Dealing with missing data under stratified sampling designs where strata are study domains.

A quick count seeks to estimate the voting trends of an election and communicate them to the population on the evening of the same day of the election. In quick counts, the sampling is based on a stratified design of polling stations. Voting information is gathered gradually, often with no guarantee of obtaining the complete sample or even information in all the strata. However, accurate interval estimates with partial information must be obtained. Furthermore, this becomes more challenging if the strata are additionally study domains. To produce partial estimates, two strategies are proposed: (1) a Bayesian model using a dynamic post-stratification strategy and a single imputation process defined after a thorough analysis of historic voting information; additionally, a credibility level correction is included to solve the underestimation of the variance and (2) a frequentist alternative that combines standard multiple imputation ideas with classic sampling techniques to obtain estimates under a missing information framework. Both solutions are illustrated and compared using information from the 2021 quick count. The aim was to estimate the composition of the Chamber of Deputies in Mexico.

Bayesian nonparametric dynamic hazard rates in evolutionary life tables.

In the study of life tables the random variable of interest is usually assumed discrete since mortality rates are studied for integer ages. In dynamic life tables a time domain is included to account for the evolution effect of the hazard rates in time. In this article we follow a survival analysis approach and use a nonparametric description of the hazard rates. We construct a discrete time stochastic processes that reflects dependence across age as well as in time. This process is used as a bayesian nonparametric prior distribution for the hazard rates for the study of evolutionary life tables. Prior properties of the process are studied and posterior distributions are derived. We present a simulation study, with the inclusion of right censored observations, as well as a real data analysis to show the performance of our model.

Bayesian regression with spatiotemporal varying coefficients.

To study the impact of climate variables on morbidity of some diseases in Mexico, we propose a spatiotemporal varying coefficients regression model. For that we introduce a new spatiotemporal-dependent process prior, in a Bayesian context, with identically distributed normal marginal distributions and joint multivariate normal distribution. We study its properties and characterise the dependence induced. Our results show that the effect of climate variables, on the incidence of specific diseases, is not constant across space and time and our proposed model is able to capture and quantify those changes.

Scalable Bayesian nonparametric measures for exploring pairwise dependence via Dirichlet Process Mixtures.

In this article we propose novel Bayesian nonparametric methods using Dirichlet Process Mixture (DPM) models for detecting pairwise dependence between random variables while accounting for uncertainty in the form of the underlying distributions. A key criteria is that the procedures should scale to large data sets. In this regard we find that the formal calculation of the Bayes factor for a dependent-vs.-independent DPM joint probability measure is not feasible computationally. To address this we present Bayesian diagnostic measures for characterising evidence against a "null model" of pairwise independence. In simulation studies, as well as for a real data analysis, we show that our approach provides a useful tool for the exploratory nonparametric Bayesian analysis of large multivariate data sets.

A time-series DDP for functional proteomics profiles.

Using a new type of array technology, the reverse phase protein array (RPPA), we measure time-course protein expression for a set of selected markers that are known to coregulate biological functions in a pathway structure. To accommodate the complex dependent nature of the data, including temporal correlation and pathway dependence for the protein markers, we propose a mixed effects model with temporal and protein-specific components. We develop a sequence of random probability measures (RPM) to account for the dependence in time of the protein expression measurements. Marginally, for each RPM we assume a Dirichlet process model. The dependence is introduced by defining multivariate beta distributions for the unnormalized weights of the stick-breaking representation. We also acknowledge the pathway dependence among proteins via a conditionally autoregressive model. Applying our model to the RPPA data, we reveal a pathway-dependent functional profile for the set of proteins as well as marginal expression profiles over time for individual markers.

Analysis of Partially Incomplete Tables of Breast Cancer Characteristics with an Ordinal Variable.

Our goal is to model the joint distribution of a series of 4×2×2×2 contingency tables for which some of the data are partially collapsed (i.e., aggregated in as few as two dimensions). More specifically, the joint distribution of 4 clinical characteristics in breast cancer patients is estimated. These characteristics include estrogen receptor status (positive/negative), nodal involvement (positive/negative), HER2-neu expression (positive/negative), and stage of disease (I, II, III, IV). The joint distribution of the first three characteristics is estimated conditional on stage of disease and we propose a dynamic model for the conditional probabilities that let them evolve as the stage of disease progresses. The dynamic model is based on a series of Dirichlet distributions whose parameters are related by a Markov prior structure (called dynamic Dirichlet prior). This model makes use of information across disease stage (known as "borrowing strength") and provides a way of estimating the distribution of patients with particular tumor characteristics. In addition, since some of the data sources are aggregated, a data augmentation technique is proposed to carry out a meta-analysis of the different datasets.

Rubbery Polya Tree.

Polya trees (PT) are random probability measures which can assign probability 1 to the set of continuous distributions for certain specifications of the hyperparameters. This feature distinguishes the PT from the popular Dirichlet process (DP) model which assigns probability 1 to the set of discrete distributions. However, the PT is not nearly as widely used as the DP prior. Probably the main reason is an awkward dependence of posterior inference on the choice of the partitioning subsets in the definition of the PT. We propose a generalization of the PT prior that mitigates this undesirable dependence on the partition structure, by allowing the branching probabilities to be dependent within the same level. The proposed new process is not a PT anymore. However, it is still a tail-free process and many of the prior properties remain the same as those for the PT.

Bayesian Random Segmentation Models to Identify Shared Copy Number Aberrations for Array CGH Data.

Array-based comparative genomic hybridization (aCGH) is a high-resolution high-throughput technique for studying the genetic basis of cancer. The resulting data consists of log fluorescence ratios as a function of the genomic DNA location and provides a cytogenetic representation of the relative DNA copy number variation. Analysis of such data typically involves estimation of the underlying copy number state at each location and segmenting regions of DNA with similar copy number states. Most current methods proceed by modeling a single sample/array at a time, and thus fail to borrow strength across multiple samples to infer shared regions of copy number aberrations. We propose a hierarchical Bayesian random segmentation approach for modeling aCGH data that utilizes information across arrays from a common population to yield segments of shared copy number changes. These changes characterize the underlying population and allow us to compare different population aCGH profiles to assess which regions of the genome have differential alterations. Our method, referred to as BDSAcgh (Bayesian Detection of Shared Aberrations in aCGH), is based on a unified Bayesian hierarchical model that allows us to obtain probabilities of alteration states as well as probabilities of differential alteration that correspond to local false discovery rates. We evaluate the operating characteristics of our method via simulations and an application using a lung cancer aCGH data set.

A semi-parametric Bayesian analysis of survival data based on Lévy-driven processes.

In the presence of covariate information, the proportional hazards model is one of the most popular models. In this paper, in a Bayesian nonparametric framework, we use a Markov (Lévy-driven) process to model the baseline hazard rate. Previous Bayesian nonparametric models have been based on neutral to the right processes, which have a number of drawbacks, such as discreteness of the cumulative hazard function. We allow the covariates to be time dependent functions and develop a full posterior analysis via substitution sampling. A detailed illustration is presented.