1. Calculate the sample size needed given these factors:one-tailed t-test with two independent groups of equal sizesmall effect size (see Piasta, S.B., & Justice, L.M., 2010)alpha =.05beta = .2Assume that the result is a sample size beyond what you can obtain. Use the compromise function to compute alpha and beta for a sample half the size. Indicate the resulting alpha and beta. Present an argument that your study is worth doing with the smaller sample (include peer-reviewed journal articles as needed to support your response). 2. Calculate the sample size needed given these factors:ANOVA (fixed effects, omnibus, one-way)small effect sizealpha =.05beta = .23 groupsAssume that the result is a sample size beyond what you can obtain. Use the compromise function to compute alpha and beta for a sample approximately half the size. Give your rationale for your selected beta/alpha ratio. Indicate the resulting alpha and beta. Give an argument that your study is worth doing with the smaller sample.3. In a few sentences, describe two designs that can address your research question. The designs must involve two different statistical analyses. For each design, specify and justify each of the four factors and calculate the estimated sample size you’ll need. Give reasons for any parameters you need to specify for G*Power.Support your paper with a minimum of 5 resources. In addition to these specified resources, other appropriate scholarly resources, including older articles, may be included.Length: 5-7 pages not including title and reference pagesReferences: Minimum of 5 scholarly resources.The first two charts in the word document will pertain to the first question and the last two charts will pertain to the second questions. Please let me know if there are any questions.

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Encyclopedia of Research Design

Cohen’s d Statistic

Contributors: Shayne B. Piasta & Laura M. Justice

Edited by: Neil J. Salkind

Book Title: Encyclopedia of Research Design

Chapter Title: “Cohen’s d Statistic”

Pub. Date: 2010

Access Date: October 26, 2017

Publishing Company: SAGE Publications, Inc.

City: Thousand Oaks

Print ISBN: 9781412961271

Online ISBN: 9781412961288

DOI: http://dx.doi.org/10.4135/9781412961288.n58

Print pages: 181-185

©2010 SAGE Publications, Inc.. All Rights Reserved.

This PDF has been generated from SAGE Knowledge. Please note that the pagination of

the online version will vary from the pagination of the print book.

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Cohen’s d statistic is a type of effect size. An effect size is a specific numerical nonzero value

used to represent the extent to which a null hypothesis is false. As an effect size, Cohen’s d is

typically used to represent the magnitude of differences between two (or more) groups on a

given variable, with larger values representing a greater differentiation between the two

groups on that variable. When comparing means in a scientific study, the reporting of an

effect size such as Cohen’s d is considered complementary to the reporting of results from a

test of statistical significance. Whereas the test of statistical significance is used to suggest

whether a null hypothesis is true (no difference exists between Populations A and B for a

specific phenomenon) or false (a difference exists between Populations A and B for a specific

phenomenon), the calculation of an effect size estimate is used to represent the degree of

difference between the two populations in those instances for which the null hypothesis was

deemed false. In cases for which the null hypothesis is false (i.e., rejected), the results of a

test of statistical significance imply that reliable differences exist between two populations on

the phenomenon of interest, but test outcomes do not provide any value regarding the extent

of that difference. The calculation of Cohen’s d and its interpretation provide a way to estimate

the actual size of observed differences between two groups, namely, whether the differences

are small, medium, or large.

Calculation of Cohen’s d Statistic

Cohen’s d statistic is typically used to estimate between-subjects effects for grouped data,

consistent with an analysis of variance framework. Often, it is employed within experimental

contexts to estimate the differential impact of the experimental manipulation across conditions

on the dependent variable of interest. The dependent variable must represent continuous

data; other effect size measures (e.g., Pearson family of correlation coefficients, odds ratios)

are appropriate for non-continuous data.

General Formulas

Cohen’s d statistic represents the standardized mean differences between groups. Similar to

other means of standardization such as z scoring, the effect size is expressed in standard

score units. In general, Cohen’s d is defined as

where d represents the effect size, μ1 and μ2 represent the two population means, and σ∊

represents the pooled within-group population standard deviation. In practice, these

population parameters are typically unknown and estimated by means of sample statistics:

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The population means are replaced with sample means (

,j) and the population standard deviation is replaced with Sp, the pooled standard deviation

from the sample. The pooled standard deviation is derived by weighing the variance around

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each sample mean by the respective sample size.

Calculation of the Pooled Standard Deviation

Although computation of the difference in sample means is straightforward in Equation 2, the

pooled standard deviation may be calculated in a number of ways. Consistent with the

traditional definition of a standard deviation, this statistic may be computed as

where nj represents the sample sizes for j groups and s2j represents the variance (i.e.,

squared standard deviation) of the / samples. Often, however, the pooled sample standard

deviation is corrected for bias in its estimation of the corresponding population parameter, σ∊.

Equation 4 denotes this correction of bias in the sample statistic (with the resulting effect size

often referred to as Hedge’s g):

When simply computing the pooled standard deviation across two groups, this formula may

be reexpressed in a more common format. This formula is suitable for data analyzed with a

two-way analysis of variance, such as a treatment-control contrast:

The formula may be further reduced to the average of the sample variances when sample

sizes are equal:

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or

in the case of two groups.

Other means of specifying the denominator for Equation 2 are varied. Some formulas use the

average standard deviation across groups. This procedure disregards differences in sample

size in cases of unequal n when one is weighing sample variances and may or may not

correct for sample bias in estimation of the population standard deviation. Further formulas

employ the standard deviation of the control or comparison condition (an effect size referred

to as Glass’s Δ). This method is particularly suited when the introduction of treatment or other

experimental manipulation leads to large changes in group variance. Finally, more complex

formulas are appropriate when calculating Cohen’s d from data involving cluster randomized

or nested research designs. The complication partially arises because of the three available

variance statistics from which the pooled standard deviation may be computed: the withincluster variance, the between-cluster variance, or the total variance (combined between- and

within-cluster variance). Researchers must select the variance statistic appropriate for the

inferences they wish to draw.

Expansion beyond Two-Group Comparisons: Contrasts and Repeated Measures

Cohen’s d always reflects the standardized difference between two means. The means,

however, are not restricted to comparisons of two independent groups. Cohen’s d may also be

calculated in multigroup designs when a specific contrast is of interest. For example, the

average effect across two alternative treatments may be compared with a control. The value of

the contrast becomes the numerator as specified in Equation 2, and the pooled standard

deviation is expanded to include all j groups specified in the contrast (Equation 4).

A similar extension of Equations 2 and 4 may be applied to repeated measures analyses. The

difference between two repeated measures is divided by the pooled standard deviation across

the j repeated measures. The same formula may also be applied to simple contrasts within

repeated measures designs, as well as interaction contrasts in mixed (between- and withinsubjects factors) or split-plot designs. Note, however, that the simple application of the pooled

standard deviation formula does not take into account the correlation between repeated

measures. Researchers disagree as to whether these correlations ought to contribute to effect

size computation; one method of determining Cohen’s d while accounting for the correlated

nature of repeated measures involves computing d from a paired t test.

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Additional Means of Calculation

Beyond the formulas presented above, Cohen’s d may be derived from other statistics,

including the Pearson family of correlation coefficients (r), t tests, and F tests. Derivations

from r are particularly useful, allowing for translation among various effect size indices.

Derivations from other statistics are often necessary when raw data to compute Cohen’s d are

unavailable, such as when conducting a meta-analysis of published data. When d is derived

as in Equation 3, the following formulas apply:

and

Note that Equation 10 applies only for F tests with 1 degree of freedom (df) in the numerator;

further formulas apply when df> 1.

When d is derived as in Equation 4, the following formulas ought to be used:

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Again, Equation 13 applies only to instances in which the numerator df= 1.

These formulas must be corrected for the correlation (r) between dependent variables in

repeated measures designs. For example, Equation 12 is corrected as follows:

Finally, conversions between effect sizes computed with Equations 3 and 4 may be easily

accomplished:

and

Variance and Confidence Intervals

The estimated variance of Cohen’s d depends on how the statistic was originally computed.

When sample bias in the estimation of the population pooled standard deviation remains

uncorrected (Equation 3), the variance is computed in the following manner:

A simplified formula is employed when sample bias is corrected as in Equation 4:

Once calculated, the effect size variance may be used to compute a confidence interval (CI)

for the statistic to determine statistical significance:

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The z in the formula corresponds to the z-score value on the normal distribution

corresponding to the desired probability level (e.g., 1.96 for a 95% CI). Variances and CIs may

also be obtained through bootstrapping methods.

Interpretation

Cohen’s d, as a measure of effect size, describes the overlap in the distributions of the

compared samples on the dependent variable of interest. If the two distributions overlap

completely, one would expect no mean difference between them

. To the extent that the distributions do not overlap, the difference ought to be greater than

zero (assuming

).

Cohen’s d may be interpreted in terms of both statistical significance and magnitude, with the

latter the more common interpretation. Effect sizes are statistically significant when the

computed CI does not contain zero. This implies less than perfect overlap between the

distributions of the two groups compared. Moreover, the significance testing implies that this

difference from zero is reliable, or not due to chance (excepting Type I errors). While

significance testing of effect sizes is often undertaken, however, interpretation based solely on

statistical significance is not recommended. Statistical significance is reliant not only on the

size of the effect but also on the size of the sample. Thus, even large effects may be deemed

unreliable when insufficient sample sizes are utilized.

Interpretation of Cohen’s d based on the magnitude is more common than interpretation

based on statistical significance of the result. The magnitude of Cohen’s d indicates the extent

of nonoverlap between two distributions, or the disparity of the mean difference from zero.

Larger numeric values of Cohen’s d indicate larger effects or greater differences between the

two means. Values may be positive or negative, although the sign merely indicates whether

the first or second mean in the numerator was of greater magnitude (see Equation 2).

Typically, researchers choose to subtract the smaller mean from the larger, resulting in a

positive effect size. As a standardized measure of effect, the numeric value of Cohen’s d is

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interpreted in standard deviation units. Thus, an effect size of d =0.5 indicates that two group

means are separated by one-half standard deviation or that one group shows a one-half

standard deviation advantage over the other.

The magnitude of effect sizes is often described nominally as well as numerically. Jacob

Cohen defined effects as small (d=0.2), medium (d= 0.5), or large (d=0.8). These rules of

thumb were derived after surveying the behavioral sciences literature, which included studies

in various disciplines involving diverse populations, interventions or content under study, and

research designs. Cohen, in proposing these benchmarks in a 1988 text, explicitly noted that

they are arbitrary and thus ought not be viewed as absolute. However, as occurred with use of

.05 as an absolute criterion for establishing statistical significance, Cohen’s benchmarks are

oftentimes interpreted as absolutes, and as a result, they have been criticized in recent years

as outdated, atheoretical, and inherently nonmeaningful. These criticisms are especially

prevalent in applied fields in which medium-to-large effects prove difficult to obtain and

smaller effects are often of great importance. The small effect of d=0.07, for instance, was

sufficient for physicians to begin recommending aspirin as an effective method of preventing

heart attacks. Similar small effects are often celebrated in intervention and educational

research, in which effect sizes of d= 0.3 to d= 0.4 are the norm. In these fields, the practical

importance of reliable effects is often weighed more heavily than simple magnitude, as may

be the case when adoption of a relatively simple educational approach (e.g., discussing vs.

not discussing novel vocabulary words when reading storybooks to children) results in effect

sizes of d= 0.25 (consistent with increases of one-fourth of a standard deviation unit on a

standardized measure of vocabulary knowledge).

Critics of Cohen’s benchmarks assert that such practical or substantive significance is an

important consideration beyond the magnitude and statistical significance of effects.

Interpretation of effect sizes requires an understanding of the context in which the effects are

derived, including the particular manipulation, population, and dependent measure(s) under

study. Various alternatives to Cohen’s rules of thumb have been proposed. These include

comparisons with effects sizes based on (a) normative data concerning the typical growth,

change, or differences between groups prior to experimental manipulation; (b) those obtained

in similar studies and available in the previous literature; (c) the gain necessary to attain an a

priori criterion; and (d) cost–benefit analyses.

Cohen’s d in Meta-Analyses

Cohen’s d, as a measure of effect size, is often used in individual studies to report and

interpret the magnitude of between-group differences. It is also a common tool used in metaanalyses to aggregate effects across different studies, particularly in meta-analyses involving

study of between-group differences, such as treatment studies. A meta-analysis is a statistical

synthesis of results from independent research studies (selected for inclusion based on a set

of predefined commonalities), and the unit of analysis in the meta-analysis is the data used

for the independent hypothesis test, including sample means and standard deviations,

extracted from each of the independent studies. The statistical analyses used in the metaanalysis typically involve (a) calculating the Cohen’s d effect size (standardized mean

difference) on data available within each independent study on the target variable(s) of

interest and (b) combining these individual summary values to create pooled estimates by

means of any one of a variety of approaches (e.g., Rebecca DerSimonian and Nan Laird’s

random effects model, which takes into account variations among studies on certain

parameters). Therefore, the methods of the meta-analysis may rely on use of Cohen’s d as a

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way to extract and combine data from individual studies. In such meta-analyses, the reporting

of results involves providing average d values (and CIs) as aggregated across studies.

In meta-analyses of treatment outcomes in the social and behavioral sciences, for instance,

effect estimates may compare outcomes attributable to a given treatment (Treatment X) as

extracted from and pooled across multiple studies in relation to an alternative treatment

(Treatment Y) for Outcome Z using Cohen’s d (e.g., d =0.21, CI = 0.06, 1.03). It is important to

note that the meaningful-ness of this result, in that Treatment X is, on average, associated

with an improvement of about one-fifth of a standard deviation unit for Outcome Z relative to

Treatment Y, must be interpreted in reference to many factors to determine the actual

significance of this outcome. Researchers must, at the least, consider whether the one-fifth of

a standard deviation unit improvement in the outcome attributable to Treatment X has any

practical significance.

effect size

standard deviations

statistical significance

repeated measures

equations

significance testing

sampling bias

Shayne B.Piasta, and Laura M.Justice

http://dx.doi.org/10.4135/9781412961288.n58

See also

Analysis of Variance (ANOVA)

Effect Size, Measures of

Mean Comparisons

Meta-Analysis

Statistical Power Analysis for the Behavioral Sciences

Further Readings

Cohen, J. (1988). Statistical power analysis for the behavioral sciences (

2nd ed.

). Mahwah, NJ: Lawrence Erlbaum.

Cooper, H., & Hedges, L. V. (1994). The handbook of research synthesis. New York: Russell

Sage Foundation.

Hedges, L. V.Effect sizes in cluster-randomized designs. Journal of Educational & Behavioral

Statistics32 (2007). 341–370. http://dx.doi.org/10.3102/1076998606298043

Hill, C. J., Bloom, H. S., Black, A. R., & Lipsey, M. W. (2007, July). Empirical benchmarks for

interpreting effect sizes in research. New York: MDRC.

Ray, J. W., and Shadish, W. R.How interchangeable are different estimators of effect size.

Journal of Consulting & Clinical

Psychology64

(1996).

1316–1325.

http://dx.doi.org/10.1037/0022-006X.64.6.1316

Wilkinson, L.APA Task Force on Statistical Inference. Statistical methods in psychology

journals: Guidelines and explanations. American Psychologist54 (1999). 594–604.

http://dx.doi.org/10.1037/0003-066X.54.8.594

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1. Calculate the sample size needed given these factors:

2. Compromise function

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