Generalized Gehan Test

Definition

For the $i$-th sample, let $T_{i1},T_{i2},\dots ,T_{i{n}_i}$, where $i=1,2,\dots ,K$, be i.i.d. survival time with a distribution $F_i$, and $C_{i1},C_{i2},\dots ,C_{i{n}_i}$ be i.i.d. censoring time with a distribution $G_i$. We can observe $(X_{ij},{\delta }_{ij}),j=1,2,\dots ,n_i$ where $X_{ij}=\min (T_{ij},C_{ij})$ and ${\delta }_{ij}=I(T_{ij}\le C_{ij})$ is the censoring indicator.

The generalized Gehan’s test is an extension of Gehan Test used for more than two sample case. Under the null hypothesis $H_0:F_1=F_2=\cdots =F_K$, the test statistic of generalized Gehen’s test is defined as $W_i=\sum _{i^{\prime }=1\ne i}^K\sum _{j=1}^{n_i}\sum _{j^{\prime }=1}^{n_{i^{\prime }}}U((X_{ij},{\delta }_{ij}),(X_{i^{\prime }{j}^{\prime }},{\delta }_{i^{\prime }{j}^{\prime }}))$ where $U$ is the statistic of thet Gehan Test.

Then, $W=(W_1,W_2,\dots ,W_K)\to N_K(0,N^3{\Sigma }^{\ast })$ where $N=\sum _{i=1}^K{n}_i$, ${\hat{\Sigma }}^{\ast }=\frac{1}{N(N-1)}\sum _{i=1}^K\sum _{j=1}^{n_i}(W_{ij}^{\ast }{)}^2\left(\begin{array}{ccc}{n}_1(N-n_1)& & -n_i{n}_j\\ & \ddots & \\ & & n_K(N-n_K)\end{array}\right)$, and $w_{ij}^{\ast }=\sum _{i^{\prime }=1\ne i}^K\sum _{j^{\prime }=1\ne j}^{K}U((X_{ij},{\delta }_{ij}),(X_{i^{\prime }{j}^{\prime }},{\delta }_{i^{\prime }{j}^{\prime }}))$