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 = (round to three decimal places as needed.)

Explanation:

Step1: Calculate $p_t$ and $p_c$

$p_t=\frac{21}{2048}\approx0.01025$, $p_c = \frac{21}{1719}\approx0.01221$

Step2: Calculate the odds - ratio formula

The odds - ratio formula is $\frac{\frac{p_t}{1 - p_t}}{\frac{p_c}{1 - p_c}}$.
Substitute $p_t\approx0.01025$ and $p_c\approx0.01221$ into the formula:
First, $\frac{p_t}{1 - p_t}=\frac{0.01025}{1 - 0.01025}=\frac{0.01025}{0.98975}\approx0.01036$
Second, $\frac{p_c}{1 - p_c}=\frac{0.01221}{1 - 0.01221}=\frac{0.01221}{0.98779}\approx0.01236$
Then, $\frac{\frac{p_t}{1 - p_t}}{\frac{p_c}{1 - p_c}}=\frac{0.01036}{0.01236}\approx0.838$

Answer:

The odds ratio = 0.838