*by Audrey Schnell*

Odds Ratios and Relative Risks are often confused despite being unique concepts. Why?

Well, both measure association between a binary outcome variable and a continuous or binary predictor variable.

And unfortunately, the names are sometimes used interchangeably. They shouldn’t be because they’re actually interpreted differently. So it’s important to keep them separate and to be precise in the language you use.

The basic difference is that the odds ratio is a ratio of two odds (yep, it’s that obvious) whereas the relative risk is a ratio of two probabilities. (The relative risk is also called the risk ratio). Let’s look at an example.

## Relative Risk/Risk Ratio

Suppose you have a school that wants to test out a new tutoring program. At the start of the school year they impose the new tutoring program (treatment) for a group of students randomly selected from those who are failing at least 1 subject at the end of the 1st quarter. The remaining students receive the customary academic support (control group).

At the end of the school year the number of students in each group who fail any of their classes is measured. Failing a class is considered the outcome event we’re interested in measuring. From these data we can construct a table that describes the frequency of two possible outcomes for each of the two groups.

The probability of an event in the Treatment group is a/(a+b)= R1 . It’s the number of tutored students who experienced an event (failing a class) out of the total number of tutored students. You can think of it this way, if a student is tutored, what is the probability (or risk) of failing a class?

Likewise, the probability of an event in the Control group is c/(c+d) = R2. Again, it’s just the number of untutored students experienced an event out of the total number of untutored students.

Although each of these probabilities (i.e., risks) is itself a ratio, this isn’t the risk ratio. The risk of failing in the tutored students needs be compared to the risk in the untutored students to measure the effect of the tutoring.

The ratio of these two probabilities R1/R2 is the relative risk or risk ratio. Pretty intuitive.

If the program worked, the relative risk should be smaller than one, since the risk of failing should be smaller in the tutored group.

If the relative risk is 1, the tutoring made no difference at all. If it’s above 1, then the tutored group actually had a higher risk of failing than the controls.

## Odds Ratio

The odds ratio is the ratio of the odds of an event in the Treatment group to the odds of an event in the control group. The term ‘Odds’ is commonplace, but not always clear, and often used inappropriately.

The odds of an event is the number of events / the number of non-events.

This turns out to be equivalent to the probability of an event/the probability of a non-event.

You’ll often see odds written as P/(1-P).

So for example, in the Treatment group, the odds of an event is the number of tutored students who failed a class/the number of students in the tutored group who passed all their classes.

The numerator is the same as that of a probability, but the denominator here is different. It’s not a measure of events out of all possible events. It’s a ratio of events to non-events. You can switch back and forth between probability and odds—both give you the same information, just on different scales.

If O1 is the odds of event in the Treatment group and O2 is the odds of event in the control group then the odds ratio is O1/O2. Just like the risk ratio, it’s a way of measuring the effect of the tutoring program on the odds of an event.

Compare this to RR which is the probability of an event occurring (a/a+b)/the probability of the event not occurring (c/c+d).

**References and Further Reading:**

- Case-Control Studies: Design, Conduct, Analysis (Monographs in Epidemiology and Biostatistics) 1st Edition James J. Schlesselman
- Foundations of Epidemiology 2nd Edition Lilienfeld and Lilienfeld.
- Essentials of Biostatistics. Robert C. Elston and William D. Johnson 1994
- Why use Odds Ratios in Logistic Regression
- Understanding Probability, Odds, and Odds Ratios in Logistic Regression