Question

For each of the following tables, calculate (i) the relative risk and (ii) the odds ratio. $$\begin{aligned}&\text { (a) }\\\&\begin{array}{|rr|}\hline 14 & 16 \\\322 & 412 \\\\\hline\end{array}\end{aligned}$$ $$\begin{aligned}&\text { (b) }\\\&\begin{array}{|rr|}\hline 15 & 7 \\\338 & 82 \\\\\hline\end{array}\end{aligned}$$

Short answer

For Table (a), RR = (14/(14+16)) / (322/(322+412)), OR = (14/16) / (322/412). For Table (b), RR = (15/(15+7)) / (338/(338+82)), OR = (15/7) / (338/82).

Step-by-step solution

Understand the Terms

Relative Risk (RR) is a ratio of the probability of the event occurring in the exposed group versus a non-exposed group. Odds Ratio (OR) is a ratio of the odds of an event occurring in the exposed group to the odds of it occurring in the non-exposed group. For a 2x2 contingency table like this one, where the table is: [[a, b], [c, d]], RR is calculated as (a/(a+b)) / (c/(c+d)) and OR is calculated as (a/b) / (c/d).

Calculate the Relative Risk for Table (a)

To compute the RR for Table (a), use the formula RR = (a/(a+b)) / (c/(c+d)). For Table (a), a = 14, b = 16, c = 322, d = 412. Therefore, RR = (14/(14+16)) / (322/(322+412)).

Calculate the Odds Ratio for Table (a)

To compute the OR for Table (a), use the formula OR = (a/b) / (c/d). For Table (a), a = 14, b = 16, c = 322, d = 412. Therefore, OR = (14/16) / (322/412).

Calculate the Relative Risk for Table (b)

To compute the RR for Table (b), use the same formula as in Step 2: RR = (a/(a+b)) / (c/(c+d)). For Table (b), a = 15, b = 7, c = 338, d = 82. Therefore, RR = (15/(15+7)) / (338/(338+82)).

Calculate the Odds Ratio for Table (b)

To compute the OR for Table (b), use the same formula as in Step 3: OR = (a/b) / (c/d). For Table (b), a = 15, b = 7, c = 338, d = 82. Therefore, OR = (15/7) / (338/82).

Perform the Calculations

Perform the calculations using the values plugged into the formulas from Steps 2 to 5 to obtain the numerical values for RR and OR for each table.

Key concepts

Probability

Probability is a fundamental concept that quantifies the likelihood of events. It's a value between 0 and 1, where 0 indicates an impossibility and 1 denotes certainty. In epidemiological studies, probabilities help determine how likely it is for a certain health event to occur within a specific population. For instance, when computing relative risk, you're actually comparing probabilities: the probability of an event occurring in the group exposed to a certain risk factor vs. the probability of the same event occurring in a non-exposed group.

Understanding the basics of probability is essential in interpreting data from studies, like those presented in the exercise. A high probability of an event in the exposed group compared to the non-exposed group could lead to a relative risk greater than 1, suggesting a potential causal relationship between the exposure and the outcome.

2x2 Contingency Table

A 2x2 contingency table is a valuable tool used in medical and epidemiological research to show the frequency of outcomes in two different groups. It consists of two rows and two columns, detailing the presence or absence of a specific characteristic (like a disease) in relation to the exposure status. In our exercise, we analyze two groups: exposed (a and c) versus unexposed (b and d).

These tables are critical in calculating measures like relative risk and odds ratio. For the relative risk, we're interested in the proportion of the entire row (a divided by a plus b, and c divided by c plus d). For the odds ratio, we look at the proportion within the cells (a divided by b, and c divided by d). By arranging data in this concise format, researchers can quickly calculate these statistics and interpret their studies' results.

Epidemiology

Epidemiology is study of the distribution and determinants of health-related states or events in specified populations, and the application of this study to control health problems. It often involves identifying risk factors for disease and targets for preventive healthcare.

Epidemiologists use tools like relative risk and odds ratio to assess the link between exposures and health outcomes. In the given exercise, we are in essence carrying out an epidemiological analysis by calculating these measures to evaluate the potential relationship between an exposure (like a virus or hazardous substance) and an outcome (such as a disease). Clear understanding of these concepts allows for informed decisions in public health policy and individual risk assessment.

Risk Assessment

Risk assessment involves determining the quantitative or qualitative value of risk related to a concrete situation and a recognized threat. In the context of our study, we assess the risk of an adverse health outcome by calculating and comparing the relative risk and odds ratio for the exposed vs. non-exposed groups.

Relative risk gives us a direct measure of how much more (or less) likely the exposed group is to experience the event compared to the unexposed group. On the other hand, the odds ratio provides an estimate of the odds of an event occurring in the exposed group compared to the non-exposed, which is particularly useful when dealing with rare events. By understanding these measures, healthcare professionals and policy-makers can make informed decisions about how to manage and mitigate risks in the population.