Risk, Odds and Relative Effects

Definition The risk of an outcome in a group is the probability that a member of the group has the outcome during a specified period, such as one year of follow-up. When $n$ people are each followed for the whole period and $x$ of them have the outcome, the sample proportion $\hat{p} = x/n$ estimates the risk in the population those people represent. In a comparison of two groups, the baseline risk is the risk in the reference group, the one the other group is compared with, often an unexposed, untreated or usual-care group. Risks are comparable only when the outcome definition and the period are the same in both groups.

Definition For a risk $p_A$ in group A and a risk $p_B$ in a reference group B, the risk difference is

estimated by $\widehat{\text{RD}} = \hat{p}_A - \hat{p}_B$. It lies between $-1$ and $1$, it equals $0$ exactly when the two risks are equal, and it is usually reported in percentage points. It is an absolute effect measure. Multiplied by 1000, it is the difference between the expected numbers of outcomes among 1000 people with risk $p_A$ and among 1000 people with risk $p_B$. The word effect in that label names a comparison. It does not by itself mean that belonging to group A causes the difference.

Definition For a reference risk $p_B > 0$, the risk ratio of group A relative to group B is

estimated by $\widehat{\text{RR}} = \hat{p}_A / \hat{p}_B$ when $\hat{p}_B > 0$. It is a relative effect measure. It is never negative, it equals $1$ exactly when the two risks are equal, and the relative change from the baseline is $\text{RR} - 1$. When $p_A > 0$, reversing the reference group replaces the ratio by $1/\text{RR}$. A known baseline turns a ratio back into risks, since $p_A = \text{RR} \cdot p_B$ and

Because a risk cannot exceed 1, a ratio $\text{RR}$ can hold only at a baseline with $\text{RR} \cdot p_B \le 1$.

Definition For an outcome with probability $p < 1$, the odds of the outcome are

which can be any nonnegative number, and conversely $p = o/(1 + o)$. For risks $p_A$ and $p_B$ strictly between $0$ and $1$, the odds ratio of group A relative to group B is

In a table with counts $a$ and $b$ in row A and $c$ and $d$ in row B, arranged as above, the estimate is $\widehat{\text{OR}} = (a/b)/(c/d) = ad/(bc)$ when all four counts are positive. The two ratios are linked by

So if $p_A > p_B$ then $\text{OR} > \text{RR} > 1$, if $p_A < p_B$ then $\text{OR} < \text{RR} < 1$, and both equal $1$ when the risks are equal. The odds ratio is close to the risk ratio when both risks are small, and the two can be far apart for a common outcome.

Definition A case-control study selects people who have the outcome, the cases, and people who do not, the controls, and then compares their earlier exposure or group membership. The investigator decides how many cases and controls to include, so the fraction of cases in the sample is not an estimate of a population risk. Write $a$, $b$, $c$ and $d$ for the counts in the source population, arranged as in the table above, and suppose each case is selected with probability $s_1$ and each person without the outcome with probability $s_0$, with selection unrelated to exposure among the people with the outcome and among those without it. The expected sampled counts are then $s_1 a$, $s_0 b$, $s_1 c$ and $s_0 d$, and

So the cross-product of the sample estimates the odds ratio of the source population. The same cross-product is the odds of exposure among cases divided by the odds of exposure among controls, which is what the sample measures directly. Risks, risk differences and risk ratios cannot be recovered from the case-control table alone. They need outside information, such as the sampling fractions or the baseline risk in the source population.

Definition A log-scale interval for a ratio is a confidence interval computed for the natural logarithm of the ratio and then mapped back with the exponential function. For independent binomial samples with $a$ outcomes among $n_A$ people and $c$ outcomes among $n_B$ people, where $a > 0$ and $c > 0$, a common large-sample standard error is

and the approximate 95 percent interval for the risk ratio is

For the odds ratio, with all four counts positive, the standard error of $\log \widehat{\text{OR}}$ is approximately $\sqrt{1/a + 1/b + 1/c + 1/d}$, and the interval is built the same way. This odds ratio formula also applies to a case-control sample, with the cases and the controls treated as two independent samples. A zero outcome count, or for the odds ratio a zero in any cell, makes the formula undefined, and the estimate itself can then be $0$ or infinite. An exact, score-based or model-based method suited to the design should then be chosen and reported, rather than a constant added to the cells without comment.