**Geek Box: Odds Ratios, Relative Risk, & Hazard Ratios*

**Geek Box: Odds Ratios, Relative Risk, & Hazard Ratios*

In epidemiology, the outcome of interest is typically a categorical event, for example ‘cerebrovascular stroke’ which falls under the umbrella of ‘cardiovascular disease’. In order to determine whether a particular exposure – for example fish intake – increases or decreases the risk of a particular categorical event occurring, statistical analysis is concerned with proportions, e.g., the proportion of events which occurred in a total sample of people.

How this risk is expressed may differ based on the particular needs of the analysis, for example whether the timing of events is relevant or whether the outcome is rare. Risk may be expressed as a hazard ratio, relative risk, or odds ratio. Often these are conflated, and it is assumed that each are communicating the same quantification of risk. However, each are different in subtle but important ways. For example, hazard ratios are used to express risk of an event occurring at/after a specific time point in a study’s follow-up period. Conversely, relative risk is concerned with the total number of events which occurred at the end of the study, and does not consider whether that risk was different at 5yrs or 10yrs.

So, how does odds differ? The main distinction is that odds is the probability of an event occurring divided by the probability of an event __not occurring__, i.e., events vs. non-events. Risk on the other hand divides the number of events in an exposure group by the total number of participants in the exposure group. For example, let’s say we have 100 participants in whom 30 had a stroke and 70 did not. The risk in this case would be 30/100 = 0.30; the odds would be 30/70 = 0.43. You can see these are different.

To calculate relative risk here, we would calculate the risk in an exposed group [as we just did] and divide this by the risk in the comparison group. The ‘relative risk’ is therefore the ratio of risk in Group A vs. ratio of risk in Group B. Conversely, the odds ratio is the odds of an event in Group A vs. odds in Group B. The exact same data set would yield different values for relative risk and odds ratios. Generally, when an association between an exposure and outcome is positive [>1.0] or negative [<1.0], the odds ratio will be higher or lower than the RR, respectively. So why use odds ratio over relative risk? Primarily, odds ratio is preferable where the outcome event is rare, i.e., there are a low number of events in the comparison groups. If this is the case, then the odds ratio and relative risk will be similar. However, if not, the odds ratio will be higher or lower than the relative risk value, thus potentially being interpreted as a greater risk.