# Difference Between Odds Ratio and Relative Risk

**Odds Ratio Vs Relative Risk**

When two groups are under study or observation, you can use two measures to describe the comparative likelihood of an event happening. These two measures are the odds ratio and relative risk. Both are two different statistical concepts, although so much related to each other.

Relative risk (RR) is simply the probability or relationship of two events. Let’s say A is event 1 and B is event 2. One can get the RR by dividing B from A or A/B. This is exactly how experts come up with popular lines like ‘Habitual alcoholic beverage drinkers are 2-4 times more at risk of developing liver problems than non-alcoholic beverage drinkers!’ This means that the likelihood of variable A which is the risk of developing liver disease for habitual alcoholic beverage drinkers is relative to the same exact risk being talked about for variable B which includes the non-alcoholic beverage drinkers. In this regard, if you belong to group B and that you are only 10% at risk for dying then it must be true that those from group A are 20-40% more at risk of dying.

The other measure ‘“ odds ratio (OR) is a term that already speaks of what it describes. Instead of using pure percentages (like in RR), OR uses ratio of odds. Take note, OR explains ‘odds’ not in its colloquial definition (i.e. chance) but rather on its statistical definition which is the probability of an event over (divided by) the probability of a certain event not happening.

A good example is the tossing of a coin. When you happen to land the coin with its tails up 60% of the time (obviously it lands with heads 40% of the time), the odds of tails in your case is 60/40=1.5 (1.5 times more likely to get tails than heads). But ordinarily, there’s really a 50 percent chance of landing on either heads or tails. So the odds are 50/50=1. So the question is on how likely this event will not happen compared to it happening. The straightforward answer is that you are just equally likely to get either way. In written formula, with A being the likelihood for group 1 while B being the likelihood for group 2, the formula to get the OR is [A/(1-A)]/[B/(1-B)].

So if the probability of having liver disease among habitual alcoholic beverage drinkers is 20% and among non-alcoholic beverage drinkers is 2% the OR will be = [20%/(1-20%)] / [2%/(2-1%/)]=12.25 and the RR of having liver disease when drinking alcoholic beverages will be = 20%/2%=10.

The RR and OR often have close results, but in some other situations they have very far numerical values most especially if the risk of occurrence is really very high to begin with. This scenario gives a high OR while the RR is kept at a minimum.

1. The RR is much simpler to interpret and is most likely consistent with everyone’s intuition. It is the risk of a situation relative (in relation) to exposure. The formula is A/B.

2. OR is a bit more complicated and uses the formula [A/(1-A)]/[B/(1-B)].

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It appears that in the example of relative risk, there is a requirement that the A being counted is a subset of B.

For example, if there are 100 habitual drinkers in the world,

and 50 people with liver problems in the world,

clearly not all habitual drinkers have liver problems;

and, in fact, because we know the RR is 2-4, not all people with liver problems are habitual drinkers.

So, A can only be considered when it is a subset of B. If there 37 habitual drinkers WITH liver problems, then 37 would be our event 1 occurrence value, as variable A, not 100. In other words, A as “event 1” and B as “event 2” requires event 1 and 2 not be exclusive. Or to put it yet another way: B is event 2, and A is event 2 sub A.

Please correct if this is wrong or validate if it is correct, so as to be of aide to others.