Summarising dichotomous data

In studies of treatment interventions we aim to describe what is happening to the participants we are studying so we can predict what is likely to happen to others. In order to do this we take an observation about a particular outcome for each participant. If that outcome is dichotomous, then each participant can be in one of two states. For example if we have a new drug thought to save lives in high risk patients, we might test it first in a group of people representative of these high risk patients and observe the number dead or alive at the end of the intervention. There are a couple of ways of summarising the information we get about the whole of our observed sample in a form that can be applied to others.

Risk

The word risk is fundamental to epidemiology and evidence-based health. Risk is the chance, or probability, of having a specific event. As it relates to clinical trials and systematic reviews, risk is not always of a bad event, we can talk about the 'risk' of a good outcome (such as cure) as well as the 'risk' of a bad outcome (such as death). We can use the word 'risk' to describe the chance of the outcome whether it's good or bad. Given a single group of people, and knowledge of how many have 'the event', we can express the risk of the event by dividing the number with the event by the number of people. For example, of 133 women taking an antibiotic for the treatment of urinary tract infection (UTI), 14 had the event 'still infected' after 6 weeks. The risk of remaining infected was 14/133 = approximately 0.1.

Odds

An alternative measure of describing how likely an event is to happen is called odds. The odds of an event is the ratio of events to non-events. Equivalently (and more formally) it's the risk of having an event divided by the risk of not having it. If we look at the 133 women taking the antibiotic for UTI, the ratio of events (still infected) to non-events (cured) is 14/119 = approximately 0.1. The more formal formula gives 14/133 (risk of having the event) divided by 119/133 (risk of not having the event), which also works out as 14/119 = approximately 0.1.

In this example the risk and odds are both similar (approximately 0.1), so why bother to have two alternatives? In this example the 133 women taking antibiotics were the treated group in a clinical trial. In this trial there was also a placebo group with another 148 women. Of the 148 receiving placebos, 128 still had a UTI after 6 weeks. So in this group, what's the risk of staying infected? It's 128 (number with the event of 'still infected') /148 (total number in the group) = 0.86. What are the odds? They are 128 (number still infected) /20 (number cured) = 6.4. So in this case the risk and odds are very different.

In fact odds and risk are never identical, but they can be similar. They are similar when they are both small - i.e. when an event is rare. Taking antibiotics, the women rarely stayed infected (the event was rare), so the risk and the odds were similar. Not taking them, they mostly stayed infected (the event was common) so the odds and risk were different.

As values of odds and risks can differ for the same data, it is important to be careful and precise when using statistical summaries of dichotomous outcomes.