Effect size

This is “cool. It changed something.. but like.. how much did it move the needle?”

We can compute this two ways:

• Cohen’s d (how much does this move the needle?)
• proportion of variance accounted for (does it have a consistent effect?)

Cohen’s D

This measures the size of the changes in the scores.

Independent tests use $d=\frac{\overline{X_1}-\overline{X_2}}{\sqrt{s_{pool}^2}}$

Related tests use $d=\frac{\bar{D}}{\sqrt{s_D^2}}$

Cohen proposed these guidelines:

d	effect
.2	small
.5	medium
.8	large

Proportion of variance accounted for

This measures how consistently the scores change.

Given this data from a related-sample study:

Before therapy	After
10	5
11	6
12	7

We can see that there is some variance b/c of the independent variable (people b/c 10 != 11).

We describe this via “square point-biserial correlation coefficient”

if $t_{obt}=+2.93;df=30$,

$$\begin{array}{rl}{r}_{pb}^2& =\frac{t_{obt}^2}{t_{obt}^2+df}\\ & =\frac{2.93^2}{2.93^2+30}\\ & =\frac{8.585}{38.585}\\ & =.22\end{array}$$

Guidelines for interpreting:

Range	Interpretation
< .09	Small effect
.10 - .25	Moderate / relatively common
> .25	Large / rare

ANOVA

When calculating effect size for ANOVA, we’re looking for eta squared ${\eta }^2=\frac{S{S}_{bn}}{S{S}_{tot}}$.