Stratified baselines

In some cases the proportional hazards assumption for different groups (levels of a factor variable) is violated. One approach to resolve this problem is to fit a so called stratified Cox model, where each level \(k=1,\ldots,K\) of factor variable \(z\) will have its own baseline-hazard:

\(\lambda(t|z, x)=\lambda_{0k}(t, z)\exp(x'\beta)\).

For illustration we again use the veteran data (censored at day 400) from the survival package, where the cell type of the lung cancer had four levels:

veteran <- survival::veteran # load veteran data
veteran <- veteran %>%
    trt   = 1*(trt == 2),
    prior = 1*(prior==10)) %>%
  filter(time < 400) # restriction for illustration
##  squamous smallcell     adeno     large 
##        30        48        27        26

Stratified Cox model

Fitting a stratified Cox model using the coxph function from the survival package is simple, including a strata term in the model formula.

cph  <- coxph(Surv(time, status) ~ strata(celltype), data=veteran)
base <- basehaz(cph)

The different baselines are visualized below:

baseline_gg <- ggplot(base, aes(x=time)) +
  geom_step(aes(y=hazard, group=strata)) +
    ylab(expression(hat(Lambda)(t))) + xlab("t")
baseline_gg + aes(col=strata)

Stratified PAM

We can obtain similar results with PAMs by fitting one smooth baseline estimate for each category of celltype in the data. Technically, we specify the by argument in mgcv s(...) function.

Note that in contrast to the model specification in coxph we need to include the celltype factor variable in the model specification:

ped <- veteran %>% as_ped(Surv(time, status)~., id="id")
pam <- gam(ped_status ~ celltype + s(tend, by=celltype), data=ped,
    family=poisson(), offset=offset)

Comparing the estimated baselines we get very similar results between the stratified Cox and stratified PAM models.

pinf <- ped %>%
  group_by(celltype) %>%
  ped_info() %>%
  add_cumu_hazard(pam) %>%
    rename(strata=celltype) %>%
  filter(!(strata == "adeno" & tend > 200))
baseline_gg +   aes(col="Nelson-Aalen") +
    geom_line(data=pinf, aes(x=tend, y=cumu_hazard, group=strata, col="PAM")) +
  geom_ribbon(data=pinf, aes(x=tend, ymin=cumu_lower, ymax = cumu_upper,
    group=strata), fill = "black", col = NA, alpha = .2) +
    facet_wrap(~strata, nrow=1) +
    scale_color_manual(name="Method", values=c(Set1[1], "black")) +
  theme(legend.position = "bottom")

Stratified Proportional Hazards Model

Both models can be easily extended to incorporate further covariates. If we assume that the covariate effects are not time-varying, this extension constitutes a so called stratified proportional hazards model. For example, including effects for treatment (trt), age (age) and the Karnofsky-Score (karno), we get the model

\[ \lambda(t|z, \mathbf{x}) = \lambda_{0k}(t)\exp(\beta_1 I(trt=1) + \beta_2\cdot x_{karno} + \beta_3\cdot x_{age}) \] where \(\lambda_{0k}(t)\) is the baseline for observation with cell type \(k\), \(k\in \{squamos,\ smallcell,\ adeno,\ large\}\):

cph2 <- update(cph, .~. + trt + age + karno)
pam2 <- update(pam, .~. + trt + age + karno)
cbind(pam=coef(pam2)[5:7], cox=coef(cph2))
##               pam         cox
## trt    0.31096123  0.32764309
## age   -0.01642337 -0.01679495
## karno -0.03670427 -0.03696870

Again, covariate estimates are very similar between the two models and can be interpreted in the same way.