Integrated Schmid Score Survival Measure

Calculates the Integrated Schmid Score (ISS), aka integrated absolute loss.

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:

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

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

The re-weighted ISS (RISS) is:

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

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():

MeasureSurvSchmid$new()
mlr_measures$get("surv.schmid")
msr("surv.schmid")

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	-
remove_obs	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 (truncate $S(t)$). Mutually exclusive with p_max or times. It's recommended to set t_max to avoid division by eps, see "Time Cutoff Details" section. 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, see Sonabend et al. (2024). If FALSE then weights scores by the Graf method which is the more common usage but the loss is not proper. See "Properness" section for more details.

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.

remove_obs (logical(1))
Only effective when t_max or p_max is provided. Default is FALSE. If TRUE, then we remove test observations for which the observed time (event or censoring) is strictly larger than t_max. See "Time Cutoff Details" section for more details.

Properness

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

See Sonabend et al. (2024) for more details. The use of proper = TRUE is considered experimental and should be used with caution.

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 using all observations from the train set, otherwise the test set is used. Using the train set is likely to reduce any bias caused by calculating parts of the measure on the test data it is evaluating. Also usually it means that more data is used for fitting the censoring distribution $G(t)$ via the Kaplan-Meier. The training data is automatically used in scoring resamplings.

Time Cutoff Details

If t_max or p_max is given, then the predicted survival function $S(t)$ is truncated at the time cutoff for all observations.

Also, if remove_obs = TRUE, observations with observed times $t > t_{max}$ are removed. This data preprocessing step mitigates issues that arise when using IPCW in cases of administrative censoring, see Kvamme et al. (2023). Practically, this step, along with setting a time cutoff t_max, helps mitigate the inflation of the score observed when an observation is censored at the final time point. In such cases, $G(t) = 0$, triggering the use of a small constant eps instead. This inflation particularly impacts the proper version of the score, see Sonabend et al. (2024) for more details. Note that the t_max and remove_obs parameters do not affect the estimation of the censoring distribution, i.e. always all the observations are used for estimating $G(t)$.

If remove_obs = FALSE, inflated scores may occur. While this aligns more closely with the definitions presented in the original papers, it can lead to misleading evaluation and poor optimization outcomes when using this score for model tuning.

References

Schemper, Michael, Henderson, Robin (2000). “Predictive Accuracy and Explained Variation in Cox Regression.” Biometrics, 56, 249–255. doi:10.1002/sim.1486 .

Schmid, Matthias, Hielscher, Thomas, Augustin, Thomas, Gefeller, Olaf (2011). “A Robust Alternative to the Schemper-Henderson Estimator of Prediction Error.” Biometrics, 67(2), 524–535. doi:10.1111/j.1541-0420.2010.01459.x .

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/2212.05260v2.

Kvamme, Havard, Borgan, Ornulf (2023). “The Brier Score under Administrative Censoring: Problems and a Solution.” Journal of Machine Learning Research, 24(2), 1–26. ISSN 1533-7928, http://jmlr.org/papers/v24/19-1030.html.

Super classes

mlr3::Measure -> mlr3proba::MeasureSurv -> MeasureSurvSchmid

Methods

Inherited methods

Method `new()`

Creates a new instance of this R6 class.

Usage

MeasureSurvSchmid$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

MeasureSurvSchmid$clone(deep = FALSE)

Arguments

deep: Whether to make a deep clone.

Examples

library(mlr3)

# Define a survival Task
task = tsk("lung")

# Create train and test set
part = partition(task)

# Train Cox learner on the train set
cox = lrn("surv.coxph")
cox$train(task, row_ids = part$train)

# Make predictions for the test set
p = cox$predict(task, row_ids = part$test)

# ISS, G(t) calculated using the test set
p$score(msr("surv.schmid"))
#> surv.schmid 
#>   0.2901703 

# ISS, G(t) calculated using the train set (always recommended)
p$score(msr("surv.schmid"), task = task, train_set = part$train)
#> surv.schmid 
#>   0.2568166 

# ISS, ERV score (comparing with KM baseline)
p$score(msr("surv.schmid", ERV = TRUE), task = task, train_set = part$train)
#> surv.schmid 
#>  0.01762828 

# ISS at specific time point
p$score(msr("surv.schmid", times = 365), task = task, train_set = part$train)
#> surv.schmid 
#>   0.4244478 

# ISS at multiple time points (integrated)
p$score(msr("surv.schmid", times = c(125, 365, 450), integrated = TRUE), task = task, train_set = part$train)
#> surv.schmid 
#>   0.3551183 

# ISS, use time cutoff
p$score(msr("surv.schmid", t_max = 700), task = task, train_set = part$train)
#> surv.schmid 
#>   0.3236368 

# ISS, use time cutoff and also remove observations
p$score(msr("surv.schmid", t_max = 700, remove_obs = TRUE), task = task, train_set = part$train)
#> surv.schmid 
#>   0.2893284 

# ISS, use time cutoff corresponding to specific proportion of censoring on the test set
p$score(msr("surv.schmid", p_max = 0.8), task = task, train_set = part$train)
#> surv.schmid 
#>   0.3300005 

# RISS, G(t) calculated using the train set
p$score(msr("surv.schmid", proper = TRUE), task = task, train_set = part$train)
#> surv.schmid 
#>   0.2003059

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

Examples

Method `new()`

Method `clone()`