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A two-sample test for high-dimensional data with applications to gene-set testing
Song Xi Chen, Ying-Li Qin
TL;DR
In large-p, small-n settings, Hotelling’s T² may be unusable, motivating a two-sample mean test that does not impose an explicit relationship between dimension and sample size. The paper establishes asymptotic normality under covariance conditions and demonstrates gene-set testing with per-set P-values.
Problem
Method
Results
The proposed test is demonstrated for gene-set significance testing and readily produces a P-value for each set.
Source: §1 · p. 3Takeaways & Limitations
Per-set P-values allow Bonferroni or false-discovery procedures to account for multiplicity across gene sets.
Source: §1 · pp. 2–3When p is bounded, the stated asymptotic normality may no longer hold.
Source: §3 · p. 10
Abstract
from arXiv · showhide
We propose a two-sample test for the means of high-dimensional data when the data dimension is much larger than the sample size. Hotelling's classical $T^2$ test does not work for this "large $p$, small $n$" situation. The proposed test does not require explicit conditions in the relationship between the data dimension and sample size. This offers much flexibility in analyzing high-dimensional data. An application of the proposed test is in testing significance for sets of genes which we demonstrate in an empirical study on a leukemia data set.
1. Introduction.
High-dimensional settings often pair large data dimensions with small samples, complicating conventional multivariate analysis and simultaneous testing. The paper proposes a two-sample mean test and demonstrates its use for gene-set significance testing with per-set P-values.
1. Introduction.
- The test targets simultaneous two-sample tests of means when the number of hypotheses is large, a setting motivated by identifying significant genes.The hypotheses concern means across data dimensions.Source: §1 · p. 1
- Earlier simultaneous-testing results allow p to grow at an exponential rate in n, with bounded-support assumptions for some marginal distributions.The cited rates are e^{αn^β} for positive constants α and β.Source: §1 · p. 2
- Bai and Saranadasa’s test replaces Hotelling’s inverse-covariance quadratic form with a Euclidean norm, but its p/n restriction is too limiting for large-p, small-n settings.Their result assumes equal covariance matrices.Source: §1 · p. 2
- The proposed test allows p to be arbitrarily large, independent of sample size, under a covariance condition, and its statistic is asymptotically normal.For common covariance, the condition is trivially satisfied if all eigenvalues are bounded or if the largest eigenvalue grows sufficiently slowly.Source: §1 · p. 2
- Gene-set testing needs P-values for all sets so Bonferroni or false-discovery procedures can account for multiplicity.The paper demonstrates the proposed test for assessing gene-set significance.Source: §1 · pp. 2–3
2. Test statistic.
The section formulates the two-sample mean-testing problem and explains why covariance-inverse methods are problematic when dimensions are high. It motivates a Euclidean-norm approach and describes the trace adjustment used in the related Bai–Saranadasa statistic.
2. Test statistic.
- The problem concerns whether two high-dimensional populations have the same mean, with each population described by its mean and covariance.The section also frames the question as simultaneous testing across marginal means.Source: §2 · p. 3
- When p exceeds n, Hotelling’s T² may be undefined because the sample covariance can be noninvertible; its power also decreases as p/n grows.The inverse covariance matrix is identified as a source of difficulty in high dimensions.Source: §2 · pp. 3–4
- The proposed test is motivated by replacing Hotelling’s inverse-covariance term with a Euclidean norm of the difference between sample means.The cited Bai–Saranadasa approach assumes equal covariance matrices.Source: §2 · p. 4
- The Bai–Saranadasa trace subtraction offsets diagonal terms, but using tr(Sn) itself imposes dimensionality demands.The subtraction makes the expected statistic equal to the squared norm of the population mean difference.Source: §2 · p. 4
3. Main results.
The proposed statistic is asymptotically normal under covariance trace conditions without a direct p–n relationship restriction, enabling a two-sample test across high dimensions. Its variance estimators and power theory support unequal covariances, while sparsity and component-identification remain practical boundaries.
3. Main results.
- The framework allows unequal covariance matrices and uses a linear-transformation model with flexible covariance structure.The model permits m ≥ p; its innovation vectors have zero means and identity covariance, with the stated moment and dependence conditions.Source: §3 · p. 5
- Theorem 1 establishes asymptotic normality of Tn without an explicit restriction between p and n; the dimension must instead satisfy condition (3.6) or (3.7), and Tn is univariate.When all eigenvalues of Σ are bounded, (3.7) is trivially true.Source: §3 · pp. 6–7
- The proposed trace estimators are ratio-consistent, supporting variance estimation for the test statistic.Cross-validation-style sample means exclude the observations whose deviations are being formed; subtracting one sample mean per observation also avoids a difficult-to-control fourth-norm term.Source: §3 · pp. 7–8
- The proposed test has nontrivial power under the two alternative cases; the discussion identifies (3.4) as local and describes (3.5) as a version of fixed alternatives.With bounded covariance eigenvalues and equal componentwise mean differences, (3.4) implies δ = o(n−1/2), smaller than δ = O(n−1/2) for fixed p.Source: §3 · pp. 6, 8, 9
- Unlike tests combining marginal statistics, the proposed test avoids their stated dimension-dependent convergence conditions, but a multivariate rejection does not identify individual components.The authors note that follow-up procedures are needed for component identification.Source: §3 · p. 9
- Under sparsity, the test may lose power when p is very large relative to sample size and the number of nonzero mean components.The paper also reports that simulations found robust power and greater power than Bonferroni- or FDR-based multiple-comparison tests in the studied settings.Source: §3 · p. 9
- For paired before-and-after measurements, differencing each pair reduces the problem to a one-sample high-dimensional test.The proposed paired-data modification has a corresponding statistic and variance estimation under analogous conditions.Source: §3 · p. 10
- When p is bounded, the stated asymptotic normality may fail, but the test remains usable with estimated critical values.The bounded-dimension limit is a weighted sum of chi-square variables with constants estimated for the critical values.Source: §3 · p. 10
4. Gene-set testing.
The paper applies a high-dimensional two-sample test to gene sets, where genes may act together and sets can overlap. In leukemia data, many gene sets were significant, while random-group back-testing produced distributions closer to their null expectations.
4. Gene-set testing.
- For large gene sets, the proposed two-sample test supplies set-level P-values for Bonferroni or false-discovery-rate procedures.Hotelling’s test may be used for low-dimensional sets.Source: §4 · pp. 10–11
- The analysis compared BCR/ABL and NEG leukemia subclasses across 575 BP, 221 MF, and 154 CC gene ontology terms.After gene filtering, 2,391 genes remained; the largest sets contained 2,059 BP, 2,112 MF, and 2,078 CC genes.Source: §4 · p. 11
- 259 BP, 110 MF, and 53 CC gene sets were significant under Bonferroni correction at family-wise error rate 0.05.Figure 1 shows P-values clustered near zero and average Qn-values above zero across the three categories.Source: §4 · p. 11, Figure 1 · p. 12
- Gene sets can overlap, and significant-set overlaps averaged 198.9 for BP, 55.6 for MF, and 41.6 for CC, indicating strong dependence among sets and their P-values.The paper notes that overlap helps explain the large number of significant sets.Source: §4 · pp. 10–12
- In random splits of the NEG class, P-value distributions were closer to uniform and Qn-value histograms were centered near zero and closer to normal than in Figure 1.This back-test used two randomly assigned NEG groups.Source: §4 · pp. 12–13, Figure 2 · p. 13
5. Simulation studies.
The simulations compare the proposed test with the Bai–Saranadasa test and Bonferroni and FDR procedures under dependent and sparse alternatives. The proposed test generally performs better in the reported settings, while its power depends on sparsity and dependence.
5.1. Moving average model.
- Moving-average simulations compare centered Gamma(4,1) and N(0,1) innovations under weaker two-dependence and full-dependence structures; both samples have equal covariance matrices, with tr(Σ²)=0.1.They use p=500 and 1000, with n=124 and 138, respectively.Source: §5.1 · p. 14
- The proposed test is more powerful than Bai–Saranadasa across the moving-average cases and remains more powerful than Bonferroni and FDR even with 95% or 99% true nulls.All tests lose power under full dependence; the proposed estimator of tr(Σ2) also has smaller bias and standard deviation than Bai–Saranadasa’s in the reported cases.Source: §5.1 · pp. 15–16, Table 1 · p. 15, Table 2 · p. 16
5.2. Sparse model.
- The sparse-model simulations vary the fraction of nonzero alternatives through c and compare the proposed test with FDR and Bonferroni procedures.The model has q=[pc] nonzero alternatives; smaller c means a sparser alternative.Source: §5.1 · p. 14, §5.2 · p. 17
- For the sparse model, the proposed test has lower power than FDR and Bonferroni at c=0.25, similar power at c=0.35, and begins to surpass them at c=0.45, with a wider gap at c=0.55.The test’s power is expected to be much reduced when c < 1/2 with respect to p.Source: §5.2 · p. 17, Table 4 · p. 18
- The proposed variance estimators show very little bias and variance even when both sample sizes are 10.This result is reported for the sparse model.Source: §5.2 · p. 18
6. Technical details.
The technical section derives the mean and variance of the test statistic and establishes its asymptotic normality. The proof uses martingale and independent-average arguments under the stated conditions.
6.1. Derivations for E(Tn) and Var(Tn).
- The derivations show E(Tn1)=0 and E(Tn2)=∥µ1−µ2∥2, with the mean difference expressible as µ1′µ1+µ2′µ2−2µ1′µ2.The variance expression includes terms involving each sample covariance and the mean difference.Source: §6.1 · pp. 18–19, §6.2 · p. 19
6.2. Asymptotic normality of Tn.
- Under condition (3.4), the proof establishes asymptotic normality of Tn using a zero-mean, square-integrable martingale sequence.The appendix identifies the sequence as a martingale and concludes that the standardized statistic converges in distribution to N(0,1).Source: §6.2 · pp. 20, 25
- For the alternative condition (3.5), the proof obtains asymptotic normality by treating Tn2 as a sum of two independent averages.The proof also assumes µ1=µ2=0 without loss of generality when deriving the asymptotic normality of Tn1.Source: §6.2 · pp. 20, 25