Gehan Test

Definition

For the first sample, let $T_1,T_2,\dots ,T_m$ be i.i.d. survival time with a distribution $F_1$, and $C_1,C_2,\dots ,C_m$ be i.i.d. censoring time with a distribution $G_1$. We can observe $(X_i,{\delta }_i),i=1,2,\dots ,m$ where $X_i=\min (T_i,C_i)$ and ${\delta }_i=I(T_i\le C_i)$ is censoring indicator.

For the second sample, let $U_1,U_2,\dots ,U_m$ be i.i.d. survival time with a distribution $F_2$, and $D_1,D_2,\dots ,D_n$ be i.i.d. censoring time with a distribution $G_2$. We can observe $(Y_i,{ϵ}_i),i=1,2,\dots ,n$ where $Y_i=\min (U_i,D_i)$ and ${ϵ}_i=I(U_i\le D_i)$ is censoring indicator.

Gehan’s test is an extension of Signed-Rank Wilcoxon Test. The test statistic of Gehen’s test is defined as $U=\sum _{i=1}^m\sum _{j=1}^n{U}_{ij}$ where $U_{ij}=\{\begin{array}{ll}1& t_i>u_j:(x_i>y_j,{ϵ}_j=1)\text{or}(x_i=y_j,{\delta }_i=0,{ϵ}_j=1)\\ 0& \text{otherwise}\\ -1& t_i<u_j:(x_i<y_j,{\delta }_i=1)\text{or}(x_i=y_j,{\delta }_i=1,{ϵ}_j=0)\end{array}$.

Under the null hypothesis $H_0:F_1=F_2,G_1=G_2$, let $(z_i,{\zeta }_i),i=1,2,\dots ,n+m$ be the combined samples and define $U_{kl}$, $U_k^{\ast }$, and $U$ $U_{kl}=\{\begin{array}{ll}1& (z_k>z_l,{\zeta }_l=1)\text{or}(z_k=z_l,{\zeta }_k=0,{\zeta }_l=1)\\ 0& \text{otherwise}\\ -1& (z_k<z_l,{\zeta }_k=1)\text{or}(z_k=z_l,{\zeta }_k=1,{\zeta }_l=0)\end{array}$ $U_k^{\ast }=\sum _{l=1\ne k}^{m+n}{U}_{kl}$ $U=\sum _{k=1}^{m+n}{U}_k^{\ast }I(k\in I_1)$ where $I_1$ is the indicator set for sample 1.

Then, the variance of $U$ is calculated as $\mathrm{Var}(U)=\frac{mn}{(m+n)(m+n-1)}\sum _{k=1}^{m+n}(U_k^{\ast }{)}^2$ and $\frac{U}{\sqrt{\mathrm{Var}(U)}}\stackrel{D}{\to }N(0,1)$