James M. Flegal

Markov Chain Monte Carlo: Can We Trust the Third Significant Figure?

Current reporting of results based on Markov chain Monte Carlo computations could be improved. In particular, a measure of the accuracy of the resulting estimates is rarely reported. Thus we have little ability to objectively assess the quality of the reported estimates. We address this issue in that we discuss why Monte Carlo standard errors are important, how they can be easily calculated in Markov chain Monte Carlo and how they can be used to decide when to stop the simulation. We compare their use to a popular alternative in the context of two examples.

Batch means and spectral variance estimators in Markov chain Monte Carlo

Calculating a Monte Carlo standard error (MCSE) is an important step in the statistical analysis of the simulation output obtained from a Markov chain Monte Carlo experiment. An MCSE is usually based on an estimate of the variance of the asymptotic normal distribution. We consider spectral and batch means methods for estimating this variance. In particular, we establish conditions which guarantee that these estimators are strongly consistent as the simulation effort increases. In addition, for the batch means and overlapping batch means methods we establish conditions ensuring consistency in the mean-square sense which in turn allows us to calculate the optimal batch size up to a constant of proportionality. Finally, we examine the empirical finite-sample properties of spectral variance and batch means estimators and provide recommendations for practitioners.

Influence on longevity of blueberry, cinnamon, green and black tea, pomegranate, sesame, curcumin, morin, pycnogenol, quercetin, and taxifolin fed iso-calorically to long-lived, F1 hybrid mice.

Phytonutrients reportedly extend the life span of Caenorhabditis elegans, Drosophila, and mice. We tested extracts of blueberry, pomegranate, green and black tea, cinnamon, sesame, and French maritime pine bark (Pycnogenol and taxifolin), as well as curcumin, morin, and quercetin for their effects on the life span of mice. While many of these phytonutrients reportedly extend the life span of model organisms, we found no significant effect on the life span of male F1 hybrid mice, even though the dosages used reportedly produce defined therapeutic end points in mice. The compounds were fed beginning at 12 months of age. The control and treatment groups were iso-caloric with respect to one another. A 40% calorically restricted and other groups not reported here did experience life span extension. Body weights were un-changed relative to controls for all but two supplemented groups, indicating most supplements did not change energy absorption or utilization. Tea extracts with morin decreased weight, whereas quercetin, taxifolin, and Pycnogenol together increased weight. These changes may be due to altered locomotion or fatty acid biosynthesis. Published reports of murine life span extension using curcumin or tea components may have resulted from induced caloric restriction. Together, our results do not support the idea that isolated phytonutrient anti-oxidants and anti-inflammatories are potential longevity therapeutics, even though consumption of whole fruits and vegetables is associated with enhanced health span and life span.

A Practical Sequential Stopping Rule for High-Dimensional Markov Chain Monte Carlo

A current challenge for many Bayesian analyses is determining when to terminate high-dimensional Markov chain Monte Carlo simulations. To this end, we propose using an automated sequential stopping procedure that terminates the simulation when the computational uncertainty is small relative to the posterior uncertainty. Further, we show this stopping rule is equivalent to stopping when the effective sample size is sufficiently large. Such a stopping rule has previously been shown to work well in settings with posteriors of moderate dimension. In this article, we illustrate its utility in high-dimensional simulations while overcoming some current computational issues. As examples, we consider two complex Bayesian analyses on spatially and temporally correlated datasets. The first involves a dynamic space-time model on weather station data and the second a spatial variable selection model on fMRI brain imaging data. Our results show the sequential stopping rule is easy to implement, provides uncertainty estimates, and performs well in high-dimensional settings. Supplementary materials for this article are available online.

Markov chain Monte Carlo estimation of quantiles

We consider quantile estimation using Markov chain Monte Carlo and establish conditions under which the sampling distribution of the Monte Carlo error is approximately Normal. Further, we investigate techniques to estimate the associated asymptotic variance, which enables construction of an asymptotically valid interval estimator. Finally, we explore the finite sample properties of these methods through examples and provide some recommendations to practitioners.

Lifespan effects of simple and complex nutraceutical combinations fed isocalorically to mice

Present data suggest that the consumption of individual dietary supplements does not enhance the health or longevity of healthy rodents or humans. It might be argued that more complex combinations of such agents might extend lifespan or health-span by more closely mimicking the complexity of micronutrients in fruits and vegetables, which appear to extend health-span and longevity. To test this hypothesis we treated long-lived, male, F1 mice with published and commercial combinations of dietary supplements and natural product extracts, and determined their effects on lifespan and health-span. Nutraceutical, vitamin or mineral combinations reported to extend the lifespan or health-span of healthy or enfeebled rodents were tested, as were combinations of botanicals and nutraceuticals implicated in enhanced longevity by a longitudinal study of human aging. A cross-section of commercial nutraceutical combinations sold as potential health enhancers also were tested, including Bone Restore®, Juvenon®, Life Extension Mix®, Ortho Core®, Ortho Mind®, Super K w k2®, and Ultra K2®. A more complex mixture of vitamins, minerals, botanical extracts and other nutraceuticals was compounded and tested. No significant increase in murine lifespan was found for any supplement mixture. Our diverse supplement mixture significantly decreased lifespan. Thus, our results do not support the hypothesis that simple or complex combinations of nutraceuticals, including antioxidants, are effective in delaying the onset or progress of the major causes of death in mice. The results are consistent with epidemiological studies suggesting that dietary supplements are not beneficial and even may be harmful for otherwise healthy individuals.

Nordihydroguaiaretic Acid Extends the Lifespan of Drosophila and Mice, Increases Mortality-Related Tumors and Hemorrhagic Diathesis, and Alters Energy Homeostasis in Mice

Mesonordihydroguaiaretic acid (NDGA) extends murine lifespan. The studies reported here describe its dose dependence, effects on body weight, toxicity-related clinical chemistries, and mortality-related pathologies. In flies, we characterized its effects on lifespan, food consumption, body weight, and locomotion. B6C3F1 mice were fed AIN-93M diet supplemented with 1.5, 2.5, 3.5, or 4.5g NDGA/kg diet (1.59, 2.65, 3.71 and 4.77mg/kg body weight/day) beginning at 12 months of age. Only the 3.5mg/kg diet produced a highly significant increase in lifespan, as judged by either the Mantel–Cox log-rank test (p = .008) or the Gehan–Breslow–Wilcoxon test (p = .009). NDGA did not alter food intake, but dose-responsively reduced weight, suggesting it decreased the absorption or increased the utilization of calories. NDGA significantly increased the incidence of liver, lung, and thymus tumors, and peritoneal hemorrhagic diathesis found at necropsy. However, clinical chemistries found little evidence for overt toxicity. While NDGA was not overtly toxic at its therapeutic dosage, its association with severe end of life pathologies does not support the idea that NDGA consumption will increase human lifespan or health-span. The less toxic derivatives of NDGA which are under development should be explored as anti-aging therapeutics.

Minimum Size Survival Analysis Sampling Plans for Comparing Multiple Treatment Groups to a Single Control Group

We develop a sample size methodology that achieves specified Type-1 and Type-2 error rates when comparing the survivor functions of multiple treatment groups versus a control group. The designs will control family-wise Type-1 error rate. We assume the family of Weibull distributions adequately describes the underlying survivor functions, and we separately consider three of the most common study scenarios: (a) complete samples; (b) Type-1 censoring with a common censoring time; and (c) Type-1 censoring with an accrual period. A mice longevity study comparing the effect on survival of multiple low-calorie diets is used to motivate our work on this problem.

Exact sampling for intractable probability distributions via a Bernoulli factory

Many applications in the field of statistics require Markov chain Monte Carlo methods. Determining appropriate starting values and run lengths can be both analytically and empirically challenging. A desire to overcome these problems has led to the development of exact, or perfect, sampling algorithms which convert a Markov chain into an algorithm that produces i.i.d. samples from the stationary distribution. Unfortunately, very few of these algorithms have been developed for the distributions that arise in statistical applications, which typically have uncountable support. Here we study an exact sampling algorithm using a geometrically ergodic Markov chain on a general state space. Our work provides a significant reduction to the number of input draws necessary for the Bernoulli factory, which enables exact sampling via a rejection sampling approach. We illustrate the algorithm on a univariate Metropolis-Hastings sampler and a bivariate Gibbs sampler, which provide a proof of concept and insight into hyper-parameter selection. Finally, we illustrate the algorithm on a Bayesian version of the one-way random effects model with data from a styrene exposure study.

Strong Consistency of Multivariate Spectral Variance Estimators in Markov Chain Monte Carlo

Markov chain Monte Carlo (MCMC) algorithms are used to estimate features of interest of a distribution. The Monte Carlo error in estimation has an asymptotic normal distribution whose multivariate nature has so far been ignored in the MCMC community. We present a class of multivariate spectral variance estimators for the asymptotic covariance matrix in the Markov chain central limit theorem and provide conditions for strong consistency. We also show strong consistency of the eigenvalues of the estimator. Finally, we examine the finite sample properties of the multivariate spectral variance estimators and its eigenvalues in the context of a vector autoregressive process of order 1.