SurvivalMetric

SurvivalMetric is used for experimental endpoints reported as survival curves or, more generally, as survival levels associated with ordered values. Typical examples include time-to-event data, progression-free survival, or any endpoint where the probability of remaining below or above a threshold is reported.

The metric does not assume a specific time scale or hazard model. Survival levels are treated as cumulative probabilities that define intervals over an ordered outcome axis.

See API reference: DigiPopData.SurvivalMetric.

Concept

Experimental survival data are typically reported as decreasing survival levels (e.g. 0.9, 0.8, 0.7) associated with increasing values (e.g. time points or threshold values).

These survival levels define a partition of the outcome space into disjoint intervals. Each interval corresponds to the fraction of the population whose outcome falls between two consecutive survival levels.

Simulated individual outcomes are assigned to these intervals, and the resulting interval counts are compared with the expected probabilities. This reduces survival comparison to the same categorical framework used by CategoryMetric and QuantileMetric.

Example

Assume experimental data report survival levels at increasing time points for a cohort of 120 patients:

• Survival 0.9 at time 5
• Survival 0.7 at time 10
• Survival 0.4 at time 20

This can be encoded as a SurvivalMetric:

using DigiPopData

s_metric = SurvivalMetric(
    120,                # experimental sample size
    [0.9, 0.7, 0.4],    # survival levels (descending)
    [5.0, 10.0, 20.0]   # corresponding values (ascending)
)

Survival levels must be sorted in descending order, while the associated values must be sorted in ascending order.

Comparison with simulated data

Assume a virtual population where each individual produces a single event time (or an ordered outcome value).

Simulation results are provided as a numeric vector:

using Random
sim_times = randexp(1000)

Each simulated value is assigned to one of the survival intervals defined by the experimental data. The resulting interval counts are used to compute the mismatch.

To compute the loss, we can use the mismatch function:

loss = mismatch(sim_times, s_metric)

Mathematics

Let reported survival levels be

\[1 \ge s_1 > s_2 > \dots > s_m \ge 0\]

with corresponding ordered values

\[v_1 < v_2 < \dots < v_m.\]

These values define m+1 disjoint intervals:

\[(-\infty, v_1),\; [v_1, v_2),\; \dots,\; [v_m, +\infty).\]

The associated interval probabilities are derived as differences of successive survival levels:

\[p_1 = 1 - s_1,\quad p_i = s_{i-1} - s_i\; (i=2,\dots,m),\quad p_{m+1} = s_m.\]

For a virtual population of size N, simulated values produce counts

\[\vec{k} = (k_1, \dots, k_{m+1}), \quad \sum_i k_i = N.\]

Under this construction, interval counts follow a multinomial distribution. As in CategoryMetric and QuantileMetric, a Gaussian approximation of the multinomial likelihood is used, leading to the quadratic loss:

\[\Lambda \approx (\vec{k} - N\vec{p})^T \Sigma'^{-1} (\vec{k} - N\vec{p}),\]

where

\[\Sigma' = N (\mathrm{diag}(\vec{p}) - \vec{p}\vec{p}^T).\]

One interval is removed to eliminate linear dependence, and the reduced quadratic form is evaluated.

Practical notes

• Survival information is treated as cumulative probability data, without assuming a specific hazard or time model.
• The resulting loss is a Gaussian approximation of a likelihood and can be combined with other likelihood-based metrics.
• The quadratic form enables use in optimization and mixed-integer quadratic programming workflows.