Abstract

This article provide a brief background about power and sample size analysis. Then, power and sample size analysis is computed for the Z test.
Next articles will describe power and sample size analysis for:

• one sample and two samples t test;,
• p test, chi-square test, correlation;
• one-way ANOVA;
• DOE $2^k$.

Finally, a PDF article showing both the underlying methodology and the R code here provided, will be published.

Background

Power and sample size analysis are important tools for assessing the ability of a statistical test to detect when a null hypothesis is false, and for deciding what sample size is required for having a reasonable chance to reject a false null hypothesis.

The following four quantities have an intimate relationship:

1. sample size
2. effect size
3. significance level = P(Type I error) = probability of finding an effect that is not there
4. power = 1 - P(Type II error) = probability of finding an effect that is there

Given any three, we can determine the fourth.

Z test

The formula for the power computation can be implemented in R, using a function like the following:

In the same way, the function to compute the sample size can be built.

The above code is provided for didactic purpose. In fact, the pwr package provide a function to perform power and sample size analysis.

The function pwr.norm.test() computes parameters for the Z test. It accepts the four parameters see above, one of them passed as NULL. The parameter passed as NULL is determined from the others.

Some examples

Power at $\mu = 105$ for $H0: \mu = 100$ against $H1: \mu>100$.
$\sigma = 15$, $n = 20$, \alpha = 0.05\$

This is the result with the self-made function:

And here the same with the pwr.norm.test() function:

The sample size of the test for power equal to 0.80 can be computed using the self-made function

or with the pwr.norm.test() function:

The power function can be drawn: