09. Causal Effects and Adjusted Marginal Contrasts (RMST)

Moving Beyond the Hazard Ratio

In clinical trials and observational studies, researchers often wish to compare survival outcomes between two groups. A Cox proportional hazards model summarizes relative hazards under its model assumptions. Conditional and marginal hazard ratios can differ even without confounding, and a single hazard ratio may be difficult to interpret under non-proportional hazards. These issues motivate complementary summaries on an absolute survival-time scale.

SuperSurv provides a model-based standardized contrast in Restricted Mean Survival Time (RMST) using the fitted survival ensemble. This does not by itself establish a causal effect or remove model-estimation uncertainty.

RMST calculates the area under the survival curve up to a specific time horizon, \(\tau\). By comparing the expected RMST if everyone in the dataset belonged to Group 1 versus if everyone belonged to Group 0, we obtain a robust, absolute measure of the difference:

\[ \Delta \text{RMST} = E[Y(1)] - E[Y(0)] = \text{RMST}_{\text{Group 1}}(\tau) - \text{RMST}_{\text{Group 0}}(\tau) \]

Philosophy: “Causal Effect” vs. “Marginal Contrast”

How you interpret this \(\Delta \text{RMST}\) depends entirely on the nature of your exposure variable. The math of G-computation is identical for both, but the statistical terminology must be used responsibly.

  1. Causal Average Treatment Effect (ATE): A causal interpretation requires a well-defined intervention, consistency, conditional exchangeability, positivity, appropriate censoring assumptions, and adequate estimation of the relevant survival functions. A manipulable exposure alone is not sufficient.
    • Interpretation: “Administering this drug causally adds an average of 4.2 months of life over a 5-year period compared to the placebo.”
  2. Adjusted Marginal Contrast: Without justified causal identification, report a covariate-standardized model-based group contrast. Standardization over the recorded covariates does not guarantee that all confounding has been controlled.
    • Interpretation: “After adjusting for all baseline clinical covariates, the presence of this biomarker is marginally associated with 4.2 additional months of survival over a 5-year period.”

Estimating the Effect with SuperSurv

Let’s demonstrate this using the built-in metabric dataset. We will evaluate the effect of the binary biomarker x4 (1 = present, 0 = absent). Because x4 is a biomarker, we will interpret the result as an Adjusted Marginal Contrast.

For a quick teaching example, we sample 200 patients and use three cross-validation folds and a 50-tree forest when available. The numerical results describe this small example, not the full METABRIC analysis.

library(SuperSurv)
set.seed(123)

# Load built-in data
data("metabric", package = "SuperSurv")
metabric <- metabric[sample(seq_len(nrow(metabric)), 200), ]

# Define predictors and time grid
X <- metabric[, grep("^x", names(metabric))]
new.times <- seq(10, 150, by = 10)

1. Train the Super Learner

First, train the ensemble. We must set control = list(saveFitLibrary = TRUE) so the models are saved for the G-computation prediction phase.

event_library <- c("surv.coxph", "surv.weibull")
if (has_rfsrc) {
  rf_library <- create_grid("surv.rfsrc", list(ntree = 50))
  event_library <- c("surv.coxph", rf_library)
}

fit <- SuperSurv(
  time = metabric$duration,
  event = metabric$event,
  X = X,
  newdata = X,
  new.times = new.times,
  event.library = event_library,
  cens.library = c("surv.coxph"),
  nFolds = 3,
  verbose = FALSE,
  control = list(
    saveFitLibrary = TRUE,
    event.t.grid = seq(0, max(metabric$duration[metabric$event == 1]), length.out = 30),
    cens.t.grid = seq(0, max(metabric$duration[metabric$event == 0]), length.out = 30)
  )
)

2. Estimate the Adjusted Marginal RMST Contrast

We use the estimate_marginal_rmst() function to compute an adjusted marginal contrast on the RMST scale. The function sets the binary grouping variable x4 to 1 for all patients, predicts their survival curves, integrates those predictions up to the restriction horizon tau, and then repeats the same procedure with x4 set to 0. The difference between these two standardized averages yields the adjusted RMST contrast.

# Estimate the adjusted difference up to tau = 100 months
results <- estimate_marginal_rmst(
  fit = fit, 
  data = metabric, 
  trt_col = "x4", 
  times = new.times, 
  tau = 100
)
#> Adjusted Delta RMST at tau = 100: -0.38 time units



print(results$ATE_RMST)
#> [1] -0.3796394

Interpretation: If the resulting \(\Delta\)RMST value is -1.24, this indicates that, after standardizing over the observed covariate distribution using the fitted Super Learner ensemble, the group with x4 = 1 is predicted to have approximately 1.24 fewer months of restricted mean survival than the group with x4 = 0 over a 100-month horizon.

Uncertainty: This function reports a model-based point contrast. Formal inference would need to account for tuning, cross-validation, learner fitting, censoring estimation, and ensemble-weight estimation using a justified inference procedure. This version does not provide such a procedure.

3. Visualizing the Effect Over Time

The difference between groups might be near zero early on but substantial later. We can visualize how the adjusted RMST point contrast evolves across different restriction times using plot_marginal_rmst_curve().

The plotting examples below run only when the optional ggplot2 package is installed; the numerical RMST estimates above do not require it.

# Plot the Delta RMST across a sequence of tau values
tau_grid <- seq(20, 140, by = 30)
plot_marginal_rmst_curve(
  fit = fit, 
  data = metabric, 
  trt_col = "x4", 
  times = new.times, 
  tau_seq = tau_grid
)
#> Adjusted Delta RMST at tau = 20: 0.039 time units
#> Adjusted Delta RMST at tau = 50: 0.019 time units
#> Adjusted Delta RMST at tau = 80: -0.027 time units
#> Adjusted Delta RMST at tau = 110: -0.596 time units
#> Adjusted Delta RMST at tau = 140: -1.314 time units

4. Diagnostic: Predicted RMST vs. Observed Time

This descriptive plot compares predicted conditional mean restricted survival with observed follow-up. A realized event time need not lie near its conditional mean, and censored follow-up is not the true event time. The plot is not a formal calibration test.

plot_rmst_vs_obs(
  fit = fit, 
  data = metabric, 
  time_col = "duration", 
  event_col = "event", 
  times = new.times, 
  tau = 100
)