hazardous.metrics.integrated_brier_score_incidence

hazardous.metrics.integrated_brier_score_incidence(y_train, y_test, y_pred, times, event_of_interest='any')

Time-integrated Brier score of a cause-specific cumulative incidence estimate.

\[\mathrm{IBS}_k = \frac{1}{t_{max} - t_{min}} \int^{t_{max}}_{t_{min}} \mathrm{BS}_k(u) du\]

This scheme was introduced in [Graf1999] for survival analysis and extended to competing events in [Kretowska2018].

Note that this assumes independence between censoring and the covariates. When this assumption is violated, the IPCW weights are biased and the Brier score is not a proper scoring rule anymore. See [Gerds2006] for a study of this bias.

Parameters:

y_trainrecord-array, dictionnary or dataframe of shape (n_samples, 2)

The target, consisting in the ‘event’ and ‘duration’ columns. This is used to fit the IPCW estimator.

y_testrecord-array, dictionnary or dataframe of shape (n_samples, 2)

The ground truth, consisting in the ‘event’ and ‘duration’ columns. In the “event” column, 0 indicates censoring, and any other values indicate competing event types.

y_predarray-like of shape (n_samples, n_times)

Incidence probability estimates predicted at times. In the binary event settings, this is 1 - survival_probability.

timesarray-like of shape (n_times)

Times at which the survival probabilities y_pred has been estimated and for which we compute the Brier score.

event_of_interestint or “any”, default=”any”

The event to consider in a competing events setting. When an integer, this should be one of the non-zero values in the “event” column of y_train and y_test.

"any" indicates that all events except the censoring marker 0 are considered collapsed together as a single event. In a single event setting, "any" and 1 are equivalent.

Returns:

ibsfloat

See also

brier_score_incidence

Time-dependent Brier score for the kth cause of event.

References

[Graf1999]

E. Graf, C. Schmoor, W. Sauerbrei, M. Schumacher, “Assessment and comparison of prognostic classification schemes for survival data”, 1999

[Kretowska2018]

M. Kretowska, “Tree-based models for survival data with competing risks”, 2018

[Gerds2006]

T. Gerds and M. Schumacher, “Consistent Estimation of the Expected Brier Score in General Survival Models with Right-Censored Event Times”, 2006