Bioinformatics Advance Access originally published online on July 27, 2007
Bioinformatics 2007 23(18):2463-2469; doi:10.1093/bioinformatics/btm359
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Robust smooth segmentation approach for array CGH data analysis
1Statistical Laboratory, Department of Statistics, University College Cork, Ireland, 2MRC Biostatistics Unit, Institute of Public Health, Cambridge CB2 2SR, UK, 3Department of Oncology, University Hospital, SE-221 85 Lund and 4Department of Medical Epidemiology and Biostatistics, Karolinska Institutet, Stockholm, Sweden
*To whom correspondence should be addressed.
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ABSTRACT |
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 TOP ABSTRACT 1 INTRODUCTION2 METHODOLOGY3 SIMULATION STUDY4 REAL DATA EXAMPLES5 DISCUSSIONACKNOWLEDGEMENTSREFERENCES |
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Motivation: Array comparative genomic hybridization (aCGH) provides a genome-wide technique to screen for copy number alteration. The existing segmentation approaches for analyzing aCGH data are based on modeling data as a series of discrete segments with unknown boundaries and unknown heights. Although the biological process of copy number alteration is discrete, in reality a variety of biological and experimental factors can cause the signal to deviate from a stepwise function. To take this into account, we propose a smooth segmentation (smoothseg) approach.
Methods: To achieve a robust segmentation, we use a doubly heavy-tailed random-effect model. The first heavy-tailed structure on the errors deals with outliers in the observations, and the second deals with possible jumps in the underlying pattern associated with different segments. We develop a fast and reliable computational procedure based on the iterative weighted least-squares algorithm with band-limited matrix inversion.
Results: Using simulated and real data sets, we demonstrate how smoothseg can aid in identification of regions with genomic alteration and in classification of samples. For the real data sets, smoothseg leads to smaller false discovery rate and classification error rate than the circular binary segmentation (CBS) algorithm. In a realistic simulation setting, smoothseg is better than wavelet smoothing and CBS in identification of regions with genomic alterations and better than CBS in classification of samples. For comparative analyses, we demonstrate that segmenting the t-statistics performs better than segmenting the data.
Availability: The R package smoothseg to perform smooth segmentation is available from http://www.meb.ki.se/~yudpaw
Contact: yudi.pawitan{at}ki.se
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1 INTRODUCTION |
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TOPABSTRACT1 INTRODUCTION2 METHODOLOGY3 SIMULATION STUDY4 REAL DATA EXAMPLES5 DISCUSSIONACKNOWLEDGEMENTSREFERENCES |
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The DNA-sequence copy number is the number of copies of DNA at a region of a genome. In humans, the normal copy number is two for all the autosomes. DNA copy number alterations are key genetic events in the development and progression of human cancers (Lengauer et al., 1998). Recently, array comparative genomic hybridization (aCGH) (Snijders et al., 2001) was developed to measure DNA copy-number variations across a whole genome. In aCGH, genomic DNA isolated from tumor and normal cells are differentially labeled by two fluorescent dyes and co-hybridized to a microarray whose spots contain DNA sequences that map to chromosomal regions within the human genome. Subsequently, the ratio of the intensities of the two fluorochrome is computed at each spot. An obvious way to visualize copy-number variation is to plot the spot fluorescence ratios as a function of their location within the genome.
Statistical methods for analyzing copy number data are aimed at identifying regions of genomic alteration that correlate with clinical phenotypes such as survival. Here, one faces hypothesis testing problems with thousands or tens of thousands of regions to be considered simultaneously. Some work has been done to address this problem (e.g. Pawitan et al., 2005; Pollack et al. 2002). However, the immediate problem of applying these methods to aCGH data is that the genomic spatial structure has not been taken into consideration.
Segmentation is one approach to make use of the spatial information. Various segmentation methods have been proposed for analyzing aCGH data. Olshen et al. (2004) proposed a circular binary segmentation (CBS) algorithm that identifies the change points through successive comparison of segments of the chromosome, with evaluation of local significance via permutation. This is followed by a pruning algorithm to control the number of change points. Picard et al. (2005) proposed a likelihood-based method to identify the change points for the sequence of log2-ratios. Hupe et al. (2004) likewise presented a Gaussian-based likelihood approach (GLAD). Change points are identified using a piecewise-constant regression model that employs adaptive weights smoothing (AWS), in which the errors are assumed to be independent and normally distributed. Fridlyand et al. (2004) proposed a discrete-state hidden Markov Model (HMM) approach. This model assumes that, given the genetic states at all previous regions, the genetic state at a given region depends only on the true state at the immediately previous region. Wang et al. (2005) used an agglomerative clustering technique (CLAC) to identify the change points.
All of those segmentation methods are based on modeling data as a series of discrete segments, with unknown boundaries and unknown heights. However, it is obvious from observed data (e.g. Fig. 1) that most segments are not discrete. Although the underlying biological process is discrete, in reality a variety of biological and experimental factors cause the signal under study to deviate from a stepwise function (Engler et al. 2006; Picard et al. 2005). One possible factor is the cell-to-cell variability in genomic instability (Kronenwett et al., 2004), so the observed pattern is only an average of a large number of cells with potentially different copy-number patterns. This averaging process tends to smooth out the discrete boundaries.
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To allow for the possibility of smooth transition between segments, we propose a smooth segmentation (smoothseg) method to analyze aCGH data. Another promising algorithm is a denoising by wavelets (Hsu et al., 2005). To get a robust segmentation, our model is based on a doubly heavy-tailed random-effect model, where the first heavy-tailed structure deals with outliers and the second with jumps in the underlying pattern associated with different segments. Smoothness is achieved because the jumps are regularized by correlated random-effect assumption. Using simulated and real data sets, we demonstrate how smoothseg is utilized to aid in identifying regions of genomic alteration that correlate with clinical phenotypes and in classification of samples.
The performance of smoothseg in identification and classification is evaluated using the false discovery rates (FDR) and classification error rates. For the real data, the classification error rates are calculated using the leave-one-out cross-validation procedure. For comparison, we also computed the FDR and classification error rates based on ‘DNAcopy’, which is the R package implementing the circular binary segmentation (CBS) algorithm. A recent comparison study (Lai et al., 2005) of CBS and 10 other approaches concluded that CBS is one of the two methods that perform consistently well. Willenbrock and Fridlyand (2005) in another comparison study also concluded that DNAcopy has the best operating characteristics in terms of sensitivity and FDR.
To summarize our novel contributions: (i) we developed a method to perform robust segmentation of aCGH data, and implemented this in a software package. (ii) We showed that smoothseg compares favorably with one of the best existing methods. For the real data sets, smoothseg leads to smaller false discovery and classification error rates than the CBS algorithm. In a realistic simulation setting, smoothseg is better than CBS. Last but not least, smoothseg performs much faster than DNAcopy (version 1.8.1). (iii) For comparative analyses, we demonstrated that segmenting the t-statistics performs better than segmenting the data.
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2 METHODOLOGY |
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TOPABSTRACT1 INTRODUCTION2 METHODOLOGY3 SIMULATION STUDY4 REAL DATA EXAMPLES5 DISCUSSIONACKNOWLEDGEMENTSREFERENCES |
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Consider
regions located on a genome at which the relative copy number is measured by the log2-ratios of fluorescence intensities between tumor and reference (normal) samples. Denote by 
the observed log2-ratios of the 
th region at location 
for 
, and define the vector 
. The genomic locations 
are fixed and known, with 
.
Our statistical model is based on correlated random-effects for the unobserved pattern. As shown in Ruppert et al. (2003) and more recently in Lee et al. (2006), the random-effect model allows flexible specification for the unobserved pattern. Particularly, a heavy-tailed random-effect model would provide a good fit to data containing jump discontinuities as would be expected from a segmented pattern, but at the same time allowing for a smooth transition. Additionally, to achieve robustness against outliers we also specify a heavy-tailed model for the error term. So, overall we have a doubly heavy-tailed model. Specifically, we model
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(1,...,n) are iid t-distribution with location 0, and unknown dispersion 
and k degrees of freedom. Hence, the problem is how to estimate f(xi) based on observations y.
In matrix form, we can write a general model
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f (xi) is an unknown random-effect parameter. The error is assumed independent of f (f1,...,fn).
The smoothness of f can be imposed on the scaled differences 
fi/xi (fi – fi – 1) / (xi – xi – 1). It is most common to specify that the scaled second-order differences 
2fi / (xi)2 are iid with some distribution. Because f is mostly smooth, the size of ai 2fi = (xi)2 is very small relative to the local noise. This means there will be little difference whether we specify the model on or ai. For convenience we shall use the latter. Traditionally, the normal distribution is assumed, but smoothing based on the normal distribution converts jumps into gradual changes and tends to round plateaus, so the normal distribution is not a good choice for aCGH data. It is well known that the normal distribution leads to a quadratic loss function. Eilers and Menezes (2003) have also demonstrated that a smoothing algorithm based on the quadratic loss function does not work well for aCGH data.
To adapt to segmented data, the model must allow jumps, which means we must consider a heavy-tailed distribution for ai. We propose a smooth segmentation (smoothseg) approach based on the assumptions that the second-order differences ai are iid Cauchy, a heavy-tailed model with a simple parametrization. For a quick illustration, Figure 1 shows the smoothseg segmentation of CGH data from chromosomes 1 to 4 from a breast cancer sample. The biggest difference between this and the standard smoothing is that in the latter the choice of smoothing parameter is paramount, and in its application to aCGH data it is easy to oversmooth the data.
The estimation of the random effects is carried out by the maximum likelihood method. The log-likelihood based on the observation vector y and the random effects f is
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, and the random effects contribution comes from the Cauchy model with location 0 and scale factor 
: |
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2.1 Estimation of f given and f
Maximizing the likelihood of the non-Gaussian random-effect model is described in Pawitan (2001, pp. 464–466), and we will derive an iterative weighted least-squares (IWLS) algorithm. The first derivative of the 
with respect to f can be written as
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2 the (n – 2) x n matrix that represents the second-order difference operator of n-dimension vectors, and |
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Combining the above results, the first derivative of 
with respect to f is
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To solve S(f) = 0, we can apply a simple iterative scheme that works like IWLS: using a starting value f 0, compute the matrices W and D, and solve the updating equation
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| (3) |
. Given an updated f, recompute W and D, and continue to iterate; at convergence we have the estimate 
.
Note that finding the solution of (3) is potentially highly demanding due to its huge dimension. To overcome this problem, we exploit the band-limited property of (W + 
(2)'D– 12). We use well-tested fortran subroutines available in Linpack (e.g. Dongarra et al., 1979, Chapter 2) and get a very fast inversion procedure.
2.2 Estimation of and f
The dispersion parameter can be estimated directly from the error term 
. For simplicity, we estimate robustly by the median absolute deviation
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Given , there is a single parameter that controls the smoothness of the model, hence the commonly used smoothing parameter estimation methods such as the AIC can be used to estimate it. First, the degrees of freedom parameter is defined as (Pawitan, 2001, p. 448)
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. Using the same method as in Pawitan (1996), the degree of freedom can be approximated as |
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and 
are the average diagonals of W and D, and 
is the jth eigenvalue of the second derivative matrix 2. Then, we estimate by minimizing the AIC criterion |
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2.3 Analysis of comparative studies
In many CGH studies, we will be interested not only in the copy-number alterations in single samples, but rather in the comparison of groups of samples, i.e. whether there is a consistent change across the samples. All our examples will deal with comparison of two independent groups; extension to several groups is straightforward. In principle, we can extend the previous model by allowing an extra term to capture group effects:
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| (4) |
ijk is the error term. We then proceed by specifying heavy-tailed correlated random-effect models for gis and fijs. Joint estimation of these parameters can be done using the iterative back-fitting algorithm (e.g. Pawitan 2001, page 445). On the first step, set fij 0, so the estimation of gi(xk) involves segmentation of the mean profile. Then define the corrected data |
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The estimation of individual fijs can use the method we have developed, but the estimation of the mean profile gi (xk) from a collection of aCGH arrays requires a non-trivial extension. Furthermore, for a medium to large number of samples, the joint computation becomes unwieldy and likely slow to converge. Instead, here we make a compromise by separating the segmentation and testing steps. It is not immediately obvious whether we should segment the data prior to testing, or vice versa. In our simulations we will compare two strategies:
- Segmenting original data for each array followed by calculating t-statistics.
- Calculating t-statistics from the original (unsegmented) data, followed by segmenting the t-statistics.
An example of the t-statistics pattern from the breast cancer data set is given in Figure 2, showing that segmenting the data and segmenting the t-statistics can produce different results.
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In both strategies, the standard t-statistic with pooled variance is computed for each probe. To deal with the multiple comparison problem, we asses the significance of the test statistics using the FDR (Benjamini and Hochberg, 1995; Storey, 2002). The FDR computation using the original log2-ratio, segmented log2-ratio or segmented t, is calculated by adding the segmentation step to the EOC function in the R package OCplus (Pawitan et al., 2005).
In this computation, P-values are obtained using the permutation test. To account for the dependence between the probes, a single permutation of the group labels is applied to the whole array, so the data from an array is always kept together and the dependence structure is preserved. Let 
be the ordered P-values from the permutation test. For the top k most significant probes, the FDR is computed as
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0 is the proportion of probes with non-differential copy number between the two groups, and it is estimated by |
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is typically stable for a wide range of c between 0.6 and 0.9, and in our computations we use c = 0.7.
2.4 Software
We have built a public-domain software smoothseg to perform the procedures described in this article. It can be accessed in http://www.meb.ki.se/~yudpaw. The function smoothseg performs the smooth segmentation for a collection of aCGH data, and the function FDRcgh() performs the estimation of FDR for comparative studies, including paired-sample, two-sample and multi-sample comparisons.
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3 SIMULATION STUDY |
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TOPABSTRACT1 INTRODUCTION2 METHODOLOGY3 SIMULATION STUDY4 REAL DATA EXAMPLES5 DISCUSSIONACKNOWLEDGEMENTSREFERENCES |
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We conducted a simulation study to evaluate the performance of smoothseg in identifying regions of genomic alterations and classifying samples. To get a realistic simulation, Willenbrock and Fridlyand (2005) proposed a scheme that generates genomic profiles of comparable complexity to real data. The first step is choosing a template from the empirical profile constructed from the CBS segmentation of a primary breast tumor data set of 145 patients. Each template consists of 500 regions placed on one chromosome, containing segments of normal and altered copy numbers, and there is one probe associated with each region. It is then assumed that genetic alteration regions in a tissue can be measured with 70 % chance, so we apply 30 % chance for the segments with copy-number changes to be re-assigned a normal copy number of 2. Finally, to take into account the sample contamination, each sample was assigned a proportion Pt of tumor cells and (1 – Pt) normal cells. Consequently, the expected log2-ratio profile for each sample was calculated as
, where c was the assigned copy number in the sample. The log2-ratio profile data was generated by adding more realistic t-distribution noise of mean 0 and dispersion SD to the expected log2-ratios. In Willenbrock and Fridlyand's; simulation setting, Pt and SD are drawn from a uniform distribution (0.3, 0.7) and (0.1, 0.2), respectively.
3.1 Identification of regions with genomic alterations
In this simulation study, we generated 150 samples from one genomic template. The ‘true regions’ with copy-number gains or losses were recorded, allowing the computation of sensitivity and specificity. The performance of smoothseg, wavelets smoothing and DNAcopy were evaluated by the operating characteristic (OC) curve. We calculated the sensitivity as the number of regions with copy-number gains or losses whose absolute value are above the threshold level divided by the number of regions with copy-number gains or losses. We calculated the specificity as the number of regions with copy-number 2 whose absolute values are below the threshold level divided by number of regions with copy-number 2. In order to calculate OC curve, we varied the threshold value from 4.5 to 0.0.
Figure 3a shows that the resulting OC curves for smoothseg, wavelets smoothing and DNAcopy are comparable. The problem is perhaps too easy as the signal-to-noise ratio is very high, so all procedures have areas under curve close to one, although smoothseg performs marginally better. We then modified Willenbrock and Fridlyand's; setting to make the problem harder by increasing the error rate. The modification consisted of dividing the length of each segment with copy-number alteration by 3 and using fixed noise level SD = 0.3. In this setting, the data are more noisy and have smaller copy-number alteration segments. Now, smoothseg clearly performs better than wavelet smoothing and DNAcopy segmentation.
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3.2 Testing for differential copy number
To simulate realistic data sets with samples from two tumor classes, two genomic templates were used and 20 samples were randomly drawn from each of the two templates. The proportion of regions with differential copy number between the two templates is determined implicitly by the choice of templates. On average, Willenbrock and Fridlyand's; setting generates 42 % regions with differential copy number.
The simulation was repeated 500 times. For each simulated data set, as described in the Methodology Section, segmenting t and segmenting data was carried out. From the 500 simulations, we calculated the sensitivity as the number of regions with copy-number difference between two templates whose P-values are below the threshold level (i.e. significant regions) divided by the number of regions with copy-number difference between two templates. We calculated the specificity as the number of regions with the same copy number between two templates whose P-values are above the threshold level divided by number of regions with same copy number between two templates. In order to calculate the OC curve, we varied the threshold value from 0.0 to 0.6.
Figure 4a shows the resulting OC curves for the different procedures. For comparison, the OC curves based on the DNAcopy segmentation are also shown in Figure 4a. Compared to unsegmented analysis, both smoothseg and DNAcopy using both segmentation strategies have improved identification of regions with differential copy number, which is evident by higher sensitivity for any given specificity. For smoothseg, segmenting the t-statistic is better than segmenting the data. In this simulation setting, DNAcopy performs better than smoothseg. This is probably not surprising, given that the simulation model is based on empirical distribution constructed from DNAcopy segmentation. Again, in this original setting, all procedures perform with relatively little errors compared to what we might expect with real data.
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For the modified setting, the resulting OC curves in Figure 4b show that the level of error rate is now more realistic, where the naive unsegmented method performs poorly. We now observe that smoothseg performs better than DNAcopy. Figure 4b also shows that segmenting the t-statistics is better than segmenting the data.
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4 REAL DATA EXAMPLES |
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TOPABSTRACT1 INTRODUCTION2 METHODOLOGY3 SIMULATION STUDY4 REAL DATA EXAMPLES5 DISCUSSIONACKNOWLEDGEMENTSREFERENCES |
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In this section, we demonstrate the performance of smoothseg for real data analysis. The performance in identifying the differential copy number is evaluated using the FDR, while the performance in supervised classification is evaluated using the classification error rates, calculated using the leave-one-out-cross validation (LOOCV).
The first data set (unpublished data) consists of 14 BRCA1 and 8 BRCA2 mutation-classified breast cancer patients, analyzed on tiling-32K BAC arrays. For each patient, fluorescence ratios are collected from 31 890 BAC clones across the genome. The data were collected at Department of Oncology, Lund University, Sweden. The second data set consists of 75 oral squamous-cell carcinoma (OSCC) samples from Snijders et al. (2001), comprising 14 p53-mutant and 61 wild-type samples. The data set was downloaded from http://www.cbs.dtu.dk/~hanni/aCGH.
4.1 Identification of differential copy number
Regions with differential copy number between two cancer subtypes are identified by testing if a region had a significantly different log2-ratio in samples from one subtype compared to the log2-ratio for the same region in samples from the other subtype. For the breast-cancer data, the profile of the t-statistics across the chromosomes is given in Figure 5.
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Existence of genomic regions with differential gain or loss between BRCA1 and BRCA2 tumors have previously been reported by conventional CGH (van Beers et al., 2005) and aCGH (Jonsson et al., 2005). Applying FDR cutoff of 5 % yielded 5851 significant BAC clones, and the segmented t-statistic profile generated several distinct genomic regions differentiating between the tumor types (Fig. 5). Identified regions included 4p15.32–p14, 4q31.3, 4q32.1–q34, 5q, 17q23.3–q24.2 and 20q12–q13.12, previously reported as highly typical for the BRCA1 or BRCA2 genotype (Jonsson et al., 2005). These differential alterations suggest different development pathways for BRCA1 and BRCA2, and might explain any differences in the clinical progression.
The FDR corresponding to varying proportion of declared significant regions from 0 to 10 % are shown in Figure 6, both for the breast cancer and OSCC data. For comparison, FDR curves based on the DNAcopy segmentation are also calculated and shown in Figure 6. We can see that segmentation-based procedures yield smaller FDR than the segmentation-free procedure. Overall, consistent with the simulated data, segmenting t-statistic performs better than segmenting the data. Finally, for the breast cancer data, smoothseg performs better than DNAcopy, but for the OSCC data they are comparable.
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4.2 Classification
We carried out classification using original log2-ratios and segmented data as input variables to the classifier. For simplicity, we considered a diagonal linear discriminant classifier (DLDA), which was previously demonstrated to have good performance in microarray studies (Dudoit et al., 2002). The classification was carried out using the R package sma (Dudoit et al., 2002). The classification error rates were calculated using the leave-one-out cross-validation (LOOCV) for a varying number of predictors. The predictors were ranked by the ratio of between-group to within-group sum of squares of the training data within each cross-validation run. Molinaro et al. (2005), in the context of classification of genomics data, performed extensive simulations that showed that the LOOCV has the smallest bias and mean-square error for linear discriminant analysis.
The classification error rates based on the breast cancer and OSCC data sets are shown in Figure 7. We can see that classification using segmented data has smaller classification error rates than those found using the original unsegmented data. Furthermore, for both data sets, smoothseg performs better than DNAcopy.
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Consistent with Jonsson et al. (2005), the BRCA1 and BRCA2 breast tumors can be distinguished extremely well, achieving zero error rate in the classification even with very few predictor genes. The biological differences between these two groups are described more meaningfully in the previous subsection. However, this result is based on a small sample and needs to be validated in a larger sample. The separation between the OSCC groups are not as successful. In fact the best error rate (12 %) is only slightly better than the rate of a naive classification of all cases as wild-type (14/75 = 18 %). This could be partly due to the low resolution of the aCGH platform, containing only 1979 regions.
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5 DISCUSSION |
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TOPABSTRACT1 INTRODUCTION2 METHODOLOGY3 SIMULATION STUDY4 REAL DATA EXAMPLES5 DISCUSSIONACKNOWLEDGEMENTSREFERENCES |
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We have described a smooth segmentation (smoothseg) approach to the analysis of aCGH data, and illustrated its performance using simulated and real data sets. Smoothseg is derived from the Cauchy distribution assumption, as it is expected to handle jumps in the copy-number pattern, while at the same time allowing some smooth transition. The latter can occur naturally because of cellular variability in genomic instability (e.g. Kronenwett et al., 2004) or because of other factors such as variation in GC content. However, while our method is motivated by this possibility, in principle it does not rely on it for its performance. In the simulation studies presented above, the templates have discrete segments.
For processing a large number of today's; high-resolution CGH arrays, speed is an important issue, especially when segmentation is used as pre-data-processing for downstream analyses such as testing and classification. Compared to the current implementation of DNAcopy (version 1.8.1), smoothseg segmentation is about 500 times faster (as measured by the R function system.time).
Using simulated and real data, we demonstrated how smoothseg is utilized to aid in the downstream analysis. Use of segmented data can improve the performance of testing and classification, as demonstrated by Willenbrock and Fridlyand (2005). From their comparative studies, they concluded that DNAcopy has the best operating characteristics in terms of its sensitivity and FDR. Our simulation studies indicate that the relative performance of DNAcopy and smoothseg depends on the data structure: for data with smaller genomic aberrations smoothseg gives better results than DNAcopy, while for larger aberrations the performance is about the same. The comparative study of Lai et al. (2005) also showed that discrete segmentation performs poorly for data with small alterations and high noise level. For the real data sets used in this article smoothseg performs better than DNAcopy.
When using segmentation for group comparisons, a natural question arises: should we segment the data or the t-statistics? Willenbrock and Fridlyand (2005) considered this problem, and in their simulation setting the two segmentation strategies performed similarly (see Fig. 4a in this article, or Fig. 4 in Willenbrock and Fridlyand, 2005). However, the results from a more realistic simulation setting and the real data analyses (Fig. 4b and Fig. 6 in this article) indicate that segmenting t-statistics performs better than segmenting the log-ratios. We described briefly a model-based estimation of group effects, but its implementation would require an extension of the existing computations.
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ACKNOWLEDGEMENTS |
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TOPABSTRACT1 INTRODUCTION2 METHODOLOGY3 SIMULATION STUDY4 REAL DATA EXAMPLES5 DISCUSSIONACKNOWLEDGEMENTSREFERENCES |
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This research is partially supported by the Swedish Cancer Society.
Conflict of Interest: none declared.
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FOOTNOTES |
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Associate Editor: Thomas Lengauer
Received on January 23, 2007; revised on June 12, 2007; accepted on July 9, 2007
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