrete variables and may be computed for any value of r or c. However, as chi-squared values tend to increase with the number of cells, the greater the difference between r and c, the more likely V will tend to 1 without strong evidence of a meaningful correlation. \omega = \sqrt{ \sum_{i=1}^m \frac{ (p_{1i} - p_{0i})^2 }{p_{0i}} } where p 0i is the proportion of the i th cell under H 0 , p 1i is the proportion of the i th cell under H 1 and m is the number of cells. The odds ratio (OR) is another useful effect size. It is appropriate when the research question focuses on the degree of association between two binary variables. For example, consider a study of spelling ability. In a control group, two students pass the class for every one who fails, so the odds of passing are two to one (or 2/1 = 2). In the treatment group, six students pass for every one who fails, so the odds of passing are six to one (or 6/1 = 6). The effect size can be computed by noting that the odds of passing in the treatment group are three times higher than in the control group (because 6 divided by 2 is 3). Therefore, the odds ratio is 3. Odds ratio statistics are on a different scale than Cohen's d, so this '3' is not comparable to a Cohen's d of 3. The risk difference (RD), sometimes called absolute risk reduction, is simply the difference in risk (probability) of an event between two groups. It is a useful measure in experimental research, since RD tells you the extent to which an experimental interventions changes the probability of an event or outcome. Using the example above, the probabilities for those in the control group and treatment group passing is 2/3 (or 0.67) and 6/7 (or 0.86), respectively, and so the RD effect size is 0.86 − 0.67 = 0.19 (or 19%). RD is the superior measure for assessing effectiveness of interventions. h = 2 ( \arcsin \sqrt{p_1} - \arcsin \sqrt{p_2}) where p 1 and p 2 are the proportions of the two samples being compared and arcsin is the arcsine transformation. d = \frac{\sum_{i,j} [x_i > x_j] - [x_i where the two distributions are of size n and m with items x_i and x_j , respectively, and [\cdot] is the Iverson bracket, which is 1 when the contents are true and 0 when false. d is linearly related to the Mann–Whitney U statistic; however, it captures the direction of the difference in its sign. Given the Mann–Whitney U , d is: d = \frac{2U}{mn} - 1 One of simplest effect sizes for measuring how much a proportion differs from 50% is Cohen's g. It measures how much a proportion differs from 50%. For example, if 85.2% of arrests for car theft are males, then effect size of sex on arrest when measured with Cohen's g is g = 0.852-0.5=0.352 . In general: g = P - 0.50 \text{ or } 0.50 - P \quad (\text{directional}), g = |P - 0.50| \quad (\text{nondirectional}). Units of Cohen's g are more intuitive (proportion) than in some other effect sizes. It is sometime used in combination with Binomial test. t := \frac{M - \mu_{\text{baseline}}}{\text{SE}} = \frac{M- \mu_{\text{baseline}}}{\text{SD}/\sqrt{n}}=\frac{\sqrt{n} \left( \frac{M-\mu}{\sigma} \right) + \sqrt{n} \left( \frac{\mu-\mu_\text{baseline}}{\sigma}\right) }{\frac{\text{SD}} \sigma} ncp=\sqrt{n} \left( \frac{\mu-\mu_\text{baseline}}{\sigma} \right) and Cohen's d := \frac{M-\mu_\text{baseline}}{\text{SD}} is the point estimate of \frac{\mu-\mu_\text{baseline}} \sigma. So, \tilde{d}=\frac{ncp}{\sqrt n}. t:=\frac{M_1-M_2}{\text{SD}_\text{within}/\sqrt{\frac{n_1 n_2}{n_1+n_2}}}, wherein \text{SD}_\text{within}:=\sqrt{\frac{\text{SS}_\text{within}}{\text{df}_\text{within}}} = \sqrt{\frac{(n_1-1)\text{SD}_1^2+(n_2-1) \text{SD}_2^2}{n_1+n_2-2}}. ncp=\sqrt{\frac{n_1 n_2}{n_1+n_2}}\frac{\mu_1-\mu_2} \sigma and Cohen's d:=\frac{M_1-M_2}{SD_\text{within}} is the point estimate of \frac{\mu_1-\mu_2} \sigma. So, \tilde{d}=\frac{ncp}{\sqrt{\frac{n_1 n_2}{n_1+n_2}}}. F := \frac{\frac{\text{SS}_\text{between}}{\sigma^2}/\text{df}_\text{between}}{\frac{\text{SS}_\text{within}}{\sigma^2}/\text{df}_\text{within}} For each j-th sample within i-th group X i,j , denote M_i (X_{i,j}) := \frac{\sum_{w=1}^{n_i} X_{i,w}}{n_i};\; \mu_i (X_{i,j}) := \mu_i. While, \begin{align} \text{SS}_\text{between}/\sigma^2 & = \frac{\text{SS}\left(M_i (X_{i,j}); i=1,2,\dots,K,\; j=1,2,\dots,n_{i}\right)}{\sigma^2}\\ & = \text{SS}\left(\frac{M_i(X_{i,j}-\mu_i)}{\sigma}+\frac{\mu_i}{\sigma};i=1,2,\dots,K,\; j=1,2,\dots,n_i \right)\\ & \sim \chi^2\left(\text{df}=K-1,\; ncp=SS\left(\frac{\mu_i(X_{i,j})}{\sigma};i=1,2,\dots,K,\; j=1,2,\dots,n_i\right)\right) \end{align} So, both ncp(s) of F and \chi^2 equate \text{SS}\left(\mu_i(X_{i,j})/\sigma;i=1,2,\dots,K,\; j=1,2,\dots,n_i \right). In case of n:=n_1=n_2=\cdots=n_K for K independent groups of same size, the total sample size is N := n·K. \text{Cohens }\tilde{f}^2 := \frac{\text{SS}(\mu_1,\mu_2, \dots ,\mu_K)}{K\cdot\sigma^2} = \frac{\text{SS} \left(\mu_i(X_{i,j})/\sigma; i=1,2,\dots,K,\; j=1,2,\dots,n_i \right)}{n\cdot K} = \frac{ncp}{n\cdot K}=\frac{ncp}N. The t-test for a pair of independent groups is a special case of one-way ANOVA. Note that the noncentrality parameter ncp_F of F is not comparable to the noncentrality parameter ncp_t of the corresponding t. Actually, ncp_F = ncp_t^2 , and \tilde{f} = \left|\frac{\tilde{d}}{2}\right| . ==See also== Estimation statistics Statistical significance Z-factor, an alternative measure of effect size ==References== Aaron, B., Kromrey, J. D., & Ferron, J. M. (1998, November). Equating r-based and d-based effect-size indices: Problems with a commonly recommended formula. Paper presented at the annual meeting of the Florida Educational Research Association, Orlando, FL. (ERIC Document Reproduction Service No. ED433353) Lipsey, M. W., & Wilson, D. B. (2001). Practical meta-analysis. Sage: Thousand Oaks, CA. ==External links== Further explanations Effect Size (ES) EffectSizeFAQ.com EstimationStats.com Web app for generating effect-size plots. Measuring Effect Size Computing and Interpreting Effect size Measures with ViSta effsize package for the R Project for Statistical Computing

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