Effect size

In statistics, effect size is a measure of the strength of the relationship between two variables. In scientific experiments, it is often useful to know not only whether an experiment has a statistically significant effect, but also the size of any observed effects. In practical situations, effect sizes are helpful for making decisions. Effect size measures are the common currency of meta-analysis studies that summarize the findings from a specific area of research.


The concept of effect size appears in everyday language. For example, a weight loss program may boast that it leads to an average weight loss of 30 pounds. In this case, 30 pounds is an indicator of the claimed effect size. Another example is that a tutoring program may claim that it raises school performance by one letter grade. This grade increase is the claimed effect size of the program.

An effect size is best explained through an example: if you had no previous contact with humans, and one day visited England, how many people would you need to see before you realize that, on average, men are taller than women there? The answer relates to the effect size of the difference in average height between men and women. The larger the effect size, the easier it is to see that men are taller. If the height difference were small, then it would require knowing the heights of many men and women to notice that (on average) men are taller than women. This example is demonstrated further below.

In inferential statistics, an effect size helps to determine whether a statistically significant difference is a difference of practical concern. In other words, given a sufficiently large sample size, it is always possible to show that there is a difference between two means being compared out to some decimal position. The effects size helps us to know whether the difference observed is a difference that matters. Effect size, sample size, critical significance level (alpha), and power in statistical hypothesis testing are related: any one of these values can be determined, given the others. In meta-analysis, effect sizes are used as a common measure that can be calculated for different studies and then combined into overall analyses.

The term effect size more often refers to a statistic, which relies on a replication of samples. However, just like the term variance, whether it means a population parameter or a samples' statistic, is contextual. In inferential statistics, the population parameter does not vary across replications or experiments, and has a confidence interval for each replication of samples, while the samples' statistic varies replication by replication, and usually converges to one corresponding population parameter as sample size increases infinitely. Conventionally, Greek letters denote population parameters and Latin letters denote samples' statistics. Currently, most named effect sizes do not make an explicit distinction. So, Cumming & Finch (2001) advised Cohen's delta to denote the corresponding population parameter of Cohen's d.

The term effect size is most commonly used to describe standardized measures of effect (e.g., r, Cohen's d, odds ratio, etc.). However, unstandardized measures (e.g., the raw difference between group means, unstandardized regression coefficients, etc.) can equally be effect size measures. Standardized effect size measures are typically used when the metrics of variables being studied do not have intrinsic meaning to the reader (e.g., a score on a personality test on an arbitrary scale), or when results from multiple studies are being combined when some or all of the studies use different scales. Some students mistook the recommendation of Wilkinson & APA Task Force on Statistical Inference (1999, p. 599)--Always present effect sizes for primary outcomes--as that reporting standardized measures of effect like Cohen's d is the default requirement. Actually, just following the sentence the authors added that -- If the units of measurement are meaningful on a practical level (e.g., number of cigarettes smoked per day), then we usually prefer an unstandardized measure (regression coefficient or mean difference) to a standardized measure (r or d).


Presentation of effect size and confidence interval is highly recommended in biological journals . Biologists should ultimately be interested in biological importance, which can be assessed using the magnitude on an effect, not statistical significance. Combined use of an effect size and its confidence interval enables to assess the relationship within data more effectively than the use of p values, regardless of statistical significance. Also, routine presentation of effect size will encourage researchers to view their results in the context of previous research and facilitate the incorporation of results in future meta-analysis. However, issues surrounding publication bias towards statistically significant results, coupled with inadequate statistical power will lead to an overestimation of effect sizes, consequently affecting meta-analyses and power-analyses.


Pearson r correlation

Pearson's r correlation, introduced by Karl Pearson, is one of the most widely used effect sizes. It can be used when the data are continuous or binary; thus the Pearson r is arguably the most versatile effect size. This was the first important effect size to be developed in statistics. 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 (1988, 1992) gives the following guidelines for the social sciences: small effect size, r = 0.1; medium, r = 0.3; large, r = 0.5.

Another often-used measure of the strength of the relationship between two variables is the coefficient of determination (the square of r, referred to as "r-squared"). This is a measure of the proportion of variance shared by the two variables, and varies from 0 to 1. An of 0.21 means that 21% of the total variance is shared by the two variables.

Effect sizes based on means

A (population) effect size θ based on means usually considers the standardized mean diffence between two populations
theta = frac{mu_1 - mu_2}{sigma},
where mu_1 is the mean for one population and mu_2 is the mean for the other population. σ is the standard deviation, — either of the second population or a standard deviation pooled from the two groups.

In the practical setting the population values are typically not known and must be obtained 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.

Cohen's d

Cohen's d is defined as the difference between two means divided by a standard deviation for the data
d = frac{bar{x}_1 - bar{x}_2}{s},
What precisely the standard deviation s is was not originally made explicit by Jacob Cohen as he defines it (using the symbol "σ") as "the standard deviation of either population (since they are assumed equal)". Other authors make the computation of the standard deviation more explicit with the following definition for a pooled standard deviation
s = sqrt{frac{(n_1-1)s^2_1 + (n_2-1)s^2_2}{n_1+n_2} },
with bar{x}_k and s_k as the mean and standard deviation for group k, for k = 1, 2.
s_1^2 = frac{1}{n_1-1} sum_{i=1}^{n_1} (x_{1,i} - bar{x}_1)^2
This definition of "Cohen's d" is termed the maximum likelihood estimator by Hedges and Olkin, and it is related to Hedges' g (see below) by a scaling
g = sqrt{frac{n_1+n_2-2}{n_1+n_2}} d

Glass's Δ

In 1976 Gene V. Glass proposed an estimator of the effect size that uses only the standard deviation of the second group
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 variations.

Under an assumption of equal population variances a pooled estimate for σ is more precise.

Hedges' g

Hedges' g, suggested by Larry Hedges in 1981, is like the other measures based on a standardized difference
g = frac{bar{x}_1 - bar{x}_2}{s^*}
but its pooled standard deviation s^* is computed slightly different from Cohen's d
s^* = sqrt{frac{(n_1-1)s_1^2 + (n_2-1)s_2^2}{n_1+n_2-2}}
As an estimator for the population effect size θ it is biased. However, this bias can be corrected for by multiplication with a factor
g^* = J(n_1+n_2-2) g approx left(1-frac{3}{4(n_1+n_2)-9}right) g
In their 1985 book Hedges and Olkin refer to this unbiased 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)}

Distribution of effect sizes based on means

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 n_1+n_2-2 degrees of freedom. Likewise, the scaled Glass' Δ is distributed with n_2-1 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

hat{sigma}^2(g^*) = frac{n_1+n_2}{n_1 n_2} + frac{(g^*)^2}{2(n_1 + n_2)}

Cohen's f^{2}

Cohen's f^{2} is an appropriate effect size measure to use in the context of an F-test for ANOVA or multiple regression. The f^{2} effect size measure for multiple regression is defined as:

f^{2} = {R^{2} over 1 - R^{2}}

where R^{2} is the squared multiple correlation.

The f^{2} effect size measure for hierarchical multiple regression 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 B.

By convention, f^{2} effect sizes of 0.02, 0.15, and 0.35 are termed small, medium, and large, respectively (Cohen, 1988).

Cohen's hat{f} can also be found for factorial analysis of variance (ANOVA, aka the F-test) working backwards using :

hat{f}_{Effect} = {sqrt{(df_{Effect}/N) (F_{Effect}-1)}}.
In a balanced design (equivalent sample sizes across groups) of ANOVA, the corresponding population parameter of f^2 is : {SS(mu_1,mu_2,cdots,mu_K)}over{K times sigma^2}, wherein mu_j denotes the population mean within the jth group of the total K groups, and sigma the equivalent population standard deviations within each groups. SS is the sum of squares manipulation in ANOVA.

φ, Cramer's φ, or Cramer's V

 phi = sqrt{ frac{chi^2}{N}}   phi_c = sqrt{ frac{chi^2}{N(k - 1)}} 
Phi (φ) Cramer's Phi (φc)
The best measure of association for the chi-square test is phi (or Cramer's phi or V). Phi is related to the point-biserial correlation coefficient and Cohen's d and estimates the extent of the relationship between two variables (2 x 2). Cramer's Phi may be used with variables having more than two levels.

Phi can be computed by finding the square root of the chi-square statistic divided by the sample size.

Similarly, Cramer's phi can be found through a slightly more complex formula that takes the number of rows or columns into account (k).

Odds ratio

The odds ratio is another useful effect size. It is appropriate when both variables are binary. For example, consider a study on spelling. In a control group, two students pass the class for every one who fails, so the odds of passing are two to one (or more briefly 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. However, odds ratio statistics are on a different scale to Cohen's d. So, this '3' is not comparable to a Cohen's d of '3'.

"Small", "medium", "big"

Some fields using effect sizes apply words such as "small", "medium" and "big" to the size of the effect. Whether an effect size should be interpreted small, medium, or big depends on its substantial context and its operational definition. Cohen's (1988) conventional criterions small, medium, or big are never ubiquitous across all fields. Power analysis or sample size planning requires an assumed population parameter of effect sizes. Many researchers adopt Cohen's standards as default alternative hypotheses. Russell Lenth criticized them as T-shirt effect sizes

This is an elaborate way to arrive at the same sample size that has been used in past social science studies of large, medium, and small size (respectively). The method uses a standardized effect size as the goal. Think about it: 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. "Medium" is definitely not the message!

For Cohen's d an effect size og 0.2 to 0.3 might be a "small" effect, around 0.5 a "medium" effect and 0.8 to 1.0 a "large" effect.


Further reading

  • 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)
  • Cohen, J. (1992). A power primer. Psychological Bulletin, 112, 155-159.
  • Cumming, G. and Finch, S. (2001). A primer on the understanding, use, and calculation of confidence intervals that are based on central and noncentral distributions. Educational and Psychological Measurement, 61, 530–572.
  • Lipsey, M.W., & Wilson, D.B. (2001). Practical meta-analysis. Sage: Thousand Oaks, CA.
  • Wilkinson, L., & APA Task Force on Statistical Inference. (1999). Statistical methods in psychology journals: Guidelines and explanations. American Psychologist, 54, 594-604.

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