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690 E. Schuster, S. Kropf, and I. Roeder: Multiple Procedures for Micro Array Based Gene Expression hypotheses H i : m ð1Þ i ¼ m ð2Þ i diagonal elements w i ¼ P2 (i ¼ 1, ..., p) are tested in the corresponding two-sample t-tests and the P n m m¼1 j¼1 ðx ðmÞ ji x i Þ 2 (x i is the total mean of the i-th variable over all n ¼ n 1 þ n 2 sample elements) of the sums of products matrix W ¼ðX XÞ 0 ðX XÞ with X ¼ 1 n x 0 and x ¼ 1 n 10 n X ðn ¼ n 1 þ n 2 Þ are used. The remaining parts of Procedures I and II are identical to the one-sample case. To determine the expression level of individual genes, the Affymetrix-type GeneChip 1 technology uses several different (about 16–20) oligo-nucleotide (oligo) measurements. For each oligo, two different sequences (which differ in one base only), the so called perfect-match (PM) and the miss-match (MM), are determined (Affymetrix, 2002). To estimate the corresponding expression level of the genes, one usually contrasts these two types of matches and thus summarizes in some sense over the oligos belonging to a given gene. Herein, all genes are treated according to one common rule. Subseqently, the obtained expression values are used to ascertain differentially expressed genes. In contrast to this approach, we propose here a test procedure which incorporates the total information of all individual PMs into the test for differential gene expression, using individual, gene specific rules for the construction of a multivariate expression score. Therefore, Procedures I and II are transferred into another multivariate context. Keeping the control over the FWE, we will use scores from multivariate tests (e.g. principle component test or standardised sum test) proposed by Låuter et al. (1996) as input for the above introduced multiple procedures. Summarizing the above statements, one can say that the essential novelty of our approach is the use of data dependent, gene specific weights rather than one common, gene independent rule in the construction of expression scores. In the following section the derivation of the amended multiple procedures is described in detail. Moreover, we provide the proof for its strong control of the claimed FWE criteria. Two examples using published data are presented in Section 3, followed by a discussion in the last section. A technical description of two R-routines for the application of the proposed procedures is outlined in the appendix. 2 Multiple Test Procedures with Parametric Multivariate Tests per Gene Our approach is based on Affymetrix raw data values including the information of all oligos per gene (i.e. cel-file level). It is assumed that the PMs follow a common multivariate normal distribution. Let k be the number of genes and p i the oligos number of gene i. From this, it follows that the total number of PMs is given by p ¼ Pk p i . i¼1 Firstly, we consider the one-sample situation, where the data usually represent difference values from two conditions in the same individual. The column vector containing all p i PMs of individual j and gene i is denoted by x ji (j =1,..., n; i =1,..., k). Thus, the whole vector for individual j is composed as 0 1 x j ¼ B @ x j1 . . x jk with expectation m ¼ C A N p ðm; SÞ 0 B @ m 1 . . m k 1 ðj ¼ 1; ...; nÞ C A and covariance matrix S ¼ 0 1 S 11 S 1k B . . . . . . C @ A. S k1 S kk Again, the n sample vectors are summarized to the n p data matrix X ¼ðX 1 X k Þ with the transposed vectors x j as rows, and the total sums of products matrix W is given by 0 1 W 11 W 1k B . . W ¼ . . . C @ A ¼ðX 1 X k Þ 0 ðX 1 X k Þ¼X 0 X : W k1 W kk # 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
Biometrical Journal 46 (2004) 6 691 Now, the local hypotheses H i : m i ¼ 0 (i = 1,..., k) are multivariate and are treated according to Theorem 1. The scores derived in these tests are then used in Procedures I and II in the same manner as the single variables previously. Here, we will consider four different choices for the weight vectors from Theorem 1 (for details see Låuter et al., 1996 or Kropf, 2000). The weight vector for gene i (denoted by c i , i ¼ 1; ...; k) is obtained for 1. the non standardized principle component (NPC) test as the eigenvector c i corresponding to the largest eigenvalue of the eigenvalue problem W ii c i ¼ lc i with c 0 i c i ¼ 1; 2. the standardized principle component (SPC) test as the eigenvector c i of the largest eigenvalue of the eigenvalue problems W ii c i ¼ Diag ðW ii Þ c i with c 0 i Diag ðW iiÞ c i ¼ 1; 3. the covariance sum (CS) test as c i ¼½Diag ðW ii ÞŠ 1 W ii ½Diag ðW ii ÞŠ 1=2 1 pi ; 4. the standardised sum (SS) test as c i ¼½Diag ðW ii ÞŠ 1=2 1 pi : To ensure that genes with balanced increased and decreased oligo measurements are not counted as differentially expressed, the weights c i are transformed to the modified form d i by: d i ¼ðd li Þ l¼1; ...; pi with d li ¼ jc lij P p ; l ¼ 1; ...; p i i ; i ¼ 1; ...; k : jc si j s¼1 Using the transformed weights, the p i -dimensional data subvectors x ji are subsumed into a score by z ji ¼ d 0 i x ji. The one-sample t-test with the score values z ji (j ¼ 1, ..., n) now yields the unadjusted P-value P i for gene i (i ¼ 1, ..., k). Using the following theorem one can formulate generalized versions of the multiple test Procedures I and II. Theorem 2 Procedures I and II strictly meet a given FWE criteria if applied to left-spherically distributed scores z ij . To indicate the different assumptions (mulivariate normal and left-spherical), Procedures I and II will be denoted as Procedure I 0 and II 0 , respectively, if applied to left-spherically distrubuted scores. The subsequent proofs are extensions of the proofs given for Procedures I and II in Kropf and Låuter (2002) and Westfall et al. (2004), respectively. Proof of Theorem 2, Procedure I 0 Let M 0 ¼fi j m i ¼ 0; i 2f1; ...; kgg ¼ fi 1 ; ...; i k0 g of size k 0 be the index set of the true local null hypotheses, i.e., of all those genes with expectation zero in all oligos, and let X 0 and W 0 be the corresponding submatrices of X and W, respectively. The case of empty M 0 can be neglected because no type I errors can occur then. All corresponding weight vectors d i1 ; ...; d ik0 are summarized into the weight matrix 0 1 D 0 ¼ B @ d i1 0 . . . . . . . 0 d ik0 C A : As the matrix X 0 consists of iid multivariate normal row vectors with expectation zero, and all four weight versions (incl. transformation from c i to d i ) use the data only through the corresponding sums of products matrix W 0 , we are in the framework of Theorem 1. Therefore, the whole matrix Z 0 ¼ X 0 D 0 is left-spherical as well as each of its columns which correspond to the scores in Procedures I 0 and II 0 . Thus, as shown in Låuter et al. (1998), Theorem 1 can be applied to the scores in the same way as to the raw data before. Let i 0 denote the index of the score with maximum w i from all scores from M 0 (uniquely determined with probability 1). Then a test according to Theorem 1 with the scores from M 0 and with the weight vector ( d ¼ðd i Þ i¼1;...; k0 ; where d i ¼ 1 for w i ¼ max w l l¼1;...; k 0 0 else # 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
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