The POWER Procedure

Overview of Power Concepts

In statistical hypothesis testing, you typically express the belief that some effect exists in a population by specifying an alternative hypothesis $H_1$. You state a null hypothesis $H_0$ as the assertion that the effect does not exist and attempt to gather evidence to reject $H_0$ in favor of $H_1$. Evidence is gathered in the form of sample data, and a statistical test is used to assess $H_0$. If $H_0$ is rejected but there really is no effect, this is called a Type I error. The probability of a Type I error is usually designated alpha or $\alpha $, and statistical tests are designed to ensure that $\alpha $ is suitably small (for example, less than 0.05).

If there really is an effect in the population but $H_0$ is not rejected in the statistical test, then a Type II error has been made. The probability of a Type II error is usually designated beta or $\beta $. The probability $1-\beta $ of avoiding a Type II error—that is, correctly rejecting $H_0$ and achieving statistical significance—is called the power. ( Note: Another more general definition of power is the probability of rejecting $H_0$ for any given set of circumstances, even those corresponding to $H_0$ being true. The POWER procedure uses this more general definition.)

An important goal in study planning is to ensure an acceptably high level of power. Sample size plays a prominent role in power computations because the focus is often on determining a sufficient sample size to achieve a certain power, or assessing the power for a range of different sample sizes.

Some of the analyses in the POWER procedure focus on precision rather than power. An analysis of confidence interval precision is analogous to a traditional power analysis, with CI Half-Width taking the place of effect size and Prob(Width) taking the place of power. The CI Half-Width is the margin of error associated with the confidence interval, the distance between the point estimate and an endpoint. The Prob(Width) is the probability of obtaining a confidence interval with at most a target half-width.