Multiple Testing in the Survival Analysis of Microarray Data José A. Correa, Florida Atlantic University Sandrine Dudoit, Univ. California Berkeley Darlene R. Goldstein, École Polytechnique Fédérale de Lausanne Contact: linkage@stat.berkeley.edu Software: http://www.math.fau.edu/correa/

cDNA gene expression data Data on m genes for n samples mRNA samples sample1 sample2 sample3 sample4 sample5 … 1 0.46 0.30 0.80 1.51 0.90 ... 2 -0.10 0.49 0.24 0.06 0.46 ... 3 0.15 0.74 0.04 0.10 0.20 ... 4 -0.45 -1.03 -0.79 -0.56 -0.32 ... 5 -0.06 1.06 1.35 1.09 -1.09 ... Genes Gene expression level of gene i in mRNA sample j 3 = (normalized) Log( Red intensity / Green intensity)

Multiple Testing Problem Simultaneously test m null hypotheses, one for each gene j Hj: no association between expression level of gene j and the covariate or response Because microarray experiments simultaneously monitor expression levels of thousands of genes, there is a large multiplicity issue Would like some sense of how ‘surprising’ the observed results are

Hypothesis Truth vs. Decision # not rejected # rejected totals # true H U V (F +) m0 # non-true H T S m1 m - R R m Decision Truth

Type I (False Positive) Error Rates Per-family Error Rate PFER = E(V) Per-comparison Error Rate PCER = E(V)/m Family-wise Error Rate FWER = p(V ≥ 1) False Discovery Rate FDR = E(Q), where Q = V/R if R > 0; Q = 0 if R = 0

Strong vs. Weak Control All probabilities are conditional on which hypotheses are true Strong control refers to control of the Type I error rate under any combination of true and false nulls Weak control refers to control of the Type I error rate only under the complete null hypothesis (i.e. all nulls true) In general, weak control without other safeguards is unsatisfactory

Comparison of Type I Error Rates In general, for a given multiple testing procedure, PCER  FWER  PFER, and FDR  FWER, with FDR = FWER under the complete null

Adjusted p-values (p*) If interest is in controlling, e.g., the FWER, the adjusted p-value for hypothesis Hj is: pj* = inf {: Hj is rejected at FWER } Hypothesis Hj is rejected at FWER  if pj*   Adjusted p-values for other Type I error rates are similarly defined

Some Advantages of p-value Adjustment Test level (size) does not need to be determined in advance Some procedures most easily described in terms of their adjusted p-values Usually easily estimated using resampling Procedures can be readily compared based on the corresponding adjusted p-values

A Little Notation For hypothesis Hj, j = 1, …, m observed test statistic: tj observed unadjusted p-value: pj Ordering of observed (absolute) tj: {rj} such that |tr1|  |tr2|  …  |trm| Ordering of observed pj: {rj} such that |pr1|  |pr2|  …  |prm| Denote corresponding RVs by upper case letters (T, P)

Control of the FWER Bonferroni single-step adjusted p-values pj* = min (mpj, 1) Holm (1979) step-down adjusted p-values prj* = maxk = 1…j {min ((m-k+1)prk, 1)} Hochberg (1988) step-up adjusted p-values (Simes inequality) prj* = mink = j…m {min ((m-k+1)prk, 1) }

Control of the FWER Westfall & Young (1993) step-down minP adjusted p-values prj* = maxk = 1…j { p(maxl{rk…rm} Pl  prk H0C )} Westfall & Young (1993) step-down maxT adjusted p-values prj* = maxk = 1…j { p(maxl{rk…rm} |Tl| ≥ |trk| H0C )}

Westfall & Young (1993) Adjusted p-values Step-down procedures: successively smaller adjustments at each step Take into account the joint distribution of the test statistics Less conservative than Bonferroni, Holm, or Hochberg adjusted p-values Can be estimated by resampling but computer-intensive (especially for minP)

maxT vs. minP The maxT and minP adjusted p-values are the same when the test statistics are identically distributed (id) When the test statistics are not id, maxT adjustments may be unbalanced (not all tests contribute equally to the adjustment) maxT more computationally tractable than minP maxT can be more powerful in ‘small n, large m’ situations

Control of the FDR Benjamini & Hochberg (1995): step-up procedure which controls the FDR under some dependency structures prj* = mink = j…m { min ([m/k] prk, 1) } Benjamini & Yuketieli (2001): conservative step- up procedure which controls the FDR under general dependency structures prj* = mink = j…m { min (m [1/j]/k] prk, 1) } Yuketieli & Benjamini (1999): resampling based adjusted p-values for controlling the FDR under certain types of dependency structures

Identification of Genes Associated with Survival Data: survival yi and gene expression xij for individuals i = 1, …, n and genes j = 1, …, m Fit Cox model for each gene singly: h(t) = h0(t) exp(jxij) For any gene j = 1, …, m, can test Hj: j = 0 Complete null H0C: j = 0 for all j = 1, …, m The Hj are tested on the basis of the Wald statistics tj and their associated p-values pj

Datasets Lymphoma (Alizadeh et al.) 40 individuals, 4026 genes Melanoma (Bittner et al.) 15 individuals, 3613 genes Both available at http://lpgprot101.nci.nih.gov:8080/GEAW

Results: Lymphoma

Results: Melanoma

Other Proposals from the Microarray Literature ‘Neighborhood Analysis’, Golub et al. In general, gives only weak control of FWER ‘Significance Analysis of Microarrays (SAM)’ (2 versions) Efron et al. (2000): weak control of PFER Tusher et al. (2001): strong control of PFER SAM also estimates ‘FDR’, but this ‘FDR’ is defined as E(V|H0C)/R, not E(V/R)

Controversies Whether multiple testing methods (adjustments) should be applied at all Which tests should be included in the family (e.g. all tests performed within a single experiment; define ‘experiment’) Alternatives Bayesian approach Meta-analysis

Situations where inflated error rates are a concern It is plausible that all nulls may be true A serious claim will be made whenever any p < .05 is found Much data manipulation may be performed to find a ‘significant’ result The analysis is planned to be exploratory but wish to claim ‘sig’ results are real Experiment unlikely to be followed up before serious actions are taken

Discussion (I) Lack of significant findings Small sample sizes FWER-controlling procedures may be too stringent in microarray applications FDR could perhaps be made even more powerful by taking into account the joint distribution of gene expression levels

Discussion (II) Computational considerations All computing done in the R statistical environment (Ihaka and Gentleman) For max T, Cox model analysis was repeated for each of 100,800 random permutations of survival times Exact maximum likelihood calculation took about 60 hours per machine in cluster of 24 PCs, each with 1 GHz Pentium III and 256 MB memory Time can be reduced substantially by using a score approximation to obtain parameter estimates, and by calling C language code from within R

References Alizadeh et al. (2000) Distinct types of diffuse large B-cell lymphoma identified by gene expression profiling. Nature 403: 503-511 Benjamini and Hochberg (1995) Controlling the false discovery rate: a practical and powerful approach to multiple testing. JRSSB 57: 289-200 Benjamini and Yuketieli (2001) The control of false discovery rate in multiple hypothesis testing under dependency. Annals of Statistics Bittner et al. (2000) Molecular classification of cutaneous malignant melanoma by gene expression profiling. Nature 406: 536-540 Efron et al. (2000) Microarrays and their use in a comparative experiment. Tech report, Stats, Stanford Golub et al. (1999) Molecular classification of cancer. Science 286: 531-537

References Hochberg (1988) A sharper Bonferroni procedure for multiple tests of significance. Biometrika 75: 800-802 Holm (1979) A simple sequentially rejective multiple testing procedure. Scand. J Statistics 6: 65-70 Ihaka and Gentleman (1996) R: A language for data analysis and graphics. J Comp Graph Stats 5: 299-314 Tusher et al. (2001) Significance analysis of microarrays applied to transcriptional responses to ionizing radiation. PNAS 98: 5116 -5121 Westfall and Young (1993) Resampling-based multiple testing: Examples and methods for p-value adjustment. New York: Wiley Yuketieli and Benjamini (1999) Resampling based false discovery rate controlling multiple test procedures for correlated test statistics. J Stat Plan Inf 82: 171-196

Acknowledgements Debashis Ghosh Erin Conlon