What does it mean when odds ratio and risk ratio aren't approximately equal ? $2 \times 2$ contingency table

Question

Consider this $2 \times 2$ contingency table

\begin{array}{|c|c|cc|} \hline \text{Treatment} & \text{Cured} & \text{Not Cured} \\ \hline \text{Tablet A}& 30&70 &\\ \hline \text{Tablet B}& 20& 100&\\ \hline \end{array}

We are looking at the relationship between Tablet A and Tablet B on patients with some specific disease.

The risk ratio is $$RR =\frac{30/100}{20/120} = 1.8$$

And the odds ratio is

$$OR = \frac{30 \times 100}{20 \times 70} = 2.14$$

in my notes for rare outcomes its usually $$OR \approx RR$$

But in this case 1.8 isn't approximately 2.14. What does this mean? How do they compare ?

Answer 1

Let's write the risk ratio and the odds ratio in comparable forms with the numerators and denominators each representing a row in your table:

$$RR = \dfrac{30/(30+70)}{20/(20+100)} =1.8\\ OR = \dfrac{30/70}{20/100}\approx 2.143 $$ so they are not the same, as you say.

If one of them had been $1$ then the other would also be $1$, as for example in $$RR = \dfrac{30/(30+120)}{20/(20+80)} =1\\ OR = \dfrac{30/120}{20/80}=1$$ though this is really saying that if the two probabilities are equal then the two odds are also equal and vice versa.

As your notes say, for rare outcomes the risk ratio and odds ratio are close. This is because small probabilities are close to the corresponding odds, i.e. $p \approx \frac{p}{1-p}$ when $1-p \approx 1$. For example

$$RR = \dfrac{3/(3+70)}{2/(2+100)} \approx 2.096\\ OR = \dfrac{3/70}{2/100}\approx 2.143$$

By contrast, for common outcomes they can be more different, for example (keeping the odds ratio the same)

$$RR = \dfrac{30/(30+7)}{20/(20+10)} \approx1.216\\ OR = \dfrac{30/7}{20/10}\approx 2.143$$

and your original example is somewhere between these last two examples of rare and common outcomes. As Jeremy Miles commented, your original example did not involve events which were rare enough to make the two ratios close enough for you.