A novel family of beta mixture models for the differential analysis of DNA methylation data: An application to prostate cancer

Koyel Majumdar; Romina Silva; Antoinette Sabrina Perry; Ronald William Watson; Andrea Rau; Florence Jaffrezic; Thomas Brendan Murphy; Isobel Claire Gormley

doi:10.1371/journal.pone.0314014

Abstract

Identifying differentially methylated cytosine-guanine dinucleotide (CpG) sites between benign and tumour samples can assist in understanding disease. However, differential analysis of bounded DNA methylation data often requires data transformation, reducing biological interpretability. To address this, a family of beta mixture models (BMMs) is proposed that (i) objectively infers methylation state thresholds and (ii) identifies differentially methylated CpG sites (DMCs) given untransformed, beta-valued methylation data. The BMMs achieve this through model-based clustering of CpG sites and by employing parameter constraints, facilitating application to different study settings. Inference proceeds via an expectation-maximisation algorithm, with an approximate maximization step providing tractability and computational feasibility. Performance of the BMMs is assessed through thorough simulation studies, and the BMMs are used for differential analyses of DNA methylation data from a prostate cancer study. Intuitive and biologically interpretable methylation state thresholds are inferred and DMCs are identified, including those related to genes such as GSTP1, RASSF1 and RARB, known for their role in prostate cancer development. Gene ontology analysis of the DMCs revealed significant enrichment in cancer-related pathways, demonstrating the utility of BMMs to reveal biologically relevant insights. An R package betaclust facilitates widespread use of BMMs.

Citation: Majumdar K, Silva R, Perry AS, Watson RW, Rau A, Jaffrezic F, et al. (2024) A novel family of beta mixture models for the differential analysis of DNA methylation data: An application to prostate cancer. PLoS ONE 19(12): e0314014. https://doi.org/10.1371/journal.pone.0314014

Editor: Nikos Kavallaris, Karlstad University: Karlstads Universitet, SWEDEN

Received: March 19, 2024; Accepted: November 4, 2024; Published: December 11, 2024

Copyright: © 2024 Majumdar et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

Data Availability: All data and code are available via the R package called betaclust which is freely available at https://github.com/koyelucd/betaclust and on CRAN at https://cran.r-project.org/web/packages/betaclust/index.html.

Funding: KM was funded with the financial support of Science Foundation Ireland (www.sfi.ie) under Grant number 18/CRT/6049. The funder played no role in the study design, data collection and analysis, decision to publish, or preparation of the manuscript.

Competing interests: The authors have declared that no competing interests exist.

Introduction

Epigenetics is the study of heritable changes in gene activity that do not involve explicit changes to the DNA sequence [1]. DNA methylation is an epigenetic process where a methyl group is added to or removed from the 5′ carbon of the cytosine ring [2]. This process assists in regulating gene expression and is essential for the development of an organism, but irregular changes in DNA methylation patterns can lead to damaging health effects [3]. A cytosine-guanine dinucleotide (CpG) site is hypomethylated if neither of the DNA strands in a diploid individual are methylated, hemimethylated if either of the DNA strands are methylated or hypermethylated if both the strands are methylated. A differentially methylated CpG site (DMC) is a CpG site that has different methylation states between DNA samples collected from different biological conditions, which may have been taken from tissues of an individual over time, different tissues from the same individuals or distinct individuals.

The DNA methylation process has been extensively studied in the context of cancer, and its treatment [4]. CpG islands that remain unmethylated in normal cells can become methylated in abnormal cells such as cancer cells [5], and it has been shown that tumour suppressor genes are silenced by hypermethylation of their promoter regions [6, 7]. For example, in prostate cancer, the fifth major cause of cancer-related mortality globally [8], hypermethylation of certain tumour suppressor genes, such as GSTP1, RARB, APC and RASSF1, has been observed during the early stages of the disease [9–11]. A better understanding of disease can therefore be achieved by identifying regions that are differentially methylated between benign and tumour samples.

The Illumina MethylationEPIC BeadChip microarray [12] is used to interrogate over 850,000 CpG sites and retrieve methylation profiling of the CpG sites in the human genome. The Illumina microarray produces a sample of methylated (Methylated) and unmethylated (Unmethylated) light signal intensities, and the level of methylation, or the beta value, is beta = max(Methylated)/(max(Methylated) + max(Unmethylated) + χ), where χ is a constant offset added for regularisation in case of very low Methylated and Unmethylated values [13]. The methylation level at a CpG site is quantified by this beta value and is constrained to lie between 0 and 1. The beta values are continuous with a value close to 1 suggesting that a site is hypermethylated, while values close to 0 represent hypomethylation. The two probe intensities are assumed to be gamma-distributed as they can take only positive values, and their ratio results in beta distributed variables. Thus, the beta values can be modelled using a beta distribution.

The beta values in general have higher variance in the center of the [0,1) interval than towards its endpoints. This leads to heteroscedasticity, which imposes challenges for analyses as assumptions for the ubiquitous Gaussian models are violated. Hence, beta values are usually converted to M-values using a logit transformation as these values are statistically more convenient; Gaussian models can be used as the transformed data lie within (−∞, ∞) [13]. However, such transformations make inference less biologically interpretable and hence there is a need to model the beta values in their innate form.

In many methylation array studies, thresholds of beta values are subjectively selected to identify the three methylation states. For instance, [14] deemed a CpG site to be hypomethylated if its beta value was <0.2 and hypermethylated if its beta value was >0.8, while [15] employed 0.3 and 0.7 as thresholds. Such subjective selection of thresholds may increase the likelihood of false positives and negatives, leading to incorrect inference and necessitating an objective approach to determining methylation state thresholds.

Mixture models for transformed beta values have been proposed in several studies to find biologically meaningful clusters. For instance, [16, 17] use mixture models to model a subset of CpG sites and cluster samples into latent groups of biologically related samples. Additionally, methods such as the variational Bayes beta mixture model [18] and the Dirichlet process beta mixture model [19] analyse untransformed beta values; the former addresses the feature selection problem in the context of DNA methylation data, whereas the latter models the beta values to identify DNA methylation subgroups. The Methylmix [20] R [21] package uses a univariate beta mixture model to uncover patient subgroups with similar DNA methylation levels for a specific CpG site, with Wilcoxon rank sum tests used to establish hypermethylated and hypomethylated genes. Beta mixture models have also been proposed for intra-array quantile normalization [22] and for clustering individual DNA samples into the three methylation states which can then be used to classify cancer tissue type [23]. The use of a beta mixture model has been extended for classifying the methylation states of CpG sites; the approach accounts for boundary values and employs a method-of-moments approach to inference, but considers only small numbers of CpG sites [24]. While mixture models for DNA methylation data have been used for a range of purposes, they have not been utilised to uncover differential methylation across the genome using untransformed beta values.

Several methods have been developed for detecting DMCs in different DNA sample types. For instance, the PanDM method [25] leverages joint modeling to perform methylation site clustering, differential methylation detection, and pan-cancer pattern discovery by modelling the transformed p-values associated with each CpG site for a given cancer type. A principal component analysis and tensor decomposition approach involving unsupervised feature selection [26] was proposed where principal component scores were associated with each CpG site and used to identify DMCs. Another approach, termed FastDMA [27], employs an analysis of covariance to perform both single probe analysis and differentially methylated region scanning while modelling the M-values. The popular limma method [28] identifies DMCs by modelling the M-values using an empirical Bayesian approach. Other studies identify the DMCs by modelling the beta values via multiple moderated t-tests or Wilcoxon rank-sum tests [29, 30]. Additionally, a multiple hypothesis testing approach, combined with multivariate permutation tests, has been proposed to detect group differences in epigenetic data [31], as has a nonparametric test to identify DMCs between multiple treatments [32]; while this approach can analyze smaller arrays with e.g., 28,000 CpG sites, it is computationally intensive for modern, larger-scale arrays. These approaches to DMC identification use subjective thresholds, transformed values, moderated t-tests and/or nonparameteric methods [33]. Crucially, such approaches lack biological interpretability and often face reproducibility and computational scalability challenges when considering data from different studies, of the scale resulting from current microarray technologies.

Several mixture models for bounded data are available. For instance, in the context of semiparametric density estimation, [34] fit a Gaussian mixture model to range-power transformed bounded data, from which the density for the original data is obtained. Mixture models of bounded Laplace distributions also allow for modelling bounded data by truncation of the Laplace distribution, but are computationally expensive for large datasets [35]. Similarly, bounded support asymmetric generalized Gaussian mixture models are adaptable to different distributional shapes but can be computationally expensive as inference requires numerical optimisation [36]. A beta mixture model is an appropriate choice for bounded DNA methylation data: the support of the beta distribution is congruent with the beta values, its flexibility allows for skew and symmetric distributional shapes, and it is computationally feasible to work with given its parsimony. Importantly, the beta distribution parameters provide relevant biological interpretations enabling biologically intuitive and meaningful inference.

Here we propose a family of beta mixture models (BMMs), which address specific research questions arising in the context of differential analysis of DNA methylation data, by introducing a range of constraints on the parameters of a BMM. The resulting novel family of BMMs facilitates a model-based approach to clustering CpG sites given their innate beta-valued methylation data to (i) objectively identify methylation state thresholds and (ii) identify DMCs between different sample types. The BMMs are capable of clustering the entire microarray of CpG sites, from DNA samples collected from multiple tissues from each of several patients, in a computationally efficient manner. Performance is assessed through simulation studies, and the BMMs are used to analyse a motivating prostate cancer (PCa) dataset. The capability of the BMMs is demonstrated to appositely model the beta values, to objectively identify thresholds and to identify existing and novel DMCs, including those related to genes implicated in prostate cancer, such as GSTP1, RARB and RASSF1. An R package, betaclust, freely available on github and CRAN, facilitates widespread use of the BMMs.

Methods

Prostate cancer data

A prostate cancer study [37], which involved collection of DNA methylation samples from four patients with metastatic prostate cancer disease, motivated the development of the BMMs. Tissue samples from matched biopsy cores (tumour and histologically matched normal—herein benign) were collected from each patient, and DNA was extracted from the samples. Methylation profiling of the DNA samples was conducted using the Infinium MethylationEPIC Beadchip [38]. The raw DNA methylation data are freely available for download on the Gene Expression Omnibus (GEO) repository (GSE119260); datasets GSM3362390-GSM3362397 were analysed here and were accessed on 26^th of January, 2021 for research purposes. The authors had no access to information that could identify individual participants.

Observed beta values for each of 694,923 CpG sites for the two DNA sample types were collected from each of the four patients. Raw methylation array data was quality controlled and pre-processed as in [37], where the data were normalized, and probes overlapping with SNPs, probes with the highest fraction of unreliable measurements, probes lying outside of CpG sites and those on the sex chromosome were removed. The resulting dataset had 103 CpG sites (< 0.014% of the total number of CpG sites) with missing beta values. While imputation techniques exist for DNA methylation data, missing values were not imputed here due to their very low percentage, and the high uncertainty associated with imputed values in diseased samples due to their heterogeneity [39]; here the 103 CpG sites with missing data were therefore removed. No observed beta values were equal to 0. The resulting dataset contained beta values for C = 694, 820 CpG sites from each of R = 2 DNA sample types collected from each of N = 4 patients. Here, these data are appositely modeled in their innate beta form to (i) objectively identify methylation state thresholds and (ii) uncover DMCs between two sample types using a model-based clustering approach.

A beta distribution

The beta distribution has support on [0, 1] and is parameterized by two positive shape parameters, α and δ. Given the properties of the beta values, the beta distribution is used here to appositely model the methylation level x_cnr of the c^th CpG site (c = 1, …, C), from the n^th patient (n = 1, …, N), from their r^th DNA sample type (r = 1, …, R) i.e., for 0 ≤ x_cnr ≤ 1, where B(α, δ) = (Γ(α)Γ(δ))/Γ(α + δ), where Γ(⋅) is the gamma function. The DNA methylation data are collected in the C × NR dimensional dataset X where each of the NR columns contains the methylation levels of the C CpG sites in one of the R sample types from each of the N patients.

A beta mixture model

A mixture model assumes the observed data have been generated from a heterogeneous population composed of K groups or clusters. In the context of DNA methylation data, there are G = 3 possible methylation states: hypomethylation, hemimethylation or hypermethylation. Hence, when analysing methylation data from a single DNA sample type (i.e., where R = 1) each CpG site exhibits one of K = G^R = 3 methylation states characterised by each of the K clusters. Here, interest lies in objectively inferring thresholds between these K = 3 methylation states.

When analysing methylation data across multiple (i.e., R > 1) DNA sample types to identify DMCs, each CpG site will exhibit one of a possible K = G^R combinations of methylation states, here characterised by each of K clusters in a mixture model. For example, given the three methylation states and considering a CpG site across R = 2 sample types (e.g., across benign and tumour samples), the CpG site can potentially exhibit any of the K = 3² = 9 combinations of these three states (for example hypermethylated in both samples, hypermethylated in one sample and hypomethylated in the other, etc). Therefore, in this scenario, K = 3² = 9 with each cluster characterising one of the possible methylation state combinations.

We propose a beta mixture model for the methylation data for all cases R ≥ 1. Here θ is used to denote the shape parameters in a beta mixture model, i.e., θ = (α₁, δ₁, …, α_K, δ_K), where α_k and δ_k are the shape parameters of cluster k. The shape parameters are allowed to vary among the clusters, patients and sample types. The mixing proportions τ = (τ₁, …,τ_K) lie between 0 and 1, , and denote the probability of belonging to cluster k ∀k = 1, …, K. Independence is assumed across patients and samples, given a CpG site’s cluster membership, leading to the probability density function for such a beta mixture model (BMM): (1)

Computation of maximum likelihood estimates (MLEs) of τ and θ from the associated log likelihood function is complex, and an incomplete data approach is therefore used here. The latent vector z_c = (z_c1, …, z_cK) is introduced for each CpG site c, where z_ck is 1 if CpG site c belongs to the k^th group and 0 otherwise. The C × K matrix Z is combined with the beta values to form the complete data (X, Z). The complete data log-likelihood function is (2)

The complete data log-likelihood function (2) can be used to find the MLEs and using the expectation-maximisation (EM) algorithm [40]; on convergence a probabilistic clustering solution is also available from the expected value of z_ck, the posterior probability of CpG site c belonging to cluster k.

A family of BMMs.

The most generalised BMM is defined in (1) which models the CpG sites as belonging to K latent groups. By introducing a variety of constraints on the parameters of this generalised BMM, a family of three beta mixture models is proposed. Each model serves a specific purpose e.g., to cluster the CpG sites into the 3 methylation states allowing objective inference of methylation state thresholds, or to facilitate the identification of DMCs between different sample types.

The K⋅⋅ model.

The K⋅⋅ model facilitates objective inference of thresholds between methylation states by clustering C CpG sites into one of K = G = 3 methylation states, based on a single sample type (R = 1) from each of N patients. Under the K⋅⋅ model the shape parameters of each cluster are constrained to be equal for each patient, but allowed to vary across clusters. The complete data log-likelihood function is therefore

The KN⋅ model.

The KN⋅ model facilitates objective inference of methylation state thresholds by clustering each of the C CpG sites into one of K = G = 3 methylation states, based on data from a single sample type (R = 1) from each of N patients. While the KN⋅ model has a similar purpose to the K⋅⋅ model, it differs in that it is less parsimonious as it allows cluster and patient-specific shape parameters. The complete data log-likelihood function is therefore

The K⋅R model.

The K⋅R model facilitates identification of differentially methylated CpG sites between R > 1 DNA sample types collected from each of N patients. The K⋅R model assumes conditional independence between CpG sites from paired samples from the same patient, given the CpG sites’ cluster membership. The K⋅R model also assumes each of the K clusters characterises a different combination of the G methylation states across the R biological conditions where K = G^R = 9 here. Under the K⋅R model the shape parameters are allowed to vary for each sample type and for different clusters but are constrained to be equal for each patient. The complete data log-likelihood function for the K⋅R model is therefore

This family of beta mixture models enables the objective inference of methylation state thresholds (via the K⋅⋅ and/or KN⋅ models) and the identification of DMCs between R DNA sample types (via the K⋅R model), as illustrated in the simulation studies and applications that follow.

Parameter estimation

The parameters of the BMMs are estimated and the cluster membership for each CpG site inferred using the EM algorithm. Here, we delineate this for the generalised BMM (1). Derivations for the K⋅⋅, KN⋅ and K⋅R models are detailed in Appendices S1–S3 in S1 File.

The EM algorithm consists of two steps: in the expectation step the expected value of the complete data log-likelihood function is obtained, conditional on the observed data and current parameter estimates. The maximisation step maximises the expected complete data log-likelihood with respect to the parameters. To obtain and , the expectation and maximisation steps are iterated until convergence to at least a local optimum of the log-likelihood function.

An initial clustering of CpG sites is obtained using k-means clustering and the method of moments is used to calculate initial values of τ and θ. The two steps proceed as follows:

Expectation-step: the expected value of z_ck is calculated, i.e., the posterior probability of CpG site c belonging to cluster k, conditional on current parameter estimates. At iteration t + 1
Maximisation-step: estimates of the parameters τ and θ are calculated by maximising the expected complete data log-likelihood function, given the values from the E-step.

For the maximisation-step, the expected complete data log-likelihood function is maximised by differentiating it w.r.t the parameters. Closed form solutions for the mixing proportions are available as For the shape parameters, the expected complete data log-likelihood function to be maximized is (3)

Differentiating (3) w.r.t α_knr yields (4) where ψ is the logarithmic derivative of the gamma function known as the digamma function, ψ(α_knr) = ∂logΓ(α_knr)/∂α_knr. Similarly, the derivative of w.r.t δ_knr is (5)

Closed form solutions for and are not available due to the presence of the digamma function. To obtain the MLEs, numerical optimisation algorithms such as BFGS [41] and BHHH [42] could be used. However, for the large datasets considered here, use of these algorithms proved to be computationally infeasible.

A digamma approximation.

Here an approximation to the digamma function is used to allow for closed form solutions for the shape parameters. The lower bound for the digamma function for all y > 1/2 is ψ(y) > log(y − 1/2) [43]. Given the context, we assume that the beta distributions in the family of BMMs are unimodal and bounded, meaning the shape parameters are >1. Thus, the lower bound approximation holds and was empirically observed to be a very close approximation of the digamma function. The lower bound is used in (4) and (5) to give (6) and (7)

Equating Eqs (6) and (7) to zero, we get closed-form, approximate estimates as and where and .

Utilising the digamma function approximation brings notable computational gains with run times, for example, reducing from 65 hours (when using numerical optimisation at the maximisation step) to 15 minutes (when using the digamma approximation) when analysing the PCa data on a computer equipped with an Intel Core i7 CPU with 2.70GHz speed, 6 physical cores and 16 GB of RAM.

Inferring methylation state thresholds

To objectively infer thresholds between methylation states, without loss of generality, we denote by clusters 1 and 2 the clusters representing hypomethylated and hypermethylated CpG sites respectively. The ratio of fitted density estimates ω_j for cluster j = 1, 2 is

The threshold separating e.g., the hypomethylated and hemimethylated clusters is calculated as the minimum beta value at which ω₁ ≥ 1. Similarly, the threshold dividing the hemimethylated and hypermethylated clusters is the maximum beta value at which ω₂ ≥ 1.

In the K⋅⋅ model, as the shape parameters are constrained to be equal for each patient, a single set of thresholds is calculated for all patients. In the KN⋅ model, the shape parameters vary for each patient, so a set of thresholds is calculated for each patient. Unless one model is appropriate given the question of interest, to choose the optimal model between the K⋅⋅ and KN⋅ models, here the well utilised model selection tools of the Akaike information criterion (AIC) [44], Bayesian information criterion (BIC) [45] and the integrated complete log-likelihood criterion (ICL) [46] are examined.

Identifying the most differentially methylated clusters

While the K⋅R model clusters the CpG sites into K clusters, subsequent quantification of the degree of differential methylation of CpG sites in each cluster is required. Here, this is quantified by comparing the R = 2 beta distributions associated with the two sample types in each cluster (e.g., benign and tumour) using the area under the curve (AUC) of the receiver operating characteristic (ROC) curve as a measure of separability, which has an advantage of being widely used in biology.

The ROC curve for a cluster is generated by drawing 1000 samples from each of the two fitted beta distributions, corresponding to two sample types, after convergence of the EM algorithm. By varying threshold values from 0 to 1, the sampled beta values are used to compute sensitivity and specificity for discriminating between sample types, which are in turn used to construct the ROC curve and associated AUC. Higher AUC values indicate greater separation between the fitted beta distributions for two sample types within a cluster, and therefore that the CpG sites within that cluster are more differentially expressed between the two. In the case where R > 2, the beta distributions linked to each sample type within a cluster are compared to one another, and the maximum AUC across pairs of sample types is selected as the cluster’s AUC. A large AUC would then indicate distinct methylation patterns between at least two sample types. We also consider the Wasserstein distance (WD) [47], which computes the disparity between cumulative distributions, as an additional approach to quantifying the degree of differential methylation of CpG sites between sample types in each cluster.

A process flow diagram that summarises the overall approach to inferring subjective thresholds and identifying DMCs using the proposed BMMs is given in Fig 1.

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Fig 1. Process flow diagram for the differential analysis of beta-valued DNA methylation data using a novel family of beta mixture models.

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Results

Simulated data results

Simulated data.

One hundred simulated datasets consisting of methylation values for C = 600, 000 CpG sites are generated using R [21]. Each simulated dataset consists of Beta values from two biological sample types (sample A and sample B) from each of N = 4 patients. Hypomethylated CpG sites were generated from a Beta(2,20) distribution, with hemi- and hyper- values generated from Beta(4,3) and Beta(20,2) distributions respectively. To emulate the noisy data observed in real settings, zero-centred Gaussian noise with standard deviation 0.01 was added to the beta-generated data; resulting values outside [0, 1] were replaced by the closest minimum or maximum value from the beta-generated data. Reflecting typical behaviour in DNA methylation data, 35%, 35% and 30% of CpG sites in a single sample were simulated as hypomethylated, hemimethylated and hypermethylated, respectively. This resulted in, on average, 64% of the CpG sites being differentially methylated between the two sample types. Of these DMCs, on average, 30% were hypomethylated in one sample and hypermethylated in the other, or vice versa; such highly differentially methylated CpG sites are of prime interest. Fig 2 illustrates a single simulated data set, with clusters of CpG sites ordered from most to least differentially methylated between samples, according to AUC.

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Fig 2. Violin plots of a simulated dataset.

Each panel illustrates the simulated beta-distributed values in samples A and B. The proportion of CpG sites in each cluster is detailed in the panel title. Clusters are ordered numerically from most to least differentially methylated, according to AUC.

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Estimating methylation state thresholds.

To cluster the CpG sites in a sample type into 3 clusters representing the 3 methylation states and to infer the thresholds between these states, the K⋅⋅ and KN⋅ models were fitted to the data from sample type A in each of the 100 simulated datasets. The true K⋅⋅ generating model was selected by AIC, BIC and ICL to be optimal in each case. Fig 3 illustrates the density estimates under the clustering solution of the K⋅⋅ model for a single simulated dataset. The hemimethylated CpG sites are clustered in cluster 1, while the hypomethylated and hypermethylated CpG sites are in clusters 2 and 3 respectively. The estimated mixing proportion of CpG sites for each cluster (see Fig 3) are notably similar to the true mixing proportions. As parameters are constrained to be equal for each patient in the K⋅⋅ model, a single set of thresholds is inferred for all 4 patients. The threshold (see Fig 3) of 0.258 indicates that any CpG site with a lower beta value is likely to be hypomethylated. Similarly any CpG site with a beta value greater than the second threshold of 0.802 is likely to be hypermethylated. These objectively inferred thresholds are very close to the true thresholds of 0.244 and 0.808.

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Fig 3. Fitted density estimates under the K⋅⋅ model on a simulated dataset from sample type A.

The thresholds between methylation states are illustrated by black dotted lines. The estimated mixing proportions are also displayed.

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The adjusted Rand index (ARI) [48] gives a measure of agreement between two clustering solutions, where an ARI of 1 indicates full agreement. The mean ARI across the 100 simulated datasets for the K⋅⋅ model was 0.9949 (s.d. 0.0002) and for the KN⋅ model was 0.9949 (s.d. 0.0002), demonstrating accurate and stable clustering solutions. The mean ARI when comparing the K⋅⋅ and KN⋅ clustering solutions was 0.999 (s.d. 0.00001). A summary of the parameter estimates and kernel density plots under the K⋅⋅ model are available in Appendices S4–S5 in S1 File.

Identifying DMCs.

To identify differentially methylated CpG sites between multiple DNA sample types in the simulated data, the K⋅R model is fitted to each of the C × NR dimensional datasets. For each CpG site, as there are R = 2 sample types, G^R = 9 different combinations of the three methylation states are possible across sample types A and B. The CpG sites that are e.g., hypomethylated in one sample type and hypermethylated in the other are of interest as they indicate potential epigenetic changes in the genome.

Under the K⋅R model, for each simulated data set, the AIC, BIC and ICL criteria were non-informative as they consistently decreased for K = 2, …, 30 (see Appendix S6 in S1 File). Thus, the biologically motivated K⋅R model with K = G^R = 9 was considered here. The AUC and WD metrics were employed to assess the similarity between the R = 2 probability distributions within each cluster, giving insight to the degree of differential methylation within clusters. Table 1 shows the mean and standard deviation of the AUC and WD values for each cluster across the 100 simulated datasets. Throughout, clusters are presented in descending order of their degree of differential methylation, based on decreasing AUC and, in the case of ties, WD values. The six differentially methylated clusters i.e., those in which the methylation state was different between the two sample types, are correctly highlighted as the most differentially methylated.

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Table 1. Mean and standard deviation (s.d.) of the AUC and WD metrics for each cluster across the 100 simulated datasets.

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The graph in Fig 4 shows the fitted density estimates of the clustering solution under the K⋅R model for a single simulated dataset. The associated dissimilarity metrics correctly indicate clusters 1–6 as the most differentially methylated clusters, with 65.1% of CpG sites belonging to these clusters, which is very close to the true mixing proportions. The density estimates show that, for example, cluster 1 captures DMCs which are hypomethylated in sample type A and hypermethylated in sample type B while cluster 2 contains DMCs which are hypermethylated in sample type A and hypomethylated in sample type B. Mean standard performance metrics across the 100 simulated datasets signify an accurate and stable clustering process, with a mean false discovery rate (FDR) of 0.0041 (s.d. 0.0121), mean sensitivity of 0.9742 (s.d. 0.0563), mean specificity of 0.9921 (s.d. 0.0244) and mean ARI of 0.9758 (s.d. 0.0370). A summary of parameter estimates under the K⋅R model is available in Appendix S4 in S1 File with kernel density estimates in Appendix S7 in S1 File.

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Fig 4. Fitted density estimates under the K⋅R model on a simulated dataset.

The estimated mixing proportions are displayed in the relevant panel.

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All computations were conducted using R [21] on a Windows 11 operating system equipped with an Intel Core i7 CPU with 2.70GHz speed and 16GB RAM. In terms of computational cost, for example, fitting the K⋅R model to a single simulated dataset took 55.86 seconds on a computer with 6 cores. To explore the impact of an increasing value of N on the computational cost, further simulation studies in which N = {8, …, 60}, where N increased in increments of 4, demonstrated a linear increase in computational cost. This is intuitive given the form of the model’s likelihood function with respect to N. Further details on computational cost are provided, for all three BMM models, in Appendix S8 in S1 File.

The ability of the K⋅R model to detect DMCs was compared with that of the state-of-the-art limma method [28], which requires the beta values to be transformed into M-values for analysis. Results indicate limma also performs well but with lower accuracy than the BMM approach: limma had a mean FDR of 0.0080 (s.d. 0.0001), mean sensitivity of 0.9074 (s.d. 0.0008), mean specificity of 0.9864 (s.d. 0.0003) and mean ARI of 0.7562 (s.d. 0.0018). The lower mean sensitivity value in particular suggests that limma identified fewer DMCs than the BMM. Boxplots displaying the performance metrics across the 100 simulations for the K⋅R model and limma are available in Appendix S9 in S1 File.

Finally, to explore the robustness of the BMM approach to model misspecification, the same simulation settings were considered but where the data were simulated from a t-distribution with 8 degrees of freedom. Both the K⋅R model and limma demonstrated mixed ability to detect DMCs when applied to the expit-transformed and logit-transformed data respectively. The BMM and limma approaches attained, respectively, a mean FDR of 0.4918 (s.d. 0.076) and 0.7018 (s.d. 0.0006), a mean sensitivity of 0.4558 (s.d. 0.0908) and 0.9995 (s.d. 0.0002) and a mean specificity of 0.875 (s.d. 0.0203) and 0.3279 (s.d. 0.0019). The BMM’s low sensitivity and high specificity suggests that the BMM identified the non-DMCs correctly but failed to detect a large portion of true DMCs, while limma demonstrated the opposite ability. Boxplots of the performance metrics are provided in Appendix S10 in S1 File.

Prostate cancer data results

Estimating methylation state thresholds.

For the PCa data, to cluster the CpG sites into the 3 methylation states and objectively infer the methylation state thresholds in the benign and tumour sample types, the K⋅⋅ and KN⋅ models were fitted. The AIC, BIC and ICL suggest the KN⋅ model as optimal for both the benign and tumour sample types. This is intuitive, particularly for the tumour sample types where the degree of disease varies for each patient, as the KN⋅ model allows for patient specific shape parameters.

The fitted density estimates and inferred thresholds for patient 1 are discussed here; those for patients 2, 3 and 4 are available in Appendix S11 in S1 File.

In the benign sample type from patient 1, the estimated mixing proportions were 0.244 for hypomethylation, 0.363 for hemimethylation, and 0.393 for hypermethylation. The inferred methylation state thresholds are 0.258 and 0.747 for the benign sample type, and 0.19 and 0.751 for the tumour sample type. The hypermethylation state thresholds in the benign and tumour sample types are very close; in contrast, the hypomethylation state thresholds are quite different. While these objective thresholds are close to the subjective values suggested in the literature of 0.2 and 0.8, the difference results in more hypo- and hypermethylated CpG sites being identified by the BMM as DMCs.

Patient 1 was known to have a greater degree of disease severity than the other patients with a matched normal DNA methylation profile more tumour-like than benign. For patient 1, the hypermethylation threshold (0.747) was lower than that for the other patients (0.774, 0.766 and 0.814 for patients 2, 3 and 4 respectively) suggesting more hypermethylated CpG sites in patient 1’s benign sample type than in the other patients’ samples. A similar pattern was observed in the methylation thresholds inferred from the patients’ tumour sample types (see Appendix S11 in S1 File) in that the threshold was lower for patient 1.

The ARIs of 0.94 and 0.96 between the KN⋅ and K⋅⋅ solutions for the benign and tumour sample types respectively indicate good clustering agreement. Summaries of parameter estimates and the kernel density estimates under the KN⋅ model are available in Appendices S4 and S12 in S1 File.

Identifying DMCs in the PCa data.

To identify differentially methylated CpG sites in the PCa data, the K⋅R model was fitted to the C × NR dimensional dataset. Similar to the simulation study, the AIC, BIC and ICL were non-informative and consistently decreased across models with K = 2, …, 30 (see Appendix S13 in S1 File). Thus, motivated by the G^R = 9 unique methylation state combinations that could be biologically present across the benign and tumor sample types, a model with K = G^R = 9 was fitted to the PCa data.

Fig 5 illustrates fitted density estimates of the clustering solution, ordered by AUC or WD in the case of equal AUC values. Kernel density estimates are in Appendix S14 in S1 File. Table 2 summarises the parameter estimates and details the AUC and WD metrics for each cluster. As the extent of disease progression varied across patients, the PCa data were noisier than the simulated data, and the AUC and WD metrics were lower in general. Both the AUC and WD metrics suggest that clusters 1 and 2 contain the CpG sites that are most differentially methylated in nature. Inspection of the density and parameter estimates of clusters 1 and 2 provides insight: cluster 1 captures CpG sites exhibiting a downward trend in methylation values for tumor samples with increased methylation values in the benign samples. On the other hand, CpG sites in cluster 2 tend to have higher methylation levels in tumor samples than in benign samples. While there are visual differences between the benign and tumor density estimates in clusters 3–5, the density estimates in later clusters are almost visually indistinguishable between the two samples, particularly in the case of clusters 6, 7 and 9. This is intuitive as clusters with smaller AUC (and WD) values contain the least differentially methylated CpG sites between the benign and tumour samples and their respective density estimates within such clusters will be very similar. Fig 6 shows the empirical cumulative distribution functions (ECDFs) for the DMCs within clusters 1 and 2, for both benign and tumor sample types. The ECDF for cluster 1 shows an increase in beta values within the benign samples, relative to the tumor samples while the ECDF of the CpG sites in cluster 2 indicates an elevation in beta values in the tumor samples, compared to the benign samples. The K⋅R model identifies 102,757 CpG sites, belonging to clusters 1 and 2, as being mostly differentially methylated.

Download:

Fig 5. Fitted density estimates under the clustering solution of the K⋅R model.

The model estimates parameters for K = 9 clusters for the DNA methylation data from benign and tumour prostate sample types. The estimated mixing proportions are displayed in the relevant panel.

https://doi.org/10.1371/journal.pone.0314014.g005

Download:

Fig 6. Empirical cumulative distribution functions for DMCs.

Empirical cumulative distribution functions for DMCs in (A) cluster 1 and (B) cluster 2 for all patients and sample types.

https://doi.org/10.1371/journal.pone.0314014.g006

Download:

Table 2. Beta distributions’ parameter estimates for the benign and tumour samples, and the AUC and WD metrics, for the PCa data under the K⋅R model.

https://doi.org/10.1371/journal.pone.0314014.t002

On performing gene ontology analysis [49], CpG sites in cluster 2 were found to be related to known genes e.g., RARB, GSTP1, RASSF1, SFRP2, which are implicated in prostate cancer. For example, hypermethylation of RARB promoter genes is a significant biomarker in diagnosing prostate cancer [50]. The methylation levels of the DMCs in cluster 2 that belong to the RARB genes suggest the median beta value is higher in the tumour sample type than in the benign sample type for all patients (see Appendix S15 in S1 File). Through non-parametric tests, the beta values were shown to be significantly higher in the tumour samples than in the benign samples for the CpG sites related to these genes (p < 0.05). Further, the ECDF for DMCs related to the RARB genes for benign and tumour sample types illustrated that the DMCs have increased beta values in the tumour samples compared to the benign samples (see Appendix S16 in S1 File). Analysis of genes linked to CpG sites in clusters 3–9, the less differentially methylated clusters, did identify some genes previously implicated in prostate cancer e.g., AKT1 [51]. However, non-parametric tests also suggested no statistically significant difference (p > 0.05) in beta values between benign and tumour samples for CpG sites in clusters 3–9 linked to the gene AKT1, as did associated box and ECDF plots (see Appendix S17 in S1 File).

Gene ontology analysis of the DMCs in cluster 1 also unveiled approximately 16 noteworthy biological processes. These processes, distinct from cancer-related pathways, encompass vital functions such as nervous system processes. A substantial count of 1001 significant biological processes were revealed among the DMCs in cluster 2 (FDR-adjusted p-value <0.05). Further, considering the KEGG pathways, the DMCs in cluster 1 were associated with one significant pathway, while the DMCs within cluster 2 exhibited involvement in a noteworthy 61 significant pathways. Of these significant pathways, many were cancer related e.g., the proteoglycans in cancer pathway was the second most enriched pathway.

Given the BMM’s model-based approach to clustering, the uncertainty in CpG site c’s clustering is available as , with a maximum possible uncertainty of 1 − 1/K = 8/9. All CpG sites have clustering uncertainties well below this maximum, demonstrating that the CpG sites are clustered with high certainty (see Appendix S18 in S1 File).

To demonstrate the general applicability of the approach, the BMMs were also fitted to a publicly available DNA methylation dataset from an esophageal squamous cell carcinoma study (ESCC); full details are available in Appendix S19 in S2 File.

Discussion

DNA methylation is widely studied for disease diagnosis and treatment. Technology advancements have led to the development of microarrays that can assay e.g., 850,000 CpG sites from a DNA sample [12], but the analysis of these large arrays has been limited by a lack of appropriate statistical methods for the bounded and heteroskedastic nature of beta-valued DNA methylation data. The methylation states of CpG sites are often of interest and are typically identified using thresholds which are defined in the literature based on intuition [14] rather than using an objective approach. Additionally, to detect DMCs, it is common practice to apply a logit transformation to beta values, and subsequently model them as Gaussian-distributed [13, 28]. Alternatively, comparisons between untransformed methylation levels among sample types are often conducted using multiple moderated t-tests or Wilcoxon rank sum tests [29, 30]. The approach proposed here advocates against transforming the data and instead proposes modelling the data in its innate form when inferring methylation state thresholds and DMCs.

In the context of prostate cancer, a family of beta mixture models is proposed which employs novel constraints on the model parameters to cluster CpG sites based on untransformed beta values to objectively identify methylation state thresholds and DMCs between benign and tumour samples. The BMMs use a model-based clustering approach and inference is computationally efficient through the use of a digamma approximation. The objective inference of methylation thresholds demonstrated that the thresholds of 0.2 and 0.8 or 0.3 and 0.7 defined in literature are not appropriate for every scenario. The thresholds inferred from each patient’s data showed variability, reflecting the different stages of disease among patients. The proposed K⋅R model clusters CpG sites from multiple DNA sample types to determine the CpG sites with differential methylation. Gene ontology enrichment analysis of the genes associated with CpG sites in the most differentially methylated clusters revealed several significant biological processes, cancer-related pathways and genes implicated in prostate cancer, opening new avenues of research. The results illustrate the ability of the BMMs to analyse large microarrays consisting of samples from multiple conditions from several patients and to reveal biologically relevant methylation patterns, thus contributing to advances in the field of quantitative DNA methylation analysis.

In terms of the family of BMMs developed here, there are several potential future research directions. For example, while DNA methylation can be influenced by environmental and clinical variables, the proposed BMMs do not incorporate such covariates. However, the BMMs could be extended, for example using a mixture of experts approach [52] where the parameters of the BMM are modeled as functions of the covariates, to offer a richer modelling framework. Further, a key assumption of the proposed BMMs is that the methylation states of adjacent CpG sites are conditionally independent given their cluster membership. However, methylation levels of adjacent CpG sites are often highly correlated [53]. This phenomenon gives rise to the emergence of biologically meaningful regions with discernible patterns. Expanding the scope of the BMM family to encompass the spatial dependencies within the data would present an opportunity to incorporate these structural nuances and ultimately facilitate the identification of particularly relevant differentially methylated regions. While the scale of missing data in the prostate cancer data considered here was almost negligible, it could be more prevalent in other settings. Such cases would motivate the development of imputation approaches that are cognisant of the heterogeneity typical of DNA methylation datasets. Finally, the methylation state of a human genome changes over time depending on clinical conditions. Longitudinal methylation data are often collected to study the effect of environmental changes or treatments on disease progression. Such data are vast and current approaches struggle to handle these extensive data in their innate form. In order to analyze methylation changes over time in multiple patients, similar to [54], the BMMs could be further enhanced to model dependency over time.

Supporting information

S1 File. Supporting information 1 for ‘A novel family of beta mixture models for the differential analysis of DNA methylation data: An application to prostate cancer’ data by Majumdar et al.

Supporting Information (Appendices S1 to S18) includes BMM derivations, parameter estimates and density estimate plots when BMM is applied to simulated and prostate cancer datasets.

https://doi.org/10.1371/journal.pone.0314014.s001

(PDF)

S2 File. Supporting information 2 for ‘A novel family of beta mixture models for the differential analysis of DNA methylation data: An application to prostate cancer’ data by Majumdar et al.

Supporting Information (Appendix S19) contains analysis of BMMs applied to the additional esophageal squamous cell carcinoma dataset.

https://doi.org/10.1371/journal.pone.0314014.s002

(PDF)

Acknowledgments

The authors wish to thank members of the Working Group in Model-based Clustering for their discussions on this work.

References

1. Berger SL, Kouzarides T, Shiekhattar R, Shilatifard A. An operational definition of epigenetics. Genes & Development. 2009 Apr 1; 23: 781–783. pmid:19339683
- View Article
- PubMed/NCBI
- Google Scholar
2. Moore LD, Le T, Fan G. DNA methylation and its basic function. Neuropsychopharmacology. 2013 Jan; 38: 23–38. pmid:22781841
- View Article
- PubMed/NCBI
- Google Scholar
3. Jin Z, Liu Y. DNA methylation in human diseases. Genes & Diseases. 2018; 5(1): 1–8. pmid:30258928
- View Article
- PubMed/NCBI
- Google Scholar
4. Das PM, Singal R. DNA methylation and cancer. Journal of Clinical Oncology. 2004 Nov 15; 22(22): 4632–4642. pmid:15542813
- View Article
- PubMed/NCBI
- Google Scholar
5. Bird A. DNA methylation patterns and epigenetic memory. Genes & Development 2002 Jan 1; 16: 6–21. pmid:11782440
- View Article
- PubMed/NCBI
- Google Scholar
6. de Almeida BP, Apolónio JD, Binnie A, Castelo-Branco P. Roadmap of DNA methylation in breast cancer identifies novel prognostic biomarkers. BMC Cancer. 2019 Mar; 19: 219. pmid:30866861
- View Article
- PubMed/NCBI
- Google Scholar
7. Kim JH, Dhanasekaran SM, Prensner JR, Cao X, Robinson D, Kalyana-Sundaram S, et al. Deep sequencing reveals distinct patterns of DNA methylation in prostate cancer. Genome Research. 2011 Jul 1; 21: 1028–1041. pmid:21724842
- View Article
- PubMed/NCBI
- Google Scholar
8. Sung H, Ferlay J, Siegel RL, Laversanne M, Soerjomataram I, Jemal A, et al. Global cancer statistics 2020: GLOBOCAN estimates of incidence and mortality worldwide for 36 cancers in 185 countries. CA: A Cancer Journal for Clinicians. 2021 May; 71(3): 209–249. pmid:33538338
- View Article
- PubMed/NCBI
- Google Scholar
9. Li LC, Okino ST, Dahiya R. DNA methylation in prostate cancer. Biochimica et Biophysica Acta (BBA)—Reviews on Cancer. 2004 Sep 20; 1704(2): 87–102. pmid:15363862
- View Article
- PubMed/NCBI
- Google Scholar
10. Daniunaite K, Jarmalaite S, Kalinauskaite N, Petroska D, Laurinavicius A, Lazutka JR, et al. Prognostic value of RASSF1 promoter methylation in prostate cancer. The Journal of Urology. 2014 Dec 1; 192(6): 1849–1855. pmid:24980613
- View Article
- PubMed/NCBI
- Google Scholar
11. Moritz R, Ellinger J, Nuhn P, Haese A, Müeller SC, Graefen M, et al. DNA hypermethylation as a predictor of PSA recurrence in patients with low-and intermediate-grade prostate cancer. Anticancer Research. 2013 Dec 1;33(12):5249–5254. pmid:24324057
- View Article
- PubMed/NCBI
- Google Scholar
12. Pidsley R, Zotenko E, Peters TJ, Lawrence MG, Risbridger GP, Molloy P, et al. Critical evaluation of the Illumina MethylationEPIC BeadChip microarray for whole-genome DNA methylation profiling. Genome Biology. 2016 Oct; 17: 208. pmid:27717381
- View Article
- PubMed/NCBI
- Google Scholar
13. Du P, Zhang X, Huang CC, Jafari N, Kibbe WA, Hou L, et al. Comparison of Beta-value and M-value methods for quantifying methylation levels by microarray analysis. BMC Bioinformatics. 2010 Nov; 11: 587. pmid:21118553
- View Article
- PubMed/NCBI
- Google Scholar
14. Chen X, Zhang Q, Chekouo T. Filtering high-dimensional methylation marks with extremely small sample size: an application to gastric cancer data. Frontiers in Genetics. 2021 Jul 12; 12: 705708. pmid:34322159
- View Article
- PubMed/NCBI
- Google Scholar
15. Men C, Chai H, Song X, Li Y, Du H, Ren Q. Identification of DNA methylation associated gene signatures in endometrial cancer via integrated analysis of DNA methylation and gene expression systematically. Journal of Gynecologic Oncology. 2017 Sep; 28(6):e83. pmid:29027401
- View Article
- PubMed/NCBI
- Google Scholar
16. Siegmund KD, Laird PW, Laird-Offringa IA. A comparison of cluster analysis methods using DNA methylation data. Bioinformatics. 2004 Aug; 20(12): 1896–1904. pmid:15044245
- View Article
- PubMed/NCBI
- Google Scholar
17. Koestler DC, Christensen BC, Marsit CJ, Kelsey KT, Houseman EA. Recursively partitioned mixture model clustering of DNA methylation data using biologically informed correlation structures. Statistical Applications in Genetics and Molecular Biology. 2013 Mar 5; 12(2): 225–240. pmid:23468465
- View Article
- PubMed/NCBI
- Google Scholar
18. Ma Z, Teschendorff AE. A variational Bayes beta mixture model for feature selection in DNA methylation studies. Journal of Bioinformatics and Computational Biology. 2013 Aug 14; 11(4): 1350005. pmid:23859269
- View Article
- PubMed/NCBI
- Google Scholar
19. Zhang L, Meng J, Liu H, Huang Y. A nonparametric Bayesian approach for clustering bisulfate-based DNA methylation profiles. BMC Genomics. 2012 Oct 26; 13 (Suppl 6): S20. pmid:23134689
- View Article
- PubMed/NCBI
- Google Scholar
20. Gevaert O, Tibshirani R, and Plevritis SK. Pancancer analysis of DNA methylation-driven genes using MethylMix. Genome Biology. 2015 Jan; 16: 17. pmid:25631659
- View Article
- PubMed/NCBI
- Google Scholar
21. R Core Team. R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria. 2024; https://www.R-project.org/.
22. Teschendorff AE, Marabita F, Lechner M, Bartlett T, Tegner J, Gomez-Cabrero D, et al. A beta-mixture quantile normalization method for correcting probe design bias in Illumina Infinium 450 k DNA methylation data. Bioinformatics. 2013 Jan 15; 29(2): 189–196. pmid:23175756
- View Article
- PubMed/NCBI
- Google Scholar
23. Laurila K, Oster B, Andersen CL, Lamy P, Orntoft T, Yli-Harja O, et al. A beta-mixture model for dimensionality reduction, sample classification and analysis. BMC Bioinformatics. 2011 May; 12: 215. pmid:21619656
- View Article
- PubMed/NCBI
- Google Scholar
24. Schröder C, Rahmann S. A hybrid parameter estimation algorithm for beta mixtures and applications to methylation state classification. Algorithms for Molecular Biology. 2017 Aug; 12: 21. pmid:28828033
- View Article
- PubMed/NCBI
- Google Scholar
25. Shi M, Tsui SK, Wu H, Wei Y. Pan-cancer analysis of differential DNA methylation patterns. BMC Medical Genomics. 2020 Oct;13 (Suppl 10): 154. pmid:33087120
- View Article
- PubMed/NCBI
- Google Scholar
26. Taguchi YH, Turki T. Principal component analysis-and tensor decomposition-based unsupervised feature extraction to select more suitable differentially methylated cytosines: Optimization of standard deviation versus state-of-the-art methods. Genomics. 2023 Mar; 115(2):110577. pmid:36804268
- View Article
- PubMed/NCBI
- Google Scholar
27. Wu D, Gu J, Zhang MQ. FastDMA: an infinium humanmethylation450 beadchip analyzer. PloS ONE. 2013 Sep 5;8(9):e74275. pmid:24040221
- View Article
- PubMed/NCBI
- Google Scholar
28. Ritchie ME, Phipson B, Wu D, Hu Y, Law CW, Shi W, et al. limma powers differential expression analyses for RNA-sequencing and microarray studies. Nucleic Acids Research. 2015 Apr 20; 43(7): e47. pmid:25605792
- View Article
- PubMed/NCBI
- Google Scholar
29. Wang D, Yan L, Hu Q, Sucheston LE, Higgins MJ, Ambrosone CB, et al. IMA: an R package for high-throughput analysis of Illumina’s 450K Infinium methylation data. Bioinformatics. 2012 Mar; 28(5): 729–730. pmid:22253290
- View Article
- PubMed/NCBI
- Google Scholar
30. Warden CD, Lee H, Tompkins JD, Li X, Wang C, Riggs AD, et al. COHCAP: an integrative genomic pipeline for single-nucleotide resolution DNA methylation analysis. Nucleic Acids Research. 2013 Jun 1; 41(11): e117. pmid:23598999
- View Article
- PubMed/NCBI
- Google Scholar
31. Schildknecht K, Olek S, Dickhaus T. Simultaneous statistical inference for epigenetic data. PLoS ONE. 2015 May 12; 10(5): e0125587. pmid:25965389
- View Article
- PubMed/NCBI
- Google Scholar
32. Chen Z, Huang H, Liu Q. Detecting differentially methylated loci for multiple treatments based on high-throughput methylation data. BMC Bioinformatics. 2014 May; 15: 142. pmid:24884464
- View Article
- PubMed/NCBI
- Google Scholar
33. Wang Z, Wu X, Wang Y. A framework for analyzing DNA methylation data from Illumina Infinium HumanMethylation450 BeadChip. BMC Bioinformatics. 2018 Apr 11; 19:115. pmid:29671397
- View Article
- PubMed/NCBI
- Google Scholar
34. Scrucca L. A transformation‐based approach to Gaussian mixture density estimation for bounded data. Biometrical Journal. 2019 Jul; 61(4): 873–888. pmid:30983031
- View Article
- PubMed/NCBI
- Google Scholar
35. Azam M, Bouguila N. Multivariate bounded support Laplace mixture model. Soft Computing. 2020 Sep; 24: 13239–13268.
- View Article
- Google Scholar
36. Azam M, Bouguila N. Multivariate bounded support asymmetric generalized Gaussian mixture model with model selection using minimum message length. Expert Systems with Applications. 2022 Oct 15; 204:117516.
- View Article
- Google Scholar
37. Silva R, Moran B, Russell NM, Fahey C, Vlajnic T, Manecksha RP, et al. Evaluating liquid biopsies for methylomic profiling of prostate cancer. Epigenetics. 2020 Jan; 15(6-7): 715–727. pmid:32000564
- View Article
- PubMed/NCBI
- Google Scholar
38. Moran S, Arribas C, Esteller M. Validation of a DNA methylation microarray for 850,000 CpG sites of the human genome enriched in enhancer sequences. Epigenomics. 2016; 8(3): 389–399. pmid:26673039
- View Article
- PubMed/NCBI
- Google Scholar
39. Di Lena P, Sala C, Prodi A, Nardini C. Missing value estimation methods for DNA methylation data. Bioinformatics. 2019 Oct; 35(19): 3786–3793. pmid:30796811
- View Article
- PubMed/NCBI
- Google Scholar
40. Dempster AP, Laird NM, Rubin DB. Maximum likelihood from incomplete data via the EM algorithm. Journal of the Royal Statistical Society, Series B (Methodological). 1977; 39(1): 1–22.
- View Article
- Google Scholar
41. Nocedal J, Wright SJ. Quasi-Newton methods. Numerical Optimization. Springer Series in Operations Research and Financial Engineering. 2006; Chapter 6: 136–163.
- View Article
- Google Scholar
42. Berndt EK, Hall BH, Hall RE, Hausman JA. Estimation and inference in non-linear structural models. Annals of Economic and Social Measurement. 1974; 3(4): 653–665.
- View Article
- Google Scholar
43. Diamond HG, Straub A. Bounds for the logarithm of the Euler gamma function and its derivatives. Journal of Mathematical Analysis and Applications. 2016 Jan 15; 433(2): 1072–1083.
- View Article
- Google Scholar
44. Akaike H. Information theory and an extension of the maximum likelihood principle. In: Parzen E., Tanabe K., Kitagawa G. (eds) Selected papers of Hirotugu Akaike. New York, NY: Springer New York. 1998; 199–213.
45. Schwarz G. Estimating the dimension of a model. The Annals of Statistics. 1978 Mar; 6(2): 461–464.
- View Article
- Google Scholar
46. Biernacki C, Celeux G, Govaert G. Assessing a mixture model for clustering with the integrated completed likelihood. IEEE Transactions on Pattern Analysis and Machine Intelligence. 2000 Jul; 22(7): 719–725.
- View Article
- Google Scholar
47. Ponti A, Giordani I, Mistri M, Candelieri A, Archetti F. The “Unreasonable” Effectiveness of the Wasserstein Distance in Analyzing Key Performance Indicators of a Network of Stores. Big Data and Cognitive Computing. 2022 Nov 15; 6(4): 138.
- View Article
- Google Scholar
48. Hubert L, Arabie P. Comparing Partitions. Journal of Classification. 1985; 2: 193–218.
- View Article
- Google Scholar
49. Maksimovic J, Oshlack A, Phipson B. Gene set enrichment analysis for genome-wide DNA methylation data. Genome Biology. 2021 Jun; 22: 173. pmid:34103055
- View Article
- PubMed/NCBI
- Google Scholar
50. Ameri A, Alidoosti A, Hosseini Y, Parvin M, Emranpour MH, Taslimi F, et al. Prognostic value of promoter hypermethylation of Retinoic Acid Receptor Beta (RARB) and CDKN2 (p16/MTS1) in prostate cancer. Chinese Journal of Cancer Research. 2011 Dec; 23: 306–311. pmid:23358881
- View Article
- PubMed/NCBI
- Google Scholar
51. Herberts C, Murtha AJ, Fu S, Wang G, Schönlau E, Xue H, et al. Activating AKT1 and PIK3CA mutations in metastatic castration-resistant prostate cancer. European Urology. 2020 Dec; 78(6): 834–844. pmid:32451180
- View Article
- PubMed/NCBI
- Google Scholar
52. Gormley IC, Frühwirth-Schnatter S. Mixture of experts models. In Handbook of mixture analysis. 2018 Dec; 1st ed.: 271–307.
53. Hodges E, Smith AD, Kendall J, Xuan Z, Ravi K, Rooks M, et al. High definition profiling of mammalian DNA methylation by array capture and single molecule bisulfite sequencing. Genome Research. 2009 Sep 1; 19(9): 1593–1605. pmid:19581485
- View Article
- PubMed/NCBI
- Google Scholar
54. Nyamundanda G, Gormley IC, Brennan L. A dynamic Probabilistic principal components model for the analysis of longitudinal metabolomics data. Journal of the Royal Statistical Society, Series C (Applied Statistics). 2014 Nov; 63(5): 763–782.
- View Article
- Google Scholar

[ref1] 1. Berger SL, Kouzarides T, Shiekhattar R, Shilatifard A. An operational definition of epigenetics. Genes & Development. 2009 Apr 1; 23: 781–783. pmid:19339683
View Article
PubMed/NCBI
Google Scholar

[2] View Article

[3] PubMed/NCBI

[4] Google Scholar

[ref2] 2. Moore LD, Le T, Fan G. DNA methylation and its basic function. Neuropsychopharmacology. 2013 Jan; 38: 23–38. pmid:22781841
View Article
PubMed/NCBI
Google Scholar

[6] View Article

[7] PubMed/NCBI

[8] Google Scholar

[ref3] 3. Jin Z, Liu Y. DNA methylation in human diseases. Genes & Diseases. 2018; 5(1): 1–8. pmid:30258928
View Article
PubMed/NCBI
Google Scholar

[10] View Article

[11] PubMed/NCBI

[12] Google Scholar

[ref4] 4. Das PM, Singal R. DNA methylation and cancer. Journal of Clinical Oncology. 2004 Nov 15; 22(22): 4632–4642. pmid:15542813
View Article
PubMed/NCBI
Google Scholar

[14] View Article

[15] PubMed/NCBI

[16] Google Scholar

[ref5] 5. Bird A. DNA methylation patterns and epigenetic memory. Genes & Development 2002 Jan 1; 16: 6–21. pmid:11782440
View Article
PubMed/NCBI
Google Scholar

[18] View Article

[19] PubMed/NCBI

[20] Google Scholar

[ref6] 6. de Almeida BP, Apolónio JD, Binnie A, Castelo-Branco P. Roadmap of DNA methylation in breast cancer identifies novel prognostic biomarkers. BMC Cancer. 2019 Mar; 19: 219. pmid:30866861
View Article
PubMed/NCBI
Google Scholar

[22] View Article

[23] PubMed/NCBI

[24] Google Scholar

[ref7] 7. Kim JH, Dhanasekaran SM, Prensner JR, Cao X, Robinson D, Kalyana-Sundaram S, et al. Deep sequencing reveals distinct patterns of DNA methylation in prostate cancer. Genome Research. 2011 Jul 1; 21: 1028–1041. pmid:21724842
View Article
PubMed/NCBI
Google Scholar

[26] View Article

[27] PubMed/NCBI

[28] Google Scholar

[ref8] 8. Sung H, Ferlay J, Siegel RL, Laversanne M, Soerjomataram I, Jemal A, et al. Global cancer statistics 2020: GLOBOCAN estimates of incidence and mortality worldwide for 36 cancers in 185 countries. CA: A Cancer Journal for Clinicians. 2021 May; 71(3): 209–249. pmid:33538338
View Article
PubMed/NCBI
Google Scholar

[30] View Article

[31] PubMed/NCBI

[32] Google Scholar

[ref9] 9. Li LC, Okino ST, Dahiya R. DNA methylation in prostate cancer. Biochimica et Biophysica Acta (BBA)—Reviews on Cancer. 2004 Sep 20; 1704(2): 87–102. pmid:15363862
View Article
PubMed/NCBI
Google Scholar

[34] View Article

[35] PubMed/NCBI

[36] Google Scholar

[ref10] 10. Daniunaite K, Jarmalaite S, Kalinauskaite N, Petroska D, Laurinavicius A, Lazutka JR, et al. Prognostic value of RASSF1 promoter methylation in prostate cancer. The Journal of Urology. 2014 Dec 1; 192(6): 1849–1855. pmid:24980613
View Article
PubMed/NCBI
Google Scholar

[38] View Article

[39] PubMed/NCBI

[40] Google Scholar

[ref11] 11. Moritz R, Ellinger J, Nuhn P, Haese A, Müeller SC, Graefen M, et al. DNA hypermethylation as a predictor of PSA recurrence in patients with low-and intermediate-grade prostate cancer. Anticancer Research. 2013 Dec 1;33(12):5249–5254. pmid:24324057
View Article
PubMed/NCBI
Google Scholar

[42] View Article

[43] PubMed/NCBI

[44] Google Scholar

[ref12] 12. Pidsley R, Zotenko E, Peters TJ, Lawrence MG, Risbridger GP, Molloy P, et al. Critical evaluation of the Illumina MethylationEPIC BeadChip microarray for whole-genome DNA methylation profiling. Genome Biology. 2016 Oct; 17: 208. pmid:27717381
View Article
PubMed/NCBI
Google Scholar

[46] View Article

[47] PubMed/NCBI

[48] Google Scholar

[ref13] 13. Du P, Zhang X, Huang CC, Jafari N, Kibbe WA, Hou L, et al. Comparison of Beta-value and M-value methods for quantifying methylation levels by microarray analysis. BMC Bioinformatics. 2010 Nov; 11: 587. pmid:21118553
View Article
PubMed/NCBI
Google Scholar

[50] View Article

[51] PubMed/NCBI

[52] Google Scholar

[ref14] 14. Chen X, Zhang Q, Chekouo T. Filtering high-dimensional methylation marks with extremely small sample size: an application to gastric cancer data. Frontiers in Genetics. 2021 Jul 12; 12: 705708. pmid:34322159
View Article
PubMed/NCBI
Google Scholar

[54] View Article

[55] PubMed/NCBI

[56] Google Scholar

[ref15] 15. Men C, Chai H, Song X, Li Y, Du H, Ren Q. Identification of DNA methylation associated gene signatures in endometrial cancer via integrated analysis of DNA methylation and gene expression systematically. Journal of Gynecologic Oncology. 2017 Sep; 28(6):e83. pmid:29027401
View Article
PubMed/NCBI
Google Scholar

[58] View Article

[59] PubMed/NCBI

[60] Google Scholar

[ref16] 16. Siegmund KD, Laird PW, Laird-Offringa IA. A comparison of cluster analysis methods using DNA methylation data. Bioinformatics. 2004 Aug; 20(12): 1896–1904. pmid:15044245
View Article
PubMed/NCBI
Google Scholar

[62] View Article

[63] PubMed/NCBI

[64] Google Scholar

[ref17] 17. Koestler DC, Christensen BC, Marsit CJ, Kelsey KT, Houseman EA. Recursively partitioned mixture model clustering of DNA methylation data using biologically informed correlation structures. Statistical Applications in Genetics and Molecular Biology. 2013 Mar 5; 12(2): 225–240. pmid:23468465
View Article
PubMed/NCBI
Google Scholar

[66] View Article

[67] PubMed/NCBI

[68] Google Scholar

[ref18] 18. Ma Z, Teschendorff AE. A variational Bayes beta mixture model for feature selection in DNA methylation studies. Journal of Bioinformatics and Computational Biology. 2013 Aug 14; 11(4): 1350005. pmid:23859269
View Article
PubMed/NCBI
Google Scholar

[70] View Article

[71] PubMed/NCBI

[72] Google Scholar

[ref19] 19. Zhang L, Meng J, Liu H, Huang Y. A nonparametric Bayesian approach for clustering bisulfate-based DNA methylation profiles. BMC Genomics. 2012 Oct 26; 13 (Suppl 6): S20. pmid:23134689
View Article
PubMed/NCBI
Google Scholar

[74] View Article

[75] PubMed/NCBI

[76] Google Scholar

[ref20] 20. Gevaert O, Tibshirani R, and Plevritis SK. Pancancer analysis of DNA methylation-driven genes using MethylMix. Genome Biology. 2015 Jan; 16: 17. pmid:25631659
View Article
PubMed/NCBI
Google Scholar

[78] View Article

[79] PubMed/NCBI

[80] Google Scholar

[ref21] 21. R Core Team. R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria. 2024; https://www.R-project.org/.

[ref22] 22. Teschendorff AE, Marabita F, Lechner M, Bartlett T, Tegner J, Gomez-Cabrero D, et al. A beta-mixture quantile normalization method for correcting probe design bias in Illumina Infinium 450 k DNA methylation data. Bioinformatics. 2013 Jan 15; 29(2): 189–196. pmid:23175756
View Article
PubMed/NCBI
Google Scholar

[83] View Article

[84] PubMed/NCBI

[85] Google Scholar

[ref23] 23. Laurila K, Oster B, Andersen CL, Lamy P, Orntoft T, Yli-Harja O, et al. A beta-mixture model for dimensionality reduction, sample classification and analysis. BMC Bioinformatics. 2011 May; 12: 215. pmid:21619656
View Article
PubMed/NCBI
Google Scholar

[87] View Article

[88] PubMed/NCBI

[89] Google Scholar

[ref24] 24. Schröder C, Rahmann S. A hybrid parameter estimation algorithm for beta mixtures and applications to methylation state classification. Algorithms for Molecular Biology. 2017 Aug; 12: 21. pmid:28828033
View Article
PubMed/NCBI
Google Scholar

[91] View Article

[92] PubMed/NCBI

[93] Google Scholar

[ref25] 25. Shi M, Tsui SK, Wu H, Wei Y. Pan-cancer analysis of differential DNA methylation patterns. BMC Medical Genomics. 2020 Oct;13 (Suppl 10): 154. pmid:33087120
View Article
PubMed/NCBI
Google Scholar

[95] View Article

[96] PubMed/NCBI

[97] Google Scholar

[ref26] 26. Taguchi YH, Turki T. Principal component analysis-and tensor decomposition-based unsupervised feature extraction to select more suitable differentially methylated cytosines: Optimization of standard deviation versus state-of-the-art methods. Genomics. 2023 Mar; 115(2):110577. pmid:36804268
View Article
PubMed/NCBI
Google Scholar

[99] View Article

[100] PubMed/NCBI

[101] Google Scholar

[ref27] 27. Wu D, Gu J, Zhang MQ. FastDMA: an infinium humanmethylation450 beadchip analyzer. PloS ONE. 2013 Sep 5;8(9):e74275. pmid:24040221
View Article
PubMed/NCBI
Google Scholar

[103] View Article

[104] PubMed/NCBI

[105] Google Scholar

[ref28] 28. Ritchie ME, Phipson B, Wu D, Hu Y, Law CW, Shi W, et al. limma powers differential expression analyses for RNA-sequencing and microarray studies. Nucleic Acids Research. 2015 Apr 20; 43(7): e47. pmid:25605792
View Article
PubMed/NCBI
Google Scholar

[107] View Article

[108] PubMed/NCBI

[109] Google Scholar

[ref29] 29. Wang D, Yan L, Hu Q, Sucheston LE, Higgins MJ, Ambrosone CB, et al. IMA: an R package for high-throughput analysis of Illumina’s 450K Infinium methylation data. Bioinformatics. 2012 Mar; 28(5): 729–730. pmid:22253290
View Article
PubMed/NCBI
Google Scholar

[111] View Article

[112] PubMed/NCBI

[113] Google Scholar

[ref30] 30. Warden CD, Lee H, Tompkins JD, Li X, Wang C, Riggs AD, et al. COHCAP: an integrative genomic pipeline for single-nucleotide resolution DNA methylation analysis. Nucleic Acids Research. 2013 Jun 1; 41(11): e117. pmid:23598999
View Article
PubMed/NCBI
Google Scholar

[115] View Article

[116] PubMed/NCBI

[117] Google Scholar

[ref31] 31. Schildknecht K, Olek S, Dickhaus T. Simultaneous statistical inference for epigenetic data. PLoS ONE. 2015 May 12; 10(5): e0125587. pmid:25965389
View Article
PubMed/NCBI
Google Scholar

[119] View Article

[120] PubMed/NCBI

[121] Google Scholar

[ref32] 32. Chen Z, Huang H, Liu Q. Detecting differentially methylated loci for multiple treatments based on high-throughput methylation data. BMC Bioinformatics. 2014 May; 15: 142. pmid:24884464
View Article
PubMed/NCBI
Google Scholar

[123] View Article

[124] PubMed/NCBI

[125] Google Scholar

[ref33] 33. Wang Z, Wu X, Wang Y. A framework for analyzing DNA methylation data from Illumina Infinium HumanMethylation450 BeadChip. BMC Bioinformatics. 2018 Apr 11; 19:115. pmid:29671397
View Article
PubMed/NCBI
Google Scholar

[127] View Article

[128] PubMed/NCBI

[129] Google Scholar

[ref34] 34. Scrucca L. A transformation‐based approach to Gaussian mixture density estimation for bounded data. Biometrical Journal. 2019 Jul; 61(4): 873–888. pmid:30983031
View Article
PubMed/NCBI
Google Scholar

[131] View Article

[132] PubMed/NCBI

[133] Google Scholar

[ref35] 35. Azam M, Bouguila N. Multivariate bounded support Laplace mixture model. Soft Computing. 2020 Sep; 24: 13239–13268.
View Article
Google Scholar

[135] View Article

[136] Google Scholar

[ref36] 36. Azam M, Bouguila N. Multivariate bounded support asymmetric generalized Gaussian mixture model with model selection using minimum message length. Expert Systems with Applications. 2022 Oct 15; 204:117516.
View Article
Google Scholar

[138] View Article

[139] Google Scholar

[ref37] 37. Silva R, Moran B, Russell NM, Fahey C, Vlajnic T, Manecksha RP, et al. Evaluating liquid biopsies for methylomic profiling of prostate cancer. Epigenetics. 2020 Jan; 15(6-7): 715–727. pmid:32000564
View Article
PubMed/NCBI
Google Scholar

[141] View Article

[142] PubMed/NCBI

[143] Google Scholar

[ref38] 38. Moran S, Arribas C, Esteller M. Validation of a DNA methylation microarray for 850,000 CpG sites of the human genome enriched in enhancer sequences. Epigenomics. 2016; 8(3): 389–399. pmid:26673039
View Article
PubMed/NCBI
Google Scholar

[145] View Article

[146] PubMed/NCBI

[147] Google Scholar

[ref39] 39. Di Lena P, Sala C, Prodi A, Nardini C. Missing value estimation methods for DNA methylation data. Bioinformatics. 2019 Oct; 35(19): 3786–3793. pmid:30796811
View Article
PubMed/NCBI
Google Scholar

[149] View Article

[150] PubMed/NCBI

[151] Google Scholar

[ref40] 40. Dempster AP, Laird NM, Rubin DB. Maximum likelihood from incomplete data via the EM algorithm. Journal of the Royal Statistical Society, Series B (Methodological). 1977; 39(1): 1–22.
View Article
Google Scholar

[153] View Article

[154] Google Scholar

[ref41] 41. Nocedal J, Wright SJ. Quasi-Newton methods. Numerical Optimization. Springer Series in Operations Research and Financial Engineering. 2006; Chapter 6: 136–163.
View Article
Google Scholar

[156] View Article

[157] Google Scholar

[ref42] 42. Berndt EK, Hall BH, Hall RE, Hausman JA. Estimation and inference in non-linear structural models. Annals of Economic and Social Measurement. 1974; 3(4): 653–665.
View Article
Google Scholar

[159] View Article

[160] Google Scholar

[ref43] 43. Diamond HG, Straub A. Bounds for the logarithm of the Euler gamma function and its derivatives. Journal of Mathematical Analysis and Applications. 2016 Jan 15; 433(2): 1072–1083.
View Article
Google Scholar

[162] View Article

[163] Google Scholar

[ref44] 44. Akaike H. Information theory and an extension of the maximum likelihood principle. In: Parzen E., Tanabe K., Kitagawa G. (eds) Selected papers of Hirotugu Akaike. New York, NY: Springer New York. 1998; 199–213.

[ref45] 45. Schwarz G. Estimating the dimension of a model. The Annals of Statistics. 1978 Mar; 6(2): 461–464.
View Article
Google Scholar

[166] View Article

[167] Google Scholar

[ref46] 46. Biernacki C, Celeux G, Govaert G. Assessing a mixture model for clustering with the integrated completed likelihood. IEEE Transactions on Pattern Analysis and Machine Intelligence. 2000 Jul; 22(7): 719–725.
View Article
Google Scholar

[169] View Article

[170] Google Scholar

[ref47] 47. Ponti A, Giordani I, Mistri M, Candelieri A, Archetti F. The “Unreasonable” Effectiveness of the Wasserstein Distance in Analyzing Key Performance Indicators of a Network of Stores. Big Data and Cognitive Computing. 2022 Nov 15; 6(4): 138.
View Article
Google Scholar

[172] View Article

[173] Google Scholar

[ref48] 48. Hubert L, Arabie P. Comparing Partitions. Journal of Classification. 1985; 2: 193–218.
View Article
Google Scholar

[175] View Article

[176] Google Scholar

[ref49] 49. Maksimovic J, Oshlack A, Phipson B. Gene set enrichment analysis for genome-wide DNA methylation data. Genome Biology. 2021 Jun; 22: 173. pmid:34103055
View Article
PubMed/NCBI
Google Scholar

[178] View Article

[179] PubMed/NCBI

[180] Google Scholar

[ref50] 50. Ameri A, Alidoosti A, Hosseini Y, Parvin M, Emranpour MH, Taslimi F, et al. Prognostic value of promoter hypermethylation of Retinoic Acid Receptor Beta (RARB) and CDKN2 (p16/MTS1) in prostate cancer. Chinese Journal of Cancer Research. 2011 Dec; 23: 306–311. pmid:23358881
View Article
PubMed/NCBI
Google Scholar

[182] View Article

[183] PubMed/NCBI

[184] Google Scholar

[ref51] 51. Herberts C, Murtha AJ, Fu S, Wang G, Schönlau E, Xue H, et al. Activating AKT1 and PIK3CA mutations in metastatic castration-resistant prostate cancer. European Urology. 2020 Dec; 78(6): 834–844. pmid:32451180
View Article
PubMed/NCBI
Google Scholar

[186] View Article

[187] PubMed/NCBI

[188] Google Scholar

[ref52] 52. Gormley IC, Frühwirth-Schnatter S. Mixture of experts models. In Handbook of mixture analysis. 2018 Dec; 1st ed.: 271–307.

[ref53] 53. Hodges E, Smith AD, Kendall J, Xuan Z, Ravi K, Rooks M, et al. High definition profiling of mammalian DNA methylation by array capture and single molecule bisulfite sequencing. Genome Research. 2009 Sep 1; 19(9): 1593–1605. pmid:19581485
View Article
PubMed/NCBI
Google Scholar

[191] View Article

[192] PubMed/NCBI

[193] Google Scholar

[ref54] 54. Nyamundanda G, Gormley IC, Brennan L. A dynamic Probabilistic principal components model for the analysis of longitudinal metabolomics data. Journal of the Royal Statistical Society, Series C (Applied Statistics). 2014 Nov; 63(5): 763–782.
View Article
Google Scholar

[195] View Article

[196] Google Scholar

Figures

Abstract

Introduction

Methods

Prostate cancer data

A beta distribution

A beta mixture model

A family of BMMs.

The K⋅⋅ model.

The KN⋅ model.

The K⋅R model.

Parameter estimation

A digamma approximation.

Inferring methylation state thresholds

Identifying the most differentially methylated clusters

Results

Simulated data results

Simulated data.

Estimating methylation state thresholds.

Identifying DMCs.

Prostate cancer data results

Estimating methylation state thresholds.

Identifying DMCs in the PCa data.

Discussion

Supporting information

S1 File. Supporting information 1 for ‘A novel family of beta mixture models for the differential analysis of DNA methylation data: An application to prostate cancer’ data by Majumdar et al.

S2 File. Supporting information 2 for ‘A novel family of beta mixture models for the differential analysis of DNA methylation data: An application to prostate cancer’ data by Majumdar et al.

Acknowledgments

References