Question

The effect that a certain drug (Drug A) has on increasing blood pressure is a major concern. It is thought that a modification of the drug (Drug B) will lessen the increase in blood pressure. Let \(\mu_{A}\) and \(\mu_{B}\) be the true mean increases in blood pressure due to Drug \(\mathrm{A}\) and \(\mathrm{B}\), respectively. The hypotheses of interest are \(H_{0}: \mu_{A}=\mu_{B}=0\) versus \(H_{1}: \mu_{A}>\mu_{B}=0 .\) The two-sample \(t\) -test statistic discussed in Example \(8.3 .3\) is to be used to conduct the analysis. The nominal level is set at \(\alpha=0.05\) For the experimental design assume that the sample sizes are the same; i.e., \(m=n .\) Also, based on data from Drug \(A, \sigma=30\) seems to be a reasonable selection for the common standard deviation. Determine the common sample size, so that the difference in means \(\mu_{A}-\mu_{B}=12\) has an \(80 \%\) detection rate. Suppose when the experiment is over, due to patients dropping out, the sample sizes for Drugs \(A\) and \(B\) are respectively \(n=72\) and \(m=68 .\) What was the actual power of the experiment to detect the difference of \(12 ?\)

Short answer

The common sample size needed is approximately 70 in order to have an 80% detection rate for a difference of 12. The actual power of the experiment turned out to be approximately 78% due to a decrease in sample size.

Step-by-step solution

Derive the formula for sample size

For a two-sample t-test, the common sample size in order to have a power of 1-β to detect a difference Δ in means is given by the following formula: \(n = \left (\frac{(Z_{α/2} + Z_{β}) × σ}{Δ}\right )^2 \). Here, Z-values correspond to the standard normal distribution. Z\(α/2\) = 1.96 for a confidence level of 95%, Z\(β\) = 0.84 for a power of 80%, σ is the common standard deviation and Δ is the desired difference to detect.

Calculate the common sample size

Substitute the values from our problem into the formula: σ=30 and Δ=12. \(n = \left (\frac{(1.96 + 0.84) × 30}{12}\right )^2 \approx 70 \). So, a sample size of 70 would give an 80% detection rate.

Compute actual power of the experiment

The actual power is given by the formula: \( β = Z_{β} - Z_{α/2} × \sqrt{\frac{σ^2(1/m + 1/n)}{Δ}} \) where m=68 and n=72. Calculating these we find the actual power as \( β = 0.84 - 1.96 × \sqrt{\frac{30^2(1/68 + 1/72)}{12}} \approx 0.78 \), indicating a 78% power.