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Chapter 8: Problem 102

Suppose a similar study is planned among women who use exogenous hormones. How many participants need to be enrolled if the mean change in \(\log _{10}\) (plasma estradiol) is \(-0.08,\) the standard deviation of change is 0.20 and we want to conduct a two-sided test with an \(\alpha\) level of .05 and a power of .80?

Short Answer

Expert verified
Two participants are needed.

Step by step solution

01

Define the Problem Parameters

We are given that the mean change in log-transformed estradiol levels is \(-0.08\), the standard deviation of the change is \(0.20\), and we want to conduct a two-sided test with a significance level \(\alpha = 0.05\) and a desired power of \(0.80\). Our task is to compute the required sample size.
02

Determine the Z-scores for Alpha and Power

For a two-sided test with \(\alpha = 0.05\), the critical z-value is found by looking for the z-score that corresponds to \(1 - \alpha/2 = 0.975\). This gives us \(z_{\alpha/2} \approx 1.96\). For a power of \(0.80\), we want \(\beta = 0.20\), so \(z_{1-\beta} \approx 0.84\).
03

Calculate the Effect Size

The effect size \(d\) is calculated as the ratio of the mean change to the standard deviation: \[ d = \frac{-0.08}{0.20} = -0.40. \]
04

Use the Sample Size Formula

The sample size \(n\) for a two-sided test is given by the formula: \[ n = \left( \frac{(z_{\alpha/2} + z_{1-\beta}) \times \sigma}{\text{effect size}} \right)^2. \] Substituting the values: \[ n = \left( \frac{(1.96 + 0.84) \times 0.20}{0.40} \right)^2 = \left( \frac{2.8 \times 0.20}{0.40} \right)^2 = \left( \frac{0.56}{0.40} \right)^2 = 1.4^2 = 1.96. \]
05

Round Up to the Nearest Whole Number

Since sample size must be a whole number, we round up 1.96 to the nearest whole number. Therefore, the required sample size is 2.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding a Two-Sided Test in Biostatistics
A two-sided test is a type of statistical analysis used when researchers want to determine if the mean of a sample significantly differs from a hypothetical value, without specifying the direction of the difference. In the context of our exercise, this means that we want to see if there is a change in estradiol levels, but we are open to the possibility that this change could be an increase or a decrease. This is why it's called "two-sided"鈥攚e are considering both tails of the distribution.
  • A two-sided test examines the extremes on both ends of the distribution curve.
  • It's more conservative than a one-sided test, as it requires more evidence to reject the null hypothesis.
  • Most biostatistical tests in clinical studies are two-sided to account for any kind of deviation from the null hypothesis, regardless of direction.
When conducting a two-sided test, researchers calculate a critical z-value that covers both potential shifts in the mean. This approach effectively doubles the level of scrutiny, which is crucial in maintaining a rigorous statistical analysis.
The Role of the Significance Level
The significance level, denoted as \(\alpha\), is a threshold set by researchers to determine when to reject the null hypothesis. In our example, the significance level is 0.05, meaning there is a 5% risk of concluding that a difference exists when there is no actual difference. This is often referred to as a Type I error, or false positive.
  • A lower \(\alpha\) value indicates a stricter criterion for rejecting the null hypothesis.
  • Common choices for \(\alpha\) levels are 0.05, 0.01, and 0.10.
  • Choosing the right significance level is crucial, as it impacts both the validity and reliability of the test.
The level \(\alpha = 0.05\) is widely used in biostatistics and provides a balance between being too lenient and overly strict. It dictates the critical region on the distribution curve where observed results are deemed statistically significant. This critical region translates into something real-world, like adjustments in medical treatments, so clear, decisive criteria are necessary.
Understanding Statistical Power
Statistical power is the likelihood that a test will correctly reject a false null hypothesis, thus avoiding a Type II error, or false negative. In our sample size calculation, a power of 0.80 implies an 80% probability of detecting a true effect if it exists.
  • High power is an indication that the test is efficient and sensitive.
  • A common guideline in biostatistics is to aim for a power of at least 80%.
  • The higher the power, the larger the sample size generally needs to be.
Using power in sample size determination ensures that the study's findings are robust and meaningful. Having a power of 0.80 means there's a good chance that the test will uncover an effect, helping substantiate findings in clinical research. It's an essential component in planning a study that is both feasible and scientifically impactful.

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