# Relative Risk and Odds Ratio

RELATIVE RISK AND ODDS RATIO

Risk and Odds just seemed the same to me for a long time. Since then, I have come to understand to important difference. Lets start with Relative Risk.

Relative Risk can be addressed by asking the following question: How many times more likely is an “exposed” group to develop a disease over a certain period of time as compared to a “non-exposed” group?

Here’s the key: Relative Risk looks to the future for the effect of a particular cause hence it is used in prospective studies say a cohort study.

Lets compare the above with Odds Ratio. The Odds Ratio can be addressed by asking te following question: How many times more likely is a diseased group to have been exposed to a risk factor as compared to a non-diseased group?

Here’s the key: Odds Ratio looks to the past for the cause of a particular effect hence it is used in retrospective studies such as a case-control study.

Lets go through some examples so we can get a better picture.

Using the table above as our 2×2 contingency table lets first consider the following case. A group of 70 individuals decide to begin a new therapeutic drug X, however the drug X has been known to cause cancer. They are compared to a control group of 60 individuals that takes a placebo instead. Question: What is the Relative Risk of developing cancer from Drug X compared to the control group? Here we need to consider whether we are looking at a case-control study or a cohort study. This is more of a cohort study, meaning the study is looking to the future to see if Drug X leads to cancer. Remember that a case-control study looks to the past.

TO CALCULATE THE RELATIVE RISK

Take the number of individuals who developed cancer (disease in the table) after having been exposed to the drug (40 or A in the table) and divide that number by the total of individuals exposed (70). Consider this value to be more of a percentage of the total exposed. We then divide this value (A/A+C) by the number of non-exposed who developed cancer over the total number of non-exposed (B/B+D). Therefore Relative Risk = the ratio (A/A+C)/(B/B+D).

To understand Odds Ratio now, lets go through another but similar example. A group of 60 individuals with cancer are being evaluated to see they were exposed to a particular toxin X. They are compared to a group of 70 individuals that do not have cancer and is similarly being evaluated for exposure to Toxin X. It is found that 40 of the 60 cancer individuals were indeed exposed and that 30 of the non-cancer individuals have also been exposed to Toxin X. Lets pause for a moment and realize that we are looking at a studying that is taking people who ALREADY have a disease and looking to the past to see if they were exposed to a Toxin thereby possibly drawing some association between the toxin and cancer.

TO CALCULATE THE ODDS RATIO

You could just memorize the shortcut AD/BC. For those of you who want to understand why this is the case read on. To calculate this lets first take the diseased group (with cancer) and compare the odds of having been exposed to not having been exposed. Not here that we are NOT dividing by the total amount in the group as we did in Relative Risk (ie. it is not a percentage of the total rather it is a comparison between two values in this case having been exposed and not having been exposed). In the table above it would be A/B. This new value is now divided by the odds of having been exposed versus not having been exposed in the non-diseased group (C/D). This comes out to (A/B)/(C/D). If we remember from basic math dividing two fractions by each other is the same as multiplying one fraction by the reciprocal of the other (A/B)*(D/C) and multiply across, which is now (AD/BC).

To recap:

Odds Ratio – Look to the past, Case-control study

Relative Risk – Look to the future, Cohort study