Odds. In Exercises 41–44, answer the given questions that involve odds.

Relative Risk and Odds Ratio In a clinical trial of 2103 subjects treated with Nasonex, 26 reported headaches. In a control group of 1671 subjects given a placebo, 22 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 role="math" localid="1644405830274" $p_c$, the relative risk is $\frac{p_t}{p_c}$. The relative risk is a measure of the strength of the effect of the Nasonex 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 the following:$\frac{p_t/\left(1-p_t\right)}{p_c/\left(1-p_c\right)}$

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 Nasonex treatment?

In a treatment group, Nasonex is given. Out of 2103 subjects, 26 reported headaches.

In the control group, out of 1671 subjects, 22 reported headaches.

A treatment group includes subjects that receive the treatment.

A control group includes subjects that do not receive the treatment.

Relative risk ratio is the ratio of the risk of event A in the treatment group to the control group.

Let be the proportion of subjects at risk in the treatment group and be the proportion of subjects at risk in the control group.

The formula for relative risk (RR) is given below:

$RR=\frac{p_t}{p_c}$

Let “developing headache” be event A.

The risk of event A is considered.

The total number of subjects in the treatment group is 2103.

The number of subjects who reported headaches is 26.

The proportion of subjects developing the risk in the treatment group is calculated below:

$$\begin{gathered} p_t=\frac{\text{Number}\text{of}\text{subjects}\text{who}\text{developed}\text{the}\text{risk}}{\text{Total}\text{number}\text{of}\text{subjects}\text{in}\text{the}\text{treatment}\text{group}} \\ =\frac{26}{2103} \\ =0.0124 \end{gathered}$$

The total number of subjects in the control group is 1671.

The number of subjects who reported headaches is 22.

The proportion of subjects developing the risk in the control group is calculated below:

$$\begin{gathered} p_c=\frac{\text{Number}\text{of}\text{subjects}\text{who}\text{developed}\text{the}\text{risk}}{\text{Total}\text{number}\text{of}\text{subjects}\text{in}\text{the}\text{control}\text{group}} \\ =\frac{22}{1671} \\ =0.0132 \end{gathered}$$

The relative risk ratio is calculated as follows:

$$\begin{gathered} RR=\frac{p_t}{p_c} \\ =\frac{0.0124}{0.0132} \\ =0.939 \end{gathered}$$

Thus, the relative risk ratio is 0.939.

The odds ratio is the ratio of the odds in favor of risk for the treatment group to the odds in favor of risk for the control group.

The formula for the odds ratio (OR) is given as follows:

$OR=\frac{\frac{p_t}{\left(1-p_t\right)}}{\frac{p_c}{\left(1-p_c\right)}}$

The odds ratio is calculated below:

$$\begin{gathered} OR=\frac{\frac{p_t}{\left(1-p_t\right)}}{\frac{p_c}{\left(1-p_c\right)}} \\ =\frac{\frac{0.0124}{\left(1-0.0124\right)}}{\frac{0.0132}{\left(1-0.0132\right)}} \\ =0.939 \end{gathered}$$

Thus, the odds ratio is 0.939.

The relative risk ratio and the odds ratio are approximately equal and close to 1.

Relative risk implies the risk of headache is almost the same for both groups, and Nasonex does not significantly reduce headaches.

Similarly, the odds ratio signifies that the odds of headaches in the treatment group are the same as those in the control group.

Thus, it can be said thatthere is no major difference in the risk of developing headaches between the treatment group and the control group.