# 6.3 - Risk, Relative Risk and Odds

6.3 - Risk, Relative Risk and Odds

In this section, we will introduce some other measures we can find using a contingency table. One of the most straightforward measures to find is the risk of any given event.

Risk
The probability that an event will occur.

In simple terms, a risk for a group is the same as the proportion of "success" for a particular group.

Have you ever heard a doctor tell you or a family member something similar to the following: "If you do not lose weight or get your cholesterol under control you are about five times more likely to suffer a heart attack than if you had these numbers in the normal range." If so, how alarmed should one be? "Five times" sounds alarming!

First off, this "five times" represents what is called relative risk.

Relative Risk
Relative risk is a ratio of the risks of two groups.

In the example described above, it would be the risk of heart attack for a person in their current condition compared to the risk of heart attack if that person were in the normal ranges. However, to truly interpret the severity of a relative risk we have to know the baseline risk.

Baseline Risk
The baseline risk is the denominator of relative risk, i.e., the risk of the group being compared to.

In our example, this would be the risk of heart attack for the normal range. If this baseline risk is high, then a relative risk of 5 would be alarming; if the baseline risk is small, then a relative risk of 5 may not be too serious.

For instance, if the risk of a heart attack for someone in the normal range was 1 out of 10, then the risk of a heart attack for a person with the above average numbers would be five times this or 5 out of 10. That is, the person would have roughly a 50/50 chance of suffering a heart attack if they didn't get their weight and cholesterol in check. However, if the risk of a heart attack for the normal range group was 1 out of 500, then the risk of a heart attack for a person with above average numbers would be 5 out of 500 or 0.01. The person would have about a 1% chance of a heart attack if they didn't improve their health. In both cases the relative risk was 5, but with entirely different levels of impact. Please note this example is not meant to be interpreted that taking care of your health is not important!!!

Another measure we can find is odds.

Odds
Odds is a ratio of the number of “success” over the number of “failures.” It can be reported as a fraction or as “number of success: number of failures.” When we reflect back on Donna's contingency table, the odds are calculated using the interior cells of the table.
Low Entrepreneurialism High Entrepreneurialism All
Northeast 300 460 760
377.9 382.1
Midwest 249 95 344
171.1 172.9
All 549 555 1104

Let's say we want to calculate the "odds" that the Northeast is high entrepreneurialism. We would simply take the number of observations of high entrepreneurialism divided by the number of low entrepreneurialism for the Northeast. In this example that would be 460/300 or an odds of 1.5. An observation from the Northeast is 1.5 times more likely to be high entrepreneurialism than low.

Let's try this for the Midwest where we end up with an odds of 95/249 or .38. This brings up a really important point about odds. An odds of 1 are "equal odds" or 50/50. Any odds more than 1 means the numerator category (in our example high entrepreneurialism) is more likely, and any odds less than 1 means the numerator category is less likely. For the Midwest being high entrepreneurialism is actually less likely than being low entrepreneurialism. This gets hard to interpret so typically what we do is take the inverse (249/95) and say that an observation from the Midwest is 2.6 times more likely to be low entrepreneurialism.

We can also calculate an "odds ratio" which is the ratio of two odds if we want to compare two groups. In Donna's example, we could calculate the odds of high entrepreneurialism in the Northeast compared to the odds of high entrepreneurialism in the Midwest. We need to make sure we are dealing with both odds the same format (so both with high entrepreneurialism in the numerator). Now we simply take the odds of the two odds!

(460/300)/(95/249) or an odds ratio of 3.94. Now we can say that the odds of an observation from the Northwest being high entrepreneurialism are 3.94 times more likely than an observation from the Midwest!

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