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# ### Odds vs. Risk

In many clinical papers, probabilities are reported both as ‘risk ratios’ and ‘odds ratios’ to indicate similar things. However, they are not calculated similarly. In this blog, I’ll review the basic concepts behind these and examine how they differ with an example.

Let’s start with a simple example of one event. If you asked a group of people if they had had a drink last month, how many would say yes? This is straightforward and can range from 0 to 100% of the people or a probability of 0 to 1. Instead of speaking in terms of probability, we can also ask the question as ‘what is the risk of picking someone who has had a drink last month’ and that is the same as the probability and ranges from 0 to 1 (this is the example from Wikipedia article on Odds Ratio).

Now we ask ‘what are the odds of picking someone who drank last month’? That is defined as p/(1-p). That is if p is the probability of the event of interest and (1-p) is the probability that it doesn’t happen. So even from this simple case we see that the risk varies from 0 to 1 and the odds vary from 0 to infinity.

In fact, if Risk (or p) < 0.5, OR < 1; If Risk = 0.5, OR = 1; If Risk = 1, OR = Infinity. In fact, the relationship between Risk and OR is shown in the figure and they are similar (within the same order of magnitude) when p < 0.5, but quickly, as p increases greater than 0.5, OR expands that from ~O(1) to infinity. However, odds are often in the context of 2 possible events and their relative probability. For example, if we expand the example above and ask a group of men if they had a drink the last month p(D/M) and a group of women the same p(D/F), then how much more likely are we to pick a male drinker compared to a female drinker? So if for instance of 50% men and 50% of women have had a drink last month, then the relative risk of picking a male drinker over a female drinker is 1 (equally likely). And if p(D/M) = 0.75 and p(D/F)=0.25, then the relative risk of picking a male over a female drinker is 3.

 Probability of having had a drink in the past month Probability of NOT having had a drink in the past month If Male p(D/M) 1-p(D/M) If Female p(D/F) 1-p(D/F)

Relative Risk (RR) is simply calculated as p(D/M)/p(D/F). For a range of values of these probabilities, the ln(RR) is shown as radii of the circles (if ln(RR) < 0, it is not filled and no circles when ln(RR)=0). Ln(RR) = 0 along the diagonal when p(D/M)=p(D/F).  Now, to Odds Ratio. That is calculated as OR = [p(D/M)/(1-p(D/M))]/[p(D/F)/(1-p(D/F))]. And similarly we show the calculated and graphical ln(OR) for a range of probabilities.

As you can readily tell, the OR expands the range of the RR by several orders of magnitude while not really changing the understanding (similar to the single event example). In fact, while risk is readily understandable in terms of probability, OR is much less intuitive. To compare how different the RR and OR are, here are OR and RR values in the same chart. While OR and RR are both sensitive to changes in p(D/M)/p(D/F), OR is very sensitive when one or both of the ‘p’s are high (> 0.5). If I were writing a paper, I would stick to using RR and not use OR. RR seems to give a reasonably intuitive sense of what is the increased probability of p(D/M) over p(D/F), whereas OR doesn’t give any greater information when the events are less likely and exaggerates the truth, if the reader is interpreting the number based on probability or risk. (This is the case for a ‘raw result’. However, OR seems to be more useful in the context of regression models with other non-binary predictors).

Thanks to Deepak Maran for initiating the discussion at Vantage!