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Log-Rank and Wilcoxon Algorithm

These tests are for comparison of two or more strata of survival data.

Suppose that we have data from stratum 0 and stratum 1. Denote the ordered observed failure times by $t_0<t_1<t_2<...<t_k$. At time $t_i$, let $w(t_i)$ be a positive weight function, and $n_i{}_j$ and $d_i{}_j$ are the size of the risk set and the number of events in the jth stratum, respectively. Let $n_i = n_i{}_0+n_i{}_1$, $d_i = d_i{}_0+d_i{}_1$.

1. Log-Rank and Wilcoxon statistic :

$E = \sum w(t_i) (d_i{}_0 - \frac{n_i{}_0 d_i}{n_i})$
  • $w(t_i)$ is 1.0 for log-rank test
  • $w(t_i)$ is $n_i$ for wilcoxon test

2. covariance for Log-Rank and Wilcoxon statistic :

$V = \sum w(t_i) (d_i \frac{n_i{}_0 n_i{}_1 (n_i-d_i)}{n_i^2 (n_i-1)})$
  • $w(t_i)$ is 1.0 for log-rank test
  • $w(t_i)$ is $n_i$ for wilcoxon test

3. Computation of Chi-Square and prob-Chi-Square :

  • Chi-Square = $\frac{E^2}{V}$
  • prob-Chi-Square comes from function of pchisq(Chi-Square, 1, false, false)
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Topic revision: r5 - 13 Sep 2005, ColeBeck
 

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