Gene expression data frequently contain missing values, however, most down-stream analyses for microarray experiments require complete data. In the literature many methods have been proposed to estimate missing values via information of the correlation patterns within the gene expression matrix. Each method has its own advantages, but the specific conditions for which each method is preferred remains largely unclear. In this report we describe an extensive evaluation of eight current imputation methods on multiple types of microarray experiments, including time series, multiple exposures, and multiple exposures × time series data. We then introduce two complementary selection schemes for determining the most appropriate imputation method for any given data set.We found that the optimal imputation algorithms  are all highly competitive with each other, and that no method is uniformly superior in all the data sets we examined. The success of each method can also depend on the underlying "complexity" of the expression data, where we take complexity to indicate the difficulty in mapping the gene expression matrix to a lower-dimensional subspace. We developed an entropy measure to quantify the complexity of expression matrixes and found that, by incorporating this information, the entropy-based selection (EBS) scheme is useful for selecting an appropriate imputation algorithm. We further propose a simulation-based self-training selection (STS) scheme. This technique has been used previously for microarray data imputation, but for different purposes. The scheme selects the optimal or near-optimal method with high accuracy but at an increased computational cost.Our findings provide insight into the problem of which imputation method is optimal for a given data set. Three top-performing methods  are competitive with each other. Global-based imputation methods  performed better on mcroarray data with lower complexity, while neighbour-based methods  performed better in data with higher complexity. We also found that the EBS and STS schemes serve as complementary and effective tools for selecting the optimal imputation algorithm. As with many types of experimental data, expression data obtained from microarray experiments are frequently peppered with missing values (MVs) that may occur for a variety of reasons. Randomly scattered MVs may be due to spotting problems, poor hybridization, inadequate resolution, fabrication errors, or contaminants on the chip including scratches, dust, and fingerprints. Because many down-stream microarray analyses such as classification methods, clustering methods, and dimension reduction procedures require complete data, researchers must either remove genes with one or more MVs, or, preferably, estimate the MVs before such procedures can be employed. Consequently, many algorithms have been developed to accurately impute MVs in microarray experiments -6.The first evaluation of MV estimation methodology in microarray data was reported by Troyanskaya et al. , who comIn this study, we present a comprehensive evaluation of the performance of current imputation methods across a wide variety of types and sizes of microarray data sets, to assess their performance under different conditions and establish guidelines for their appropriate use. In addition, we develop and test two selection procedures for determining the most appropriate imputation method for a given data set. To this end, we have implemented and tested existing methods for MV imputation, to assess the performance of each of these methods under various conditions and determine the circumstances for which different imputation procedures are preferred.Specifically, we tested eight different algorithms from the literature that have been shown to perform well at imputing MVs in microarray data sets: KNN.e (Euclidean based neighbor selection), KNN.c (correlation based neighbor selection), SVD, ordinary least squares (OLS) ,9, partik values varied greatly (between 210 and 2397), the number of tested k values in this range was typically around 6 or 7. Neither the LSA nor BPCA methods required parameter optimization, as the parameter settings are predetermined in both cases.Optimal parameter values for the eight methods under investigation, for each data set, are reported in Table et al.[Figure et al. have preThe average LRMSE values for each method in all nine data sets are given in Figure The entropy of each data set, calculated using (4), is given in Table As indicated in Figure p-values on the order of 10-10 to 10-12.The accuracy of the STS scheme in selecting the top performing method is given in Table et al.[We performed an extensive evaluation of existing methods for imputing missing values in microarray data. In contrast to the recent comprehensive comparative study in gene clustering of microarray data , where tet al. found thrd best in SP.AFA and SP.ELU), and thus has the overall advantage. In particular, the difference between LSA and LLS occurs mainly on the high entropy data sets ROS and ALI, where the performance of LLS drops considerably (to 4th best out of 8), while LSA has the lowest LRMSE. However, we again emphasize that the overall differences between these top three  methods is slim, and that there is no conclusively best method. Several extensions and improvements from these methods have been proposed in the literature [Overall, the LSA, LLS, and BPCA imputation algorithms performed the best in our simulation study. Both the LSA and LLS algorithms are based on selection of gene neighbors for imputation, but they also each have features which resemble global based imputation. LSA uses array-based imputation in addition to gene-based imputation, and LLS allows for the selection of a very large number of genes  to use for imputation. Since LLS solves the least squares system using the Moore-Penrose pseudoinverse of the neighbor gene expression matrix, singular matrices can be used. The pseudoinverse is determined by the SVD of the expression matrix, and this method is similar to the supervised principal components procedure , which hterature ,4,11,12 For each imputation method, we used the optimal parameter settings for each data set. Generally, the parameter settings were similar across all data sets. For KNN and OLS, the number of neighbors selected was between 5 and 15, although in one case 200 neighbors was the optimal choice. For PLS, the number of components was between 8 and 15, and for SVD the percentage of eigenvalues was typically in the range of 0.15 to 0.25. The variation in the optimal parameter setting was greatest for LLS, though this range actually corresponded to only a small number of actual choices, and further the LRMSE curve was relatively flat over that range of values. It should be noted that neither BPCA nor LSA required parameter optimization, and LLS has parameter selection built into the algorithm internally, making these algorithms attractive choices for automated imputation of MVs.et al.[While we have evaluated a broad spectrum of imputation algorithms in the literature, our coverage is by no means exhaustive. Some methods we excluded include , which uet al. compare et al.[Our assessment of the imputation algorithms and selection schemes is based on an overall average of the LRMSE. Other authors have examined the distribution of squared imputation errors  as a function of the true gene expression values to determine if certain algorithms are more accurate for imputing high (or low) expression levels . For exaet al. indicateet al.,16,17. TPrevious studies have shown that the performance of an imputation algorithm depends on the underlying correlation structure of the gene expression matrix . We havee(D) < 0.9), we may restrict our attention to imputation methods like LSA, LLS, and BPCA, while with high complexity data (e(D) > 0.94) we may instead shift our attention to local imputation methods like LSA, KNN.c, and OLS..The entropy measure can be formally used to select an appropriate imputation algorithm for a particular data set, via the EBS scheme (6). It has an appealing advantage of fast computation since, for each new data set, the preferred method is selected only according to its entropy measure. The EBS scheme does have a practical limitation, in that it requires imputing MVs from multiple other data sets in order to fit the regression model (5). In addition its performance is dependent on which data sets were used to fit the model, although the cross validation results using the leave-one-out data sets demonstrate that the method performs well despite these potential limitations. One use of the EBS scheme is to reduce the number of imputation methods under consideration. For example, with low complexity data , multiple exposure (ME), and time series × multiple exposure (TS × ME). Each of these experimental designs, in general, exhibit different types of expression patterns. As examples of TS experiments, we selected the yeast cell-cycle data from Spellman et al.  and correlation (KNN.c) based neighbor selection approaches in our study..The widely used KNN procedure finds the distance . Becausek most similar neighbor genes. MVs are imputed as the weighted average of the predicted values from the regression of the gene with MVs onto each neighbor gene [In this neighboring-based approach, the gene with the MV is regressed over each of the bor gene ,9. Neighryx is the correlation between y, the gene with MVs, and the potential neighbor gene x. As in KNN, MVs are omitted from the distance calculation and the simple linear regressions.where k nearest neighbors. The MVs are imputed based on the least squares estimates, determined using the pseudoinverse of the k nearest neighbors expression matrix. If the percentage of MVs is relatively small, then neighbor genes with MVs are excluded from the least squares system, otherwise MVs are initially estimated by the row (gene-wise) average.The LLS procedure of  selects The LSA procedure of  combinesc components are selected. The missing values are then imputed by regressing the target gene onto the PLS components. MVs are first imputed by row average prior to PLS imputation [PLS regression selects linear combinations of genes  exhibiting high covariance with the gene having the MV (the target gene). The first linear combination has the highest covariance with the target gene, and subsequent components have the greatest covariance with the target gene in a direction orthogonal to the previously selected components until a total number of putation .p of eigengenes which correspond to the largest eigenvalues are then used to reconstruct the MVs in the expression matrix. An expectation-maximization (EM) approach is used to iteratively improve the imputed MVs and expression covariance matrix until total change in the matrix falls below a prescribed threshold (here taken to be 0.01) [This approach initially sets MVs to the row average, and then uses singular value decomposition of the gene expression matrix to create orthogonal principle components, called "eigengenes." The proportion This method uses Bayesian estimation to fit a probabilistic PCA model . A variayij = log(xij), where xij is the expression intensity of gene i and sample j), we calculate the LRMSE as the ordinary RMSE of the log-transformed expression matrix (where yij):We employed the log-transformed root mean squared error (LRMSE) as the metric by which we assessed the performance of our imputation methods. This statistic has the important property of scale-invariance, which allows direct comparisons of imputation accuracy across different data sets. Therefore, it is scientifically more reasonable than other RMSE metrics. By log-transforming the expression matrices prior to imputation D, we create N data sets with 5% random missing values, denoted by Dk)     in the denominator is to standardize for different k and represents the summand in the case where pi = 1/k, 1 ≤ i ≤ k. It is easy to show that 0 ≤ e(D) ≤ 1. Intuitively, a low entropy measure reflects low complexity of the data, represented by relatively few eigenvalues that are distinctively larger than the other eigenvalues. This indicates that the data matrix can be effectively reduced to a low-dimensional space. Conversely, a large entropy measure indicates a similar magnitude of all the eigenvalues and that the data cannot be reduced to a low-dimensional space. In the case of chaotic data, all the eigenvalues are the same and (4) gives a maximum entropy of one regardless of the rank.Note that the term log is the entropy level of Dj, αi and βi are the intercept and regression slope for imputation method Mi, i = 1, ...,8 yj is a fixed effect representing the intrinsic imputation difficulty of data set Dj, and εijk are random noises. Essentially, the above model is an analysis of covariance model, with separate regression lines fitted for each method of the LRMSE values on the entropy values of the data sets. Note that the parameter yj in model (5) is necessary because different data sets Dj have different levels of intrinsic imputation difficulty. We fit the model in equation (5) using all nine data sets and denote the estimated model by L, with Mi performs better  for data sets with low entropy measure. Conversely, if Mi is better for more complex (with higher entropy) data sets.where D with missing values, we can determine the best imputation method in Mi from the above estimated linear model. We denote by D with missing values imputed by Mi. Since the entropy of D, e(D), cannot be calculated with missing values, it is estimated by MAve is the imputation method based on the average expression level of a gene. From (5), the best imputation method for data set D is selected byGiven a new data set D based on the model fitted from the nine selected data sets, L.i.e., the imputation method with the lowest predicted LRMSE of data Dj, let Dj)  using the set Dj)  and accuracy (3) measures defined previously.et al.[et al.[The STS procedure explicitly determines the optimal imputation algorithm for a particular data set by simulating MVs in the subset of the expression matrix which is complete , imputing these simulated MVs, and comparing these imputed values to the known expression values. This strategy has also been employed by others, though for different purposes. Jornsten et al. used thel.[et al. found thn replicates to generate data sets l = 1,..., n. For each method Mi, we calculate the rank-sum statisticMore specifically, we randomly remove an additional 5% of expression values from each The STS scheme is then defined asn = 10 replicates we used was sufficient for determining the preferred imputation method, the null hypothesis that all methods are equally effective  was tested using Friedman's test [Again, the difference (2) and accuracy (3) measures defined previously can be applied for evaluation. To assess whether the n's test .k between 5 and 200, while for SVD we varied the proportion p of eigengenes between 0.1 and 0.5. For PLS we tested the number of components c between 2 and 25. LLS has parameter optimization built into the algorithm. The number of neighbor genes evaluated depended on the data set, but generally varied from 10 to upwards of 2000, with spacing between adjacent k values increasing as k increased. For our study, we ran the algorithm several times to determine the optimal k value, then held this fixed for the subsequent simulations. BPCA and LSA do not require parameter optimization.To evaluate our imputation methods and selection schemes on missing data, we randomly removed known expression values from the complete matrices, imputed these intentionally created MVs, and assessed imputation performance using the LRMSE. For our first set of simulations (Simulation I), we empirically tested 10 simulations for each data set under different parameter values for each of the methods to optimize their performance, with the percentage of MVs set at 5%. For the KNN and OLS methods, we tested values of We next performed 50 simulations using the optimized parameters on different data sets with 5% of the expression values removed to compare the accuracy of different algorithms across a wide range of microarray data (Simulation II). These 50 simulations were also used to test the EBS scheme via the leave-one-out data sets cross validation. To test the STS scheme, we ran an additional set of simulations (Simulation III). Simulation III consisted of 10 initial "first tier" simulations per data set each having 5% MVs. For each of these first tier simulations, 10 "second tier" simulations having an additional 5% MVs each were generated. The second tier simulations were used to determine the STS selected algorithm, while the first tier simulations were used to evaluate the performance of STS. Lastly, we ran an additional 20 simulations at 2%, 10%, and 15% missing for the ALO, CAU, GOL, SP.AFA, and SP.ELU data sets, to evaluate the performance of each algorithm with varying percentage of MVs.KNN, OLS, PLS, SVD, and BPCA were all coded in the R language . The BPCThe author(s) declares that there are no competing interests.GNB wrote the code for the imputation algorithms and the selection schemes, analyzed the results, conducted the simulation studies, and drafted the manuscript. JRS also conducted the simulation studies, tabulated the results, and drafted the manuscript. REB helped code the imputation algorithms. MJL performed the data collection. GCT conceived of the study and the selection schemes, participated in its design and coordination, and helped to draft the manuscript. All authors read and approved the final manuscript.


While in principle a seemingly infinite variety of combinations of mutations could result in tumor development, in practice it appears that most human cancers fall into a relatively small number of "sub-types," each characterized a roughly equivalent sequence of mutations by which it progresses in different patients. There is currently great interest in identifying the common sub-types and applying them to the development of diagnostics or therapeutics. Phylogenetic methods have shown great promise for inferring common patterns of tumor progression, but suffer from limits of the technologies available for assaying differences between and within tumors. One approach to tumor phylogenetics uses differences between single cells within tumors, gaining valuable information about intra-tumor heterogeneity but allowing only a few markers per cell. An alternative approach uses tissue-wide measures of whole tumors to provide a detailed picture of averaged tumor state but at the cost of losing information about intra-tumor heterogeneity.The present work applies "unmixing" methods, which separate complex data sets into combinations of simpler components, to attempt to gain advantages of both tissue-wide and single-cell approaches to cancer phylogenetics. We develop an unmixing method to infer recurring cell states from microarray measurements of tumor populations and use the inferred mixtures of states in individual tumors to identify possible evolutionary relationships among tumor cells. Validation on simulated data shows the method can accurately separate small numbers of cell states and infer phylogenetic relationships among them. Application to a lung cancer dataset shows that the method can identify cell states corresponding to common lung tumor types and suggest possible evolutionary relationships among them that show good correspondence with our current understanding of lung tumor development.Unmixing methods provide a way to make use of both intra-tumor heterogeneity and large probe sets for tumor phylogeny inference, establishing a new avenue towards the construction of detailed, accurate portraits of common tumor sub-types and the mechanisms by which they develop. These reconstructions are likely to have future value in discovering and diagnosing novel cancer sub-types and in identifying targets for therapeutic development. One of the great contributions of genomic studies to human health has been to dramatically improve our understanding of the biology of tumor formation and the means by which it can be treated. Our understanding of cancer biology has been radically transformed by new technologies for probing the genome and gene and protein expression profiles of tumors, which have made it possible to identify important sub-types of tumors that may be clinically indistinguishable yet have very different prognoses and responses to treatments -4. A deeoncogenetic trees) using evolutionary distances estimated from the presence or absence of specific mutation events [More sophisticated computational models of tumor evolution, drawn from the field of phylogenetics, have provided an important tool for identifying and characterizing novel cancer sub-types . The prin events , global n events , or micrn events . More inn events . These aAn alternative approach to tumor phylogenetics, developed by Pennington et al. ,16, reliEach of these two approaches to cancer phylogenetics has advantages, but also significant limitations. The tumor-by-tumor approach has the advantage of allowing assays of many distinct probes per tumor, potentially surveying expression of the complete transcriptome or copy number changes over the complete genome. It does not, however, give one access to the information provided by knowledge of intratumor heterogeneity, such as the existence of transitory cell populations and the patterns by which they co-occur within tumors, that allow for a more detailed and accurate picture of the progression process. The cell-by-cell approach gives one access to this heterogeneity information, but at the cost of allowing only a small number of probes per cell. It thus allows for only relatively crude measures of state using small sets of previously identified markers of progression.components. Unmixing comes up in numerous contexts in the analysis and visualization of complex datasets and has been independently studied under various names in different communities, including unmixing, "the cocktail problem," "mixture modeling," and "compositional analysis." In the process, it has been addressed by many methods. One common approach relies on classic statistical methods, such as factor analysis [simplex) that encloses the observed points [One potential avenue for bridging the gap between these two methodologies is the use of computational methods for mixture type separation, or "unmixing," to infer sample heterogeneity from tissue-wide measurements. In an unmixing problem, one is presented with a set of data points that are each presumed to be a mixture of unknown fractions of several fundamental analysis ,24, prinanalysis , multidianalysis , or moreanalysis ,28. Mixtanalysis , such asd points  making uThe use of similar unmixing methods for tumor samples was pioneered by Billheimer and colleagues  for use In the present work, we develop a new approach using unmixing of tumor samples to assist in phylogenetic inference of cancer progression pathways. Our unmixing method adapts the geometric approach of Ehrlich and Full  to repred genes in n tumor samples. These data are collectively encoded as a d × n gene expression matrix M, in which each column corresponds to expression of one tumor sample and each row to a single gene in that sample. We make no assumptions about whether the sample is representative of the whole patient population or biased in some unspecified way, although we would expect the methods to be more effective in separating states that constitute a sufficiently large fraction of all cells sampled across the patient population. The fraction of cells needed to give sufficiently large representation cannot be specified precisely, however, as it would be expected to depend on data quality, the number of components to be inferred, and the specific composition of each component. We define mij to be element  of M. Note that it is assumed that M is a raw expression level, possibly normalized to a baseline, and not the more commonly used log expression level. This assumption is necessary because our mixing model assumes that each input expression vector is a linear combination of the expression vectors of its components, an assumption that is reasonable for raw data but not for logarithmic data. We further assume that we are given as input a desired number of mixture components, k. The algorithm proceeds in two phases: unmixing and phylogeny inference.We assume that the input to our methods consists primarily of a set of gene expression values describing activity of d ×k matrix C, in which each column corresponds to one of the k components to be inferred and each row corresponds to the expression level of a single gene in that component. The mixture fractions are encoded in an n × k matrix F, in which each row corresponds to the observed mixture fractions of one observed tumor sample and each column corresponds to the amount of a single component attributed to all tumor samples. We define fij to be the fraction of component j assigned to tumor sample i and i. We assume that ∑i fij = 1 for all j. The overall task of the unmixing step, then, is to infer C and F given M and k.The output of the unmixing step is assumed to consist of a set of mixture components, representing the inferred cell types from the microarray data, and a set of mixture fractions, describing the amount of each observed tumor sample attributed to each mixture component. Mixture components, then, represent the presumed expression signatures of the fundamental cell types of which the tumors are composed. Mixture fractions represent the amount of each cell type inferred to be present in each sample. The degree to which different components co-occur in common tumors according to these mixture fractions provides the data we will subsequently use to infer phylogenetic relationships between the components. The mixture components are encoded in a M, C, and F for k = 3. In the example, we see two data points, M1 and M2, meant to represent primary tumor samples derived from three mixture components, C1, C2, and C3. For this example, we assume data are assayed on just two genes, G1 and G2. The matrix M provides the coordinates of the observed mixed samples, M1 and M2, in terms of the gene expression levels G1 and G2. We assume here that M1 and M2 are mixtures of the three components, C1, C2, and C3, meaning that they will lie in the triangular simplex that has the components as its vertices. The matrix C provides the coordinates of the three components in terms of G1 and G2. The matrix F then describes how M1 and M2 are generated from C. The first row of F indicates that M1 is a mixture of equal parts of C1 and C2, and thus appears at the midpoint of the line between those two components. The second row of F indicates that M2 is a mixture of 80% C3 with 10% each C1 and C2, thus appearing internal to the simplex but close to C3. In the real problem, we get to observe only M and must therefore infer the C and F matrices likely to have generated the observed M.The unmixing problem is illustrated in Fig. C1, C2, and C3 from a sample of tumors and, further, have inferred that one tumor is composed of component C1 alone, another of components C1 and C2, and another of components C1 and C3. Then we could infer that C1 is the parent state of C2 and C3 based on the fact that the presence of C2 or C3 implies that of C1 but not vice-versa. This purely logical model of the problem cannot be used directly on unmixed data because imprecision in the mixture assignments will lead to every tumor being assigned some non-zero fraction of every component. We therefore need to optimize over possible ancestry assignments using a probability model that captures this general intuition but allows for noisy assignments of components. This model is described in detail under the subsection "Phylogeny" below.The output of the phylogeny step is presumed to be a tree whose nodes correspond to the mixture components inferred in the unmixing step. The tree is intended to describe likely ancestry relationships among the components and thus to represent a hypothesis about how cell lineages within the tumors collectively progress between the inferred cell states. We assume for the purposes of this model that the evidence from which we will infer a tree is the sharing of cell states in individual tumors, as in prior combinatorial models of the oncogenetic tree problem -13. For We perform cell type identification by seeking the most tightly fitting bounding simplex enclosing the observed point set, assuming that this minimum-volume bounding simplex provides the most plausible explanation of the observed data as convex combinations of mixture components. Our method is inspired by that of Ehrlich and Full , who proWe adopt a similar high-level approach of sampling candidate simplices and iteratively expanding boundaries to generate possible component sets. There are, however, some important complications raised by gene expression data, especially with regard to its relatively high dimension, that lead to substantial changes in the details of how our method works. While the raw data has a high literal dimension, though, the hypothesis behind our method is that the data has a low intrinsic dimension, essentially equivalent to the number of distinct cell states well represented in the tumor samples. To allow us to adapt the geometric approach to unmixing to these assumed data characteristics, our overall method proceeds in three phases: an initial dimensionality reduction step, the identification of components through simplex-fitting as in Ehrlich and Full, and assignment of likely mixture fractions in individual samples using the inferred simplex.k - 1 . For this purpose, we use principal components analysis (PCA) [M into a set of orthogonal basis vectors of maximum variance, and then use the k - 1 components of highest variance. This operation has the effect of transforming the d × n expression matrix M into a linear combination PV + A, where V is the matrix of principal components of M, P is the weighting of the first k - 1 components of V in each tumor sample, and A is a d × n matrix in which each element aij contains the mean expression level of gene d across all n tumor samples. The matrix P then represents a maximum variance encoding of M into dimension k - 1. P serves as the principal input to the remainder of the algorithm, with V and A used in post-processing to reconstruct the inferred expression vectors of the components in the original dimension d.For ease of computation, we begin our calculations by transforming the data into dimension is (PCA) , which dk dimensions, something that is not guaranteed for the common alternatives to PCA. It is also of practical value in solving the simplex-fitting problem to avoid using dimensions with very little variance, an objective PCA will accomplish.Note that although PCA is itself a form of unmixing method, it would not by itself be an effective method for identifying cell states. We would not in general expect cell types to yield approximately orthogonal vectors since distinct cell types are likely to share many modules of co-regulated genes, and thus similar expression vectors, particularly along a single evolutionary lineage. Furthermore, the limits of expression along each principal component are not sufficient information to identify the cell type mixture components, each of which would be expected to take on some portion of the expression signature of several components. For the same reasons, we would not be able to solve the present problem by any of the other common dimension-reduction methods similar to PCA, such as independent components analysis (ICA) , kernel M into the reduced-dimension matrix P, the core of the algorithm then proceeds to identify mixture components from P. For this purpose, we seek a minimum-volume polytope with k vertices enclosing the point set of P. The vertices will represent the k mixture components to be inferred. Intuitively, we might propose that the most plausible set of components to explain a given data set is the most similar set of components such that every observed point is explainable as a mixture of those components. Seeking a minimum volume polytope provides a mathematical model of this general intuition for how one might define the most plausible solution to the problem. The minimum volume polytope can also be considered a form of parsimony model for the observed data, providing a set of components that can explain all observed data points while minimizing the amount of empty space in the simplex, in which data points could be, but are not, observed.Once we have transformed the input matrix P. We use a heuristic biased sampling procedure designed to favor points far from one another, and thus likely to enclose a large fraction of the data points. The method first samples among all pairs of observed data points  weighted by the distance between the points raised to the kth power: ||k. It then successively adds additional points to a growing set of candidate vertices. Sampling of each successive point is again weighted by the volume of the simplex defined by the new candidate point and the previously selected vertices raised to the kth power. Simplex volume is determined using the Matlab convhulln routine. The process of candidate point generation terminates when all k candidate vertices have been selected, yielding a guess as to the simplex vertices that we will call K, which will in general bound only a subset of the point set of P.Component inference begins by chosing a candidate point set that will represent an initial guess as to the vertices of the polytope. We select these candidate points from within the set of observed data points in P are enclosed in the simplex. This step begins by measuring the distance from each observed point to each face of the simplex. A face is defined by any k - 1 of the k candidate vertices, so we can refer to face fi as the face defined by K/{ki}. This distance is assigned a sign based on whether the observed point is on the same side of the face as the missing candidate vertex (negative sign) or the opposite side of the face (positive sign). The method then identifies the largest positive distance from among all faces fi and observed points pj, which we will call dij. dij represents distance of the point farthest from the simplex. We then transform K to enclose pj by translating all points in K/{ki} by distance dij along the tangent to fi, creating a larger simplex K that now encloses pj. This process of simplex expansion repeats until all observed points are within the simplex defined by K. This final simplex represents the output of one trial of the algorithm. We repeat the method for n trials, selecting the simplex of minimum volume among all trials, Kmin, as the output of the component inference algorithm.The next step of the algorithm uses an approach based on that of Ehrlich and Full  to move Kmin, we must explain all elements of M as convex combinations of the vertices of Kmin. We can find the best-fit matrix of mixture fractions F by solving for a linear system expressing each point as a combination of the mixture components in the k - 1-dimensional subspace. To find the relative contributions of the mixture components to a given tumor sample, we establish a set of constraints declaring that for each gene i and tumor sample t:Once we have selected We also require that the mixture components sum to one for each tumor sample:ftj values optimally satisfying the constraints then define the mixture fraction matrix F.Since there are generally many more genes than tumor samples, the resulting system of equations will usually be overdetermined, although solvable assuming exact arithmetic. We find a least-squares solution to the system, however, to control for any arithmetic errors that would render the system unsolvable. The Kmin back from the reduced dimension into the space of gene expressions. We can perform that transformation using the matrices V and A produced by PCA as follows:We must also transform our set of components C and mixture fractions F are the primary outputs of the code. The full inference process is summarized in the following pseudocode:The resulting mixture components M and desired number of mixture components k:Given tumor samples Kmin to be an arbitrary simplex of infinite volume1. Define k - 1-dimension approximation M ≈ PV + A2. Apply PCA to yield the i = 1 to n3. For each ka. Sample two points j = 3 to kb. For each P weighted by volume <volume(Kmin) then Kmin ← Kd. If i4. For each tumor sample ftj of F defined by the constraints:i. Solve for the elements j ftjkij = pit ∀i, t∑j ftj = 1 ∀ t∑C ← KminV + A5. Find the component matrix C, F) as the inferred components and mixture fractions6. Return  is assigned weight -s. We solve this problem with the Matlab graphminspantree routine.The model makes several assumptions about the available data. We assume that we have inferred all states present in the data and that our states therefore represent both internal and leaf nodes of the phylogeny. This assumption follows from the evidence that tumor samples maintain remnant populations of their earlier progression states -19, leadWe first validated the method using two protocols for simulated data generation. Simulated data is essential for validation because the ground truth components and their representation in particular tumors are not known for real tumor data sets. In addition, it allows us to explore how performance of the method varies with assumptions about the data set. We began by applying a simple simulation protocol for generating uniformly sampled mixtures, in which each component is simulated as an independent vector of unit normal random variables and each observed tumor passed as input to the data set is simulated as a uniformly random mixture of this common set of components (see Methods). We developed a second simulation protocol meant to better mimic the substructure expected from true tumor samples due to the evolutionary relationships among sub-types. In this protocol, we assume that mixture components correspond to nodes in a binary tree and that each observed tumor represents a mixture of components along a random path in that tree (see Methods). In both protocols, we add log normal noise to all simulated expression measurements.Fig. Fig. Figs. k-node tree is (k - 1)/k = 3 to 29% for k = 7. We can thus conclude that the performance, while not perfect, was substantially better than would be observed by chance.Fig. In order to demonstrate the applicability of the methods to real tumor data, we next examined a dataset of lung tumor expression measurements from Jones et at. . This daFig. Fig. Table We next examined performance of the method with six components, which we label ts Table  and geneComparing gene expression vectors from Additional files Fig. The application to real lung cancer data provides a further opportunity to evaluate the approach, in addition to suggesting some novel hypotheses about the molecular evolution of lung cancers. Both 4- and 6-component inferences suggest that the tumor types examined here evolve into two major groups early in their progression: one consisting of large cell neuroendocrine, small cell, and carcinoid tumors and the other consisting of adenocarcinoma and large cell carcinoma. This subdivision is supported by many lines of evidence on specific genetic abnormalities frequently found in the different sub-types, which suggest that small cell and large cell neuroendocrine carcinomas arise from common progenitors  and thatThe 6-component tree appears to be an elaboration on the 4-component tree, supporting a common model of the two major pathways but adding some additional features beyond that. One intriguing feature is the insertion of the  patient  but othe patient . The infs et al.  dataset Hungarian.m, a third-party Matlab routine for performing weighted bipartite maximum matching [All code for this project was written in Matlab and executed with Matlab v.7 on a Linux PC. Matlab was also used for visualization of component inferences. The validation code also required matching . Unmixinhttp://www.cs.cmu.edu/~russells/software/unmixing/as Matlab ".m" files. Other custom code used in data analysis and generation of simulated data sets will be provided upon request.We have developed a novel approach to tumor phylogenetics combining unmixing methods with a cell-by-cell strategy for phylogeny inference. The method is an attempt to gain the advantages of both intratumor heterogeneity information available to cell-by-cell methods and the large probe sets available to tumor-by-tumor methods. The application to simulated data sets suggests that the method is effective at making component and mixture fraction inferences from large, noisy datasets for limited numbers of components. The method does, however, degrade in performance quickly with increasing numbers of mixture components. Phylogeny inference, which depends on the quality of the mixture fraction inference, similarly shows high tolerance for noise, although a loss of quality with increasing numbers of components. Nonetheless, phylogeny inferences show good reconstruction accuracy for as many as seven components on simulated tree-embedded samples. The methods are generally more effective at component inference from these tree-embedded samples, suggesting that they can effectively exploit some features of the geometric substructure we would expect an evolutionary process to produce. Application to a real lung cancer data set shows the method to be effective at inferring components consistent with known lung cancer sub-types and with grouping these components into phylogenies generally consistent with the prior literature on the evolution of lung tumors. The method does, however, make some apparent mistakes and provide low confidence to some seemingly correct predictions, suggesting room for improvement.The approach, at least as presently realized, does make a number of assumptions about the data and tumor progression in general that one might reasonably question. The primary purpose of this paper is to lay out the general concept of how unmixing methods can inform tumor phylogenetics and this concept in itself implies certain assumptions. Some assumptions concern the biology of tumor development: that there are reproducible cancer sub-types that co-occur in the population, that tumors accumulate remnant cell populations as they progress, that these remnant progression states are themselves reproducible across patients, and that these remnant states persist at high enough levels to measureably influence overall expression. There is considerable literature supporting all of these points, as discussed in the Introduction, although they might reasonably be debated. A further assumption is that the states we wish to observe differ sufficiently in expression profile as to be separable by unmixing methods. While there is strong evidence in the literature that distinct sub-types can be separated by their expression profiles, there is no direct experimental basis from which to argue that individual cell states along a single progression pathway are similarly separable. It remains to be seen how precisely one can sub-divide a progression pathway and which mutations or combinations of mutations will or will not lead to discernible changes in expression profile.The specific implementation of the model in the present work adds additional assumptions that underlie the results here but might conceivably be relaxed in future work. The present model assumes that individual components of a mixture contribute linearly to the mixture, i.e., that the expression level of each gene in a tumor sample is a derived from the weighted sum of the expression level of the gene in each component of the sample. This assumption might break down due to limitations of the microarray technology or for biological reasons, e.g., if intracellular communication leads to radically different expression in mixtures of cell types than in the cell types independently. We would expect the dimensionality reduction step to partially correct for such problems by extracting a linear subset of the full expression space. Non-linear dimensionality reduction methods -38, howeThere are also several potentially debatable assumptions underlying our current phylogeny methods. The most obvious assumption is that all of the cell types we detect are in fact evolutionarily related. The ultimate product of our method will be a phylogeny connecting all of the inferred cell types, which will not in general be a meaningful outcome if the cell types are not related. At some level, of course, all cells in a given individual are related. Distinct tumors may, however, arise from different populations of healthy cells. Furthermore, stromal contamination will in general lead to tumor samples containing mixtures of healthy cells that may not be ancestral to any of the tumor types. We chose to accept this assumption in the model, even knowing it to be imperfect, in the belief that it is better to have the predictions of the method about ancestry available and account for the assumptions after the fact in interpreting the meaning of the inferred ancestry relationships. One might alternatively seek to explicitly separate healthy from cancerous cells prior to phylogenetic inference, using the mixture model to remove stromal contaminants as in Etzioni et al. . A relatA related question raised by these assumptions is whether we can tell, either in advance or post-hoc, whether the assumptions of the model are in fact satisfied by a given data set. One can in principal assess how well the raw input data is explained by a linear mixture of a small number of components, for example by examining the rate at which singular values of the matrix decay. Such a validation does not tell us whether information not accounted for by the linear model reflects genuinely non-linear contributions, the need for a more complex linear model, or noise in the true expression data or the experimental measurements. Neither does it tell us whether the linear model extracted carries sufficient information to characterize the cell states and their relationships. One can also apply a generic post-hoc validation based on how reproducible the results are to sub-sampling, as in our bootstrapping for the lung phylogenies. If the assumptions of the model are violated, then we would expect the data to only weakly support a defined tree topology. The lung cancer example suggests mixed success, with some features of the inferred trees strongly supported across subsamples but others nearly arbitrary. We can also, after the fact, examine the degree to which the inferred mixture fractions conform to the phylogeny. Healthy cells, when available, should be assigned minimal fractions of non-healthy components and tumors on a given progression pathway should exhibit minimal contamination with components characteristic of other pathways. While the lung data does show a clear partitioning of mixture components by sub-type, it nonetheless shows frequent contamination suggestive of poor fitting of the simplex. All of these measures suggest room for improvement in the model and algorithms and will provide a basis for determining whether any alternative methods do in fact improve on those proposed here. Perhaps the ultimate test of the method is how well it recapitulates what we already know about a given real data set. Our comparison of the lung cancer results to prior knowledge again suggests that the method works sufficiently well to recapitulate our prior understanding about the classification and origin of the major lung tumor classes, but with less precision than we might like and with some apparent mistakes. Further post-hoc validation might also be conducted directly on the derived components by testing whether they meet the expression signatures of known tumor sub-types or healthy cell populations.We have presented a novel method for identifying likely pathways of tumor progression using computational unmixing methods to interpret expression measurements from tumor samples as mixtures of fundamental components. Validation on simulated data demonstrates good effectiveness at inferring mixture components, assigning mixture fractions to samples, and inferring phylogenies provided noise levels and numbers of components are sufficiently small. The prototype methods presented here do appear to suffer from insufficiently precise fits of polytopes to the data, especially as the number of components increases, which can in turn result in spurious identification of components in samples that lack them and inaccurate phylogeny inferences. Nonetheless, the effectiveness of the models across a variety of scenarios and in the presence of relatively high levels of noise suggests that the approach has good promise for improving our ability to identify phylogenetic relationships among tumor cells. Likewise, application of the method to real lung cancer microarray data shows that the method to be effective at identifying components corresponding to known clinical sub-types and at inferring progression pathways largely consistent with current knowledge about the molecular evolution of lung tumors. These inferences also suggest some novel hypotheses about the genesis of lung cancers. The methods developed here thus represent a promising new computational model for phylogenetic studies of tumors that can provide many of the competing advantages of the two major paradigms for tumor phylogeny inference: tissue-wide and cell-by-cell. This new model is likely to benefit in the future from further methodological insights from both the unmixing and the phylogenetics fields.d, a number of samples n, a number of mixture components k, and a noise fraction σ. We then simulate k components by constructing a d × k matrix Ctrue)  for each data point i, defining a mixture fraction matrix Ftrue) Zij is a unit normal random variable, implementing a log normal noise model. The resulting matrix M is then processed by the algorithm described in "Cell type identification by unmixing" above to infer some Cinferred) (and Finferred) (and Cinferred) (and Finferred) (and Ftrue) (and Finferred)(4) due to its exclusive use of publicly available, anonymized patient data.RS and SS both participated in conceiving the approach used in this work, conceiving and designing validation experiments, and interpreting the results. RS carried out the algorithm development, coding, and execution of the experiments in this work. Both authors contributed to writing the manuscript.Marker genes for the lung cancer four-component inference. This supplementary table provides relative expression levels inferred for each annotated microarray probe for the Jones et al. lung cancer data set for each of the four inferred mixture components. Columns of the table correspond to the component number, the gene ID, gene description, and relative expression level. Values are sorted by component and in decreasing order of relative expression level within each component.Click here for fileMarker genes for the lung cancer six-component inference. This supplementary table provides relative expression levels inferred for each annotated microarray probe for the Jones et al. lung cancer data set for each of the six inferred mixture components. Columns of the table correspond to the component number, the gene ID, gene description, and relative expression level. Values are sorted by component and in decreasing order of relative expression level within each component.Click here for file


Microarray technology is a popular means of producing whole genome transcriptional profiles, however high cost and scarcity of mRNA has led many studies to be conducted based on the analysis of single samples. We exploit the design of the Illumina platform, specifically multiple arrays on each chip, to evaluate intra-experiment technical variation using repeated hybridisations of universal human reference RNA (UHRR) and duplicate hybridisations of primary breast tumour samples from a clinical study.A clear batch-specific bias was detected in the measured expressions of both the UHRR and clinical samples. This bias was found to persist following standard microarray normalisation techniques. However, when mean-centering or empirical Bayes batch-correction methods (ComBat) were applied to the data, inter-batch variation in the UHRR and clinical samples were greatly reduced. Correlation between replicate UHRR samples improved by two orders of magnitude following batch-correction using ComBat (ranging from 0.9833-0.9991 to 0.9997-0.9999) and increased the consistency of the gene-lists from the duplicate clinical samples, from 11.6% in quantile normalised data to 66.4% in batch-corrected data. The use of UHRR as an inter-batch calibrator provided a small additional benefit when used in conjunction with ComBat, further increasing the agreement between the two gene-lists, up to 74.1%.In the interests of practicalities and cost, these results suggest that single samples can generate reliable data, but only after careful compensation for technical bias in the experiment. We recommend that investigators appreciate the propensity for such variation in the design stages of a microarray experiment and that the use of suitable correction methods become routine during the statistical analysis of the data. DNA microarray technology has rapidly seduced scientists and clinicians with the ability to simultaneously measure the expression of tens of thousands of transcripts, enabling data-driven, holistic comparisons of groups or populations of cells, subtyping tissues, or predicting prognosis ,2. HowevEfforts to promote the routine formalisation and control of all stages of the experimental workflow have seen success and are increasingly promoted by journals and microarray data repositories . More reDespite all efforts towards standardisation, it is still not possible to account for all potential sources of variation in the experiment workflow; identical experiments performed at different sites have produced significantly different results -9. InconThe common practice of hybridising samples with no technical replication (i.e. one replicate of each sample per experiment) is a result of the relatively high cost of arrays, the perceived improvement in array manufacturing quality, and the difficulties of obtaining sufficient amounts of high quality mRNA from some clinical samples. This practice is fundamentally reliant on the assumption that the intra-experiment variability is of a small enough magnitude not to undermine the power of the assay to resolve interesting biological differences that may exist between predefined groups of samples. There is, however, mounting evidence ,12,14-18A large amount of effort has been expended in assessing the reliability, reproducibility, and compatibility of results generated by a number of array platforms within and between laboratory sites. The microarray quality control (MAQC) project, a US Food and Drug Administration initiative , exploreIt is slowly becoming accepted that batch effects are to be expected when combining data generated across different labs, by different researchers, or using different platforms ,10-13. TSome researchers are now aware of bias arising due to analysis of samples at different sites or the use of different microarray platforms. The MAQC project , for exaWe investigated intra-experiment batch-processing variability on the Illumina BeadChip  platformA qualitative measure of the performance of the BeadChips used in this study is provided by a measurement of the fraction of probes that are consistently called to be detected or undetected over all arrays. Analysis of detection consistency in the UHRR data in this study was comparable with the MAQC results  with 60-A clear batch-specific effect was observed in the raw data when the correlations of identical UHRR samples were assessed over all available pairs across the five runs processed on different days as illustrated in Figure The probe-wise standard deviations of the raw expressions were found to be consistently small across the UHRR arrays (mean = 0.28). Using the nested analysis of variance described in methods, 60%  of the variability was due to that between runs and less than 40% to that within each run. The magnitude of the variation was marginally increased by detection-filtering (mean = 0.31), which would be expected due to the preferential filtering of probes with low signal. The application of quantile normalisation had a positive effect, decreasing the standard deviation to half that of the raw data. However both after detection filtering and quantile normalisation the relative contributions of the inter- and intra-run components to the total standard deviation remained approximately unchanged. Of a selection of other normalisation methods, loess, and cubic-spline performed similarly to quantile and all of these methods out-performed simple median normalisation (Supplementary File 1). In all cases a further correction step is required after normalisation to correct for the batch effect.Both mean-centring and ComBat reduced inter-run variation to such an extent that it could no longer be accurately detected by the nested-Anova method Figure . The onlThe differences in measured expression between all combinations of pairs of UHRR samples that straddled the five runs 128 pairs) were calculated for raw, quantile-normalised, mean-centred, and ComBat corrected data Figure . The dis8 pairs wThe sixty-three duplicate clinical samples provided a means to assess inter- and intra-run variation using samples more representative of those commonly analysed using microarray technology. The differences in the measured expressions between each of the duplicate pairs of the clinical samples that straddled the five runs (56 pairs) were calculated for raw, quantile-normalised, mean-centred, and ComBat corrected data Figure . As withPearson rank-products were calculated to assess the correlation between the duplicate samples. As with the UHRR the clinical samples were generally very highly correlated (>98%), although the samples on BeadChips 13/15 and 14/16 were found to be less similar than the others . Using these samples we further assessed the impact of the intra-experiment variation in terms of the differences between lists of differentially expressed genes reported by each half of the duplicate samples. The hybridisation plan for the 21 pairs of pre- and post-treatment samples in each duplicate-group is illustrated in Figure limma and siggenes, were used to perform the statistical analyses (see methods).The 'A' and 'B' duplicate sample groups, containing the 'a' and 'b' hybridisations of each sample, respectively, were considered as two completely independent datasets (as they were processed on completely separate BeadChips) in order to assess the extent to which run-specific processing bias can influence the identification of differentially expressed genes. These datasets were independently filtered by detection calls, quantile-normalised, and, where stated, batch corrected by mean-centering or ComBat before generating lists of differentially expressed genes. Two BioConductor packages, Using the same stringency in the assessment of differential expression  and using quantile normalised data, many more probes were found to be differentially expressed between pre- and post-treatment samples in sample group A (192) than in group B 30). Following batch correction with ComBat the number of differentially expressed genes identified in the two groups was more consistent 260 and 211) and the overlap, in terms of probes reported in both groups, increased from just 11.6% to 66.4%, however the use of mean-centred data only increased the overlap marginally to 15.2%  p-value << 0.001).Similar results were seen with less stringent criteria (fold-change ±1.2), which consequently led to larger numbers of probes, but similar proportions of overlapping probes were reported (data not shown). The analysis was repeated using significance analysis of microarrays (SAM) at a predicted false discovery rate of 5% and generated very similar results to those obtained using limma as differentially expressed in all three groups after ComBat correction was 41.3% compared to 11.2% after quantile normalisation alone and 12.0% after mean-centering  p-value <<< 0.001).These analyses were repeated using the UHRR as inter-batch calibrator, designating it as a covariate in both the mean-centring and ComBat corrections. The inclusion of UHRR during quantile normalisation produced only a small difference in the number of differentially expressed genes identified in each of the three sample groups. However, the inclusion of the UHRR as a covariate in the mean-centring and ComBat corrections gave very different results. In both methods there was a large reduction in the total number of genes reported in each list, in terms of the consensus between the A and B groups, the agreement dropped to 7.5% following mean-centring, but increased to 74.1% after correction by ComBat Table . The depBatch-processing effects in microarray experiments are commonly encountered when combining datasets from different studies, different labs, or different technologies. In this study we have demonstrated that batch effects can arise within a single study, at a single lab, using a single technology and that these can have a significant impact on reported gene-lists.The magnitude of the variation in the observed expression of replicate samples derived from the UHRR in this study is consistent with that reported in other studies assessing the quality of microarray data, such as the MAQC . We haveThere are many stages of sample-processing prior to conducting any gene-expression experiment and each is vulnerable to the introduction of systematic processing errors . OpportuIn the context of primary breast tumour samples, which have been repeatedly shown to have highly heterogeneous mRNA expression profiles, there is much greater variation between the RNA profiles from different individuals than within tumours ; either Using the duplicate-experiment approach we were able to demonstrate that single samples can generate reliable data, particularly when batch correction is performed to minimise processing bias. However the genes reported to be differentially expressed in the pooled duplicate samples in group C were more robust in terms of their agreement with those identified in groups A and B, especially following batch-correction.In addition to the technical replicates commonly used in other assays, in cases where the execution of the assay is split into several runs, it is very common for an inter-run calibrator to be used to quantify the variation introduced by the splitting of the experiment and to normalise for it. Despite the UHRR samples used in this study showing very similar variation and correlation to that previously reported, we found that the samples were of limited utility as predictors of the batch variation amongst the clinical samples. However the replicate UHRR samples were found to slightly improve the consensus between the results of the duplicate experiments when used in conjunction with the ComBat correction.Although the UHRR has been reported to be useful as a standard for microarray experiments and suitable for monitoring the performance of genome-wide expression platforms ,9,32, itThere is no reason to believe that the batch-processing effects observed here are limited to the Illumina BeadChip platform. Many previous investigations of other platforms have postulated potential factors responsible for the introduction of processing errors in microarray experiments ,33-35. ORegardless of the platform chosen, it is clear that compensation for processing variation is beneficial and can only be achieved by incorporating the design of the experiment into the downstream data analyses. If all pre-treatment samples had been processed in one batch and all post-treatment samples in a second batch, it would not be possible to rule out confounding differences between treatment and batch processing. 'Real' differences due to the common variable of interest may have been partially or completely obscured by the batch effect. Design oversights of this type are beginning to be highlighted  and demoOur results support the notion that analysis of gene expression data should begin with an evaluation of batch effects. If the possibility of batch effects has been anticipated and confounding factors separated, then it should be possible to remove the bias to generate more robust results. As with other studies, ours is limited by the samples that were used in the evaluation of processing variation. We would have liked to test the applicability of our findings in other published datasets, however we were unable to find comparable datasets that include technical replicates and details of hybridisation 'batches' in the existing data repositories. In terms of cost and practicalities it is understandable why most researchers do not perform replicates in clinical studies, our results indeed suggest they may not be necessary; however providing a hybridisation plan along with the raw data, would make the processing of data more transparent.In summary, intra-experiment bias can distort the findings of gene expression studies. Replicate samples were found to be beneficial in both the identification and reduction of processing bias and lead to increased consensus in reported gene-lists, especially following specialised batch-corrections. We conclude that single samples can generate reliable data, although an appreciation for sources of intra-experiment variation during the design of the experiment is required to maximise the efficacy of specialised corrections in order to minimise susceptibility to potentially confounding intra-experiment batch-effects. Finally, based on the discrepancy between the lists of differentially expressed genes in each group of duplicate tumour samples, the observed rate of falsely-reported genes was consistently and significantly larger than that predicted by SAM. Therefore, based on the results of this study, a healthy degree of skepticism is advised when interpreting published results of microarray experiments that do not include validation by technical replication or, preferably, by another technique such as qPCR. In the absence of large numbers of biological replicates, it is our opinion that technical replication should be encouraged in order to provide robust, reliable, and credible expression-profiles.® TotalPrep RNA Amplification Kit (Ambion) and quantified on a Bioanalyser 2100. 750 ng cRNA per sample was hybridized to Illumina HumanRef-8 v2 Expression BeadChips  using Whole-Genome Expression Direct Hybridisation kit (Illumina) and scanned with a BeadStation 500GX (Illumina). Full details of the sample biopsies taken at diagnosis and at surgery were as previously described . The duplication of the clinical samples was performed after labelling and labelled samples were stored as per the manufacturer's recommendations.In order to compare the consistency of gene expression profiles between and within processing runs a single sample of Universal Human Reference RNA  was added to eighteen Illumina HumanRef-8 v2 Expression BeadChips. The remaining seven arrays on each chip were used to analyse the response to an mTOR inhibitor, Everolimus, in pre-operatively treated post-menopausal women with oestrogen receptor-positive breast cancer. From each extraction 100 ng RNA was amplified and biotinylated using Illuminahttps://catissuesuite.ecmc.ed.ac.uk/caarray/.All raw gene expression files, clinical annotation and R scripts used to perform the analysis are publicly available from the caBIG supported Edinburgh Clinical Research Facility Data Repository limma [limma package, genes were defined as being differentially expressed after satisfying a minimum fold-change of ±1.5 and a maximum, Benjamini-Hochberg adjusted, p-value of 0.01. For the SAM analysis (using the siggenes package), the differentially expressed genes were selected at a maximum predicted false discovery rate of 5% and the same minimum fold-change of ±1.5. Paired statistical tests were performed in both the limma and the SAM analyses. Hierarchical clustering of samples and probes for the creation of all heatmaps was performed using complete linkage and similarities calculated according to the method described in [A summary work flow of the analysis approach is given in figure limma  and SAM limma . For theribed in .Data were filtered, where specified, using the detection confidence reported by Illumina's BeadStudio software- determined for each bead based on the expressions of internal control probes, local background intensity, and the uniformity of the reported intensity of the bead. The filtering was performed prior to normalisation such that probes with a detection confidence less than or equal to 80% in more than 25% of the samples were removed from further analysis.Xij, of each probe is defined asWe applied a linear additive model to UHRR expression data on the log-scale to estimate the inter- and intra-batch variance contributions. These contributions are assumed to be independent and randomly drawn from log-normal distributions. As all factors meet in unique combinations a nested, or hierarchical, variance model is individually applied for each gene such that the model of the measured expression, μ is the geometric-mean expression of the gene from the UHRR population, Ai is the random effect attributed to the ith batch, and εij is the random measurement error attributed to the jth array in the ith batch. Finally, b is the total number of batches and n the number of replicate samples in the corresponding batch. The variance of any given observation, Xij, is σ2A + σ2; these components represent the inter-batch and intra-batch variance respectively. The estimation of σ2A and σ2 is performed independently for each gene as stated in [where tated in .VSS conceived the study and processed the samples. RRK and AHS analysed the data. AHS, JIvH, and JMSB supervised the work. EJM, LR, and JMD performed biopsies and collected the clinical material. JST quality-assured the diagnostic material. RRK and AHS drafted the manuscript. All authors read and approved the final manuscript.Coefficient of variation amongst replicate UHRR samples. Two box and whiskers plots of the coefficient of variation (CV) of the replicate UHRR samples. The first plot (A) shows the experiment-wide CV of the UHRR samples. The left-most of the four main sections shows the CV of the raw (detection filtered) data, to the right of this is the CV after four popular normalisation algorithms; quantile, loess, cubic-spline (qspline), and median. The final two segments show the CV after batch-correcting each normalised dataset using either mean-centring or ComBat. In the second plot (B), from the left, the first four segments contain five box-plots illustrating the CV within each of the five runs; the four segments containing raw (white), quantile-normalised (dark-blue), mean-centred (lighter-blue), and ComBat-corrected  data respectively. All data were detection-filtered prior to analysis. The right-most segment shows the experiment-wide CV of the UHRR (coloured as the previous segments) calculated with no consideration of the individual runs.Click here for fileUHRR inter-run pairwise differences. Pairwise differences between each of the five runs calculated using UHRR samples for raw, quantile-normalised, mean-centred, and ComBat-corrected data.Click here for fileNumber of differentially expressed genes identified in replicate analyses. Numbers of genes reported to be differentially expressed after standard analysis  (left), after a standard analysis with mean-centring (middle), and after a standard analysis augmented with the ComBat batch correction (right). A and B refer to the results from independent analyses of the duplicate sample groups while C refers to the results from the pooled duplicate samples. The rows of Venn diagrams illustrate the results with (i) limma, (ii) SAM, (iii) limma using UHRR, and (iv) SAM using UHRR.Click here for fileHeatmaps. Full heatmaps of quantile normalised, quantile normalised plus mean-centred and quantile normalised plus ComBat data, including probe and sample annotations.Click here for file


We introduce Approximate Entropy as a mathematical method of analysis for microarray data. Approximate entropy is applied here as a method to classify the complex gene expression patterns resultant of a clinical sample set. Since Entropy is a measure of disorder in a system, we believe that by choosing genes which display minimum entropy in normal controls and maximum entropy in the cancerous sample set we will be able to distinguish those genes which display the greatest variability in the cancerous set. Here we describe a method of utilizing Approximate Sample Entropy (ApSE) analysis to identify genes of interest with the highest probability of producing an accurate, predictive, classification model from our data set.In the development of a diagnostic gene-expression profile for cervical intraepithelial neoplasia (CIN) and squamous cell carcinoma of the cervix, we identified 208 genes which are unchanging in all normal tissue samples, yet exhibit a random pattern indicative of the genetic instability and heterogeneity of malignant cells. This may be measured in terms of the ApSE when compared to normal tissue. We have validated 10 of these genes on 10 Normal and 20 cancer and CIN3 samples. We report that the predictive value of the sample entropy calculation for these 10 genes of interest is promising .The success of the Approximate Sample Entropy approach in discerning alterations in complexity from biological system with such relatively small sample set, and extracting biologically relevant genes of interest hold great promise. Genomic heterogeneity is a characteristic feature of nearly all solid tumors, appearing early in tumor progression. Over time, early genomic instability evolutionarily leads to a molecularly heterogeneous population of cells naturally selected for their abilities to proliferate and invade, while simultaneously evading host defenses. The selection process, influenced by the unique to the tumor host environment, results in a further diverse intrapatient tumor population which is responsible for the clinical heterogeneity of the disease.Concepts of tumorigenesis include the stochastic model, disputing the existence of genomic destabilization in tumor cells, instead arguing that mutations occurring during differentiation, development, and proliferation combine to create a cell with the perquisites for malignancy; according to this model, the cell then simply proliferates to create the cancer. On the oThe evidence for genomic heterogeneity has been found in the extensive, progressive, and diverse genomic damage observed within the tumor, pre-neoplastic lesion, -5 and tuWhile a fortuitous random process is created by genomic destabilization and natural selection, most are searching for a coherent pattern within this extremely diverse event. It is likely that the observed clinical heterogeneity of cervical cancer is a reflection of the dissimilar genomic events between the patients. Since the advent of microarray technology, data generated from these experiments have most often been analyzed with the aim to identify specific patterns in the expression levels, such as periodicity or monotonous increases or decreases. -11 TheseWe introduce Approximate Sample Entropy as a mathematical method of analysis for microarray data. Approximate entropy is traditional utilized in analysis of temporal patterns; such as heart rate variability.,13 The uCervical cancer is an excellent model for evaluation of this analysis for multiple reasons. Aside from being the second leading cancer among women worldwide; with 470,000 new cases occurring annually; cervical cancer uniquely has an easily accessible, well defined, distinct pre-neoplastic lesion.. The proCervical biopsy samples were obtained from radical hysterectomy specimens of patients with Stage 1B1 squamous cell carcinoma of the cervix. Lesional and perilesional samples were obtained from 4 individual patients. Control biopsy samples were obtained from hysterectomy specimens of patients with documented normal cervix. All biopsies were histologically confirmed. Perilesional biopsies were normal cervical epithelium adjacent to the tumors.Cervical biopsy samples from an additional 20 patients were utilized for real-time pcr validation. These biopsies were obtained from patients with either the preneoplasitc lesion (CIN 3), or with Stage 1A-1B squamous cell carcinoma of the cervix. As for the training set, normal cervical samples obtained from hysterectomy specimens of patients were utilized as controls. State of disease was histologically confirmed on an adjacent biopsy.Total RNA was extracted from each sample using the RNAqueous RNA Isolation Kit  following manufacturer's instructions. The extracted RNA is DNase treated prior to quantification. The RNA quality was assessed by detecting the 28S/18S peaks with an Agilent Bioanalyzer 2100 . Only RNA with the highest fidelity  is used for microarray analysisTwenty μg of DNA-free RNA from each biopsy was applied to the GeneChip Human HG_U133 A & B GeneChip (Affymetrix). Microarray gene chip analysis was performed on all samples using the Affymetrix HU 133 A & B human 33,000-GeneChip  for identification of disease specific expression patterns. Labeling of cRNA, hybridization, and scanning of the microarrays were done according to the manufactures protocols .Quantitative reverse transcription-PCR (qRT-PCR) was performed to validate differential expression of genes in an independent set of 20 cervical tissue biopsies. QRT-PCR was done with a 7500 Fast Real-time PCR system (Applied Biosystems) using gene-specific primer/probe mix  method was used in comparing the level of mRNA expression of the genes of interest to that of the internal control gene, GAPDH. Each reaction was set up in triplicate, with final expression levels in cancer samples compared to non-diseased, normal controls.Initially a normal baseline was established, 966 genes were found to be unchanging between patients with normal cervical biopsies when limited to a variance of less then 0.5% of the log2 expression values, a variance below the technical noise level of the Affymetrix array. Figure Half of the normal patient's biopsies were used in the reference sample set while the other half was treated as experimental samples similar to the cancer and perilesional biopsy samples. Changes in gene expression were calculated for each of the 966 genes asexperimental denotes expression in either the non-reference normal, perilesional or cancer samples. For use during the clustering phase, the ΔE values were averaged over subjects within each classification group  so that for any gene k, the mean ΔE is a 3-element vectorwhere with superscripts 1, 2, and 3 representing normal, perilesional and cancer sample groups, respectively.E values were input into a Bayesian classification algorithm  in MATLAB  and the 966 genes were partitioned into subsets (classes) based on expression change patterns. The clustering partitioned the data in 3-space, where each dimension corresponded to one of the sample groups types. Genes were assigned to the cluster with the smallest Bayesian distanceThe 3 by 966 array of mean Δd were calculated aswhere the distances Ci are the estimated mean and covariance matrix for cluster ωi and p(ωi) is the probability of membership in cluster ωi and |Ci | is the determinant of the covariance matrix.a-priori (or prior) probability p(ωi) of 1. On subsequent iterations, this quantity was based on the fraction of genes in each potential cluster.On the first pass through the clustering algorithm, all points were considered to be members of a single cluster located at a center given by the mean of each group over all 966 genes with an 2 statistic of hypothetical clusters to determine if the data set represents a single or multiple Gaussian distributions. If the computed chi2 value for any cluster exceeded a set threshold  and if the product of the number of members of a class and the amount by which the chi-squared test exceeded the threshold was greater than a second criterion, the data were considered for further subsets. Subdivision of existing clusters continued until these conditions were no longer met. By varying the second criterion, it is possible to stabilize at a greater or lesser total number of clusters.The number of subsets was determined using a process of blind clustering which estimated the χIf an existing cluster was considered for further subdivision, two new cluster centers were located symmetrically along the principal axis (eigenvector) by a distance determined by the magnitude of the principal component  computed from that cluster's covariance matrix. Bayesian distances were then estimated relative to the new clusters and the process was repeated. Figure ApSE values were computed for the genes within each cluster.  as the negative logarithm of the conditional probability P(x) that a dataset of sample x, falling within a tolerance range (R) for point k, will also repeat itself for each point in the set. Suppose | (ΔE)n | represents the absolute value of ΔE for subject n belonging to one of the three biopsy groups. Let SR denote the set of genes whose ΔE value for that subject falls within a neighborhood R of zero:After dividing the 966 genes into 10 groups based on cluster membership, . Figure  The entrR is a real number ranging from 0.1 to 5.5 in steps of 0.1.where A(R) as the number of genes in set SR. If N is the total number of genes in the cluster being considered, then the approximate sample entropy can be written as:[We can then define itten as:ApSE scores and known biopsy classes of the subjects in the training set and 2) application of the decision rules to the ApSE values of subjects in the validation set to predict biopsy class. The agreement between the actual and predicted biopsy class was tabulated and scored.Classification consisted of two stages: 1) determination of decision rules based on the Decision rules were determined separately for each subset of genes. ApSE values from the training set were input as independent/explanatory variables into a regression tree . . in the downloadable "tree" library. As with the "tree" routine in S-Plus, this methodology builds upon the CART model developed by Breitman et al.. [The tree-fitting algorithm used in Insightful Miner is based on the recursive partitioning code called RPART. Similar  et al..  A summar et al..  A compre et al.. Unlike traditional regression techniques which use a weighted sum of independent values to estimate the dependent variable, regression trees output a hierarchical series of rules or logical if-then conditions that can be used to classify the cases based on the values of the predictor variables.Figure  Tree-basTen genes were chosen for initial quantitative real-time validation of the gene array, for the two gene subsets (subset 1 and 2) which best represent our progressive model of disease. RNA was extracted from fresh frozen biopsies of normal, cervical intraepithelial neoplasia grade 3 (CIN 3) and stage IB-IIB squamous cell carcinoma (SCC) for Quantitative Real-Time PCR (qRT-PCR) using the RNAqueous RNA Isolation Kit . Using the 7500 FAST Real-Time System (Applied Biosystems), cDNA from the samples was amplified in a single-plex reaction, for the following genes of interest , Collagen typeVI alpha 3(Col6A3), Fibronectin 1 (FN1), Growth arrest specific 6 (GAS6), Human leukocyte antigen (HLA-C), High-mobility group nucleosome binding domain 1 (HMGN1), Heat shock 60 kDa protein 1 (HSPD1), Interferon gamma-inducible protein 16 (IFI16), Matrix metallopeptidase 2 (MMP2), Proteasome subunit beta-2 PSMB2, Tenascin C (TNC))We believe that the observed clinical heterogeneity of most cancers is a reflection of the gross genomic heterogeneity and subsequently the molecular heterogeneity also found within the tumors, also that the more heterogeneous the cell population, the more likely the development of a metastatic, aggressive phenotype. Genomic destabilization, evolution, and selection for invasive, proliferating populations of cells encompass the fundamental nature of cancer. Cancer cells represent a diversely complex evolutionary life within the host. The resultant heterogeneous tumor is linked to its genetic state and represented in its gene expression profile. We have shown here that encompassing this heterogeneity in the data analysis aids in capturing the diversity within the cancer subgroups and in the identification of subgroup specific gene expression profiles.Since the advent of microarray technology, data generated from these experiments have most often been analyzed with the aim to identify specific patterns in the expression levels, such as periodicity or monotonous increases or decreases. -11 TheseScientists have struggled with theoretical means to detect alterations in system complexities. Entropy, a measure of disorder or randomness has been used to classify complex systems externally viewed as chaotic. A limited number of studies have utilized the concept of entropy in gene expression analysis; all to enhance clustering techniques by minimizing entropy and thus the amount of disorder in the groups. -24 ConveApproximate sample entropy analysis of microarray data has proven to be a reliable method of differentiating subgroups of patient samples. With the ability to classify the perilesional biopsy sample from normal with 75% accuracy, and to classify cancer with 80% accuracy with an 81 gene set (subset 1). , and stages I and II squamous cell cancer. We find in Table ApSE between cancer and perilesional (dApSE). (Figure The sample entropy analysis has the ability to detect abnormality with a predictive probability of 88%, with region R set at the maximum value for difference in . Figure  The abilSKB conceived of the study, participated in design, supervised the molecular studies and carried out the analysis. JBT assisted in development of analysis and draft of the manuscript. SAS carried out the molecular studies. MFB assisted in patient recruitment, sample acquisition and molecular studies. CDA participated in design and supervised all aspects of the study. All authors contributed to the writing of this manuscript. All authors read and approved the final manuscript.