In a case-control study, it is said that we can approximate relative risk(RR) by odds ratio(OR) when the disease is rare.

$OR$ = $\frac{\frac{p_1}{1-p_1}}{\frac{p_2}{1-p_2}}$ = $\frac{p_1}{p_2}$$\frac{1-p_2}{1-p_1}$ = $ RR $ $\frac{1-p_2}{1-p_1}$

But there is one thing that I do not understand. In a case-control study, it is impossible to approximate RR because the study is done in a retrospective way. Thus, the value $\frac{p_1}{p_2}$ cannot represent $RR$ even though the formula looks the same! Besides, if $\frac{p_1}{p_2}$ = $RR$, why should we use $OR$ to estimate $RR$? We can just use this value $\frac{p_1}{p_2}$ = $RR$ to estimate RR!


The rare event assumption means that the probability of being included in the sample as a healthy exposed individual is very close to the probability of being included in the sample as a healthy unexposed individual, both of which are very close to 1.

One strength of the odds ratio is that with outcome dependent sampling, it estimates the same quantity that is estimated in a prospective study. That means, where the rare outcome assumption is met, and the study is well designed, reasonable estimates to the RR can be obtained by conducting much smaller, and much cheaper studies.

Show this by constructing the two-by-two contingency table of sampling frequencies:

$$ \begin{array}{c|ccc} & E & \bar{E} & \\ \hline D & a & b \\ \bar{D} & c & d \\\end{array}$$

So the $p_1 = a/(a+c)$ (probability of disease exposed) and $p_2 = b/(b+d)$ (probability of disease unexposed). The arithmetic is easy but basically leads to $\frac{p_1/(1-p_1)}{p_2/(1-p_2)} = ad/(bc)$. Now in outcome dependent sampling, it is the exposure which is considered a random variable with unknown distribution, so defining a new quantity $q_1 = P(E|D)$ and $q_2 = P(E|\bar{D})$, you can see that $\frac{q_1/(1-q_1)}{q_2/(1-q_2)} = ad/(bc)$ again. It turns out that in univariate logistic regression, it matters not which variable is coded as exposure and which is coded as outcome, the same estimate is obtained with the same SE both ways, unlike linear regression.

There are some extensions to multivariate logistic regression/stratified categorical analysis, but I'll leave that as a cliffhanger for now.

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