# 8.3 - Risk, Relative Risk and Odds

## Risk Section

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.

## Relative Risk Section

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.”

## Example 8-1 Cont'd: Risk and Relative Risk Section

If we return to our Political Party and Opinion survey data, find the risk for either party favoring the tax bill and use these risks to find and interpret a relative risk. Also, find the odds of a democrat favoring the bill.

favor indifferent opposed total
democrat 138 83 64 285
republican 64 67 84 215
total 202 150 148 500

From the table, the risk of democrats favoring the bill: $$\dfrac{138}{285}=0.484$$

The risk of republicans favoring the bill: $$\dfrac{64}{215}=0.298$$

The relative risk that democrats favor the bill compared to republicans: $$\dfrac{0.484}{0.298}=1.62$$

We would interpret this relative risk as "Democrats are about 1.6 times more likely than Republicans to favor the bill (i.e.: Democrats are 60% more likely to support the bill than Republicans)."

The odds of a democrat favoring the tax bill is $$\frac{138}{147}$$ or $$138:147$$.

## Try it! Section

Consider again our previous example comparing gender and preferred condiments. The summary table is shown below for convenience.

Condiment
Gender   Ketchup Mustard Total
Male 15 23 38
Female 25 19 44
Total 40 42 82

Find the risk of either gender preferring ketchup and use those risks to find and interpret the relative risk.

The risk of males preferring ketchup is $$\frac{15}{38}=0.395$$.

The risk of females preferring ketchup is $$\frac{25}{44}=0.567$$.

The relative risk that females prefer ketchup compared to males is: $$\frac{0.567}{0.395}=1.435$$

We can interpret the relative risk as...

“Females are about 1.435 times more likely to prefer ketchup on hot dogs than males.”