In adibender/pamm: Functions and Utilities for fitting Piece-wise Exponential (Additive) models

library(knitr)
opts_chunk$set(
  fig.align = "center",
  fig.width = 4,
  fig.height = 4,
  crop=TRUE)

library(ggplot2)
theme_set(theme_bw())
library(gridExtra)
library(magrittr)
library(tidyr)
library(dplyr)
library(pamm)
library(survival)
library(mgcv)
Set1    <- RColorBrewer::brewer.pal(9, "Set1")
Greens  <- RColorBrewer::brewer.pal(9, "Greens")
Purples <- RColorBrewer::brewer.pal(9, "Purples")

In this vignette we show examples of how to fit time-varying effects of time-constant continuous covariates. Note that time-varying effects of time-constant categorical variables are analogous to stratified proportional hazards models, where observations from different levels of the categorical variable have different baseline hazards. That setting is described in vignette("strata", package="pam"). Note that all time-varying effects in a PAM are still assumed to be piece-wise constant over the intervals used to specify the PAM!

Possible specifications of time-variation

In the following we denote the continuous time-constant covariate with $x$ and time with $t$. A time-varying effect of $x$ can then be specified as an interaction term between $x$ and $t$, where different levels of complexity and flexibility for this interaction are possible:

a) $\beta_x\cdot x+\beta_{x:t}\cdot(x\cdot g(t))$: Linear effect of $x$ with time-variation given by $g(t)$, where $g(\cdot)$ is a known or pre-specified transformation of time $t$, e.g. the $\log$-function.

b) $f_x(x)\cdot g(t)$: Non-linear effect of $x$, (linearly) time-varying with $g(t)$, where $g(t)$ is a known or pre-specified transformation of time

c) $f_t(t)\cdot x$: A varying coefficient model in $x$, where time-variation is non-linear and estimated from the data. If $x$ is a dummy variable coding for levels of a categorical variable this constitutes a stratified model with a different "baseline" hazard for each category, see vignette("strata", package="pam").

d) $f_{x,t}(x,t)$: A non-linear effect of $x$ that varies non-linearly over time $t$.

Veteran Data example

For illustration and comparison we use the veteran data presented in the vignette of the survival package (vignette("timedep", package = "survival")). Besides information on survival, the data set contains the Karnofsky performance scores karno (the higher the better), age and whether prior therapy occurred, along with some additional covariates, see help("veteran", package = "survival") for details:

# for some reason the prior variable is coded 0/10 instead of 0/1
data("veteran", package="survival")
veteran %<>%
  mutate(
    trt   = 1*(trt == 2),
    prior = 1*(prior==10)) %>%
  filter(time < 400) # restriction for illustration
head(veteran)

Extended Cox Model (with known shape of time-variation function)

To fit a time-varying effect of karno the authors suggest to use the function $$ f(x_{\text{karno}},t) = \beta_{\text{karno}}\cdot x_{\text{karno}} + \beta_{\text{karno},t} \cdot x_{\text{karno}} \cdot \log(t+20). $$ This is an instance of case a) above with $g(t) = \log(t+20).$

vfit <- coxph(
  formula = Surv(time, status) ~ trt + prior + karno + tt(karno),
  data    = veteran,
  tt      = function(x, t, ...) x * log(t + 20))
coef(vfit)

ttcoef <- round(coef(vfit), 3)[3:4]

Thus the time-varying component of the effect becomes $\beta_{\text{karno}}+\beta_{\text{karno},t}\cdot\log(t+20) = r ttcoef[1] + r ttcoef[2]\cdot\log(t+20)$:

t <- seq(0, 400, by = 10)
plot(x = t, y = coef(vfit)["karno"] + coef(vfit)["tt(karno)"]*log(t + 20),
  type = "l", ylab = "Beta(t) for karno", las = 1, ylim=c(-.1, .05), col=Set1[1])

PAM (with known shape of time-variation function)

To fit a PAM with equivalent model specification (except for the baseline hazard) we can use

# data transformation
ped <- split_data(Surv(time, status)~., data = veteran, id = "id") %>%
    mutate(logt20 = log(tstart+(tstart-tend)/2 + 20))
head(ped) %>% select(interval, ped_status, trt, karno, age, prior, logt20)

# fit model
pam <- gam(ped_status ~ s(tend) + trt + prior + karno + karno:logt20,
    data = ped, offset = offset, family = poisson())
cbind(
  pam = coef(pam)[2:5],
  cox = coef(vfit))
# compare fits
plot(x = t, y = coef(vfit)["karno"] + coef(vfit)["tt(karno)"]*log(t + 20),
  type = "l", ylab = "Beta(t) for karno", ylim = c(-.1, .05), las = 1,
  col = Set1[1])
t_pem <- int_info(ped)$tend
lines(x = t_pem, y = coef(pam)["karno"] + coef(pam)["karno:logt20"]*log(t_pem + 20),
  col = Set1[2], type = "s")

Both methods yield very similar estimates of the time-varying effect of the Karnofsky-Score, with a reduced hazard for higher-scoring patients at the beginning of the follow-up that diminishes over time and turns into an increased hazard for higher-scoring patients after about day 150.

PAM (penalized estimation of time-variation function)

In case we don’t want to pre-specify which shape the time-dependency should have, we can specify the effect of karno as $f(x_{\text{karno}},t) = f(t)\cdot x_{\text{karno}}$, where $f(t)$ is estimated from the data:

# no need to specify main effect for karno here
pam2 <- gam(ped_status ~ s(tend) + trt + prior + s(tend, by = karno),
    data = ped, offset = offset, family = poisson())
term.df <- ped %>% ped_info() %>% add_term(pam2, term = "karno") %>%
    mutate_at(c("fit", "low", "high"), funs(./.data$karno)) %>%
    mutate(
        cox.fit = coef(vfit)["karno"] + coef(vfit)["tt(karno)"]*log(tend + 20),
        pam.fit = coef(pam)["karno"] + coef(pam)["karno:logt20"]*log(tend + 20))
ggplot(term.df, aes(x = tend, y = fit)) +
    geom_step(aes(col = "PAM with penalized spline")) +
    geom_stepribbon(aes(ymin = low, ymax = high), alpha = 0.2) +
    geom_line(aes(y = cox.fit, col = "Cox with log-transform")) +
    geom_step(aes(y = pam.fit, col = "PAM with log-transform")) +
    scale_color_manual(name="Method", values = c(Set1[1:2], 1)) +
    xlab("t") + ylab(expression(hat(f)(t)))

The semi-parametric PAM model estimate for $f(t)$ increases fairly linearly up to day 150 and flattens out at about 0 (i.e., no effect of Karnofsky scores on the hazard) afterwards.

PAM with smooth, smoothly time-varying effect of the Karnosky Score

To fit a non-linear, non-linearly time-varying effect we can specify a two-dimensional interaction between the covariate of interest (here the Karnofsky-Score and a variable that represents time in the respective interval, e.g. interval end-points) using tensor product terms.

In mgcv::gam such two-dimensional effects can be directly used either via te or ti terms in the model specification. The later is especially useful for disentangling the marginal (time-constant) and interaction (time-varying) effects of the respective covariate.

Below we first fit a model using the te specification. Note that we did not include a s(tend) term here, as the time-variable tend is already present in the te term, thus the effect te(tend, karno) also includes the shape of the baseline hazard as well. The level of the baseline log hazard is given by the intercept of the model.

## Non-linear, non-linearly time-varying effects
pam3 <- gam(
  formula = ped_status ~ trt + prior + s(age) + te(tend, karno),
  data   = ped,
  family = poisson(),
  offset = offset)

The summary of the model indicates that the estimated bivariate function $\hat{f}(x_{\text{karno}}, t)$ is highly non-linear ($edf \approx 8.9$):

summary(pam3)

The 3D perspective plot can aid interpretation, where y- and x-axes depict the Karnosky-Score and the time respectively and the z-axis displays the contribution of the effect to the log-hazard for each combination of $x_{\text{karno}}$ and $t$.^[Note that the graphical representation in the 3D wireframe plot as well as the heatmap/contour plots below not exact -- these effects are actually step functions over time, with steps at the interval end points tend, since a PAM implies that all time-varying effects are piece-wise constant over the intervals used for the fit. In practice, this subtle difference can be neglected if the intervals are small enough, as in this case.]

plot(pam3, select = 2, pers = T, theta = 150, ticktype = "detailed")

Such 3D plots are sometimes difficult to interpret, thus we also provide a heat-/contourplot (left panel) with respective slices for fixed values of the Karnofsky-Score (middle panel) and fixed time-points/intervals (right panel) below.

The left panel again depicts the Karnosky Score on the y-axis and the time on the x-axis. The value of $\hat{f}(x_{\text{karno}}, t)$ is visualized using a color gradient, where blue colors indicate log-hazard decrease and red colors a log-hazard increase. The grayed out areas depict combinations of karno and tend that were not present in the data. Dotted horizontal and vertical lines indicate slices that are displayed in the middle and right panel.

For fixed $t=0$, we obtain the effect of the Karnofsky-Score on the log-hazard at the beginning of the follow-up (red line left panel), which decreases strongly from low to high values of $x_{\text{karno}}$.

Holding the Karnofsky Score constant, we can see how the log hazard changes over time for different given values of $x_{\text{karno}}$ (middle panel). For higher values $x_{\text{karno}} >60$ the log-hazard is smaller at the beginning and increases over the course of the follow-up (orange and blue lines middle panel). This is consistent with the time-varying effect of the Karnofsky-Score estimated in the previous section. However, the bivariate function indicates that the log hazard may have a parabolic shape for low $x_{\text{karno}}$ values, decreasing at the beginning and increasing again for later time-points (green line middle panel).

Expand here for detailed R-Code

## heat map/contour plot
te_gg <- gg_tensor(pam3) +
  geom_vline(xintercept = c(1, 51, 200), lty = 3) +
  geom_hline(yintercept = c(40, 75, 95), lty = 3) +
  scale_fill_gradient2(
    name = expression(hat(f)(list(x[plain(karno)],t))),
    low  = "steelblue", high = "firebrick2") +
  geom_contour(col="grey30") +
  xlab("t") + ylab(expression(x[plain(karno)])) +
  theme(legend.position  = "bottom")



## plot f(karno, t) for specific slices
karno_df <- combine_df(
  int_info(ped),
  select(sample_info(ped), -karno),
  data.frame(karno = c(40, 75, 95)))

karno_df <- karno_df %>% add_term(pam3, term="karno")

karno_gg <- ggplot(karno_df, aes(x=tend, y=fit)) +
  geom_step(aes(col=factor(karno)), lwd=1.1) +
  geom_stepribbon(aes(ymin = low, ymax = high, fill=factor(karno)), alpha=.2) +
  scale_color_manual(
    name   = expression(x[plain(karno)]),
    values = Greens[c(4, 7, 9)]) +
  scale_fill_manual(
    name   = expression(x[plain(karno)]),
    values = Greens[c(4, 7, 9)]) +
  ylab(expression(hat(f)(list(x[plain(karno)],t)))) +
  xlab("t")+  coord_cartesian(ylim= c(-4, 3))+
  theme(legend.position  = "bottom")

time_df <-ped %>%
  make_newdata(tend=c(1, 51, 200), karno=seq(20, 100, by=5)) %>%
  add_term(pam3, term="karno")

time_gg <- ggplot(time_df, aes(x=karno)) +
  geom_line(aes(y=fit, col=factor(tend)), lwd=1.1) +
  geom_ribbon(aes(ymin = low, ymax = high, fill=factor(tend)), alpha=.2) +
  scale_color_manual(name="t", values=Purples[c(4, 6, 8)]) +
  scale_fill_manual(name="t", values=Purples[c(4, 6, 8)]) +
  ylab(expression(hat(f)(list(x[plain(karno)],t)))) +
  xlab(expression(x[plain(karno)])) + coord_cartesian(ylim= c(-4, 3))+
  theme(legend.position  = "bottom")

grid.arrange(te_gg, karno_gg, time_gg, nrow=1)

The following figure shows the estimated effect (middle panel) along with a pointwise upper (right) and lower (left) CI. Note that we have to be somewhat cautious with interpretation, considering the large uncertainty of the effect estimate, especially for lower Karnofsky-Scoresand later time-points. Also note that the estimate does not include the estimated average time-constant log-hazard (coefficients(pam3)["(Intercept)"]=``r round(coefficients(pam3)["(Intercept)"], 3)) and its uncertainty:

gg_tensor(pam3, ci=TRUE) +
  xlab("t") + ylab(expression(x[plain(karno)]))

adibender/pamm documentation built on May 14, 2019, 5:22 p.m.