Probability vs. Odds

The terms odds and probability are used interchangeably in everyday life.  However, in the setting of Biostats they are two different things. Generally speaking they both represent how likely something is, but they are calculated differently and used in different situations. Probability is essentially the same things as percentage. You are comparing the number of occurrences of a certain outcome to the number of total events measured. Probability ranges between zero and one. Odds is a ratio of the likelihood of an event happening compared to the likelihood of an event not happening. Odds can be zero or any positive number (not just values between 0 and 1).

So the probability of rolling a 4 on one attempt with one six faced die is 1/6. The odds of rolling a 4 are 1/5. Here is another example. If 13 people of a 60 person sample have lung cancer the probability of a person in that group having lung cancer is 13/60 and the odds of a person in that group having lung cancer is 13/ 47.

When we are talking about common events the difference between odds and probability is high. For example, flipping a coin one time gives you pretty different results. You have 1/1 odds of getting head and 1/2 probability of getting heads. However, as an event gets more and more rare the difference between odds and probability gets very small. Pretend there is a drawing with one winner and 10,000 people entered. The odds of winning are 1/9,999 (0.0001) and the probability of winning is 1/10,000 (0.0001). In this case, odds and probability are essentially identical.

Relative Risk (RR) & Odds Ratio (OR)

The difference between odds and probability is important because Relative Risk is calculated with probability and Odds Ratio is calculated with odds. Relative Risk (RR) is a ratio of probabilities or put another way it is one probability divided by another. Odds Ratio (OR) is a ratio or proportion of odds. I just remember that odds ratio is a ratio of odds and probability isn’t a ratio of odds (AKA it is the other option).

Relative Risk = Probability / Probability

Odds Ratio = Odds / Odds

Now that you have a general idea of what odds ratio and relative risk are you need to know when to use them. They don’t always just ask you to calculate one or the other. Sometimes questions on Step 1 also require you to figure out which type of calculation is needed based on the situation. In clinical trials and cohort studies we use relative risk to compare the incidence of health outcomes between groups of differing exposure or treatment. For case-control trials we use odds ratio to compare the “incidence” of past exposures or treatments.

                             Cohort Studies (and clinical trials) –> Relative Risk

Case-Control studies –> Odds Ratio 

I remember this by thinking about a group of pirates (group = cohort) all saying “aRRrrr!”. That reminds you that cohort studies use RR and the “other one” uses OR.

Now that we understand the research setting for each term we can redefine RR & OR. I should note that I think memorizing these definitions is unnecessary because if you understand the simpler definitions you should be able to create these based on the scenario presented in the question.

An RR or OR of 1 means there is no difference between the two groups being compared with respect to what you are measuring. In this case the treatment or risk factor being study has no effect on the rate of outcome development. Similarly, an OR or RR of 2 means whatever you are measuring is two times as likely to occur in the group being studied when compared with the control group. 0.5 means it is half as likely and so on. Later in the chapter we will cover how confidence intervals are applied to RR & OR.

ERRATA: At about the 3:00 mark the slide says “10,00” when it is really supposed to say “10,000.” I added a pop up box to fix it. Thanks to Mehdi Hedjazi for pointing this typo out in a youtube comment!

Now that you have finished this video you should check out the next video in the Biostats & Epidemiology section which covers Number Needed to Harm & Attributable Risk.