Today, it has become a hot issue in cancer research to make precise prognostic prediction for breast cancer patients, which can not only effectively avoid overtreatment and medical resources waste, but also provide scientific basis to help medical staff and patients family members to make right medical decisions. As well known, cancer is a partly inherited disease with various important biological markers, especially the gene expression profile data and clinical data. Therefore, the accuracy of prediction model can be improved by integrating gene expression profile data and clinical data. In this paper, we proposed an end-to-end model, Attention-based Multi-NMF DNN (AMND), which combines clinical data and gene expression data extracted by Multiple Nonnegative Matrix Factorization algorithms (Multi-NMF) for the prognostic prediction of breast cancer. The innovation of this method is highlighted through using clinical data and combining multiple feature selection methods with the help of Attention mechanism. The results of comprehensive performance evaluation show that the proposed model reports better predictive performances than either models only using data of single modality, e.g., gene or clinical, or models based on any single NMF improved methods which only use one of the NMF algorithms to extract features. The performance of our model is competitive or even better than other previously reported models. Meanwhile, AMND can be extended to the survival prediction of other cancer diseases, providing a new strategy for breast cancer prognostic prediction. Nowadays, cancer has become the leading cause of morbidity and mortality worldwide, in which breast cancer is one of the most common malignant tumors, especially among women –9. Accor via combining multimodality data, it is still challenging with the fusion of multiple feature extraction methods to obtain better feature representation and consider the relationship between multimodality data.During the past decades, gene expression profiling has become a powerful instrument for studying the biology of breast cancer. With these techniques, many prognostic features in gene expression can predict breast cancer recurrence risk . Van de Recently, the Attention method is proposed and it is argued that it is able to adaptively consider the importance of a single feature to the final global feature representation. It assigns different weights to each part of the feature sequence, extracting more critical and important information, allowing the model to make more accurate judgments. Based on this attribute, it has been widely used in machine translation and speech recognition. For example, Bahdanau et al.  proposedIn this paper, we propose a deep neural network model (AMND) based on the Attention mechanism which fuses the patients gene expression data and clinical data for the breast cancer prognosis. The preprocessed gene expression profile data is decomposed by AMND with five algorithms, NMF_mu , NMF_als , NMF_alsobs , NMF_pg , and PNMF (probabilistic nonnegative matrix factorization), respectively. Through the five algorithms, five characteristic matrices can be obtained. In order to individualize the importance of representations obtained from different NMF methods, Attention mechanism is introduced to calculate the weight of these representations of each sample according to its clinical data. After that, the weighted summating of these representations obtained by five single NMF methods is concatenated with clinical data, which serves as the final feature representation. This representation is input into the deep neural network for the classification task. This method not only takes into account the multimodality data, but also integrates multiple feature extraction methods, which can fully extract the high-level feature expression of gene expression data and clinical data, so as to improve the prognostic performance of breast cancer. Importantly, this method can be extended to survival prediction studies of other tumors, which provides a new strategy for other diseases prognosis.Wm×r and Hr×n given a nonnegative matrix Vm×n and a positive integer r, . It can be presented as follows:r is smaller than m or n, forcing the dimensions of W and H to be less than the dimensions of the original matrix. If Vm×n represents the pretreatment gene expression data matrix, m represents the number of samples, and n represents the number of genes, the NMF algorithm is to decompose the pretreatment genetic data matrix into feature matrix Wm×r and coefficient matrix Hr×n, so as to achieve dimensionality reduction. In general, the objective function is used to guarantee the approximation effect before and after NMF factorization. Lee and Seung i,i is the th element in the inverse matrix. In the OBS process, the weight of the minimum eigenvalue will be deleted, and the remaining weight will be corrected according to ((c) The corresponding optimal value of Lagrange operator S for element rding to .NMF is based on Optimal Brain Surgery and Alternate Least Square algorithms, which is simply noted as NMF_ alsobs in this paper. NMF_alsobs is based on OBS algorithm to iteratively optimize W and H in  and 9)..9). The (a) Based on the iterative optimization problem of alternate least squares, a local model of the error surface is constructed to analyze the impact of negative perturbations in the matrix.(b) Construct Lagrange operator to solve the constraint optimization problem.(c) Get the optimal W or H.n independent nonlinear optimization problems on convex sets. The following nonlinear optimization problems can be solved using the projection gradient method:f(x) is the differentiable function defined on Rn. Lin [Because of Rn. Lin  proposedt method:minx∈R+nfβ and σ, where 0 < β < 1, 0 < σ < 1; initial feasible point x1.(a) Input: constant k = 1,2, 3,4…,αk = βtk, tk takes the values 1,2, 3… in turn. When αk satisfies formula  until the conditions satisfied.(b) For the number of iterations, that is,  formula , the valSince the gene expression profile data contains fixed noise, it is necessary to take the random characteristics of the data into account to conduct systematic processing and analysis. Belhassen Bayar et al.  proposedThe objective function of PNMF is follows:Wij←WijVHWHHT+αWijHij←HijWTRW,H=V−WHThe nonnegative matrix factorization algorithm is a nonconvex optimization process in the iteration. Furthermore, the result of the iteration depends to some extent on the initial value, which is generated randomly. As a result, the selection of the initial values of W and H will directly affect the iterative results of the decomposition algorithm. Most NMF algorithms in the literature use random nonnegative initialization for . Iterates converge to a local minimum, so it becomes necessary to run several instances of the algorithm using different random initializations and then select the best solution. This obviously reduces the efficiency and real-time performance of the algorithm. Therefore, we use the method of Nonnegative Double Singular Value Decomposition  as an inLet the singular value decomposition of matrix Y beYk consists of the largest K pairs of singular values i=1k of matrix Y,m1 and n1 are also nonnegative vectors, so the first column of the matrix W can bem2 and n2 may contain negative elements due to the orthogonality of singular value decomposition. For the matrix X = σ2m2n2, the negative element in X is replaced by 0, and the remaining elements are unchanged, that is, taking the positive part X+ of matrix Y. As an approximate matrix of X, X+ is subjected to singular value decomposition, and W is initialized by the first singular value vector of X+. The initialization of other columns is the same. In this way, NNDSVD initial matrix is constructed as the initial value of nonnegative matrix.From the singular value decomposition of matrix Y, it can be seen that the matrix The NMF algorithm decomposes the nonnegative matrix into two matrixes in multiplication forms without changing the original data structure. Because there is no negative value in the process of NMF and the factorization results are highly interpretable, NMF analysis of gene expression data makes the research results more valuable . CompareAttention mechanism was originally used to deal with alignment problems in machine translation, because each word in the original sentence may contribute to a word in the target sentence with different contribution. Attention mechanism, however, can adaptively consider the importance of each word in the original sentence to the word in the target sentence. Similarly, the eigenvectors obtained by different NMF algorithms are only an approximation of the original data matrix and can not fully represent the information of the original data. Therefore, the concept of Attention mechanism applied in RNN is used to generate a more biological feature vector by adaptively summing the eigenvectors obtained by different NMF algorithms. In this way, not only was the correlation between clinical data and gene expression data considered, but also the feature vectors obtained by multiple NMF improved algorithm were combined to better express the original data and further improve the prediction performance.Fj_xi is used to represent the eigenvector of the i-th sample in the eigenvalue matrix obtained by the j-th NMF decomposition method, and the clinical feature vector of the sample is denoted as C_xi. So, Weight calculation formula is as follows:Fj_xi and C_xi. wji is the weight corresponding to the feature vector of the i-th sample in the j-th NMF decomposition algorithm. According to formula  and used the dataset's gene expression and clinical data. The METABRIC dataset used in this study included 1980 samples. Moreover, according to the research work of Khademi et al. [We downloaded the METABRIC breast cancer dataset from the website Synapsei et al. , five-yei et al.  were useFor the performance evaluation of AMND, we plot the ROC curve to show the interaction between True Positive (TP) and False Positive (FP) by changing the threshold, and calculate the AUC. In addition to AUC, Accuracy (Acc), Precision (Pre), F1-score, and recall are also used for performance evaluation. The following experimental results are derived from the average of the results obtained from 100 repartitioning data sets.In order to verify the effectiveness of Multi-NMF algorithm fused by Attention mechanism, we compare the results of our model with that of algorithms using single NMF algorithm on the dataset. Here, we choose the model of the three most effective single NMF improved algorithm to draw ROC curve together with AMND. As shown in In addition, the Acc, Pre, F1-score, and recall performance indicators of the five deep neural network models using single NMF improved algorithm were compared with that of AMND, and the results were shown in Clinical data are only used for weight calculation of Attention mechanismIn this experiment, clinical data only provide supervised information for computing the weights of Attention mechanism, and the eigenvectors obtained by the five NMF algorithms are weighted and summed using the obtained weights. The final eigenvectors obtained by weighted summation are directly input into the neural network for classification. The purpose of this experiment is mainly to verify the effectiveness of the Attention mechanism. The corresponding model is named clinical_first here.Clinical data are only used to fuse multimodality dataIn this experiment, the eigenvectors obtained by the five NMF algorithms are respectively assigned with weights of 0.2 and summed . Then this middle representation is concatenated with clinical data to obtain the final representation, which is then put into neural network for classification. In this variant, the Attention mechanism is not used and this variant is named clinical_second here.Neural network model based on clinical dataMany clinical features are directly related to prognosis. Therefore, clinical data are directly input into the neural network for training and prediction to verify the validity of the fusion of multimodality data. The corresponding model is named only_clinical here.Neural network model based on gene expression profile dataIn this experiment, the eigenvectors obtained by the five NMF algorithms are, respectively, assigned with weights of 0.2, and the final eigenvectors obtained by weighted summation are input into the neural network for classification. The corresponding model is named only_exp here.In order to verify the effectiveness of Attention mechanism in AMND model and the significance of fusing multimodality data, we designed the following four comparative experiments:The experimental results are shown in In order to further verify the good effect of the proposed method on the prediction of breast cancer survival, this paper also compared AMND with SVM, LR, and RF. This paper also analyzes the values of Acc, Pre, F1-score, and recall of different methods. The corresponding results are shown in In order to verify the performance of the model, it is compared with the results obtained from similar studies. Sun et al.  conducteIn conclusion, the AMND model proposed in this paper improves the prediction accuracy of breast cancer prognosis prediction research. It can not only help patients understand their life expectancy, but also provide a theoretical support for clinicians in making medical decisions and avoid wasting medical resources. Firstly, NMF algorithm is used to extract features from the original gene expression profile data, which can be high-dimensional and hard to be directly used. Therefore, using NMF algorithm can reduce the dimension of the gene data. From a biological point of view, each line in matrix W obtained by NMF can be regarded as a combination of the different features within the original gene data for each sample. Therefore, the decomposed W is the characteristic matrix and H is the coefficient matrix. W not only reduces the dimension based on the original gene matrix, but also achieves the purpose of feature extraction. Secondly, based on five NMF decomposition algorithms, five low-dimensioned eigenvectors are obtained, which are then fused by Attention mechanism to generate a more biologically meaningful feature representation, which can greatly help the downstream classification task. Finally, we use multimodality data and deep learning methods in our proposed model. Not only can better low-dimensional representation of the original data be obtained, but also higher classification performance can be achieved.In a summary, a deep neural network model based on Attention mechanism (AMND) was proposed for prediction of breast cancer. To effectively extract useful information within the gene profile data, clinical data is first used to compute the weights of five eigenvectors obtained by five NMF algorithms. Then the weighted summation of five eigenvectors is concatenated with clinical data to generate the final representation, which is put into deep neural networks for classification. The AMND method is a preliminary attempt to study the prediction of the prognosis of breast cancer by the Attention mechanism. The results show that the use of the Attention mechanism can better consider the connection between patients' clinical data and gene expression data; furthermore, the results also demonstrate that the use of multimodality data can improve the representative ability of the final feature vector. We also compare our performance with an existing method, namely, MDNNMN. Our results show that the proposed model is superior to MDNNMN on multiple evaluation indexes. The most important success of this work is the improvements for the in-depth understanding breast cancer omics data and the development of relevant prediction methods for survival. Moreover, this method can be extended to predict the survival time of other cancer diseases, providing a new strategy for cancer prognosis.


Quantification of gene expression levels at the single cell level has revealed that gene expression can vary substantially even across a population of homogeneous cells. However, it is currently unclear what genomic features control variation in gene expression levels, and whether common genetic variants may impact gene expression variation. Here, we take a genome-wide approach to identify expression variance quantitative trait loci (vQTLs). To this end, we generated single cell RNA-seq (scRNA-seq) data from induced pluripotent stem cells (iPSCs) derived from 53 Yoruba individuals. We collected data for a median of 95 cells per individual and a total of 5,447 single cells, and identified 235 mean expression QTLs (eQTLs) at 10% FDR, of which 79% replicate in bulk RNA-seq data from the same individuals. We further identified 5 vQTLs at 10% FDR, but demonstrate that these can also be explained as effects on mean expression. Our study suggests that dispersion QTLs (dQTLs) which could alter the variance of expression independently of the mean can have larger fold changes, but explain less phenotypic variance than eQTLs. We estimate 4,015 individuals as a lower bound to achieve 80% power to detect the strongest dQTLs in iPSCs. These results will guide the design of future studies on understanding the genetic control of gene expression variance. Common genetic variation can alter the level of average gene expression in human tissues, and through changes in gene expression have downstream consequences on cell function, human development, and human disease. However, human tissues are composed of many cells, each with its own level of gene expression. With advances in single cell sequencing technologies, we can now go beyond simply measuring the average level of gene expression in a tissue sample and directly measure cell-to-cell variance in gene expression. We hypothesized that genetic variation could also alter gene expression variance, potentially revealing new insights into human development and disease. To test this hypothesis, we used single cell RNA sequencing to directly measure gene expression variance in multiple individuals, and then associated the gene expression variance with genetic variation in those same individuals. Our results suggest that effects on gene expression variance are smaller than effects on mean expression, relative to how much the phenotypes vary between individuals, and will require much larger studies than previously thought to detect. Robustness, or the ability to maintain a stable phenotype despite genetic mutations and environmental perturbations, is an important property of many key biological processes, such as those underlying embryogenesis and development , 2. ConvTo make progress in understanding the balance between robustness and evolvability, we need to characterize the mechanisms that underlie robustness. Robustness can arise through a number of different mechanisms: for example, redundancy of regulatory elements or feedback loops in regulatory circuits. In these different scenarios, we hypothesize evolvability could be maintained through different selective pressures. If we are able to characterize gene-specific robustness to expression variability, we can begin to ask about the balance between natural selection of gene function and the ability to maintain evolvability.In model organisms, robustness and evolvability can be studied using experimental evolution approaches. These approaches typically quantify robustness as the change in trait variation after applying an experimental perturbation , 6. HoweSaccharomyces cerevisiae as a model system, studies have shown that heterogeneity in the expression of certain genes across cells is highly heritable and placed under complex genetic control, suggesting that the level of noise in gene regulation may also differ between individuals of multicellular organisms depending on their genetic background [Using the yeast ckground . Follow-ckground . HoweverHere, we take an unbiased, genome-wide approach to identify quantitative trait loci associated with gene expression variance across cells (vQTLs). We study human induced pluripotent stem cells (iPSCs), which offer a homogeneous population of cells allowing a relatively simple statistical model. Investigating iPSCs also provides the possibility to study gene expression variance across cells during differentiation in follow-up studies. To directly measure the mean and variance of gene expression within cell populations as phenotypes, we generated single cell RNA-seq (scRNA-seq) data from cells derived from multiple individuals.Using the Fluidigm C1 platform, we isolated and collected scRNA-seq from 7,585 single cells from iPSC lines of 54 Yoruba in Ibadan, Nigeria (YRI) individuals. We used unique molecular identifiers (UMIs) to tag RNA molecules and account for amplification bias in the single cell data . To estiWe excluded data from one individual (NA18498) with evidence of contamination, then filtered poor quality samples as previously described . After qTo ensure that our measurements are comparable across samples, we first sought to assess the impact of observed technical variation on the data and to identify unobserved technical confounders. To this end, we performed principal components analysis (PCA) on the matrix of log counts per million (log CPM).We found that across samples, the loading on the top principal component (PC) was correlated with gene detection rate (the proportion of genes with at least one molecule detected), but not with the biological variable of interest  or the expected technical confounders  after Bonferroni correction  or Fano factor (ratio of variance to mean) as quantitative phenotypes. However, prior work shows these quantities have predictable relationships with the mean, and therefore effects could still be explained away , 19. Then = 58), and still recovered 1,136 eQTLs (FDR 10%) after subsampling to n = 53. Our results therefore suggest that eQTL discovery in scRNA-seq (as opposed to replication of previously discovered eQTLs) loses power compared to equal-sized studies in bulk RNA-seq, likely due to increased experimental noise.We found 235 eQTLs, 5 vQTLs, 0 CV-QTLs, and 0 Fano-QTLs FDR 10%; . To valiWe found 85% of the eQTLs were also discovered as vQTLs (when restricting to testing only at the eQTL SNP), and 100% of vQTLs were discovered as eQTLs . We thenOur goal in this study was to find QTLs which alter the variance of gene expression independently of altering the mean expression. Under our model, these QTLs should explain variation in the dispersion parameter across individuals; however, we failed to find dQTLs. Further, all of the vQTLs we were able to identify could be explained by mean effects. In contrast, we were able to discover eQTLs, but fewer than expected based on bulk RNA-seq in matched samples.To understand why we failed to discover dQTLs, and why we discovered fewer eQTLs than expected, we first derived the power function in terms of effect size (log fold change), sample size, noise ratio , and significance level . We thenth percentile eQTL effect size is 0.022, but is 0.090 for dQTLs. Given this result and the power function we derived, there are two possible explanations for why we still failed to find dQTLs: (1) the noise ratio of dispersion is large (measurement error reduced power), or (2) the residual variance of dispersion is large (genetic variation explains little phenotypic variance).To estimate the distribution of QTL effect sizes, we fit a flexible unimodal distribution for the true effect sizes which maximizes the likelihood of the observed effect sizes and standard errors . SurprisTo estimate the typical noise ratio, we developed a two-step procedure to estimate the measurement error variance and residual variance per gene . BrieflyUsing our approach, we estimated that the typical noise ratio of the dispersion is 2.99, compared to 4.18 for the mean . This reth percentile dQTL effect size, assuming the typical noise ratio estimated above. We found that our study had only 0.001% power to detect that effect size at Bonferroni–corrected level α = 5 × 10−6  point-Gamma distribution. This mixture of ZINB model suggests an expectation-maximization approach where each cell is assigned to a subpopulation, and then the distributions of the subpopulations are re-estimated.Second, we made a strong assumption that latent gene expression is point-Gamma distributed. In this study, we directly assessed whether or not this was true using a simple statistical diagnostic, and did not find any gross violations of this assumption in the data. However, it is likely that this assumption will be violated in heterogenous populations of cells. One possible extension of our method to this case would be to assume there are Finally, we took a modular approach to map QTLs in this study: (1) we estimated parameters for each individual using only the scRNA-seq data, and then (2) we mapped QTLs using phenotypes derived from the estimated parameters. An alternative approach would be to include genotype in the count model for the data, and jointly learn the mean, dispersion, proportion of excess zeros, and genetic effect sizes for mean and dispersion. Such an approach could borrow information across cells with common genotypes to improve power, holding the experiment size fixed. However, further development will be needed to efficiently fit the models at QTL mapping scale.We stress that our power calculation is only a rough guideline for designing QTL mapping studies using scRNA-Seq. Intuitively, some minimum number of cells per individual is required to adequately estimate means and variances. However, having achieved that lower bound, the most important quantity to maximize is the number of individuals. In support of this argument, we estimate thousands of individuals would be required to detect a dQTL no matter how many cells were collected per individual.We based our power calculations on typical values of the noise ratio for the mean expression and dispersion, and chose a conservative significance level. However, we found considerable variation in the noise ratio across genes, suggesting that our results may not generalize even across genes. Overall, our results suggest that the technical noise introduced by scRNA-seq greatly reduces the power to discover eQTLs. Our results also suggest that, for iPSC lines, dramatically larger studies will be required to map both eQTLs and dQTLs from scRNA-seq.The cell lines used in this study were obtained from the NHGRI Sample Repository for Human Genetic Research at the Coriell Institute for Medical Research. All samples were collected by the Coriell Institute for Medical with written informed consent and with IRB approval.We cultured YRI iPSCs  in feedeSubjunc [UMI-tools [featureCounts [verifyBamID [We mapped the reads to human genome GRCh37 (including the ERCC spike-ins) with Subjunc , dedupliMI-tools , and coureCounts . We thenifyBamID .We filtered samples on the following criteria, derived as previously described :Only one cell observed per wellValid identificationAt least 1,011,612 mapped readsLess than 49% ERCC readsAt least 4,730 genes with at least one readLinear discriminant analysis predicts one cellWe filtered genes for QTL mapping on the following criteria:6 = 4096Number of molecules less than 4Median log CPM at least 3X of log counts per million (log CPM), using the pseudocount proposed in edgeR [We applied principal component analysis (PCA) to the matrix in edgeR .X =  be observed p-vectors, and let  be latent k-vectors where k ≪ p. Then, PCA corresponds to maximum likelihood estimation in the following latent variable model [We corrected for gene detection rate by simultaneously regressing out quantiles of gene expression, correcting for sample-specific and gene-specific means, and performing PCA. Let le model :xi∼N. However, as previously reported [In this parameterization, reported , we addiu is an n-vector of per-sample means, v is a p-vector of per-gene means, and Q =  is a n × k matrix of expression quantiles.Our approach is based on the latent variable model:We fit the model as follows:βj via least squares estimation of the following linear model:Estimate u, v via coordinate descent:Construct the residual matrix xij ≔ xij − ui − vj, then estimate W via maximum likelihood. The MLE k singular vectors of residual matrix X [Construct the residual matrix matrix X .We estimated the squared correlation between each PC and categorical covariates  by recoding each category as a binary indicator, fitting a multiple linear regression of the PC loadings against the binary indicators, and then estimating the coefficient of determination of the model.We assume the count data are generated by a zero-inflated negative binomial (ZINB) distribution . Let:rijk be the number of molecules for individual i, cell j, gene kRij be a size factor for each cellμik be proportional to relative abundanceϕik be the variance of expression noiseπik be the proportion of excess zerosxij be a q-vector of confounders per cellβk be a q-vector of confounding effects on gene kThen, we assume:Under this model, the mean and variance of gene expression are:πik: Considering just the non-zero component, marginalizing out λ yields the negative binomial (NB) log likelihood, weighted by 1 − Then, marginalizing over the mixture yields the ZINB log likelihood:μik > 0, ϕik > 0, 0 ≤ πik ≤ 1. To make the problem easier, we re-parameterized in terms of ln μik, ln ϕik, logit(πik) and performed unconstrained optimization.To estimate the model parameters, we maximized the ZINB log likelihood. The parameters must satisfy the constraints μik, ln ϕik, initializing from zero. In the second stage, we used the NB solution and logit(πik) = −8  as the initialization and optimized the ZINB log likelihood. In both stages, we used batch gradient descent for 30,000 iterations with fixed learning rate 10−3, accelerated by RMSProp [The ZINB log likelihood is nonconvex; therefore, we used a two stage optimization procedure. In the first stage, we optimized the NB log likelihood with respect to ln  RMSProp . We impl RMSProp .We defined the size factor of each cell as the total number of molecules detected (before excluding genes in QC). To correct for technical confounders, we included C1 chip as an observed confounder, recoded as binary indicator variables and centered. This approach is sufficient to also correct for batch, because in our experimental design, batch is a linear combination of C1 chip. Intuitively, if there were a batch effect independent of C1 chip, then we could add the batch effect to each chip effect and set the batch effect to 0.x1, …, xn are continuous random variables generated from a continuous CDF F, then F(xi) ∼ Uniform. Then, to test for goodness of fit of an estimated x1, …, xn, we apply the Kolmogorov-Smirnov (KS) test to test whether the values To assess the goodness of fit of the method, we used a diagnostic test based on the following simple fact: if the data F is not continuous. To address this issue, we used randomized quantiles [xi ∼ F then ui ∼ Uniform.Here, we have to modify this simple procedure to account for the fact that our data are discrete counts, so uantiles : we sampxijk comes from a different distribution Fijk, because it depends on the library size which is cell-specific. We therefore draw i and gene k, we apply the KS test to whether the randomized quantiles uijk across cells j are uniformly distributed.In our model, each observed UMI count We imputed dosages for 120 Yoruba individuals from the HapMap project  as previously described . We restQTLtools [For each single cell expression phenotype tested, we standardized and quantile-normalized the phenotype matrix to a standard normal as previously described . We callQTLtools . We inclQTLtools  using thWe performed replication testing by taking each SNP-gene pair from the discovery cohort, and testing that pair in the replication cohort. We defined a hit as replicating if it passed the Benjamini–Hochberg procedure at level 10% (restricted to the set of SNP-gene pairs tested) and had the same effect size direction.i and gene k, we assume the generative model:yik is the true phenotype, xi is the genotype at the SNP of interest, For individual ϵik ∼ N. From this model, we estimate n is the number of individuals. Under the working model, the power function is:α denotes the significance level, SE(⋅) denotes standard error, and Φ(⋅) denotes the standard Gaussian CDF. Under the assumed generative model, the power function equals:b/σr, and δ, the power function implies useful reference points; for example, δ = 1 is equivalent to cutting the sample size in half.To perform QTL mapping, we fit a working model which ignores measurement error:b, we estimate the distribution of true effect sizes b given observed effect sizes g is a unimodal mixture of Gaussians. We estimate g using adaptive shrinkage (ash) [b to be the 99th percentile of the fitted distribution.To determine the effect size ge (ash) . We tookAlthough we assumed a single measurement error variance δ, we estimate a measurement error variance per gene g is a unimodal mixture of uniforms, estimated using ash. To fit the model, we centered the k, concatenated them across genes, and assumed a common prior.To estimate the typical noise ratio  problem :y˜ik∣yiThen, the required estimates are:S1 Text(PDF)Click here for additional data file.S1 FigNumber of cells per individual, and number of molecules per cell after applying quality control filters.(PDF)Click here for additional data file.S2 Figμ) and latent mean are displayed for logit(π) < 0. Estimates of ln(ϕ) and latent variance are displayed for ln(μ) > −10, logit(π) < 0. Estimates of logit(π) are displayed over the entire range of parameter values. In each trial, simulated molecule counts for 95 cells are drawn from the model assuming 114,026 molecules per cell, matching the median number of cells, and molecules per cell in the observed data.Estimates of ln((PDF)Click here for additional data file.S3 FigFor each simulated data set, we use Kolmogorov-Smirnov test to test for departure of randomized quantiles of the data (based on the fitted ZINB distribution) from the uniform distribution.(PDF)Click here for additional data file.S4 FigFor the set of observed UMI counts for each individual, for each gene, we use Kolmogorov-Smirnov test to test for departure of randomized quantiles of the data (based on the fitted ZINB distribution) from the uniform distribution.(PDF)Click here for additional data file.S5 FigQQ plots are shown for dispersion, mean, variance, coefficient of variation (CV), and Fano factor.(PDF)Click here for additional data file.S6 FigWe fit a unimodal mixture of Gaussians to the distribution of observed eQTL (QTL) effect sizes (in terms of log fold change) using Empirical Bayes.(PDF)Click here for additional data file.S7 FigNoise ratios (ratio of measurement error variance to phenotypic variance) are estimated for 200 randomly chosen genes using a two-step empirical Bayes procedure.(PDF)Click here for additional data file.


Most of the subtyping schemes used in clinics today are derived from analyses of microarray data from thousands of different tumors together with clinical data for the patients from which the tumors were isolated. However, RNA sequencing (RNA‐Seq) is gradually replacing microarrays as the preferred transcriptomics platform, and although transcript abundances measured by the two different technologies are largely compatible, subtyping methods developed for probe‐based microarray data are incompatible with RNA‐Seq as input data. Here, we present an RNA‐Seq data processing pipeline, which relies on the mapping of sequencing reads to the probe set target sequences instead of the human reference genome, thereby enabling probe‐based subtyping of breast cancer tumor tissue using sequencing‐based transcriptomics. By analyzing 66 breast cancer tumors for which gene expression was measured using both microarrays and RNA‐Seq, we show that RNA‐Seq data can be directly compared to microarray data using our pipeline. Additionally, we demonstrate that the established subtyping method CITBCMST (Guedj  CITBCMSTCIT breast cancer molecular subtypesFPKMfragments per kilobase per millionMADmedian absolute deviationPCAprincipal component analysisPREBSprobe region expression estimation based on sequencingQCquality controlRMArobust multi‐array averageRNABCRNA breast cancerRNA‐SeqRNA sequencingSCCSpearman's rank correlation coefficientTDMtraining distribution matchingTPMtranscripts per million1et al., et al., et al., et al., et al., et al., et al., et al., et al., Breast cancer is a highly heterogeneous disease with several clinical subtypes defined by transcriptomic expression profiles that correlate with pathogenesis, clinical features, and prognosis  subtyping method employed at our clinic is a machine learning‐based model constructed on 355 selected samples from primary breast carcinomas, which were collected in France in the Cartes d'Identité des Tumeurs (CIT) program and analyzed on Affymetrix HG‐U133 Plus 2.0 arrays  was downloaded from ArrayExpress (accession: E‐MTAB‐365), preprocessed as described by the authors, and reduced to the 355 core samples used to train the model. We also removed the AFFX control probe sets before any further application.In order to visualize the subtype calls, CITBCMST produces two principal component analysis (PCA) plots of PCs 1 and 2: one for the training data and one for the test samples. In this study, PCA plots have been generated using a slightly modified version of the CITBCMST source code, producing a single plot containing both the training data and the classified samples. Furthermore, we modified the CITBCMST script to perform the PCA on the training data and then subsequently projecting new data into this PC space by scaling and multiplying the vectors with the rotation, in order to produce comparable PCA plots for each run of the script.2.2et al. , the samples comprise a balanced mix of the four IHC breast cancer subtypes: 17 triple negative, 14 HER2 positive, 16 luminal A, and 10 luminal B patients. In this work, we reclassified the Bordet samples using the CITBCMST six‐class subtyping scheme (Guedj In the latter dataset (henceforth referred to as the ‘RH’ dataset), tumor specimens originated from nine women diagnosed with breast cancer undergoing primary surgical procedures (during 2015 and 2016) at RH. The study was approved by The Danish Data Protection Agency (jr. no.: 2012‐58‐0004) and Danish Breast Cancer Group (jr. no.: DBCG‐2015‐14), meaning that tumor material was obtained with the informed consent of the patients and the study conforms to the standards established by the Declaration of Helsinki. Following surgical resection, fresh tumor specimens were evaluated by designated pathologists and tumor biopsies (~ 100 mg) were stored in RNAlater . RNA was isolated using the AllPrep DNA/RNA purification kit  and the QIACube workstation according to the manufacturer's instructions. The integrity of the RNA was measured using the Agilent RNA 6000 Nano Kit on an Agilent 2100 Bioanalyzer .2.3GSE43358.For the Bordet dataset, the transcriptomic profiles were obtained using the Affymetrix HG‐U133 Plus 2.0 microarray. The raw Affymetrix cell intensity files (.CEL files) files are available on the NCBI Gene Expression Omnibus under accession number agcc; Affymetrix, Thermo Fisher Scientific). For both datasets, quality control (QC) was performed using the R/Bioconductor package affyQCReport. After passing QC, the raw. CEL‐files were processed using robust multi‐array average (RMA) from the R/Bioconductor package affy  as described in  to gain an average output of 50–100 Andrews,  was run 2.52.5.1et al., et al., et al., et al., r package plyr .Raw reads were trimmed using Trimmomatic . Each target sequence represents a probe set on the microarray, and individual probes in a given probe set are selected as subsequences of the target sequence.Reads were mapped using kallisto  between the training data and the centroid for the given subtype, the sample is classified as an ‘outlier’. n is calculated as the maximum between the six subtypes of the value (maxdistances to centroid ‐ meddistances to centroid)/maddistances to centroid  and class mismatch (between either one of the subtypes)—the latter of course being the more severe error.R2 value for the least squares regression for microarray probe set intensities vs. RNA‐seq FPKM/TPM and the Spearman's Rho for a rank‐based measure of correlation, as we cannot expect probe set intensities and FPKM/TPM to correlate linearly for all preprocessing methods.Additionally, we calculated the 33.1R2 = 0.9445 and Spearman's ρ = 0.9638) and highly similar subtyping  compared with the subtyping on the corresponding microarray data. The RNABC pipeline is implemented in the r programing language , followed by quantile normalization to a target distribution consisting of the mean probe intensities in the microarray training data, followed by batch correction with the microarray training data Fig.  resulted3.2The RNABC pipeline resulted in 51 out of the 57 paired samples matching on predicted CITBCMST subtype, while the remaining six samples were mismatches. For the six discordant samples, additional analyses were performed to examine the reason for the mismatch. A CITBCMST prediction for each of six samples was performed and the results visualized Fig. . In the 3.3To ensure more general applicability, a second dataset of nine samples was used to validate the performance of the pipeline. Beyond these being data generated at our clinic, validating on a smaller dataset is relevant since in clinical settings data are typically submitted in smaller batches of samples, which may affect the results of the batch correction step. For the RH data, the Spearman's Rho between the data types after RNABC transformation was 0.9734 (0.5722 before running the RNABC pipeline) and seven out the nine samples were predicted to be of the same subtype for both data types  is also near‐instantaneous, bringing the runtime down from hours to minutes per sample on a standard laptop computer.44.1Some of the samples are almost equally close to two subtypes. Because of this uncertainty, these borderline cases receive a less meaningful class assignment than the core samples clearly belonging in a single subtype. Even minor differences in true and estimated probe abundance can contribute to class switching in the mixed cases, and clinically speaking, these classifications are not used. Samples that do not fit well into any of the subtypes also represent classifications without clinical utility. Both the outlier and the mixed cases point out a weakness of the entire subtyping paradigm in precision medicine. Every cancer patient is unique and even two patients that are predicted to be of the exact same subtype may be quite different on a molecular level. This lack of robustness should certainly be improved on, but this should be addressed by improving subtyping methods as well as patient stratification and treatment in general.4.2R2 = 0.0119, SCC = 0.6982), while quantile normalization alone following probe set mapping results in five matches (R2 = 0.5540) in the CITBCMST subtyping for the Bordet dataset, and batch correction following probe set mapping yields only mismatches (R2 = 0.0118). These results indicate that the application of mapping and both postmapping steps is necessary to ensure high accuracy of the classification. Interestingly, each of the two postmapping steps alone yields Spearman's rank correlation coefficients (SCC) of 0.6977 and 0.8796, respectively, which corresponds well to the observation in section The RNABC pipeline consists of three main components: the mapping of RNA‐Seq reads to probe set target sequences, quantile normalization to the CITBCMST training dataset, and batch correction to the CITBCMST training dataset. The first step, the mapping of the RNA‐Seq reads directly to the probe set target sequences, allows for subtyping including all the 375 probe sets used as the original CITBCMST subtype signatures. This makes the subtyping more robust, than applying the CITBCMST classifier on the 256 HUGO symbol signature. With the goal of reproducing the microarray‐based subtyping from the RNA‐Seq data, both quantile normalization and batch correction are essential components of the pipeline. Probe set mapping alone results in no matches, as a result of poor correlation between microarray and RNA‐Seq data  as each run will produce a different plot. Instead, we suggest projecting new samples in to a PC space precalculated on the training data, in order to ensure consistency in the output. In general, the robustness of the algorithm could also be improved upon. For this, a complete reworking of the subtyping scheme is probably necessary—preferably using RNA‐Seq data as a starting point. Until such time, our RNABC pipeline enables the utility of legacy microarray‐based subtyping methods.4.5It should be noted that many different breast cancer subtyping schemes exist besides CITBCMST. The choice of subtyping scheme is not standardized, and the arguments for using one or the other may be based on experience with different approaches in a particular clinic. The lack of robustness of many existing subtyping methods calls for creating entirely new schemes for cancer subtyping based on high‐throughput sequencing, possibly including biomarkers from other omics data types, and ideally some form of patient immune profiling as well. The subtypes of the future algorithms can of course remain the same, and the training data could be switched using paired samples of training data, possibly with multiple replicates to ensure reliability. Having clinics phase out microarray analysis altogether requires strong scientific evidence that RNA‐Seq‐based subtyping can perform equally well or better than current methodologies. Despite the current challenges, more studies demonstrating the utility of RNA‐Seq are continuously conducted indicating a promising future for RNA‐Seq in clinical medicine. Hopefully, the increasing use of RNA‐Seq and this contribution to its utility will lead to new insights and inspire novel treatment options.5Molecular subtyping of cancer for clinical decision‐making is common practice for many cancer types. The majority of these tools are built from DNA microarray profiling of large cohorts of patients—in many cases through massive collaborative efforts—and the resulting classifications serve as international clinical standards. Gene expression profiling is to an increasing degree being performed using RNA‐Seq, but while the two data types are comparable in nonparametric analyses of transcript abundance, RNA‐Seq count data are not directly compatible with established probe intensity‐based subtyping methods.R2 = 0.9445, Spearman's ρ = 0.9638) and near perfectly matched breast cancer subtype assignments from the microarray data using the corresponding RNA‐Seq data. We compared our method to three state‐of‐the‐art methods for comparison of the two data types and outperform them all significantly, as no existing tool is able to generate data comparable enough to enable accurate clinical subtyping. Furthermore, since we utilize pseudoalignment directly to the probe sequences, we circumvent traditional read alignment and thereby cut the processing time down from more than 3 h to < 3 min per sample on a standard laptop. The tool is freely available at https://bitbucket.org/cbligaard/rnabc/.We here present a method that enables fast and accurate subtyping of tumor samples for which gene expression is measured using RNA‐Seq. We tested our method on 66 breast cancer samples for which we have measured transcript abundance using both microarrays and RNA‐Seq, and were able to achieve near‐perfect correlation between probe intensities from microarrays and the probe‐level expression inferred from RNA‐Seq data (LRO and MR conceived and designed the project. MR and FCN acquired the data. CBP and LRO analyzed and interpreted the data developed the computational framework. CBP and LRO wrote the manuscript.


To consider all possible disease models, we extend Omnibus-Fisher to an optimal test by using perturbations. In our simulations, a usual Fisher’s method has inflated type I error rates when directly applied to correlated omics data. In contrast, Omnibus-Fisher preserves the expected type I error rates. Moreover, Omnibus-Fisher has increased power compared to its optimal version when the true disease model involves all types of omics data. On the other hand, the optimal Omnibus-Fisher is more powerful than its regular version when only one type of data is causal. Finally, we illustrate our proposed method by analyzing whole-genome genotyping, DNA methylation data, and RNA sequencing data from a study of childhood asthma in Puerto Ricans.The development of high-throughput biotechnologies allows the collection of omics data to study the biological mechanisms underlying complex diseases at different levels, such as genomics, epigenomics, and transcriptomics. However, each technology is designed to collect a specific type of omics data. Thus, the association between a disease and one type of omics data is usually tested individually, but this strategy is suboptimal. To better articulate biological processes and increase the consistency of variant identification, omics data from various platforms need to be integrated. In this report, we introduce an approach that uses a modified Fisher’s method (denoted as Omnibus-Fisher) to combine separate  p-values of association testing for a trait and SNPs, DNA methylation markers, and RNA sequencing, calculated by kernel machine regression into an overall gene-level p-value to account for correlation between omics data. We further extended the method to an optimal version in order to consider all possible disease models.In this research, we developed a statistical approach using a modified Fisher’s method (denoted as Omnibus-Fisher) to combine separate  Because of major advances in high-throughput biotechnologies, large amounts of omics data have been collected to study the biological mechanisms underlying complex diseases at different levels, such as genomics, epigenomics, and transcriptomics. Such different types of omics data can help us understand a disease from several perspectives. However, each of the arrays or sequencing technologies is designed to collect a specific type of omics data, such as SNPs, DNA methylation markers, and RNA sequencing. Thus, the association between a complex disease and one type of omics data is usually tested individually, but this strategy is suboptimal and has some disadvantages. Researchers often find that only a small proportion of disease variation can be explained by one type of omics  data, leading to “missing heritability” . Moreovep-values can be computed analytically and quickly without using resampling techniques. Although gene-based tests were originally developed for genetic studies, the same concept can be applied to studies of multi-omics data. Another issue is small sample size, especially for epigenomic or transcriptomic data. For example, large-scale genome wide association studies (GWASs) have been widely conducted for genetic studies for many years, so researchers usually have hundreds or thousands of genotyped samples. However, genome- wide methylation studies are more recent, and thus researchers often have a small number of samples. Moreover, incomplete samples may be wasted when using methods requiring complete samples . In this scenario, methods combining multiple p-values can be applied to make full use of data. For example, the p-values for association testing of a disease and SNPs, DNA methylation markers, and RNA sequencing data are be calculated separately, and then these separate p-values can be appropriately combined into one final p-value.Although integrative analysis of omics data is clearly needed, the complexity of disease mechanisms, the large number of collected molecular variables, and relatively small datasets can make such analysis quite challenging. Bersanelli et al.  summarizp-values for association testing of a disease or trait and SNPs, methylation markers, and RNA sequencing data calculated by KM regression into an overall gene-level p-value accounting for correlation between omics data. This method can be applied to either samples with all three types of omics data or samples with one or two types. To account for all possible disease models, we further extend the modified Fisher’s method to an optimal test by using perturbations. In our simulation studies, we show that a usual Fisher’s method has inflated type I error rates when directly applied to correlated omics data. In contrast, our Omnibus-Fisher test preserves the expected type I error rates when employed in correlated omics data. Moreover, the Omnibus-Fisher method has increased power compared to its optimal version when the true disease model involves all of SNPs, methylation markers, and RNA sequencing data. On the other hand, the optimal Omnibus-Fisher method is more powerful than its regular version when only one type of data is causal. Finally, we illustrate our proposed methodology by analyzing whole-genome genotyping, DNA methylation, and RNA sequencing data from a study of childhood asthma in Puerto Ricans.In order to test for overall gene-level significance, we here present an approach to use a modified Fisher’s method (denoted as Omnibus-Fisher) to combine separate p-value covariance [Omnibus-Fisher and usual Fisher], and the optimal test with p-values from Omnibus-Fisher as inputs [optimal Omnibus-Fisher]) had empirical Type I error rates close to the nominal level . When the usual Fisher’s method without considering covariance was applied to G and E correlated data, the Type I error rate was inflated . In contrast, the optimal and regular Omnibus-Fisher methods with considering covariance retained the desired Type I error rates as evidenced by the patterns observed in the QQ plots shown in Fig 1B and Table 1. Similar results were observed when extending to 100,000 datasets for evaluation (When applied to samples with independent SNPs (G), methylation markers (M), and RNA sequencing data (E), all of the methods used , the power of optimal Omnibus-Fisher was consistent higher than that of the regular Omnibus-Fisher method when G was the only causal factor, but when G, M and E were all causal factors, the optimal methods had lower power. This was expected because the optimal methods automatically searched for the appropriate disease model; in contrast, the regular Omnibus-Fisher assumed that G, M and E were all causal factors. Thus, when the simulation matched the assumption of the regular version method, they performed better than the optimal version and vice versa. However, when G and M were causal factors, no methods were consistently better than another. Furthermore, similar patterns were observed, when evaluated using the samples with G and E correlated . Since the causal SNPs in G were correlated with E, GM causal was equivalent to G, M and E causal. Note that the usual Fisher’s method is not included in Fig 3 because of its inflated Type I error rate with correlated data.When we compared the power of the statistics on the samples with independent G, M and E  and regular (P = 3.39×10−5) Omnibus-Fisher tests, although it didn’t reach a Bonferroni corrected significance level (P = 3.38×10−6) . The ZPBP2 region from chromosome 17q21 has been consistently replicated as an asthma-susceptibility locus across diverse ethnic groups ) from the regular Omnibus-Fisher test. Of these genes, KAT2A had moderate effects for SNPs, DNA methylation, and RNA expression separately, which could be overlooked by a single type of data analysis. The results also indicate that the optimal Omnibus-Fisher test was more powerful than its regular version when the significance was driven by one type of data. Conversely, when statistical significance was driven by two or three types of data, the regular Omnibus-Fisher had overall better power than the optimal version. These observations were generally consistent with the simulation results. Here, the optimal Omnibus-Fisher test does not outperform its regular version that assumes all types of omics data are in the disease model. Since only three types of omics data were analyzed in this study, it was still fine to assume they all were in the disease model. However, when more omics data are analyzed, the optimal test could be more useful than simply assuming all types of data are causal. We additionally output the p-value correlation for each gene across the whole genome  regression framework, which can be employed to test overall gene-level significance by combining separate p-value. Because we only consider three types of omics data in this study, assuming they all are causal could be still acceptable. However, when more omics data are analyzed, we believe that the optimal test would be more powerful for most genes than simply assuming all types of data are causal. Nevertheless, both the optimal and regular Omnibus-Fisher tests are able to detect genes with moderate separate effects, which could be overlooked by single type of data analyses.In the simulation studies, we showed that using a usual Fisher’s method on correlated omics data results in an inflated Type I error rate, while the modified Fisher’s method, Omnibus-Fisher, had the correct Type I error rate because it considered the omics data correlation in the model. The Omnibus-Fisher method achieves better power performance compared to its optimal version when the true disease model involves all of SNPs, RNA expressions and DNA methylations. On the other hand, the optimal Omnibus-Fisher method has better power than its regular version assuming all types of data are causal when only one type of data is actually causal. Our real data study also shows that the regular Omnibus-Fisher test has better power than the optimal test, when two or three types of data contribute to the combined p-values) so as to save computation times when calculating large p-values. Thus, the majority of the computation time is used by genes with small p-values. However, if a large number of genes are highly associated with the phenotype, the optimal test may be infeasible due to computational intensity. In such case, the regular test is recommended. Although the regular test also involves resampling technique to calculate covariances between different types of omics data, it only requires a small number of resampling .Although the optimal Omnibus-Fisher test uses perturbation to consider the correlation between omics data and search for the optimal disease model, the genome-wide data analysis could be completed within hours using multiple CPUs . We adapt a stepwise manner to implement perturbation  and the R package (https://cran.r-project.org/web/packages/OmnibusFisher/index.html) is available.Our method framework is general and flexible. Both continuous and binary traits for independent samples can be analyzed. Covariates can be easily incorporated into the model and different covariates can be used for different omics data. The regular and optimal version of Omnibus-Fisher algorithms were implemented in R , G is an n × q genotype matrix for the q genetic variants of interest where an additive genetic model is assumed  for illustration, γ is a q × 1 vector for the random effects of the q genetic variants, and ε is an n × 1 vector for the random error. The random effect γj for variant j is assumed to be normally distributed with mean zero and variance τwj; thus, the null hypothesis H0: γ = 0 is equivalent to H0: τ = 0, which can be tested with a variance component score test  where λl,1 ≥ λl,2 ≥⋯≥ λl,ml are the ml positive eigenvalues of vl,1,⋯,vl,ml, where Dl ∈ {omics data matrices G, M and E}. For example, the aforementioned G.2. For G, M and E, the same G, M and E, respectively. Thus, whether G, M and E come from the same subjects or different subjects are considered.3. Generate l ∈ {G, M and E}, rotate rb)-(5) pG, pM and pE by using b ∈ {0, 1,…, B}.7. Calculate the covariance between p-values of SNPs, DNA methylation and RNA expression  G; (2) G and M; or (3) G, M and E. Thus, we had the scenarios for G, M and E are independent: (1.1) continuous traits and no causal factors (denoted as null-c-ind), (1.2) continuous traits and G is causal , (1.3) continuous traits and G and M are causal , (1.4) continuous traits and G, M and E are causal , (1.5) binary traits and no causal factors (denoted as null-b-ind), (1.6) binary traits and G is causal , (1.7) binary traits and G and M are causal , and (1.8) binary traits and G, M and E are causal ; and we had the scenarios for G and E are correlated: (2.1) continuous traits and no causal factors (denoted as null-c-cor), (2.2) continuous traits and G is causal , (2.3) continuous traits and G and M are causal , (2.4) binary traits and no causal factors (denoted as null-b-cor), (2.5) binary traits and G is causal , and (2.6) binary traits and G and M are causal . As shown in i was generated via the model:i1 is a continuous covariate generated from a normal distribution with a mean of 20 and a standard deviation of 1; Xi2 is a binary covariate generated from a Bernoulli distribution with a probability of 0.5; Gi1,Gi2,…,Gi5 are the genotypes of randomly selected causal SNPs. We saved the first 500 cases and 500 controls to form a dataset, and we simulated 1,000 such datasets  causal-G-b-ind. Following the flow in datasets .i1, Xi2 and Gij are the same as described above; εi follows a standard normal distribution. We further permuted the continuous traits 10 times for each of the 1,000 (1.2) datasets to get 10,000 datasets for the scenario of (1.1) null-c-ind (After obtaining (1.6), we permuted the binary traits 10 times for each of the 1,000 (1.6) datasets to get 10,000 datasets for the scenario of (1.5) null-b-ind . Furtherll-c-ind .i1, Xi2 and Gij are the same as described above; Mi1 and Mi2 are the randomly selected causal methylated sites. We further used the same G, M and E data as generated in (1.7) to simulate 1,000 sets of continuous phenotypes for (1.3) causal-GM-c-ind (To simulate (1.7) causal-GM-b-ind, we followed a similar procedure to simulating (1.6) causal-G-b-ind via the model:GM-c-ind  via the i1, Xi2, Gij and Mij are the same as described above; Ei is the RNA expression and 7.45 is the global mean of RNA expression in the pool.Similarly, (1.8) causal-GME-b-ind and (1.4) causal-GME-c-ind were simulated via the following models for binary and continuous traits respectively:i is the aforementioned RNA expression used in the setting of G, M and E independent, Gi1 is the first causal SNP in the simulated gene, and Finally, all the scenarios in the setting of assuming G and E are correlated  ~ (2.6)) were simulated in the analogous ways to the first setting and used the same models. The only difference is that one causal SNP in the gene is correlated with its RNA expression:Subject recruitment and study procedures have been described in detail , 44. In Children with and without asthma (aged 9–20 years) were recruited in San Juan (PR) from February 2014 to May 2017, using a similar approach to that used in a previous study . Whole-gFig 5).In summary, we acquired multiple-dimensional data from microarray platforms for SNP genotyping and DNA methylation from white blood cells (WBCs), and RNA sequence data from WBCs in PR children. 10,994,111 imputed and genotyped SNPs with no missing genotypes were selected and they were assigned to 22,332 genes. 368,529 methylated sites with M values were selected and they were assigned to 20,596 genes. 16,188 genes with average TPM value greater than 0.05 were included for RNA expression. In the final analysis, we used the 14,808 genes with all SNP, DNA methylation and RNA expression information, and 1,116 subjects. Note that 471 out of the 1,116 subjects have all three types of omics data (S1 Table(DOCX)Click here for additional data file.S1 Fig(A) between SNPs and DNA methylation markers, (B) between SNPs and expression genes, and (C) between DNA methylation markers and expression genes.(PDF)Click here for additional data file.S2 Fig(A) SNPs, DNA methylation and RNA expression are independent; (B) SNPs and RNA expression are correlated.(PDF)Click here for additional data file.S3 Fig(PDF)Click here for additional data file.