# Odds Ratio Vs. Risk Ratio

Relative risk, odds ratio, risk ratio, risk difference - these are all measures of the direction and the strength of the association between two categorical variables.

Can I use any of these four measures? Or, is there any condition to use the measures?

These are all different ways of describing the difference between two probabilities. Mathematically they are all equally valid - none is more "right" or "wrong" than the other. But they all quantify the same thing in different ways, and thus some are more useful than others in different purposes.

Let's say one the probability of a outcome Y is 20% for the control group and 40% for the treatment group. How big is the effect of the treatment?

To calculate the "risk ratio" or "relative risk" you DIVIDE the two probabilities by one another. .40/.20=2. This tells you that people who got the treatment are "200 PERCENT as likely" (or 100 percent more likely) to have experienced Y.

To calculate the risk difference you SUBTRACT the probabilities: .40-.20=.2. This tells you that people who got the treatment are "20 PERCENTAGE POINTS more likely to have experienced Y.

To calculate odds ratios....well it's a huge pain and the result isn't intuitive. See in stats an "odds" is already a ratio - it's the probability of a thing happening divided by the probability of the thing not happening. For the treatment group the odds of Y are 40% divided by 60% so 0.666. For the control group the odds of Y are 20% divided by 80% so 0.25.The odds ratio is the ratio of these two odds, so .666/.25=2.65. That means that the ODDS (not probability!) of Y are 265% as high for the treatment group compared to the control group. Of course, since no one knows what an "odds" means (people often use the term to mean "percentage phrased as a ratio - 25% as "1 in four chance") this value doesn't really tell you much.

The only reason we actually talk about odds ratios is that (unlike risk ratios and differences) the odds of a given "effect size" stay constant as the base rate (in this case the % for the control group) goes up and down. For this reason a logit model can only express its results in terms of odds ratios, unless you add some extra information about the base rate. But in even peer-reviewed papers both authors and readers frequently misinterpret odds ratios as risk ratios, thinking that an OR of 2.65 means that the treatment group is "165% more likely to experience Y," which is (in most cases) wrong. So be very careful about ever using them.

Any of these three measures are valid comparisons of the probability of disease between two groups. (I say three because relative risk and risk ratio are the same thing.)

For case-control studies, you can only use the odds ratio because you do not have a valid estimate of risk. In case-control studies you choose the number of cases and number of controls, so you cannot provide an estimate of risk.

Risk ratios can be misleading as to the importance of the difference when the risk in the comparison group is very small. For example, if the risk in the unexposed group is 0.01 and the risk in the exposed group is 0.05, that leads to a risk ratio of 5, but a risk difference of only 0.04. On the other hand, if the risk in the unexposed group is 0.05, but the risk in the exposed group is 0.25, you still have a risk ratio of 5. But the risk difference is now 0.2! Therefore, the importance of risk ratios depends upon the baseline risk for the disease in the unexposed group.

Yes, as Graham says, odds ratios (ORs) are not intuitive and frequently misinterpreted as risk ratios (RRs). ORs are always farther from 1 than RRs, unless both are equal to 1. Even if you use a logistic regression model, you can present your results with RRs or risk differences (RDs), but it's tricky. Here's how to do it in Stata.

RDs are absolute risk differences. If the exposure is a treatment, the absolute risk reduction (ARR) is -RD, and the number needed to treat is 1/ARR.

The relative risk difference $$RRD = RR-1$$. This can be generalized to other populations with different baseline risk in the unexposed $$R_u$$. $$RD = RRD \times R_u$$

If you know the baseline risk in the unexposed $$R_u$$ you can calculate the RR from the OR. $$RR = \frac{OR}{1-R_{u} + R_{u}(OR)}$$ $$R_{u}(OR - 1) = \frac{OR - RR}{RR}$$ The first equation is called Levin's Equation. I prefer the second equation, which provides the percent by which the OR overestimates/underestimates the RR. If $$R_u$$ is small and OR is close to 1, then OR $$\approx$$ RR, but not otherwise.

$$R_{e} =$$ Risk in the exposed

$$R_{u} =$$ Risk in the unexposed

\begin{align*} OR &=\frac{\dfrac{R_{e}}{1-R_{e}}}{\dfrac{R_{u}}{1-R_{u}}} = \text{Odds Ratio}\\ RR &= \frac{R_{e}}{R_{u}} = \text{Risk Ratio}\\ &= OR \left(\frac{1-R_{e}}{1-R_{u}} \right)\\ &= OR \left(\frac{1-R_{u}(RR)}{1-R_{u}} \right)\\ RR(1-R_{u}) &= OR \left({1-R_{u}(RR)}\right)\\ RR(1-R_{u}) &= OR-R_{u}(OR)(RR)\\ RR(1-R_{u}) + R_{u}(OR)(RR) &= OR\\ RR(1-R_{u} + R_{u}(OR)) &= OR\\ RR &= \frac{OR}{1-R_{u} + R_{u}(OR)} &&\text{(Levin's Equation)} \\ RR + (RR)R_{u}(OR) - (RR)R_{u} &= OR \\ RR + (RR)R_{u}(OR - 1) &= OR \\ (RR)R_{u}(OR - 1) &= OR - RR\\ R_{u}(OR - 1) &= \frac{OR - RR}{RR}\\ \end{align*}

The $$OR$$ mis-estimates the $$RR$$ by a percentage equal to $$R_{u}\cdot(OR -1)$$, where $$R_{u} =$$ risk in the unexposed (control) group.

If $$OR = 3$$ and $$R_{u} = 10\%$$, then the $$OR$$ is 20% higher than the $$RR$$, because $$0.10\cdot(3-1) = 0.2$$.
That means the $$RR$$ is $$3/1.2 = 2.5$$.

If $$OR = 0.33$$ and $$R_{u} = 10\%$$, then the $$OR$$ is 6.7% lower than the $$RR$$, because $$0.10\cdot(0.33-1) = -0.067$$.

That means the $$RR$$ is $$0.33/0.933 = 0.357$$.