Source-linked AI summary
Unbiased split variable selection for random survival forests using maximally selected rank statistics
Marvin N. Wright, Theresa Dankowski, Andreas Ziegler
TL;DR
The paper addresses split-variable bias and limited nonlinear-effect detection in survival forests. It proposes MSR-RF, which uses maximally selected rank statistics and p-value approximations for split selection. Across simulated and real benchmarks, it improves on RSF or CIF in the stated settings, with faster runtime when using a simple approximation.
Problem
Random survival forests favor variables with many possible split points, while conditional inference forests use linear rank tests that may miss nonlinear covariate effects.
Method
MSR-RF selects binary splits with maximally selected rank statistics, compares candidate variables on the p-value scale, and uses p-value approximations instead of conditional Monte Carlo.
Results
MSR-RF performs better than RSF with informative dichotomous and uninformative multi-category variables, better than CIF with nonlinear effects, and faster than both alternatives with a simple approximation.
Takeaways & Limitations
The method provides a useful survival-forest alternative when reducing split-variable bias, modeling nonlinear effects, and computational efficiency are jointly relevant.
Takeaways & Limitations
Unbiased split-variable selection was achieved only with computationally intensive permutation or maximally selected Gauss approaches, creating a trade-off with runtime.
Abstract
from arXiv · showhide
The most popular approach for analyzing survival data is the Cox regression model. The Cox model may, however, be misspecified, and its proportionality assumption may not always be fulfilled. An alternative approach for survival prediction is random forests for survival outcomes. The standard split criterion for random survival forests is the log-rank test statistics, which favors splitting variables with many possible split points. Conditional inference forests avoid this split variable selection bias. However, linear rank statistics are utilized by default in conditional inference forests to select the optimal splitting variable, which cannot detect non-linear effects in the independent variables. An alternative is to use maximally selected rank statistics for the split point selection. As in conditional inference forests, splitting variables are compared on the p-value scale. However, instead of the conditional Monte-Carlo approach used in conditional inference forests, p-value approximations are employed. We describe several p-value approximations and the implementation of the proposed random forest approach. A simulation study demonstrates that unbiased split variable selection is possible. However, there is a trade-off between unbiased split variable selection and runtime. In benchmark studies of prediction performance on simulated and real datasets the new method performs better than random survival forests if informative dichotomous variables are combined with uninformative variables with more categories and better than conditional inference forests if non-linear covariate effects are included. In a runtime comparison the method proves to be computationally faster than both alternatives, if a simple p-value approximation is used.
1 Introduction
Survival forests offer an alternative when Cox-model assumptions or scalability are problematic, but existing forest split rules have complementary weaknesses. The paper motivates MSR-RF to reduce split-variable bias while detecting nonlinear covariate effects.
- Cox models may be misspecified, violate proportional hazards, and be poorly suited to large-scale omics or image-analysis problems.
- Random survival forests select the largest log-rank statistic across variables and split points, favoring covariates with many possible split points.
- Conditional inference forests reduce split-variable bias by separating variable selection from split-point selection, but default linear rank tests may not detect nonlinear effects.
- Maximally selected rank statistics adjust for multiple candidate splits and compare variables on the p-value scale, while p-value approximation introduces a runtime trade-off.
2 Maximally selected rank statistics for survival endpoints
The method chooses a cutpoint that maximizes separation between two survival groups using standardized rank statistics. Variables are compared through p-values adjusted for searching across possible cutpoints.
- The optimal cutpoint is the one that maximizes separation between the survival curves of the two resulting groups.
- For censored survival observations, the procedure uses log-rank scores and forms a rank statistic from scores among observations with X_i ≤ µ.
- The null hypothesis states that the survival-outcome distribution is identical on both sides of a candidate cutpoint.
- The maximally selected statistic searches cutpoints between covariate quantiles ε1 and ε2, restricting group sizes while maximizing the standardized test statistic.
- Because searching over split points can bias variable selection, the method obtains a p-value for each variable's maximally selected statistic and compares variables on that scale.
3 Survival forests with maximally selected rank statistics
MSR-RF extends conditional-inference forest ideas with maximally selected rank statistics in a two-step survival-forest splitting procedure. It estimates forest survival functions by averaging tree-specific Kaplan–Meier predictions and supports computational implementations for larger datasets.
- 3.1 Core elements of the algorithm: The algorithm retains random-forest sampling and random covariate selection, growing one tree for each bootstrap sample or subsample.
- 3.1 Core elements of the algorithm: MSR-RF uses a two-step split procedure that selects each covariate's best binary split, obtains its adjusted p-value, and chooses variables using those p-values.Multiple-split adjustments are unnecessary for dichotomous variables but matter for ordinal and continuous covariates.
- 3.2 Estimation: In each terminal node, MSR-RF estimates survival with a Kaplan–Meier estimator and averages predictions across trees for each observation and time point.
- 3.2 Estimation: Prediction error is estimated from out-of-bag observations using the censored-data Brier score and its integrated form.
- 3.3 Implementation: The fast ranger implementation uses the minimum of two Lausen-based p-value approximations and is reported to provide high prediction accuracy.
4 Simulation study 1: split variable selection
The first simulation study assessed split-variable selection bias under null associations and compared p-value approximations, resampling schemes, and computational runtimes.
- The simulation varied null covariate cardinalities, comparing unadjusted splitting with six p-value approximations under subsampling and bootstrapping.Covariates had 2, 4, 10, or 20 ordered categories, or were continuous.
- Subsampling produced unbiased splits for condMC and exactGauss, whereas bootstrapping induced preferential selection of covariates with many categories.HL instead favored covariates with few categories, particularly under subsampling.
- HL could not produce results for n = 200 with bootstrapping because its approximation cannot handle sample sizes greater than 171.The limitation arises from computing the number of permutations with the gamma function in double precision.
- RSF was biased toward variables with many categories, while standard CIF was unbiased; CIF with maximally selected rank statistics was unbiased under subsampling but biased under bootstrapping.
- Lau92 and Lau94 were approximately equally fast and faster than the other approximations, while HL runtime grew exponentially with sample size.condMC and exactGauss scaled approximately linearly, with a larger slope for exactGauss; minLau combined the Lau92 and Lau94 runtimes.
5 Simulation study 2: prediction performance
The second simulation study compared prediction error across forest methods and p-value approximations in scenarios emphasizing categorical selection bias and nonlinear effects.
- Scenario 1 combined six strongly predictive dichotomous covariates with 100 normally distributed variables without effect, whereas Scenario 2 included strong nonlinear and moderate linear covariate effects.
- Prediction error was evaluated with nested cross-validation using integrated Brier scores relative to the Cox model, under both subsampling and bootstrapping.The analyses used ten outer folds, five inner folds, 500 trees, and mtry = p/2.
- MSR-RF outperformed RSF and standard CIF in the benchmark scenarios, especially when informative dichotomous variables coexisted with many uninformative variables or nonlinear effects were present.In Scenario 1, MSR-RF performed best with Lau92; in Scenario 2, it performed best with all approximations except HL.
- RSF suffered from biased split selection in Scenario 1, while standard CIF missed nonlinear covariates in Scenario 2 because its split-variable test used a linear rank statistic.
6 Real data applications: survival prediction of breast cancer patients
The study evaluates MSR-RF, RSF, and CIF variants on three breast cancer survival datasets, including clinical, gene-expression, and GWAS data. MSR-RF achieved comparable or better prediction performance while remaining computationally feasible on the larger datasets.
- The datasets covered clinical covariates, 24,481 gene-expression measurements, and a pruned GWAS dataset retaining 151,346 SNPs.The three analyses used recurrence-free, metastasis-free, and relapse-free survival endpoints, respectively.
- Prediction accuracy was estimated with nested cross-validation using integrated Brier scores across 500-tree forests and both resampling schemes.MSR-RF used ε1 = 0.1 and the minimum of two p-value approximations; analyses used mtry = p/2.
- MSR-RF outperformed RSF on the nki dataset and was the only method to complete the success GWAS analysis.Both CIF variants failed to complete the nki analysis within four days, while CIF and RSF aborted with memory errors on the success dataset.
- On GBSG2, all methods performed similarly, but MSR-RF was fastest while CIF maxstat required over 3 hours.The other methods analyzed GBSG2 in only a few seconds.
- Subsampling and bootstrapping produced no substantial prediction differences across the three datasets.
7 Discussion
MSR-RF reduces split-variable selection bias while handling nonlinear covariate effects, but its implementation balances statistical unbiasedness against computational efficiency and depends on modeling choices.
- Real-data performance: Real-data analyses showed comparable performance in a clinical dataset, lower prediction error than RSF in gene expression data, and completion of the genome-wide association analysis only with fast MSR-RF.Runtime was lowest in the clinical dataset, and MSR-RF was substantially faster than RSF in the gene expression example.
- Runtime and bias: The ranger implementation is a compromise: only permutation and maximally selected Gauss approaches achieved unbiased selection, but both were computationally intensive.The minimum of the Lausen et al. and Lausen–Schumacher approximations was identified as the best larger-forest trade-off.
- Extensions: The method can extend beyond survival outcomes to continuous, binary, and categorical endpoints through maximally selected statistics and related p-value procedures.Continuous outcomes use rank scores, while categorical outcomes can use maximally selected χ2 statistics or proposed approximations.
- Split-point restrictions: Restricting split points to an interval supports p-value approximations by suppressing extreme end-cut splits, although relaxing minprop may improve prediction in high-noise settings.The authors note that smaller minprop values, including zero, may be reasonable because approximations performed well for small samples.
- Scope and assumptions: The approach assumes continuous marginal covariate distributions; unrestricted unordered categorical splits would increase computation and the multiple-testing burden exponentially.Ordering categories before splitting is discussed as a possible alternative for categorical covariates.
- Performance: MSR-RF performed better than RSF with mixed covariate types and better than CIF when covariate effects were nonlinear.Across the considered scenarios, no random forest method failed completely, and random forest methods performed equally well or better than the standard Cox model in simulations.