One use of effect-size is as a standardized index that is independent of
sample size and quantifies the magnitude of the difference between populations or the relationship
between explanatory and response variables. Another use of effect size is its use in performing
power analysis.

Effect size formulas

Effect size for difference in means

Effect size for differences in means is given by
Cohen’s d is defined in terms of population means (μs) and a population standard
deviation (σ), as shown below.

There are several different ways that one could estimate σ from sample data which
leads to multiple variants within the Cohen’s d family.

Using the root mean square standard deviation

Using the pooled standard deviation (Hedges’ g)

This version of Cohen’s d uses the pooled standard deviation and is also known as Hedges’ g.

You can easily obtain this value from an anova program by taking the square root of the mean
square error which is also known as the root mean square error.

Using the control group standard deviation (Glass’ Δ)

Another version of Cohen’s d using the standard deviation for the control group is also known as
Glass’ Δ.

More then two groups

When there are more then two groups use the difference between the largest and smallest
means divided by the square root of the mean square error.

Effect size for F-ratios in regression analysis

For OLS regression the measure of effects size is F which is defined by Cohen as follows.

Once again there are several ways in which the effect size can be computed from sample data. Note
that η² is another name for R².

Effect size for F-ratios in analysis of variance

The effect size used in analysis of variance is defined by the ratio of population standard
deviations.

Although Cohen’s f is defined as above it is usually computed
by taking the square root of f².

Effect size for χ² from contingency tables

Once again we start off with the definitional formula in terms of population values.
Effect size w is the square root of the standardized chi-square statistic.

And here is how w is computed using sample data.

Here is a table of suggested values for low, medium and high effects (Cohen, 1988). These values should not
be taken as absolutes and should interpreted within the context of your research program. The values for
large effects are frequently exceeded in practice with values Cohen’s d greater than 1.0 not uncommon.
However, using very large effect sizes in prospective power analysis is probably not a good idea as
it could lead to under powered studies.

                           small   medum   large
t-test for means      d     .20     .50     .80
t-test for corr       r     .10     .30     .50
F-test for regress    f2    .02     .15     .35
F-test for anova      f     .10     .25     .40
chi-square            w     .10     .30     .50

Comparable effect sizes when there are two groups

        d          r         r2          f         f2
        2       .707    .499849   .9996981   .9993963
      1.9       .689    .474721   .9506578   .9037502
      1.8       .669    .447561   .9000859   .8101547
      1.7       .648    .419904   .8507953   .7238526
      1.6       .625    .390625   .8006408   .6410257
      1.5         .6        .36   .7500001   .5625001
      1.4       .573    .328329   .6991596   .4888242
      1.3       .545    .297025   .6500198   .4225257
      1.2       .514    .264196   .5992141   .3590576
      1.1       .482    .232324   .5501208   .3026329
        1       .447    .199809   .4997016   .2497016
       .9        .41      .1681   .4495192   .2020676
       .8       .371    .137641    .399512   .1596099
       .7        .33      .1089   .3495834   .1222085
       .6       .287    .082369   .2996042   .0897627
       .5       .243    .059049   .2505087   .0627546
       .4       .196    .038416   .1998768   .0399507
       .3       .148    .021904    .149648   .0223945
       .2         .1        .01   .1005038    .010101
       .1        .05      .0025   .0500626   .0025063
        0          0          0          0          0

Some conversion formulas

Convert d to f²

where k = number of groups

Convert t to d for two independent groups

Convert r to d for two independent groups

Noncentrality estimates

Power analysis using analytic methods require an estimate of noncentrality which is basically
the effect size multiplied by a sample size factor. Here are some formulas for estimating
noncentrality.

Example power analysis

Here is an example that brings together effect size and noncentrality in a power analysis.

Consider a one-way analysis of variance with three groups (k = 3). If we expect and
eta² to equal .12 in which case the effect size will be

effect size f = sqrt(eta2/(1-eta2)) = sqrt(.12/(1-.12)) = .369

With a projected sample size of 60 the estimate of noncentrality is

noncentrality coefficient lambda = N*f = 60*.369^2 = 60*.136 = 8.17

The numerator degrees of freedom is k-1 = 3-1 = 2 while the denominator df is N-k = 60-3 = 57.
The critical value of F with 2 and 57 degrees of freedom is 3.16. Which results in a power of

power = noncentralFtail(df1,df2,lambda,Fcrit(2,57)) = noncentralFtail(2,57,8.17,3.16) = .703

thus, an N of 60 and effect size of .369 yields a projected power of about .7.

We can improve on the power of .7 by using a projected sample size of 75 instead of 60. With the same effect size of
.369, we get a new noncentrality estimate of

noncentrality coefficient lambda = N*f = 75*.369^2 = 75*.136 = 10.2

The numerator degrees of freedom remain the same while the denominator df now equal N-k = 75-3 = 72.
The critical value of F with 2 and 72 degrees of freedom of 3.12. This time the power is

power = noncentralFtail(df1,df2,lambda,Fcrit(2,72)) = noncentralFtail(2,72,10.2,3.12) = .807

which is within acceptable research limits.

Please note that different stat packages use different names and a different order of
arguments in
the function that we have call noncentralFtail. You will need to read the documentation that
comes with your software.

References

Buchner,A., Erdfelder,E. and Faul,F. How to use G*power. Viewed 19 Jun 2009 <http://www.psycho.uni-duesseldorf.de/aap/projects/gpower/how_to_use_gpower.html>

Becker, L. Psychology 590 course notes. Viewed 19 Jun 2009 <http://web.uccs.edu/lbecker/Psy590/es.htm>

Cohen, J. 1988. Statistical power analysis for the behavioral sciences. Hillsdale, New Jersey:
Lawrence Erlbaum Associates