# How to Understand a Risk Ratio of Less than 1

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When a model has a binary outcome, one common effect size is a risk ratio. As a reminder, a risk ratio is simply a ratio of two probabilities. (The risk ratio is also called relative risk.)

Risk ratios are a bit trickier to interpret when they are less than one.

A predictor variable with a risk ratio of less than one is often labeled a “protective factor” (at least in Epidemiology). This can be confusing because in our typical understanding of those terms, it makes no sense that a risk be protective.

### So how can a RISK be protective?

Well, by indicating lower risk.

For example, let’s say you’re running a model where the outcome is Conviction of a Felony (yes/no) and among your predictors are Previous Criminal Activity (yes/no) and Graduation from High School (yes/no).

We would expect that a Yes on Previous Criminal Activity is related to an increase in the risk of committing a felony. Likewise, we would expect that a Yes on Graduation from High School is related to a decrease in the risk of committing a felony.

In other words, Previous Criminal Activity would be a risk factor and Graduation from High School would be a protective factor. Yet the effect of both factors would be measured with a risk ratio.

The risk ratio is always defined as the ratio of the comparison category’s probability to the reference category’s probability.

A risk ratio greater than one means the comparison category indicates increased risk.

A risk ratio less than one means the comparison category is protective (i.e., decreased risk).

Say we have the following data for a group of defendants:

 Felony Conviction Graduation Yes No Total No 300 100 400 Yes 225 175 400 Total 525 275 800

From this table, we can calculate the probability that either a graduate or a dropout is convicted of a felony.

P(Felony conviction|Dropout) = 300/400 = .75
P(Felony conviction|Graduate) = 225/400 = .5625

And from those, we can calculate the risk ratio for graduates compared to dropouts.

As you can see, the probability of a felony conviction is lower for graduates (.5625) than it is for dropouts (.75). Likewise, the risk ratio of felony convictions for graduates compared to dropouts is less than one (.75).

So one interpretation is that graduation is protective — it is associated with a lower risk of conviction.

How much lower? By a factor of .75, or 25% lower risk.

Now if we reversed this comparison, we could say that dropping out of high school increases risk and therefore is a risk factor. We would do this by swapping the comparison and recalculating the risk ratio:

RR  Dropouts/Graduates = .75/.56 = 1.33

Here we conclude that dropouts are 33% more likely than graduates to be convicted of a felony.

Some references will advise re-coding the data so that the relative risk is always greater than 1. However, it is important to take into consideration the message you want to deliver. In the example above, it may make sense to drive home the message that graduates are 25% less likely to be convicted.

If, after your initial analysis, you find the risk ratios counterintuitive, you can recode the reference group so that the interpretation makes sense. Binary, Ordinal, and Multinomial Logistic Regression for Categorical Outcomes
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