Landmark event probabilities in ertte

library(ertte)
library(survival)

This article covers ertte_landmark(), ertte‘s simplest scalar exposure-response (E-R) reduction of a time-to-event (TTE) endpoint. It’s the shorter sibling of the RMST article, which covers a more involved reduction (restricted mean survival time) built on the same idea – turning a full survival curve into a single number that can be plotted with erplots’ ordinary E-R grammar. If you haven’t already, start with the overview article for background on ertte_aft()/ ertte_coxph().

Why a landmark?

A fitted TTE model describes an entire survival curve $S(t \mid x)$ , but many exposure-response questions are really about one particular point on it: what fraction of subjects with a given exposure profile are expected to have had the event by some fixed, clinically meaningful time $t^*$ ? (“Landmark” here refers to that fixed time, not to landmark-based dynamic prediction from repeated measurements, a different technique with the same name.)

Reducing to a single time point converts a TTE endpoint into an ordinary binary response, $P(\text{event by } t^*) = 1 - S(t^* \mid x)$ , which erplots’ existing er_plot()/er_vpc() grammars – built for scalar exposure-response summaries – can visualise with no TTE-specific plotting code at all.

How it’s computed

ertte_landmark() doesn’t implement any new survival-curve machinery of its own. It calls [ertte_predict()] at time = landmark_time and transforms the result:

$P(\text{event by } t^*) = 1 - S(t^*).$

Because this is a decreasing, monotonic transform, the confidence interval bounds simply swap – the upper bound on survival becomes the lower bound on event probability, and vice versa – with no separate calculation needed. Whatever validity ertte_predict()’s interval has for a given engine (a Wald interval on the linear predictor for ertte_aft(), or survival::survfit()’s own log-transform interval for ertte_coxph()) carries through unchanged. This also means ertte_landmark() inherits all of ertte_predict()’s existing edge-case handling for free – e.g. the all-censored-Cox-model guard, or NA propagation for an aliased covariate.

One consequence of delegating this way: landmark_time must be a single fixed value, unlike ertte_predict()’s time argument (which accepts a vector). A landmark is by definition one specific point in time; if you want event probabilities at several different landmarks, call ertte_landmark() once per landmark.

A worked example

mod_aft <- ertte_aft(Surv(time, event) ~ aucss, ertte_data)
mod_cox <- ertte_coxph(Surv(time, event) ~ aucss, ertte_data)

nd <- ertte_data[c(1, 50, 100), ]
nd[, c("id", "aucss")]
#>      id    aucss
#> 1     1 1114.089
#> 50   50  148.240
#> 100 100  427.802

ertte_landmark(mod_aft, nd, landmark_time = 90)
#> # A tibble: 3 × 14
#>      id sex      age weight  dose treatment aucss cmaxss event admin_censor
#>   <int> <fct>  <int>  <dbl> <dbl> <fct>     <dbl>  <dbl> <dbl>        <dbl>
#> 1     1 Female    27     70   200 Drug      1114.  187.      1          180
#> 2    50 Male      28     58   100 Drug       148.   10.4     0          180
#> 3   100 Female    26     48   200 Drug       428.   35.1     0          180
#> # ℹ 4 more variables: landmark_time <dbl>, fit_resp <dbl>, ci_lower <dbl>,
#> #   ci_upper <dbl>

ertte_landmark(mod_cox, nd, landmark_time = 90)
#> # A tibble: 3 × 14
#>      id sex      age weight  dose treatment aucss cmaxss event admin_censor
#>   <int> <fct>  <int>  <dbl> <dbl> <fct>     <dbl>  <dbl> <dbl>        <dbl>
#> 1     1 Female    27     70   200 Drug      1114.  187.      1          180
#> 2    50 Male      28     58   100 Drug       148.   10.4     0          180
#> 3   100 Female    26     48   200 Drug       428.   35.1     0          180
#> # ℹ 4 more variables: landmark_time <dbl>, fit_resp <dbl>, ci_lower <dbl>,
#> #   ci_upper <dbl>

As with ertte_predict(), the two engines’ point estimates and intervals should generally agree closely when both fit the data reasonably well (as here), but aren’t expected to match exactly, since the underlying survival curves and confidence interval constructions differ.

Using a landmark in a visual predictive check

Passing landmark_time through simulate_args to erplots::er_vpc_add_simulated() (or directly to er_simulate()/ simulate.ertte_model()) turns each simulated replicate’s raw sim_time/sim_event into a landmark-binary outcome for comparison against the observed data:

sim <- simulate(mod_aft, nd, nsim = 500, seed = 3157)
sim[1:3, c("id", "sim_id", "sim_time", "sim_event")]
#> # A tibble: 3 × 4
#>      id sim_id sim_time sim_event
#>   <int>  <int>    <dbl>     <dbl>
#> 1     1      1    127.          1
#> 2    50      1      0.3         0
#> 3   100      1    115.          1

Internally, this reduces each replicate to 1 (an event on or before landmark_time), 0 (known to still be event-free at landmark_time), or NA if the replicate was censored strictly before landmark_time – a genuinely ambiguous outcome, since it’s not known whether that replicate would have had the event by $t^*$ had follow-up continued. This is the same complete-case convention used for RMST-based VPCs (see the RMST article’s “Using RMST in a visual predictive check” section for the full reasoning); erplots::er_vpc_add_simulated()’s mean(..., na.rm = TRUE) aggregation excludes these NAs automatically.

Limitations

A landmark answers a question about one time point only. It’s simpler and more directly interpretable than RMST, but it discards everything the fitted model says about the survival curve away from $t^*$ – two exposure profiles with very different survival curves can have the same landmark probability if the curves happen to cross near $t^*$ .
The choice of $t^*$ matters, and isn’t estimated from the data. Different landmark times can tell different (both valid) stories about the same fitted model, the same caveat RMST’s choice of tau carries.
A replicate censored before $t^*$ contributes no information to a landmark VPC and is dropped, not imputed or reweighted – see “Using a landmark in a visual predictive check” above.

Why a landmark?

How it’s computed

A worked example

Using a landmark in a visual predictive check

Limitations

Further reading