Estimating shared copy number aberrations for array CGH data: the linear-median method

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1 University of Wollongong Research Online Faculty of Informatics - Paers (Archive) Faculty of Engineering and Information Sciences 2010 Estimating shared coy number aberrations for array CGH data: the linear-median method Yan-Xia Lin University of Wollongong, yanxia@uow.edu.au Veera Baladandayuthaani The University of Texas M.D. Anderson Cancer Centre, veera@mdanderson.org V Bonato Pfizer Global Research and Develoment, vinibonato@yahoo.com.br K.-A. Do The University of Texas M.D. Anderson Cancer Centre, kimdo@mdanderson.org Publication Details Lin, Y., Baladandayuthaani, V., Bonato, V. & Do, K. (2010). Estimating shared coy number aberrations for array CGH data: the linear-median method. Cancer Informatics, Research Online is the oen access institutional reository for the University of Wollongong. For further information contact the UOW Library: research-ubs@uow.edu.au

2 Estimating shared coy number aberrations for array CGH data: the linear-median method Abstract Motivation: Existing methods for estimating coy number variations in array comarative genomic hybridization (acgh) data are limited to estimations of the gain/loss of chromosome regions for single samle analysis. We roose the linear-median method for estimating shared coy numbers in DNA sequences across multile samles, demonstrate its oerating characteristics through simulations and alications to real cancer data, and comare it to two existing methods. Results: Our roosed linear-median method has the ower to estimate common changes that aear at isolated single robe ositions or very short regions. Such changes are hard to detect by current methods. This new method shows a higher rate of true ositives and a lower rate of false ositives. The linear-median method is non-arametric and hence is more robust in estimating coy number. Additionally the linear-median method is easily comutable for ractical acgh data sets comared to other coy number estimation methods. Keywords Estimating, shared, coy, number, aberrations, for, array, CGH, data, linear, median, method Discilines Physical Sciences and Mathematics Publication Details Lin, Y., Baladandayuthaani, V., Bonato, V. & Do, K. (2010). Estimating shared coy number aberrations for array CGH data: the linear-median method. Cancer Informatics, This journal article is available at Research Online: htt://ro.uow.edu.au/infoaers/1616

3 Cancer Informatics Original Research Oen Access Full oen access to this and thousands of other aers at htt:// Estimating Shared Coy Number Aberrations for Array CGH Data: The Linear-Median Method Y.-X. Lin 1, V. Baladandayuthaani 2, V. Bonato 3 and K.-A. Do 2 1 Centre for Statistical and Survey Methodology, School of Mathematics and Alied Statistics, University of Wollongong NSW 2522, Australia. 2 Deartment of Biostatistics, Box 1411, The University of Texas M.D. Anderson Cancer Center, 1515 Holcombe Boulevard, Houston, Texas , USA. 3 NonClinical Statistics Deartment, Pfizer Global Research and Develoment 445 Eastern Point Road, Groton, CT , USA. Corresonding author yanxia@uow.edu.au Abstract Motivation: Existing methods for estimating coy number variations in array comarative genomic hybridization (acgh) data are limited to estimations of the gain/loss of chromosome regions for single samle analysis. We roose the linear-median method for estimating shared coy numbers in DNA sequences across multile samles, demonstrate its oerating characteristics through simulations and alications to real cancer data, and comare it to two existing methods. Results: Our roosed linear-median method has the ower to estimate common changes that aear at isolated single robe ositions or very short regions. Such changes are hard to detect by current methods. This new method shows a higher rate of true ositives and a lower rate of false ositives. The linear-median method is non-arametric and hence is more robust in estimating coy number. Additionally the linear-median method is easily comutable for ractical acgh data sets comared to other coy number estimation methods. Keywords: array CGH, coy number alterations, common coy number alterations regions Cancer Informatics 2010: doi: /CIN.S5614 This article is available from htt:// the author(s), ublisher and licensee Libertas Academica Ltd. This is an oen access article. Unrestricted non-commercial use is ermitted rovided the original work is roerly cited. Cancer Informatics 2010:9 229

4 Lin et al 1. Introduction During cell division, a cell relicates its genome by synthesizing a new coy of each chromosome, using the original DNA as a temlate. The exected coy number of 2, may be less/greater than 2 when alterations occur during the relication rocess. Research has suggested that such abnormalities in the number of DNA coies in a cell are associated with the develoment and rogression of disease, including cancer. 1 Laboratory research to estimate the altered coy numbers in a DNA sequence often uses acgh. The technology used to roduce acgh data, however, may result in data that contain uncontrollable noise. 2 The use of aroriate statistical methods to normalize the data and roduce meaningful estimates of coy number variation in a DNA sequence is integral to this research. Develoing imroved statistical methods for this alication is the focus of this aer. Different statistical methods have been suggested for use with acgh data to estimate coy numbers in DNA sequences. Methods to analyze coy numbers in terms of identifying the locations of gains or losses of chromosome regions have been develoed. Assuming that there is a connection between coy number changes in a cancer cell and the develoment/rogression of the cancer, there must exist some common change regions in DNA sequences collected from different atients with the same cancer diagnosis. Techniques for analyzing shared coy number regions have been develoed. 3,4 For detecting coy number regions in a single samle, Olshen et al 5 and Venkatraman et al 6 had develoed a widely used method, the faster circular binary segmentation (CBS) method. In this aer, we roose a new method, the linear-median method, for estimating shared coy number alterations in DNA sequences collected from the same tye of cancer cells. The linear-median method is able to otimally use the information available across indeendent DNA sequences. This aer is organized as follows. In Section 2.1, we discuss current existing statistical models used to assess acgh data and describe a new model for analyzing multile indeendent acgh data sets. We introduce the linear-median method in Section 2.2. In Section 3.1, we resent three simulation studies. We study how much extra information on coy number aberration can be obtained by using the linear-median method comared to the comarative genomic hybridization minimal common region (cgh) method and the CBS algorithm. We resent an alication of the linear-median method to real data in Section 3.2. Suorting figures and tables are available online as Sulementary Material. 2. Methods 2.1. Modeling DNA coy number alterations in acgh data acgh emloys the comarative hybridization of genomic DNA that is differentially labeled according to its source in a cancer cell versus a normal cell. The ratio of the hybridization intensities along the chromosomes rovides a measure of the relative coy number of sequences in the genomes that hybridize to each location on the chromosomes. Estimating coy numbers and identifying the locations of gains and losses in a DNA sequence are two main challenges in the analysis of acgh data. We label the normal genomic sequences as reference samle and the genomic sequences from cancer cells as the test samle. Let T denote the test coy number at robe osition and R denote the reference coy number at robe osition. We briefly describe two current methods for modeling acgh data. Let us denote by Y the acgh data (the logarithm intensity ratio) observed at robe osition. Model 1: Y = log 2 (T /R ) + ε, (1) where ε are i.i.d. with normal distribution N( 0, σ ε ). This Gaussian model forms the basis of many models for acgh data. 4,6 8 Model 2: Y T = log 2 R + ε + η, (2) where ε and η are i.i.d with a normal distribution N(0,σ 2 ). 9,10 In ractice, R is assumed to be 2. Given the logarithm intensity ratio observations, {Y }, we want to estimate the true coy number at osition or to estimate if the coy number at is greater/less than Cancer Informatics 2010:9

5 The linear-median method Models 1 and 2 assume very different robability structures to describe the system. The variance of the log intensity ratios given by Model 1 is a constant, whereas the variance of the log intensity ratios given by Model 2 is a function of T. We consider which of the two models is a more aroriate model for the analysis of acgh data. Although Model 1 looks simler, it is not an aroriate model for acgh data. The main reason for this is that acgh data rovide the ratio of the coy number variations, not the ratio of the coy numbers. Furthermore, emirical studies show that the standard error of the logarithm of the intensity ratios increases as the coy number increases. Additionally, the distribution of the logarithm of intensity ratios is skewed. 9 Thus, the distribution of ε should not be assumed to be normal if Model 1 is adoted. Comared to Model 1, Model 2 is a more aroriate model for acgh data, as it takes into account the ratio of the coy number variations. However, this model can be imroved further. The normality assumtions on the distributions of ε and η can imly that negative values of ε and η will lead to log 2 (T + ε/2 + η) being ill-defined. Theoretically, this will cause roblems for statistical inference methods based on such an assumtion. In Model 2, the errors ε and η lay the role of measurement errors. Given the fact that the acgh technique is maturing, it might be reasonable to suggest that both ε and η follow a uniform distribution U(-a, a), where a can assume any value between 0 and 2, deending on the nature of the underlying acgh technique. If a takes a value close to 2, this may mean that the underlying acgh technique is not very accurate, ossibly leading to a very large variation in the observations of the intensity ratios. If a takes a value close to 0, we may assume that the underlying acgh technique is very accurate and that there is less variation in the observations of the intensity ratios. For exlicit technical considerations see wikiedia. 2 For our urose, we restrict a to be less than 2. We aly this restriction to real data analysis in Section 3.2. The outut of the real data analysis shows the restriction is accetable. Therefore, we consider a third model: Model 3: X T = R + ε + η, (3) where ε and η are indeendent and have uniform distribution U(-a, a) with constant a (0, 2), and X is the observed intensity ratio at robe osition. To allow the model to be more flexible, we can assume that the uniform distributions for ε and η are not necessarily the same. Model 3 is used to model one acgh rofile from one samle/atient. However, if there is a grou of indeendent samles of acgh data (eg, multile atients) and their data share coy number change regions, we can extend Model 3 to such data. Consider the following scenario. A grou of n atients suffer from a common cancer. For each atient a samle of acgh data is collected from a cancer cell. Let X i, be the observed intensity ratio for the ith samle at robe osition. We use t to denote the theoretical true value of the shared coy number at robe osition for the test and let T i, be the true coy number for the ith atient at robe osition. T i, is not necessarily equal to t because, for different atients, the coy number at osition might be affected by different uncontrollable random factors. We use T to denote the observed coy number for test at osition. T is a random variable and T i, is a samle from T. Let R i, be the true coy number for the ith reference at the osition. In this aer, we always assign R i, = 2 because the true coy number for the reference (normal) genome is 2 (For the urose of this study we ignore some secial cases). For multile indeendent acgh data, the extended model can be considered as Ti, + εi, Model 4 : Xi, =, Ri, + ηi, 1# # M, i = 1, 2,, n, (4) where M is the total number of robe ositions; n is the number of indeendent samles in the grou; ε i, and η i, are mutually indeendent random variables; T i, has distribution P(T i, = t ) = π and P(T i, = 2) = 1 π, if t 2, ie, if at robe osition the shared true coy number is not 2, then the coy number given by the ith samle at the robe osition will follow a Bernoulli distribution with mean π; ε i, and η i, will have uniform distributions U(-a,a), as defined in Model 3. (Different uniform distributions are allowed for ε i, and η i, ; however, such alications are beyond the scoe of this aer.) Cancer Informatics 2010:9 231

6 Lin et al Model 4 rovides a flexible way to model multile indeendent acgh data in terms of the following arguments: i. The robability distributions of ε i, and η i, are allowed to be different. This means that the robability distribution of the measurement errors for the test and reference are allowed to be different. ii. The true shared coy number at osition is no longer a constant. T is a random variable. This means that the coy number (if it were observable) at osition could be different from atient to atient. Hereafter, we consider multile indeendent acgh data and assume Model 4 as the basis for develoing a method to estimate the shared coy number t, = 1,, M The linear-median method Currently, all raw data used for coy number analysis are resented in the format of a log 2 intensity of the ratios of the test to the reference. From the current literature, we know that a linear format refers to using the intensity of the ratios of the test to the reference, and a nonlinear format refers to using a log 2 intensity of the ratios of the test to the reference, as the log 2 (ratio) is not linearly related to the coy number. The variance of a linear format tends to be larger than the variance of a nonlinear format when the relative coy number is far away from This may exlain why the nonlinear format is widely used. It is exected that the log 2 of the true relative coy number, ie, log 2 (t /R ), can be well estimated using the observations of the log 2 intensity of the ratios of the test to the reference, ie, log 2 [(T i, + ε i, )/ (R i, +η i, )], through the samle mean. Unfortunately, this is generally not true. A simle reason for this is that, in general, T + ε ET E log2 log2 R + [ + ε ] η ER [ + η ] ET [ ] = log 2. R Further, the robability distribution of log 2 [(T + ε )/ (R + η )] is not symmetric. Therefore, the samle mean of log 2 [(T i, + ε i, )/(R i, + η i, )] might be biased from E [log 2 ((T +ε )/(R +η ))] for smaller samles. Figure 1 shows a histogram of simulated data drawn from the oulation log 2 [(1 + ε)/(2 + η)], with ε and η i.i.d. uniformly distributed U(-1.8, 1.8) (the function will not be defined if 1 + ε # 0). For the estimating rocedure we roose, we will use linear format data rather than nonlinear format data to estimate the shared coy number at robe osition, 0 # # M. As defined in Model 4, X i, is a random variable of the intensity of the ratios of the test to the reference given by the ith samle at robe osition, 1 # # M, and satisfies the model X i, T = R + ε i, i, + η i, i,, = 12,,, M, i = 12,,, n, where i denotes the ith samle/atient; ε i, and η i, are i.i.d. with uniform distribution U(-a, a); T i, and R i, are the test intensity and reference intensity, resectively, for robe for the ith samle. As stated in Section 2.2, we always assign R i, = 2, which is the information given by the reference genome. The true shared coy number t at osition needs to be estimated. The estimate of t is denoted by tˆ,1 M. Let x i, be the observed values of X i,, i = 1, 2,, n, = 1,, M. Herein, we assume that arameter a is unknown but has a value within (0, 2) and that arameter π (defined in Model 4) is known or can be estimated from emirical knowledge. Frequency Figure 1. Histogram of log 2 [(1 + ε )/(2 + η)]. 232 Cancer Informatics 2010:9

7 The linear-median method The estimation of t, = 1,, M, consists of three stes: Ste 1 Calculate the median of {x i, } i = 1,2,...,n for each, denoted by M. Ste 2 Calculate 2(M -1 + π)/π for each. Ste 3 Determine the estimate of t, = 1,, M, Thus 1 = ( tπ + 21 ( -π)) E 2 + η t π + 21 ( - π) 2 + a = log a a tˆ = 2( M π) π,if 2( M - 1+ π) π 2( M - 1+ π) π + 0.5; 2a t = 2 + a log 2 - a EX ( )-21 ( -π) / π. (5) tˆ = 2( M - 1+ π) π + 1,if 2( M - 1+ π) π > 2( M - 1+ π) π + 0.5, where [c] denotes the integer art of the real number c. We call this 3-ste method the linear-median method. Linear indicates that the data (the intensity of the ratios of the test to the reference) are in a linear format. Median indicates that the median of the data is emloyed by this method. Next, we exlain theoretically why coy numbers can be accurately estimated by this 3-ste method. Let X be the intensity of the ratios of the test to the reference at robe osition, X T + ε = 2 + η where ε and η are i.i.d. with uniform distribution U[-a, a]; and T is a random variable indeendent of ε and η, and has distribution P(T = t ) = π and P(T = 2) = 1 - π, if the shared coy number t 2. As exlained in Section 2.1, we assume 0, a, 2. Following the definition of X and assuming the indeendence of T + ε and η, we have EX E T + ε ( ) = 2 + η 1 = ET ( + ε ) E 2 + η, Equation (5) gives the exact relationshi between t and E(X ). For each robe osition, if the mean of the intensity of the ratios of the test to the reference is known, and the system arameters a and π are known, the shared coy number at the robe osition can be correctly identified. However, E(X ) is unknown in ractice and the robability distribution of X is not usually symmetric. It is inaroriate to estimate E(X ) by using the samle mean X when the samle size is not aroriately large. Therefore, it is difficult to evaluate t directly from (5) in ractice. To overcome this difficulty, we suggest the following way to evaluate t : 2a t = EX ( ) -21 ( -π) / π 2 + a log - a 2 2a EX ( ) = mx -21 ( -π) / π, 2 + a m log X 2 - a where m X is the median of X. It is technically ossible to directly evaluate the ratio ae( X ) a log a m X (6) and rove that the ratio is close to 1, for any a (0, 2) and any π (0, 1). Cancer Informatics 2010:9 233

8 Lin et al We use the Monte Carlo method to indirectly show that the value of (6) is close to 1 for a = 0.1, 0.2,, 1.9 and π = 0.1, 0.2,, 1. (see Aendix A and Sulementary Tables 1 and 2 in the online materials for details). Therefore, t 2( mx -( 1-π )). π 3. Imlementation and results 3.1. Simulation studies The linear-median method is designed for estimating shared coy number aberrations and mainly focuses on the information across the samle for each robe osition. Therefore, this method ignores the deendency within each individual samle. Our focus is two-fold: i) to determine the extent of information of shared coy number aberrations that can be detected, regardless of the imact of deendency, and ii) to assess the differences in detection outcomes obtained from the linear-median method versus other methods. In a recent review of methods for detecting recurrent coy number alterations, Rueda and Diaz- Uriarte evaluated the CGHregions method, Master HMMs, cgh, GISTIC, MSA, RAE, and others. 12 In this subsection, we comare the linear-median method to the cgh method and the CBS method. We resent three simulation studies to highlight the erformance of our roosed linear-median method. Examle 1: A sequence of integers test at robe osition 1 is 2; t 11 = 3 means that the true gain in the shared coy number by the test at robe osition 11 is 3. We simulated a grou of indeendent realizations {X i, } from model (t + ε i, )/(2+η i, ), = 1, 2,, 100 and i = 1, 2,, n, where ε i, and η i, are i.i.d. with uniform distribution U[-a, a]. Subsequently, we generated 1000 relicates. For the kth relicate, k = 1, 2,, 1000, let d(k) be the ercentage of t - t 0 out of the 100 robe ositions; d(k) is used to measure the error rate in the estimation of t. The mean and standard error of {d(k)} are resented in Table 1. Table 1 shows that the error rate increases with a. This is obvious because a larger value of a is equivalent to a larger measurement error in the data. However, the error rate will be reduced when the number of indeendent samles in the grou increases. In general, the mean error rate calculated for the linearmedian method is reasonably low: the mean error rate was less than 10%, as exected, for all three cases of varying a. Although the underlying model involves the arameter a, Examle 1 shows, in general, that the imact of the value of a on the estimation of the coy number is not significant in terms of the mean of d(k), excet for a very large value of a(.1). (Further demonstrations are resented in the Sulementary Material.) In summary, the value of a (0, 2) has minimal effect on the estimation of the shared coy number when the samle size is reasonable large. As a result, the linear-median method can be emloyed without knowing the value of a, as long as a (0, 2) serves as a sequence of the true shared coy number t, = 1, 2,, 100, obtained from the exerimental samle, ie, the test. To simlify, we assume π = 1. Thus, for examle, t 1 = 2 means that the true shared coy number shown by the Examle 2: In Table 1 of their review of 15 estimation methods, Rueda and Diaz-Uriarte indicate that only the cgh method both uses an inut of the log2 ratio and roduces estimations of the differences in the states of two successive robes Cancer Informatics 2010:9

9 The linear-median method Table 1. The samle mean and samle standard error of the estimated error rate {d(k)} given by different combinations of a and n, where a is the arameter of the uniform distribution U[-a, a] and n is the number of the indeendent sequences in the realizations. a n ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) True coy number t Figure 2. Plot of the sequence of the true coy numbers. The cgh method is designed to identify the minimal common coy number alteration regions among a grou of indeendent samles; thus it is analogous to the linear-median method and is an aroriate method to comare to the linear-median method. Using segmented data (ie, smoothed data), the cgh algorithm first identifies altered segments within each subject (those above the 97th or below the 3rd ercentile of the data) and then joins adjacent segments searated by a user-defined arameter. The R ackage for the cgh method is available at the following URL: htt:// 2.6/bioc/html/cgh.html. See the work of Aguirre et al for exlicit details and a comlete review of the cgh method. 3 We use simulated data to comare the erformance of the linear-median method to that of the cgh method. The data were simulated by assuming non deendency between the intensity ratios across robe ositions, which is a very simle situation. Consider a sequence of true shared coy number {t } lotted in Figure 2. The sequence t consists of four abnormal shared coy number regions, corresonding to coy numbers 1, 3, 4 and 5. Some of the abnormal shared coy regions are very short, involving only 1 or 4 robe ositions. Using this examle, we comare the linearmedian method to the cgh method in terms of each methods caability of correctly assessing the information of gains/losses in shared coy numbers. X i, We simulated data from the following model 2 + εi,, 2 + ηi, B(, 1π) i, + 2( B(, 1π) i, - 1) + εi, 2 + ηi, 2 + ε i,, 2 + ηi, 4B(, 1π), + 2( B(, 1π), - 1) + ε, 2 + η i, 2 + εi,, 2 + ηi, = 5B(, 1π) i, + 2( B(, 1π ) i, -1) + ε, 2 + ηi, 2 + εi,, 2 + ηi, 3B(, 1π) i, + 2( B( 1, π) i, - 1) + ε, 2 + ηi, 2 + εi,, 2 + ηi, i i i i i,,, 1# # 10, 11 # # 50, 51# # 98, 99 # # 102, 103 # # 109, = 110, 111# # 150, 151# # 200, 201# # 250, (7) i = 1, 2,, n, where ε i, and η i, are i.i.d. with uniform distribution U(-a, a). Let B(1, π) be a random variable with a Bernoulli distribution such that E[B(1, π)] = π. We considered different combinations for (a, π, n), where a = 0.5, 1, 1.5, π = 0.2, 0.4, 0.6, 0.8, 1 and n = 20, 50, 100. We alied the linear-median method and the cgh- method to each grou of indeendent samles with size n for different airs of arameters (a, π), Cancer Informatics 2010:9 235

10 Lin et al resectively. Then, for each trilet (a, π, n), we calculated the true ositive (TP) rates and the false ositive (FP) rates roduced by each model. TP rate = P(the method shows coy number changed coy number is changed). FP rate = P(the method shows coy number changed coy number is not changed). The linear-median method is able to rovide an estimate of the shared coy number at each robe osition. Therefore, when we say that a correct detection of the shared coy number was roduced by the linear- median method at osition, it means that t^ = t. In contrast, the cgh method rovides information on only the shared coy number gain/ loss at each robe osition. It does not rovide information on how many coy numbers were gained/lost. Therefore, when we say that a correct detection was roduced by the cgh method at osition, it means only that a gain/loss was correctly identified at osition. Finally, we carried out 250 relicates for the case where n = 20; 100 relicates for the case where n = 50, and 50 relicates for the case where n = 100. The resulting TP and FP rates, means, and standard errors obtained from both methods are shown in Sulementary Tables 3 5. In terms of the TP rates, the linear-median method worked reasonably well in each case and erformed vastly better than the cgh method, which showed oor erformance, esecially when a was larger and π was smaller. In this articular examle of a true shared coy number sequence, the cgh method tended to give a lower FP value, ie, it did not call as many gains/losses, and hence was very conservative. Comared to the cgh method, the linear-median method gave a lower FP value when a was not close to 2 or π was greater than 0.5. In summary, two advantages of using the linear-median method include: 1. The ability to estimate the actual shared coy number at each osition. The estimation accuracy of the linear-median method is very high, as reflected by the values of the TP and FP rates. 2. Better ower in identifying shorter alternating regions. For examle, considering the data simulated from (7) with a = 1.5, π = 1 and n = 20, we can comare the means of the estimated coy numbers given by both methods. Since a = 1.5, the variance for U(-a, a) is relatively large and the simulated data involve a lot of random noise. By choosing π = 1, there is no variation on the true coy numbers shared across the indeendent samles. Technically, one exects that the linear-median method and the cgh method will erform at the same level. However, it turns out that the linear-median method dominates the cgh method. At almost every robe osition, the samle mean and median of the estimated shared coy number given by the linear-median method was the same as the true shared coy number. In contrast, the cgh method did not accurately identify the gain/loss regions (see Sulementary Figures 1 3). This simulation examle (Examle 2) illustrates that the cgh method erforms very oorly in high-noise scenarios, for examle, a = 1.5, and the cgh method is not robust for large values of a. We believe this is due to the fact that the cgh- method erforms segmentation and calling functions indeendently of one other; whereas the linear-median method borrows strength from all the samles. Examle 3: In this examle we consider data X i,, simulated from the following model: X i, t + ε = 2 + η i, i, 2 + ε 2 + η 3 + ε 2 + η 4 + ε, 2 + η, 3 + ε, = 2 + η, 1+ ε, 2 + η, 2 + ε, 2 + η, 1+ ε, 2 + η, i, i, i, i, i i i i i i i i i i 1# # 100, l = # # 150, l = # # 152, l = 2 153# # 200, l = # # 202, l = 2 203# # 204, l = # # 300, l = Cancer Informatics 2010:9

11 The linear-median method where ε i, and η i, are i.i.d uniformly distributed in [ 1, 1], i = 1, 2, 60. In this examle we continue to assume π = 1. The abnormal coy number regions are [101, 150] for t = 3; [151, 152] for t = 4; [153, 200] for t = 3; [201, 202] and [205, 300] for t = 1. Segments of [101, 150], [153, 200] and [205, 300] are relatively longer. Segments of [151, 152] and [201, 202] are relatively shorter. In this examle, we comare the linear-median method to the circular binary segmentation (CBS) method, which was develoed by Olshen et al. 6 An R ackage descrition for the CBS method is available at the following URL: htt://bioconductor. org/ackages/ 2.6/bioc/manuals/DNAcoy/man/DNAcoy.df. The CBS method is emloyed to find segments along the chromosome that share constant DNA coy numbers. Technically, it is inaroriate to directly comare the analytical results obtained by these two methods because the CBS method is designed for alication to a single samle of data, whereas the linear-median method is alicable to a grou of indeendent samles. To aly the CBS algorithm to observations {x i, }, i = 1,, 60, = 1,, 300, we make the following adjustment. We calculate log 2 (x i, ) for all i and, since the CBS method is designed for data in a nonlinear format. Then, for each fixed, we calculate the median of {log 2 (x i, )}, forming a new sequence. Finally, we aly the CBS method to this sequence. We justify this comarison with the following argument: If there are common coy number alteration regions among the grou of indeendent samles, the new sequence must contain the information on shared common regions. We consider the new sequence as if it were a single samle of data from a atient. Thus, if the information of a shared common region is strong enough, the CBS method should be able to detect the region based on the data of the new sequence. We used the default arameters in our alication of the R ackage to the simulation data in this examle. Figure 3 shows the lot of the medians of {log 2 (x i, )} and the estimate of log 2 (t /2) (in red), obtained by the CBS method (to anel), and the lot of the estimation of t obtained by the linear-median method (bottom anel). We see that the linear-median method is able to detect all the changes in the coy number. Comaring the lots in Figures 3, both aroaches, the linear-median method and the CBS method, were able to detect all the longer regions of alternations. However, all the shorter regions of alterations, [151, 152], [201, 202] and [203, 204], were missed by the CBS method. This indicates that the linear-median method has more ower than the CBS method to detect shorter segments of alterations or narrow gas between segments Alication to real data We alied the linear-median method to a subset of acgh data from 39 well-studied lung cancer cell lines. log 2.Ratio hatt Index Index Figure 3. Alication of the CBS method to the sequence of the median of the logarithm of the ratios (to anel). The red bars show the values of the estimation of log 2 (t /2). Alication of the linear-median method to the data in Examle 3 (bottom anel), showing the estimates of t at each robe osition. Cancer Informatics 2010:9 237

12 Lin et al The data, originally ublished by Coe et al 13 and Garnis et al 14 are available for downloading from htt://sigma.bccrc.ca/. For this study, we used data from only the subgrou with the largest samle size, that of non-small cell adenocarcinoma (NA), which included 18 samles. As both the linear-median method and the cgh- method are designed for alication to multile acgh data, the samle size is a critical issue. Data with more indeendent samles are able to rovide more information on the commonalities across all samles. Accurately identifying the locations of coy number aberrations has many imortant medical alications. As far as we know, the cgh method is one of the methods used to estimate the shared coy number for multile acgh data. Many other methods give an estimation of only the robability of gain/loss at each robe osition. 4,13 Information on the exact shared coy number(s) at each robe osition is not available for the data we have analyzed (the NA data). Therefore, based on only the analytic oututs of the linear-median method and the cgh method, it is difficult for us to claim which method is better in terms of the accuracy of estimating the true coy numbers. As a result, we comared the similarities between the analytic oututs of the two methods and determined which method rovides more information on the changes in the coy numbers in the NA data. As a reference for this comarison, we used the robability of gain/loss at each robe osition that was reorted by Shah et al. 4 The total number of robe ositions in the NA data (chromosome 9) is Recalling Model 4 in Section 2.1, in order to estimate the shared coy numbers in a test DNA sequence, we need to know the arameter π. This tye of information is also required for the cgh method. The value of π might be estimated based on the researcher s emirical knowledge. For the NA data, emirical knowledge on the value of π is not available. Therefore, we alied the cgh method and the linear-median method to the data for different values of π, 0.2, 0.4, 0.6, 0.8 and 1. Then we comared the results from both methods and also comared those results to findings reorted by Shah et al. 4 We exected to find little difference in the results obtained from the three methods. Shah et al found a loss of the shared coy number in a significant ortion of the NA data (see Figure 7 in their aer). 4 However, for π = 0.4, 0.6, 0.8 or 1, both the cgh method and the linearmedian method rovided high roortions of neutral states, ie, where the shared coy number equals 2. Therefore, it is reasonable to use π = 0.2 when analyzing the NA data. We limit our reort of the analytic results to the case where π = 0.2. Combining all the results given by the linearmedian method and the cgh method for π = 0.2, 0.4, 0.6, 0.8 and 1, we were able to identify a common trend in the oututs of the two methods for all robe ositions as the value π moves from 1 to 0.2 (data not shown). For the NA data, both the linearmedian method and the cgh method give neutral states to all robe ositions when π is assigned as 1, with the excetion of a few robe ositions identified as gain/loss by the linear-median method. In our emirical study of the NA data, if a robe osition a is more likely to lose coy number(s), then the shared coy number estimation given by both methods will decrease as π moves from 1 to 0.2; if a robe osition a is more likely to gain coy number(s), then the shared coy number estimation given by both methods will increase as π moves from 1 to 0.2. One imortant henomenon we observed from the oututs of the two methods is that once a robe osition has been identified as having a shared coy number change when π = π 0, the observation remains the same for any π. π 0. Comaring the results of the two methods, we found that the estimation of the shared coy number at each robe osition given by the cgh- method is reluctant to change as the value of π decreases. In contrast, the linear-median method can show changes in the estimated shared coy number as π decreases. This may reflect the later detection of an aberration by the cgh method comared to the linear-median method when the true shared coy number at a robe osition is gained/lost, and as the value of π decreases. Based on our analysis of the NA data, the linear-median method was able to reort the estimated shared coy number at each robe osition; whereas the cgh method reorted only the state of the shared coy number, ie, wether there was a gain, loss or no change (neutral state), in the shared coy number. To simlify the comarison between the results given by the two methods, we reort only the gain, loss, or neutral states of the shared coy 238 Cancer Informatics 2010:9

13 The linear-median method number for the linear-median method. A lot of the states for both methods is given in Figure 4. In the lot, we use 1, 0 and -1 to indicate a shared coy number gain, neutrality, or loss, resectively. We summarize the results as follows. From robe ositions 1 to 500 and 1235 to 1249, both the cgh method and the linear-median method rovide similar results, excet for some isolated rob ositions. This is what we exect to find because our simulation studies demonstrated that the linear-median method can identify those isolated regions. From robe ositions 501 to 1234, the results obtained from the linear-median method and the cgh- method are quite different. The cgh method claims that all the robe ositions are neutral, in contrast to the findings of the linear-median method, which identifies gains/losses at these robe ositions. One ossible exlanation for the large difference between the two sets of results in this rob region is that the π used in the estimation for this region may be too high. A lower value of π should be used to accurately estimate coy numbers in this interval. These results suggest that the arameter π might vary over sequences of NA data. If this is true, then, detecting the change in π will be an interesting challenge for future studies. Information on the true shared coy numbers for the NA data is not available; hence, we cannot be certain which method would best estimate the shared coy number variations in these data. However, through our comarison of the two methods and taking into account the results given by Shah et al 4 we can claim that the linear-median method has some caability to reasonably estimate shared coy numbers in DNA sequences. As shown in our simulation studies, the linear-median method can easily identify isolated robe ositions with shared coy number changes or short shared alternating segments. These changes are often missed by the cgh aroach. The 1249 robe sets we studied target the shared coy number status of 1262 genes resent in the chromosome 9. In order to classify these genes as one of three general categories, we erformed a search of the OMIM database (htt:// The three categories we used were not related to/unknown cancer henotye (NR/U), cancer-related henotye, excet for lung cancer (CR), and lung cancerrelated henotye (LCR). The results are resented in Tables 2 and 3. Identifying altered regions where imortant cancer-related genes are located aids the biological interretation of our findings and works as an emirical form of validation. Detailed locations of Probes 1 to cgh linear-median adj Figure 4. The outut of the linear-median adjusted method is shown in red and that of the cgh method is in green. Cancer Informatics 2010:9 239

14 Lin et al Table 2. Number of genes identified by the linear-median method (LM) and the cgh method in the regions of shared coy number aberrations with the status of coy number loss, neutrality or gain. NR/U is not cancer-related or unknown function henotye, CR is cancer-related henotye (excet for lung cancer), and LCR is lung cancerrelated henotye. NR/U CR LCR Total LM cgh LM cgh LM cgh LM cgh Losses Neutral Gains Table 3. List of lung cancer-related genes for each henotyic grou identified by the linear-median method (LM) and the cgh method. LM cgh Loss PSIP1, CDKN2A PSIP1, CDKN2A TUSC1, IGFBPL1 TUSC1, IGFBPL1 TLE1, FRMD3 DAPK1, MIRLET7A1 PTPN3 Neutral PHF19, DAB2IP PHF19, DAB2IP RPL12 RPL12, TLE1 FRMD3, DAPK1 MIRLET7A1, PTPN3 GAS1 Gain GAS1 the genes categorized as NR/U, CR and LCR are resented in Sulementary Aendix B. From Tables 2 and 3 we can see that the linear-median method is able to reort more CR and LCR with coy number losses/gains than the cgh method. We were able to find additional information of interest from the outut of the linear-median method. Focusing on the robe ositions at which the estimated shared coy number given by the linear-median method was,1 or.3 when π = 0.2, we identified 145 such robe ositions out of 1249 (see Figure 5). Among those 145 robe ositions, 22 robe ositions showed an estimated coy number $4 or #-1. These results rovided a more serious warning of coy number aberrations a warning that was not obtained from the cgh method. 4. Conclusion We develoed a new model for acgh data analysis, the linear-median method, which estimates shared coy numbers in DNA sequences. Using simulated data, we found the linear-median method to be more owerful than the cgh method in terms of achieving a higher rate of true ositives and a lower i = Coy number Probe ositions where coy number <1 or >3 1 = 0.2 Figure 5. The lot of the estimated coy numbers (,1 or.3) given by the linear-median method for π = Cancer Informatics 2010:9

15 The linear-median method rate of false ositives. In addition to estimating the common gain/loss of chromosome regions, the linear-median method estimates the number of DNA coies. In other words, analytic results roduced by the linear-median method allow us to extract additional information on the tested DNA sequences. In articular, the linear-median method has the ower to estimate common changes that aear at isolated single robe ositions or very short regions. The only drawback of the linear-median method is that it ignores the deendency information in samles. However, based on our alication of the roosed method to real data, we find that most information on shared coy number aberrations can be catured by the linear-median method using only the information across indeendent samles. Acknowledgement V. Baladandayuthaani was artially suorted by US National Science Foundation grant IIS K.-A. Do was artially suorted by the University of Texas SPORE grants in Prostate Cancer P50 CA140388, Breast Cancer P50 CA116199, Brain Cancer P50 CA127001, and the Cancer Center Suort Grant P30 CA We would also like to acknowledge LeeAnn Chastain (UTMDACC) for her editorial contributions to the manuscrit. Disclosure This manuscrit has been read and aroved by all authors. This aer is unique and is not under consideration by any other ublication and has not been ublished elsewhere. The authors and eer reviewers of this aer reort no conflicts of interest. The authors confirm that they have ermission to reroduce any coyrighted material. References 1. Cauzzo F, Hirsch FR, Rossi E, et al. Eidermal growth factor recetor gene and rotein and gefitinib sensitivity in non-small-cell lung cancer. J Nat Cancer Inst. 2005;97: htt://en.wikiedia.org/wiki/array_comarative_genomic_hybridization 3. Aguirre AJ, Brennan C, Bailey G, et al. High-resolution characterization of the ancreatic adenocarcinoma genome. Proc Nat Acad Sci U S A. 2004;101: Shah SP, Xuan X, deleeuw RJ, et al. Integrating coy number olymorhisms into array CGH analysis using a robust HMM. Bioinformatics. 2006; 22:e Venkatraman ES, Olshen AB. A faster circular binary segmentation algorithm for the analysis of array CGH data. Bioinformatics. 2007;23: Olshen AB, Venkatraman ES, Lucito R, Wigler M. Circular binary Segmentation for the analysis of array-based DNA coy number data. Bio Statistics. 2004;5: Molinaro AM, van der Laan MJ, Moore DH. Comarative Genomic Hybridization Array Analysis. U.C. Berkeley Division of Bio-statistics Working Paer Series. Working Paer Series. Working Paer 106. htt:// Guha S, Li Y, Neuberg D. Bayesian hidden Markov modeling of array CGH data. J Am Stat Assoc. 2008;103: Pinkel D, Albertson DG. Comarative genomic hybridization. Ann Rev Genom Hum Genet. 2005;6: Pinkel D, Albertson DG. Array comarative genemic hybrization and its alication in cancer. Nat Genet. 2005;37 Sul:S Pinkel D, Davis R, Albertson D. Detection of gene dosage abnormalities using comarative genomic hybridization. htt://cancer.ucsf.edu/array/ nccls_inkel.df Rueda OM, Diaz-Uriarte R. Finding recurrent coy number alteration regions: a review of methods. Current Bioinformatics. 2010;5: Coe BP, Lockwood WW, Girard L, et al. Differential disrution of cell cycle athways in small cell and non-small cell lung cancer. Br J Cancer. 2006;94: Garnis C, Lockwood WW, Vucic E, et al. High resolution analysis of nonsmall cell lung cancer cell lines by whole genome tiling ath array CGH. Int J Cancer. 2006;118: Cancer Informatics 2010:9 241

16 Lin et al Sulementary Material Aendix A Use Monte Carlo method to indirectly show that the value of ae(x )/{log [(2+a)/(2 a)] m X } is close to 1 for a = 0.1, 0.2,, 1.9 and π = 0.1, 0.2,, 1. The simulation is conducted as follows. For each trilet (a, π, t ), 5000 indeendent samles are simulated from model T + ε X X ( a, π ) =, 2 + η where random variables T, ε and η are indeendent; T has a distribution such that P(T = t ) = π and P(T = 2) = 1 - π; ε and η have uniform distribution U(-a, a), a = 0.1, 0.2, 0.3,, 1.9 and π = 0.1,, 1 with increments of 0.1 resectively; t = 1, 2,, 9 with increments of 1. The mean and median of X (a, π) are estimated by its samle mean X ( a, π ) and samle median median(x )(a, π) resectively. Then ae(x (a,π))/{log [(2+a)/(2 a)] m X (a,π)} is estimated and evaluated by ax ( a, π ). 2 + a log a median ( X )( a, π ) 2 - For each π and t fixed, the samle mean m(π, t ) and samle variance s 2 (π, t ) of ax ( a, π ), a = 01., 02.,, 19., 2 + a log a median ( X )( a, π ) 2 - are calculated by the following formulae: 19. ax ( a, π) m( π, t ) = / 19, a= a log a median ( X a )(, π) ax (, ) 2 a π s ( π, t ) = - m( π, t 2 + a ) / 19, a= 01. log ( )( 2 - a median X a, π) and reorted in Tables 1 and 2, which follow, where s 2 (π) is given within the arentheses. The Monte Carlo simulation results clearly show that all the samle means m(π, t ) are close to 1 and the samle variance s 2 (π, t ) are close to 0. Therefore, it is reasonable to accet that ae(x (a,π))/{log [(2+a)/ (2 a)] m X (a,π)}, for any a (0, 2), π (0, 1) and t {l,, 9}. Table S1. The values of m(π, t ) and s 2 (π, t ) (Part A). π = 1 t = 1 t = 2 t = 3 t = 4 t = ( e-05) ( e-06) ( e-06) ( e-05) ( e-05) t = 6 t = 7 t = 8 t = ( e-06) ( e-06) ( e-06) ( e-06) π = 0.9 t = 1 t = 2 t = 3 t = 4 t = ( e-04) ( e-06) ( e-05) t = 6 t = 7 t = 8 t = ( e-04) ( e-04) ( e-04) ( e-04) ( e-04) ( e-04) (Continued) 242 Cancer Informatics 2010:9

17 The linear-median method Table S1. (Continued) π = 0.8 t = 1 t = 2 t = 3 t = 4 t = ( e-03) ( e-06) ( e-04) t = 6 t = 7 t = 8 t = ( e-03) π = ( e-03) ( e-03) ( e-04) ( e-03) ( e-03) t = 1 t = 2 t = 3 t = 4 t = ( e-03) ( e-05) ( e-04) ( e-03) ( e-03) t = 6 t = 7 t = 8 t = ( e-03) ( e-03) ( e-03) ( e-03) π = 0.6 t = 1 t = 2 t = 3 t = 4 t = ( e-03) ( e-06) ( e-04) ( e-03) ( e-03) t = 6 t = 7 t = 8 t = ( e-03) ( e-03) ( e-03) ( e-02) Table S2. The values of m(π, t ) and s 2 (π, t ) (Part B). π = 0.5 t = 1 t = 2 t = 3 t = 4 t = ( e-03) ( e-06) ( e-03) ( e-03) ( e-02) t = 6 t = 7 t = 8 t = ( e-02) π = ( e-02) ( e-02) ( e-02) t = 1 t = 2 t = 3 t = 4 t = ( e-03) ( e-06) ( e-03) ( e-02) t = 6 t = 7 t = 8 t = ( e-02) ( e-02) ( e-01) ( e-01) ( e-02) π = 0.3 t = 1 t = 2 t = 3 t = 4 t = ( e-03) ( e-06) ( e-03) ( e-03) ( e-02) t = 6 t = 7 t = 8 t = ( e-02) ( e-02) ( e-02) ( e-02) (Continued) Cancer Informatics 2010:9 243

18 Lin et al Table S2. (Continued) π = 0.2 t = 1 t = 2 t = 3 t = 4 t = ( e-04) ( e-06) ( e-04) ( e-03) ( e-03) t = 6 t = 7 t = 8 t = ( e-03) ( e-02) ( e-02) ( e-02) π = 0.1 t = 1 t = 2 t = 3 t = 4 t = ( e-04) ( e-05) ( e-04) ( e-04) ( e-03) t = 6 t = 7 t = 8 t = ( e-03) ( e-03) ( e-03) ( e-03) a = 1.5, i = 1, n = 20 mcoy t Figure S1. The lot of the mean of gains/losses obtained at each robe osition using the cgh method. 244 Cancer Informatics 2010:9

19 The linear-median method Aendix B The locations of the genes of NR/U, CR and LCR in non-small cell adenocarci-noma (NA) and related references. Probe ositions from 1 to 295: A total of 200 genes are found in this region, 28 of them (14%) are genes related to cancer henotye while 3 (1.5%) are related to lung cancer henotye. All LCR genes are located in chromosomal regions identified as losses by both methods (LM and cgh). The LCR genes located at this region are PSIP1, CDKN2A, and TUSC1. PSIP1 and CDKN2A, a well-known lung cancer suressor 1 are both located in a region frequently found deleted in lung cancer atients. 2 In addition, TUSC1 is found mutated and silent in nonsmall cell lung carcinoma cell lines. 3 Probe ositions from 296 to 331: A total of 12 NR/U genes are found in this region. Probe ositions from 332 to 341: Only 3 genes are located in this region with one of them being classified as CR (ACO1). Both methods identify the region where this gene is located as loss. Probe ositions from 342 to 375: A total of 113 genes are located in this regions with 14 of them being classified as CR. Probe ositions from 376 to 500: A total of 171 genes are located in this region. Four of them are CR and only one (IGFBPL1, classified as loss by both methods) is classified as LCR. IGFBPL1 has already been shown to be downregulated in lung tumor samles. 4 Probe ositions from 501 to 1234: A total of 744 genes are located in this region, 90 of them being classified as CR, and 9 as LCR. The cgh method does not identify any region containing LCR as altered. On the other hand, the LM method identifies five of the LCR genes in chromosomal regions of loss (TLE1, FRMD3, DAPK1, MIRLET7A1, PTPN3) and, consequently, are exected to have lower exression in lung tumor samles. In fact, TLE1 is frequently found altered in squamous cell carcinomas and adenocarcinomas 5 while FRMD3 exression is usually silenced in rimary nonsmall cell lung carcinomas. 6 Likewise, mouse lung carcinoma clones characterized by highly aggressive metastatic behavior did not exress Dak1. 7 Also, MIRLET7A1 and PTPN3 exressions are downregulated in lung cancer. 8,9 The LM indeti-fies one gene located in a gain region (GAS1), and therefore, it is exected to be overexressed in lung cancer samles. Surrisingly, Gas1 exression is known by its caacity of suressing metastasis in lung, 10 therefore, a = 1.5, i = 1, n = mcoy t Figure S2. The lot of the mean of coy numbers obtained at each robe osition using the linear-median method. Cancer Informatics 2010:9 245

20 Lin et al a = 1.5, i = 1, n = 20 Mediancoy t Figure S3. The lot of the median of coy numbers obtained at each robe osition using the linear-median method. we hyothesize that the this gene might be regulated eigenetically or it is a false ositive identified by the LM method. Again, the cgh method does identifies this region as neutral. In addition, 3 genes are found by both methods in neutral regions (PHF19, DAB2IP, RPL12) and, therefore, we believe that their regulation is being erformed by eigenetic factors. In fact, PHF19 mrna is known to be overexressed in lung cancers 9 as well as methylation of the romoter of DAB2IP is associated with the lung cancer henotye. 11 Likewise, RPL12 slice variant are frequently found in human lung carcinoma cell. 12 Probe ositions from 1235 to 1249: A total of 17 genes are located in this regions with only one of them (ABL1) being classified as CR and identified as a gain by both methods. Aendix C R code for the linear-median function x is an n T matrix, the elements of y are acgh observations in linear format n denotes the number of indeendent samles T denotes the size of each individual samle At any robe osition, if the true shared coy number is not 2, the robability of having coy number changed is rob Function Linear_Median gives the estimate of shared coy number at each robe osition. Linear_Median = function(x,n,t,rob){ medianx = c() for (i in 1:T){ medianx[i] = median(x[i,]) } justx = c() justx = 2*(medianx-1+rob)/rob xx = c() xx = floor(justx) for(i in 1:T){ if (justx[i].= xx[i]+0.5) xx[i] = xx[i]+1 } xx } 246 Cancer Informatics 2010:9

21 The linear-median method Table S3. The true ositive (TP) rates and false ositive (FP) rates for the linear-median method and the cgh method, where n = 20. n = 20 L-M cgh L-M cgh π α = 0.5 α = 1 α = TP (0.0496) (0.1101) (0.0406) (0.0188) (0.0414) (0) FP e (0.0384) (0.0154) (0.0384) (0.0041) (0.0357) (0) 0.4 TP (0.0429) (0.1830) (0.0413) (0.0779) (0.0453) (0) FP (0.0248) (0.0302) (0.0402) (0.0081) (0.0408) (0) 0.6 TP (0.0227) (0.0224) (0.0359) (0.1129) (0.0410) (0) FP e (0.0090) (0.0004) (0.0310) (0.0114) (0.0419) (0) 0.8 TP (0.0060) (0.0206) (0.0204) (0.1308) (0.0331) (0) FP (0.0028) (0) (0.0917) (0) (0.0358) (0) 1 TP (0) (0.0147) (0.0147) (0.1561) (0.0287) (0) FP 7.74e (0.0007) (0) (0.0154) (0) (0.0314) (0) L-M cgh Table S4. The true ositive (TP) rates and false ositive (FP) rates for the linear-median method and the cgh method, where n = 50. n = 50 L-M cgh L-M cgh π α = 0.5 α = 1 α = TP (0.0547) (0.0626) (0.0499) (0) (0.0455) (0) FP e (0.0346) (0.0065) (0.0488) (0) (0.0437) (0) 0.4 TP (0.0347) (0.1542) (0.0357) (0.0109) (0.0420) (0) FP (0.0070) (0.0297) (0.0365) (0) (0.0416) (0) 0.6 TP (0.0072) (0.0149) (0.0212) (0.0842) (0.0358) (0) FP 6.45e (0.0006) (0) (0.0189) (0) (0.0364) (0) L-M cgh (Continued) Cancer Informatics 2010:9 247

22 Lin et al Table S4. (Continued) n = 50 L-M cgh L-M cgh π a = 0.5 a = 1 a = TP (0) (0.0118) (0.0100) (0.0416) (0.0209) (0) FP (0) (0) (0.0082) (0) (0.0238) (0) 1 TP (0) (0.0154) (0.0029) (0.1679) (0.0107) (0) FP (0) (0) (0.0038) (0) (0.0153) (0) L-M cgh Table S5. The true ositive (TP) rates and false ositive (FP) rates for the linear-median method and the cgh method, where n = 100. n = 100 L-M cgh L-M cgh π α = 0.5 α = 1 α = TP (0.0539) (0.0203) (0.0505) (0) (0.0461) (0) FP (0.0187) (0) (0.0412) (0) (0.0381) (0) 0.4 TP (0.0233) (0.1317) ( ) (0.0030) (0.0341) (0) FP (0.0013) (0.0270) (0.0196) (0) (0.0340) (0) 0.6 TP (0.0015) (0.0108) ( ) (0.0706) (0.0239) (0) FP (0) (0) (0.0075) (0) (0.0224) (0) 0.8 TP (0) (0.0087) (0.0015) (0.0193) (0.0099) (0) FP (0) (0) (0.0013) (0) (0.0125) (0) 1 TP (0) (0.0177) (0.0015) (0.1156) (0.0044) (0) FP (0) (0) (0.0013) (0) (0.0056) (0) L-M cgh 248 Cancer Informatics 2010:9

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