4.1 Comparing two risks

With risks \(p_1\) among the exposed and \(p_0\) among the unexposed, the three measures of Section 1.2 are \[ \RR = \frac {p_1}{p_0}, \qquad \RD = p_1 - p_0, \qquad \OR = \frac {p_1/(1 - p_1)}{p_0/(1 - p_0)} . \] No association means \(\RR = \OR = 1\) and \(\RD = 0\). The ratios lie in \((0, \infty )\), the difference in \((-1, 1)\).

The measures answer different questions. The ratios measure the strength of an association: how many times more frequent the disease is among the exposed. Strength is what bears on causation, since a large ratio is hard to produce by confounding alone. The difference measures the impact: how many cases per person are added by the exposure, which is what matters for deciding whether to remove it. Smoking raises the risk of lung cancer about tenfold and of heart disease about twofold, yet because heart disease is so much more common it causes more extra deaths from heart disease than from lung cancer. Section 5 develops this into the attributable fractions.

The odds ratio has two properties the risk ratio lacks. It is the only one of the three that a case-control study can estimate (Theorem 1.4.2). And it is symmetric in the outcome: the odds ratio for not getting the disease is exactly \(1/\OR \), while the risk ratio for not getting it, \((1 - p_1)/(1 - p_0)\), is not \(1/\RR \) (Problem 4.7.2). Its drawback is that it is not what most people mean by “how many times more likely”; reported for a common outcome as if it were a risk ratio, it exaggerates (Section 1.4).

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