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Difficulty: MediumStatistical Inference and Hypothesis Testing

A pharmacovigilance research group is conducting a retrospective analysis of clinical trials investigating various homeopathic remedies for the treatment of seasonal allergic rhinitis. The researchers identify a total of 140 independent, randomized controlled trials that compared a specific homeopathic intervention to a matching placebo. It is firmly established by comprehensive biochemical analysis that this homeopathic remedy contains only water and is biologically completely inert, meaning it has the exact same true clinical effect as the placebo.

Each of the 140 clinical trials was rigorously designed and executed with a pre-specified significance level (α\alpha) of 0.05 and a statistical power of 80%.

Based on the principles of hypothesis testing, how many of these 140 trials are expected to report a statistically significant difference in efficacy between the homeopathic remedy and the placebo purely by chance?

Answer: 7 trials

Answer

7
The null hypothesis states that there is no true difference between the intervention and the placebo. Because the homeopathic remedy is completely inert and identical to water, the null hypothesis is absolutely true. A Type I error occurs when researchers incorrectly reject a true null hypothesis, concluding that a statistically significant difference exists when none actually does. The probability of making a Type I error is determined entirely by the pre-specified significance level (α\alpha), which is set to 0.05. Therefore, in 140 independent trials where the intervention has no true clinical effect, we expect 5% of them to report a statistically significant difference purely due to random chance. The calculation is 140×0.05=7140 \times 0.05 = 7. Statistical power (80%) and Type II error (20%) are irrelevant in this scenario because they only apply when the alternative hypothesis is true (i.e., when the drug actually works).

Step-by-Step Solution

1
Determine the true state of the intervention to identify which statistical errors apply.
The intervention is biologically inert and identical to placebo, meaning the null hypothesis of no difference is true in reality.
To distinguish whether a Type I error (false positive) or Type II error (false negative) is statistically possible in this scenario.
2
Identify the probability of concluding a statistically significant difference when the null hypothesis is true.
This probability is defined by the significance level (α\alpha), which is given as 0.05 (or 5%).
A Type I error occurs when the null hypothesis is incorrectly rejected due to random sampling variation.
3
Calculate the expected number of trials that will show a statistically significant difference.
140 trials ×0.05=7\times 0.05 = 7 expected trials.
The expected frequency of an event is the probability of the event multiplied by the total number of independent trials.

Key Concept

Type I Error (α\alpha) Expected Frequency
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Topics

Statistical Inference and Hypothesis Testing

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