Question

For each of the following tables, calculate (i) the relative risk and (ii) the odds ratio. $$\begin{aligned}&\text { (a) }\\\&\begin{array}{|rr|}\hline 25 & 23 \\\492 & 614 \\\\\hline\end{array}\end{aligned}$$ $$\begin{aligned}&\text { (b) }\\\&\begin{array}{|cr|}\hline 12 & 8 \\\93 & 84 \\\\\hline\end{array}\end{aligned}$$

Short answer

For table (a), the relative risk is (25*1106) / (48*492) and the odds ratio is (25*614) / (492*23). For table (b), the relative risk is (12*177) / (20*93) and the odds ratio is (12*84) / (93*8).

Step-by-step solution

Define the Terms

Define the key terms: Relative risk is a ratio of the probability of an event occurring in an exposed group versus a non-exposed group. The odds ratio is a measure of association between an exposure and an outcome; it is the ratio of the odds of an event occurring in an exposed group to the odds of it occurring in a non-exposed group.

Calculate the Relative Risk for Table (a)

Use the formula for relative risk: RR = (a/(a+b)) / (c/(c+d)), where a=25, b=23, c=492, and d=614. So, RR = (25/(25+23)) / (492/(492+614)) = (25/48) / (492/1106) = (25*1106) / (48*492).

Calculate the Odds Ratio for Table (a)

Use the formula for the odds ratio: OR = (a/c) / (b/d), where a=25, b=23, c=492, d=614. So, OR = (25/492) / (23/614) = (25*614) / (492*23).

Calculate the Relative Risk for Table (b)

Using the same formula RR = (a/(a+b)) / (c/(c+d)), for table (b) where a=12, b=8, c=93, and d=84, RR = (12/(12+8)) / (93/(93+84)) = (12/20) / (93/177) = (12*177) / (20*93).

Calculate the Odds Ratio for Table (b)

Using the same formula OR = (a/c) / (b/d), for table (b) where a=12, b=8, c=93, d=84, OR = (12/93) / (8/84) = (12*84) / (93*8).

Key concepts

Probability in Exposed vs Non-Exposed Group

Understanding the probability of an event occurring in groups that have been exposed to a particular factor compared to those that have not is a cornerstone of epidemiological research. In the context of our exercise, the exposed group refers to individuals who have been subject to a specific condition or treatment (represented by the first column of numbers), whereas the non-exposed group consists of those who have not (the second column of numbers).

For instance, in Table (a) with numbers 25 (exposed and event occurred) and 492 (exposed and event did not occur), against 23 (non-exposed but event occurred) and 614 (non-exposed and event did not occur), the probabilities we calculate are essential in determining the potential impact of the exposure on the likelihood of the event. In Table (b), the groups are smaller, yet the method remains the same. To convey the difference in these probabilities more accurately to students, it might be helpful to provide real-life context or examples on how exposure might affect the outcome, like the effect of a vaccine on the likelihood of getting a disease.

Measure of Association

The measure of association is a crucial concept that helps us quantify the relationship between an exposure and an outcome. It tells us how much more (or less) likely an event is to occur in one group versus another. In our exercise, two common measures of association are computed: relative risk (RR) and odds ratio (OR).

The relative risk compares the risk of developing an outcome in the exposed group to the non-exposed group, providing a direct measure of risk increase or decrease due to exposure. On the other hand, the odds ratio compares the odds (not the probability) of an event occurring in both groups, which can be particularly useful when dealing with rare events. It's important to clarify, however, that while OR can approximate RR well when the event is rare, they are not the same and can lead to different interpretations, especially in cases where the event is common.

Statistical Analysis in Life Sciences

Statistical analysis plays a vital role in the life sciences for validating hypotheses and making informed decisions based on data. It applies to a variety of fields, from medicine and pharmacy to environmental and agricultural sciences. In the given exercise, statistical methods are used to discern the strength of association between exposure and outcomes, such as diseases or physiological reactions.

Students should be aware that these calculations are part of a larger process that typically includes data collection, data cleaning, analysis, and interpretation. For clarity, we use relative risk and odds ratio as they are particularly suited to studies where conditions or treatments are compared. The real-life impact of correctly applying these statistical tools can directly influence public health policies and medical practices. Additionally, emphasizing the ethical implications of research and the importance of statistical integrity could enhance students' understanding of the field's significance.