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 the stratification vignette. 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:

  • \(\beta_x\cdot x+\beta_{x:t}\cdot(x\cdot g(t))\): 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.

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

  • \(f_t(t)\cdot x\): 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 the strata vignette.

  • \(f_{x,t}(x,t)\): 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:

##   trt celltype time status karno diagtime age prior
## 1   0 squamous   72      1    60        7  69     0
## 2   0 squamous  228      1    60        3  38     0
## 3   0 squamous  126      1    60        9  63     1
## 4   0 squamous  118      1    70       11  65     1
## 5   0 squamous   10      1    20        5  49     0
## 6   0 squamous   82      1    40       10  69     1

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 the “known time-variation function” case 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)
##         trt       prior       karno   tt(karno) 
##  0.07914694  0.12051224 -0.15466404  0.02930082

Thus the time-varying component of the effect becomes \(\beta_{\text{karno}}+\beta_{\text{karno},t}\cdot\log(t+20) = -0.155 + 0.029\cdot\log(t+20)\):

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 <- veteran %>% as_ped(Surv(time, status)~., id = "id") %>%
    mutate(logt20 = log(tstart + (tstart - tend) / 2 + 20))
head(ped) %>% select(interval, ped_status, trt, karno, age, prior, logt20)
##   interval ped_status trt karno age prior   logt20
## 1    (0,1]          0   0    60  69     0 2.970414
## 2    (1,2]          0   0    60  69     0 3.020425
## 3    (2,3]          0   0    60  69     0 3.068053
## 4    (3,4]          0   0    60  69     0 3.113515
## 5    (4,7]          0   0    60  69     0 3.113515
## 6    (7,8]          0   0    60  69     0 3.277145
# 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))
##                      pam         cox
## trt           0.04795137  0.07914694
## prior         0.11689785  0.12051224
## karno        -0.15921723 -0.15466404
## karno:logt20  0.03049036  0.02930082
# 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:

The figure below visualizes the estimate of the time-varying effect of karno for all three models

Expand here to see R-Code

term.df <- ped %>% ped_info() %>% add_term(pam2, term = "karno") %>%
    mutate_at(c("fit", "ci_lower", "ci_upper"), 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))
## Warning: funs() is soft deprecated as of dplyr 0.8.0
## Please use a list of either functions or lambdas: 
## 
##   # Simple named list: 
##   list(mean = mean, median = median)
## 
##   # Auto named with `tibble::lst()`: 
##   tibble::lst(mean, median)
## 
##   # Using lambdas
##   list(~ mean(., trim = .2), ~ median(., na.rm = TRUE))
## This warning is displayed once per session.
gg_tv_karno <- ggplot(term.df, aes(x = tend, y = fit)) +
    geom_step(aes(col = "PAM with penalized spline")) +
    geom_stepribbon(aes(ymin = ci_lower, ymax = ci_upper), 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], "black")) +
    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 Karnofsky 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.

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)
## 
## Family: poisson 
## Link function: log 
## 
## Formula:
## ped_status ~ trt + prior + s(age) + te(tend, karno)
## 
## Parametric coefficients:
##             Estimate Std. Error z value Pr(>|z|)    
## (Intercept) -4.96563    0.16877 -29.422   <2e-16 ***
## trt          0.11795    0.19881   0.593    0.553    
## prior        0.01467    0.20946   0.070    0.944    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Approximate significance of smooth terms:
##                  edf Ref.df Chi.sq p-value    
## s(age)         1.003  1.006  0.371   0.546    
## te(tend,karno) 8.859 11.182 60.369   1e-08 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## R-sq.(adj) =  0.00423   Deviance explained = 6.92%
## UBRE = -0.83464  Scale est. = 1         n = 5392

The 3D perspective plot can aid interpretation, where y- and x-axes depict the Karnofsky-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 \(t\).1

plot(pam3, select = 3, scheme = 1, theta = 120, 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 depicts the Karnofsky-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 \(t=1\), we obtain the effect of the Karnofsky-Score on the log-hazard at the beginning of the follow-up (see also right panel for \(t=1\)), 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 \(x_{\text{karno}}\) (middle panel). For larger values (\(x_{\text{karno}} \in \{75, 90\}\)) the log-hazard is smaller at the beginning and increases over the course of the follow-up, while for small values (\(x_{\text{karno}} \in \{40\}\)) the log-hazard is positive and decreases toward later time points. This could indicate that the effect of the Karnofsky-Score tends towards 0 over time as the information collected at the beginning of the follow-up becomes outdated (but see uncertainty).

Expand here to see 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")
## Scale for 'fill' is already present. Adding another scale for 'fill', which
## will replace the existing scale.
# plot f(karno, t) for specific slices
karno_df <- ped %>%
  make_newdata(tend = unique(tend), karno = c(40, 75, 95)) %>%
  add_term(pam3, term = "karno")

# shortcut
# gg_slice(ped, pam3, "karno", tend = unique(tend), karno = c(40, 75, 95))
karno_gg <- ggplot(karno_df, aes(x = tend, y = fit)) +
  geom_step(aes(col = factor(karno)), lwd = 1.1) +
  geom_stepribbon(aes(ymin = ci_lower, ymax = ci_upper, 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 = ci_lower, ymax = ci_upper, 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")

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-Scores and later time-points. Also note that the estimate does not include the estimated average time-constant log-hazard (coefficients(pam3)["(Intercept)"]=-4.966) and its uncertainty.

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


  1. Note that the graphical representation in the 3D wireframe plot as well as the heatmap/contour plots below are 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.