Hypothesis Tests

Sample Size and Power Calculations

Hypothesis Tests

The power for the one-sample t-test, the paired t-test, and the two-sample t-test is computed in the usual fashion. That is, power is the probability of correctly rejecting the null hypothesis when the alternative is true. The sample size is the number per group; these calculations assume equally sized groups. To compute the power of a t-test, you make use of the noncentral t distribution. The formula (O'Brien and Lohr 1984) is given by

${Power} = {Prob}(t \gt t_{crit}, \nu, NC)$

for a one-sided alternative hypothesis and

${Power} = {Prob}(t \gt t_{critu}, \nu, NC)+ {Prob}(t\lt t_{critl}, \nu, NC)$

for a two-sided alternative hypothesis where t is distributed as noncentral $t(NC,\nu)$ .

$t_{crit} = t_{(1-\alpha,\nu)}$ is the ( $1-\alpha$ ) quantile of the t distribution with $\nu$ df
$t_{critu} = t_{(1-\alpha/2,\nu)}$ is the (1- $\alpha/2$ ) quantile of the t distribution with $\nu$ df
$t_{critl} = t_{(\alpha/2,\nu)}$ is the ( $\alpha/2$ ) quantile of the t distribution with $\nu$ df

For one sample and paired samples,

$\nu = n - 1 & {is the df} \ NC = \delta\sqrt{n} & {is the noncentrality parameter}$

For two samples,

$\nu = 2(n - 1) & {is the df} \ NC = \frac{\delta}{\sqrt{2/n}} & {is the noncentrality parameter} \end {array}$

Note that n equals the sample size (number per group).

The other parameters are

$\delta = \{ \frac{|\mu_a-\mu_0|}s & {for one-sample} \ \frac{(\mu_1-\mu_2)}s & {for two-sample and paired samples} .$

$s = \{ {standard deviation for one-sample} \ {standard deviation of the differences for paired samples} \ {pooled standard deviation for two samples} .$

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