6.2 Several groups, trends and strata
With \(k\) groups the vector of \(O_g - E_g\) for \(k - 1\) of them, with its hypergeometric covariance matrix \(\mathbf V\), gives \(\chi ^2 = (\mathbf O - \mathbf E)^{\mathsf T}\mathbf V^{-1}(\mathbf O - \mathbf E)\) on \(k - 1\) degrees of freedom. When the groups are ordered, such as the stages of a disease or levels of a performance score, a test for trend with scores \(w_g\) uses \(U = \sum _gw_g(O_g - E_g)\) and \(\chi ^2_{\text {trend}} = U^2/\mathbf w^{\mathsf T}\mathbf V\mathbf w\) on one degree of freedom, which is more powerful than the general test when the effect is monotone.
Example 6.2.1. In lung, patients with ECOG performance scores \(0\), \(1\) and \(2\) (fully active, restricted, ambulatory but unable to work) had \(37\), \(82\) and \(44\) deaths against \(53.9\), \(83.1\) and \(26.0\) expected. The test of equality gives \(\chi ^2_2 = 18.0\), \(p = 0.0001\), and the trend test \(\chi ^2_1 = 16.4\), \(p = 0.00005\): risk rises steadily with the score, and almost all of the difference is the trend (Software 6.4.3).
A stratified log-rank test compares the groups within strata, for example within cell types or centres, and adds the stratum totals: \(\chi ^2 = \bigl (\sum _s(O_s - E_s)\bigr )^2/\sum _sV_s\). It removes the effect of a factor that changes the hazard but is not of interest, as long as the treatment effect is similar across strata. In the veteran trial, the two chemotherapy regimens give \(\chi ^2 = 0.008\) unstratified and \(0.70\) stratified by tumour cell type: no evidence of a difference either way, the stratification having removed the large differences between cell types from the comparison.
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