Deriving Power Equations

This is a brief summary of deriving statistical power equations based on the text of Hogg, McKean, and Craig (Section 5.5, see references).

Table of Contents

  1. Hypotheses
  2. Statistical Conclusions
  3. Critical Regions
  4. Power
  5. References

1. Hypotheses

Hypothesis testing is based on the idea of evaluating the likelihood of two hypotheses relative to each other. These hypotheses are based on a parameter, $\theta$.

Let us assume the following:

  1. $\theta \in \Omega$
  2. $X$ is a random variable with density function $f(x; \theta)$
  3. $\omega_0 \in \Omega$
  4. $\omega_1 \in \Omega$
  5. $\omega_0 \cup \omega_1 = \Omega$

We define the hypotheses as $$H_0: \theta \in \omega_0 \mbox{ vs. } H_1: \theta_1 \in \omega_1$$

$H_0$ is commonly referred to as the null hypothesis and $H_1$ the alternative hypothesis. In practice, we often assign the null hypothesis a known or established values (such as would be typical of our current belief about the parameter) and the alternative hypothesis is assigned a value we believe to represent a change or difference from the current belief.

2. Statistical Conclusions

Based on a sample of data, we may either Reject $H_0$ or we may Fail to Reject $H_0$, and our decision rules must be designed so that only one of those conditions may occur. Also, based on any given sample, we run the risk of making an incorrect decision.

|-------|-------|-------| | | $H_0$ is TRUE | $H_0$ is FALSE | | Reject $H_0$ | Type I Error | Correct Decision | | Fail to Reject $H_0$ | Correct Decision | Type II Error |

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We wish to design statistical tests in a way that minimizes the possibility of errors. Since we generally view Type I Error to be more severe, we often define the test in a manner that fixes the probability of that error. We label this probability $\alpha$ and also refer to is as the significance level or the test. The Type II Error, denoted $beta$ is often then derived as a function of the distribution function and the sample size.

3. Critical Regions

Decision rules can be written in terms of Critical Regions of a probability distribution. If we let $X$ be a random sample with a sample space $D = space{(X_1, \ldots, X_n)}$, and $\theta$ is a statistic derived from $X$, we choose a critical region $C$ where $C \subset D$ and $$P_\theta[(X_1, \ldots, X_n) \in C] = \alpha$$

4. Power

Statistical Power is generally referred to as $1 - \beta$ or the complement of the probability of a Type II Error. In it's most generic form, it is defined

$$\gamma(\theta) = 1 - P_\theta[\mbox{Type II Error}] = P_\theta[(X_1, \ldots, X_n) \in C]$$

where $\theta \in \omega_1$.

To illustrate the derivation of a power equation, we will derive the power equation for a test of the mean. We will assume that $X$ is a normally distributed random variable with mean $\mu$ and variance $\sigma^2$. Let $X_1$, $\ldots$, $X_n$ be a random sample of size $n$ from $X$. Our sample has mean $\bar X$ and variance $S^2$. To simplify for the purpose of illustration, we will assume a one-sided test where $$H_0: \mu \leq \mu_0 \mbox{ vs. } H_1: \mu > \mu_0$$

For a chosen $\alpha$, we know we will reject $H_0$ when $T = \frac{\bar X - \mu_0}{S/\sqrt{n}} \geq t_{\alpha, n-1}$.

$$\begin{align} \gamma(\mu) &= P_\mu(\bar X \geq \mu_0 + t_{\alpha, n-1} \cdot S / \sqrt{n})\ &=P_\mu(\bar X - \mu \geq \mu_0 + t_{\alpha, n-1} \cdot S/\sqrt{n} - \mu)\ &=P_\mu\Big(\frac{\bar X - \mu}{S/\sqrt{n}} \geq \frac{\mu_0 + t_{\alpha, n-1} \cdot S/\sqrt{n} - \mu}{S/\sqrt{n}}\Big)\ &=P_\mu\Big(T \geq \frac{\mu_0}{S/\sqrt{n}} + \frac{t_{\alpha, n-1} \cdot S/\sqrt{n}}{S/\sqrt{n}} + \frac{\mu}{S/\sqrt{n}}\Big)\ &=P_\mu\Big(T \geq \frac{\mu_0 - \mu}{S/\sqrt{n}} + t_{\alpha, n-1} \Big)\ &=P_\mu\Big(T \geq t_{\alpha, n-1} + \frac{\mu_0 - \mu}{S/\sqrt{n}}\Big)\ &=P_\mu\Big(T \geq t_{\alpha, n-1} + \frac{\sqrt{n}(\mu_0 - \mu)}{S}\Big) \end{align}$$

Where $t_{\alpha, n-1} + \frac{\sqrt{n}(\mu_0 - \mu)}{S}$ has a non-central T distribution with noncentrality parameter $\frac{\sqrt{n}(\mu_0 - \mu)}{S}$.

5. References

Hogg RV, McKean JW, Craig AT, Introduction to Mathematical Statistics, Pearson Prentice Hall 6th ed., 2005. ISBN: 0-13-008507-3



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