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Classification and clustering of sequencing data using a Poisson model

Daniela M. Witten

arXiv:1202.6201v1stat.AP

TL;DR

Sequencing data require classification and clustering methods suited to nonnegative counts, because normality-based tools may be inappropriate. The paper develops Poisson-model approaches for both tasks and evaluates them on simulations and sequencing datasets. The methods work well under Poisson or mild overdispersion, while performance degrades with severe overdispersion.

  • Problem

    Sequencing data are nonnegative counts, but common classification and clustering methods assume Gaussian data or Euclidean distances.

  • Method

    The paper develops Poisson log linear methods for classification and a Poisson-based dissimilarity measure for clustering.

  • Results

    The proposed clustering performs well across the simulated overdispersion range, and Poisson dissimilarities best distinguish tumor from nontumor cervical-cancer samples.

  • Takeaways & Limitations

    The approaches provide sequencing-data methods for classification and clustering and can likely extend beyond RNA sequencing to DNA and ChIP sequencing.

  • Takeaways & Limitations

    Performance degrades under severe overdispersion, although the methods work well when the true model is Poisson or only mildly overdispersed.

Abstract

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In recent years, advances in high throughput sequencing technology have led to a need for specialized methods for the analysis of digital gene expression data. While gene expression data measured on a microarray take on continuous values and can be modeled using the normal distribution, RNA sequencing data involve nonnegative counts and are more appropriately modeled using a discrete count distribution, such as the Poisson or the negative binomial. Consequently, analytic tools that assume a Gaussian distribution (such as classification methods based on linear discriminant analysis and clustering methods that use Euclidean distance) may not perform as well for sequencing data as methods that are based upon a more appropriate distribution. Here, we propose new approaches for performing classification and clustering of observations on the basis of sequencing data. Using a Poisson log linear model, we develop an analog of diagonal linear discriminant analysis that is appropriate for sequencing data. We also propose an approach for clustering sequencing data using a new dissimilarity measure that is based upon the Poisson model. We demonstrate the performances of these approaches in a simulation study, on three publicly available RNA sequencing data sets, and on a publicly available chromatin immunoprecipitation sequencing data set.

1. Introduction.

Sequencing produces nonnegative count data with sample- and feature-specific variation, motivating Poisson-based classification and clustering rather than methods built on normality or Euclidean distance.

  • 1.1. An overview of RNA sequencing data.: RNA sequencing measures nonnegative read counts mapped to genes, exons, or other genomic regions.Reads are sequenced from cDNA, mapped to a reference when available, and pooled into regions of interest.
  • 1.1. An overview of RNA sequencing data.: Compared with microarrays, RNA sequencing avoids cross-hybridization and can discover transcripts without pre-specified probes.Microarrays can suffer background noise from cross-hybridization and cannot measure transcripts lacking array probes.
  • 1.2. Statistical models for RNA sequencing data.: Sequencing data present sample-depth variation and feature-level count variation, requiring models that accommodate both sources of variability.Samples can have vastly different total read counts, while counts for genomic regions are nonnegative.
  • 1.2. Statistical models for RNA sequencing data.: The paper addresses supervised class prediction and unsupervised subgroup identification for sequencing observations.The two tasks are classification when class labels are available and clustering when samples are unlabeled.
  • 1.2. Statistical models for RNA sequencing data.: Poisson log linear models are used for classification and clustering because conventional methods often assume normally distributed data.A two-dimensional Poisson example compares Bayes-optimal boundaries obtained under normal and Poisson assumptions.

2. A Poisson log linear model for multiple-class sequencing data.

The model represents sequencing counts through sample and feature effects, class-specific expression effects, and a Poisson framework that can be adapted for overdispersion.

  • 2. A Poisson log linear model for multiple-class sequencing data.: Sequencing counts are modeled with a Poisson log linear model whose sample and feature effects capture variation in sequencing depth and regional abundance.The model assumes independence across observations and features.
  • 2. A Poisson log linear model for multiple-class sequencing data.: The baseline mean is estimated without class labels, after which class effects are estimated using the baseline as an offset.This separates sample and feature variability from class differences as much as possible.
  • 2. A Poisson log linear model for multiple-class sequencing data.: The estimated class-feature effect exceeds one for over-expression and falls below one for under-expression relative to baseline.The class effects allow feature expression to depend on biological condition.
  • 2. A Poisson log linear model for multiple-class sequencing data.: A Gamma(β,β) prior smooths class-effect estimates when a feature has zero total counts in a class.Without smoothing, the maximum likelihood estimate is zero, which precludes a nonzero count for that feature in the class.
  • 2. A Poisson log linear model for multiple-class sequencing data.: A power transformation is used to make overdispersed data approximately Poisson, avoiding the need to fit a more complicated negative binomial model.The transformed values need not be integers, but are nevertheless modeled using the Poisson distribution.

3. A proposal for classifying sequencing data.

The paper develops Poisson linear discriminant analysis for sequencing counts, using independent-feature Poisson modeling and estimated size factors, with sparse PLDA providing feature selection through shrinkage.

  • Poisson linear discriminant analysis: Size factors for training and test observations are estimated using total count, quantile, or median-ratio approaches before fitting the classifier.The test-observation estimate directly extends the corresponding training-data normalization method.
  • Poisson linear discriminant analysis: Class priors may be set equally or estimated from training-class proportions, with equal priors used in the paper’s examples.The equal-prior choice sets each class prior to 1/K.
  • Poisson linear discriminant analysis: PLDA replaces normality with a Poisson log linear model while retaining a classification rule linear in the test observation’s feature counts.The model assumes independent features and uses class-specific parameters together with size-factor normalization.
  • Sparse PLDA: Sparse PLDA shrinks estimates toward 1 through soft thresholding, and increasing ρ makes more estimates exactly equal to 1.When ρ = 0, the estimator reduces to the unshrunk estimate; greater shrinkage occurs for small class sizes or small class-feature means.
  • Sparse PLDA: Sparse PLDA is the Poisson-model counterpart of nearest shrunken centroids, which provides a natural comparison method.Both methods shrink class-specific mean-related quantities using soft thresholding, but sPLDA uses a Poisson rather than normal model.

4. A proposal for clustering sequencing data.

The paper constructs a Poisson-model dissimilarity for sequencing observations and uses it with hierarchical clustering, while positioning the approach against Gaussian, multinomial, and negative-binomial alternatives.

  • Poisson dissimilarity: Squared Euclidean distance has a likelihood-ratio interpretation under a Gaussian model, motivating a Poisson-based alternative for sequencing counts.The authors note that the Gaussian model underlying Euclidean distance is not appropriate for sequencing data.
  • Poisson dissimilarity: The proposed dissimilarity tests whether two observations share equal feature-specific scaling parameters under a Poisson model.The method estimates nuisance quantities, uses posterior means under Gamma(β,β) priors, and forms a modified log likelihood ratio.
  • Poisson dissimilarity: The resulting Poisson dissimilarity is nonnegative, equals 0 for identical observations, and forms an n × n matrix for pairwise clustering.The matrix entry is the modified log likelihood ratio for the corresponding observation pair.
  • Poisson clustering: Hierarchical clustering can operate directly on the Poisson dissimilarity matrix without prespecifying the number of clusters.The resulting procedure is called Poisson clustering.
  • Poisson clustering: The framework can also reverse the roles of observations and features to obtain a p × p feature dissimilarity matrix.That matrix can be used for Poisson clustering of features.
  • Alternative approaches: The paper compares Poisson clustering with methods based on Poisson, multinomial, negative-binomial, and variance-stabilized approaches.These comparisons include SAGE-related Poisson clustering, multinomial testing, edgeR dissimilarities, and transformed-data Euclidean clustering.

5. A simulation study.

The simulation study evaluates nine classification methods and ten clustering methods for sequencing data under varying overdispersion. The proposed sPLDA and Poisson clustering perform well near the Poisson setting, while EdgeR is stronger under substantial overdispersion but more computationally demanding.

  • Classification: Nine classification methods combined three classifiers with total-count, quantile, and median-ratio size-factor estimates.The classifiers included NSC, transformed NSC, and sPLDA.
  • Classification: sPLDA performs well when dispersion is small but deteriorates relative to NSC under very high overdispersion.The study evaluated φ values of 0.01, 0.1, and 1, representing slight, substantial, and very high overdispersion.
  • Classification: The choice among total-count, quantile, and median-ratio size factors has little effect on classifier performance.The authors note that these estimates can differ more substantially for differential-expression detection than for classification.
  • Clustering: Ten clustering proposals were compared using complete-linkage hierarchical clustering, with methods evaluated by clustering error rate.The hierarchical dendrograms were cut to produce three clusters, while Cai et al.’s method used K = 3 directly.
  • Clustering: Poisson clustering performs well across the simulated overdispersion range, whereas EdgeR performs better under moderate or severe overdispersion but is much slower.For n = 50 and p = 10,000, computing the dissimilarity matrix took 14 seconds for Poisson clustering versus 6 minutes for EdgeR.

6. Application to sequencing data sets.

The authors evaluated nine classification methods and eight clustering dissimilarities across four sequencing datasets. Classification was error-free on two datasets, while clustering performance varied by dataset and method.

  • Data sets: Four datasets comprised three RNA sequencing studies and one chromatin immunoprecipitation sequencing study.
  • Classification: Nine classification methods were compared using three size-factor estimates for each classifier.
  • Classification: All methods achieved 0 cross-validation errors on the yeast and liver-and-kidney datasets across the tuning parameters considered.
  • Clustering: Poisson dissimilarities best distinguished tumor from nontumor cervical samples, but no method convincingly separated ADC from SCC.
  • Clustering: The transcription-factor-binding data were most complex, with ten individual-level groups and no clear clustering winner.
  • Clustering: On yeast data, Euclidean- and Poisson-based methods generally produced accurate dendrograms by grouping the dT samples together.

7. Discussion.

The discussion frames the methods as Poisson-model approaches for sequencing data while identifying overdispersion and feature-count bias as important scope considerations. The proposed clustering dissimilarity may also support other matrix-based analyses.

  • Contributions: The paper proposes classification and clustering approaches designed for RNA sequencing and applied to ChIP sequencing data.
  • Limitations: Performance is strongest under Poisson or mild-overdispersion conditions and degrades with severe overdispersion.
  • Future directions: A negative-binomial extension may improve performance when sequencing data are overdispersed.
  • Interpretation: Normalization had little apparent effect on classification and clustering results, despite its importance for differential-expression analysis.
  • Limitations: The methods’ dependence on total counts per feature may be undesirable for differential-expression analysis, but its desirability for classification or clustering remains unresolved.
  • Clustering: The Poisson dissimilarity matrix can be supplied to hierarchical clustering and other techniques such as multidimensional scaling.

APPENDIX A: PROPERTIES OF THE POISSON DISSIMILARITY MEASURE

The appendix establishes that the proposed Poisson dissimilarity is nonnegative and vanishes when two observations are identical.

  • Nonnegativity: The proof establishes nonnegativity by applying concavity to the component function g(d_ij).
  • Identity: When x_i = x_i′, the fitted ratios equal one and the dissimilarity in (17) equals zero.

APPENDIX B: EQUIVALENCE OF LOG LIKELIHOOD RATIO STATISTICS UNDER POISSON MODEL AND MULTINOMIAL MODEL

The appendix shows that the Poisson and multinomial log likelihood ratio statistics are equivalent under appropriate estimates, while practical dissimilarity measures can differ because they use different priors and estimators.

  • Under appropriate estimates of Nij, Ni′j, dij, and di′j, the Poisson log likelihood ratio statistic is identical to the multinomial statistic.
  • The Poisson comparison models observations with feature-specific Poisson means under a null constraint dij = di′j = 1 and unconstrained alternatives.
  • Under the null Poisson model, maximum-likelihood estimates allocate the combined feature count according to each observation’s total count.
  • The multinomial formulation compares shared feature probabilities under the null with unconstrained probabilities under the alternative.
  • In practice, Section 4.1 and Berninger et al. use different priors and dissimilarity constructions, and Berninger et al.’s proposal performs substantially worse in the simulation study.
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