Search Our Scientific Publications & Authors

Am Stat 2019 4;73(4):350-359. Epub 2018 Jun 4.

Department of Health Outcomes and Policy, University of Florida.

Despite the popularity of the general linear mixed model for data analysis, power and sample size methods and software are not generally available for commonly used test statistics and reference distributions. Statisticians resort to simulations with homegrown and uncertified programs or rough approximations which are misaligned with the data analysis. For a wide range of designs with longitudinal and clustering features, we provide accurate power and sample size approximations for inference about fixed effects in linear models we call reversible. Read More

Am Stat 2019 4;73(2):117-125. Epub 2018 Jun 4.

Biostatistics Branch, Division of Cancer Epidemiology and Genetics National Cancer Institute, Rockville, MD.

Group testing has its origin in the identification of syphilis in the U.S. army during World War II. Read More

Am Stat 2019 20;73(Suppl 1):271-280. Epub 2019 Mar 20.

La Trobe University, Plenty Road, Bundoora, Melbourne VIC, 3086, Australia.

The "New Statistics" emphasizes effect sizes, confidence intervals, meta-analysis, and the use of Open Science practices. We present 3 specific ways in which a New Statistics approach can help improve scientific practice: by reducing over-confidence in small samples, by reducing confirmation bias, and by fostering more cautious judgments of consistency. We illustrate these points through consideration of the literature on oxytocin and human trust, a research area that typifies some of the endemic problems that arise with poor statistical practice. Read More

Am Stat 2019 20;73(Suppl 1):82-90. Epub 2019 Mar 20.

Department of Biostatistics, Harvard University.

As statisticians and scientists consider a world beyond < 0.05, it is important to not lose sight of how we got to this point. Although significance testing and -values are often presented as prescriptive procedures, they came about through a process of refinement and extension to other disciplines. Read More

Am Stat 2019 20;73(Suppl 1):152-156. Epub 2019 Mar 20.

Department of Health Care Policy, Harvard Medical School, Boston, MA 01201 and Research Associate at NBER, Cambridge, MA, 02138.

Stepwise regression building procedures are commonly used applied statistical tools, despite their well-known drawbacks. While many of their limitations have been widely discussed in the literature, other aspects of the use of individual statistical fit measures, especially in high-dimensional stepwise regression settings, have not. Giving primacy to individual fit, as is done with p-values and R, when group fit may be the larger concern, can lead to misguided decision making. Read More

Am Stat 2019 20;73(Suppl 1):129-134. Epub 2019 Mar 20.

Texas A&M University.

This article examines the evidence contained in statistics that are marginally significant in 5% tests. The bases for evaluating evidence are likelihood ratios and integrated likelihood ratios, computed under a variety of assumptions regarding the alternative hypotheses in null hypothesis significance tests. Likelihood ratios and integrated likelihood ratios provide a useful measure of the evidence in favor of competing hypotheses because they can be interpreted as representing the ratio of the probabilities that each hypothesis assigns to observed data. Read More

Am Stat 2019 20;73(1):56-68. Epub 2019 Mar 20.

Department of Biostatistics and Bioinformatics, Duke University, Durham, NC.

This article resulted from our participation in the session on the "role of expert opinion and judgment in statistical inference" at the October 2017 ASA Symposium on Statistical Inference. We present a strong, unified statement on roles of expert judgment in statistics with processes for obtaining input, whether from a Bayesian or frequentist perspective. Topics include the role of subjectivity in the cycle of scientific inference and decisions, followed by a clinical trial and a greenhouse gas emissions case study that illustrate the role of judgments and the importance of basing them on objective information and a comprehensive uncertainty assessment. Read More

Am Stat 2019 10;73(1):22-31. Epub 2018 May 10.

Area of Information Systems and Quantitative Sciences, Texas Tech University.

Many Bayes factors have been proposed for comparing population means in two-sample (independent samples) studies. Recently, Wang and Liu (2015) presented an "objective" Bayes factor (BF) as an alternative to a "subjective" one presented by Gönen et al. (2005). Read More

Am Stat 2019 10;73(1):43-49. Epub 2018 May 10.

Vaccine and Infectious Disease Division, Fred Hutchinson Cancer Research Center.

The Wilcoxon-Mann-Whitney (WMW) test is a popular rank-based two-sample testing procedure for the strong null hypothesis that the two samples come from the same distribution. A modified WMW test, the Fligner-Policello (FP) test, has been proposed for comparing the medians of two populations. A fact that may be underappreciated among some practitioners is that the FP test can also be used to test the strong null like the WMW. Read More

Am Stat 2018 30;72(2):199-205. Epub 2017 Oct 30.

Biostatistics and Bioinformatics, Duke University School of Medicine Duke Cancer Institute, Duke University Medical Center.

The score statistic continues to be a fundamental tool for statistical inference. In the analysis of data from high-throughput genomic assays, inference on the basis of the score usually enjoys greater stability, considerably higher computational efficiency, and lends itself more readily to the use of resampling methods than the asymptotically equivalent Wald or likelihood ratio tests. The score function often depends on a set of unknown nuisance parameters which have to be replaced by estimators, but can be improved by calculating the efficient score, which accounts for the variability induced by estimating these parameters. Read More

Am Stat 2018 26;72(2):121-129. Epub 2018 Jan 26.

Alan D. Hutson is Professor of Oncology, Roswell Park Cancer Institute, Department of Biostatistics & Bioinformatics, Elm & Carlton Streets, Buffalo, NY 14623. Albert Vexler is Associate Professor, University at Buffalo, Department of Biostatistics, 706 Kimball Tower, 3435 Main St., Buffalo, NY 14214-3000.

In this note we examine the four parameter beta family of distributions in the context of the beta-normal and beta-logistic distributions. In the process we highlight the concept of numerical and limiting alias distributions, which in turn relate to numerical instabilities in the numerical maximum likelihood fitting routines for these families of distributions. We conjecture that the numerical issues pertaining to fitting these multiparameter distributions may be more widespread than has originally been reported across several families of distributions. Read More

Am Stat 2018 14;72(4):382-391. Epub 2018 Nov 14.

Department of Biostatistics and Computational Biology, Dana-Farber Cancer Institute, Boston, MA.

Demand for data science education is surging and traditional courses offered by statistics departments are not meeting the needs of those seeking training. This has led to a number of opinion pieces advocating for an update to the Statistics curriculum. The unifying recommendation is that computing should play a more prominent role. Read More

Am Stat 2017;71(2):171-176. Epub 2017 Feb 28.

Department of Urology, Weill Cornell Medical College, New York, NY.

We consider settings where it is of interest to fit and assess regression submodels that arise as various explanatory variables are excluded from a larger regression model. The larger model is referred to as the full model; the submodels are the reduced models. We show that a computationally efficient approximation to the regression estimates under any reduced model can be obtained from a simple weighted least squares (WLS) approach based on the estimated regression parameters and covariance matrix from the full model. Read More

Am Stat 2016 10;70(3):285-292. Epub 2016 Aug 10.

Department of Epidemiology, Johns Hopkins University, Baltimore, MD, USA.

Statistical prediction models inform decision-making processes in many real-world settings. Prior to using predictions in practice, one must rigorously test and validate candidate models to ensure that the proposed predictions have sufficient accuracy to be used in practice. In this paper, we present a framework for evaluating time series predictions that emphasizes computational simplicity and an intuitive interpretation using the relative mean absolute error metric. Read More

Am Stat 2017 11;71(4):344-353. Epub 2018 Jan 11.

Department of Statistics, College of Arts and Sciences, University of Kentucky.

Correlated data are commonly analyzed using models constructed using population-averaged generalized estimating equations (GEEs). The specification of a population-averaged GEE model includes selection of a structure describing the correlation of repeated measures. Accurate specification of this structure can improve efficiency, whereas the finite-sample estimation of nuisance correlation parameters can inflate the variances of regression parameter estimates. Read More

Am Stat 2016 Oct 21;70(4):385-394. Epub 2016 Nov 21.

MRC Biostatistics Unit, University of Cambridge , Cambridge , UK.

The funnel plot is a graphical visualization of summary data estimates from a meta-analysis, and is a useful tool for detecting departures from the standard modeling assumptions. Although perhaps not widely appreciated, a simple extension of the funnel plot can help to facilitate an intuitive interpretation of the mathematics underlying a meta-analysis at a more fundamental level, by equating it to determining the center of mass of a physical system. We used this analogy to explain the concepts of weighing evidence and of biased evidence to a young audience at the Cambridge Science Festival, without recourse to precise definitions or statistical formulas and with a little help from Sherlock Holmes! Following on from the science fair, we have developed an interactive web-application (named the Meta-Analyser) to bring these ideas to a wider audience. Read More

Am Stat 2016 9;70(2):187-194. Epub 2016 Jun 9.

Department of Epidemiology and Population Health, Albert Einstein College of Medicine, Yeshiva University, Bronx, NY, USA.

Given the increasing level and scope of biostatistics expertise needed at academic health centers today, we developed best practices guidelines for biostatistics units to be more effective in providing biostatistical support to their institutions, and in fostering an environment in which unit members can thrive professionally. Our recommendations focus on the key areas of: 1) funding sources and mechanisms; 2) providing and prioritizing access to biostatistical resources; and 3) interacting with investigators. We recommend that the leadership of biostatistics units negotiate for sufficient long-term infrastructure support to ensure stability and continuity of funding for personnel, align project budgets closely with actual level of biostatistical effort, devise and consistently apply strategies for prioritizing and tracking effort on studies, and clearly stipulate with investigators prior to project initiation policies regarding funding, lead time, and authorship. Read More

Am Stat 2016 Jan 31;70(1):108-113. Epub 2016 Mar 31.

The Rao-Blackwell theorem offers a procedure for converting a crude unbiased estimator of a parameter θ into a "better" one, in fact unique and optimal if the improvement is based on a minimal sufficient statistic that is complete. In contrast, behind every minimal sufficient statistic that is not complete, there is an improvable Rao-Blackwell improvement. This is illustrated via a simple example based on the uniform distribution, in which a rather natural Rao-Blackwell improvement is uniformly improvable. Read More

Am Stat 2016 Mar 14;70(1):47-54. Epub 2015 Dec 14.

Department of Biostatistics, Harvard T.H. Chan School of Public Health.

Although propensity scores have been central to the estimation of causal effects for over 30 years, only recently has the statistical literature begun to consider in detail methods for Bayesian estimation of propensity scores and causal effects. Underlying this recent body of literature on Bayesian propensity score estimation is an implicit discordance between the goal of the propensity score and the use of Bayes theorem. The propensity score condenses multivariate covariate information into a scalar to allow estimation of causal effects without specifying a model for how each covariate relates to the outcome. Read More

Am Stat 2016 Jan 31;70(1):33-38. Epub 2016 Mar 31.

There is growing frustration with the concept of the -value. Besides having an ambiguous interpretation, the value can be made as small as desired by increasing the sample size, . The -value is outdated and does not make sense with big data: Everything becomes statistically significant. Read More

Am Stat 2015 Oct 29;69(4):371-386. Epub 2015 Dec 29.

Bootstrapping has enormous potential in statistics education and practice, but there are subtle issues and ways to go wrong. For example, the common combination of nonparametric bootstrapping and bootstrap percentile confidence intervals is less accurate than using -intervals for small samples, though more accurate for larger samples. My goals in this article are to provide a deeper understanding of bootstrap methods-how they work, when they work or not, and which methods work better-and to highlight pedagogical issues. Read More

Am Stat 2016 1;70(4):413-423. Epub 2014 Jun 1.

Professor, Department of Statistics, University of South Carolina, Columbia, SC 29208.

In this pedagogical article, distributional properties, some surprising, pertaining to the homogeneous Poisson process (HPP), when observed over a possibly random window, are presented. Properties of the gap-time that covered the termination time and the correlations among gap-times of the observed events are obtained. Inference procedures, such as estimation and model validation, based on event occurrence data over the observation window, are also presented. Read More

Am Stat 2014 May;68(2):125-126

Johns Hopkins Medical Institution, In the Public Domain.

Am Stat 2015 Feb;69(1):34-40

Biostatistics and Data Management Core, University of Hawaii John A. Burns School of Medicine, Honolulu, Hawaii 96813.

Statistical principles and methods are critical to the success of biomedical and translational research. However, it is difficult to track and evaluate the monetary value of a biostatistician to a medical school (SoM). Limited published data on this topic is available, especially comparing across SoMs. Read More

Am Stat 2014;68(3):191-195

Peter H. Westfall is Horn Professor in the Area of Information Systems and Quantitative Sciences, Texas Tech University, Lubbock, TX 79409 ( ).

The incorrect notion that kurtosis somehow measures "peakedness" (flatness, pointiness or modality) of a distribution is remarkably persistent, despite attempts by statisticians to set the record straight. This article puts the notion to rest once and for all. Kurtosis tells you virtually nothing about the shape of the peak - its only unambiguous interpretation is in terms of tail extremity; i. Read More

Am Stat 2014 Oct;68(4):271-276

Professor in the Department of Mathematics and Statistics, University of New Mexico, Albuquerque, NM 8713.

This paper presents a counterintuitive result regarding the estimation of a regression slope co-efficient. Paradoxically, the precision of the slope estimator can deteriorate when additional information is used to estimate its value. In a randomized experiment, the distribution of baseline variables should be identical across treatments due to randomization. Read More

Am Stat 2015 27;69(3):254-255. Epub 2015 Aug 27.

University of New Mexico

Am Stat 2015 17;69(1):45-52. Epub 2014 Nov 17.

Chief and Senior Investigator, Biostatistics and Bioinformatics Branch, Division of Intramural Population Health Research, Eunice Kennedy Shriver National Institute of Child Health and Human Development, Bethesda, MD 20892.

Group testing is an active area of current research and has important applications in medicine, biotechnology, genetics, and product testing. There have been recent advances in design and estimation, but the simple Dorfman procedure introduced by R. Dorfman in 1943 is widely used in practice. Read More

Am Stat 2014;48(3):158-169

Department of Biostatistics, The State University of New York at Buffalo, Buffalo, NY 14214, U.S.A.

We develop a novel nonparametric likelihood ratio test for independence between two random variables using a technique that is free of the common constraints of defining a given set of specific dependence structures. Our methodology revolves around an exact density-based empirical likelihood ratio test statistic that approximates in a distribution-free fashion the corresponding most powerful parametric likelihood ratio test. We demonstrate that the proposed test is very powerful in detecting general structures of dependence between two random variables, including non-linear and/or random-effect dependence structures. Read More

Am Stat 2014;68(2):83-84

Department of Statistics, University of California, Irvine.

Am Stat 2013;67(4):235-241

Department of Nutritional Sciences, The University of Texas at Austin, Austin, TX 78712.

The aim of this paper is to address issues in research that may be missing from statistics classes and important for (bio-)statistics students. In the context of a case study, we discuss data acquisition and preprocessing steps that fill the gap between research questions posed by subject matter scientists and statistical methodology for formal inference. Issues include participant recruitment, data collection training and standardization, variable coding, data review and verification, data cleaning and editing, and documentation. Read More

Am Stat 2013 Aug;67(3)

Institute for Health Research and Policy, University of Illinois at Chicago, 1747 W. Roosevelt Rd., Chicago, IL 60608.

Several statistical packages are capable of estimating generalized linear mixed models and these packages provide one or more of three estimation methods: penalized quasi-likelihood, Laplace, and Gauss-Hermite. Many studies have investigated these methods' performance for the mixed-effects logistic regression model. However, the authors focused on models with one or two random effects and assumed a simple covariance structure between them, which may not be realistic. Read More

Am Stat 2013 Jan;67(1):42-43

Department of Biostatistics, Columbia University, 722 West 168th Street 6 Floor, New York, NY 10032, USA.

We demonstrate the algebraic equivalence of two unbiased variance estimators for the sample grand mean in a random sample of subjects from an infinite population where subjects provide repeated observations following a homoscedastic random effects model. Read More

Am Stat 2012;66(1):34-41. Epub 2012 Mar 21.

Department of Statistics, Box 354322, University of Washington, Seattle, WA 98195-4322.

We consider the problem of estimating the correlation in bivariate normal data when the means and variances are assumed known, with emphasis on the small sample case. We consider eight different estimators, several of them considered here for the first time in the literature. In a simulation study, we found that Bayesian estimators using the uniform and arc-sine priors outperformed several empirical and exact or approximate maximum likelihood estimators in small samples. Read More

Am Stat 2012;66(1):16-24. Epub 2012 Jun 12.

Statistical Methods Group, Edwards Air Force Base Edwards, CA 93524

Plausibility of high variability in treatment effects across individuals has been recognized as an important consideration in clinical studies. Surprisingly, little attention has been given to evaluating this variability in design of clinical trials or analyses of resulting data. High variation in a treatment's efficacy or safety across individuals (referred to herein as treatment heterogeneity) may have important consequences because the optimal treatment choice for an individual may be different from that suggested by a study of average effects. Read More

Am Stat 2011 Nov;65(4):223-228

Department of Epidemiology & Biostatistics, School of Rural Public Health, Texas A&M Health Science Center, 1266 TAMU, College Station, TX 77843-1266.

When employing model selection methods with oracle properties such as the smoothly clipped absolute deviation (SCAD) and the Adaptive Lasso, it is typical to estimate the smoothing parameter by m-fold cross-validation, for example, m = 10. In problems where the true regression function is sparse and the signals large, such cross-validation typically works well. However, in regression modeling of genomic studies involving Single Nucleotide Polymorphisms (SNP), the true regression functions, while thought to be sparse, do not have large signals. Read More

Am Stat 2012 Jan;65(4):274-282

Institute for Social Research, Center for Statistical Consultation and Research, University of Michigan-Ann Arbor, Ann Arbor, MI, 48109.

At present, there are many software procedures available enabling statisticians to fit linear mixed models (LMMs) to continuous dependent variables in clustered or longitudinal data sets. LMMs are flexible tools for analyzing relationships among variables in these types of data sets, in that a variety of covariance structures can be used depending on the subject matter under study. The explicit random effects in LMMs allow analysts to make inferences about the variability between clusters or subjects in larger hypothetical populations, and examine cluster- or subject-level variables that explain portions of this variability. Read More

Am Stat 2012;66(3):173-179

University of Tennessee Health Science Center, Department of Preventive Medicine 66 N. Pauline Street, Memphis, TN 38105.

Cox proportional hazards (PH) models are commonly used in medical research to investigate the associations between covariates and time to event outcomes. It is frequently noted that with less than ten events per covariate, these models produce spurious results, and therefore, should not be used. Statistical literature contains asymptotic power formulae for the Cox model which can be used to determine the number of events needed to detect an association. Read More

Am Stat 2012;66(3):155-162

Department of Health Studies, University of Chicago, 5841 S. Maryland Ave., MC2007, Chicago, IL 60637.

The randomized discontinuation trial (RDT) design is an enrichment-type design that has been used in a variety of diseases to evaluate the efficacy of new treatments. The RDT design seeks to select a more homogeneous group of patients, consisting of those who are more likely to show a treatment benefit if one exists. In oncology, the RDT design has been applied to evaluate the effects of cytostatic agents, that is, drugs that act primarily by slowing tumor growth rather than shrinking tumors. Read More

Am Stat 2011 Oct;65(4):229-238

José Zubizarreta is a Doctoral Student, and Paul Rosenbaum is a Professor, Department of Statistics, The Wharton School, University of Pennsylvania, 473 Jon M. Huntsman Hall, 3730 Walnut Street, Philadelphia, PA 19104-6340. Caroline E. Reinke is an Instructor of Surgery, Rachel R. Kelz is an Assistant Professor of Surgery, and Jeffrey H. Silber is a Professor of Pediatrics at the University of Pennsylvania School of Medicine, Philadelphia, PA 19104.

Matching for several nominal covariates with many levels has usually been thought to be difficult because these covariates combine to form an enormous number of interaction categories with few if any people in most such categories. Moreover, because nominal variables are not ordered, there is often no notion of a "close substitute" when an exact match is unavailable. In a case-control study of the risk factors for read-mission within 30 days of surgery in the Medicare population, we wished to match for 47 hospitals, 15 surgical procedures grouped or nested within 5 procedure groups, two genders, or 47 × 15 × 2 = 1410 categories. Read More

Am Stat 2011 Feb;65(1):16-20

Duke University, Durham, NC 27708-0251.

Effective component relabeling in Bayesian analyses of mixture models is critical to the routine use of mixtures in classification with analysis based on Markov chain Monte Carlo methods. The classification-based relabeling approach here is computationally attractive and statistically effective, and scales well with sample size and number of mixture components concordant with enabling routine analyses of increasingly large data sets. Building on the best of existing methods, practical relabeling aims to match data:component classification indicators in MCMC iterates with those of a defined reference mixture distribution. Read More

Am Stat 2010 Aug;64(3):263-267

University of Notre Dame.

This paper shows that, when variables with missing values are linearly related to observed variables, the normal-distribution-based pseudo MLEs are still consistent. The population distribution may be unknown while the missing data process can follow an arbitrary missing at random mechanism. Enough details are provided for the bivariate case so that readers having taken a course in statistics/probability can fully understand the development. Read More

Am Stat 2011 Jan;65(3)

Case Western Reserve University Cleveland, OH 44106-7281, USA.

It is an obvious fact that the power of a test statistic is dependent upon the significance (alpha) level at which the test is performed. It is perhaps a less obvious fact that the performance of two statistics in terms of power is also a function of the alpha level. Through numerous personal discussions, we have noted that even some competent statisticians have the mistaken intuition that relative power comparisons at traditional levels such as = 0. Read More

Am Stat 2011;65(1):21-30. Epub 2012 Jan 1.

Division of Biostatistics, College of Public Health, The Ohio State University, B110 Starling-Loving Hall, 320 West 10th Avenue, Columbus, OH 43210.

Matching is a powerful statistical tool in design and analysis. Conventional two-group, or bipartite, matching has been widely used in practice. However, its utility is limited to simpler designs. Read More

Am Stat 2011;65(4):213-221. Epub 2012 Jan 24.

Department of Statistics, North Carolina State University, Raleigh, NC 27695-8203.

P-values are useful statistical measures of evidence against a null hypothesis. In contrast to other statistical estimates, however, their sample-to-sample variability is usually not considered or estimated, and therefore not fully appreciated. Via a systematic study of log-scale p-value standard errors, bootstrap prediction bounds, and reproducibility probabilities for future replicate p-values, we show that p-values exhibit surprisingly large variability in typical data situations. Read More

Am Stat 2009 Nov;63(4):297-307

Post-Doctoral Fellow, Department of Human Genetics, University of California, Los Angeles, CA 90095 ( ).

The movie distribution company Netflix has generated considerable buzz in the statistics community by offering a million dollar prize for improvements to its movie rating system. Among the statisticians and computer scientists who have disclosed their techniques, the emphasis has been on machine learning approaches. This article has the modest goal of discussing a simple model for movie rating and other forms of democratic rating. Read More

Am Stat 2009 Nov;63(4):378-388

Department of Methodology and Statistics, Maastricht University, Peter Debyeplein 1, 6229 HA Maastricht, the Netherlands.

While marginal models, random-effects models, and conditional models are routinely considered to be the three main modeling families for continuous and discrete repeated measures with linear and generalized linear mean structures, respectively, it is less common to consider non-linear models, let alone frame them within the above taxonomy. In the latter situation, indeed, when considered at all, the focus is often exclusively on random-effects models. In this paper, we consider all three families, exemplify their great flexibility and relative ease of use, and apply them to a simple but illustrative set of data on tree circumference growth of orange trees. Read More

Am Stat 2009 May;63(2):147-154

Department of Biophysics and Department of Biostatistics, Johns Hopkins University.

Equivalence testing is growing in use in scientific research outside of its traditional role in the drug approval process. Largely due to its ease of use and recommendation from the United States Food and Drug Administration guidance, the most common statistical method for testing equivalence is the two one-sided tests procedure (TOST). Like classical point-null hypothesis testing, TOST is subject to multiplicity concerns as more comparisons are made. Read More

Am Stat 2008 Nov;62(4):307-313

Cancer Statistical Center, Duke University Medical Center, Hock Plaza, 2424 Erwin Road, Suite 802, Durham, NC 27705 (E-mail: ).

We propose two innovations in statistical sampling for controls to enable better design of population-based case-control studies. The main innovation leads to novel solutions, without using weights, of the difficult and long-standing problem of selecting a control from persons in a household. Another advance concerns the drawing (at the outset) of the households themselves and involves random-digit dialing with atypical use of list-assisted sampling. Read More

Am Stat 2009 May;63(2):155-162

Department of Biostatistics, Vanderbilt University, Nashville, TN 37232.

Statistical experiments, more commonly referred to as Monte Carlo or simulation studies, are used to study the behavior of statistical methods and measures under controlled situations. Whereas recent computing and methodological advances have permitted increased efficiency in the simulation process, known as variance reduction, such experiments remain limited by their finite nature and hence are subject to uncertainty; when a simulation is run more than once, different results are obtained. However, virtually no emphasis has been placed on reporting the uncertainty, referred to here as Monte Carlo error, associated with simulation results in the published literature, or on justifying the number of replications used. Read More