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.
RR: Graduates/Dropouts =.5625/.75 = .75
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.
{ 2 comments… read them below or add one }
Great example given. Really appreciate this article. Helped me a lot!
I actually think of it in a different manner. First in my mind initially there are only factors which will then prove to be a risk or a protective depending on the association. In theory a factor could be a risk in one context but protective in another context. What I think of is the natural state of the subject. The process involved is what happens if I were to pour a factor into a subject that would alter their natural state. Hence people were not born to be smokers. Then the natural unexposed group is a non-smoker. What happens if I were to load a person with smoking (exposed group) and then examine the association relative to the natural state. In this case, we tend to see a risk or odds ratio greater than unity. In that case, I say that the factor poses a risk hence becomes a risk factor. On the other hand, people are not born to be educated. It is a conscious effort that societies have learnt (since the Industrial Revolution) to be desirable. Then the natural state is a non-educated person (unexposed) but the more education poured into that person (exposed) tends to yield a risk or odds ratio less than one. Then the conclusion is that adding education to the person or to populations tend to yield a protective result hence labelled a protective factor. It is very rare that across studies factors are risk in one context and protective in others. Across most studies, the relationship is generally invariant. One possible long time held such cross over was smoking — being a risk factor for most diseases but apparently a protective factor for dementia or Alzheimer’s disease. However this has been debunked in the WHO “Tobacco Knowledge Summary on Tobacco and Dementia” which argued that a meta analysis of studies funded by the Tobacco Industry yielded smoking as a protective factor whereas a meta analysis of non-Tobacco Industry funded studies yielded smoking as a risk factor.