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Difficulty: MediumHypothesis Testing and Statistical Power

A medical director at a rural community health center is planning a randomized controlled trial to evaluate a novel text-message adherence protocol for patients with hypertension. Initially, statistical modeling determined that 300 patients are needed to achieve 80%80\% power at a two-tailed significance level of α=0.05\alpha = 0.05. This calculation assumed a minimum clinically important difference in mean systolic blood pressure of 5 mmHg5\text{ mmHg} and a population standard deviation of 8 mmHg8\text{ mmHg}.

A week before enrollment begins, a comprehensive regional health survey is published, revealing that the true population standard deviation for systolic blood pressure in this demographic is actually 14 mmHg14\text{ mmHg}. If the investigator proceeds with the originally planned sample size of 300 patients and an alpha of 0.050.05, which of the following is the most accurate statement regarding the study's design and statistical parameters?

  1. The statistical power of the study will decrease, leading to a higher probability of failing to reject a false null hypothesis.Answer
  2. B
    The increased variance will act as a confounder between the intervention and outcome, requiring stratified analysis to prevent effect modification.
  3. C
    The investigator must change the primary analysis from an independent t-test to a Chi-square test to properly handle the larger continuous variance.
  4. D
    The wider dispersion of blood pressure values increases the underlying population prevalence, which will lower the positive predictive value of any statistically significant trial result.
  5. E
    The investigator's reliance on the initial standard deviation estimate of 8 mmHg8\text{ mmHg} despite the new evidence is a manifestation of the availability heuristic.

Answer

The statistical power of the study will decrease, leading to a higher probability of failing to reject a false null hypothesis.
Statistical power represents the probability of correctly rejecting the null hypothesis when the alternative hypothesis is true. Power is directly proportional to sample size, effect size, and significance level (α\alpha), but inversely proportional to the standard deviation (variance) of the underlying population. Because the actual standard deviation (14 mmHg14\text{ mmHg}) is much larger than the originally estimated standard deviation (8 mmHg8\text{ mmHg}), there will be more 'noise' in the data. This increased noise makes it harder to detect the true 'signal' (the 5 mmHg5\text{ mmHg} difference), increasing the Type II error rate (β\beta)—which is the probability of failing to reject a false null hypothesis. Consequently, the power of the study will fall below the desired 80%80\%.

Step-by-Step Solution

1
Identify the primary relationship being tested in the stem.
The standard deviation (variance) of the population increased from 8 mmHg8\text{ mmHg} to 14 mmHg14\text{ mmHg}, while sample size and alpha remained fixed.
Understanding which statistical parameters are changing is required to determine the impact on the study.
2
Evaluate the effect of increased standard deviation on statistical power.
Higher standard deviation creates wider distribution curves. This increases the overlap between the null hypothesis curve and the alternative hypothesis curve.
Statistical power (1−β1 - \beta) is the ability to detect a true difference. More noise (variance) obscures the signal (the 5 mmHg5\text{ mmHg} difference).
3
Correlate the loss of power with specific error types.
As power decreases, the Type II error rate (β\beta) increases. A Type II error is failing to reject a false null hypothesis.
Power and Type II error are mathematically inversely related (Power=1−βPower = 1 - \beta). The Type I error (α\alpha) was fixed at 0.050.05 by the investigator.

Key Concept

Increases in population variance (standard deviation) decrease statistical power and increase the probability of a Type II error (β\beta) if sample size and alpha remain constant.
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Topics

Hypothesis Testing and Statistical Power

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