Question

in a clinical trial of 2048 subjects treated with a certain drug, 21 reported headaches. in a control group of 1719 subjects given a placebo, 21 reported headaches. denoting the proportion of headaches in the treatment group by $p_t$ and denoting the proportion of headaches in the control (placebo) group by $p_c$, the relative risk is $p_t/p_c$. the relative risk is a measure of the strength of the effect of the drug treatment. another such measure is the odds ratio, which is the ratio of the odds in favor of a headache for the treatment group to the odds in favor of a headache for the control (placebo) group, found by evaluating $\frac{p_t/(1 - p_t)}{p_c/(1 - p_c)}$. the relative risk and odds ratios are commonly used in medicine and epidemiological studies. find the relative risk and odds ratio for the headache data. what do the results suggest about the risk of a headache from the drug treatment?
find the relative risk for the headache data.
the relative risk = 0.840 (round to three decimal places as needed.)
find the odds ratio for the headache data.
the odds ratio = 0.838 (round to three decimal places as needed.)
what do the results suggest about the risk of a headache from the drug treatment?
a. the drug does not appear to pose a risk of headaches because $p_t$ is slightly less than $p_c$.
b. the drug appears to pose a risk of headaches because $p_t$ is slightly less than $p_c$.
c. the drug appears to pose a risk of headaches because $p_t$ is greater than $p_c$.
d. the drug has no risk because the relative risk and odds ratio are almost equal.

Explanation:

Step1: Calculate treatment - group proportion

The proportion of headaches in the treatment group $p_t$ is calculated as the number of headache - reporting subjects in the treatment group divided by the total number of subjects in the treatment group. So, $p_t=\frac{21}{2048}\approx0.01025$.

Step2: Calculate control - group proportion

The proportion of headaches in the control group $p_c$ is calculated as the number of headache - reporting subjects in the control group divided by the total number of subjects in the control group. So, $p_c = \frac{21}{1719}\approx0.01221$.

Step3: Calculate relative risk

The relative risk is $RR=\frac{p_t}{p_c}=\frac{\frac{21}{2048}}{\frac{21}{1719}}=\frac{1719}{2048}\approx0.840$.

Step4: Calculate odds in treatment group

The odds in favor of a headache in the treatment group is $O_t=\frac{p_t}{1 - p_t}=\frac{\frac{21}{2048}}{1-\frac{21}{2048}}=\frac{21}{2048 - 21}=\frac{21}{2027}$.

Step5: Calculate odds in control group

The odds in favor of a headache in the control group is $O_c=\frac{p_c}{1 - p_c}=\frac{\frac{21}{1719}}{1-\frac{21}{1719}}=\frac{21}{1719 - 21}=\frac{21}{1698}$.

Step6: Calculate odds ratio

The odds ratio is $OR=\frac{O_t}{O_c}=\frac{\frac{21}{2027}}{\frac{21}{1698}}=\frac{1698}{2027}\approx0.838$.

Step7: Interpret results

Since the relative risk ($0.840$) and odds ratio ($0.838$) are both less than 1, it means that the proportion of headaches in the treatment group is less than that in the control group. So the drug does not appear to pose a risk of headaches.

Answer:

A. The drug does not appear to pose a risk of headaches because $p_t$ is slightly less than $p_c$.