# Effect Size

The purpose of the various measures of **effect size** is to provide a statistically valid reflection of the *size* of the *effect* of some feature of an experiment. As such it is a rather loose concept. However there is an underlying assumption that this is taking place in some parametric design, and that the effect of the feature of interest (or *manipulation*) can be measured by some estimable function of the parameters.

This is certainly the case in the paradigmatic model for the evaluation of effect size, namely the **two-conditions, two-groups** design. Suppose that a test $$\mathbf{T}$$ is administered to two groups of sizes $$n_A$$ and $$n_B$$ in two conditions $$A$$ and $$B$$.

The samples are assumed to be independently and normally distributed with the same variance:

- $${a_i|i=1 \ldots n_A} \qquad ~ \qquad \textrm{(i.i.d.)} \qquad N(\mu_A,\sigma^2)$$

and

- $${b_i|i=1 \ldots n_B} \qquad ~ \qquad \textrm{(i.i.d.)} \qquad N(\mu_B,\sigma^2)$$.

Effect Size $$d$$ was defined by Cohen (1988)^{1} as the difference between the two condition means divided by the common standard deviation:

- $$d = \frac{\mu_A - \mu_B}{\sigma}.$$

That is to say it is the **Signal to Noise Ratio**. There are obvious connections with the definition of the classical Signal Detection Theory parameter $$d'$$.

Let $$\bar{a}=\frac{\sum_{i=1}^{{n_A} a_i}{n_A}$$, $$\bar{b}=\frac{\sum_{i=1}}{n_B} b_i}{n_B}$$ and $$SS_A=\sum_{i=1}^{{n_A}(a_i-\bar(a))}2$$, $$SS_B=\sum_{i=1}^{{n_B}(b_i-\bar(b))}2$$. Then $$\hat{\sigma}^{2 = \frac{SS_A + SS_B}{n_A+n_B-2}$$ is the conventional estimator of $$\sigma}2$$, and $$\hat{d}=\frac{\bar{a}-\bar{b}}{\hat{sigma}}$$ serves at an estimator for $$d$$.

It is instructive to compare the estimator $$\hat{d}$$ with the standard $$T$$-test statistic. In fact

- $$\hat{d} = t \sqrt{\frac{1}{n_A} + \frac{1}{n_B}}$$.

The effect size measure $$d$$ deliberately ignores design aspects that relate to sample size, and expresses things in terms of a the variance of a single observation from eith of the two underlying distributions. It is instructive to rewite the above relationship as

$$\hat{d} = t / \sqrt{n_h/2}$$ where $$n_h$$ is the

**harmonic mean**of $$n_A$$ and $$n_B$$.

Cohen, J. (1988). Statistical power analysis for the behavioral sciences (2nd ed.). New York:Academic Press (1)