Journal of Data Science

We propose a differentially private Bayesian framework for envelope regression, a technique that improves estimation efficiency by modelling the response as a function of a low-dimensional subspace of the predictors. Our method applies the analytic Gaussian mechanism to privatize sufficient statistics from the data, ensuring formal $(\epsilon ,\delta )$-differential privacy. We develop a tailored Gibbs sampling algorithm that performs valid Bayesian inference using only the noisy sufficient statistics. This approach leverages the envelope structure to isolate the variation in predictors that is relevant to the response, reducing estimation error compared to standard regression under the same privacy constraints. Through simulation studies, we demonstrate improved estimation accuracy and tighter credible intervals relative to a differentially private Bayesian linear regression baseline.

Abstract: The present article discusses and compares multiple testing procedures (MTPs) for controlling the family wise error rate. Machekano and Hubbard (2006) have proposed empirical Bayes approach that is a resampling based multiple testing procedure asymptotically controlling the familywise error rate. In this paper we provide some additional work on their procedure, and we develop resampling based step-down procedure asymptotically controlling the familywise error rate for testing the families of one-sided hypotheses. We apply these procedures for making successive comparisons between the treatment effects under a simple-order assumption. For example, the treatment means may be a sequences of increasing dose levels of a drug. Using simulations, we demonstrate that the proposed step-down procedure is less conservative than the Machekano and Hubbard’s procedure. The application of the procedure is illustrated with an example.

Abstract: The detection of slope change points in wind curves depends on linear curve-fitting. Hall and Titterington’s algorithm based on smoothing is adapted and compared to a Bayesian method of curve-fitting. After prior spline smoothing of the data, the algorithms are tested and the errors between the split-linear fitted wind and the real one are estimated. In our case, the adaptation of the edge-preserving smoothing algorithm gives the same good performance as automatic Bayesian curve-fitting based on a Monte Carlo Markov chain algorithm yet saves computation time.

Survival analysis is the widely used statistical tool for new intervention comparison in presence of hazards of follow up studies. However, it is difficult to obtain suitable survival rate in presence of high level of hazard within few days of surgery. The group of patients can be directly stratified into cured and non-cured strata. The mixture models are natural choice for estimation of cure and non-cure rate estimation. The estimation of cure rate is an important parameter of success of any new intervention. The cure rate model is illustrated to compare the surgery of liver cirrhosis patients with consenting for participation HFLPC (Human Fatal Liver Progenitor Cells) Infusion vs. consenting for participation alone group in South Indian popula-tion. The surgery is best available technique for liver cirrhosis treatment. The success of the surgery is observed through follow up study. In this study, MELD (Model for End-Stage Liver Disease) score is considered as response of interest for cured and non-cured group. The primary efficacy of surgery is considered as covariates of interest. Distributional assumptions of the cure rate are solved with Markov Chain Monte Carlo (MCMC) techniques. It is found that cured model with parametric approach allows more consistent estimates in comparison to standard procedures. The risk of death due to liver transplantation in liver cirrhosis patients including time dependent effect terms has also been explored. The approach assists to model with different age and sex in both the treatment groups.

Compositional data consist of known compositions vectors whose components are positive and defined in the interval (0,1) representing proportions or fractions of a “whole”. The sum of these components must be equal to one. Compositional data is present in different knowledge areas, as in geology, economy, medicine among many others. In this paper, we propose a new statistical tool for volleyball data, i.e., we introduce a Bayesian anal- ysis for compositional regression applying additive log-ratio (ALR) trans- formation and assuming uncorrelated and correlated errors. The Bayesian inference procedure based on Markov Chain Monte Carlo Methods (MCMC). The methodology is applied on an artificial and a real data set of volleyball.