4.4 Implementation using the EM-algorithm
4.4.1 The E-step
Denote the parameter-triplet (b, β, βt) by Θ, and the observed information for each observation (λi,j, δi, ti) by O. The conditional expectation of the
complete-data log likelihood (formula (4.6)) in the (r + 1)th E-step is given by
Q(Θ(r+1)| Θ(r)) = E[log L
c(Θ(r+1)| Y ; z, x(t)) | O, Θ(r)]. (4.9)
It can easily be seen that these functions are linear in Yi, which reduces
the problem to find an expression for the conditional expectation of Yi,
which is given by wi(r) = E(Yi | O, Θ(r)) = π(zi)S(ti| Yi= 1; zi, xi(ti)) π(zi)S(ti | Yi= 1; zi, xi(ti)) + (1 − π(zi)) for δi= 0 1 for δi= 1. (4.10)
Note that E(Yi | O, Θ(r)) takes one value per iteration for each observation.
The weights w(r)i are computed using the value of the TVCs at time of censoring, and can be interpreted as the probability that individual i will be susceptible to the event.
4.4.2 The M-step
In the M-step, the expected complete-data log likelihood in (4.9) is max- imized with respect to the unknown parameters. The conditional expec- tation of the incidence log likelihood is straightforward, replacing Yi’s in
(4.7) by w(r)i . The conditional expectation of the latency log likelihood (4.8) can then be written, using λi,jlog wi(r) = 0 and λi,jwi(r) = λi,j (Cai
et al., 2012a) as E[log Llat(β, βt| Y ; z, x(t)) | O, Θ(r)] = n X i=1 ki X j=1 λi,jlog wi(r)h(tj | Yi = 1; zi, xi(tj)) +w(r)i log S(tj | Yi = 1; zi, xi(tj)) = log n Y i=1 ki Y j=1 wi(r)h0(tj) exp(β0zi+ β0txi(tj)) λi,j × S0(tj)exp(β 0z i+β0txi(tj))w(r)i = log n Y i=1 ki Y j=1 h0(tj) exp(β0zi+ β0txi(tj) + log wi(r)) λi,j × S0(tj)exp(β 0z i+β0txi(tj)+log w(r)i ).
When (4.9) is maximized, the baseline survival function of the rth M- step should be updated in order to proceed with the next E-step. This is done non-parametrically using the Breslow-type estimator for S0(t) and
combining the results of Andersen (1992) and Cai et al. (2012a). Denote R(tj) the individuals at risk in the interval (Bj−1, Bj], then
ˆ S0(t) = exp − X j:tj≤t Pn iR(tj)λi,j Pn iR(tj)w (r) i exp(β 0(r) zi+ β 0(r) t xi(tj)) ! . (4.11)
4.5. Computational scheme 103
The E-step and the M-step are repeated until parameter convergence.
4.4.3 Variance estimation
When estimating parameters through the EM-algorithm, standard errors of these parameter estimates are not directly available. A widespread method for estimating variances in the mixture cure context is bootstrap (e.g. Peng, 2003; Cai et al., 2012a; Tong et al., 2012). Though the bootstrap technique is easy to implement, this method is computationally extensive, especially with big datasets and resulting slow convergence of the EM-algorithm. In this work we use the supplemented EM (SEM)-algorithm introduced by Meng and Rubin (1991), as in Section 2.3.3. While other approximation methods exist, see, among others Sy and Taylor (2000a); Peng and Dear (2000), the advantage of SEM is that it can be applied to any problem to which EM is applied, assuming that there is access to the complete-data asymptotic variance-covariance matrix, which is indeed the case here.
4.5
Computational scheme
4.5.1 Data structure
Including TVCs in the survival part of the mixture cure model requires rearrangement of the data. To make TVCs computationally feasible in a Cox PH model, each time period (Bj−1, Bj] with j = 1, . . . , ki for each
individual i is represented as a single row in the data set (Fox, 2002). Note that the number of rows for each observation depends on the observation itself as Bki = ti. The advantage of this data structure is that one can
use the coxph-function in package survival in R (Therneau, 2014), using preamble “Surv(start, stop, default)” instead of the more familiar “Surv(time, default)”.
The mixing proportions of the mixture cure model modeled by the bi- nomial logit do not include TVCs, and using several lines per observation in this model part would lead to wrong b-parameter estimates. As a result, for the mixture cure model with TVCs, different data set structures are
δi ti z1 z2 obs 1 1 1 -1 2 obs 2 0 2 0.3 3 obs 3 1 3 0.4 2.3 Bj−1 Bj λi,j δi ti z1 z2 x1(t) x2(t) obs 1 0 1 1 1 1 -1 2 0.3 -0.7 obs 2 0 1 0 0 2 0.3 3 0.2 0.4 obs 2 1 2 0 0 2 0.3 3 0.7 -0.1 obs 3 0 1 0 1 3 0.4 2.3 0.5 -1 obs 3 1 2 0 1 3 0.4 2.3 0.2 -0.3 obs 3 2 3 1 1 3 0.4 2.3 0.4 0.2
Table 4.1: Example of the incidence versus latency model data structure. The left hand table represents the data structure for the binomial logit part of the mixture cure model, where no TVCs are present. The right hand table is the long data structure needed for incorporation of TVCs in the survival part of the model.
being used depending on whether the respective calculations are performed on the latency versus the incidence part of the model. An example of the ‘short’ (incidence) data structure versus the ‘long’ (latency) data structure is represented in Table 4.1. To transform the general short form of sur- vival data into the long structure, Fox and Carvalho (2012) introduced the “unfold” function in the R-package RcmdrPlugin.survival.
4.5.2 Procedure
The procedure consists of three main steps: the initialization step, the E-step and the M-step.
Initialization
1) Initialize w: Initialize w(0)i by taking wi(0) = δi. Each observation
has one w(0)i .
2) Initialize b: Fit a binomial logit model to w(0)i using the ‘short’ data set and covariate vector z, in order to retrieve an initial estimate ˆ
b(0).
3) Initialize β and βt: Obtain ˆβ(0)and ˆβ(0)t -estimates using the coxph- function for the (long) survival data including TVCs. Use wi’s as
4.5. Computational scheme 105
weights in the model, matching wi with each line that corresponds
with observation i.
4) Initialize S0(t): Compute ˆS0(0)(t) using formula (4.11).
Expectation step
1) Compute πi(1)(zi) for each i, using Formula (4.4) and ˆb (0)
.
2) Compute wi(1) for each i, using Formula (4.10) ˆβ(0). Note that the survival estimates used here,
ˆ S(0)(ti | Yi = 1; zi, xi(ti)) = ˆS0(0)(ti)exp( ˆβ 0(0) zi+ ˆβ 0(0) t xi(ti)),
correspond for each observation with the estimate at time of the last observation, hence the linear predictor consists of the TVC-values at time ti.
Maximization step
1) Update b: Obtain a new estimate ˆb(1) using the w(1)i of the E-step when fitting the binomial logit model.
2) Update β and βt: Obtain a new estimate of ˆβ(1)t and ˆβ(1) including the w(1)i ’s as weights.
3) Update S: Obtain a new estimate of ˆS(0)(t) using formula (4.11).
The E- and M-step will be repeated with all updated estimates, until pa- rameter convergence. The algorithm stops when the sum of the squared differences between βˆ(r+1)t , ˆβ(r+1)t , ˆb(r+1) and ˆβ(r)t , ˆβ(r)t , ˆb(r) is smaller than a pre-specified value.
4.6
Simulation study
4.6.1 Simulating survival times with time-dependent co-