Sunday, January 11, 2015

Effect Size For Repeated-Measures ANOVA in Excel

Effect Size For Repeated-

Measure ANOVA in Excel

This is one of the following four articles on Repeated-Measures ANOVA

Single-Factor Repeated-Measures ANOVA in 4 Steps in Excel

Sphericity Testing For Repeated-Measures ANOVA in 9 Steps in Excel

Effect Size For Repeated-Measures ANOVA in Excel

Friedman Testing For Repeated-Measures ANOVA in 3 Steps in Excel

Overview of Effect Size

Effect size is a way of describing how effectively the method of data grouping allows those groups to be differentiated. A simple example of a grouping method that would create easily differentiated groups versus one that does not is the following.

Imagine a large random sample of height measurements of adults of the same age from a single country. If those heights were grouped according to gender, the groups would be easy to differentiate because the mean male height would be significantly different than the mean female height. If those heights were instead grouped according to the region where each person lived, the groups would be much harder to differentiate because there would not be significant difference between the means and variances of heights from different regions.

Because the various measures of effect size indicate how effectively the grouping method makes the groups easy to differentiate from each other, the magnitude of effect size tells how large of a sample must be taken to achieve statistical significance. A small effect can become significant if a larger enough sample is taken. A large effect might not achieve statistical significance if the sample size is too small.

The Most Common Measure of Effect Size

The most common measure of effect size of single-factor ANOVA is the following:

η2 – eta squared

(Greek letter “eta” rhymes with “beta”)

Eta Square (η2)

Eta square quantifies the percentage of variance in the dependent variable (the variable that is measured and placed into groups) that is explained by the independent variable (the method of grouping). If eta squared = 0.35, then 35 percent of the variance associated with the dependent variable is attributed to the independent variable (the method of grouping).

Eta square provides an overestimate (a positively-biased estimate) of the explained variance of the population from which the sample was drawn because eta squared estimates only the effect size on the sample. The effect size on the sample will be larger than the effect size on the population. This bias grows smaller is the sample size grows larger.

Eta square is affected by the number and size of the other effects.

η2 = SS Between_Groups / SSTotal These two terms are part of the ANOVA calculations found in the Single-factor ANOVA output.

Magnitudes of eta-squared are generally classified exactly as magnitudes of r2 (the coefficient of determination) are as follows: = 0.01 is considered a small effect. = 0.06 is considered a medium effect. = 0.14 is considered a large effect. Small, medium, and large are relative terms. A large effect is easily discernible but a small effect is not.

Calculating Eta Squared (η2) in Excel

Eta squared is calculated with the formula

η2 = SSBetween_Groups / SSTotal

and is implemented in Excel on the data set as follows:

Repeated-Measures ANOVA - Effect Size Generic Example for Single-Factor ANOVA
(Click On Image To See a Larger Version)

An eta-squared value of 0.104 would be classified as a medium-size effect.

Magnitudes of eta-squared are generally classified exactly as magnitudes of r2 (the coefficient of determination) are as follows: = 0.01 is considered a small effect. = 0.06 is considered a medium effect. = 0.14 is considered a large effect. Small, medium, and large are relative terms. A large effect is easily discernible but a small effect is not.

The raw data matrix for the repeated-measures ANOVA example used in these blog articles is the following:

Repeated-Measures ANOVA - Raw Data Matrix
(Click On Image To See a Larger Version)

If single-factor ANOVA were performed on this data and eta-squared was calculated from this data, the result would be the following:

Repeated-Measures ANOVA - Effect Size Performed on Single-factor ANOVA of That Data
(Click On Image To See a Larger Version)

After conducting a repeated-measure ANOVA test and then applying a correction for lack of sphericity, eta-squared is calculated in the same fashion as follows:

Repeated-Measures ANOVA - Effect Size Calculation
(Click On Image To See a Larger Version)

The variance attributed to the subjects, SSsubjects, has been entirely removed from the Repeated-Measure ANOVA test and should not added back into the total Variance, SStotal, to calculate eta square.

This represents a very large effect size. This means that the difference between the group means is very noticeable if the variation attributed to the subjects is removed and the only remaining variance is between groups and statistical noise (SSerror).

 

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