Jul
10

## Reverse engineering contingency (2×2) table from Odds Ratio (OR)

Given the odds ratio (OR), we will calculate the individual cells in the contingency table (a,b,c,d).

In yellow, I’ve highlighted what is known.
a,b,c, and d are unknown and what we want to calculate.

Odds Ratio = (a/c) / (b/d)

Cases Controls Total
Exposed a b total_exposed
(a+b)
Unexposed c d total_unexposed
(c+d)
Total total_cases
(a+c)
total_controls
(b+d)
total_participants

If you’re getting the OR from a paper, the paper usually has total_exposed, total_unexposed, total_cases,and total_participants.

In that case, you can derive a, b, c, and d.

Solving for a:

Cases Controls Total
Exposed a total_exposed – a total_exposed (a+b)
Unexposed total_cases – a total_unexposed – total_cases + a total_unexposed (c+d)
Total total_cases (a+c) total_controls (b+d) total_participants

So now, the equation for OR can be written in terms of a and the known numbers :

OR = (a * d) / (b * c)
OR = (a * (total_unexposed – total_cases + a)) / ((total_exposed – a) * (total_cases – a))

If you have the values for OR, total_exposed, total_unexposed, total_cases, and total_controls, you can solve for a</i> using the quadratic formula.

Once you solve for a, solving for b, c, and d is trivial.

## Try it out!

Deriving cells of 2×2 Contingency Table from Odds Ratio:

Enter values in yellow cells

 Condition Odds Ratio Absent Present Totals Group 1 Group 2 Totals

I came across this problem when reading an Alzheimer’s paper.

Looking at ApoE ε4 carriers (n=452), smokers have an OR of 1.97 for dementia compared to non-smokers.

Because this was a population study, I wanted to know how many smokers got dementia, and how many non-smokers got dementia. If I got the individual cells, I could calculate this.

Out of the 452 ApoE ε4 carriers, 207 were smokers (45.8% of 452) and 31 had dementia (6.9% of 452).

From this,

• OR = 1.97
• total_exposed = 207
• total_unexposed = 245
• total cases (those with dementia) = 31
• total controls (without dementia) = 421

I plugged in the above calculator to get:

Dementia Non-Dementia Total
Smoking 19 188 207
Non-smoking 12 233 245
Total 31 421 452

In this population-based study, 9% (19/207) of the smokers had dementia while 5% (12/245) of the nonsmokers had dementia.