In connection with a meta analysis I helped with recently we wanted to compare some relative risk results from another analysis with the odds ratios that came from ours. This made me wonder to what extent the two measurements are comparable. The short story is that they aren’t generally comparable, but when dealing with small probabilities they are quite similar. The long story follows below.

Suppose we are comparing the probability of falling ill between two groups. People
who belong to group one have a \(p_1\) probability of falling ill; people who belong
to group two have a \(p_2\) probability of falling ill. The *relative risk* between
the two groups is the ratio of the two probabilities \(RR = p_1/p_2\), while the
*odds ratio* is the ratio of the odds in the two groups. The odds of some event that
has probability \(p\) is \(p/(1-p),\) so the odds ratio is \(OR = \frac{p_1/(1-p_1)}{p_2/(1-p_2)} = \frac{p_1(1-p_2)}{p_2(1-p_1)}.\)

As usual my first instinct is to simulate some numbers and see what happens. Below I generate pairs of probabilities uniformly between 0 and 1 and plot the relationship between their ORs and RRs.

```
# generate probabilities
probs1 <- runif(100000)
probs2 <- runif(100000)
# calculate RR and OR
RR <- probs1/probs2
OR <- (probs1/(1-probs1))/(probs2/(1-probs2))
plot(OR, RR, xlim=c(0,15), ylim=c(0,15), pch=20, col="grey", bty="n")
abline(0,1, col="black", lwd=2)
abline(1,0, col="black", lwd=2)
```

The plot shows OR-RR pairs for \((p_1, p_2)\) pairs. There is a black line horizontally at RR=1, and a black line with zero intercept, unity slope. The general pattern here is fairly uninteresting. Focussing on the case where \(p_1 > p_2:\) For a given RR the OR is always greater. This is because \(OR = RR\cdot\frac{1-p_2}{1-p_1},\) where \(\frac{1-p_2}{1-p_1} \gt 1.\) it is the other way around when \(p_2 > p_1\). Other than this there isn’t much reason to believe that we can compare these quantities if we know nothing else about them. What if we know that at least one of the probabilities is quite small? Let’s say less than 5%?

```
one_small <- probs1 < .05 | probs2 < .05
plot(OR[one_small], RR[one_small], xlim=c(0,15), ylim=c(0,15), pch=20,
col="grey", bty="n", main="At least one probability < .05")
abline(0,1, col="black", lwd=2)
```

OK! This is pretty good, at least, say, below five. If we look at it the other way, and focus on the case where one probability is greater than 95% we are pretty much guaranteed not to have comparable RR and OR.

```
one_large <- probs1 > .95 | probs2 > .95
plot(OR[one_large], RR[one_large], xlim=c(0,15), ylim=c(0,15), pch=20,
col="grey", bty="n", main="At least one probability > .95")
abline(0,1, col="black", lwd=2)
```

So we should definitely be careful there. What if both probabilities are small though?

```
small <- probs1 < .05 & probs2 < .05
plot(OR[small], RR[small], xlim=c(0,15), ylim=c(0,15), pch=20,
col="grey", bty="n", main="Both probabilities < .05")
abline(0,1, col="black", lwd=2)
```

This is great! Dealing with rare events you can safely compare an odds ratio with a relative risk. Incidentally we were dealing with some very small death rates in our meta analysis and so a comparison should be possible. However we got an odds ratio of 6 and the other analysis reported a relative risk of somehting like 12, which as we saw is impossible if the base probabilities are the same.

We can rewrite the odds ratio once again to get \(OR = \frac{p_1 - p_1p_2}{p_2 - p_1p_2}\). Now if we let \(p_1\) and \(p_2\) approach zero, the term \(p_1p_2\) approaches zero much faster and so \(\frac{p_1 - p_1p_2}{p_2 - p_1p_2} \approx \frac{p_1}{p_2}=RR.\) If we let \(p_1 \to 1\) we get \(\frac{p_1 - p_1p_2}{p_2 - p_1p_2} \to \infty\). This explains both the low-probability figure and the high-probability figure.