Dirichlet-multinomial Distribution Research Articles

Efficient Analysis of Overdispersed Data Using an Accurate Computation of the Dirichlet Multinomial Distribution.

Modeling count data using suitable statistical distributions has been instrumental for analyzing the patterns it conveys. However, failing to address critical aspects, like overdispersion, jeopardizes the effectiveness of such an analysis. In this paper, overdispersed count data is modeled using the Dirichlet Multinomial (DM) distribution by maximizing its likelihood using a fixed-point iteration algorithm. This is achieved by estimating the DM distribution parameters while comparing the recent Languasco-Migliardi (LM), and the Yu-Shaw (YS) procedures, which address the well-known computational difficulties of evaluating its log-likelihood. Experiments were conducted using multiple datasets from different domains spanning polls, images, and IoT network traffic. They all showed the superiority of the LM procedure as it succeeded at estimating the DM parameters at the designated level of accuracy in all experiments, while the YS procedure failed to produce sufficiently accurate results (or any results at all) in several experiments. Moreover, the LM procedure achieved a speedup that ranged from 2-fold to 20-fold over YS.

Seamless phase 2/3 design for trials with multiple co-primary endpoints using Bayesian predictive power

Seamless phase 2/3 design has become increasingly popular in clinical trials with a single endpoint. Trials that define success based on the achievement of all co-primary endpoints (CPEs) encounter the challenge of inflated type 2 error rates, often leading to an overly large sample size. To tackle this challenge, we introduced a seamless phase 2/3 design strategy that employs Bayesian predictive power (BPP) for futility monitoring and sample size re-estimation at interim analysis. The correlations among multiple CPEs are incorporated using a Dirichlet-multinomial distribution. An alternative approach based on conditional power (CP) was also discussed for comparison. A seamless phase 2/3 vaccine trial employing four binary endpoints under the non-inferior hypothesis serves as an example. Our results spotlight that, in scenarios with relatively small phase 2 sample sizes (e.g., 50 or 100 subjects), the BPP approach either outperforms or matches the CP approach in terms of overall power. Particularly, with n1 = 50 and ρ = 0, BPP showcases an overall power advantage over CP by as much as 8.54%. Furthermore, when the phase 2 stage enrolled more subjects (e.g., 150 or 200), especially with a phase 2 sample size of 200 and ρ = 0, the BPP approach evidences a peak difference of 5.76% in early stop probability over the CP approach, emphasizing its better efficiency in terminating futile trials. It’s noteworthy that both BPP and CP methodologies maintained type 1 error rates under 2.5%. In conclusion, the integration of the Dirichlet-Multinominal model with the BPP approach offers improvement in certain scenarios over the CP approach for seamless phase 2/3 trials with multiple CPEs.

Inferring single- and multi-species distributional aggregation using quadrat sampling

Ecologists often employ cost-effective strategies such as quadrat sampling for field-based biodiversity assessments. While the negative binomial distribution (NBD) is widely used to characterize quadrat sampling-derived biodiversity data in single-species settings, an equivalent model for accurately modeling community-level distributional aggregation levels, especially for quadrat sampling data, is still lacking. Here, we recommend the application of symmetric Dirichlet-multinomial (SDM) distribution to characterize aggregation patterns for single- or multi-species. Theoretically, we proved that the likelihood formulae for the SDM and NBD models were nearly equivalent, and consequently, the shape parameters of both models were almost identical. Numerical simulations demonstrated that the SDM model consistently outperformed the NBD model except for cases with large numbers of small-sized quadrats. Empirical tests using the SDM model showed that trees of the forest plot on the Barro Colorado Island shifted from aggregated to even and then back to aggregated again from 1982 to 2015 because of stand competition dynamics driven by tree mortality and recruitment balance. Functionally, the aggregation parameter of the SDM model reflects more purely this biological mechanism-driven aggregation trend, whereas classical nonparametric aggregation metrics failed. In summary, the proposed SDM can be an indispensable tool for inferring species- and community-level distributional aggregation patterns.

Estimating effective reproduction number revisited

Accurately estimating the effective reproduction number is crucial for characterizing the transmissibility of infectious diseases to optimize interventions and responses during epidemic outbreaks. In this study, we improve the estimation of the effective reproduction number through two main approaches. First, we derive a discrete model to represent a time series of case counts and propose an estimation method based on this framework. We also conduct numerical experiments to demonstrate the effectiveness of the proposed discretization scheme. By doing so, we enhance the accuracy of approximating the underlying epidemic process compared to previous methods, even when the counting period is similar to the mean generation time of an infectious disease. Second, we employ a negative binomial distribution to model the variability of count data to accommodate overdispersion. Specifically, given that observed incidence counts follow a negative binomial distribution, the posterior distribution of secondary infections is obtained as a Dirichlet multinomial distribution. With this formulation, we establish posterior uncertainty bounds for the effective reproduction number. Finally, we demonstrate the effectiveness of the proposed method using incidence data from the COVID-19 pandemic.

Sccomp: Robust differential composition and variability analysis for single-cell data

Cellular omics such as single-cell genomics, proteomics, and microbiomics allow the characterization of tissue and microbial community composition, which can be compared between conditions to identify biological drivers. This strategy has been critical to revealing markers of disease progression, such as cancer and pathogen infection. A dedicated statistical method for differential variability analysis is lacking for cellular omics data, and existing methods for differential composition analysis do not model some compositional data properties, suggesting there is room to improve model performance. Here, we introduce sccomp, a method for differential composition and variability analyses that jointly models data count distribution, compositionality, group-specific variability, and proportion mean-variability association, being aware of outliers. sccomp provides a comprehensive analysis framework that offers realistic data simulation and cross-study knowledge transfer. Here, we demonstrate that mean-variability association is ubiquitous across technologies, highlighting the inadequacy of the very popular Dirichlet-multinomial distribution. We show that sccomp accurately fits experimental data, significantly improving performance over state-of-the-art algorithms. Using sccomp, we identified differential constraints and composition in the microenvironment of primary breast cancer.

A Bayesian zero-inflated Dirichlet-multinomial regression model for multivariate compositional count data.

The Dirichlet-multinomial (DM) distribution plays a fundamental role in modern statistical methodology development and application. Recently, the DM distribution and its variants have been used extensively to model multivariate count data generated by high-throughput sequencing technology in omics research due to its ability to accommodate the compositional structure of the data as well as overdispersion. A major limitation of the DM distribution is that it is unable to handle excess zeros typically found in practice which may bias inference. To fill this gap, we propose a novel Bayesian zero-inflated DM model for multivariate compositional count data with excess zeros. We then extend our approach to regression settings and embed sparsity-inducing priors to perform variable selection for high-dimensional covariate spaces. Throughout, modeling decisions are made to boost scalability without sacrificing interpretability or imposing limiting assumptions. Extensive simulations and an application to a human gut microbiome dataset are presented to compare the performance of the proposed method to existing approaches. We provide an accompanying R package with a user-friendly vignette to apply our method to otherdatasets.

Clustering multivariate count data via Dirichlet-multinomial network fusion

Clustering of multivariate count data has widespread applications in areas such as text analysis and microbiome studies. The need to account for overdispersion generally results in a nonconvex loss function, which does not fit into the existing convex clustering framework. Moreover, prior knowledge of a network over the samples, often available from citation or similarity relationships, is not taken into account. We introduce Dirichlet-multinomial network fusion (DMNet) for clustering multivariate count data, which models the samples via Dirichlet-multinomial distributions with individual parameters and employs a weighted group L1 fusion penalty to pursue homogeneity over a prespecified network. To circumvent the nonconvexity issue, we present two exponential family approximations to the Dirichlet-multinomial distribution, which are amenable to efficient optimization and theoretical analysis. We derive an ADMM algorithm and establish nonasymptotic error bounds for the proposed methods. Our bounds involve a trade-off between the connectivity of the network and its fidelity to the true parameter. The usefulness of our methods is illustrated through simulation studies and two text clustering applications.

Activity discovery using Dirichlet multinomial mixture models from discrete sensor data in smart homes

To develop smart home technology designed to analyze the activity of residents based on the logs of installed sensors, an activity model tailored to individuals must be constructed from less privacy-invasive sensors to avoid interference in daily life. Unsupervised machine learning techniques are desirable to automatically construct such models without costly data annotation, but their application has not yet been sufficiently successful. In this study, we show that an activity model can be effectively estimated without activity labels via the Dirichlet multinomial mixture (DMM) model. The DMM model assumes that sensor signals are generated according to a Dirichlet multinomial distribution conditioned on a single unobservable activity and can capture the burstiness of sensors, in which even sensors that rarely fire may fire repeatedly after being triggered. We demonstrate the burstiness phenomenon in real data using passive infrared ray motion sensors. For such data, the assumptions of the DMM model are more suitable than the assumptions employed in models used in previous studies. Moreover, we extend the DMM model so that each activity depends on the preceding activity to capture the Markov dependency of activities, and a Gibbs sampler used in the model estimation algorithm is also presented. An empirical study using publicly available data collected in real-life settings shows that the DMM models can discover activities more correctly than the other models and expected to be used as a primitive activity extraction tool in activity analysis.

Wind farm layout optimization with uncertain wind condition

Due to the uncertainty of wind condition, the annual energy production (AEP) of a wind farm during operation time is varying from year to year and is difficult to be estimated. A risk management method based on conditional value-at-risk (CVaR) is introduced for increasing the robustness of the AEP under uncertain wind conditions. To implement the risk management in the wind farm layout optimization, uncertainty analysis method for the discretized wind distribution in the future is proposed, and the CVaR of the AEP as the objective function is calculated by the proposed normal approximation method with the uncertainty analysis results. The discrete probability distribution of future wind condition is denoted by the wind relative frequencies. The wind relative frequencies itself are treated as random variables. A Dirichlet-Multinomial distribution is applied to model the uncertainty of the wind relative frequencies in the future. The approximated normal distribution of AEP is established based on the probability distribution of the wind relative frequencies, and the CVaR can be calculated analytically based on the mean and variance of the normal distribution. The simulation verifies that the optimization method achieves a trade-off between maximizing the expected value of AEP with historical wind condition, and minimizing the potential variation of the AEP due to a changing wind condition in the future.

Dealing with overdispersion in multivariate count data

The problem of overdispersion in multivariate count data is a challenging issue. It covers a central role mainly due to the relevance of modern technology-based data, such as Next Generation Sequencing and textual data from the web or digital collections. A comprehensive analysis of the likelihood-based models for extra-variation data is presented. Particular attention is paid to the models feasible for high-dimensional data. A new approach together with its parametric-estimation procedure is proposed. It can be viewed as a deeper version of the Dirichlet-Multinomial distribution and it leads to important results allowing to get a better approximation of the observed variability. A significative comparison of the proposed model and existing strategies is made through two different simulation studies and an empirical data set, that confirm a better capability to describe overdispersion.

Sparse Count Data Clustering Using an Exponential Approximation to Generalized Dirichlet Multinomial Distributions.

Clustering frequency vectors is a challenging task on large data sets considering its high dimensionality and sparsity nature. Generalized Dirichlet multinomial (GDM) distribution is a competitive generative model for count data in terms of accuracy, yet its parameters estimation process is slow. The exponential-family approximation of the multivariate Polya distribution has shown to be efficient to train and cluster data directly, without dimensionality reduction. In this article, we derive an exponential-family approximation to the GDM distributions, and we call it (EGDM). A mixture model is developed based on the new member of the exponential-family of distributions, and its parameters are learned through the deterministic annealing expectation-maximization (DAEM) approach as a new clustering algorithm for count data. Moreover, we propose to estimate the optimal number of EGDM mixture components based on the minimum message length (MML) criterion. We have conducted a set of empirical experiments, concerning text, image, and video clustering, to evaluate the proposed approach performance. Results show that the new model attains a superior performance, and it is considerably faster than the corresponding method for GDM distributions.

1064. Treatment Success in Reducing Recurrent Clostridioides difficile Infection with Investigational Live Biotherapeutic RBX2660 Is Associated with Microbiota Restoration: Consistent Evidence from a Phase 3 Clinical Trial

BackgroundSeveral investigational microbiota-based live biotherapeutics are in clinical development for reducing recurrence of Clostridioides difficile infection (rCDI), including RBX2660 a liquid suspension of a broad consortium of microbiota, which includes Bacteridetes and Firmicutes. RBX2660 has been evaluated in >600 participants in 6 clinical trials. Here we report that RBX2660 induced significant shifts to the intestinal microbiota of treatment-responsive participants in PUNCH CD3—a Phase 3 randomized, double-blinded, placebo-controlled trial.MethodsPUNCH CD3 participants received a single dose of RBX2660 or placebo between 24 to 72 hours after completing rCDI antibiotic treatment. Clinical response was the absence of CDI recurrence at eight weeks after treatment. Participants voluntarily submitted stool samples prior to blinded study treatment (baseline), 1, 4 and 8 weeks, 3 and 6 months after receiving study treatment. Samples were extracted and sequenced using shallow shotgun methods. Operational taxonomic unit (OTU) data were used to calculate relative taxonomic abundance, alpha diversity, and the Microbiome Health Index (MHI)—a biomarker of antibiotic-induced dysbiosis and restoration.ResultsClinically, RBX2660 demonstrated superior efficacy versus placebo (70.4% versus 58.1%). From before to after treatment, RBX2660-treated clinical responders’ microbiome diversity shifted significantly (Mann-Whitney), and so did microbiome composition (Generalized Wald Test). Post-treatment changes were characterized by increased Bacteroidia and Clostridia and decreased Gammaproteobacteria and Bacilli, changes and were durable to at least 6 months. Repeated measures analysis confirmed that shifts were greater among RBX2660 responders compared to placebo responders (DMRepeat). The majority of responders’ MHI values shifted from a range common to antibiotic dysbiosis to a range common in healthy populations.Figure 1 Left panel. Mean relative abundance taxonomic class level at timepoints for participants in PUNCH CD3 before and after RBX2660 treatment, and for doses of RBX2660 administered in PUNCH CD3. The four taxonomic classes that change most from before to after treatment are shown with the mean and confidence intervals based on fitting OTU data to a Dirichlet multinomial distribution. Right panel, MHI biomarker for the same time points and investigational product groups, shown as median (red) and individual samples. A previously calculated threshold of MHI = 7.2 is shown (dotted line), above which MHI values predict healthy, below which MHI values predict antibiotic-induced dysbiosis.ConclusionAmong PUNCH CD3 clinical responders, RBX2660 significantly restored microbiota from less to more healthy compositions, and this restoration was durable to at least 6 months. These clinically-correlated microbiome shifts are highly consistent with results from multiple prior trials of RBX2660.Disclosures Ken Blount, PhD, Rebiotix Inc., a Ferring Company (Employee) Dana M. Walsh, PhD, Rebiotix (Employee)

1039. Rapid Restoration of Bile Acid Compositions After Treatment with RBX2660 for Recurrent Clostridioides difficile Infection—Results from the PUNCH CD3 Phase 3 Trial

BackgroundMicrobiota-based treatments are increasingly evaluated as a strategy to reduce recurrence of Clostridioides difficile infection (rCDI), and their proposed mechanisms include restoration of the microbiota and microbiota-mediated functions, including bile acid metabolism. RBX2660—a broad-consortium investigational live biotherapeutic—has been evaluated in >600 participants in 6 clinical trials, with consistent reduction of rCDI recurrence. Here we report that fecal bile acid compositions were significantly restored in treatment-responsive participants in PUNCH CD3—a Phase 3 randomized, double-blinded, placebo-controlled trial of RBX2660.MethodsPUNCH CD3 participants received a single dose of RBX2660 or placebo between 24 to 72 hours after completing rCDI antibiotic treatment. Clinical response was the absence of CDI recurrence at eight weeks after treatment. Participants voluntarily submitted stool samples prior to blinded study treatment (baseline), 1, 4 and 8 weeks, 3 and 6 months after receiving study treatment. A liquid chromatography tandem mass spectrometry method was developed to extract and quantify 33 bile acids from all participant fecal samples received up to the 8-week time point. Mean bile acid compositions were fit to a Dirichlet multinomial distribution and compared across time points and between RBX2660- and placebo-treated participants.ResultsClinically, RBX2660 demonstrated superior efficacy versus placebo (70.4% versus 58.1%). RBX2660-treated clinical responders’ bile acid compositions shifted significantly from before to after treatment. Specifically, primary bile acids predominated before treatment, whereas secondary bile acids predominated after treatment (Figure 1A). These changes trended higher among RBX2660 responders compared to placebo responders. Importantly, median levels of lithocholic acid (LCA) and deoxycholic acid (DCA) showed large, significant increases after treatment (Figure 1B). A. Bile acid compositions before (BL) and up to 8 weeks after RBX2660 treatment among treatment responders. Compositions are shown as the fraction of total bile acids classified as primary or secondary conjugated or deconjugated bile acids. B. Concentrations of lithocholic acid (LCA) and deoxycholic acid (DCA) among RBX2660 treatment responders, shown with individual samples and time point group median with interquartile ranges.ConclusionAmong PUNCH CD3 clinical responders, RBX2660 significantly restored bile acids from less to more healthy compositions. These clinically correlated bile acid shifts are highly consistent with results from a prior trial of RBX2660.Disclosures Romeo Papazyan, PhD, Ferring Research Institute (Employee) Bryan Fuchs, PhD, Ferring Pharmaceuticals (Employee) Ken Blount, PhD, Rebiotix Inc., a Ferring Company (Employee)

Studying the Third Cumulant of the Mixture of Dirichlet-multinomial Distributions

Traditionally, the overdispersion parameter ϕ is estimated by using Pearson’s lack of fit statistic X2or the Deviance statistic D, which do not perform well in the case of sparse data. This paper particularly focuses on an estimator ϕnew of overdispersion parameter which was proposed for sparse multinomial data. The estimator was derived on the basis of an assumption on the 3rd cumulant of the response variable.When the data comes from the Dirichlet-multinomial distribution ϕnew is known to have the lowest root mean squared error comparing to the other three estimators. In this paper the 1st to 3rd order raw moments of the finite mixture of Dirichlet-multinomial distributions are derived, which results in complicated mathematical expressions. Furthermore, it is found that the 3rd cumulant of this mixture does not satisfy the assumption which is considered in the derivation of ϕnew . Dhaka Univ. J. Sci. 69(2): 96-100, 2021 (July)

Emotion recognition: A smoothed Dirichlet multinomial solution

Multinomial-based models have been extensively used for count data modeling and challenging applications such as image processing, text recognition, and behavioral sciences. Despite the good performance obtained with those models, they still suffer from challenging issues that require continuous exploring of other alternative approaches. In this work, we address the issue of smoothing language modeling. To the best of our knowledge, distributions defined in a smoothed simplex were not considered before as conjugate priors for the multinomial. We propose a smoothed Dirichlet multinomial (SDM) distribution and a mixture of SDMs with a likelihood-based learning. We evaluate the proposed approach on three challenging applications related to emotion recognition: depression on social media, happiness analysis, and pain estimation. The smoothed Dirichlet multinomial solution presents the best results comparing to the related works and the multinomial-based models such as Dirichlet compound multinomial and the multinomial model.

The Dirichlet model in marketing

AbstractWe present a review of Gerald Goodhardt's most famous contribution to marketing science—the NBD‐Dirichlet model. This provides a powerful illustration of the complex pathway and useful associated discoveries that over 25 years lead to the specification and application of a key marketing model. We identify the process that started with the negative binomial distribution (NBD) applied to purchase incidence, examined alternatives such as the logarithmic series distribution (LSD) and the beta binomial distribution (BBD), and along the way developed conditional trend analysis (CTA). We then explore the development of brand choice beginning with duplication of purchase as a method for understanding the underlying patterns, before moving into various approaches to modelling choice including a multivariate NBD, and eventually a multivariate BBD—the Dirichlet multinomial distribution (DMD). We discuss key events in modelling consumer behaviour and outline the model's implications for how marketers should think about consumers, or more specifically the Dirichlet consumer. Finally, we provide a survey of the model's applications uncovering a rich recent history and bright future for Goodhardt's legacy.

Model-based ordination of pin-point cover data: Effect of management on dry heathland

Recently, there has been an increasing interest in model-based approaches for the statistical modelling of the joint distribution of multi-species abundances. The Dirichlet-multinomial distribution has been proposed as a suitable candidate distribution for the joint species distribution of pin-point plant cover data and is here applied in a model-based ordination framework. Unlike most model-based ordination methods, both fixed and random effects are in our proposed model structured as p-dimensional vectors and added to the latent variables before multiplying with the species-specific coefficients. This changes the interpretation of the parameters, so that the fixed and random effects now measure the relative displacement of the vegetation by the fixed and random factors in the p-dimensional latent variable space. This parameterization allows statistical inference of the effect of fixed and random factors in vector space, and makes it easier for practitioners to perform inferences on species composition in a multivariate setting. The method was applied on plant pin-point cover data from dry heathlands that had received different management treatments (burned, grazed, harvested, unmanaged), and it was found that treatment have a significant effect on heathland vegetation both when considering plant functional groups or when the taxonomic resolution was at the species level.

A Conway–Maxwell-multinomial distribution for flexible modeling of clustered categorical data

Categorical data are often observed as counts resulting from a fixed number of trials in which each trial consists of making one selection from a prespecified set of categories. The multinomial distribution serves as a standard model for such data but assumes that trials are independent and identically distributed. Extensions such as the Dirichlet-multinomial and random-clumped multinomial distribution can express positive association, where trials are more likely to result in a common category due to membership in a common cluster. This work considers a Conway–Maxwell-multinomial (CMM) distribution for modeling clustered categorical data exhibiting positively or negatively associated trials. The CMM distribution features a dispersion parameter which allows it to adapt to a range of association levels and includes several recognizable distributions as special cases. We explore properties of CMM, illustrate its flexible characteristics, identify a method to efficiently compute maximum likelihood (ML) estimates, present simulations of small sample properties under ML estimation, and demonstrate the model via data analysis examples.

DIM: Adaptively Combining User Interests Mined at Different Stages Based on Deformable Interest Model

User interest mining is widely used in the fields of personalized search and personalized recommendation. Traditional methods ignore the formation of user interest which is a process that evolves over time. This leads to the inability to accurately describe the distribution of user interest. In this paper, we propose the interest tracking model (ITM). To add the timing, ITM uses Dirichlet distribution and multinomial distribution to describe the evolutional process of interest topics and frequent patterns, which well adapts to the evolution of user interest hidden in short texts between different time slices. In addition, it is well known that user interest is composed of long-term interest and situational interest including short-term interest and social hot topics. State-of-the-art methods simply regard the users’ long-term interest as the users’ final interest, which makes those unable to completely describe the user interest distribution. To solve this problem, we propose the deformable interest model (DIM) which designs an objective function to combine users’ long-term interest and situational interest and more comprehensively and accurately mine user interest. Furthermore, we present the degree of deformation which measures the subinterest's degree of influence on final interest and propose in DIM the influence real-time update mechanism. The mechanism adaptively updates the degree of deformation through the linear iteration and reduces the degree of dependence of the interest model on training sets. We present results via a dataset consisting of Flickr users and their uploaded information in three months, a dataset consisting of Twitter users and their tweets in three months, and a dataset consisting of Instagram users and their uploaded information in three months, showing that the perplexity is reduced to 0.378, the average accuracy is increased to 94%, and the average NMI is increased to 0.20, which prove better interest prediction.

New statistics to test log-linear modeling hypothesis with no distributional specifications and clusters with homogeneous correlation

Traditionally, the Dirichlet-multinomial distribution has been recognized as a key model for contingency tables generated by cluster sampling schemes. There are, however, other possible distributions appropriate for these contingency tables. This paper introduces new statistics capable of testing log-linear modeling hypotheses with distributional unspecification, when the individuals of the clusters are possibly homogeneously correlated. An estimator for the intracluster correlation coefficient, valid for different cluster sizes, plays a crucial role in the construction of the goodness-of-fit test-statistics.