6.1 The test
The null hypothesis is that the two groups have the same survival function, \(H_0: S_1(t) = S_2(t)\) for all \(t\), equivalently the same hazard. At each distinct event time \(t_{(j)}\), arrange the data in a \(2 \times 2\) table:
| events | survivors | at risk | |
| group \(1\) | \(d_{1j}\) | \(n_{1j} - d_{1j}\) | \(n_{1j}\) |
| group \(2\) | \(d_{2j}\) | \(n_{2j} - d_{2j}\) | \(n_{2j}\) |
| total | \(d_j\) | \(n_j - d_j\) | \(n_j\) |
Under \(H_0\), given the margins, the \(d_j\) events fall at random among the \(n_j\) at risk, so \(d_{1j}\) has a hypergeometric distribution with \[ E_{1j} = d_j\frac {n_{1j}}{n_j}, \qquad V_j = d_j\frac {n_{1j}}{n_j}\Bigl (1 - \frac {n_{1j}}{n_j}\Bigr )\frac {n_j - d_j}{n_j - 1} . \] Summing over the event times, the observed events in group \(1\), \(O_1 = \sum d_{1j}\), are compared with the expected \(E_1 = \sum E_{1j}\): \[ \chi ^2_{\text {LR}} = \frac {(O_1 - E_1)^2}{V}, \qquad V = \sum _jV_j , \] which is approximately \(\chi ^2\) with one degree of freedom under \(H_0\). The tables at different times are not independent, but the differences \(d_{1j} - E_{1j}\) are uncorrelated (they form a martingale), which justifies adding the variances.
The test uses only the ranks of the times, which groups the events fall in and who is at risk, not the times themselves, hence its name. It is most powerful when the hazards are proportional, \(h_2(t) = ch_1(t)\), the alternative it is designed for. A useful summary of the size of the difference is the ratio \[ \widehat {\text {HR}} = \frac {O_2/E_2}{O_1/E_1} , \] an estimate of the hazard ratio of group \(2\) to group \(1\); the Cox model of the next section estimates it properly, with a confidence interval.
Example 6.1.1. The ovarian trial compared two treatments in \(26\) women, \(13\) in each arm; \(12\) died. Working through the \(12\) death times, group \(1\) had \(7\) deaths against \(5.23\) expected, with \(V = 2.94\), so \(\chi ^2_{\text {LR}} = (7 - 5.23)^2/2.94 = 1.06\), \(p = 0.30\) (Software 6.4.1). The difference between the treatments is well within chance: with so few deaths, only a large effect could be detected.
Example 6.1.2. For men and women in lung (Figure 4), \(\chi ^2_{\text {LR}} = 10.3\), \(p = 0.0013\): women’s longer survival is unlikely to be chance. The ratio of observed to expected gives an estimated hazard ratio of \(0.59\) for women relative to men: at any time, a woman’s risk of death was about \(40\%\) lower.
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