Complex biological systems are described as a multitude of cell–cell interactions (CCIs). Recent single-cell RNA-sequencing studies focus on CCIs based on ligand–receptor (L–R) gene co-expression but the analytical methods are not appropriate to detect many-to-many CCIs. In this work, we propose scTensor, a novel method for extracting representative triadic relationships (or hypergraphs), which include ligand-expression, receptor-expression, and related L–R pairs.Through extensive studies with simulated and empirical datasets, we have shown that scTensor can detect some hypergraphs that cannot be detected using conventional CCI detection methods, especially when they include many-to-many relationships. scTensor is implemented as a freely available R/Bioconductor package.The online version contains supplementary material available at 10.1186/s12859-023-05490-y. Complex biological systems and processes such as tissue homeostasis , 2, neurPrevious studies have investigated CCIs using technologies such as fluorescence microscopy –13, micrDespite their wide usage, the analytical methods based on L–R pairs are still not mature; such methods implicitly assume that CCIs consist of one-to-one relationships between two cell types and that the corresponding L–R co-expression is observed in a cell-type-specific manner. One study even removed ligand and receptor genes expressed in multiple cell types from their data matrix, assuming one-to-one CCIs . In realWe developed a novel simulator to model the CCIs as hypergraphs and quantitatively evaluate the performance of scTensor and other L–R detection methods.We re-implement some L–R detection methods from scratch in order to analyze the same L–R database with all of these methods and focus on only the performance of L–R detection methods, not the slight differences in data pre-processing and the L–R database used.We show that scTensor’s performance with respect to its accuracy of many-to-many CCI detection, computation time, and memory usage are superior to the other L–R detection methods.We describe the implementation of scTensor as an R/Bioconductor package to enable the reproducibility of data analyses as well as continuous maintenance and improvements. We provide some original visualization functions and a function to generate an HTML report in scTensor to enable detailed interpretation of the results. We have extended our framework to work with 125 species.For the above reason, we propose scTensor, which is a novel CCI prediction method based on a tensor decomposition algorithm. Our method regards CCIs as hypergraphs and extracts some representative triadic relationships consisting of ligand-expression, receptor-expression, and related L–R pairs. The main contributions of this article are summarized as follows.One of the simplest CCI representations is a directed graph, where each node represents a cell type and each edge represents the co-expression of all L–R pairs Fig. a, left. approach , 52, 55.The drawback of using an adjacency matrix to describe CCIs is that multiple L–R co-expression scores are collapsed into a single value by summation or averaging. Because the average is simply a constant multiple of the sum, here we discuss only the sum. The summed value has no meaning in which L–R pairs are related to the CCI, and therefore CCIs and the related L–R pair lists cannot be detected simultaneously.In contrast to an adjacency matrix , the triadic relationship of CCIs also can be described as directed hypergraphs , where each node is a cell type but the edges are distinguished from each other by the different related L–R pair sets Fig. a, right.Here we introduce the procedure of scTensor. Firstly, a tensor data is constructed through the following steps Fig. b. A scRNJ is the number of cell types and K is the number of L–R pairs, is generated as the merged matrices. In this work, this tensor is called the “CCI-tensor”.Next, the data matrix is converted to a cell-type-level average matrix according to the cell type labels. Combined with an L–R database, two corresponding row-vectors of an L–R pair are extracted from the matrix. The outer product (direct product) of the two vectors is calculated, and a matrix is generated. The matrix can be considered as the similarity matrix of all possible cell-type combinations for each L–R pair. Finally, for each L–R pair, the matrix is calculated, and the tensor R1 and R2 are the NTD-2 rank parameters. The factor matrix R1 of ligand gene expression patterns in each cell type and the factor matrix R2 of receptor gene expression patterns in each cell type, and core tensor After the construction of the CCI-tensor, we use the non-negative Tucker2 decomposition (NTD-2) algorithm , 60. NTDr1 , two CCI styles (one-to-one or many-to-many including one-to-many and many-to-one), three numbers of CCI types , and three threshold values  for recognition of differentially expressed genes (DEGs). According to these conditions, ground truth CCIs were determined , each oTo differentiate significant CCIs from non-significant CCIs, many CCI methods introduce a label permutation test with a random permutation of cell-type labels to simulate the null distribution of CCIs. This process is considered a kind of binarization . For scTensor, binarization was realized by median absolute deviation (MAD) thresholding against each column vector in factor matrices calculated by tensor decomposition.To quantitatively evaluate how selectively each CCI method was able to detect the ground truth CCIs before and after binarization, nine evaluation measures were introduced. Four of them were applied both before and after binarization, and the remaining five were applied to the results only after binarization.The area under the curve of precision–recall (AUCPR) and Matthews correlation coefficient (MCC) values of 30 datasets with an E10 threshold value are shown in Fig. Figure The comprehensive validation described that the three methods (scTensor (NTD-2), Halpern’s score, and sum score) performed better than the others under certain conditions. To further examine the characteristics and trends of each method, we aggregated the number of CCIs detected in three datasets in which each of the three methods excelled Fig. .scTensor (NTD-2): this method performed well when the CCI style was many-to-many. For example, in Fig. Sum score: This method performed well when the number of cell types was small and the style of CCI was restricted to one-to-one Fig. b. Even tHalpern’s score: In most data sets, Halpern’s score was found to be too conservative, with many FN CCIs, but in a very specific situation, that is, when the CCI-style was one-to-all , it outperformed the other methods Fig. c. In conNext, we applied these CCI methods to real empirical datasets Table –12. As eN is the number of cell types and L is the number of L–R pairs. The label permutation tests combining any L–R scores require P is the number of random shuffles of cell-type labels. In many cases, P is typically set as a large value greater than 1000 [We also assessed the orders of computational complexity and memory usage of all the CCI methods Table . All thehan 1000 , 69, makR1 and R2 are the number of columns or “rank” parameters for the first- and second-factor matrices, respectively. The rank parameters are typically set as small numbers , this leads to a substantial computational advantage compared with the label permutation test. The computation time and memory usage when analyzing the simulated and real empirical datasets show that scTensor has an advantage in computational complexity compared with the label permutation test  does not perform the label permutation; instead, it simply utilizes the factor matrices after the decomposition of the CCI-tensor. Hence, the order of computational complexity is reduced to The first method similar to scTensor is CellChat. This method uses the communication probabilities (3rd-order tensor), which is a CCI tensor constructed with the authors’ original score and is normalized so that the sum in the second mode is 1. In addition to the label permutation test on each L–R pair in the third-order tensor, NMF on the matrix data summarized by the second mode of the 3rd-order tensor is performed to detect global CCI patterns. However, this summarization reduces the order of tensor  and loses information on which L–R pairs contributed to the CCI. In particular, as this study has shown, the label permutation test tends to detect one-to-one CCI, whereas NMF may also detect many-vs-many CCI, making it difficult to consider many-vs-many CCI and the L–R pairs that contribute to it simultaneously, even when the two methods are combined. Therefore, models like scTensor that can handle higher-order data as is are preferable.Inspired by our method, another method Tensor-cell2cell  extendedSecond, there are differences in tensor decomposition models; Tensor-cell2cell performs CANDECOMP/PARAFAC-type non-negative tensor decomposition but scTensor’s model is NTD-2. The difference between these models is the number of rank parameters. Tensor-cell2cell has only one rank parameter, while scTensor has rank parameters for the number of tensor orders (i.e. 3). This difference can be an advantage or a disadvantage; a small number of ranks reduces the computational time required to estimate the optimal ranks but it might make the model too simple.Third, there are differences in the related tools. Tensor-cell2cell assumes their text file for input, it only supports major species such as mouse and human, and it seems to assume to be used with LIANA , 73, anohttps://bioconductor.org/packages/release/bioc/vignettes/scTensor/inst/doc/scTensor_1_Data_format_ID_Conversion.html#case-iii-umi-count).scTensor is implemented as an R/Bioconductor package that is freely available. Both a scRNA-seq dataset and L–R database are required for scTensor execution. The default format for a scRNA-seq dataset is SingleCellExperiment, in which the gene IDs correspond to NCBI’s Gene database to allow links with other databases Fig. a. A scRNhttps://github.com/rikenbit/lrbase-workflow/blob/master/sample_sheet/sample_sheet.csv). The details of the data processing pipeline are summarized in the README.md of lrbase-workflow (https://github.com/rikenbit/lrbase-workflow), which is a workflow for constructing the LRBase for each of the species. For data sustainability, we offer the data files, including older versions, on the AnnotationHub server. The data files are bi-annually updated in conjunction with Bioconductor updates and are provided using lrbase-workflow. Users can specify which version of the data is used for analysis, thus ensuring data reproducibility.LRBase, which is the L–R database for scTensor, is stored on a remote server called AnnotationHub and is downloaded to the user’s machine on demand, only when called by the user Fig. a. To extNTD-2 was implemented as within the function of nnTensor  R/CRAN pTo enhance the biological interpretation of CaH results, we implemented some visualization functions Fig. a and theTo validate that the detected the co-expression of L–R gene pairs is also consistently detected in the other data including tissue- or cell-type-level transcriptome data, the hyperlinks to RefEx , ExpressIn this work, we regarded CCIs as CaHs, which represent the triadic relationships of ligand-expressing cell types, receptor-expressing cell types, and the related L–R pairs. We implemented a novel algorithm scTensor based on a tensor decomposition algorithm for detecting such CaHs. Our evaluations using both simulated and real empirical datasets suggest that scTensor can detect many-to-many CCIs more accurately than the other conventional CCI methods. Additionally, the calculation time and memory usage performances of scTensor are also superior to those of the other CCI methods.To extend the use of scTensor to a wide range of organisms, we also created multiple L–R datasets for 125 organisms. scTensor has been published as an R/Bioconductor package, facilitating the reproducibility of data analysis and the maintainability of datasets. We also implemented an HTML report function that simplifies checking the analysis results of scTensor. Like many CCI tools, scTensor can import an external L–R database.In the development of many CCI tools, the authors also develop their own L–R databases and investigate the differences among various L–R databases, particulaly when comparing their method with other conventional methods . This maWe were also able to examine several strengths and weaknesses of the methods other than scTensor. For example, Halpern’s score was found to be too conservative with many FN CCIs, but it was superior to the other methods with respect to the detection of one-to-all  CCIs. A possible reason for this is that since the formula for this score includes the square root of the chi-square distribution with two degree of freedom , and these distributions are known to be heavy-tailed to some extent, thus potentially inflating the number of significant L–R pairs.The permutation test implicitly assumes that the interactions occur between very few cell types because the larger the observed L–R score is than the empirical distribution computed by label permutation, the more significant the test result is. However, if the expression levels of ligand and receptor genes are high in any cell of any cell type, the L–R scores calculated by the label permutation are will also be high, and thus, the observed value of the L–R score will be regarded as not a particularly high value in the empirical distribution; consequently, such a test result will be not significant. Therefore, detection of many-to-many CCIs by label permutation test is difficult in principle. In the extreme case of all-to-all CCIs, the current approaches  cannot avoid FN CCIs.There are still some plans to improve scTensor to build on the advantages of this current framework. For example, the algorithm can be improved by utilizing acceleration techniques such as randomized algorithm/sketching methods , incremeTo reduce the memory usage of scTensor, we are developing DelayedTensor , which iCCI inference via receptor-receptor and extracellular matrix data , 99.Consideration of multi-subunit complexes .Comparison of CCIs across multiple conditions , 100–106Consideration of downstream transcriptional factors, target genes, and signaling pathways , 107–109Integration with bulk RNA-Seq or other type of omics datasets , 107–110Integration with pseudo-time , 112.Integration with spatial transcriptome data .In particular, in a recent benchmark study [Tensor data formats are very flexible ways to represent heterogeneous biological data structures , becauserk study , the prork study  and Addirk study –116.Although it is beyond the scope of the present paper to cover all of the above-mentioned topics, considering these in the framework of tensor decomposition is a promising research direction, so we aim to continuously work on these through the development of updates and releases of scTensor for Bioconductor.In this work, we present and evaluate scTensor, a new method for detecting CCIs based on L–R co-expression in scRNA-seq datasets. We also revealed that the widely used label permutation test has a bias that impedes the detection of many-to-many CCIs and demonstrated that the proposed method is a viable alternative.g and cell type c, and g, respectively.The simulated single-cell gene expression data were sampled from the negative binomial distribution The artz-Seq , and thea and b, we detected the DEGs using edgeR. By setting the threshold values  of edgeR as a and b values for each threshold were estimated as , , and , respectively.For the determination of differentially expressed genes (DEGs) and non-DEGs, For genes identified as DEGs based on a threshold according to the non-linear relationship above, the estimated c=1), and the expression values were randomly converted to 0 according to the dropout probability.To simulate the “dropout” phenomenon of scRNA-seq experiments, we also introduced the dropout probability  in ZIFA  , the style of CCIs (one-to-one or many-to-many including one-to-many and many-to-one), the number of types of CCIs , and the DEG threshold value ; in total, 90 synthetic CCI tensors were generated.http://pklab.med.harvard.edu/scw2014/subpop_tutorial.html) with low The gene expression matrix of human FetalKidney data was retrieved from the GEO database (GSE109205), and only highly variable genes , and only HVGs with low The gene expression matrix and the cell-type labels of human HeadandNeckCancer data were retrieved from the GEO database (GSE103322), and only HVGs with low The gene expression matrix of mouse Uterus data was retrieved from the GEO database (GSE118180), and only HVGs with low The gene expression matrix and the cell-type labels of mouse VisualCortex data were retrieved from the GEO database (GSE102827), and only HVGs with low The gene expression values of each cell are normalized by CPMED –58 and lI is the number of genes and H is the number of cells. Matrix J is the number of cell types. The cell-type labels are assumed to be specified by the user’s prior analysis, such as clustering or confirmation of marker gene expression. The relationship between the X and each element of j’s cell type.Here, data matrix X, and if both IDs are found, corresponding J-length row-vectors of the ligand and receptor genes   as8\docuing rule  as folloTo binarize each column vector of the factor matrices obtained by NTD-2, median absolute deviation (MAD), which is the median version of standard deviation (SD), was applied. Because we are only interested in the outliers of the elements of each vector in the positive direction, not the negative one, we focused only on the elements that deviate from the median in the positive direction as follows:Several methods have been proposed to score the degree of co-expression of a given L–R pair between two cell types, as described below.l can be averaged over cells belonging to the sth cell type within J cell types as follows:The gene expression of a ligand gene r is averaged over cells belonging to the tth cell type within J cell types as follows:Likewise, the gene expression of a receptor gene Using these values, the sum score is calculated as follows:lPhoneDB , Giotto lPhoneDB , CrossTalPhoneDB , and SqulPhoneDB , essentiIn some studies, the degree of co-expression is expressed as a product instead of a summation.as NATME , FunRes as NATME , ICELLNEas NATME , and Traas NATME , essentiDerived from the sum score, Halpern et al. proposed a score described below Table .Z-scaling is firstly applied to both J cell types as follows:In this score, Derived from the product score, Cabello mentclass2pt{minimmentclass2pt{minimlSignalR .To quantify the deviation of the observed scores obtained from real data, many studies employ Here, we consider assigning a The CCIs detected by the various methods tested in this paper were compared with ground truth CCIs to quantitatively evaluate the performance of each method. To evaluate the results, we used the metrics below.Each CCI method uses each corresponding L–R score to quantify the degree of co-expression of a given L–R pair between two cell types. To quantitatively evaluate the performance of each score, area under the curve of receiver operating characteristic (AUCROC) and area under the curve of precision–recall (AUCPR) were used.A receiver operating characteristic (ROC) curve is a plot of the true positive rate  versus precision .The calculation time and memory usage were evaluated by using the benchmark rules of Snakemake , false positive rate (FPR), and false negative rate (FNR) values of all the methods.https://bioconductor.org/packages/devel/bioc/html/scTensor.htmlscTensor: https://cran.r-project.org/web/packages/nnTensor/index.htmlnnTensor: https://bioconductor.org/packages/release/bioc/html/AnnotationHub.htmlAnnotationHub: https://bioconductor.org/packages/release/bioc/html/LRBaseDbi.htmlLRBaseDbi: Operating system: Linux, Mac OS X, WindowsProgramming language: R : https://github.com/rikenbit/lrbase-workflowlrbase-workflow : Operating system: Linux, Mac OS X, WindowsProgramming language: Python (vLicense: MITAny restrictions to use by non-academics: For non-profit use onlyAdditional file 1. List of existing L–R scoring and CCI detection methods.Additional file 2. Ground truth CCIs in simulated datasets.Additional file 3. AUCROC values of all methods.Additional file 4. AUCPR values of all methods.Additional file 5. Memory values of all methods.Additional file 6. Computational time values of all methods.Additional file 7. F-measure values of all binarization methods.Additional file 8. MCC values of all binarization methods.Additional file 9. PR values of all binarization methods.Additional file 10. FPR values of all binarization methods. Additional file 11. FNR values of all binarization methods.Additional file 12. TR of all datasets.Additional file 13. Three L-R pairs in which each of the three methods excelled.Additional file 14. HTML report of FetalKidney.Additional file 15. HTML report of GermlineFemale.Additional file 16. HTML report of HeadandNeckCancer.Additional file 17. HTML report of Uterus.Additional file 18. HTML report of VisualCortex.


Here, we present a protocol for deriving a continuum score for group 3 and 4 medulloblastoma tumor samples analyzed via RNA-sequencing or DNA methylation microarray. We describe steps for utilizing NMF-defined group 3/group 4 metagenes to calculate a continuum score between 0 and 1 that can be projected onto new sample data analyzed via RNA-sequencing. We then detail procedures for reverse engineering a continuum score for samples analyzed via DNA methylation microarray using a random forest classifier.  •A protocol to generate medulloblastoma group 3/group 4 continuum scores•Input either expression or methylation data•Run samples through R pipeline or through Shiny app•Generate a score between 1 and 0 to place patients on the continuum Publisher’s note: Undertaking any experimental protocol requires adherence to local institutional guidelines for laboratory safety and ethics. Here, we present a protocol for deriving a continuum score for group 3 and 4 medulloblastoma tumor samples analyzed via RNA-sequencing or DNA methylation microarray. We describe steps for utilizing NMF-defined group 3/group 4 metagenes to calculate a continuum score between 0 and 1 that can be projected onto new sample data analyzed via RNA-sequencing. We then detail procedures for reverse engineering a continuum score for samples analyzed via DNA methylation microarray using a random forest classifier. Determining sample subgroup is therefore an essential first step. This can be determined via the upload of DNA methylation microarray sample data (IDAT format) to the Molecular Neuropathology Classifier website . Prior knowledge of R, RStudio and data processing for tumor sample data from RNA-sequencing and DNA methylation microarray sources is also assumed: please refer to the original publicationThe protocol below describes the specific To derive a continuum score for group 3/group 4 tumors analyzed via RNA-sequencing, an NMF (Non-negative Matrix Factorization) model trained using 331 medulloblastoma tumors is projected onto incoming sample data using a procedure broadly outlined by Tamayo et al.By utilizing paired expression-methylation profiles where RNA-based continuum score was known (n = 192/331), a random forest regression model was trained using 115 CpGs. This enables reverse engineering of a continuum score for group 3/group 4 samples, with high accuracy (RMSE = 0.036) and without the explicit need for an expression profile i.e., from either an Illumina Methylation450k or MethylationEPIC profile alone. More detail on how the random forest model classifier was trained is provided in the original publication.Each step of the protocol has been tested within R Studio using R version 4.2.2. Both command line and shiny app tutorials can be executed using a standard work machine with 16-GB RAM and has been tested under Linux (running Ubuntu 20.04.5 LTS), MacOS and Windows operating systems.Timing: ∼5–15 min (Timing can vary depending on the number of samples in your dataset)https://github.com/hackingjpr/Group3-4App). We also provide sample usage data along with expected outputs in the GitHub. All in-silico analyses detailed in this section are performed in R and require the following lines of code to be run to set up the package environment and load the necessary data objects prior to continuum score assignment:Here, we project our existing NMF model onto a previously unseen group 3/group 4 medulloblastoma RNA-sequencing dataset to derive a continuum score for each sample. Several R packages, custom functions and data objects are required. Data objects and functions can be found in the associated GitHub repository > install.packages> BiocManager::install("biomaRt")# For specific package versions, seekey resources tablesection. When confronted withyes/no questions, answer yes to install dependency packages.> library(NMF)> library(MASS)# This loads the packages required into your working environment.3.Load required data objects.CRITICAL: You MUST update ‘/your/directory/’ to the location which you cloned the GitHub repository in step 1.> nmf.res <- readRDS(file = "/your/directory/Group3-4App/StarProtocols_Guide/data/nmf.res.rds")# This loads in the precalculated NMF model.4.Load the required custom functions.> source(file = "/your/directory/Group3-4App/StarProtocols_Guide/R/Project_NMF.R")Wrapper function used to project NMF model onto unseen group3/group4 sample data. A function breakdown is provided below (see)#below see5.Load sample data as a matrix object.> tpms.mat <- read.delim("/your/directory/Group3-4App/StarProtocols_Guide/data/tpms.mat.txt")CRITICAL: If you wish to use your own RNA-sequencing sample data, you must ensure that it follows the same format as tpms.mat. This object is a matrix, where columns correspond to samples and rows correspond to genes, with expression counts presented in the transcripts per million (TPM) format or equivalent. All genes (rows) must use HUGO gene nomenclature i.e., gene symbols. If your input dataset is not annotated correctly, please see Note: When projecting onto platforms other than bulk RNA-seq, appropriate filtering strategies to remove invariant genes/probes may be necessary.6.Project NMF model onto sequencing data> tpms.H <- project.NMF(input.array = as.matrix(tpms.mat),nmf.result = nmf.res)#Apply project.NMF function to input dataset. Function breakdown is provided below (see)Figure 17.Extract Group 3 and Group 4 metagenes from data and transpose matrix.> g3g4.tpms <- t,])#Rows 3 and 1 in tpms.H correspond to the metagenes for Groups 4 and 3 respectively8.Apply logistic transformation to metagenes.> logistic.g3g4.tpms <- apply{(1 / (1 + exp(-x)))})# apply a logistic transformation> logistic.g3g4.tpms.score <-apply{x[2]/(x[1]+x[2])})# calculate a ratio between logistically transformed Group3 and Group4 metagene9.Scale values between 0 and 1.> scaling.function <- function(x){(x-min(x)) / (max(x)-min(x))}# create a function to scale values between 0 and 1> logistic.g3g4.tpms.continuum.score <- scaling.function(logistic.g3g4.tpms.score)# apply the function to the unscaled g3g4 scoresCRITICAL: If you are using a small dataset or one that does not represent the full spectrum of Group 3/Group 4 medulloblastomas you may want to omit this step and present unscaled G3/G4 ratios (see Limitations below).Alternatives: You may wish to append to the precalculated G3/G4 ratios from Williamson et al. and then scale together with your new samples in which case the following alternative command should be used:> scaling.function1 <- function(x){(x - 0.3953062) / (0.5964371- 0.3953062)}# create a function to scale values between 0 and 1 using Williamson et al. data)> logistic.g3g4.tpms.continuum.score <-scaling.function1(logistic.g3g4.tpms.score)# Apply scalingNote: Theoretically, this could lead to some sample returning values under 0 or over 1. The user would need to take a considered view on such samples. They could simply be producing values close to 1 or 0 but otherwise consistent with samples at the extreme limits of the G3/G4 continuum, in which case manually assigning them the maximum 1 or minimum 0 value may be a valid approach. Should they massively exceed previous limits they may simply be outliers or technical artefacts that would be best noted but excluded from further analysis.10.Present output as data.frame for export.> logistic.g3g4.tpms.continuum.score <-as.data.frame(logistic.g3g4.tpms.continuum.score)> colnames(logistic.g3g4.tpms.continuum.score) <- 'Continuum Score'# Renaming for easier interpretation> write.csv#Export as .csv tableWithin RStudio.Timing: ∼5–15 min (Timing can vary depending on the number of samples in your dataset)https://github.com/hackingjpr/Group3-4App). We also provide example usage data along with expected outputs in the GitHub. All in-silico analyses detailed in this section are performed in R, and require the following code to be run to set up the package environment and load the necessary data objects prior to continuum score assignment:11.Install/Load required packages and their dependencies> install.packages> install.packages> install.packages# For specific package versions, seekey resources tablesection.> library(mlbench)> library(caret)> library(randomForest)# This loads each package into your working environmentCRITICAL: You MUST update ‘/your/directory/’ to the location which you cloned the GitHub repository in step 1 of Continuum score assignment (RNA-Sequencing).12.Load in the prediction object> load(file = "/your/directory/Group3-4App/StarProtocols_Guide/data/g3.g4.cont.rfe.Rdata")# This loads in the precalculated random forest model13.Load in example methylation dataset.> mvals.mat <- read.delimCRITICAL: The random forest model in this protocol requires that test data be provided as a matrix of M-values , where columns correspond to sample ID and rows correspond to probes. If your data is a matrix of beta values (object below named as “your.betas”), you can easily convert these to M-values using the following:> mvals.mat <- log2(your.betas/(1-your.betas))#logit-transformation14.Subset M-Value matrix to probes used as predictors in model> mvals.mat <- as.matrix,])#Removes probes that are not used for predictionAlternatives: If you are working with data obtained using other methylation platforms such as bisulfite sequencing (BS-Seq), it is still possible to derive a continuum score. A typical BS-Seq methylation data output will contain Chromosome, Start and End columns which can be used to ‘lift’ over CpGs that map to those covered by the Illumina Methylation Manifest. This will create a matrix of BS-Seq derived beta values that are annotated with associated Illumina probe IDs that can be used to subset to the required predictors in the random forest model. Whilst BS-Seq methylation values are known to be correlated to those obtained from the methylation array platform and in theory should produce similar predicted continuum scores, it should be noted that the random forest model in this protocol was not explicitly trained using sample data from this platform. An example script that can be used to achieve this is provided in a separate GitHub related to processing whole genome bisulfite sequencing data at https://github.com/tezla93/WGBS_MB/blob/main/InterPlatform/Liftover.R15.Apply test set to model and get predicted continuum scores using predict> pred.cont.rand.for <- as.data.frame))> write.csv#Export as .csvHere, we project our existing NMF model onto a previously unseen group 3/group 4 medulloblastoma DNA methylation dataset to derive a continuum score for each sample. Several R packages, custom functions and data objects are required, each of which can be found in the associated GitHub repository (Timing: ∼5–10 min (Timing can vary depending on number of samples in your dataset)https://github.com/hackingjpr/Group3-4App.git, an in-depth tutorial can be found in the GitHub Readme and on the app itself, there is also a tutorial video  repository’s ReadMe file (README.md).16.Select Expression or Methylation and Upload Data: Select the appropriate button on the left-hand side of app and then press “Browse...”Note: For Methylation idat files, we recommend uploading no more than 10 files at a time; this will speed up the score generation and keep the PC’s memory use low. There is no limit for expression samples, although with too many the resulting plots can begin to look crowded.CRITICAL: If using your own RNA-sequencing sample data, ensure that it follows the same format as tpms.mat. tpms.mat is a matrix, where columns correspond to samples and rows to genes. Expression counts need to be presented in the transcripts per million (TPM) format or equivalent. All genes (rows) must use HUGO gene nomenclature, i.e., gene symbols. If your input dataset differs to this, please see 17.Generate Group 3–4 Scores by clicking on the “Generate Group 3/4 Score” button.Note: The app will take you to the results tab once the scores are calculated. This tab will contain a table displaying your samples and their group 3/4 score and three different plots. These plots include placing your samples along the group 3/4 continuum and two survival curves. The first being a general survival curve based on the patient’s score and the second showing the effect that age can have on patient’s survival.18.Export Data: Navigate to the Download tab, which is above your results. You will export the results in either CSV or PDF format. CSV will be the raw table and PDF will contain the table as well as any plots. 19.Reset: If more samples are to be tested then press the reset button, this takes you back to the start and allows you to upload more samples.You may run the app on your machine using RStudio. Upon opening the “app.R″ file you will either be able to run the app or will need to run the whole block of code so that RStudio can recognize that it encodes a Shiny app. To run a block of code select it all with CTRL+ A and then run it by pressing CTRL+ENTER. Further installation information is included in the Group3-4App . These tables can be viewed and analyzed further  within R or saved to be used at a later date.Another outcome of this protocol is the ability to calculate group 3/4 continuum scores and generate plots in an interactive shiny app. This app allows users to generate group 3/4 scores without having to have knowledge of coding. The outputs of this app are then exportable for further study. For testing the app, or script, we have uploaded a number of sample files to the GitHub repository, the results generated by these files are in our GitHub repository’s Readme file.The resulting continuum scores can then be used for a number of applications. Our paperThere are a number of limitations to this protocol which we wish to highlight here. First, this protocol is, of course, only useful for group 3 or 4 medulloblastoma. In future this could be expanded to different cancer types and different data structures. This is outside the scope of this publication but could be an interesting and highly useful future piece of work.,,,https://www.molecularneuropathology.org/mnp/). For expression data, an integrated clustering with pre-existing medulloblastoma reference expression sets  and molecular pathology presented within the Williamson et al. 2022 study.When projecting my input RNA-Seq dataset onto the NMF model, I get the following error message.Likely cause: RNA-Seq dataset has not been annotated with HUGO gene IDs.Annotate your dataset with HUGO gene IDs using biomaRt.•Install biomaRt> BiocManager::install("biomaRt")> library(biomaRt)•Isolate the unique identifiers for each transcript in your dataset  - Here, we have provided an example dataset (problem1.tpms) that is incorrectly annotated with Ensembl gene identifiers to demonstrate how this can be corrected.> problem1.tpms <- read.delim(“/directory/to/Group3-4App/StarProtocols_Guide/data/problem1.tpms.txt”)> ensembl_IDs <- rownames(problem1.tpms)•Homo SapiensCreate a biomaRt connection to database for > mart<- useMart•Annotate each Ensembl identifier with its respective gene symbol:#This deals with an object labelled with hgnc symbols – If you arenot using hgnc_symbols, change this accordingly.> symbols <- getBM,     filters = ‘ensembl_gene_id’,     ensembl_IDs,     mart = mart)•Replace your current identifiers with gene symbols:> annotatedix <- match> symbols -> annotatedGenes> problem1.tpms$hgnc_symbol <- annotatedGenes$hgnc_symbol#Remove duplicated rows using:> problem1.tpms <- problem1.tpms #Remove transcripts with no gene annotation using:> problem1.tpms <- na.omit(problem1.tpms)#Set gene IDs as rownames and remove column:> rownames(problem1.tpms) <- problem1.tpms$hgnc_symbol> problem1.tpms$hgnc_symbol <- NULLStep-by-step:The app will not run on first deployment.Likely cause: If it is the first time loading up the app on your own machine it is likely the issue is due to packages wanting to be updated. The first time you run the app it has to install a number of packages that all have sub dependencies. Also. whilst it is true that the app automatically installs the relevant packages at the time of testing and running on the machines we had available, it is possible that certain CRAN/Bioconductor packages and their dependencies get updated over time in a way that prevents the packages from installing automatically on your machine.In the bottom left quadrant in RStudio there should be a question along the lines of “Update (Y/N)”. Typing Y and then pressing Return will lead to R updating the necessary packages. Also pay attention to the console in RStudio if it indicates an error installing a specific package then you may need to manually install certain packages e.g., running install.packages(“my_package”) command for CRAN packages.The app will not open after an error caused it to close.Likely cause: Efforts have been made to minimize errors that lead to the app closing itself but of course it may still happen in some situations. If it does then RStudio may still be running the script, even though you cannot see the app.In the top right corner of the bottom left quadrant box there should be a small red octagon with “STOP” written on it. If you click this the script will stop running and you can re-open the app.daniel.williamson@newcastle.ac.uk).Further information and requests for resources and reagents should be directed to and will be fulfilled by the lead contact, Dan Williamson (email: https://github.com/hackingjpr/Group3-4App.All materials required to complete this protocol are available at https://github.com/hackingjpr/Group3-4App.All data and code required to complete this protocol are available at


Drosera rotundifolia herbarium specimens, which were collected 100 years apart from different locations. Principal component and hierarchical clustering analysis enabled differentiation between three main regions on the plant , and between the different specimens. Lipids and protein spectral regions were particularly sensitive differentiators of plant tissues. Differences between the different sets of specimens were smaller. This study demonstrates that relevant information can be extracted from herbarium specimens using FTIR, with little impact on the specimens. FTIR, therefore, has the potential to be a powerful tool to unlock historic information within herbaria.Dried plant specimens stored in herbaria are an untapped treasure chest of information on environmental conditions, plant evolution and change over many hundreds of years. Owing to their delicate nature and irreplaceability, there is limited access for analysis to these sensitive samples, particularly where chemical data are obtained using destructive techniques. Fourier transform infrared (FTIR) spectroscopy is a chemical analysis technique which can be applied non-destructively to understand chemical bonding information and, therefore, functional groups within the sample. This provides the potential for understanding geographical, spatial and species-specific variation in plant biochemistry. Here, we demonstrate the use of mid-FTIR microspectroscopy for the chemical analysis of  Hundreds of millions of specimens are stored worldwide, with approximately 390 million housed in 3100 herbarium collections (https://sweetgum.nybg.org/science/). Rapid advances in cataloguing and digitization of records are revolutionizing the use of herbaria; their potential is only just starting to be realized [Herbaria are collections of dried plant specimens, which can date back hundreds of years from ca 600 and hrealized ,3. Methorealized ,5. Measurealized , climaterealized . MeasureFourier transform infrared (FTIR) spectroscopy is a rapid, inexpensive, non-destructive method for assessing chemical characteristics of biological samples. This technique measures the interaction of infrared radiation with molecular bonds to provide information on the biochemical compounds in a sample. In plant tissues FTIR has been used to measure the impact on plants of differences in, for example, nutrient availability, water stress, temperature stress, heavy metals and disease . A numbeDrosera rotundifolia L., which traps prey using adapted leaves presenting a sticky ‘fly-paper’ trap. We chose this species owing to its abundance in herbarium collections, wide distribution and because it is a sentinel species being very sensitive to environmental variation [D. rotundifolia leaves comprise clearly differentiated structures, which are functionally distinct. The lamina is the primary photosynthetic surface, and we know that it changes colour in response to differences in biotic and abiotic conditions [tentacle stem on the end of which is a tentacle head which produces a viscous muco-polysaccharide [In this study, our aim was to demonstrate the use of FTIR as a non-destructive technique for analysis of herbarium specimen chemistry. We explored the feasibility of using FTIR for identifying differences between leaf structures and variation between specimens collected at different times in different locations. We used the cosmopolitan carnivorous plant ariation . In addinditions . Tentaclccharide . Since t. 2 (a)D. rotundifolia herbarium specimens were used in this study (Four  (b)−1 resolution. Background spectra were taken in air before each sample, with 64 scans averaged at each map point; samples were suspended enabling direct transmission of IR. Data are available via the Dryad Digital Repository [Specimens were carefully removed from the herbarium sheet and placed onto the sample stage. FTIR spectra were taken for each of three tissue types uasar v.1.4.0 (https://quasar.codes). All spectra collected were normalized using SNV , to account for variations in sample thickness. Spectra were either analysed as a whole from 4000 to 650 cm−1, or specifically within the lipid region 3000–2800 cm−1; CO2 bands were removed from data analysis. Amide regions were not studied separately owing to variation in sample thickness of tentacle heads, particularly in young samples, impacting spectral band shape in that region.Spectra were obtained directly from Thermo Scientific Omnic Picta (v.9.7.0.39) with map files being split and exported as individual CSVs. Spectral analysis was carried out using Q (d)uasar v.1.4.0. Ordinations of PC axes and loadings for the different samples were created and assessed visually to establish patterns in differences between FTIR spectra of the samples. To further understand the FTIR spectra, hierarchical cluster analysis (HCA) was carried out for the raw spectra data using Euclidean distance and a max pruning level of 2 in Quasar v. 1.4.0. Linkage was derived from averages.Principal component (PC) analysis (PCA) was used to highlight key spectral differences by reducing the number of variables in the dataset, being carried out in Q (e)xcel v. 2209  on PC loadings over both the datasets using a two-tail, two-sample t-test assuming unequal variances. p-Values were adjusted using a Bonferroni correction to account for the number of t-tests done. Evidence of spectral differences between samples was considered in terms of the extent of the difference in PC loading scores and the p-value for the differences between the two groups.For statistical analysis the mean of each spectra map was used, to reduce problems with pseudo-replication. Statistical analysis was carried out in E. 3 (a)b). Peak intensity differed between the 3500–2700 cm−1 and the 1700–1100 cm−1 regions, probably owing to variation in sample thickness. This resulted in truncation of amide band peaks, particularly for young samples. The HCA dendrogram shows differentiation between sample and tissue type on the basis of the full, raw spectra data. HCA segregated the four plants into four separate groups (c). Plants A and B were linked as similar, as were plants C and D. In addition, HCA differentiated tissue types. Lamina and tentacle stalk were more similar to each other than they were to tentacle head.All spectra demonstrated broadly similar peaks but these varied in intensity and shape b. Peak ie groups c. Plantsa,b, −1/1500 cm−1 and C=O 1700–1600 cm−1), lipid (C–H ∼ 3000–2800 cm−1) and further in the fingerprint region (figure 2c). This demonstrates that the variability between tissue types is owing to differences between spectra in these regions. PC2 and PC3 show discrimination again in these regions, with PC3 highlighting protein bands as a dominant feature. This demonstrates that variability between different aged specimens is owing to differences in the spectra in these regions.PCA analysis of the whole spectra indicates that the first three PCs accounted for 90% of variance within the spectra. PC1 accounted for 74% of variance, PC2 12% and PC3 4%. The clearest differences in PC scores are between tissue types (figure 2 (b)d,e). All three tissue types were different from each other, on all three PC axes. ‘Old’ and ‘Young’ specimens are differentiated for the lamina area on all three PC axes, for the tentacle stalk on PCs 1 and 2, and not at all for the tentacle head area. For PC1 differentiation is across the entire lipid region spectra, for PC3 differentiation is owing to peaks at 2850 and 2910 cm−1. This indicates that variability between the specimens is accounted for by the range of lipids detected by FTIR analysis.PCA analysis of the lipid region alone indicated differences between samples and tissues . The larfigure 2. 4−1), and in the amide region associated with proteins (approx. 1700–1500 cm−1). Tentacle heads were quite thick in some specimens resulting in semi-truncated spectra in the amide region. Little variation was observed directly in relation to specimen age, except for the slightly less prominent band at approximately 1710 cm−1 associated with a carbonyl likely from cellulose. This type of observation has been reported before in tree bark studies, where retting may have resulted in reduction of cellular material [Our aim was to demonstrate FTIR as a non-destructive method for the analysis of herbarium specimen chemistry. We found clear spectral differences between the three plant tissue types  and between specimens collected nearly a century apart from different locations. We found evidence of differences between tissue types and specimens primarily in the lipid region associated with fats, fatty acids, waxes and lipids  [−1) [−1). A wide range of metabolites are involved in plant carnivory [et al. found that concentrations of Jasmonates increased after prey capture in both Drosera capensis [Dionaea muscipula [Our study shows that FTIR data can be used to characterize biochemical components within herbarium specimens, indicating chemical differences in different plant tissues, with limited levels of molecular degradation over time. The three tissues we measured support different functions, and so we expected their different biochemistry to result in clear separation in FTIR spectra. We found this to be the case. These differences likely result from C–O and overlapping O–H deformation associated with lignins (approx. 1350–1310 cm−1)  the suga−1) [−1) , and fatarnivory  and preyarnivory . It seemarnivory . Pavlovicapensis  and Dionuscipula . Such biet al. [Differences between the specimens which were collected from separate locations nearly 100 years apart, were less clear. Age and location confounded, so it is not possible to determine which contributes to these differences, but comparison is useful to highlight the potential for understanding temporal and/or spatial variation in plant biochemistry. The different ages contrasted primarily in amine and carbonyl spectral features, likely resulting from variation in protein or alkanoic amine chemical groups. This could suggest partial degradation of protein molecules in specimens, although spectra show band broadening rather than reduction in intensity with age, possibly indicating denaturing rather than breakdown of the protein chemistry. This change is also positively linked with changes in the lipid region supporting chemical structural change. Mendes Resende et al.  also shob), which is more evident in young specimens, and most prominent for the cellulose peak at approximately 1040 cm−1.As all samples were dried we interpret the differences found as being associated with the sample itself (i.e. it is age or location) rather than changes owing to storage. Differences in sample preparation and analysis have been shown to result in spectral variation  which wofigure 1D. rotundifolia herbarium specimens and show that the data collected differentiates within and between plant tissues, with limited change related to specimen age. Spectra obtained from age-matched similar plant regions were consistent across the sample repeats tested, indicating the robustness of the methodology and potential for further analysis of herbarium specimens. More work is required to understand the potential that FTIR analysis offers for the chemical interrogation of historical plant specimens. Further comparison of large numbers of samples, along with verification using more-sensitive approaches which may require destructive sampling is needed to ensure robust interpretation of FTIR data. Such confirmation of the potential and limitations of FTIR will enable the exploration of the environmental and climate change chemistry locked within these samples.The delicate and finite nature of herbarium specimens necessitates the use of non-destructive analysis methods, with these samples providing historical records of environmental impact/change and evolutionary change. Our measurements demonstrate the feasibility of using FTIR to extract meaningful data from https://doi.org/10.5061/dryad.jsxksn0dw [Data freely available Open Acess from the Dryad Digital Repository: sxksn0dw .Data being shared as raw spectral files as both original instruments file (Thermo Scientific Omnic Picta .map files) and as .CSV exports for wide dissemination of the source data.The data are also provided as part of the electronic supplementary material .


Due to the prevalence of complex data, data heterogeneity is often observed in contemporary scientific studies and various applications. Motivated by studies on cancer cell lines, we consider the analysis of heterogeneous subpopulations with binary responses and high-dimensional covariates. In many practical scenarios, it is common to use a single regression model for the entire data set. To do this effectively, it is critical to quantify the heterogeneity of the effect of covariates across subpopulations through appropriate statistical inference. However, the high dimensionality and discrete nature of the data can lead to challenges in inference. Therefore, we propose a novel statistical inference method for a high-dimensional logistic regression model that accounts for heterogeneous subpopulations. Our primary goal is to investigate heterogeneity across subpopulations by testing the equivalence of the effect of a covariate and the significance of the overall effects of a covariate. To achieve overall sparsity of the coefficients and their fusions across subpopulations, we employ a fused group Lasso penalization method. In addition, we develop a statistical inference method that incorporates bias correction of the proposed penalized method. To address computational issues due to the nonlinear log-likelihood and the fused Lasso penalty, we propose a computationally efficient and fast algorithm by adapting the ideas of the proximal gradient method and the alternating direction method of multipliers (ADMM) to our settings. Furthermore, we develop non-asymptotic analyses for the proposed fused group Lasso and prove that the debiased test statistics admit chi-squared approximations even in the presence of high-dimensional variables. In simulations, the proposed test outperforms existing methods. The practical effectiveness of the proposed method is demonstrated by analyzing data from the Cancer Cell Line Encyclopedia (CCLE). These efforts have resulted in valuable cancer cell line (CCL) data resources, such as the Cancer Cell Line Encyclopedia (CCLE)2. CCLE data provide responses to 24 anticancer drugs in hundreds of cell lines across multiple tumor types, along with genomic information about these cell lines, such as the expression of nearly 20,000 genes. These data are often used to build computational models to predict drug response3. For example, to predict drug response using gene expression, researchers have used several methods, including Ridge regression4, mixture regression5, support vector machine6, random forest7, and neural networks8.Significant efforts have been made in recent research to perform screening genetic profiling and drug testing in human cancer cell lines to explore how genomic backgrounds influence response to therapy9. Different tumor types may be characterized by their own tumor-specific genes, and thus tumor type interactions may need to be considered. In addition, it is often observed that patients with different types of cancer have similar expression patterns of certain genes, but the efficacy of treatments varies. For example, HER2 overexpression has been observed in subsets of patients with cancers such as breast and gastric cancer, but a HER2 inhibitor, pertuzumab, is more effective in treating HER2-positive breast cancer compared to HER2-positive gastric cancer10. Therefore, interactions with tumor type may need to be considered for more accurate statistical analysis. Several studies have attempted to develop statistical modeling methods for drug response that account for heterogeneity between different cancer types12. However, a common approach is still to analyze several tumor types simultaneously13. This approach assumes that there are no interaction effects with tumor-specific factors. Such inconsistencies in the current literature raises a critical research question regarding the uniformity of the effects of gene expression on drug response across different cancer types. To date, this question remains unanswered. This paper aims to fill this gap by developing a statistical test for homogeneity in the effects of gene expression on drug response across multiple cancer types. This task is particularly challenging due to the high-dimensional nature of CCLE data. In a different context, heterogeneity testing between studies in meta-analyses has been investigated15.Given that recent large public CCL datasets, such as CCLE, include multiple cancer types, there is a need to develop statistical inference to assess the heterogeneity of the effects of gene expression on a drug across multiple tumor types. Quantifying the heterogeneity of the effects of genes across different tumors can provide valuable information in drug response modeling, as cancer tissue heterogeneity must be considered when modeling drug response across different cancer typesjth covariate for the gth subpopulation, and c can be specified, e.g. c can be unspecified. The null hypothesis with unspecified c indicates homogeneity, i.e., equal effects of the covariate. Testing  and p-dimensional vector of covariates, e.g. gene expression variables. We consider the following logistic regression modelgth group. Let jth covariate. We consider the high-dimensional setting where the number of covariates p increases with samples sizes Here we present a logistic regression model for heterogeneous subpopulations. Suppose the data come from lth covariate across the G groups, i.e., for lth covariate is significant to at least one of the groups:For a matrix notation, letWe propose to test  or 4) b b4) baseWe propose 32, we setLet (p)⊤. In , the twomentclass2pt{minimAssume that Conditions 1–2 in the Theorem S and Assume that the conditions of Theorem j is not selected in Theorem 19, let 19 considered only linear regression, while Van de Geer et al.17 examined a generalized linear model. Van de Geer et al.17 used nodewise regression33 to estimate standard errors for the debiased Lasso estimator. While we use the bias correction technique developed by Javanmard and Montanari19, our proposed method differs from existing methods. Specifically, we use the quadratic programming considered in Javanmard and Montanari19 to estimate 24, Tian and Feng34, Caner35). In addition, most of these existing methods were developed for statistical inference about a parameter in the classical single-population setting, and thus are Lasso to estimate parameters for a single population. In contrast, we develop tests based on our fused group Lasso for inference about parameters for multiple subpopulations.In this subsection, we develop the debiased version of the fused group Lasso estimator 17, Van de Geer et al.17 assumed that each row of 36. As demonstrated in Javanmard and Montanari19, we can achieve asymptotic normality of the proposed debiased estimator without assuming exact sparsity for To derive an asymptotic normality of the debiased LassoSuppose conditions of Theorem G degrees of freedom. The notion of weak convergence is not well defined in our setting, since we consider the case where G diverges with n. Theorem Let nt}H0 in  when \doSuppose conditions of Theorem The following Corollary implies (G)=0 in .Assume Conditions of Theorem nt}H0 in , for anyBased on this result, the corresponding chi-square test statistic for the hypothesis  is \docunt}H0 in  if \docuNext, we consider testing the homogeneity hypothesis in , which cAssume Conditions of Theorem Assume the conditions of Theorem nt}H0 in , for anyCorollary βj(G) in , it holdpothesis  is \docunt}H0 in  if \docutth update in the proximal gradient method. Then the 37. Note that 38. Let G by sth update in the ADMM to compute In this section we present the computational algorithm for solving . The algIn this section, we include details of the ADMM update presented at .Update for Update for j’s. For each Update for 17 separately for each subpopulationDL (Debiased Lasso) : Chi-squared test based on applying debiased Lasso17 separately for each subpopulationDL-B: Bonferroni correction using p-values obtained by applying debiased Lasso36 separately for each subpopulation, where this bias-correction method is developed for the scenario where the sample size is greater than the number of regressorsDL-E (Debiased Lasso based on the Exact inverse of the information matrix): Chi-squared test based on applying debiased Lasso36 separately for each subpopulationDL-E-B: Bonferroni correction using p-values obtained by applying debiased LassoWe also compared the proposed method with DR-B, Bonferroni correction based on debiased Ridge39. R code (https://github.com/luxia-bios/DebiasedLassoGLMs) was used to implement DL-E, and R package hdi40 was used to implement DL and DR-B. We also used R code (https://web.stanford.edu/~montanar/ssLasso/) to solve the quadratic programming  in testing the following hypotheses:g, i) from the p-dimensional multivariate normal distribution with mean p/4 identical blocks, where the In our simulation study, we fix m, p): We set First, we present simulation results when testing for homogeneity . Next, w41.We consider mentclass2pt{minimmentclass2pt{minimmentclass2pt{minim41, which investigated debiased group Lasso for linear regression. When sample sizes are set as In this subsection, we examine the performance of the proposed significance test. We consider 42. We also apply the BH procedure to p-values from DL and those from DL-E. We don’t consider methods using Bonefrroni-correction, i.e., DL-B, DL-E-B, and DR-B, for multiple testing. This is because they are too conservative, as observed in sections “We evaluate the empirical performance of the proposed testing procedures in the context of multiple testing. We consider the following two multiple testing problems,  and 16)16), respFWER: The percentage of the cases where Power: Average of the empirical power where When considering multiple testing for significance , we measity test  and DL-Eity test .Table 3Pp-dimensional multivariate normal distribution with mean S from 17 to Pooled-Lasso, which adapts to Lasso to analyze samples from different groups together, i.e., In this section, we perform a simulation analysis to assess the performance of DFGL when the sample sizes for the groups vary. Specifically, we consider the following model parameters: We consider We consider The observed higher power of DFGL in testing for significance and homogeneity compared to Lasso-based approaches is attributed to the fused group Lasso regularization. This regularization method allows for increasing sample sizes in regression parameter estimation by combining subpopulations, leading to more accurate statistical inference. These results suggest the effectiveness of our DFGL in analyzing data characterized by limited sample sizes for groups, such as the CCLE data.In this section, we consider the two multiple testing problems  and 16)16), respIn this subsection, we discuss the computational cost of computing the DFGL estimate and compare it with the costs of computing three estimates used in DL, DL-E, and DR-B. To compute Lasso or Ridge penalized estimate, we use R package glmnet. For the penalty parameters, we consider 100 candidates for In order to assess computational efficiency, we examine a simulation scenario with the AR(1) covariance matrix for the covariates, 43, and are also used to identify predictive biomarkers for anticancer drug sensitivity44. We consider three cancer types that include at least 30 cancer cell lines: Lymphoid, Lung, and Skin in our CCLE data analysis. Our main objective is to check whether a specific gene is significant to binary drug response in at least one of these cancer types, and whether such significant genes have heterogeneous effects on a drug across the cancer types. Analyses similar to ours could be useful in two ways. First, examining the significance of the effects of a gene across different cancer types may lead to the identification of potential gene expression markers of drug response that can be used for multiple cancer types. Such versatile gene expression markers are valuable in research16. Second, studying the heterogeneity of a gene’s effects across cancer types can provide insights into understanding differences in drug sensitivity across cancer types.In this section, we present real data analyses when our method is applied to the CCLE data. The CCLE data contains information on cancer treatment responses for 24 drugs on 504 cancer cell lines of 23 cancer types, where the transcription profile of each cell line is characterized by the measured expression levels of 19,177 probes. Cancer cell lines are widely used to understand cancer biology and test the efficacy of novel therapies45, we classify cancer cell lines into two categories for each drug. If a drug response value (IC50) is less than 0.5, then the cancer cell line is assigned to the “sensitive” category; otherwise, it is assigned to the “resistant” category. Then, most cancer cell lines are either sensitive or resistant to most drugs in some of the three cancer types. For example, all lymphoid cancer cell lines are resistant to Erlotinib, and only one of the lung cancer cell lines is sensitive to Panobinostat. After removing these imbalanced drugs in our analysis, we consider the following five drugs: 17-AAG, AZD6244, Irinotecan, PD-0325901, and Topotecan. Table Following Park et al.47. Similar to sure independence screening46, we removed relatively irrelevant genes to each drug, respectively, before fitting models. The gene screening procedure is as follows: We selected the top 3,000 genes with the largest sample variances.For We selected the top 300 genes with the smallest p-values for the significance test.These screening procedures have been used in the literature on high-dimensional regressions .The analysis of ultra-high-dimensional (UHD) data, such as CCLE data analysis, is accompanied by several challenges, including high collinearity, spurious correlation, noise accumulation, and a significant computational burden. To alleviate these difficulties inherent in UHD data, it is desirable to reduce the dimensionality of the feature space17, which do not incorporate a fusion penalty, can be considered to test for heterogeneity or significance of a gene’s effects. However, it is worth noting that approaches based on lasso penalization may yield inaccurate results when applied to data characterized by limited sample sizes, as in the case of our CCLE dataset. This limitation was demonstrated in our simulation study in section “While we analyze a set of First, we identify important genes using the proposed penalized estimation . Figure mentclass2pt{minim5.We also use DL, DL-B, and DR-B for comparison with DFGL. In the simulation in section “52, and ETV4 was also identified as a related gene to PD-0325901 in the CCLE data analysis performed by Liang et al.52. In addition, expression of ETV4, detected as a significant gene expression to PD-0325901, might modulate sensitivity to a MEK inhibitor trametinib53. Hayashi et al.54 discovered that activation of MEK was induced by ligation of SIRPWhen performing a sensitivity analysis based on Bootstrap in section entclass1pt{minima5, a penalized mixture regression suggested that the effects of SLFN11 are different between some of the clusters. These results suggest that our method may uncover underlying heterogeneous effects of a gene across cancer types that are difficult to capture using Lasso or Ridge-based methods. We observed that some genes identified by the homogeneity test have positive estimated coefficients in specific cancer types. For Topotecan, the estimated effect of gene MAOA is only positive in skin cancer cell lines. Low expression of MAOA has been observed in melanoma skin cancer compared with normal samples55, but high expression of MAOA has been observed in lung cancer tissues56 and lymphoma57.Our equivalence test shows different results for different cancer types, as summarized in Table 36 or Ridge39 in that it proves higher power, while it controls type I error quite well as shown in section “In this paper, we propose two different tests: (1) testing the homogeneity of the effects of the covariate across different groups and (2) testing the significance of the covariate over groups. We develop non-asymptotic analyses for the proposed fused group Lasso and prove that the debiased test statistics admit chi-squared approximations even in the presence of high dimensional variables. The proposed tests generally outperform the existing bias-correction methods based on Lasso59.From a methodological point of view, there are some extensions to our method. First, our tests can be applied to generalized linear models, including linear regression and the Poisson regression model, although we focus on logistic regression. In addition, our theoretical analyses can be extended to the generalized linear regression (GLM) setting. Second, we expect that the performance of the proposed tests can be improved by simultaneously estimating the inverse of the information matrices across subpopulations, as in the joint estimation of precision matrices60, other omics features such as DNA copy number are available in the analysis of CCL data. It is of great interest to investigate which omics data are most predictive of drug response in a specific cancer type, or have heterogeneous effects on drug response across cancer types. Given the different characteristics of different types of omics data, it is expected that our method may have some limitations in the analysis of multi-omics data. Therefore, a sophisticated extension of our method in estimation and construction of test statistics will be needed for the analysis of multi-omics data. In CCLE data, there are cell lines with missing responses to a drug. Therefore, an extension of our method to include cell lines with missing drug responses would be beneficial for the analysis of CCLE data.From the perspective of CCLE data analysis, there are several interesting directions for future research. Although our primary focus in CCLE data analysis was on the use of gene expression, which is known to be predictive of drug responseSupplementary Information.