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Relation to relative risk

Risk Ratio vs Odds Ratio

As explained in the "Motivating Example" section, the relative risk is usually better than the odds ratio for understanding the relation between risk and some variable such as radiation or a new drug. That section also explains that if the rare disease assumption holds, the odds ratio is a good approximation to relative risk and that it has some advantages over relative risk. When the rare disease assumption does not hold, the odds ratio can overestimate the relative risk.

If the absolute risk in the unexposed group is available, conversion between the two is calculated by:

$Relative\;Risk\approx {\frac {Odds\;Ratio}{1-R_{C}+(R_{C}\times Odds\;Ratio)}}$

where R_C is the absolute risk of the unexposed group.

If the rare disease assumption does not apply, the odds ratio may be very different from the relative risk and can be misleading.

Consider the death rate of men and women passengers when the Titanic sank. Of 462 women, 154 died and 308 survived. Of 851 men, 709 died and 142 survived. Clearly a man on the Titanic was more likely to die than a woman, but how much more likely? Since over half the passengers died, the rare disease assumption is strongly violated.

To compute the odds ratio, note that for women the the odds of dying were 1 to 2 (154/308=.5). For men, the odds were 5 to 1 (709/142\approx 4.99). The odds ratio is 9.99 (4.993/0.5). Men had ten times the odds of dying as women.

For women, the probability of death was 33% (=154/462). For men the probability was 83% (=709/851). The relative risk of death is 2.5 (=.83/.33). A man had 2.5 times a woman's probability of dying.

Which number correctly represents how much more dangerous it was to be a man on the Titanic? Relative risk has the advantage of being easier to understand and of better representing how people think.

A motivating example, in the context of the rare disease assumption

Suppose a radiation leak in a village of 1,000 people increased the incidence of a rare disease. The total number of people exposed to the radiation was $V_{E}=400,$ out of which $D_{E}=20$ developed the disease and $H_{E}=380$ stayed healthy. The total number of people not exposed was $V_{N}=600,$ out of which $D_{N}=6$ developed the disease and $H_{N}=594$ stayed healthy. We can organize this in a table:

${\begin{array}{|r|cc|}\hline &{\text{ Diseased }}&{\text{ Healthy }}\\\hline {\text{ Exposed }}&{20}&{380}\\{\text{ Not exposed }}&{6}&{594}\\\hline \end{array}}$

The risk of developing the disease given exposure is $D_{E}/V_{E}=20/400=.05$ and of developing the disease given non-exposure is $D_{N}/V_{N}=6/600=.01$. One obvious way to compare the risks is to use the ratio of the two, the relative risk (another way is to look at the absolute difference, $.05-.01=.04).$

$Relative\;Risk={\frac {D_{E}/V_{E}}{D_{N}/V_{N}}}={\frac {20/400}{6/600}}={\frac {.05}{.01}}=5\,$

The odds ratio is different. The odds of getting the disease if exposed is $D_{E}/H_{E}=20/380\approx .052,$ and the odds if not exposed is $D_{N}/H_{N}=6/594\approx .010\,.$ The odds ratio is the ratio of the two,

$Odds\;Ratio={\frac {D_{E}/H_{E}}{D_{N}/H_{N}}}={\frac {20/380}{6/594}}\approx {\frac {.052}{.010}}=5.2\,$.

As you can see, in a rare-disease case like this, the relative risk and the odds ratio are almost the same. By definition, rare disease implies that $N_{E}\approx H_{E}$ and $N_{N}\approx H_{N}$. Thus, the denominators in the relative risk and odds ratio are almost the same ($400\approx 380$ and $600\approx 594)$.

Relative risk is easier to understand than the odds ratio, so why use the odds ratio? One reason is that usually we do not have data on the entire population and must use random sampling. In our example, suppose it is very costly to interview villagers and find out if they were exposed to the radiation; we have no idea of the prevalence of radition exposure, the values of $V_{E}$ or $V_{N}$. We could take a random sample of fifty villagers, but quite possibly such a random sample would not include anybody with the disease, since only 2.6% of the population are diseased. Instead, we might use a case-control study in which we interview all 26 diseased villagers and a random sample of 26 who do not have the disease. The results might turn out as follows ("might", because this is a random sample):

${\begin{array}{|r|cc|}\hline &{\text{ Diseased }}&{\text{ Healthy }}\\\hline {\text{ Exposed }}&{20}&{10}\\{\text{ Not exposed }}&{6}&{16}\\\hline \end{array}}$

The odds in this sample of getting the disease given that someone is exposed is 20/10 and the odds given that he is not exposed is 6/16. The odds ratio is thus ${\frac {2}{6/16}}\approx 5.3$. The relative risk, however, cannot be calculated, because it is the ratio of the risks of getting the disease and we would need $V_{E}$ and $V_{N}$ to figure those out. Because we selected for people with the disease, half the people in our sample have the disease and we know that is more than the population-wide prevalance.

It is standard in the medical literature to calculate the odds ratio and then use the rare-disease assumption (which is usually reasonable) to claim that the relative risk is approximately equal to it. This not only allows for the use of case-control studies, but makes controlling for confounding variables such as weight or age using regression analysis easier and has the desirable properties discussed in other sections of this article of invariance and insensitivity to the type of sampling.