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Serious side effects In \(2007,\) the FDA announced that the popular drug Zelnorm used to treat irritable bowel syndrome was withdrawn from the market, citing concerns about serious side effects. The analysis of several studies revealed that 13 of 11,614 patients receiving the drug had a serious side effect such as heart attack or stroke, compared to just 1 out of 7,031 on placebo. a. Construct a \(2 \times 2\) contingency table from these data. b. Compute the relative risk and interpret. c. Compute the odds ratio and interpret. d. In a letter to health care professionals, the drug maker quoted a P-value of 0.024 for the comparison between the probability of a serious side effect in the drug and placebo group. By entering the data into the Fisher's Exact Test web app (or other software), show that this is the P-value for Fisher's exact test with a two-sided alternative hypothesis. What are the conclusions at a 0.05 significance level?

Step by step solution

01

Construct the Contingency Table

The contingency table summarizes the outcomes for the drug and placebo groups. We have 13 patients with serious side effects in the Zelnorm group out of 11,614 total and 1 in the placebo group out of 7,031 total. The table is constructed as follows: \[\begin{array}{|c|c|c|}\hline & \text{Serious Side Effect} & \text{No Serious Side Effect} \\hline\text{Zelnorm} & 13 & 11614-13 \\hline\text{Placebo} & 1 & 7031-1 \\hline\end{array}\] This table becomes: \[\begin{array}{|c|c|c|}\hline & \text{Serious Side Effect} & \text{No Serious Side Effect} \\hline\text{Zelnorm} & 13 & 11601 \\hline\text{Placebo} & 1 & 7030 \\hline\end{array}\]

02

Compute the Relative Risk

The relative risk can be calculated as the ratio of the probability of a serious side effect occurring in the Zelnorm group to that in the placebo group. The probability for the Zelnorm group is \( \frac{13}{11614} \), and for the placebo is \( \frac{1}{7031} \).The relative risk (RR) is:\[RR = \frac{\frac{13}{11614}}{\frac{1}{7031}} = \frac{13 \times 7031}{11614} \approx 7.88\]This indicates that the risk of serious side effects is about 7.88 times higher in the Zelnorm group compared to the placebo group.

03

Compute the Odds Ratio

The odds ratio compares the odds of having a serious side effect in both groups. The odds for the Zelnorm group: \( \frac{13}{11601} \).The odds for the placebo group: \( \frac{1}{7030} \).The odds ratio (OR) is:\[OR = \frac{\frac{13}{11601}}{\frac{1}{7030}} = \frac{13 \times 7030}{11601} \approx 7.88\]Like the relative risk, this means that the odds of a serious side effect for the Zelnorm group is about 7.88 times those of the placebo.

04

Calculate and Interpret the P-value using Fisher's Exact Test

Using Fisher’s Exact Test, we enter the values from our contingency table. The test requires: \[(13, 1, 11601, 7030)\]In statistical software or a web app, inputting these data gives a two-sided P-value of approximately 0.024.Since the P-value (0.024) is less than the significance level of 0.05, there is statistically significant evidence to suggest a difference in the incidence of serious side effects between the two groups.

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Key Concepts

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

Relative Risk

Relative Risk is a fundamental concept in statistics, especially when comparing the likelihood of events between two different groups.
For this exercise, the relative risk (RR) measures how much more likely patients on Zelnorm are to experience serious side effects compared to those on a placebo.
Here's how it's calculated:

• First, determine the probability of experiencing a side effect for each group. For Zelnorm: \( \frac{13}{11614} \). For placebo: \( \frac{1}{7031} \).
• Then, find the ratio of these probabilities, \( RR = \frac{\frac{13}{11614}}{\frac{1}{7031}} \).
• The result, 7.88, implies that Zelnorm users are about 7.88 times more likely to have serious side effects than placebo users.

Relative Risk is useful as it provides a clear, intuitive measure of increased risk between two groups. It is important in making decisions about the safety of treatments or medications.

Odds Ratio

The Odds Ratio (OR) is another way to compare the frequency of outcomes between two groups. It is especially useful in studies where the outcomes are rare.
In this exercise, the odds ratio examines the odds of having a serious side effect with Zelnorm versus a placebo. The calculation steps are:

• Calculate the odds of having a side effect for Zelnorm: \( \frac{13}{11601} \) because 13 patient out of 11,614 experienced it.
• For the placebo group: \( \frac{1}{7030} \).
• The OR is \( \frac{\frac{13}{11601}}{\frac{1}{7030}} \), yielding an OR of 7.88.

An OR of 7.88 suggests that the odds of serious side effects are almost 8 times higher for Zelnorm users than those on a placebo. This metric is crucial in understanding the impact of treatments in health studies.

P-value

A P-value is a statistical metric that helps in determining the significance of observed differences in studies. It provides a way to test hypotheses.
In the exercise, the P-value from Fisher's Exact Test is 0.024, indicating the probability of observing differences in side effects just by chance, assuming no real effect exists.
To interpret it:

• A P-value less than the chosen significance level (usually 0.05) implies that the results are statistically significant.
• In this case, since 0.024 < 0.05, it suggests strong evidence against the null hypothesis of no difference between the drug and placebo groups.

A small P-value points towards a real difference in side effects, reinforcing the need to assess Zelnorm's safety.

Fisher's Exact Test

Fisher's Exact Test is a statistical technique often used in analyzing contingency tables, especially when sample sizes are small. It calculates the exact P-value for the chance of observing data like the one collected in the study.
For this exercise, Fisher's Exact Test was used to ascertain the variability in side effect incidence between Zelnorm and placebo groups. Here's what makes it unique:

• It's non-parametric, meaning it doesn't assume underlying data distribution, making it ideal for small sample sizes or rare events (like serious side effects).
• Inputting the contingency table values of serious and non-serious outcomes into a statistical tool produces an exact P-value.
• The test's result aids in understanding whether the observed outcomes could be due to chance.

Overall, Fisher's Exact Test provides a precise measure of significance, supporting the claim of a genuine difference in incidences between groups.