Chapter 6 Effect sizes

Recall that as well as determining whether a difference in mean is statistically significant, it can also be useful to determine the relative size of the effect; that is, the effect size. For \(t\)-tests, we used an effect size called Cohen's \(d\). However, there is more than one type of effect size, and the type we use largely depends on which statistical test we have carried out. For a one-way ANOVA test, we will consider the effect size called "eta squared". "Eta" is a Greek letter: \(\eta\). The eta squared (\(\eta^2\)) value is a measure of the proportion of variation in the response variable (dependent variable: flipper length in our example) that can be attributed to the independent variable (species in our example). One way of thinking about this would be to ask: how much does a penguin's species explain their flipper length? We expect that a penguin's flipper length is of course related to which species the penguin is, but there are other factors, and randomness too, which leads to variability within species. Otherwise, every penguin of the same species would have exactly the same flipper length, and we of course know that this is not true.

For interpretation of eta squared, the following conventions apply (Cohen 1988):

Guidelines for interpreting \(\eta^2\) effect sizes:

  • \(\eta^2 < 0.01\): "negligible"
  • \(0.01 \leq \eta^2 < 0.06\): "small"
  • \(0.06 \leq \eta^2 < 0.14\): "medium"
  • \(\eta^2 \geq 0.14\): "large"

The results of the effect size calculation for our example is as follows:

           eta.sq eta.sq.part
species 0.7782289   0.7782289

The above output provides both the eta squared value (eta.sq), and the "partial eta squared" value (eta.sq.part). For a one-way ANOVA, both values are equivalent. For more complicated types of ANOVAs (where there is more than one independent variable), the "partial eta squared" should be used.

As we can see, the effect size was 0.78, which is considered large.

References

Cohen, J. 1988. Statistical Power Analysis for the Behavioral Sciences. 2nd edition. New York: Academic Press.