Relative Risk vs. Odds Ratio: When to Use Which

Risk vs. Odds: A Quick Refresher

Before we dive in, let's clarify two terms that are often used interchangeably in everyday language but have precise statistical meanings.

• Risk (or Probability): The likelihood of an event occurring. It's calculated as the number of times the event of interest happens divided by the total number of people.

  $$\text{Risk}=\frac{\text{Number of events}}{\text{Total number of people}}$$

  For example, if 10 out of 100 people develop a disease, the risk is $10/100=0.1$ or 10%.

• Odds: The ratio of the probability of an event happening to the probability of it not happening.

  $$\text{Odds}=\frac{\text{Number of events}}{\text{Number of non-events}}$$

  In the same example, 10 people developed the disease and 90 did not. The odds are $10/90\approx 0.11$.

As you can see, risk and odds are related but numerically different. This distinction is the foundation for understanding relative risk and odds ratios.

Relative Risk and Odds Ratio: Calculation and Interpretation

Let's watch a video that provides an excellent walkthrough of calculating and interpreting both measures using a single, clear example. This will help solidify the difference between them.

Summary of Calculations

Let's use the standard 2x2 table format to summarize the formulas from the video:

• Relative Risk (RR): The ratio of the risk in the exposed group to the risk in the unexposed group.

  $$RR=\frac{\text{Risk in Exposed}}{\text{Risk in Unexposed}}=\frac{A/(A+B)}{C/(C+D)}$$

  Interpretation: "The risk of the disease is RR times as high in the exposed group compared to the unexposed group."

• Odds Ratio (OR): The ratio of the odds of disease in the exposed group to the odds of disease in the unexposed group.

  $$OR=\frac{\text{Odds in Exposed}}{\text{Odds in Unexposed}}=\frac{A/B}{C/D}=\frac{A\times D}{B\times C}$$

  Interpretation: "The odds of the disease are OR times as high in the exposed group compared to the unexposed group."

For both measures, a value of 1.0 is the null value, indicating no association between the exposure and the outcome. A value greater than 1.0 suggests a positive association (increased risk/odds), while a value less than 1.0 suggests a negative or protective association (decreased risk/odds).

The Deciding Factor: Study Design

So, if you can calculate both, which one should you use? The answer hinges almost entirely on the study design, because that determines whether you can legitimately calculate risk.

There are two main types of observational studies you'll encounter:

1. Cohort Studies (and RCTs): These studies start with a disease-free group of people. They identify who is "exposed" and who is "unexposed" and then follow them over time to see who develops the disease. Because you are observing new (incident) cases develop in a defined group, you can calculate the true risk of the disease for both the exposed and unexposed groups.

   • Decision: For cohort studies and Randomized Controlled Trials (RCTs), Relative Risk (RR) is the preferred measure. It is more direct and intuitive.

2. Case-Control Studies: These studies work backward. Researchers start by identifying a group of people who already have the disease ("cases") and a comparable group who do not ("controls"). Then, they look back in time (retrospectively) to determine the proportion of each group that was exposed to the factor of interest.

   • Decision: You cannot calculate risk or RR in a case-control study. The proportion of cases in your sample is fixed by you, the researcher (e.g., 100 cases and 100 controls), so it doesn't represent the true incidence in the population. Therefore, you must use the Odds Ratio (OR).

The following video segment and image perfectly illustrate this crucial distinction.

The "Rare Disease Assumption"

You might have noticed in the video that the OR (2.25) was larger than the RR (2.0) for the same data. This is always true when the association is positive. The OR will always be further from the null value of 1.0 than the RR.

However, there is a special situation where they are very close: the rare disease assumption.

• When a disease is rare in the population, the number of people with the disease is small compared to the number without it.
• This means that for the unexposed group, $C+D\approx D$, and for the exposed group, $A+B\approx B$.
• Consequently, the RR formula, $\frac{A/(A+B)}{C/(C+D)}$, becomes approximately $\frac{A/B}{C/D}$, which is the formula for the OR.

Why this matters for your exam: In a case-control study where you can only calculate an OR, if the problem states that the disease is rare, you can interpret the OR as an approximation of the RR. If the disease is common, the OR significantly overestimates the RR, and you should be cautious about this exaggeration.

Putting It All Together: A Decision Guide

For your exam, use this clear decision-making process when a question asks for a measure of association strength.

1. Identify the Study Design:
   • Does the study start with exposed/unexposed groups and follow them over time to see who gets a disease?
     • YES -> Cohort Study or RCT. Your preferred measure is Relative Risk (RR).
   • Does the study start with people who have a disease (cases) and people who don't (controls) and look backward at exposures?
     • YES -> Case-Control Study. You must use the Odds Ratio (OR).
   • Does the study take a "snapshot" in time, assessing exposure and disease status simultaneously?
     • YES -> Cross-Sectional Study. The standard measure is the Odds Ratio (OR).

This is the most critical distinction. The table below from an Oregon State University textbook summarizes this perfectly.

The flowchart below provides a final visual summary of the correct usage of these measures.

Conclusion

You have now learned to distinguish between relative risk and odds ratio, two essential measures for quantifying the strength of an association in a 2x2 table.

Key Takeaways:

• Relative Risk (RR) is a ratio of two risks (probabilities). It is the preferred measure for cohort studies and RCTs because it is more intuitive.
• Odds Ratio (OR) is a ratio of two odds. It is the necessary measure for case-control studies (where risk cannot be calculated) and the standard for cross-sectional studies.
• The choice between RR and OR is determined by the study design.
• The rare disease assumption states that when an outcome is uncommon, the OR provides a good approximation of the RR. When the outcome is common, the OR exaggerates the strength of the association compared to the RR.

This lesson forms a crucial bridge. We've completed our look at foundational contingency table analysis. The odds ratio, in particular, will reappear as a central element when we move into our next topic: an introduction to Generalized Linear Models (GLMs), starting with the motivation for why we need them and their basic structure.