Effect sizes can be used to determine the sample size for follow-up studies, or examining effects across studies. This article aims to provide a practical primer on how to calculate and report effect sizes for t -tests and ANOVA's such that effect sizes can be used in a-priori power analyses and meta-analyses. Whereas many articles about effect sizes focus on between-subjects designs and address within-subjects designs only briefly, I provide a detailed overview of the similarities and differences between within- and between-subjects designs.
I suggest that some research questions in experimental psychology examine inherently intra-individual effects, which makes effect sizes that incorporate the correlation between measures the best summary of the results. Finally, a supplementary spreadsheet is provided to make it as easy as possible for researchers to incorporate effect size calculations into their workflow. Researchers want to know whether an intervention or experimental manipulation has an effect greater than zero, or when it is obvious an effect exists how big the effect is.
Researchers are often reminded to report effect sizes, because they are useful for three reasons. First, they allow researchers to present the magnitude of the reported effects in a standardized metric which can be understood regardless of the scale that was used to measure the dependent variable.
Such standardized effect sizes allow researchers to communicate the practical significance of their results what are the practical consequences of the findings for daily life , instead of only reporting the statistical significance how likely is the pattern of results observed in an experiment, given the assumption that there is no effect in the population.
Second, effect sizes allow researchers to draw meta-analytic conclusions by comparing standardized effect sizes across studies. Third, effect sizes from previous studies can be used when planning a new study. An a-priori power analysis can provide an indication of the average sample size a study needs to observe a statistically significant result with a desired likelihood.
The aim of this article is to explain how to calculate and report effect sizes for differences between means in between and within-subjects designs in a way that the reported results facilitate cumulative science.
There are some reasons to assume that many researchers can improve their understanding of effect sizes. This practical primer should be seen as a complementary resource for psychologists who want to learn more about effect sizes for excellent books that discuss this topic in more detail, see Cohen, ; Maxwell and Delaney, ; Grissom and Kim, ; Thompson, ; Aberson, ; Ellis, ; Cumming, ; Murphy et al.
A supplementary spreadsheet is provided to facilitate effect size calculations. Reporting standardized effect sizes for mean differences requires that researchers make a choice about the standardizer of the mean difference, or a choice about how to calculate the proportion of variance explained by an effect.
I point out some caveats for researchers who want to perform power-analyses for within-subjects designs, and provide recommendations regarding the effect sizes that should be reported.
Knowledge about the expected size of an effect is important information when planning a study. Researchers typically rely on null hypothesis significance tests to draw conclusions about observed differences between groups of observations. The probability of correctly rejecting the null hypothesis is known as the power of a statistical test Cohen, If three are known or estimated , the fourth parameter can be calculated.
In an a-priori power analysis, researchers calculate the sample size needed to observe an effect of a specific size, with a pre-determined significance criterion, and a desired statistical power. A generally accepted minimum level of power is 0. This minimum is based on the idea that with a significance criterion of 0. Some researchers have argued that Type 2 errors can potentially have much more serious consequences than Type 1 errors, however Fiedler et al.
Thus, although a power of 0. Effect size estimates have their own confidence intervals [for calculations for Cohen's d , see Cumming , for F -tests, see Smithson ], which are often very large in experimental psychology. Therefore, researchers should realize that the confidence interval around a sample size estimate derived from a power analysis is often also very large, and might not provide a very accurate basis to determine the sample size of a future study. Meta-analyses can provide more accurate effect size estimates for power analyses, and correctly reporting effect size estimates can facilitate future meta-analyses [although due to publication bias, meta-analyses might still overestimate the true effect size, see Brand et al.
Given that the mean difference is the same i. There are two diverging answers to this question. One viewpoint focusses on the generalizability of the effect size estimate across designs, while the other viewpoint focusses on the statistical significance of the difference between the means.
I will briefly discuss these two viewpoints. As Maxwell and Delaney , p. Although you can exclude individual variation in the statistical test if you use a pre- and post-measure, and the statistical power of a test will often substantially increase, the effect size e. A second perspective, which I will refer to as the statistical significance viewpoint, focusses on the statistical test of a predicted effect, and regards individual differences as irrelevant for the hypothesis that is examined.
The goal is to provide statistical support for the hypothesis, and being able to differentiate between variance that is due to individual differences and variance that is due to the manipulation increases the power of the study.
Researchers advocating the statistical significance viewpoint regard the different effect sizes e. The focus on the outcome of the statistical test in this perspective can be illustrated by the use of confidence intervals.
As first discussed by Loftus and Masson , the use of traditional formulas for confidence intervals developed for between-subjects designs can result in a marked discrepancy between the statistical summary of the results and the error bars used to visualize the differences between observations. To resolve this inconsistency, Loftus and Masson , p. To summarize, researchers either focus on generalizable effect size estimates, and try to develop effect size measures that are independent from the research design, or researchers focus on the statistical significance, and prefer effect sizes and confidence intervals to reflect the conclusions drawn by the statistical test.
Although these two viewpoints are not mutually exclusive, they do determine some of the practical choices researchers make when reporting their results.
Regardless of whether researchers focus on statistical significance or generalizability of measurements, cumulative science will benefit if researchers determine their sample size a-priori, and report effect sizes when they share their results. In the following sections, I will discuss how effect sizes to describe the differences between means are calculated, with a special focus on the similarities and differences in within and between-subjects designs, followed by an illustrative example.
Effect sizes can be grouped in two families Rosenthal, : The d family consisting of standardized mean differences and the r family measures of strength of association. Conceptually, the d family effect sizes are based on the difference between observations, divided by the standard deviation of these observations. The r family effect sizes describe the proportion of variance that is explained by group membership [e.
These effect sizes are calculated from the sum of squares the difference between individual observations and the mean for the group, squared, and summed for the effect divided by the sums of squares for other factors in the design. A further differentiation between effect sizes is whether they correct for bias or not e. Population effect sizes are almost always estimated on the basis of samples, and all population effect size estimates based on sample averages overestimate the true population effect for a more detailed explanation, see Thompson, Therefore, corrections for bias are used even though these corrections do not always lead to a completely unbiased effect size estimate.
These effects sizes will be discussed in more detail in the following paragraphs. Cohen's d is used to describe the standardized mean difference of an effect. This value can be used to compare effects across studies, even when the dependent variables are measured in different ways, for example when one study uses 7-point scales to measure dependent variables, while the other study uses 9-point scales, or even when completely different measures are used, such as when one study uses self-report measures, and another study used physiological measurements.
It ranges from 0 to infinity. Cohen uses subscripts to distinguish between different versions of Cohen's d , a practice I will follow because it prevents confusion without any subscript, Cohen's d denotes the entire family of effect sizes.
Cohen refers to the standardized mean difference between two groups of independent observations for the sample as d s which is given by:. In this formula, the numerator is the difference between means of the two groups of observations.
The denominator is the pooled standard deviation. Remember that the standard deviation is calculated from the differences between each individual observation and the mean for the group. These differences are squared to prevent the positive and negative values from cancelling each other out, and summed also referred to as the sum of squares. This value is divided by the number of observations minus one, which is Bessel's correction for bias in the estimation of the population variance, and finally the square root is taken.
This correction for bias in the sample estimate of the population variance is based on the least squares estimator see McGrath and Meyer, Note that Cohen's d s is sometimes referred to as Cohen's g , which can be confusing. Cohen's d s for between-subjects designs is directly related to a t -test, and can be calculated by:. Formula 2 underlines the direct relation between the effect size and the statistical significance.
The standardized mean difference can also be calculated without Bessel's correction, in which case it provides the maximum likelihood estimate for a sample, as noted by Hedges and Olkin The difference between Cohen's d s and Cohen's d pop for the population is important to keep in mind when converting Cohen's d s to the point biserial correlation r pb which will simply be referred to as r in the remainder of this text.
Many textbooks provide the formula to convert Cohen's d pop to r , while the formula to convert Cohen's d s to r which can only be used for between-subjects designs is provided by McGrath and Meyer :. Therefore, Cohen's d s is sometimes referred to as the uncorrected effect size.
The corrected effect size , or Hedges's g which is unbiased, see Cumming, , is:. I use the same subscript letter in Hedges's g to distinguish different calculations of Cohen's d. Although the difference between Hedges's g s and Cohen's d s is very small, especially in sample sizes above 20 Kline, , it is preferable and just as easy to report Hedges's g s. There are also bootstrapping procedures to calculate Cohen's d s when the data are not normally distributed, which can provide a less biased point estimate Kelley, As long as researchers report the number of participants in each condition for a between-subjects comparison and the t -value, Cohen's d and Hedges' g can be calculated.
How should researchers interpret this effect size? However, these values are arbitrary and should not be interpreted rigidly Thompson, The only reason to use these benchmarks is because findings are extremely novel, and cannot be compared to related findings in the literature Cohen, Cohen's d in between-subject designs can be readily interpreted as a percentage of the standard deviation, such that a Cohen's d of 0.
However, the best way to interpret Cohen's d is to relate it to other effects in the literature, and if possible, explain the practical consequences of the effect. Regrettably, there are no clear recommendation of how to do so Fidler, An interesting, though not often used, interpretation of differences between groups can be provided by the common language effect size McGraw and Wong, , also known as the probability of superiority Grissom and Kim, , which is a more intuitively understandable statistic than Cohen's d or r.
It can be calculated directly from Cohen's d , converts the effect size into a percentage, and expresses the probability that a randomly sampled person from one group will have a higher observed measurement than a randomly sampled person from the other group for between designs or for within-designs the probability that an individual has a higher value on one measurement than the other.
It is based on the distribution of the difference scores, with a mean that is estimated from the mean differences between the samples, and a standard deviation that is the square root of the sum of the sample variances divided by two. Mathematically, the common language effect size is the probability of a Z-score greater than the value that corresponds to a difference between groups of 0 in a normal distribution curve.
Z can be calculated by:. The supplementary spreadsheet provides an easy way to calculate the common language effect size. Conceptually, calculating Cohen's d for correlated measurements is the same as calculating Cohen's d for independent groups, where the differences between two measurements are divided by the standard deviation of both groups of measurements.
However, in the case of correlated measurements the dependent t -test uses the standard deviation of the difference scores. Testing whether observations from two correlated measurements are significantly different from each other using a paired samples t -test is mathematically identical to testing whether the difference scores of the correlated measurements is significantly different from 0 using a one-sample t -test.
Similarly, calculating the effect size for the difference between two correlated measurements is similar to the effect size that is calculated for a one sample t -test. The standardized mean difference effect size for within-subjects designs is referred to as Cohen's d z , where the Z alludes to the fact that the unit of analysis is no longer X or Y, but their difference, Z, and can be calculated with:.
The effect size estimate Cohen's d z can also be calculated directly from the t -value and the number of participants using the formula provided by Rosenthal :. Given the direct relationship between the t -value of a paired-samples t -test and Cohen's d z , it will not be surprising that software that performs power analyses for within-subjects designs e.
Cohen's d z is only rarely used in meta-analyses, because researchers often want to be able to compare effects across within and between-subject designs. One solution which is not generally recommended is to use Cohen's d rm , where the subscript is used by Morris and DeShon to indicate this is the equivalent of Cohen's d for repeated measures.
Cohen's d rm controls for the correlation between the two sets of measurements, as explained below. An alternative formula to calculate the standard deviation of the difference scores from the standard deviations of both groups and their correlation is given by Cohen as:. As the correlation between measures increases, the standard deviation of the difference scores decreases.
In experimental psychology, correlations between measures are typically a positive non-zero value. This has two consequences. First, within-subjects designs typically have more statistical power than between-subjects designs, because the standard deviation of the difference scores is smaller than the standard deviations of the two groups of observations.
In my head, this is like saying "the mean difference did not reach statistical significance but is still of particular note because the effect size indicated from the eta squared is medium". Or, is effect size a replacement value for significance testing, rather than complementary? Sign up to join this community.
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This will give the same value as eta squared in single IV Independent Groups Designs, but a different value in single IV repeated measures designs. This causes no end of problems with my students. Add a comment. Active Oldest Votes. Measures like eta square are influenced by whether group samples sizes are equal, whereas Cohen's d is not.
I also think that the meaning of d-based measures are more intuitive when what you are trying to quantify is a difference between group means. The above point is particularly strong for the case where you only have two groups e. If you have more than two groups, then the situation is a little more complicated. I can see the argument for variance explained measures in this case. A third option is that within the context of experimental effects, even when there are more than two groups, the concept of effect is best conceptualised as a binary comparison i.
In this case, you can once again return to d-based measures. The d-based measure is not an effect size measure for the factor, but rather of one group relative to a reference group. Suppose we want to determine if exercise intensity and gender impact weight loss. To test this, we recruit 30 men and 30 women to participate in an experiment in which we randomly assign 10 of each to follow a program of either no exercise, light exercise, or intense exercise for one month.
The following table shows the results of a two-way ANOVA using exercise and gender as factors and weight loss as the response variable :. We can calculate the partial eta squared for gender and exercise as follows:. We would conclude that the effect size for exercise is very large while the effect size for gender is quite small.
The p-value for exercise 0.
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