calculate
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| Subject | Mathematics |
| Due By (Pacific Time) | 02/12/2015 10:00 am |
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Calculate the sample size needed given these factors:
- one-tailed t-test with two independent groups of equal size
- small effect size (see Piasta, S.B., & Justice, L.M., 2010)
- alpha =.05
- beta = .2
- Assume 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.
2. Calculate the sample size needed given these factors:
- ANOVA (fixed effects, omnibus, one-way)
- small effect size
- alpha =.05
- beta = .2
- 3 groups
- Assume 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 need. Give reasons for any parameters you need to specify for G*Power
Project #57112 – Calculate Project has been cancelled Solutions No Solutions were uploaded! Contact Scholar Details
| Subject | Mathematics |
| Due By (Pacific Time) | 02/12/2015 10:00 am |
| Upgrades | No Upgrades Selected Upgrade Now! |
Calculate the sample size needed given these factors:
- one-tailed t-test with two independent groups of equal size
- small effect size (see Piasta, S.B., & Justice, L.M., 2010)
- alpha =.05
- beta = .2
- Assume 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.
2. Calculate the sample size needed given these factors:
- ANOVA (fixed effects, omnibus, one-way)
- small effect size
- alpha =.05
- beta = .2
- 3 groups
- Assume 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 need. Give reasons for any parameters you need to specify for G*Power
Encyclopedia of Research Design
Cohen’s d Statistic
Contributors: Shayne B. Piasta & Laura M. Justice
Editors: Neil J. Salkind
Book Title: Encyclopedia of Research Design
Chapter Title: “Cohen’s d Statistic”
Pub. Date: 2010
Access Date: February 10, 2015
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: 181-186
©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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Statistic
http://dx.doi.org/10.4135/9781412961288.n58
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 [p. 181 ↓ ] 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.
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Statistic
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, μ
1
and μ
2
represent the two population means, and #
#
represents the pooled within-group population standard deviation. In practice, these
population parameters are typically unknown and estimated by means of sample
statistics:
The population means are replaced with sample means (
,
j
) and the population standard deviation is replaced with S
p
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Statistic
, the pooled standard deviation from the sample. The pooled standard deviation is
derived by weighing the variance around 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 n
j
represents the sample sizes for j groups and s2
j
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, #
#
. 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:
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Statistic
The formula may be further reduced to the average of the sample variances when
sample sizes are equal:
[p. 182 ↓ ]
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 within-cluster 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.
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Statistic
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 within-subjects 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.
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:
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Statistic
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:
[p. 183 ↓ ]
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:
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Statistic
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:
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
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Statistic
. 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 [p. 184 ↓ ] effect size. As a standardized measure
of effect, the numeric value of Cohen’s d is 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
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Statistic
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 meta-analyses 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
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Statistic
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 meta-analysis 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 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 [p. 185 ↓ ]
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.
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
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Statistic
• 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 Statistics 32 (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 Psychology 64 (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 Psychologist 54 (1999).
594–604. http://dx.doi.org/10.1037/0003-066X.54.8.594
