Monte Carlo P Value Estimation Adaptive

Adaptive Multi-Level Split Monte-Carlo P-Value Estimation

Summary

A method for computing arbitrarily low gene set enrichment analysis (GSEA) P-values using an adaptive multi-level split Monte-Carlo scheme, eliminating fixed lower-bound constraints by enabling precision refinement through iterative sampling. This skill is essential when gene set rankings must be compared at extreme statistical significance levels or when default P-value resolution (e.g., eps=1e-10) is insufficient for downstream filtering or interpretation.

When to use

When you have ranked gene statistics and gene set collections, and your analysis requires P-value discrimination below a fixed lower bound (e.g., distinguishing between pathways at p < 1e-10). Particularly relevant when comparing many gene sets where the default eps=1e-10 threshold collapses multiple pathways to identical P-values, or when publication-grade precision is needed for pathway ranking.

When NOT to use

• Input gene sets are very small (< 15 genes) or very large (> 500 genes) — these are filtered by minSize/maxSize and will not yield valid estimates regardless of eps setting.
• Computational resources are severely constrained — eps=0 requires iterative refinement and will be substantially slower than eps=1e-10; use default eps if runtime is critical.
• Gene ranking is unweighted or non-continuous — fgsea requires preranked, continuous statistics; if you have binary phenotype calls, compute Wilcoxon or t-test statistics first.

Inputs

• ranked gene statistics (numeric vector with gene identifiers as names)
• gene set collection (list of character vectors containing gene identifiers)
• minSize and maxSize thresholds (integer, typically 15 and 500)

Outputs

• fgsea results table with columns: pathway, pval, padj, log2err, ES, NES, size
• P-value precision estimates with associated log2err convergence metrics
• comparison table of default vs. adaptive estimates (differences, fold-changes)

How to apply

Load ranked gene statistics (exampleRanks) and pathway definitions (examplePathways). Run fgsea() with eps=0.0 to disable the lower-bound estimate and activate the adaptive multi-level split Monte-Carlo refinement scheme. This parameter switch allows the algorithm to iteratively refine P-value estimates without a fixed minimum threshold. Compare results against a default run (eps=1e-10) by joining on pathway identifier and computing P-value fold-change ratios and difference distributions (log-scale). Validate by confirming pathway rankings remain consistent between runs, no numerical errors occur, and P-values decrease monotonically as precision increases. Record log2err values to assess convergence of the multi-level scheme.

Related tools

• fgsea (Core implementation of adaptive multi-level split Monte-Carlo P-value estimation; provides fgsea() function with eps parameter control and output table generation.) — https://github.com/ctlab/fgsea
• data.table (Efficient joining and comparison of results from eps=1e-10 and eps=0 runs; grouping and aggregation of P-value differences.)
• ggplot2 (Visualization of P-value distribution changes (log-scale) and identification of pathways with greatest precision improvement.)
• R (Runtime environment for fgsea, data.table, and ggplot2 integration.)

Examples

fgseaRes <- fgsea(pathways = examplePathways, stats = exampleRanks, eps = 0.0, minSize = 15, maxSize = 500)

Evaluation signals

• P-values computed with eps=0 are ≤ those with eps=1e-10 for the same pathways (monotonic precision gain).
• log2err values are numeric and finite (not NA or Inf) for eps=0 run, indicating convergence of the multi-level scheme.
• Pathway rankings (order by p-value) remain consistent or improve (no rank inversions) between eps=1e-10 and eps=0 runs.
• No numerical errors or warnings during fgsea execution with eps=0; function completes without NaN or undefined statistics in output table.
• Comparison table shows meaningful P-value differences for pathways originally tied at eps=1e-10 boundary; fold-change ratio distribution is unimodal and right-skewed (most pathways show modest refinement, a few show extreme improvement).

Limitations

• Adaptive refinement is computationally expensive; eps=0 will significantly increase runtime compared to eps=1e-10, especially for large pathway collections or highly significant pathways.
• Extremely low P-values (< 1e-20) become numerically unstable and may be sensitive to random seed and number of Monte-Carlo iterations; reproducibility requires explicit seed control.
• The adaptive scheme assumes sufficiently many gene-set members to enable reliable split-level refinement; very small gene sets (near minSize) may show erratic log2err or unrepresentative P-value estimates.
• No changelog tracking for algorithm evolution; users upgrading fgsea versions must re-run eps=0 analyses as the underlying multi-level scheme may be refined.

Evidence

• [methods] fgsea implements an adaptive multi-level split Monte-Carlo scheme that enables calculation of arbitrarily low GSEA P-values for gene set collections without a fixed lower-bound constraint.: "fgsea implements an adaptive multi-level split Monte-Carlo scheme that enables calculation of arbitrarily low GSEA P-values"
• [readme] P-value estimation is based on an adaptive multi-level split Monte-Carlo scheme allowing arbitrarily low P-values.: "P-value estimation is based on an adaptive multi-level split Monte-Carlo scheme"
• [readme] As you can see fgsea has a default lower bound eps=1e-10 for estimating P-values. If you need to estimate P-value more accurately, you can set the eps argument to zero in the fgsea function.: "If you need to estimate P-value more accurately, you can set the eps argument to zero"
• [methods] Run fgsea with eps=0 to disable the lower-bound estimate and enable adaptive multi-level split Monte Carlo refinement.: "Run fgsea again with eps=0 to disable the lower-bound estimate and enable adaptive multi-level split Monte Carlo refinement"
• [methods] Validate that eps=0 run completes without numerical errors and produces valid P-value rankings consistent with default run.: "Validate that eps=0 run completes without numerical errors and produces valid P-value rankings consistent with default run"