# How does the risk ratio compare to the odds ratio?

#### Both the odds ratio (OR) and risk ratio (RR) are summaries of relationship between two probabilities. These plots can help clarify the relationships among odds ratios, risk ratios, and probabilities.

We'll call the two probabilities the 'baseline' probability, p1, and the 'exposed' probability, p2. The OR is (p2/(1-p2))/(p1/(1-p1)). The RR is p2/p1. Neither gives as much information as p1 and p2 together. Interpreting an odds ratio is difficult. One simple interpretation is to suppose that the odds ratio is similar to the risk ratio.

#### Dynamic plot controls

###### (Only ratios > 1 are permitted)

Note: plots may look funny for baseline probabilities > 0.6, or for very small RRs.

Made by Ken Kleinman. This app generated using Shiny software, and hosted by the generous folks at RStudio.

#### Static bias plot

This plot shows the amount of bias inherent in the odds ratio when interpreted as a risk ratio, depending on the baseline probability. The bias is defined as (OR/RR) -1, the proportion by which the OR exceeds the RR. With small baseline probability or small odds ratio, the bias is not too bad. As either or both increase, bias becomes problematic.

#### RR for fixed OR

This is the risk ratio associated with the chosen odds ratio, as the baseline probability changes. All points on the curve have the input OR. As the baseline probability (p1) changes, the exposed probability changes to maintain a fixed OR, and the RR changes as well. The risk ratio is similar to the odds ratio for 'small' baseline probabilities, but approaches 1 as the baseline probability increases. For reference, we show the exposed probability (p2) at either end of the spectrum.
The red 'B' shows the baseline probability where the OR has your 'acceptable' input bias relative to the RR. Bias is defined as (OR/RR) - 1, the proportion by which the OR exceeds the RR. Higher baseline probabilities have smaller RR and thus more bias than is acceptable; lower baseline probabilities have larger RR and acceptable bias.

#### OR and RR for your probabilities

Here we plot the RR by the baseline probability. The figure title shows the RR and OR associated with your probabilities. The red dot appears above your input baseline probability; the label shows the bias in the OR as an indicator of RR for your probabilities. The line shows the RR for other baseline probabilities with this OR. As the baseline probability (p1) changes, the exposed probability must change to maintain a fixed OR, and the RR changes as well. You can use the plot to assess the effects on the OR/RR relationship as the baseline probability changes. The risk ratio is similar to the odds ratio for 'small' baseline probabilities, but approaches 1 as the baseline probability increases. This is why it is crucial to know the basleine probability before attempting to interpret the OR.

#### RR and OR for baseline probability

One way to think about the difference between the RR and the OR is to ask how high the exposed probability would have to be, in order to get the RR to be as large as the OR. In this plot, we show the values of the baseline and exposed probabilities that have the input ratio-- the odds ratio is in red, and the risk ratio is in blue. If you look at a baseline probability, the red line shows the exposed probability that generates the odds ratio. To get the risk ratio to have that value, you need the exposed probability to have the value on the blue line. Note that for some baseline values it is not possible to achieve the the RR, since the exposed probability cannot be greater than 1.