Power

Given testing procedure, and estimate of variability, it is possible to calculate the probability of detecting effects of certain sizes, called power study.

Power measures ability to detect deviations from null hypothesis = 1 – P(type II error) = P(reject $H_0$ when $H_0$ is false), is a measure of sensitivity. Generally fix specificity ($\alpha$ level, type I error) and then estimate sensitivity.

Decision to reject $H_0$ based on size of test statistic, which under $H_0$ follows the t distribution. If our alternative hypothesis is a specific value, then we can try various values and look at the probability of rejecting $H_0$ .

How large does the test statistic have to be? $T = (\bar{Y}_{.1g.} - \bar{Y}_{.2g.}) / (\hat{sigma} / \sqrt{r})$ . We reject $H_0$ if $|T| \gt t_{\frac{\alpha}{2}}(df)$ , where $df = at(r-1)$ . This is equivalent to rejecting $H_0$ if $|\bar{Y}_{.1g.} - \bar{Y}_{.2g.}| \gt t_{\frac{\alpha}{2} \frac{\hat{sigma}}{\sqrt{r}}}$

Under the alternative hypothesis, for a particular value of $\mu_A$ , we want to calculate $P(|\bar{Y}_{.1g.} - \bar{Y}_{.2g.}|) \gt y^*$ , which will be normally distributed with mean $\mu_A$ , this implies that $(\bar{Y}_{.1g.} - \bar{Y}_{.2g.} - \mu_A) / \frac{\hat{sigma}{\sqrt{r}}} ~ t_{at(r-1)}$ .

\[ P(Reject H_0 | \mu_A) = P\left( T \gt t_{\frac{\alpha}{2}} - \frac{\mu_A}{\frac{\hat{sigma}}{\sqrt{r}}} \right) + P\left( T \gt t_{\frac{\alpha}{2}} + \frac{\mu_A}{\frac{\hat{sigma}}{\sqrt{r}}} \right) \]

For microarrays

Use common variance (gene specific would be crazy) Need to take multiple comparison technique into account – Bonferroni is easiest ($\alpha / g$ )