Question

Many over-the-counter decongestants and appetite suppressants contain the ingredient phenylpropanolamine. A study was conducted to investigate whether this ingredient is associated with strokes. The study found that 6 of 702 stroke victims had used an appetite suppressant containing phenylpropanolamine, compared to only 1 of 1,376 subjects in a control group. The following table summarizes these data. (a) Calculate the sample value of the odds ratio. (b) Construct a \(95 \%\) confidence interval for the population value of the odds ratio. (c) Upon hearing of these data, some scientists called the study "inconclusive" because the numbers of users of appetite suppressants containing phenylpropanolamine (7 total: 6 in one group and 1 in the other) are so small. What is your response to these scientists?

Short answer

The sample odds ratio is calculated as approximately 11.86. A 95% confidence interval for the odds ratio can be constructed using the natural logarithm transformation, but the precision of this estimate and interval will be affected by the small sample size, which renders the study potentially inconclusive. Addressing the scientists' concerns, the findings are indeed limited by the small number of exposed individuals.

Step-by-step solution

Define Odds Ratio

Odds Ratio (OR) is a measure of association between exposure and an outcome. It represents the odds that an outcome will occur given a particular exposure, compared to the odds of the outcome occurring without that exposure. The formula to calculate the odds ratio in this case is OR = (a/b) / (c/d), where 'a' is the number of stroke victims who used the appetite suppressant, 'b' is the number of stroke victims who did not, 'c' is the number of control subjects who used the suppressant, and 'd' is the number of control subjects who did not.

Calculate the Odds Ratio

Using the data provided, calculate the OR with the formula from Step 1. Let 'a' be the 6 stroke victims who used the suppressant, 'b' be the 696 stroke victims who did not (702 total stroke victims minus 6 who used the suppressant), 'c' be the 1 control subject who used the suppressant, and 'd' be the 1,375 control subjects who did not (1,376 total controls minus 1 who used the suppressant). Thus, OR = (6/696) / (1/1375).

Simplify the Formula for Odds Ratio

Simplify the formula: OR = (6/696) / (1/1375) = (6 * 1375) / (696 * 1) = 8250 / 696.

Compute the Odds Ratio

Compute the odds ratio by performing the division from Step 3 to obtain the sample value of the OR.

Calculate the 95% Confidence Interval for OR

Compute the natural logarithm of the OR. Then, use the formula for the standard error (SE) of the natural logarithm of OR: SE(ln(OR)) = sqrt(1/a + 1/b + 1/c + 1/d). Next, calculate the 95% confidence interval for ln(OR) using the formula: ln(OR) \( \pm \ z_{0.025} \times SE(ln(OR)) \), where \(z_{0.025} \) is the 1.96 for a 95% confidence interval. Finally, take the exponential of the lower and upper bounds to get the confidence interval for the OR.

Addressing the Scientists' Concerns

The small number of users of appetite suppressants implies that the study may not have sufficient power to detect a true association if one exists. The low occurrence of exposure also increases the variance of the odds ratio estimate, making the confidence interval wide and thus less precise. This can lead to inconclusiveness as the confidence interval could include both a substantial risk and no risk at all.

Key concepts

Confidence Interval

In statistics, the confidence interval (CI) is a range of values that's likely to contain a population parameter with a certain degree of confidence. It's used to indicate the reliability of an estimate. For example, a 95% confidence interval for the odds ratio (OR) suggests that we can be 95% certain that the true OR lies within the computed interval.

Constructing a confidence interval involves statistical methods such as the one described in the exercise. After calculating the odds ratio, the confidence interval is derived using the standard error and the z-score corresponding to the desired confidence level. However, the CI's width can vary depending on the number of cases; small numbers of events can lead to wide intervals, which indicates lower precision in the estimate and may make the results inconclusive. Hence, the CI is crucial for interpreting the statistical significance and relevance of study findings.

Exposure and Outcome Association

Understanding the link between exposure and outcome is at the heart of many epidemiological studies. The association reveals whether the presence (or absence) of a particular factor (exposure) affects the likelihood of a specified event or condition (outcome).

In the context of the exercise, researchers are interested in whether the use of appetite suppressants containing phenylpropanolamine is associated with an increased risk of strokes. The odds ratio (OR) is used here as a measure of that association. If the OR is significantly greater than 1, it may suggest that there is a positive association; that is, the exposure is related to an increased risk of the outcome. However, an OR of 1 indicates no association, while an OR less than 1 suggests a negative association, possibly a protective effect of the exposure.

Statistical Significance

Determining statistical significance is a key part of hypothesis testing. It helps researchers conclude whether the observed effect (e.g., association between an exposure and an outcome) is unlikely to have occurred by chance.

Statistical significance is typically assessed using a p-value, which is the probability of obtaining an effect at least as extreme as the one observed if the null hypothesis (no effect) were true. A commonly used significance level is 0.05 or 5%. If the p-value is less than this threshold, the result is deemed 'statistically significant' and suggests that the null hypothesis can be rejected. Nevertheless, it's important to bear in mind that 'statistical significance' does not necessarily equate to 'clinical significance,' and factors like effect size and confidence intervals should be considered when interpreting results.

Study Power

Study power refers to the probability that a study will detect an effect when there actually is one, which is termed the ability to avoid a Type II error (failing to reject a false null hypothesis). The power of a study is influenced by several factors, including the sample size, the effect size, and the significance level chosen for the test.

A study with low power may fail to detect a real association, leading to a false conclusion that the exposure has no effect. As highlighted in the exercise, the small number of subjects who had the exposure of interest (using appetite suppressants) can result in a study with low power. This situation could potentially explain the inconclusive nature of the study and the reservations expressed by the scientists. High-powered studies not only provide more reliable evidence but also contribute to a deeper ethical responsibility in research, as they use resources efficiently and respect participant involvement.