Introduction to Power and Sample Size Analysis

Equivalence and Noninferiority

Whereas the standard two-sided hypothesis test for a parameter $\mu$ (such as a mean difference) aims to demonstrate that it is significantly different than a null value $\mu _0$ :

$\begin{align*} H_{0}\colon & \mu = \mu _0\\ H_{1}\colon & \mu \ne \mu _0 \end{align*}$

an equivalence test instead aims to demonstrate that it is significantly similar to some value, expressed in terms of a range $\theta _ L, \theta _ U$ around that value:

$\begin{align*} H_{0}\colon & \mu < \theta _ L \quad \mbox{or}\quad \mu > \theta _ U\\ H_{1}\colon & \theta _ L \le \mu \le \theta _ U \end{align*}$

Whereas the standard one-sided hypothesis test for $\mu$ (say, the upper one-sided test) aims to demonstrate that it is significantly greater than $\mu _0$ :

$\begin{align*} H_{0}\colon & \mu \le \mu _0\\ H_{1}\colon & \mu > \mu _0 \end{align*}$

a corresponding noninferiority test aims to demonstrate that it is not significantly less than $\mu _0$ , expressed in terms of a margin $\delta > 0$ :

$\begin{align*} H_{0}\colon & \mu \le \mu _0 - \delta \\ H_{1}\colon & \mu > \mu _0 - \delta \end{align*}$

Corresponding forms of these hypotheses with the inequalities reversed apply to lower one-sided noninferiority tests (sometimes called nonsuperiority tests).

The POWER procedure performs power analyses for equivalence tests for one-sample, paired, and two-sample tests of normal and lognormal mean differences and ratios. It also supports noninferiority tests for a variety of analyses of means, proportions, and correlation, both directly (with a MARGIN= option representing $\delta$ ) and indirectly (with an option for a custom null value representing the sum or difference of $\mu _0$ and $\delta$ ).