Analysing a nominal and scale variable
Part 3c: Effect size
The one-way ANOVA informed us that location has an influence on the grade the students gave. To indicate how strong the influence is, it is a good habit to also report a so-called effect size. The most commonly reported effect size that goes with a one-way ANOVA is eta-squared (η2) (Bakeman, 2005; Levine & Hullett, 2002) although omega squared (ω2) might actually be preferred.
An eta square of 0 would mean no differences (and no influence), while 1 would indicate a full dependency. In the example the eta-squared is .263. Unfortunately there is no formal way to determine if 0.263 is high or low, and the rules of thumb floating around on the internet vary quite a lot, often depending on the field (e.g. biology, medicine, business, etc.). One of them is from Cohen (1988):
0.00 < 0.01 - Negligible
0.01 < 0.06 - Small
0.06 < 0.14 - Medium
0.14 < 1.00 - Large
In this example we can therefor speak of a large effect size. We can add this to our report:
The one-way ANOVA showed that Location had significant large effect on how students evaluated the course, F(2, 45) = 8.04, p = .001, η2 = .26. The Levene test showed that the variances across the three locations were significantly different, F(2, 45) = 4.68, p = .014. A Games-Howell post hoc analysis showed that there was a significant difference between Diemen and Haarlem, p = .001, and also between Diemen and Rotterdam, p = .010.
Click here to see how to determine eta squared with SPSS, R (studio), Excel, Online calculator, or Manually
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Manually (formula and example)
The formula for eta-squared is:
In this formula SS is short for 'Sum of Squares', which in turn in short for 'sum of squared deviations from the mean'. The formulas are:
In these formulas k is the number of categories, is the number of scores in category i, the mean of all scores, the mean of the scores in the i-th category, the j-th score in the i-th category, is the i-th score, and n the total sample size.
The formulas for the means are:
The formula for the SS within is:
Note: different example than used in the rest of this section, but the same as the example used in the test section example for the manual calculation.
We are given grades people gave to a brand, and grouped the people in three categories (local, regional, outside). The grades were:
The first category has 4 scores, the second 3, and the third 5. Therefore:
The SS within and between were already calculated in the example in the test section. They were:
Now for the SS total we could either simply add these two up (70.8 + 7.2 = 78), but let's use the formula. We had already determined in the test section:
So we can determine SS total with the formula:
Eta squared can now be determined:
We can now wrap things up and combine all the parts for a full report on this, which will be on the next page.
Nominal vs Scale