Wednesday, May 28, 2014

2-Factor ANOVA w/Rep Effect Size in Excel 2010 and Excel 2013

This is one of the following seven articles on Two-Factor ANOVA With Replication in Excel

Two-Factor ANOVA With Replication in 5 Steps in Excel 2010 and Excel 2013

Variance Tests: Levene’s and Brown-Forsythe For 2-Factor ANOVA in Excel 2010 and Excel 2013

Shapiro-Wilk Normality Test in Excel For 2-Factor ANOVA With Replication

2-Factor ANOVA With Replication Effect Size in Excel 2010 and Excel 2013

Excel Post Hoc Tukey’s HSD Test For 2-Factor ANOVA With Replication

2-Factor ANOVA With Replication – Test Power With G-Power Utility

Scheirer-Ray-Hare Test Alternative For 2-Factor ANOVA With Replication


Two-Factor ANOVA With

Replication Effect Size In


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 of two-factor ANOVA is the following:

Eta Square (η2)

(Greek letter “eta” rhymes with “beta”) 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 = SSBetween_Groups / SSTotal These two terms are part of the ANOVA calculations found in the Single-factor ANOVA output.

Jacob Cohen in his landmark 1998 book Statistical Analysis for the Behavior Sciences proposed that effect sizes could be generalized as follows:

η2 = 0.01 for a small effect. A small effect is one that not easily observable.

η2 = 0.05 for a medium effect. A medium effect is more easily detected than a small effect but less easily detected than a large effect.

η2 = 0.14 for a small effect. A large effect is one that is readily detected with the current measuring equipment.

Eta squared is sometimes called the nonlinear correlation coefficient because it provides a measure of strength of the curvilinear relationship between the dependent and independent variables. If the relationship is linear, eta squared will have the same value as r squared.

Calculating Eta Square (η2)

in Excel

Eta squared is calculated with the formula

η2 = SSBetween_Groups / SSTotal

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

anova, effect size, statistics, excel, excel 2010, excel 2013, cohen's d,cohens d, eta square, eta (Click On Image To See a Larger Version)

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.

η2Factor_1 = 0.198 which is considered to be a large effect.

η2Factor_2 = 0.111 which is considered to be a medium effect.

η2Interaction = 0.260 which is considered to be a large effect.

η2Error = 0.431 which is considered to be a very large effect.

Such large eta-square term for the error component of the variation indicates perhaps another independent variable that has not been included in the test accounts for a substantial part of total variation of the data.

A large eta-square error term also indicates the possibility of inaccuracy during data collection and recording.


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