Odds Ratio Does What Risk Ratio Fails to Do — an Intuitive Example
By: Atul Sharma
Tags: Machine Learning, Artificial Intelligence, Mathematics, Data Science, Data

Odds Ratio Does What Risk Ratio Fails to Do — an Intuitive Example

Understand the concept from scratch

Photo by Sydney Sims on Unsplash

Let’s begin by dropping the ultimate takeaway of this blog:

“The odds ratio is a consistent measure for both the Population & Sample statistics (Case Control studies with significant effect) where the Risk ratio shows inconsistency”

Now the question that comes to our mind is — Why is that?

I will prove this using a relatable example so that we can build an intuitive sense around it as well. But before jumping to the example, let’s see how Risk & Odds behave:

(Image by author)

Notice how Odds reach closer & closer in terms of difference to the risk measure as it gets lower. It is because of this reason Odds & Risk are interpreted similarly in the cases where the event rate(Positive here) is relatively low. In cases where the event rate is relatively high(above 0.3 or 30%), there is a significant difference in both measures.

But the point we are trying to understand here is why the odds ratio stays consistent in Case-control studies, whereas the risk ratio becomes inconsistent. Time to explain this with the help of an example — The scenario we are considering is related to the sensitive issue of depression and its dominance in Corporates:

(Image by author)

The visual you see above is the representation of the population; however, for control case studies, only samples are taken from both the affected population as well as the unaffected population. Let’s first calculate the Risk(Corporate), Risk(Non-Corporate), Odds(Corporate), Odds(Non-Corporate), Risk Ratio, and Odds ratio taking approximate figures of the population just for the sake of understanding the concepts:

(Image by author)

Risk(Corporate) = 0.8/2 = 0.40

Risk(Non-Corporate) = 2/6 = 0.33

Odds(Corporate) = 0.8/1.2 = 0.67

Odds(Non-Corporate) = 2/4 = 0.50

Risk Ratio = 0.40/0.33 = 1.21

Odds ratio = 0.67/0.50 = 1.34

You will agree that the Risk ratio measure(also known as ‘Relative risk’) is more intuitive than the Odds ratio because it conveys that a person working in corporate has 1.21 times more risk of having depression than a non-corporate one. Making intuitive sense out of the Odds ratio is not easy, but it is a consistent measure when it comes to Case-control studies(sample statistics). Let’s see how:

(Image by author)

From both the Depression(X+Z) & Depression-free population(Y+N), 1000 samples each are taken to conduct the Case-control study. The values look like this:

(Image by author)

Now we shall observe the inconsistency in the Risk ratio that we have been talking about since the start of this blog, and the sole reason for that is the under-representation of the Depression-free population in the Case-control study.

Risk(Corporate) = 285/515 = 0.55

Risk(Non-Corporate) = 715/1485 = 0.48

Odds(Corporate) = 285/230 = 1.23

Odds(Non-Corporate) = 715/770 = 0.92

Risk Ratio = 0.55/0.48 = 1.14

Odds ratio = 1.23/0.92 = 1.34

Appreciate how the Risk Ratio has changed in the Case-control study, but the Odds ratio remained the same(consistent). To find out why the Odds ratio stays consistent, let us look at its formula:

Odds ratio = (x/y)/(z/n)

A little rearrangement in the formula totally takes care of the issue of under-representation:

Odds ratio = (x/z)/(y/n)

Now the numerator reflects the split of values (Corporate & Non-corporate) in the Depression set, and the denominator reflects the split of values (Corporate & Non-corporate) in the Depression-free set, thereby computing the same value in the Case-control study as in the whole Population study.

It is because of this reason. The Odds ratio measure is preferred over the Risk ratio to make an estimate with confidence intervals in Case-control studies.

Conclusion:

When the event rate is relatively low, and there is not much significant effect between categories, only then the Risk ratio & the odds ratio behave in a similar manner. Moreover, Risk involves the process of a forward-looking scenario, i.e., After a subject is exposed to something, the likeliness of him/her developing the outcome is computed, whereas, for the odds measure, it is the opposite, i.e., Subject who has already developed an outcome, the odds of him/her being exposed to something in the past is the area of interest. I hope I am able to establish the difference/distinction between the Risk ratio and the Odds ratio with a relevant example. Make sure to visit my profile for similar intuitive and simplified blogs. Many more to come….

Thanks!!!