Multiple Comparisons with Gene Expression Arrays Using a Data Driven Ordering of Hypotheses Siegfried Kropf, Jürgen Läuter, Magdeburg, Germany Peter H. Westfall, Lubbok, Texas, USA Markus Eszlinger, Leipzig, Germany MCP 2002, Bethesda, Maryland, USA, August 5-7, 2002

MCP Introduction Two well known procedures for MCPs controlling the FWE: Testing with a-priori ordered hypotheses (without  -adjustment) Bonferroni-Holm (data dependent order, with adjustment) In analysis of high-dimensional gene expression arrays not applicable/optimal.  We are looking for a method with data dependent ordering of hypotheses but without  -adjustment.

MCP Basic method Consider one-sample situation first: data matrix from n iid p-dimensional normal data vectors Aim: test of the local hypotheses H i :  i = 0 at the strong FWE . Procedure I: sort variables for decreasing values of, in that order carry out the unadjusted one-sample t tests for the variables as long as significance is attained.

MCP Remarks: This procedure maintains the FWE for normally distributed sample vectors with arbitrary covariance structure. Proof in Kropf (2000), Kropf and Läuter (2002), based on multi- variate theorems for spherically distributed observation vectors (Läuter, Glimm and Kropf, 1996, 1998). In order to yield an efficient order of variables, variances of variables should be approximately equal because with we have.

MCP Example 1 6 patients with nodules in thyroid gland (3 hot, 3 cold) 6 blocks, each 98 genes (double spotted): 588 genes + housekeeping genes Atlas Human Cancer 1.2 Array

MCP Comparison nodules vs. surrounding (hot and cold nodules together  one-sample test vs. 0) 1. block A only (98 genes, 2 spots aver., corr. with housek. genes) gene no. sum of squares unadjusted P-value # locally sign. genes: 33 # sign. genes Westfall-Young: 0 # sign. genes Holm‘s proc.: 0 # sign. genes Procedure I: 10

MCP blocks A - F (588 genes) – very similar : unadjusted: 131, Holm: 0, Proc.I: 9, Westfall-Young: 1 Simulation experiments guided by the example with one block: n = 6,...,33 cases, p = 98 variables, normally distributed, variance 1, pairwise correl. 0.5, expectation 0 for 88 var‘s, other 10 var‘s: sample size n Average # of significant genes in Monte Carlo replications

MCP Extensions Other testing problems: –particularly comparison of two/more independent samples ordering by sums of squares, i.e., related to the variablewise total mean of all samples, then two-sample t tests or one-way ANOVA. Other subsets of variables (e.g., pairs of variables)  Kropf, Läuter (Biometrical Journal, end of 2002) „Distribution-free“ version possible

MCP Example 2 30 patients with nodules in thyroids 15 hot nodules, 15 cold nodules tissue samples of nodules and surrounding analyzed with Affymetrix ® Gene Chips Signal log ratio nodule vs. surrounding from each patient for each of genes approximately multivariate normal distribution “similar” variances for all genes,expectation 0 if unaffected

MCP Cold nodules vs. surrounding (one-sample problem) For comparison: without any adjustment: 1064 Bonferroni /Holm: 1 (gene 8104) Westfall / Young: 0 The present procedure stops already after the 2nd gene. The procedure is sensitive to disturbances. It should be smoothed (see below, hybridisation with Bonferroni / Holm) · · · · · P valuegeneno.

MCP A weighted procedure: In the notation of the one-sample problem (Westfall, Kropf, Finos, 2002) Calculate the P-values p i (i = 1, …, p) for the usual unadjusted one-sample t test for each of the p variables. For each variable, determine the sums of squares values and the weights for fixed   0. Calculate the weighted P-values q i = p i / g i and order the variables for increasing values of them. Let S j denote the set of indices of all variables following the jth ordered variable in that order (including that variable itself). Then the hypothesis H (j) for the jth ordered variable is rejected iff

MCP Basic idea of proof: We restrict the consideration to the submatrix consisting of those variables with true null hypothesis (expectation zero). This matrix is left-spherically distributed. For fixed sums of squares w ii (i = 1,...,p) and cross products, its conditional distribution is also left-spherical. As then the weights are fixed, the standard theory (Fang, Zhang, 1990; Läuter, Glimm and Kropf, 1998) can be applied and ensures that the FWE is maintained for each condition and hence unconditionally, too. Special cases:  = 0 : Then the procedure is identical to Bonferroni / Holm.   : According to Westfall and Krishen (2001), the critical function converges to the fixed order as used in Procedure I. In an application,  has to be fixed in advance!

MCP Example 2 again Cold nodules vs. surrounding unadjusted 1064, Westf./Y  Is the choice of genes stable? B/H Pr. I 

MCP Example 2, cont. hot nodules vs. surrounding  hot vs. cold nodules B/H Pr. I B/H Pr. I unadjusted 2597, Westf./Y. 93unadjusted 1290, Westf./Y 

MCP Summary A new technique for multiple testing with data-dependent ordering of hypotheses is proposed. It keeps the FWE in the strong sense for arbitrary multivariate normal data. In order to provide a high power, the variables should have approximately equal variances. The proposal is advantageous in very small samples of high- dimensional data. The method is sensitive to disturbances. Westfall‘s proposal of the weighted procedure establishes a link of the above procedure and the Bonferroni-Holm method and smoothes out for these disturbances. The weighted procedure is a real alternative to existing analysis techniques for microarray data.

MCP References Fang, K.-T. and Zhang, Y.-T., 1990: General Multivariate Analysis. Science Press Beijing and Springer-Verlag Berlin Heidelberg. Kropf, S., 2000: Hochdimensionale multivariate Verfahren in der medizinischen Statistik. Shaker Verlag, Aachen. Kropf, S., and Läuter, J., 2002: Multiple Tests for Different Sets of Variables Using a Data-Driven Ordering of Hypotheses, with an Application to Gene Expression Data. Biometrical Journal, in print. Läuter, J., Glimm, E., and Kropf, S., 1996: New Multivariate Tests for Data with an Inherent Structure. Biometrical Journal 38, Erratum: Biometrical Journal 40, Läuter, J., Glimm, E., and Kropf, S., 1998: Multivariate Tests Based on Left-Spherically Distributed Linear Scores. Annals of Statistics 26, Erratum: Annals of Statistics 27, Westfall, P.H., Kropf, S., and Finos, L., 2002: Weighted FWE-controlling methods in high-dimensional situations. Submitted for IMS Philadelphia companion volume. Westfall, P.H. and Krishen, A. (2001): Optimally weighted, fixed sequence, and gatekeeping multiple testing procedures. Journal of Statistical Planning and Inference 99, Westfall, P.H. and Young, S.S., 1993: Resampling-Based Multiple Testing. John Wiley & Sons, New York.