A Stochastic Version of the EM Algorithm for Mixture Cure Model with Exponentiated Weibull Family of Lifetimes

Abstract

In this paper, we propose a stochastic approximation to the well-studied expectation–maximization (EM) algorithm for finding the maximum likelihood (ML)-type estimates in situations where missing data arise naturally and a proportion of individuals are immune to the event of interest. A flexible family of three parameter exponentiated Weibull (EW) distributions is assumed to characterize lifetimes of the non-immune individuals as it accommodates both monotone (increasing and decreasing) and non-monotone (unimodal and bathtub) hazard functions. To evaluate the performance of the proposed algorithm, an extensive simulation study is carried out under various parameter settings. Using likelihood ratio tests, we also carry out model discrimination within the EW family of distributions. Furthermore, we study the robustness of the proposed algorithm with respect to outliers in the data and the choice of initial values to start the algorithm. In particular, we show that our proposed algorithm is less sensitive to the choice of initial values when compared to the EM algorithm. For illustration, we analyze a real survival data on cutaneous melanoma. Through this data, we illustrate the applicability of the likelihood ratio test toward rejecting several well-known lifetime distributions that are nested within the wider class of the proposed EW distributions.

Appendix: Development of the EM Algorithm

To implement the EM algorithm, we define the complete data likelihood function as:

$$\begin{aligned} L_C({\varvec{\theta }}; {\varvec{t}}, {\varvec{\delta }}, {\varvec{X}}, {\varvec{\eta }})\propto & {} \prod _{i \in \Delta _1} f_p(t_i; {\varvec{\theta }}, \delta _i, {\varvec{x}}^*_i) \times \prod _{i \in \Delta _0}\left\{ \pi _0({\varvec{x}}^*_i; {\varvec{\beta }})\right\} ^{1-\eta _i}\nonumber \\&\left\{ S_p(t_i; {\varvec{\theta }}, \delta _i, {\varvec{x}}^*_i) - \pi _0({\varvec{x}}^*_i; {\varvec{\beta }}) \right\} ^{\eta _i}, \end{aligned}$$

(32)

where ${\varvec{\eta }}=\left( \eta _1, \dots , \eta _n\right) ^{\tiny \mathrm T}$ and $\pi _0({\varvec{x}}^*_i; {\varvec{\beta }})=\left\{ 1+e^{{\varvec{x}}_i^{\tiny \mathrm T} {\varvec{\beta }}}\right\} ^{-1}$ is the cure rate. Equivalently, the expression for the complete data log-likelihood function is obtained as:

$$\begin{aligned} l_C({\varvec{\theta }}; {\varvec{t}}, {\varvec{\delta }}, {\varvec{X}}, {\varvec{\eta }})&= \text {constant} + \sum _{i \in \Delta _1} \log f_p(t_i; {\varvec{\theta }}, \delta _i, {\varvec{x}}^*_i) + \sum _{i \in \Delta _0}(1-\eta _i) \log \pi _0({\varvec{x}}^*_i; {\varvec{\beta }}) \nonumber \\ {}&\quad +\sum _{i \in \Delta _0}{\eta _i}\log \left\{ S_p(t_i; {\varvec{\theta }}, \delta _i, {\varvec{x}}^*_i) - \pi _0({\varvec{x}}^*_i; {\varvec{\beta }}) \right\} . \end{aligned}$$

(33)

For the mixture cure rate model, the expression given in (33) takes the following form:

$$\begin{aligned} l_{C}({\varvec{\theta }}; {\varvec{t}}, {\varvec{\delta }}, {\varvec{X}}, {\varvec{\eta }})&= \text {constant}+ n_1 (\log \alpha +\log k - k \log \lambda ) + (k-1)\sum _{i \in \Delta _1} \log t_i \nonumber \\&\quad - \sum _{i \in \Delta _1} \left( \frac{t_i}{\lambda }\right) ^k + \sum _{i \in \Delta _1} (\alpha -1) \log \left\{ 1 - e^{-(t_i/\lambda )^k}\right\} \nonumber \\&\quad + \sum _{i \in \Delta _1} {\varvec{x}}_i^{\tiny \mathrm T} {\varvec{\beta }}- \sum _{i=1}^n \log \left( 1+ e^{{\varvec{x}}_i^{\tiny \mathrm T} {\varvec{\beta }}} \right) \nonumber \\&\quad + \sum _{i \in \Delta _0} \eta _i e^{{\varvec{x}}_i^{\tiny \mathrm T} {\varvec{\beta }}} + \sum _{i \in \Delta _0} \eta _i \log \left\{ 1- \left[ 1 - e^{-(t_i/\lambda )^k}\right] ^{\alpha } \right\} . \end{aligned}$$

(34)

1.1 Steps Involved in the EM Algorithm

Begin the iterative process by considering an initial estimate ${\varvec{\theta }}^{(0)}=\left( {\varvec{\beta }}^{(0)}, \alpha ^{(0)}, k^{(0)}, \lambda ^{(0)}\right) ^{\tiny \mathrm T}$ of ${\varvec{\theta }}$. The choice of ${\varvec{\theta }}^{(0)}$ requires justifications based on background knowledge and some sample real-life data. For $r=1, 2, \dots$, let ${\varvec{\theta }}^{(r)}$ be the estimate of ${\varvec{\theta }}$ at the r-th step of the iteration. Then, ${\varvec{\theta }}^{(r+1)}$ is obtained using the following steps:

• 1. E-Step Find the conditional expectation $Q\left( {\varvec{\theta }}; {\varvec{\theta }}^{(r)} \right) = E\left\{ l_{C}({\varvec{\theta }}; {\varvec{t}}, {\varvec{\delta }}, {\varvec{X}}, {\varvec{\eta }})| \left( {\varvec{\theta }}^{(r)},\right. \right. \left. \left. {\varvec{t}}, {\varvec{\delta }}, {\varvec{X}} \right) \right\}$, which is given by

  $$\begin{aligned} Q\left( {\varvec{\theta }}; {\varvec{\theta }}^{(r)} \right)&= \text {constant}+ n_1 (\log \alpha +\log k - k \log \lambda ) + (k-1)\sum _{i \in \Delta _1} \log t_i \nonumber \\&\quad - \sum _{i \in \Delta _1} \left( \frac{t_i}{\lambda }\right) ^k + \sum _{i \in \Delta _1} (\alpha -1) \log \left\{ 1 - e^{-(t_i/\lambda )^k}\right\} \nonumber \\&\quad + \sum _{i \in \Delta _1} {\varvec{x}}_i^{\tiny \mathrm T} {\varvec{\beta }}- \sum _{i=1}^n \log \left( 1+ e^{{\varvec{x}}_i^{\tiny \mathrm T} {\varvec{\beta }}} \right) \nonumber \\&\quad + \sum _{i \in \Delta _0} E\left\{ \eta _i \big \vert \left( {\varvec{\theta }}^{(r)}, {\varvec{t}}, {\varvec{\delta }}, {\varvec{X}} \right) \right\} e^{{\varvec{x}}_i^{\tiny \mathrm T} {\varvec{\beta }}} \nonumber \\&\quad + \sum _{i \in \Delta _0} E\left\{ \eta _i \big \vert \left( {\varvec{\theta }}^{(r)}, {\varvec{t}}, {\varvec{\delta }}, {\varvec{X}} \right) \right\} \log \left\{ 1- \left[ 1 - e^{-(t_i/\lambda )^k}\right] ^{\alpha } \right\} , \end{aligned}$$

  (35)

  where

  $$\begin{aligned}&E\left\{ \eta _i \big \vert \left( {\varvec{\theta }}^{(r)}, {\varvec{t}}, {\varvec{\delta }}, {\varvec{X}} \right) \right\} \nonumber \\&\quad = P\left\{ \eta _i = 1\big \vert \left( {\varvec{\theta }}^{(r)}, {\varvec{t}}, {\varvec{\delta }}, {\varvec{X}} \right) \right\} \nonumber \\&\quad =P\left\{ \eta _i = 1\big \vert \left( {\varvec{\theta }}^{(r)}, Y_i>t_i, {\varvec{x}}^*_i, i \in \Delta _0 \right) \right\} \nonumber \\&\quad =\frac{P\left\{ Y_i>t_i\big \vert \left( \eta _i=1, {\varvec{\theta }}^{(r)}, {\varvec{x}}^*_i, i \in \Delta _0 \right) \right\} P\left\{ \eta _i=1 \big \vert \left( {\varvec{\theta }}^{(r)}, {\varvec{x}}^*_i, i \in \Delta _0\right) \right\} }{P\left\{ Y_i>t_i\big \vert \left( {\varvec{\theta }}^{(r)}, {\varvec{x}}^*_i, i \in \Delta _0 \right) \right\} } \nonumber \\&\quad =\frac{S_p\left( t_i; {\varvec{\theta }}^{(r)}, \delta _i, {\varvec{x}}^*_i\right) - \pi _0\left( {\varvec{x}}^*_i; {\varvec{\beta }}^{(r)}\right) }{S_p\left( t_i; {\varvec{\theta }}^{(r)}, \delta _i, {\varvec{x}}^*_i\right) }\nonumber \\&\quad =1- \frac{ \pi _0\left( {\varvec{x}}^*_i; {\varvec{\beta }}^{(r)}\right) }{S_p\left( t_i; {\varvec{\theta }}^{(r)}, \delta _i, {\varvec{x}}^*_i\right) }. \end{aligned}$$

  (36)

• 2. M-Step Find

  $$\begin{aligned} {\varvec{\theta }}^{(r+1)}=\left( {\varvec{\beta }}^{(r+1)}, \alpha ^{(r+1)}, k^{(r+1)}, \lambda ^{(r+1)}\right) ^{\tiny \mathrm T} = \underset{{\varvec{\theta }}}{{\arg \max }} \text { }Q\left( {\varvec{\theta }}; {\varvec{\theta }}^{(r)} \right) . \end{aligned}$$

  (37)

  The maximization step can be carried out using multidimensional unconstrained optimization methods such as Nelder–Mead simplex search algorithm or quasi Newton methods such as BFGS algorithm. These algorithms are available in statistical software R version 4.0.3 under General Purpose Optimization package called optimr().

• 3. Convergence Check if the stop** or convergence criterion for the iterative process is met. For our analysis, we consider that the EM algorithm has converged to a local maxima if

  $$\begin{aligned} \underset{1 \le k' \le d+4}{{\max }}\text { }{\left| \frac{\theta ^{(r+1)}_{k'}- \theta ^{(r)}_{k'}}{\theta ^{(r)}_{k'}} \right| < \epsilon }, \end{aligned}$$

  (38)

  where $\theta ^{(r)}_{k'}$ and $\theta ^{(r+1)}_{k'}$ are the $k'$-th component of ${\varvec{\theta }}^{(r)}$ and ${\varvec{\theta }}^{(r+1)}$, respectively, and $\epsilon$ is a tolerance such as 0.001.

If the condition in (38) is satisfied, then the iterative process is stopped and ${\varvec{\theta }}^{(r)}$ is considered as the ML estimate of ${\varvec{\theta }}$.