Suppose we want to design a new placebo-controlled trial to evaluate an experimental medication to increase lung capacity. The primary outcome is peak expiratory flow rate, a continuous variable measured in liters per minute. the primary outcome will be measured after 6 months on treatment. the expected peak expiratory flow rate in adults is 300 with a standard deviation of 50. How many subjects should be enrolled to ensure 80% power to detect a difference of 15 liters per minute with a two sided test and a=0.05?
To calculate the sample size required for this placebo-controlled trial, we can use a power analysis based on the given information. Here's how you can calculate the appropriate sample size:
Step 1: Determine the effect size
The effect size represents the difference in peak expiratory flow rate that we want to detect. In this case, it is 15 liters per minute.
Step 2: Determine the desired power
Power is the probability of correctly detecting a true difference if it exists. In this case, we want a power of 80%. This corresponds to a power of 0.8.
Step 3: Determine the significance level
The significance level, denoted as "α" or alpha, represents the probability of erroneously rejecting the null hypothesis when it is true. In this case, the significance level is stated as 0.05.
Step 4: Determine the standard deviation
The standard deviation (SD) of peak expiratory flow rate is given as 50 liters per minute.
Step 5: Determine the expected mean
The expected mean peak expiratory flow rate in adults is given as 300 liters per minute.
Step 6: Use a sample size calculator
Based on the above information, you can use a sample size calculator or statistical software to determine the required sample size. One commonly used method is to use a formula based on t-distribution.
The formula for sample size calculation is:
n = 2 * (Zα/2 + Zβ)^2 * σ^2 / (δ^2)
where:
- n represents the required sample size
- Zα/2 is the critical value for the desired significance level (0.05/2 = 0.025), which corresponds to a two-sided test
- Zβ is the critical value for the desired power (0.8)
- σ is the standard deviation
- δ is the effect size
Plugging in the values, we have:
n = 2 * (Z0.025 + Z0.8)^2 * 50^2 / 15^2
Step 7: Calculate the critical values
To calculate the critical values (Zα/2 and Zβ), you can refer to a standard normal distribution table or use statistical software. For a significance level of 0.05, Z0.025 is approximately 1.96. For a power of 0.8, Z0.8 is approximately 0.84.
Plugging in these critical values and the other known values into the equation, we get:
n = 2 * (1.96 + 0.84)^2 * 50^2 / 15^2
n ≈ 2 * (2.8)^2 * 50^2 / 15^2
n ≈ 2 * 7.84 * 50^2 / 15^2
n ≈ 2 * 7.84 * (50^2 / 15^2)
n ≈ 2 * 7.84 * (2500 / 225)
n ≈ 2 * 7.84 * 11.11
n ≈ 173.6344 (rounded up to the nearest whole number)
Therefore, the estimated sample size required to ensure 80% power to detect a difference of 15 liters per minute with a two-sided test and a significance level of 0.05 is approximately 174 subjects.