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Difficulty: MediumHypothesis Testing and Statistical Power

A team of cardiologists is designing a randomized controlled trial to evaluate the efficacy of a novel supervised cardiac rehabilitation protocol compared to standard care. The primary outcome is the continuous change in the 6-minute walk test distance over 12 weeks. Initial sample size calculations indicate that 400 patients are required to achieve 80% power to detect a true difference of 25 meters between the groups, using a standard α\alpha level of 0.05. However, due to unexpected budget reductions, the maximum number of patients that can be enrolled is capped at 250. The investigators still wish to maintain 80% statistical power without altering the minimum clinically important difference (25 meters) or the significance level (α=0.05\alpha = 0.05). Which of the following study modifications would best help achieve this goal?

  1. Restricting enrollment to a specific patient subgroup to ensure highly homogenous baseline exercise capacitiesAnswer
  2. B
    Broadening the inclusion criteria to capture a wider range of baseline disease severities
  3. C
    Stratifying the primary analysis to evaluate for effect modification across different age groups
  4. D
    Decreasing the accepted probability of a Type I error from 0.05 to 0.01
  5. E
    Analyzing the continuous outcome data using a Chi-square test instead of a Student's t-test

Answer

Restricting enrollment to a specific patient subgroup to ensure highly homogenous baseline exercise capacities
Statistical power is the probability of correctly rejecting a false null hypothesis (Power = 1−β1 - \beta). Power is determined by four primary factors: sample size (nn), effect size (Δ\Delta), significance level (α\alpha), and the variance of the underlying population (σ2\sigma^2). Because the investigators are forced to use a smaller sample size, and they refuse to compromise on the effect size or the α\alpha level, their only remaining option to preserve statistical power is to decrease the population variance (σ2\sigma^2). By restricting enrollment to a highly homogenous subgroup, the standard deviation of the baseline exercise capacity is minimized. A smaller standard deviation leads to narrower confidence intervals and a larger test statistic for any given difference, thereby increasing statistical power.

Step-by-Step Solution

1
Identify the primary challenge presented in the scenario
The sample size (nn) is being reduced from 400 to 250, which will inherently decrease the statistical power (1−β1-\beta) of the study.
Understanding the constraints is necessary to identify which variable can be manipulated.
2
Recall the mathematical determinants of statistical power
Statistical power depends on four main factors: Sample size (nn), Effect size (Δ\Delta), Significance level (α\alpha), and Population variance (σ2\sigma^2). Power increases as nn, Δ\Delta, and α\alpha increase, and increases as σ2\sigma^2 decreases.
This relationship dictates how study design parameters interact to affect the probability of detecting a true difference.
3
Evaluate the fixed parameters and determine the required adjustment
The scenario states that sample size (n=250n=250), effect size (Δ=25\Delta=25), and significance level (α=0.05\alpha=0.05) are strictly fixed. Therefore, to increase power back to 80%, the investigators must decrease the population variance (σ2\sigma^2).
By process of elimination, variance is the only remaining modifiable parameter in the power equation.
4
Identify which option successfully decreases population variance
Restricting enrollment to a homogenous subgroup reduces clinical variability, thereby decreasing the statistical variance of the outcome measure and effectively increasing power.
A more homogenous population yields a narrower standard deviation for the tested variable.

Key Concept

Statistical power is inversely proportional to the variance of the outcome variable; reducing population heterogeneity decreases variance and consequently increases power.
Estimated Time:1m 15s
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Topics

Hypothesis Testing and Statistical Power

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