Integrated Log-Likelihood Survival Measure

Calculates the Integrated Survival Log-Likelihood (ISLL) or Integrated Logarithmic (log) Loss, aka integrated cross entropy.

Details

This measure has two dimensions: (test set) observations and time points. For a specific individual $i$ from the test set, with observed survival outcome $(t_i, \delta_i)$ (time and censoring indicator) and predicted survival function $S_i(t)$, the observation-wise loss integrated across the time dimension up to the time cutoff $\tau^*$, is:

$$L_{ISLL}(S_i, t_i, \delta_i) = -\text{I}(t_i \leq \tau^*) \int^{\tau^*}_0 \frac{log[1-S_i(\tau)] \text{I}(t_i \leq \tau, \delta=1)}{G(t_i)} + \frac{\log[S_i(\tau)] \text{I}(t_i > \tau)}{G(\tau)} \ d\tau$$

where $G$ is the Kaplan-Meier estimate of the censoring distribution.

The re-weighted ISLL (RISLL) is:

$$L_{RISLL}(S_i, t_i, \delta_i) = -\delta_i \text{I}(t_i \leq \tau^*) \int^{\tau^*}_0 \frac{\log[1-S_i(\tau)]) \text{I}(t_i \leq \tau) + \log[S_i(\tau)] \text{I}(t_i > \tau)}{G(t_i)} \ d\tau$$

which is always weighted by $G(t_i)$ and is equal to zero for a censored subject.

To get a single score across all $N$ observations of the test set, we return the average of the time-integrated observation-wise scores: $$\sum_{i=1}^N L(S_i, t_i, \delta_i) / N$$

Dictionary

This Measure can be instantiated via the dictionary mlr_measures or with the associated sugar function msr():

MeasureSurvIntLogloss$new()
mlr_measures$get("surv.intlogloss")
msr("surv.intlogloss")

Parameters

Id	Type	Default	Levels	Range
integrated	logical	TRUE	TRUE, FALSE	-
times	untyped	-		-
t_max	numeric	-		$[0, \infty)$
p_max	numeric	-		$[0, 1]$
method	integer	2		$[1, 2]$
se	logical	FALSE	TRUE, FALSE	-
proper	logical	FALSE	TRUE, FALSE	-
eps	numeric	0.001		$[0, 1]$
ERV	logical	FALSE	TRUE, FALSE	-

Meta Information

Type: "surv"
Range: $[0, \infty)$
Minimize: TRUE
Required prediction: distr

Parameter details

integrated (logical(1))
If TRUE (default), returns the integrated score (eg across time points); otherwise, not integrated (eg at a single time point).

times (numeric())
If integrated == TRUE then a vector of time-points over which to integrate the score. If integrated == FALSE then a single time point at which to return the score.

t_max (numeric(1))
Cutoff time $\tau^*$ (i.e. time horizon) to evaluate the measure up to. Mutually exclusive with p_max or times. This will effectively remove test observations for which the observed time (event or censoring) is strictly more than t_max. It's recommended to set t_max to avoid division by eps, see Details. If t_max is not specified, an Inf time horizon is assumed.

p_max (numeric(1))
The proportion of censoring to integrate up to in the given dataset. Mutually exclusive with times or t_max.

method (integer(1))
If integrate == TRUE, this selects the integration weighting method. method == 1 corresponds to weighting each time-point equally and taking the mean score over discrete time-points. method == 2 corresponds to calculating a mean weighted by the difference between time-points. method == 2 is the default value, to be in line with other packages.

se (logical(1))
If TRUE then returns standard error of the measure otherwise returns the mean across all individual scores, e.g. the mean of the per observation scores. Default is FALSE (returns the mean).

proper (logical(1))
If TRUE then weights scores by the censoring distribution at the observed event time, which results in a strictly proper scoring rule if censoring and survival time distributions are independent and a sufficiently large dataset is used. If FALSE then weights scores by the Graf method which is the more common usage but the loss is not proper.

eps (numeric(1))
Very small number to substitute zero values in order to prevent errors in e.g. log(0) and/or division-by-zero calculations. Default value is 0.001.

ERV (logical(1))
If TRUE then the Explained Residual Variation method is applied, which means the score is standardized against a Kaplan-Meier baseline. Default is FALSE.

Properness

RISLL is strictly proper when the censoring distribution is independent of the survival distribution and when $G(t)$ is fit on a sufficiently large dataset. ISLL is never proper. Use proper = FALSE for ISLL and proper = TRUE for RISLL. Results may be very different if many observations are censored at the last observed time due to division by $1/eps$ in proper = TRUE.

Time points used for evaluation

If the times argument is not specified (NULL), then the unique (and sorted) time points from the test set are used for evaluation of the time-integrated score. This was a design decision due to the fact that different predicted survival distributions $S(t)$ usually have a discretized time domain which may differ, i.e. in the case the survival predictions come from different survival learners. Essentially, using the same set of time points for the calculation of the score minimizes the bias that would come from using different time points. We note that $S(t)$ is by default constantly interpolated for time points that fall outside its discretized time domain.

Naturally, if the times argument is specified, then exactly these time points are used for evaluation. A warning is given to the user in case some of the specified times fall outside of the time point range of the test set. The assumption here is that if the test set is large enough, it should have a time domain/range similar to the one from the train set, and therefore time points outside that domain might lead to interpolation or extrapolation of $S(t)$.

Implementation differences

If comparing the integrated graf score to other packages, e.g. pec, then method = 2 should be used. However the results may still be very slightly different as this package uses survfit to estimate the censoring distribution, in line with the Graf 1999 paper; whereas some other packages use prodlim with reverse = TRUE (meaning Kaplan-Meier is not used).

Data used for Estimating Censoring Distribution

If task and train_set are passed to $score then $G(t)$ is fit on training data, otherwise testing data. The first is likely to reduce any bias caused by calculating parts of the measure on the test data it is evaluating. The training data is automatically used in scoring resamplings.

Time Cutoff Details

If t_max or p_max is given, then $G(t)$ will be fitted using all observations from the train set (or test set) and only then the cutoff time will be applied. This is to ensure that more data is used for fitting the censoring distribution via the Kaplan-Meier. Setting the t_max can help alleviate inflation of the score when proper is TRUE, in cases where an observation is censored at the last observed time point. This results in $G(t_{max}) = 0$ and the use of eps instead (when t_max is NULL).

References

Graf E, Schmoor C, Sauerbrei W, Schumacher M (1999). “Assessment and comparison of prognostic classification schemes for survival data.” Statistics in Medicine, 18(17-18), 2529–2545. doi:10.1002/(sici)1097-0258(19990915/30)18:17/18<2529::aid-sim274>3.0.co;2-5 .

Super classes

mlr3::Measure -> mlr3proba::MeasureSurv -> MeasureSurvIntLogloss

Methods

Inherited methods

Method `new()`

Creates a new instance of this R6 class.

Usage

MeasureSurvIntLogloss$new(ERV = FALSE)

Arguments

ERV: (logical(1))
Standardize measure against a Kaplan-Meier baseline (Explained Residual Variation)

Method `clone()`

The objects of this class are cloneable with this method.

Usage

MeasureSurvIntLogloss$clone(deep = FALSE)

Arguments

deep: Whether to make a deep clone.

Details

Dictionary

Parameters

Meta Information

Parameter details

Properness

Time points used for evaluation

Implementation differences

Data used for Estimating Censoring Distribution

Time Cutoff Details

References

See also

Super classes

Methods

Public methods

Method new()

Usage

Arguments

Method clone()

Usage

Arguments

Method `new()`

Method `clone()`