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Example 9-1
Section* *

Suppose you are interested in the question: "Does one group have a prevalence percentage that is different than other groups?" For example:

Baseline prevalence of smoking in a particular community is 30%. A clean indoor air policy goes into effect. What is the sample size required to detect a decrease in smoking prevalence of at least 2 percentage points? \(\alpha=0.05\); 90% power.

We are interested in testing the following hypothesis:

Null hypothesis:

\(H_0\colon \text{prevalence}_{(Before)}\le \text{prevalence}_{(After)}\)

Alternative hypothesis:

\(H_A\colon \text{prevalence}_{(Before)}- \text{prevalence}_{(After)}=\delta\)

Where \(\delta \gt 0\)

The resulting formula for the sample size for testing a difference in prevalence using a one-sided test is as follows:

and for this example, *n *can be calculated as:

\(n=\dfrac{1}{d^{2}}\left [ z_{\alpha }\sqrt{\pi_{0}(1-\pi_{0})}+z_{\beta }\sqrt{\pi_{1}(1-\pi_{1})} \right ]^{2}\)

Replace \(z_{\alpha }\) by \(z_{\alpha/2 }\) for a two-sided test

Take a moment to look at the table below for sample size requirements for testing the value of a single proportion with a one-sided test. Prevalence can be found along the top of the table and the percentage point difference vertically on the left. How many individuals do we need to include in our study in order to meet the above criteria?

(Tables from Woodward, M. Epidemiology Study Design and Analysis. Boca Raton: Chapman and Hall:, 1999 )

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Try It!
Section* *

- Prevalence increases (\(B_0\))? Does the sample size increase or decrease?
- What happens to the sample size as effect size decreases?
- What is the minimal detectable difference if you had funds for 1,500 subjects?

- The largest sample sizes occur with baseline prevalence at 0.5
- The smaller the effect size, the larger the sample size
- About 3.6% decrease in prevalence