Calculates the right-censored log-likelihood (RCLL) or logarithmic loss, introduced by Avati et al. (2020).
Details
The observation-wise RCLL is defined by:
$$L_{RCLL}(S_i, t_i, \delta_i) = -log[\delta_i f_i(t_i) + (1 - \delta_i) S_i(t_i)]$$
where \(\delta_i\) is the censoring indicator, \(f_i\) the predicted probability density function and \(S_i\) the predicted survival function for observation \(i\). RCLL is proper given that censoring and survival distribution are independent, see Rindt et al. (2022). Simulation studies by Sonabend et al. (2024) provide strong empirical evidence supporting the properness of this score. See section Interpolation for implementation details.
To get a single score across all \(N\) observations of the test set, we return the average of the observation-wise scores:
$$\sum_{i=1}^N L_{RCLL}(S_i, t_i, \delta_i) / N$$
Dictionary
This Measure can be instantiated via the dictionary mlr_measures or with the associated sugar function msr():
Parameter details
eps(numeric(1))
Very small number to substitute near-zero values in order to prevent errors in e.g. log(0) and/or division-by-zero calculations. Default value is 1e-06.
ERV(logical(1))
IfTRUEthen the Explained Residual Variation method is applied, which means the score is standardized against a Kaplan-Meier baseline. Default isFALSE.
Interpolation
To evaluate scores involving subject-specific survival functions \(S_i(t)\), we perform linear interpolation on the discrete survival values provided in the prediction. Duplicate survival values are removed prior to interpolation to ensure strict monotonicity and non-negative density values. Therefore we are left with the distinct survival time points \(t_0 < \cdots < t_n\) and the corresponding survival values \(S(t_j)\).
Interpolation is performed using base R’s approx() with method = "linear"
and rule = 2, ensuring:
Left extrapolation (for \(t < t_0\)) assumes \(S(0) = 1\) and uses the slope from \((0, 1)\) to \((t_0, S(t_0))\).
Right extrapolation (for \(t > t_n\)) uses the slope from the last interval \((t_{n-1}, S(t_{n-1}))\) to \((t_n, S(t_n))\), with results truncated at 0 to preserve non-negativity.
This ensures a continuous, piecewise-linear survival function \(S(t)\) that satisfies \(S(0) = 1\) and remains non-increasing and non-negative across the entire domain.
The density at time point \(t_k\), with \(t_i \le t_k < t_{i+1}\), is estimated as follows:
$$ f_i(t_k) = -\frac{S_i(t_{i+1}) - S_i(t_i)}{t_{i+1} - t_i} $$
This corresponds to the (negative) slope of the \(S_i(t)\) between the closest grid point after \(t_i\) and \(t_i\) itself.
References
Avati, Anand, Duan, Tony, Zhou, Sharon, Jung, Kenneth, Shah, H N, Ng, Y A (2020). “Countdown Regression: Sharp and Calibrated Survival Predictions.” Proceedings of The 35th Uncertainty in Artificial Intelligence Conference, 115(4), 145–155. https://proceedings.mlr.press/v115/avati20a.html.
Rindt, David, Hu, Robert, Steinsaltz, David, Sejdinovic, Dino (2022). “Survival regression with proper scoring rules and monotonic neural networks.” Proceedings of The 25th International Conference on Artificial Intelligence and Statistics, 151(4), 1190–1205. https://proceedings.mlr.press/v151/rindt22a.html.
Sonabend, Raphael, Zobolas, John, Kopper, Philipp, Burk, Lukas, Bender, Andreas (2024). “Examining properness in the external validation of survival models with squared and logarithmic losses.” https://arxiv.org/abs/2212.05260v3.
See also
Other survival measures:
mlr_measures_surv.calib_alpha,
mlr_measures_surv.calib_beta,
mlr_measures_surv.calib_index,
mlr_measures_surv.chambless_auc,
mlr_measures_surv.cindex,
mlr_measures_surv.dcalib,
mlr_measures_surv.graf,
mlr_measures_surv.hung_auc,
mlr_measures_surv.intlogloss,
mlr_measures_surv.logloss,
mlr_measures_surv.mae,
mlr_measures_surv.mse,
mlr_measures_surv.nagelk_r2,
mlr_measures_surv.oquigley_r2,
mlr_measures_surv.rmse,
mlr_measures_surv.schmid,
mlr_measures_surv.song_auc,
mlr_measures_surv.song_tnr,
mlr_measures_surv.song_tpr,
mlr_measures_surv.uno_auc,
mlr_measures_surv.uno_tnr,
mlr_measures_surv.uno_tpr,
mlr_measures_surv.xu_r2
Other Probabilistic survival measures:
mlr_measures_surv.graf,
mlr_measures_surv.intlogloss,
mlr_measures_surv.logloss,
mlr_measures_surv.schmid
Other distr survival measures:
mlr_measures_surv.calib_alpha,
mlr_measures_surv.calib_index,
mlr_measures_surv.dcalib,
mlr_measures_surv.graf,
mlr_measures_surv.intlogloss,
mlr_measures_surv.logloss,
mlr_measures_surv.schmid
Super classes
mlr3::Measure -> mlr3proba::MeasureSurv -> MeasureSurvRCLL