A critical issue for users of Markov Chain Monte Carlo (MCMC) methods in applications is how to determine when it is safe to stop sampling and use the samples to estimate characteristics of the distribution of interest. Research into methods of computing theoretical convergence bounds holds promise for the future but currently has yielded relatively little that is of practical use in applied work. Consequently, most MCMC users address the convergence problem by applying diagnostic tools to the output produced by running their samplers. After giving a brief overview of the area, we provide an expository review of thirteen convergence diagnostics, describing the theoretical basis and practical implementation of each.We then compare their performance in two simple models and conclude that all the methods can fail to detect the sorts of convergence failure they were designed to identify. We thus recommend a combination of strategies aimed at evaluating and accelerating MCMC sampler convergence, including applying diagnostic procedures to a small number of parallel chains, monitoring autocorrelations and crosscorrelations, and modifying parameterizations or sampling algorithms appropriately. We emphasize, however, that it is not possible to say with certainty that a finite sample from an MCMC algorithm is representative of an underlying stationary distribution.
The ordinal probit, univariate or multivariate, is a generalized linear model (GLM) structure that arises frequently in such disparate areas of statistical applications as medicine and econometrics. Despite the straightforwardness of its implementation using the Gibbs sampler, the ordinal probit may present challenges in obtaining satisfactory convergence.We present a multivariate Hastings-within-Gibbs update step for generating latent data and bin boundary parameters jointly, instead of individually from their respective full conditionals. When the latent data are parameters of interest, this algorithm substantially improves Gibbs sampler convergence for large datasets. We also discuss Monte Carlo Markov chain (MCMC) implementation of cumulative logit (proportional odds) and cumulative complementary log-log (proportional hazards) models with latent data.
Poor adherence to medication regimens is a well-documented phenomenon in clinical practice and an ever-present concern in clinical trials. Little is known about adherence to inhaled medication regimens over extended periods. The present paper describes the 2-yr results of the Lung Health Study (LHS) program, which was developed to maintain long-term adherence to an inhaled medication regimen in 3,923 special intervention participants (as measured by self-report and medication canister weight). The LHS is a double-blind, multicenter, randomized controlled clinical trial of smoking intervention and bronchodilator therapy (ipratropium bromide or placebo) for early intervention in chronic obstructive pulmonary disease (COPD). At the first 4-mo follow-up visit, nearly 70% of participants reported satisfactory or better adherence. Over the next 18 mo, self-reported satisfactory or better adherence declined to about 60%. Canister weight classified adherence as satisfactory or better in 72% of participants returning all canisters at 1 yr, and in 70% of the participants returning all canisters at the 2-yr follow-up. Self-reporting confirmed by canister weight classified 48% of participants at 1 yr as showing satisfactory or better adherence. Overusers were 50% more likely than others to misrepresent their true smoking status, suggesting that canister weights indicating overuse may be deceptive. Results of multiple logistic regression analysis indicate that the best compliance was found in participants who were married, older, white, had more severe airways obstruction, less shortness of breath, and fewer hospitalizations, and who had not been confined to bed for respiratory illnesses.(ABSTRACT TRUNCATED AT 250 WORDS)
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