Measures of Association: Ratios and Interpretation

Setting the Stage: The 2x2 Table

Before we can compare groups, we need a standard way to organize our data. In epidemiology, this is almost always done with a 2x2 table, also known as a contingency table. It cross-classifies individuals based on their exposure status (e.g., smoker vs. non-smoker) and their outcome status (e.g., has lung cancer vs. does not).

The standard layout is:

• a: Number of people who were exposed and have the disease.
• b: Number of people who were exposed and do not have thedisease.
• c: Number of people who were not exposed and have the disease.
• d: Number of people who were not exposed and do not have the disease.

This simple table is the starting point for calculating all the measures of association we will discuss.

1. Risk Ratio (RR): Comparing Cumulative Incidence

The most intuitive way to compare two groups is to compare their risks. The Risk Ratio (RR), also called the Relative Risk, does exactly this. It's the ratio of the risk of an outcome in an exposed group to the risk in an unexposed group. Remember from our last lesson that "risk" here refers to the cumulative incidence.

The video below provides an excellent visual introduction to the concept of risk and how to compare it between groups. It cleverly avoids getting bogged down in formulas, focusing instead on the intuition.

As the video explained, the formula for the Risk Ratio is:
$\text{Risk Ratio (RR)}=\frac{\text{Risk in exposed group}}{\text{Risk in unexposed group}}=\frac{C{I}_{exposed}}{C{I}_{unexposed}}=\frac{a/(a+b)}{c/(c+d)}$

Let's walk through another example from a CDC resource to solidify your understanding.

The interpretation of the RR is straightforward:

• RR > 1: The exposure is associated with an increased risk of the outcome. An RR of 2.4 means the exposed group has 2.4 times the risk of the outcome compared to the unexposed group.
• RR = 1: There is no association between exposure and outcome. The risk is identical in both groups. This is the "null" value.
• RR < 1: The exposure is associated with a decreased risk of the outcome. It is a "protective" factor. An RR of 0.28 means the exposed group has only 28% of the risk of the unexposed group.

2. Rate Ratio: Comparing Incidence Rates

What if our data doesn't come from a closed group followed for a fixed period? As we saw in the last lesson, when follow-up times vary, we use incidence rates with person-time in the denominator. To compare incidence rates between two groups, we use the Rate Ratio.

The concept is identical to the Risk Ratio, but we substitute incidence rates for cumulative incidences.

$\text{Rate Ratio}=\frac{\text{Incidence Rate in exposed group}}{\text{Incidence Rate in unexposed group}}=\frac{I{R}_{exposed}}{I{R}_{unexposed}}=\frac{a/({\text{Person-Time}}_{exposed})}{c/({\text{Person-Time}}_{unexposed})}$

The interpretation is also analogous: a rate ratio of 2.2 means that the outcome is occurring at 2.2 times the speed or rate in the exposed group compared to the unexposed group.

Let's look at a practical example.

The key difference between the Risk Ratio and Rate Ratio lies in the data used to calculate them, which reflects different study designs—something we will explore in the next module. The RR uses cumulative incidence (risk), suitable for cohort studies with complete follow-up. The Rate Ratio uses incidence rates (person-time), which is more flexible and suitable for dynamic populations where follow-up times differ.

3. Odds Ratio (OR): A Necessary Alternative

Sometimes, we can't calculate risk or incidence rates. This is especially true in a case-control study, where we start with a group of people who already have the disease (cases) and a group who don't (controls), and then look backward to assess their prior exposures. In this design, we don't have the "total population at risk" to form the denominator for a risk or rate calculation.

For these situations, we use the Odds Ratio (OR). To understand the OR, we first need to distinguish "odds" from "risk" (or probability).

• Risk (Probability) = (Number of times an event happens) / (Total number of trials)
• Odds = (Probability an event happens) / (Probability an event does not happen) = P / (1 - P)

For example, if the risk of an event is 25% (or 0.25), the odds are $0.25/(1-0.25)=0.25/0.75=0.33$. The following video explains this distinction and introduces the Odds Ratio.

The Odds Ratio (OR) is calculated in two main ways, which are mathematically equivalent:

1. The ratio of the odds of disease in the exposed to the odds of disease in the unexposed.
2. The ratio of the odds of exposure in the cases to the odds of exposure in the controls. (This is how it's often thought of in case-control studies).

This leads to a simple formula using our 2x2 table, often called the cross-product ratio:

$\text{Odds Ratio (OR)}=\frac{a/b}{c/d}=\frac{a\times d}{b\times c}$

This calculation is explored in the CDC resource.

The interpretation of the OR is the same as for the RR: >1 indicates a harmful association, =1 indicates no association, and <1 indicates a protective association.

A critical point mentioned in the reading is that when the disease is rare, the Odds Ratio is a good approximation of the Risk Ratio. This is because if the number of cases (a, c) is very small compared to the non-cases (b, d), then $a+b\approx b$ and $c+d\approx d$. In this scenario, the RR formula, $(a/(a+b))/(c/(c+d))$, becomes approximately $(a/b)/(c/d)$, which is the formula for the OR. This property is incredibly useful, as it allows us to use the OR from a case-control study to estimate the more intuitive RR.

You may also recall from our earlier lesson on logistic regression that the exponentiated coefficient of a predictor gives you an odds ratio. This is the same odds ratio, connecting the world of regression modeling, which is common in machine learning, directly to the foundational measures of epidemiology.

Putting It All Together: A Visual Summary

The following infographics provide a concise summary of these three key measures, how to calculate them, and when to use them. They are excellent references to consolidate your understanding.

Conclusion

In this lesson, you have learned to calculate and interpret the fundamental measures that quantify the association between an exposure and an outcome. These ratios are the end-products of many epidemiological analyses and are crucial for making evidence-based decisions in public health.

Here are your key takeaways:

• Measures of association compare the frequency of disease in two or more groups to quantify the relationship between an exposure and a disease.
• The Risk Ratio (RR) compares the cumulative incidence (risk) in exposed and unexposed groups. It is calculated as ${\text{Risk}}_{\text{exposed}}/{\text{Risk}}_{\text{unexposed}}$.
• The Rate Ratio compares incidence rates, accounting for person-time. It is calculated as ${\text{Rate}}_{\text{exposed}}/{\text{Rate}}_{\text{unexposed}}$.
• The Odds Ratio (OR) compares the odds of an outcome. It is calculated as the cross-product $(ad)/(bc)$ and is the primary measure for case-control studies where risk cannot be calculated directly.
• For all three measures, a value of 1 means no association, >1 means increased risk/odds (harmful exposure), and <1 means decreased risk/odds (protective exposure).

You are now equipped with the basic tools to both measure disease frequency and assess associations. In our next module, we will dive deeper into the study designs themselves—cross-sectional, cohort, and case-control studies—and you will see exactly why the choice between a Risk Ratio and an Odds Ratio is dictated by how the study is designed.