8.1 Categorical covariates
A factor with \(K\) levels enters the model as \(K - 1\) indicator (dummy) variables, each comparing one level with a reference level. In R the reference is the first level unless changed with relevel. The coefficients are then log hazard ratios relative to the reference, and the question whether the factor matters at all is answered by a test of all \(K - 1\) coefficients together: a likelihood ratio test comparing the models with and without the factor, on \(K - 1\) degrees of freedom.
Example 8.1.1. With treatment, cell type and Karnofsky score, and squamous cell as the reference, the hazard ratios are \(2.28\) for small-cell (\(95\%\) interval \(1.35\) to \(3.87\)), \(3.17\) for adenocarcinoma (\(1.78\) to \(5.65\)) and \(1.48\) for large-cell (\(0.85\) to \(2.58\)) tumours. Removing cell type lowers the log partial likelihood from \(-474.9\) to \(-484.0\), a likelihood ratio statistic of \(18.1\) on \(3\) degrees of freedom, \(p = 0.0004\): cell type matters, although the large-cell group alone does not differ significantly from squamous. Each Karnofsky point lowers the hazard by a factor \(0.969\), so \(10\) points by \(0.969^{10} = 0.73\). The treatment hazard ratio, \(1.30\) (\(0.88\) to \(1.93\)), shows no benefit of the test regimen (Software 8.6.1).
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