Determining cutoff values of prognostic factors in survival data with competing risks

Abstract

In clinical studies, patients are often classified into high or low risk groups based on prognostic factors related to survival outcomes. Using maximally selected linear rank statistics, several methods have been developed to determine a cutoff value of the prognostic factor. We propose an extension of these methods for the circumstances that competing risks are encountered in conjunction with an event outcome of interest. A simulation study is carried out to demonstrate the performance of the proposed method using some commonly used measures such as bias, precision, and power. We also apply our method to two real datasets involving lung cancer and hepatocellular carcinoma, illustrating optimal determinations of cutoff values for binary decisions on prognosis.

Appendix

For a fixed value of $\mu ,$ the equivalence of $\tilde{U}_\mu$ and $v_\mu$ can be obtained straightforwardly as follows:

$$\begin{aligned} v_\mu= & {} \sum _{l=1}^{d_1}\left\{ 1- \xi _{l1}^{(1)\mu }\frac{d_1}{\tilde{r}_1^\mu }\right\} \varPhi _l^{(1)\mu }-\sum _{j=1}^{m_1} \xi _{1j1}^{\mu }\frac{d_1}{\tilde{r}_1^\mu }\varPhi _{1j}^\mu \\&\quad + \,\sum _{l=1}^{d_2}\left\{ 1- \xi _{l1}^{(2)\mu }\frac{d_1}{\tilde{r}_1^\mu }-\xi _{l2}^{(2)\mu }\frac{d_2}{\tilde{r}_2^\mu }\right\} \varPhi _l^{(2)\mu }-\sum _{j=1}^{m_2} \left\{ \xi _{2j1}^{\mu }\frac{d_1}{\tilde{r}_1^\mu }+\xi _{2j2}^{\mu }\frac{d_2}{\tilde{r}_2^\mu }\right\} \varPhi _{2j}^\mu \\&\quad + \, \cdots +\sum _{l=1}^{d_k}\left\{ 1- \xi _{l1}^{(k)\mu }\frac{d_1}{\tilde{r}_1^\mu }-\cdots -\xi _{lk}^{(k)\mu }\frac{d_k}{\tilde{r}_k^\mu }\right\} \varPhi _l^{(k)\mu } \\&\quad - \, \sum _{j=1}^{m_k} \left\{ \xi _{kj1}^{\mu }\frac{d_1}{\tilde{r}_1^\mu }+\cdots +\xi _{kjk}^{\mu }\frac{d_k}{\tilde{r}_k^\mu }\right\} \varPhi _{kj}^\mu \\= & {} \left\{ \sum _{l=1}^{d_1}\varPhi _l^{(1)\mu } +\sum _{l=1}^{d_2}\varPhi _l^{(2)\mu }+\cdots +\sum _{l=1}^{d_k}\varPhi _l^{(k)\mu }\right\} \\&\quad - \, d_1\frac{w_{11}^\mu }{\tilde{r}_1^\mu }\left\{ \sum _{l=1}^{d_1}\varPhi _l^{(1)\mu } +\sum _{l=1}^{d_2}\varPhi _l^{(2)\mu }+\cdots +\sum _{l=1}^{d_k}\varPhi _l^{(k)\mu } \right. \\&\quad \left. + \, \sum _{j=1}^{m_1}\varPhi _{1j}^\mu +\sum _{j=1}^{m_2}\varPhi _{2j}^\mu +\cdots +\sum _{j=1}^{m_k}\varPhi _{kj}^\mu \right\} \\&\quad - \, d_2\frac{w_{12}^\mu }{\tilde{r}_2^\mu }\left\{ \sum _{l=1}^{d_2}\varPhi _l^{(2)\mu }+\cdots +\sum _{l=1}^{d_k}\varPhi _l^{(k)\mu }+\sum _{j=1}^{m_2}\varPhi _{2j}^\mu +\cdots +\sum _{j=1}^{m_k}\varPhi _{kj}^\mu \right\} \\&\quad - \,\cdots -d_k\frac{w_{1k}^\mu }{\tilde{r}_k^\mu }\left\{ \sum _{l=1}^{d_k}\varPhi _l^{(k)\mu } +\sum _{j=1}^{m_k}\varPhi _{kj}^\mu \right\} \\= & {} \left\{ \sum _{l=1}^{d_1}\varPhi _l^{(1)\mu } +\sum _{l=1}^{d_2}\varPhi _l^{(2)\mu }+\cdots +\sum _{l=1}^{d_k}\varPhi _l^{(k)\mu }\right\} \\&\quad - \, \left\{ d_1\frac{\tilde{r}_{11}^\mu }{\tilde{r}_1^\mu }+d_2\frac{\tilde{r}_{12}^\mu }{\tilde{r}_2^\mu }+\cdots +d_k\frac{\tilde{r}_{1k}^\mu }{\tilde{r}_k^\mu }\right\} \\= & {} \sum _{i=1}^k\left\{ \sum _{l=1}^{d_i}\varPhi _l^{(i)\mu }-d_i\frac{\tilde{r}_{1i}^\mu }{\tilde{r}_i^\mu }\right\} \\= & {} \tilde{U}_\mu . \end{aligned}$$