Random Survival Forests For Integrated Omics

Why Use a Forest for Integrated Omics?

An omics predictor may relate to survival in a curved, threshold-like, or otherwise non-monotone way. The predictive value of one feature may also depend on another feature, including a feature measured in a different omics layer. A linear predictor with a prespecified functional form can represent such patterns only if those forms and interactions are built into the model. A random survival forest offers a different strategy: it combines many survival trees, each of which partitions the predictor space into regions with relatively similar observed survival experience.

This flexibility does not make the forest assumption-free. The observed outcomes are right-censored, so the method must distinguish an observed event from a subject whose event time is not observed by the end of follow-up. It also needs enough events and comparable subjects in the relevant regions of the omics feature space to estimate survival patterns usefully. Its variable importance measures are about predictive performance under a specified procedure; they are not regression coefficients, hazard ratios, or causal effects.

In this tutorial, the event of interest is a single event type. The forest uses the observed time and event indicator, not a fabricated event time for censored subjects. Competing events require a target and an appropriate competing-risk prediction procedure; ordinary single-event forest output should not be interpreted as a competing-risk cumulative incidence estimate.

How a Survival Tree Splits

At a node of a tree, candidate splits divide the subjects into two groups, such as those with a feature below a threshold and those at or above it. A split is useful when the groups have different survival experience. A common survival-tree criterion is the logrank statistic. It compares the observed number of events in one child group with the number expected if, at each event time, failures were allocated in proportion to the groups’ risk-set sizes.

The expected count \(Y_{A,t}d_t/Y_t\) follows from the risk-set proportions. The variance term is the hypergeometric variance for the number of the \(d_t\) failures assigned to \(A\), under random allocation among the \(Y_t\) people at risk. The factor \((Y_t-d_t)/(Y_t-1)\) is the finite-population correction. Summing the time-specific contributions gives the conventional logrank variance estimate. Tree software may use a closely related criterion or an implementation-specific rule for ties and degenerate splits, so the actual fitting procedure should be recorded.

A tree typically searches a random subset of predictors at each node and chooses a candidate split with a large absolute standardized score or equivalent squared score. This random candidate selection, along with resampling subjects, makes the trees differ. A variable can be considered repeatedly in different parts of a tree, enabling stepwise partitions that represent thresholds and combinations of features. The process can capture interactions without specifying a product term in advance: a split on one feature followed by a split on another creates a region whose prediction depends on both.

Terminal-Node Estimates and Forest Predictions

Once a tree is grown, a new subject’s predictors route that subject through the splits to one terminal node. A common terminal-node estimate is the Nelson–Aalen cumulative hazard estimate. For event time \(t\) within that node, it adds the fraction of people at risk who experience the event. The cumulative hazard estimate then increases across event times. A survival estimate can be obtained by exponentiating the negative cumulative hazard. Some implementations combine tree survival estimates directly instead; the convention should be specified because these aggregation methods need not give identical results.

This prediction is a model-based estimate of survival for subjects with the specified predictor profile, under the fitted forest and its assumptions. It is not a guarantee that an individual will experience that outcome. The forest’s partitions can approximate complicated patterns, but the resolution of those patterns depends on sample size, event count, the candidate-feature strategy, and stopping rules such as minimum terminal-node size.

Out-of-Bag Variable Importance

A commonly used measure is permutation variable importance. For each tree, subjects not selected in that tree’s bootstrap sample are out-of-bag (OOB) for that tree. The fitted tree predicts these subjects. The values of one feature are then permuted among the OOB subjects, breaking the observed link between that feature and their other data while leaving the other feature columns unchanged. The tree’s OOB prediction error is recalculated. Aggregating the change over trees gives an importance measure: a large increase in error after permutation suggests that the feature contributed to prediction under this forest, error measure, and resampling procedure.

Because this is a comparison of predictive error, the result depends on the error measure, the permutation scheme, the training sample, and forest settings. It is not an effect estimate. In particular, importance does not give the direction or size of a feature’s association with event time, does not estimate a hazard ratio, and does not show that changing the feature would change survival.

Assumptions, Diagnostics, and Alternatives

The forest can flexibly partition predictors, but its survival estimates still require defensible outcome and sampling conditions. Subjects should be independent sampling units, or the analysis must account for clustering at the design level. For standard right-censored estimation, censoring should be independent of the event time conditional on the information used for prediction, at least within the regions represented by the fitted trees. Informative censoring can distort estimated survival even if the split algorithm is flexible. There must also be enough follow-up and events to support the predictions of interest; no algorithm can recover late survival reliably when essentially no subjects remain under observation.

Useful diagnostics include checking event counts and risk-set sizes across follow-up, the range over which predictions are supported, and the stability of predictions and importance rankings across resamples or reasonable forest settings. Inspect whether a few very small terminal nodes drive extreme predictions. Check whether preprocessing, batch correction, feature filtering, and any supervised feature selection were performed using training data only within each validation or resampling split. If information from held-out subjects influenced these steps, OOB predictions can be optimistic despite the OOB mechanism.

In multi-omics data, high correlation and different numbers of measured features across layers complicate importance. A layer with thousands of correlated features may distribute predictive information across many substitutes, while a single isolated feature may appear more important. Feature-wise permutation can also create combinations of values that rarely occur jointly in real samples. Consider reporting the precise importance procedure and examining layer-level or grouped permutation analyses when that matches the scientific question; grouped importance answers a different question than importance for one feature.

The forest is therefore a useful predictive alternative when nonlinearities or interactions are plausible and a prespecified parametric form is not the main goal. It does not automatically solve high dimensionality, confounding, transportability, or causal interpretation. When the objective is an interpretable effect estimate, a forest’s split patterns and permutation scores should not be substituted for a model and estimand designed for that objective.

Common Pitfall

A second computational pitfall occurs in the logrank split variance at the last event time. If only one subject is at risk, substituting \(Y_t=1\) into a formula containing \(Y_t-1\) produces division by zero. The correct handling is to define that contribution as zero because there is no random allocation of the sole person at risk. Omitting this boundary case can make a hand calculation appear undefined even though the risk-set interpretation is straightforward.