Click for printer friendely version of this HowTo What Statistical Power Means Power is a term that is used quite frequently to describe statistical tests. As is often the case, the word has a rather specific definition which we will attempt to describe here. Due to their close relation to the definition of power, we will also briefly describe the various types of errors that statistical tests can make. Thus, $\alpha =$ the probability you will reject $H_0$ when it is true. This type of error is called Type I Error. $\beta =$ the probability you will accept $H_0$ when it false. This type of error is called Type II Error. Power $= 1 - \beta$, the probability the test will reject $H_0$ when it is false. Thus, the more power, the higher probability of correctly rejecting $H_0$. You can increase power by increasing the sample size, $n$, for the test. This is because the larger sample size will decrease the variance of the estimated parameters. For example, consider $\bar{X}$ as an estimate of $\mu$. By the central limit theorem, the variance of $\bar{X}$, where E$X = \mu$ and Var $(X) = \sigma^2$ for independent and identically distributed samples from any distribution, is approximately $\sigma^2/n$, which gets smaller as $n$ gets larger. An example of this is shown in Figures 3.3.1 and 3.3.2. Figure: The predicted distribution of $\bar{X}$ given by the null hypothesis, $H_0: \mu \le 0$, is depicted in the top graph. The bottom graph shows the true distribution of $\bar{X}$, since $\mu = 1$. With the current sample size, the variation in $\bar{X}$, our estimator for $\mu$, is great enough to make it more than likely that we well fail to reject $H_0$ even though it is false. Figure: The predicted distribution of $\bar{X}$ given by the hull hypothesis, $H_0: \mu \le 0$, is shown in the top graph. The bottom graph depicts the true distribution of $\bar{X}$, since $\mu = 1$. However, compared with Figure 3.3.1, the sample size has been increased enough to reduce the variation in the parameter estimate by one half. This makes it more likely that our test will reject $H_0$, and thus, the test has more power. Click for printer friendely version of this HowTo

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