Pearson r or correlation coefficient

Pearson's correlation, often denoted r and introduced by Karl Pearson, is widely used as an effect size when paired quantitative data are available; for instance if one were studying the relationship between birth weight and longevity. The correlation coefficient can also be used when the data are binary. Pearson's r can vary in magnitude from −1 to 1, with −1 indicating a perfect negative linear relation, 1 indicating a perfect positive linear relation, and 0 indicating no linear relation between two variables. Cohen gives the following guidelines for the social sciences:^[9][16]

Eta-squared (η²)

Eta-squared describes the ratio of variance explained in the dependent variable by a predictor while controlling for other predictors, making it analogous to the r². Eta-squared is a biased estimator of the variance explained by the model in the population (it estimates only the effect size in the sample). This estimate shares the weakness with r² that each additional variable will automatically increase the value of η². In addition, it measures the variance explained of the sample, not the population, meaning that it will always overestimate the effect size, although the bias grows smaller as the sample grows larger.

$${\displaystyle \eta ^{2}={\frac {SS_{\text{Treatment}}}{SS_{\text{Total}}}}.}$$

Omega-squared (ω²)

See also: Adjusted R²

A less biased estimator of the variance explained in the population is ω^2[17]

$${\displaystyle \omega ^{2}={\frac {{\text{SS}}_{\text{treatment}}-df_{\text{treatment}}\cdot {\text{MS}}_{\text{error}}}{{\text{SS}}_{\text{total}}+{\text{MS}}_{\text{error}}}}.}$$

This form of the formula is limited to between-subjects analysis with equal sample sizes in all cells.^[17] Since it is less biased (although not unbiased), ω² is preferable to η²; 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.^[18] In addition, methods to calculate partial ω² for individual factors and combined factors in designs with up to three independent variables have been published.^[18]

Cohen's f²

Cohen's f² is one of several effect size measures to use in the context of an F-test for ANOVA or multiple regression. Its amount of bias (overestimation of the effect size for the ANOVA) depends on the bias of its underlying measurement of variance explained (e.g., R², η², ω²).

The f² effect size measure for multiple regression is defined as:

$${\displaystyle f^{2}={R^{2} \over 1-R^{2}}}$$

where R² is the squared multiple correlation.

Likewise, f² can be defined as:

$${\displaystyle f^{2}={\eta ^{2} \over 1-\eta ^{2}}}$$

or

$${\displaystyle f^{2}={\omega ^{2} \over 1-\omega ^{2}}}$$

for models described by those effect size measures.^[19]

The ${\displaystyle f^{2}}$ effect size measure for sequential multiple regression and also common for PLS modeling^[20] is defined as:

$${\displaystyle f^{2}={R_{AB}^{2}-R_{A}^{2} \over 1-R_{AB}^{2}}}$$

where R²_A is the variance accounted for by a set of one or more independent variables A, and R²_AB is the combined variance accounted for by A and another set of one or more independent variables of interest B. By convention, f² effect sizes of ${\displaystyle 0.1^{2}}$, ${\displaystyle 0.25^{2}}$, and ${\displaystyle 0.4^{2}}$ are termed small, medium, and large, respectively.^[9]

Cohen's ${\displaystyle {\hat {f}}}$ can also be found for factorial analysis of variance (ANOVA) working backwards, using:

$${\displaystyle {\hat {f}}_{\text{effect}}={\sqrt {(F_{\text{effect}}df_{\text{effect}}/N)}}.}$$

In a balanced design (equivalent sample sizes across groups) of ANOVA, the corresponding population parameter of ${\displaystyle f^{2}}$ is

$${\displaystyle {SS(\mu _{1},\mu _{2},\dots ,\mu _{K})} \over {K\times \sigma ^{2}},}$$

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.

Cohen's q

Another measure that is used with correlation differences is Cohen's q. This is the difference between two Fisher transformed Pearson regression coefficients. In symbols this is

$${\displaystyle q={\frac {1}{2}}\log {\frac {1+r_{1}}{1-r_{1}}}-{\frac {1}{2}}\log {\frac {1+r_{2}}{1-r_{2}}}}$$

where r₁ and r₂ are the regressions being compared. The expected value of q is zero and its variance is

$${\displaystyle \operatorname {var} (q)={\frac {1}{N_{1}-3}}+{\frac {1}{N_{2}-3}}}$$

where N₁ and N₂ are the number of data points in the first and second regression respectively.

Standardized mean difference

Plots of Gaussian densities illustrating various values of Cohen's d.

A (population) effect size θ based on means usually considers the standardized mean difference (SMD) between two populations^[21]: 78

$${\displaystyle \theta ={\frac {\mu _{1}-\mu _{2}}{\sigma }},}$$

where μ₁ is the mean for one population, μ₂ 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 ${\displaystyle {\sqrt {n}}}$. 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.^[22]

Cohen's d

Cohen's d is defined as the difference between two means divided by a standard deviation for the data, i.e.

$${\displaystyle d={\frac {{\bar {x}}_{1}-{\bar {x}}_{2}}{s}}.}$$

Jacob Cohen defined s, the pooled standard deviation, as (for two independent samples):^[9]: 67

$${\displaystyle s={\sqrt {\frac {(n_{1}-1)s_{1}^{2}+(n_{2}-1)s_{2}^{2}}{n_{1}+n_{2}-2}}}}$$

where the variance for one of the groups is defined as

$${\displaystyle s_{1}^{2}={\frac {1}{n_{1}-1}}\sum _{i=1}^{n_{1}}(x_{1,i}-{\bar {x}}_{1})^{2},}$$

and similarly for the other group.

The table below contains descriptors for magnitudes of d = 0.01 to 2.0, as initially suggested by Cohen and expanded by Sawilowsky.^[10]

Other authors choose a slightly different computation of the standard deviation when referring to "Cohen's d" where the denominator is without "-2"^{[23][24]: 14}

$${\displaystyle s={\sqrt {\frac {(n_{1}-1)s_{1}^{2}+(n_{2}-1)s_{2}^{2}}{n_{1}+n_{2}}}}}$$

This definition of "Cohen's d" is termed the maximum likelihood estimator by Hedges and Olkin,^[21] and it is related to Hedges' g by a scaling factor (see below).

With two paired samples, we look at the distribution of the difference scores. In that case, s is the standard deviation of this distribution of difference scores. This creates the following relationship between the t-statistic to test for a difference in the means of the two groups and Cohen's d:

$${\displaystyle t={\frac {{\bar {X}}_{1}-{\bar {X}}_{2}}{\text{SE}}}={\frac {{\bar {X}}_{1}-{\bar {X}}_{2}}{\frac {\text{SD}}{\sqrt {N}}}}={\frac {{\sqrt {N}}({\bar {X}}_{1}-{\bar {X}}_{2})}{SD}}}$$

and

$${\displaystyle d={\frac {{\bar {X}}_{1}-{\bar {X}}_{2}}{\text{SD}}}={\frac {t}{\sqrt {N}}}}$$

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.^[25]

For paired samples Cohen suggests that the d calculated is actually a d', which doesn't provide the correct answer to obtain the power of the test, and that before looking the values up in the tables provided, it should be corrected for r as in the following formula:^[26]

$${\displaystyle d={\frac {d'}{\sqrt {1-r}}}.}$$

Glass' Δ

In 1976, Gene V. Glass proposed an estimator of the effect size that uses only the standard deviation of the second group^[21]: 78

$${\displaystyle \Delta ={\frac {{\bar {x}}_{1}-{\bar {x}}_{2}}{s_{2}}}}$$

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.

Hedges' g

Hedges' g, suggested by Larry Hedges in 1981,^[27] is like the other measures based on a standardized difference^[21]: 79

$${\displaystyle g={\frac {{\bar {x}}_{1}-{\bar {x}}_{2}}{s^{*}}}}$$

where the pooled standard deviation ${\displaystyle s^{*}}$ is computed as:

$${\displaystyle s^{*}={\sqrt {\frac {(n_{1}-1)s_{1}^{2}+(n_{2}-1)s_{2}^{2}}{n_{1}+n_{2}-2}}}.}$$

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

$${\displaystyle g^{*}=J(n_{1}+n_{2}-2)\,\,g\,\approx \,\left(1-{\frac {3}{4(n_{1}+n_{2})-9}}\right)\,\,g}$$

Hedges and Olkin refer to this less-biased estimator ${\displaystyle g^{*}}$ as d,^[21] but it is not the same as Cohen's d. The exact form for the correction factor J() involves the gamma function^[21]: 104

$${\displaystyle J(a)={\frac {\Gamma (a/2)}{{\sqrt {a/2\,}}\,\Gamma ((a-1)/2)}}.}$$

There are also multilevel variants of Hedges' g, e.g., for use in cluster randomised controlled trials (CRTs).^[28] 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:^[19]

$${\displaystyle \Psi ={\sqrt {{\frac {1}{k-1}}\cdot \sum _{j=1}^{k}\left({\frac {\mu _{j}-\mu }{\sigma }}\right)^{2}}}}$$

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.^[19]

Distribution of effect sizes based on means

Provided that the data is Gaussian distributed a scaled Hedges' g, ${\textstyle {\sqrt {n_{1}n_{2}/(n_{1}+n_{2})}}\,g}$, follows a noncentral t-distribution with the noncentrality parameter ${\textstyle {\sqrt {n_{1}n_{2}/(n_{1}+n_{2})}}\theta }$ and (n₁ + n₂ − 2) degrees of freedom. Likewise, the scaled Glass' Δ is distributed with n₂ − 1 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^{[21] : 86}

$${\displaystyle {\hat {\sigma }}^{2}(g^{*})={\frac {n_{1}+n_{2}}{n_{1}n_{2}}}+{\frac {(g^{*})^{2}}{2(n_{1}+n_{2})}}.}$$

Other metrics

Mahalanobis distance (D) is a multivariate generalization of Cohen's d, which takes into account the relationships between the variables.^[29]

Cohen's omega (ω)

Another measure of effect size used for chi-squared tests is Cohen's omega (${\displaystyle \omega }$). This is defined as

$${\displaystyle \omega ={\sqrt {\sum _{i=1}^{m}{\frac {(p_{1i}-p_{0i})^{2}}{p_{0i}}}}}}$$

where p_0i is the proportion of the i^th cell under H₀, p_1i is the proportion of the i^th cell under H₁ and m is the number of cells.

In Statistical Power Analysis for the Behavioral Sciences (1988, pp.224-225), Cohen gives the following general guideline for interpreting omega (see table below), but warns against its "possible inaptness in any given substantive context" and advises to use context-relevant judgment instead.

Effect Size	${\displaystyle \omega }$
Small	0.10
Medium	0.30
Large	0.50

Odds ratio

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.

Relative risk

The relative risk (RR), also called risk ratio, is simply the risk (probability) of an event relative to some independent variable. This measure of effect size differs from the odds ratio in that it compares probabilities instead of odds, but asymptotically approaches the latter for small probabilities. 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. The effect size can be computed the same as above, but using the probabilities instead. Therefore, the relative risk is 1.28. Since rather large probabilities of passing were used, there is a large difference between relative risk and odds ratio. Had failure (a smaller probability) been used as the event (rather than passing), the difference between the two measures of effect size would not be so great.

While both measures are useful, they have different statistical uses. In medical research, the odds ratio is commonly used for case-control studies, as odds, but not probabilities, are usually estimated.^[32] Relative risk is commonly used in randomized controlled trials and cohort studies, but relative risk contributes to overestimations of the effectiveness of interventions.^[33]

Risk difference

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.^[33]

Cohen's h

One measure used in power analysis when comparing two independent proportions is Cohen's h. This is defined as follows

$${\displaystyle h=2(\arcsin {\sqrt {p_{1}}}-\arcsin {\sqrt {p_{2}}})}$$

where p₁ and p₂ are the proportions of the two samples being compared and arcsin is the arcsine transformation.

Rank-biserial correlation

An effect size related to the common language effect size is the rank-biserial correlation. This measure was introduced by Cureton as an effect size for the Mann–Whitney U test.^[38] That is, there are two groups, and scores for the groups have been converted to ranks. The Kerby simple difference formula computes the rank-biserial correlation from the common language effect size.^[35] Letting f be the proportion of pairs favorable to the hypothesis (the common language effect size), and letting u be the proportion of pairs not favorable, the rank-biserial r is the simple difference between the two proportions: r = f − u. In other words, the correlation is the difference between the common language effect size and its complement. For example, if the common language effect size is 60%, then the rank-biserial r equals 60% minus 40%, or r = 0.20. The Kerby formula is directional, with positive values indicating that the results support the hypothesis.

A non-directional formula for the rank-biserial correlation was provided by Wendt, such that the correlation is always positive.^[39] The advantage of the Wendt formula is that it can be computed with information that is readily available in published papers. The formula uses only the test value of U from the Mann-Whitney U test, and the sample sizes of the two groups: r = 1 – (2U)/(n₁ n₂). Note that U is defined here according to the classic definition as the smaller of the two U values which can be computed from the data. This ensures that 2U < n₁n₂, as n₁n₂ is the maximum value of the U statistics.

An example can illustrate the use of the two formulas. Consider a health study of twenty older adults, with ten in the treatment group and ten in the control group; hence, there are ten times ten or 100 pairs. The health program uses diet, exercise, and supplements to improve memory, and memory is measured by a standardized test. A Mann-Whitney U test shows that the adult in the treatment group had the better memory in 70 of the 100 pairs, and the poorer memory in 30 pairs. The Mann-Whitney U is the smaller of 70 and 30, so U = 30. The correlation between memory and treatment performance by the Kerby simple difference formula is r = (70/100) − (30/100) = 0.40. The correlation by the Wendt formula is r = 1 − (2·30)/(10·10) = 0.40.