```
library(tidyr)
library(dplyr)
library(ggplot2)
theme_set(theme_bw())
library(survival)
library(mgcv)
library(pammtools)
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 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!**

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 \(t\).

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
veteran <- survival::veteran
veteran <- veteran %>%
mutate(
trt = 1L * (trt == 2),
prior = 1L * (prior == 10)) %>%
filter(time < 400) # restriction for illustration
head(veteran)
```

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

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 \(\beta_{\text{karno}}+\beta_{\text{karno},t}\cdot\log(t+20) = -0.155 + 0.029\cdot\log(t+20)\):

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 0.04795134 0.07914694
## prior 0.11689786 0.12051224
## karno -0.15921724 -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.

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())
```

`karno`

for all three models
```
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()` was deprecated in dplyr 0.8.0.
## ℹ 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))
## Call `lifecycle::last_lifecycle_warnings()` to see where this warning was
## generated.
```

```
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.

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)`

```
##
## 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 <2e-16 ***
## ---
## 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).

```
# 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)]))
```

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.↩︎