Consensus clustering, a new method for analyzing microarray data that takes a consensus set of clusters from various algorithms, is shown to perform better than individual methods alone. Microarray analysis using clustering algorithms can suffer from lack of inter-method consistency in assigning related gene-expression profiles to clusters. Obtaining a consensus set of clusters from a number of clustering methods should improve confidence in gene-expression analysis. Here we introduce consensus clustering, which provides such an advantage. When coupled with a statistically based gene functional analysis, our method allowed the identification of novel genes regulated by NFκB and the unfolded protein response in certain B-cell lymphomas. There are many practical applications that involve the grouping of a set of objects into a number of mutually exclusive subsets. Methods to achieve the partitioning of objects related by correlation or distance metrics are collectively known as clustering algorithms. Any algorithm that applies a global search for optimal clusters in a given dataset will run in exponential time to the size of problem space, and therefore heuristics are normally required to cope with most real-world clustering problems. This is especially true in microarray analysis, where gene-expression data can contain many thousands of variables. The ability to divide data into groups of genes sharing patterns of coexpression allows more detailed biological insights into global regulation of gene expression and cellular function.Many different heuristic algorithms are available for clustering. Representative statistical methods include k-means, hierarchical clustering (HC) and partitioning around medoids (PAM) -3. Most Recently, a number of studies have attempted to compare and validate cluster method consistency. Cluster validation can be split into two main procedures: internal validation, involving the use of information contained within the given dataset to assess the validity of the clusters; or external validation, based on assessing cluster results relative to another data source, for example, gene function annotation. Internal validation methods include comparing a number of clustering algorithms based upon a figure of merit (FOM) metric, which rates the predictive power of a clustering arrangement using a leave-one-out technique . This anweighted-kappa metric [weighted-kappa compares clusters to generate the score within the range -1 (no concordance) to +1 (complete concordance)  [With the synthetic dataset, all clustering algorithms had a 'high' t shown) . It is pt shown) ,12. Thert shown) .The partial agreement of the different clustering algorithms must reflect the clustering of highly similar gene-expression vectors regardless of the clustering methods used. Where algorithm-based inconsistency problems occur in other aspects of computational biology, such as protein secondary structure prediction, consensus algorithms are often used . These ci and j, all clustering methods must have allocated them to the same cluster in order for them to be assigned to a robust cluster. This gives a higher level of confidence to the correct assignment of genes appearing within the same cluster. Robust clustering works by first producing an upper triangular n × n agreement matrix with each matrix cell containing the number of agreements among methods for clustering together the two variables, represented by the row and column indices  for compiling the results of different clustering methods reporting only the co-clustered genes grouped together by all the different algorithms - that is, with maximum agreement across clustering methods. For two genes s Figure . This maList, which contains all the pairs where the appropriate cell in the agreement matrix contains a value equal to the number of clustering methods being combined (that is, full agreement). Starting with an empty set of robust clusters RC, where RCi is the ith robust cluster, the first cluster is created containing the elements of the first pair in List. Then the pairs in List are iterated through and checked to see if one of the members of the current pair is within any of the existing clusters, RCi.Robust clustering uses the agreement matrix to generate a list, RCi in RC, then a new cluster is added to RC containing each element of the pair. When the end of the list is reached, the set of robust clusters, RC, is the output. The robust clustering algorithm is as follows:If one element of the current pair is found and the other element of the pair is not in the same cluster, then the other element is added to that cluster. If neither element of the pair is found in an existing n × n), AInput:Agreement Matrix  in the matrix, with agreement = the number of methods(1) Set RC to be an empty list of clusters(2) Set x, y) of the first pair in List into it(3) Create a cluster and insert the two elements (i = 2 to size of List-1(4) For j = 1 to number of Clusters in RC(5)  For x or y of Listi is found within RCj(6)   If Listi is not found in RCj(7)    If the other member of the pair RCj(8)     Add the other member to (9)    End IfListi is not found in RCj(10)   Else If the other member of the pair RC containing x and y(11)    Add a new cluster to (12)   End If(13)  End For(14) End ForRCOutput:Set of Robust Clusters weighted-kappa cannot be calculated. This metric requires both clustering arrangements being compared to be drawn from the same set of items. This is not the case with robust clustering because many items will not be assigned to a cluster. However, approximately 80% of the total ASC data variables and 25% of the B-cell lymphoma variables are assigned to a robust cluster. Robust clustering further subdivides the datasets into smaller clusters, with 24 rather than 13 clusters being defined for ASC, and 154 rather than 40 being defined for the B-cell lymphoma data  in Equation (1) to score each cluster of size siConsensus clustering relaxes the full agreement requirement by taking a parameter, 'minimum agreement', which allows different agreement thresholds to be explored. Rather than grouping variables on the basis of full agreement only, consensus clustering maximizes a metric, which rewards variables in the same cluster if they have high cluster method agreement and penalizes variables in the same cluster if they have low agreement. Consensus clustering maximizes agreement using the function A is the agreement matrix, Gij is the jth element of cluster i (Gi) and β is a user-defined parameter (the agreement threshold), which determines whether the score for the cluster is increased or decreased. The score for a clustering arrangement is the sum of the scores of each cluster, which consensus clustering attempts to maximize.where β is equal to Min, the minimum value in A, then the function is maximized when all variables are placed into the same cluster (that is, a single large cluster). Alternatively, when β is equal to Max, the maximum value in A, the function is maximized when each variable is placed into its own cluster. Essentially all clusters produced by Consensus Clustering are scored by f(Gi), rewarding and preserving clusters with high agreement between members, while penalizing and discarding clusters containing low agreement between members. A value for β should lie between the minimum and the maximum agreement so as not to skew the scoring function. A suitable value for β is (Max + Min)/2, where Max is the maximum value in A and Min is the minimum. For a uniformly distributed agreement matrix, (Max + Min)/2 is the mean value; therefore we penalize values below the mean agreement and reward above it. For both the ASC and B-cell lymphoma data β was 2, as Max = 4  and Min = 0 (no agreement). In order to maximize the scoring function for consensus clustering, a search over possible cluster membership is needed. There are many methods for performing a search and it was decided that SA was best because it is an efficient search/optimization procedure that does not suffer from becoming stuck in local maxima. The consensus algorithm is as follows:If n × n), A; Maximum Number of Clusters sought, m; Number of Iterations, Iter; Agreement Threshold, β; Initial Temperature, θ0; Cooling Rate, cInput: Agreement Matrix ((1) Generate a random number of empty clusters (<m)n between the clusters(2) Randomly distribute the variables (genes) 1..(3) Score each cluster according to Equation (1)i = 1 to Iter do(4) For (5)  Either Split a cluster, Merge two clusters or Move a variable (gene) from one random cluster to anotherf to difference in score according to Equation (1)(6)  Set Δf < 0 Then(7)  If Δp, according to Equation (2)(8)  Calculate probability, p > random(0,1) then undo operator(9)  If (10) End Ifθi = cθi-1(11)  (12) End ForOutput:Set of Consensus Clustersrandom(0,1) (line 9) returns a random uniformly distributed real number between 0 and 1.Note that The 'split', 'merge' and 'move' operators (line 5) are as follows and used with equal probability:Split a cluster:g of size nInput: Cluster (1) Randomly shuffle the clusteri to be a random whole number between 1 and n-1(2) Set g1 and g2(3) Create two empty clusters i of g to g1(4) Add elements 1..n of g to g2(5) Add elements i+1..g1 and g2Output:Two new clusters Here the old cluster is deleted and the two new clusters are then added to the set of clusters.Merge two clusters:g1 and g2Input: Two Clusters g is created by forming the union of g1 and g2(1) A new cluster gOutput: A new cluster Here the old clusters are deleted and new cluster is then added to the set of clusters.Move a gene:GInput:A set of clusters g1 and g2 are chosen where the size of g1 is greater than one(1) Two random clusters g1 is moved into g2(2) A random element of GOutput:The updated set of clusters p) (line 8) is calculated by:The probability  solution will be accepted with probability 0.368 (e-1). As the temperature cools, this probability will reduce. Here we set θ0 to be the average of Δf over 1,000 trial evaluations, so that at the beginning of the algorithm, the average worse solution (Δf = θ0) will be accepted with the probability stated above.These parameter settings for SA are effectively determined by the ith stage of the SA algorithm θ0 = θ0ci. The SA algorithm works by assuming that the temperature reduces to zero over an infinite number of iterations. As it is not practical to run the SA algorithm to infinity the method is usually terminated after a fixed number of iterations, (iter). At this time the temperature will not be zero, but very small and positive, say ε. Therefore,It can be seen from the consensus algorithm that during the ε is chosen, and the algorithm is to run for a defined number of iterations (iter), then the decay constant c is calculated as above.Hence if some small positive value for As consensus clustering relaxes the 'complete agreement' criteria we would expect the majority but not necessarily all robust cluster members to be assigned to the same consensus clusters. This was indeed true for the B-cell data where consensus clustering of the datasets showed that 98.5% of the B-cell robust clusters were assigned correctly to their respective consensus clusters. With the more consistent ASC data 100% of the robust clusters were assigned to the correct consensus clusters.weighted-kappa score for each pairwise combination of individual clustering algorithms  (Table weighted-kappa score of 0.96. Importantly, there was little variation between each of the 10 runs with a standard deviation of the mean weighted-kappa of 0.0015. SA had a similar low standard deviation, but produced lower inter-run consistency (mean weighted-kappa of 0.816). CAST was the least consistent of the methods (mean weighted-kappa of 0.646). The differences in the consensus clustering mean compared to SA and CAST are significant at greater than the 99.9% confidence level, thereby showing consensus clustering identifies a reliable data partition, which is significantly better than multiple runs of single clustering methods.The advantage of consensus clustering over all single-cluster methods was evident when comparing consensus clustering to the mean 2) Table . Consensweighted-kappa metric. Cluster accuracy using the single methods ranged between a weighted-kappa of 0.505 to 0.7 (mean weighted-kappa of 0.614)  and the second using a multivariate normal distribution (MVN). The number of genes in each cluster varied from 1 to 60, with the number of conditions (arrays) set to 20. The datasets therefore contained 1,830 genes over 20 conditions. As the structure of each dataset is known, the results of each clustering method can be evaluated for accuracy using the Consensus clustering greatly improves the accuracy of identifying cluster group membership based solely on the gene-expression vector, but as with other clustering algorithms still produces essentially unannotated clusters which require further external validation by gene function analysis. To address this problem, we derived a probability score to test the significance of observing multiple genes with known function in a given cluster against the null hypothesis of this happening by chance. This identifies clusters of high functional group significance, aiding assignment of functions to unclassified genes in the cluster using the 'guilt by association' hypothesis.i, of size si, contains x genes from a defined functional group of size kj, then the chance of this occurring randomly follows a binomial distribution and is defined by:The probability score is based on the hypothesis that, if a given cluster, n is the number of genes in the dataset. As kj and x may potentially be very large, Pr from the above equation would be difficult to evaluate. Therefore the normal approximation to the binomial distribution can be used as defined by:where z mean that the probability of observing x elements from functional group j in cluster i by chance is very small, (for example z > 2.326 corresponds to a probability less than 1%). Note that we perform a one tailed test as we are only interested in the case where a significantly high number of co-clustered genes belong to the same functional group.Large positive values of et al. [This cluster function probability score was used to identify statistically significant (at the 1% level) B-cell consensus clusters containing defined genes known to be associated with 10 functional groups . To deteet al. . For eacsi) (FG/si = 1). Alternatively, single clustering methods may produce lower probability scores by increasing the cluster size, thereby pulling many genes into the cluster resulting in a FG/si ratio tending towards zero. This would also reduce the usefulness of the clusters. We determined the cluster size and functional group size for two representative functional groups where the consensus clustering probability was similar to the single method probability score, namely the endoplasmic reticulum (ER) stress response  (ER/UPR) functional group, or the markedly better nuclear factor-κB (NFκB) functional group  would equal the cluster size , and the genes for class I MHC-C and lymphotoxin α, as these have been previously characterized as NFκB responsive and were present in the initial NFκB-defined gene set. Interestingly, NFκB-binding sites were also found in six additional promoters for which accurate mapping of promoter transcription factor binding is not available  or the loss of function of the inhibitor of NFκB, namely IκB (L-428). This implies that many of these genes could be NFκB responsive. Twenty-four putative promoter regions were analyzed and NFκB-binding sites were identified in 12 of these. As expected, NFκB-binding sites were found in the CD40L receptor gene, e Figure . All bute Figure  29,30] ,30 Bfl-1N-glycan biosynthesis. This functional group was overexpressed in cell lines of plasmablast or plasma-cell tumors, where physiological upregulation of the ER is required for cellular function. This process is controlled by two transcription factors, ATF6 and XBP1 [ATF6 transcript was present as a defining signature gene in the ER/UPR functional group. This suggests that ATF6 and XBP1 may be responsible for upregulation of the calcium-ion binding and N-glycan biosynthetic genes. Two responsive elements have been defined for ATF6 and XBP1 respectively, the ER stress-response element (ESRE), comprising the binding site CCAATN9CCACG and the unfolded protein response element (UPRE), comprising the binding site TGACGTG(G) [N-glycan biosynthetic genes but no ESRE sites were found. This suggests that these two groups of genes are regulated through two distinct mechanisms by transcription factors ATF6 and XBP1 as a result of ER stress.The consensus cluster associated with the ER/UPR functional group contained genes not previously associated with ER stress-induced upregulation. The gene-expression profiles were visualized and annotated as described for the NFκB functional group Figure . Annotatand XBP1 . The ATFACGTG(G) . ATF6 anACGTG(G) . ESRE siweighted-kappa metric. This metric effectively scores how well different cluster method pairs assign the same genes to the same clusters. The weighted-kappa metric readily shows that, even for highly correlated gene-expression profiles present in the ACS dataset, no two clustering algorithms have complete agreement, although the global search methods such as SA seem to produce the most consistent results. Overall this emphasizes that no single analysis method will identify all patterns in the gene-expression data; therefore multiple analyses should be performed and compared [Grouping data into sets based on a consistent property is a common occurrence in biological analysis. This has recently increased in importance with the production of large microarray datasets. Implicit in the experimental rationale is the fact that patterns of coexpressed genes should be identifiable in a gene expression matrix and these can be linked to shared biological processes. However, different clustering algorithms are known to partition data into different groups -13. We acompared .et al. was to use a meta-consensus across different clustering algorithms rather than to re-sample over the same algorithm. Our results represent this extension and confirm the validity of consensus clustering. We have developed both robust and consensus clustering, with these methods offering specific advantages over the use of individual clustering algorithms for microarray analysis. The robust clustering algorithm is useful for creating clusters of genes with high confidence and is extremely effective for reducing the dimensionality of large gene-expression datasets. However, robust clustering can be restrictive in discarding genes that do not have full agreement. Consensus clustering overcomes this problem, requiring a minimum-agreement parameter to generate clusters based on the combined results of a number of existing clustering methods. This strategy enables the effective identification of cluster groups that are of high reliability and cluster method independent.We and others recently described the use of consensus clustering to improve confidence  or as a The choice of clustering algorithms and parameter settings is a major stumbling block for all gene array cluster analysis. The effect of varying the parameters depends on the cluster method used. The performance of cluster methods has been extensively investigated . The autet al. using a database of clusters [et al. used the hypergeometric distribution. This distribution can be formally shown to asymptotically become the binomial distribution when the population size increases. Therefore, when used on large gene-expression datasets our methods are directly analogous to Wu et al. However, consensus clustering has the advantage over a database of clusters by producing low-probability clusters containing a significant percentage of known elements from functional groups.Consensus clusters are likely to contain gene subsets that are co-regulated by common transcriptional control networks or are coexpressed to participate in cellular processes that together manifest a global phenotype of the cell or tissue. In either case, these clusters are of high biological value. To facilitate further analysis it is useful to know which clusters are involved in a given biological process. By supplying a list of genes from a given biological process or network, the use of the normal approximation to the binomial distribution of these genes over all consensus clusters, allows the identification of clusters of high functional significance. Similar statistical assignment of gene function based on cluster analysis was performed by Wu clusters . To assiBfl-2, has been experimentally verified [Two functionally significant clusters, the NFκB-responsive cluster and the ER/UPR cluster, were investigated further here. Within the NFκB-responsive cluster 50% of the putative promoters of genes investigated had canonical NFκB-binding sites within 1,000 base pairs of the transcriptional start site, suggesting that they are NFκB responsive ,30. The verified . AnalysiXBP1 mRNA is cleaved by the ER stress activated protein IRE1 to yield the transcriptionally active form of XBP1, inducing further genes of the UPR [N-glycan biosynthesis genes suggests that these genes are differentially regulated by ATF6/XBP1 and XBP1 respectively.In cells, the presence of unfolded proteins in the ER is associated with induction of the ER/UPR. However, during the maturation of B-cells to antibody-secreting plasma cells, expansion of the ER to accommodate increased secretion of immunoglobulins is thought to be coupled to the final stages of plasma-cell maturation. The induction of the ER/UPR occurs via the coordinated activation of the transcription factors ATF6 and XBP1 ,33. ATF6 the UPR . The actα-mannoside-like protein (EDEM), whose induction depends solely on IRE1/XBP1 activity [N-acetylglucosamine-1 phosphate transferase (DPAGT1) and UDP-GlcNAc:α-6-D-mannoside β-1-2-N-acetylglucosaminyltransferase II (MGAT2), which catalyze essential steps in the biosynthetic pathway of complex N-linked glycans, supporting a clear link between the dolichol pathway and the UPR [DPAGT1 and MGAT2 expression is regulated solely by the IRE1/XBP1 pathway. Altogether, these results show that consensus clustering and gene functional group analysis provide a highly accurate way of mining gene-expression data for novel insights into different genes within the cluster.The only known UPRE target gene is ER degradation-enhancing activity . Inducti the UPR . In addiRobust and consensus clustering provide a platform for more efficient microarray analysis pipelines. There is effectively no limit to the number of different clustering algorithms that can be used to feed into the consensus clusters, and each clustering algorithm could be run under different parameter sets to fully explore a microarray dataset . In addiAll areas of functional genomics that produce high-dimensional datasets with inherent patterns will require data partitioning to allow interpretation. Consensus clustering in the context of statistically defined functional groups could allow a consistent analysis platform for such diverse data types.r, to measure the linear relationship between two variables, x1 and x2, where the variable can be either discrete or continuous [We implemented and compared a representative sample of methods from the statistical, AI and data-mining communities. The methods used were average linkage HC, PAM, SA and CAST. As all the clustering techniques use correlation between variables, we used the Pearson's correlation coefficient, ntinuous . HC and ntinuous , while Cm out of n total objects that are the closest (according to a distance matrix) to the remaining (n - m) objects. The fitness of these medoids is calculated by placing the remaining (n - m) objects in a group according to the nearest medoid and summing the distances of the group members from this medoid. These m selected objects are the initial medoids. A Swapping procedure is then applied to reassort the objects until there is no improvement in the fitness of the medoids [HC is an agglomerative method that produces a hierarchical (binary) tree or dendrogram representing a nested set of data partitions. It has been applied successfully to many gene-expression datasets . Section medoids . As with medoids .θ0 = 100, c = 0.99994 and number of iterations = 1,000,000.SA  is an itCAST  is a heux1 and x2, to be expressed as binary (b). As Pearson's correlation coefficient is bounded, it provides a good basis for defining a binary relationship between two variables as defined by:Methods such as CAST and SA require the differences/relationships between a pair of observations, α ≤ 0 is a constant and  is a floor function that returns the largest integer less than or equal to the real number y.where 0 <Two datasets were used for evaluating the cluster methods. The first is a set of multiply repeated control element spots relating to the Amersham Score Card (ASC) probe set on the Human Genome Mapping Project Human Gen1 clone set array . The ASCThe second dataset consists of a series of 26 arrays (1,987 filtered genes) measuring gene-expression difference across a set of human B-cell lymphomas and leukemias . The datp is a linear multivariate time series defined byTwo synthetic datasets were generated using a vector autoregressive process (VAR) or a multivariate normal distribution (MVN). The two datasets contained 1,830 genes and 20 conditions (arrays). The VAR process of order x(t) is the n-dimensional vector of observations at time t, Ai is the n × n autoregressive coefficient matrix at time lag i, and ε(t) is the zero mean n-dimensional noise vector at time t . Therefore x(t) is a linear combination of the previous observations plus some random noise. For the synthetic dataset, each cluster was generated by a vector autoregressive model of order p = 1 and size n equal to the number of genes in the cluster.where x has a MVN distribution if every linear combination of that vector is also normal. Under such conditions we use the notation x ~ N(μ, Σ) to denote that x follows the MVN distribution, where μ is the mean vector and Σ is a positive definite matrix of covariance. The probability density function of x is given byFor the MVN dataset, a vector of random variables μ and covariance Σ.where |Σ| = det(Σ). For the synthetic dataset, each cluster was drawn from an MVN distribution with varying mean weighted-kappa was used [N, is defined in the following equation, where Countij is the number of elements in the matrix cell indexed by (i,j),To compare the resultant clusters for each method, a statistic known as was used . This men is the number of variables (genes) in the clusters as this represents the number of unique variable pairings.and weighted-kappa metric is calculated from the contigency table by:The wij is the weights for each category comparison; po(w) and pe(w) represent the observed weighted proportional agreement and the expected weighted proportional agreement; Countij is the ith, jth element of the 2 × 2 contingency table; N is the sum of the elements within this table; Row(i) and Col(i) are the row and column totals for this table respectively and Kw is the weighted-kappa value. The interpretation of weighted-kappa values indicates the strength of agreement between two observers (Table where, rs Table  is used


The objective of this study was to translate and to test the reliability and validity of the 10-item Dermatology Life Quality Index (DLQI) questionnaire in Iranian patients with vitiligo.Using a standard "forward-backward" translation procedure, the English language version of the questionnaire was translated into Persian  by two bilinguals. Seventy patients with vitiligo attending the Department of Dermatology, Saadi Hospital, Shiraz, Iran, were enrolled in this study.The reliability and internal consistency of the questionnaire were assessed by Cronbach's alpha coefficient and Spearman's correlation, respectively. Validity was performed using convergent validity.In all, seventy people entered into the study. The mean age of respondents was 28.3 (SD = 11.09) years. Scores on the DLQI ranged from 0 to 24 (mean ± SD, 7.05 ± 5.13). Reliability analysis showed satisfactory result (Cronbach's α coefficient = 0.77). There were no statistically significant differences between daily activity (DA) and personal relationship (PR) scale mean scores in generalized versus focal-segmental involvement in sufferers (P = 0.056, P = 0.053, respectively). There were also strong differences between the mean scores of the PR  scale with the involvement of covered only and covered/uncovered areas (P= 0.016) that was statistically significant in the second group.The study findings showed that the Persian version of the DLQI questionnaire has a good structural characteristic and is a reliable and valid instrument that can be used for measuring the effects of vitiligo on quality of life. Vitiligo is an important skin disease having major impact on the quality of life of patients suffering from vitiligo. Appearance of skin can condition an individual self-image, and any pathological alteration can have psychological consequences ,2. Many The DLQI is calculated by summing the score of each question resulting in a maximum of 30 and a minimum of 0. The higher the score, the more quality of life is impaired. The DLQI can also be expressed as a percentage of the maximum possible score of 30.The scores for each of these sections can also be expressed as a percentage of either 6 or 3.Since the DLQI is a brief, simple, easy to complete, and its application in research settings as a screening tool is well documented, it was decided to translate the DLQI into Persian  and to examine reliability and validity of this questionnaire in an Iranian population with vitiligo.The standard "forward-backward" procedure was applied to translate the questionnaire from English into Persian. Two independent bilinguals translated the items and two others translated the response categories and after cultural adaptation, a provisional version was provided. Subsequently, it was back translated into English and then the final version was provided. The cultural adaptation was done by the translation of the "partner" to the "spouse" in Persian language and adding of it in question 8 and 9, respectively.The final draft of the Persian version was administered to a sample of patients with vitiligo that referred to Department of Dermatology, Saadi Hospital, Shiraz, Iran. There were no restrictions on patient selection with regard to extension of lesions. The patients were introduced to the subject of this study and informed about the personal nature of the questionnaire, and all those who gave consent were given the DLQI questionnaire to complete. The questionnaires were completed by the patients whom were referred to our clinic for psoralen and UVA (PUVA) therapy. The patients were categorized by extension of lesions into covered only (vitiligo patches are covered by cloths) and covered/uncovered involvement and by severity of disease to focal-segmental  , which iValidity was performed using convergent validity to demonstrate the extent to which the DLQI correlated with global quality of life. Construct validity was checked by factor analysis.Seventy patients aged 18 to 68 (mean ± SD, 28.3 ± 11.09) years completed the questionnaire. We did not have any incomplete questionnaires; therefore we included all questionnaires in our study.Scores on the DLQI ranged from 0 to 24(mean ± SD, 7.05 ± 5.13). The reliability of the questionnaire was obtained by Cronbach's alpha coefficient (α = 0.77). There were no statistically significant differences between the sex, item scores and mean DLQI score. Table Cronbach's α coefficient by gender, marital status, severity, and extension of disease are shown in Table There were no statistically significant differences between item and mean DLQI scores of males versus females and married versus single cases.Cronbach's α reliability coefficients ranged from 0.69 to 0.78 for symptoms and feelings (SF) scale, 0.71 to 0.78 for daily activities (DA) scale, 0.67 to 0.88 for leisure (L) scale, and 0.50 to 0.99 for personal relationships (PR) scale. The reliability coefficient for work and school (WS) scale was equal to treatment (T) scale that was 0.100. Table The L scale was found to have significant relationship with the severity of the disease  .The DA and PR scales, also had no statistical association with severity factor (P = 0.056 and P = 0.053, respectively).The PR scale had strong correlation with the type of the extension of lesions (covered only versus covered/uncovered type) .There was no association between the numbers of treatment sessions with the type of disease .The number of treatment sessions and mean DLQI score was found to have a positive correlation coefficient  but this correlation was statistically significant in the generalized type only .Spearman's correlation coefficient of severity of the disease with questions 5, 6, and 8 were 0.25, 0.24, and 0.26 respectively. For question 8 and the extension of disease and also for question 1 and the stage of disease it was equal to 0.26.The result in question 4 and gender status was statistically significant (0.008) there was no statistically significance correlation between the age and each of questions.The result of question 9 and marital status was statistically significant (P = 0.002) and it was higher in married patients. The range of the paired correlations between the items was 0.17–0.68.Factor analysis is performed to determine the Persian version is a two-dimensional measure including social and psychological parameters Table .The DLQI questionnaire is a well-known instrument for measuring dermatological distress and has been translated into a variety of languages ,9 and 1.The translation process set by the international quality of life assessment (IQOLA) project was built on lessons from cross-cultural psychology and other health survey projects to develop protocols for translating, validating, and norming health status questionnaires, entails forward translation by at least two translators who were native speakers of the target language, rating of translation equality by two other bilinguals, and back translation by two translators who were native speakers of American-English or British-English . BecauseVitiligo is an acquired depigmentation disorder of great cosmetic importance affecting 1–4% of the world's population. The disease has a major impact on quality of life of patients, many of whom feel stigmatized by their condition .Porter et al. studied the effect of vitiligo on sexual relationships and found that embarrassment during sexual relationships was especially frequent for men with vitiligo . Salzer Weiss et al compared the difficulties faced by patients with vitiligo with those with leprosy in India . There mIn general, the finding of this study indicated that mental health in vitiligo patients is poor and it is strongly associated with their quality of life. Since the patients with higher DLQI scores responded less favorably to a given therapeutic modality , improviThis study reports data from a validation study of the 10-item DLQI questionnaire in Iran. In general the findings showed promising results and were comparable with other research finding throughout the world . The twoThe mean DLQI score in this study was 7.05 that is lower than that obtained by Finlay and Khan (mean 7.3)  and ParsReliability was associated by internal consistency of the questionnaire reporting Cronbach's alpha coefficient and validity was examined by convergent validity showed satisfactory results (α = 0.77). Cronbach's α was < 0.7 for males, focal/segmental, and covered vitiligo that may be related to small sample size and cultural differences.The Persian version of the DLQI questionnaire proved to be acceptable to patients and it is worth nothing that occasionally the questionnaire was administered by a trained nurse in face-to-face interviews. However this was done in illiterate patients and some ones indicated that some questions were difficult to answer, especially question 8. Perhaps this was the reason why a weaker correlation was found for this item with its corresponding subscale.The study finding showed that the Persian version of the DLQI questionnaire has a good structured characteristic and is a reliable and valid instrument that can be used for measuring the effects of the vitiligo on quality of life. Especially, the reliability of this questionnaire was high in females and patients with generalized involvement, because of the great cosmetic importance in these groups.None declared.DLQI: Dermatology Life Quality IndexSD: standard deviationIQOLA: international quality of life assessmentThe pre-publication history for this paper can be accessed here:


Enzyme-linked immunosorbent assay (ELISA) is a standard immunoassay to estimate a protein's concentration in a sample. Deploying ELISA in a microarray format permits simultaneous estimation of the concentrations of numerous proteins in a small sample. These estimates, however, are uncertain due to processing error and biological variability. Evaluating estimation error is critical to interpreting biological significance and improving the ELISA microarray process. Estimation error evaluation must be automated to realize a reliable high-throughput ELISA microarray system.In this paper, we present a statistical method based on propagation of error to evaluate concentration estimation errors in the ELISA microarray process. Although propagation of error is central to this method and the focus of this paper, it is most effective only when comparable data are available. Therefore, we briefly discuss the roles of experimental design, data screening, normalization, and statistical diagnostics when evaluating ELISA microarray concentration estimation errors.We use an ELISA microarray investigation of breast cancer biomarkers to illustrate the evaluation of concentration estimation errors. The illustration begins with a description of the design and resulting data, followed by a brief discussion of data screening and normalization. In our illustration, we fit a standard curve to the screened and normalized data, review the modeling diagnostics, and apply propagation of error.We summarize the results with a simple, three-panel diagnostic visualization featuring a scatterplot of the standard data with logistic standard curve and 95% confidence intervals, an annotated histogram of sample measurements, and a plot of the 95% concentration coefficient of variation, or relative error, as a function of concentration.This statistical method should be of value in the rapid evaluation and quality control of high-throughput ELISA microarray analyses. Applying propagation of error to a variety of ELISA microarray concentration estimation models is straightforward. Displaying the results in the three-panel layout succinctly summarizes both the standard and sample data while providing an informative critique of applicability of the fitted model, the uncertainty in concentration estimates, and the quality of both the experiment and the ELISA microarray process. Proteomic approaches are resulting in the identification of large numbers of proteins that can potentially be used as disease markers or drug targets. Unfortunately, proteomic approaches currently lack the throughput or quality metrics necessary to evaluate hundreds or thousands of samples that may be required to determine clinical usefulness of a biomarker . TraditiProcessing a ELISA microarray experiment produces large volumes of data of wide variety and high complexity. Similar to traditional 96-well ELISA data, ELISA microarray data often are perturbed by processing error -7. ProceStatistically evaluating ELISA microarray concentration estimation errors depends upon both the availability of the appropriate set of comparable measurements and the choice of data analysis methods. Sufficient replication within and across arrays is key to making precise estimates of both concentrations and errors . Hence, In this paper, we describe and illustrate a methodology for calculating the concentration estimation error of each assay in an ELISA microarray experiment based on a statistical analysis of the most likely sources of error. We expect the resulting data analysis algorithms to be a key component in a bioinformatics package for evaluating ELISA microarray data.Making concentration estimates and estimating their errors in our ELISA microarray studies involve a sequence of steps beginning with the layout of the ELISA microarray and design of the experiment. Following execution of the analytical components of the experiment, the statistical analysis proceeds with data screening, normalization, and model identification. Estimation and evaluation of the standard curves and error estimation functions come next. Finally, the standard curves and error estimation functions are applied and then evaluated using a modeling diagnostic.To estimate errors in concentration estimates, it is necessary to carefully lay out the microarray and design the experiment. Our layout features several distally separate replicates of each assay spot on each microarray to evaluate local processing effects. Our design addresses selection and application of treatments – in particular, replicate treatments – to a collection of arrays. This replication facilitates adjustments for the sources of variability that lead to ambiguous concentration estimates ,17. In aTo illustrate our technique for evaluating estimation errors in an ELISA microarray experiment, we used a subset of data from an ELISA microarray investigation of breast cancer biomarkers. The ELISA microarray experiments were performed as previously described ,3. BriefData screening, an exploratory data analysis, serves several purposes – identifying outliers, anomalous values, and experimental design shortcomings; identifying data transforms to improve curve-fitting and application; identifying measurement trends and other processing effects; and suggesting an appropriate functional form for the standard curve ,18-21. TBecause our protein arrays feature fewer spots per array than do typical gene expression microarrays, a different approach to normalization, suitable for low spot-frequency arrays, is required. This normalization is critical, given that array-to-array processing error is common and that standard curves are estimated from reference spot intensities calculated from one set of arrays and then applied to sample spot intensities estimated from a separate set of arrays.e transformations of both concentrations and spot intensities.A scatterplot of intensity estimates of standard spots versus concentration is particularly useful. First, outliers and anomalies may be readily apparent. Second, the spacing between concentration values may be assessed. If standard concentrations follow from a dilution series, then the separation between concentrations decreases significantly with the decrease in concentration. This results in spot intensities measured at higher concentrations having much more leverage on the fit of the model than may be desirable. It should also be apparent whether the variability in spot intensity is increasing with mean spot intensity. Both increasing spacing in the concentrations and heteroskedasticity in the measured intensities affect the model fit and follow-on statistical inferences . These mA scatterplot of raw or transformed standard spot intensity versus concentration also provides an indication of the appropriate model for the data. In particular, data following a sigmoid curve favor the logistic curves while data apparently lacking the horizontal asymptotes of a sigmoid curve favor a linear or power law model. Although several models may be fit and one selected based on a goodness-of-fit statistic (see next section), the scatterplot is a useful visual check on this selection.Several plots provide useful information about the quality of the fitted model. Of special importance are the scatterplot of residuals versus concentration and the scatterplot of residuals versus estimated intensity. In both cases, the variability of the residuals should be centered about zero and constant across concentration or intensity. Model bias is indicated by a systematic drift of residuals to one side of the zero line. Heteroskedasticity is indicated by a systematic change in the variation of the residuals. Both may indicate that a better model is necessary before proceeding to estimation of sample concentrations and estimation of concentration errors.An ELISA standard curve expresses protein concentration as a function of spot intensity. One standard curve is required for each assay. In an ELISA microarray experiment, the standard data are collected by fixing a set of concentrations and measuring spot intensities via imagery of the treated arrays. A standard curve is estimated by fitting an appropriate function to the set of (concentration, intensity) measurement pairs . This eqCommon parametric choices for standard curve models are multiparameter logistic functions and power law functions. For an ELISA microarray, a strictly monotone model is consistent with the belief that a monotone change in concentration should result in a monotone change in spot intensity.I as a function of concentration C and parameters P1, P2, P3 and P4, is defined asWe estimate standard curves with both logistic and power law parametric models. The four-parameter logistic model , expressI as a function of concentration C and parameters P1 and P2, in loge terms, isThe two-parameter power law model  expressie (I) = P1 + P2 loge (C) + εlogε, are independent and normally distributed with mean 0 and variance σ2. With either of these parametric models, concentration estimation errors may be estimated using propagation of error, also known as the delta method.We assume the errors, denoted by the term PRESS statistic, a direct measure of the predictive capability of each candidate model [To choose between competing candidate models, a number of measures exist for evaluating model fit when replicate observations of each assay are available. These include partitioning the mean squared error, or MSE, into components representing pure error and lack of fit , and pente model .PRESS statistic for each candidate model, suppose we exclude each poin (xj, yj) in turn and fit the model to the remaining points. We predict the value  at the excluded point xj and calculate the PRESS residual defined by ej, - j = yj - . Then, the PRESS statistic is the sum of the squared PRESS residualsTo calculate the PRESS score as the best predictive model to estimate concentrations.The candidate model with the lowest The basic approach to estimating concentration errors with the propagation of error method has three steps . First, C(I|), with , denote the inverted N parameter model expressing concentration C as a function of intensity I and the parameter estimates  Suppose  is the NxN parameter covariance matrix estimated by fitting I as a function of C, say I(C|P). Now, let Cs be the estimated concentration from the sample intensity estimate IS, say CS = C(IS/P) and  be the corresponding estimated standard error of IS. Then, the propagation of error estimate for the concentration estimate CS is the square root of the product of , the sample covariance matrix augmented with , and the Jacobian matrix J evaluated at IS and the parameter estimates . In this application, the Jacobian is the matrix of partial derivatives of C(I|P) with respect to the intensity I and the parameters P. Hence, the concentration estimation error of C(I|P) is the square root of the concentration estimation variance V(C(I|P))Let V(C(I|P)) = J(C(I|P))T ΣJ(C(I|P))where the Jacobian isJ(C(I|P))T = [∂C/∂I, ∂C/∂P1,..., ∂C/∂PN]and the augmented covariance matrix isCS isHence, the formula for estimated standard error of IS and standard error , the estimated concentration and approximate 95% confidence interval (CL95%, CU95%) areFor a given intensity estimate CS = C(IS)CL 95%= CS - 2SE[CS]     (2)andCU 95%= CS + 2SE[CS]     (3)C in terms of I and the four parametersFor example, consider the four parameter logistic model, Eqn. 1. The concentration estimation equation is obtained by solving this equation for C (Eqn. 4) with respect to I and the parameters P1, P2, P3 and P4The Jacobian matrix is obtained by taking the partial derivatives of the inverted four-parameter logistic function of A three-panel display combining a histogram of normalized sample spot intensities for a given antigen, its corresponding standard curve, and the graph of the concentration coefficient of variation, or relative error, versus concentration provides pertinent information about the conduct of the current experiment as well as information to improve future experiments. The standard curve panel presents a scatterplot of normalized standard spot intensities versus standard concentrations. The scatterplot is overlain with the estimated standard curve expressing concentration as a function of spot intensity. This panel also includes approximate 95% confidence intervals. These intervals summarize the uncertainty in concentration estimates due to both the uncertainty in estimating the standard curve and the uncertainty in the sample spot intensity estimate. Finally, a highlighted region helps distinguish concentration estimates s with acceptable errors from concentration estimates with possibly less than acceptable errors.IL, CL) and (IU, CU), are the two points such that the confidence intervals begin to increase significantly in length. This segment generally corresponds to the linear segment of a standard curve. We identify the intensity IL of the lower pair as the smallest intensity such that 95% UB(IL) is less than 95% UB(I) for intensity values I less than IL. Similarly, we identify IU as the largest intensity such that 95% LB(IU) is greater than 95% LB(I) for intensity values I greater than IU. We define CL and CU to be CL = C(IL) and CU = C(IU), respectively. We believe that this is a conservative approach to identifying intensities that generate concentration estimates with acceptable errors.The segment of the standard curve corresponding to acceptable concentration errors may be determined using the 95% confidence intervals. The lower and upper endpoints of this segment, (IL, CL) and (IU, CU). Consider the union of the two rectangular regions defined by the two sets of vertices [(IL, 0), (IL, CL), (IU, CU), (IU, 0)], and [(0, CL), (0, CU), (IU, CU), (IU, CLU)]. This union defines an L-shaped region covering the standard curve segment and bound at its extremes by the intensity and concentration segments. From this visualization, one can quickly grasp the dynamic range of acceptable intensities and the potential range of acceptable concentration estimates.An informative visualization of acceptable concentration estimates may be generated using the points /C(I), or relative error of a concentration estimate – as a function of concentration. This provides an alternative view of the error in concentration estimation over the concentration range covered by the concentration estimation equation. A standard curve modeled with a four-parameter logistic function generally will have a bathtub shape due to the increasing uncertainty in concentration estimates at the two ends of the concentration range where the curve approaches horizontal asymptotes.The second panel in this display shows the concentration coefficient of variation – that is, %The third panel in this display features an annotated histogram of sample spot intensity estimates on the intensity axis opposite the scatterplot. In this representation, it is easy to see the extent of overlap between the distribution of sample intensity estimates and the range of intensities that result in concentration s estimates with acceptable errors.e(fluorescent intensity) the difference between the mean of its slide's control spot loge (intensities) and the corresponding loess estimate.To evaluate concentration estimation errors in the example analysis, we attempted to quantify or understand those errors that we can and minimize those errors that we cannot. We began with data screening. The most significant anomaly uncovered during this exploratory analysis of the cancer biomarker data was a decreasing trend in control spot intensity as a function of array print order Figure . The treOur evaluation addressed the selection and fitting of an acceptable concentration estimation model. To that end, we examined two plots. The first displays the fluorescent intensities of the standards as a function of concentration Figure . Two chae transformation of the intensity estimates stabilizes the variability across concentrations , each must be addressed to achieve the best fit of the standard curve and resulting concentration estimation inferences. A logs Figure .With the heteroskedasticity and undue leverage addressed, we estimated a standard curve by selecting one of two models: a four-parameter logistic model (Eqn. 1) and a power curve model (Eqn. ??). We chose the logistic model as the model that fits the data best visually and in terms of the PRESS statistic Figure . The logFigure In the standard curve panel Figure , we see The histogram Figure  shows thThe concentration coefficient of variation, or the ratio of the concentration estimation error to the corresponding estimated concentration, offers an alternative expression to a confidence interval as a means to evaluate concentration estimation error. A graph of this estimation error as a function of concentration offers a comprehensive summary of the variation in the concentration coefficient of variation over the concentration range Figure .Presenting the results in this type of plot allows us to immediately look for several potential problems. First, does the fitted curve seem reasonable, given the data points to which we are fitting? We also can determine whether most of the unknown sample data fall within the acceptable range of the curve. The usable concentration range is made clear and, if it is too limited in range, it is immediately apparent. If problems are identified, several fixes are available, including changing the settings on the imager or using a different concentration range to create the standard curves.Evaluation of errors in estimating concentrations is important to establishing confidence in protein concentration estimates. Propagation of error provides a straightforward approach to estimating concentration estimation errors in ELISA microarray experiments. When presented in a simple multi-panel visualization, the propagated errors provide valuable information about individual concentration estimates, the applicability of the estimated standard curve, quality of the experiment, and the conduct of the ELISA microarray processing. The visualization provides a rapid assessment of the quality of the data, particularly in regard to the goodness of fit of the estimated standard curve and its capability to estimate concentrations over the observed range of intensities of biological samples.DSD framed evaluation of ELISA microarray concentration estimation errors as a statistical problem. KKA suggested the use of propagation of error and outlined the initial derivation. DSD and AMW derived the appropriate propagation of error equations, developed the algorithms, and designed the diagnostic visualizations. AMW encoded the algorithms, analyzed the microarray imagery, and produced the statistical results. SMV designed and printed the arrays and then carried out the ELISA microarray experiments. RCZ conceived the study, designed it with SMV, and then coordinated all our efforts. All authors submitted comments on drafts, then read and approved the final manuscript.