Walter W Piegorsch
 Professor, Mathematics
 Professor, Public Health
 Director, Statistical Research and Education
 Professor, AgriculturalBiosystems Engineering
 Professor, BIO5 Institute
 Professor, Applied Mathematics  GIDP
 Professor, StatisticsGIDP
 Member of the Graduate Faculty
 (520) 6212357
 Bioscience Research Labs, Rm. 230
 Tucson, AZ 85721
 wpiegors@arizona.edu
Biography
Walter W. Piegorsch, Ph.D., PStat(ASA), is the Director of Statistical Research & Education at the University of Arizona’s BIO5 Institute. He is also a Professor of Mathematics, a Professor of Public Health, and a Member and former Chair of the University’s Graduate Interdisciplinary Program (GIDP) in Statistics. Dr. Piegorsch studies data science for environmental problems, with emphasis on informatics for environmental hazards and risk assessment. He coordinates these interests with his research translating quantitative risk analytics to problems in public health, including geospatially referenced disaster informatics; multiple/simultaneous inferences for toxicological and genetic endpoints; and the historical development of statistical thought as prompted by problems in the biological and environmental sciences. He currently leads a team developing statistical methods for estimating benchmark dose markers in environmental hazard analyses. This research has been funded by the U.S. National Institute of Environmental Health Sciences, the U.S. Environmental Protection Agency, and the U.S. National Cancer Institute. He also has constructed statistical models for data from transgenic biotechnologies, developed guidelines for the design of bioassays in select transgenic animal systems, and has proposed retrospective designs for analyzing geneenvironment and genenutrient interactions in human population studies.
Degrees
 Ph.D. Statistics
 Cornell University, Ithaca, New York, United States
 Admissible and Optimal Confidence Bands in Linear Regression
 M.S. Statistics
 Cornell University, Ithaca, New York, United States
 A Modification of the Least Squares Join Point Estimator in Bilinear Segmented Regression
 B.A. Mathematics
 Colgate University, Hamilton, New York, United States
Work Experience
 University of Arizona, Tucson, Arizona (2006  Ongoing)
 South Carolina Cancer Center (2004  2006)
 University of South Carolina (1993  2006)
 University of North Carolina, Chapel Hill, North Carolina (1993  2002)
 North Carolina State University (1988  1993)
 National Institute of Environmental Health Sciences (1984  1993)
Awards
 Fellow
 American Statistical Association, Summer 1995
 Distinguished Achievement Medal
 American Statistical Association Section on Statistics & the Environment, Summer 1993
 University of South Carolina Educational Foundation Research Award for Science, Mathematics, and Engineering
 University of South Carolina, Spring 2000
Licensure & Certification
 Accredited Professional Statistician (PStat®), American Statistical Association (2010)
Interests
Research
Environmental Statistics/Environmetrics, Quantitative Risk Assessment, Informatics for Precision Medicine
Teaching
Statistical Computing, Supervised Learning, Statistical Learning, Environmental Statistics
Courses
202324 Courses

Dissertation
BIOS 920 (Spring 2024) 
Dissertation
BIOS 920 (Fall 2023)
202223 Courses

Dissertation
BIOS 920 (Spring 2023) 
Adv Stat Regress Analys
MATH 571A (Fall 2022) 
Adv Stat Regress Analys
STAT 571A (Fall 2022) 
Dissertation
BIOS 920 (Fall 2022) 
Dissertation
STAT 920 (Fall 2022)
202122 Courses

Dissertation
BIOS 920 (Spring 2022) 
Dissertation
STAT 920 (Spring 2022) 
Statistical Computing
STAT 675 (Spring 2022) 
Adv Stat Regress Analys
MATH 571A (Fall 2021) 
Adv Stat Regress Analys
STAT 571A (Fall 2021) 
Dissertation
STAT 920 (Fall 2021) 
Research
BIOS 900 (Fall 2021)
202021 Courses

Dissertation
STAT 920 (Spring 2021) 
Dissertation
STAT 920 (Fall 2020) 
Independent Study
STAT 599 (Fall 2020)
201920 Courses

Dissertation
STAT 920 (Spring 2020) 
Statistical Computing
STAT 675 (Spring 2020) 
Thesis
STAT 910 (Spring 2020) 
Adv Stat Regress Analys
MATH 571A (Fall 2019) 
Adv Stat Regress Analys
STAT 571A (Fall 2019) 
Independent Study
STAT 599 (Fall 2019)
201819 Courses

Thesis
STAT 910 (Spring 2019) 
Adv Stat Regress Analys
MATH 571A (Fall 2018) 
Adv Stat Regress Analys
STAT 571A (Fall 2018) 
Independent Study
STAT 599 (Fall 2018)
201718 Courses

Statistical Computing
STAT 675 (Spring 2018) 
Adv Stat Regress Analys
MATH 571A (Fall 2017) 
Adv Stat Regress Analys
STAT 571A (Fall 2017)
201617 Courses

Dissertation
STAT 920 (Summer I 2017) 
Dissertation
STAT 920 (Spring 2017) 
Adv Stat Regress Analys
MATH 571A (Fall 2016) 
Adv Stat Regress Analys
STAT 571A (Fall 2016) 
Dissertation
STAT 920 (Fall 2016)
201516 Courses

Dissertation
STAT 920 (Spring 2016) 
Intro:Stat+Biostatistics
MATH 263 (Spring 2016) 
Statistical Computing
STAT 675 (Spring 2016)
Scholarly Contributions
Books
 Piegorsch, W. W., Levine, R. A., Zhang, H., & Lee, T. C. (2022). Computational Statistics in Data Science. Chichester: John Wiley & Sons.
 Piegorsch, W. W. (2015). Statistical Data Analytics.
 Balakrishnan, N., Brandimarte, P., Everitt, B., Molenberghs, G., Piegorsch, W. W., & Ruggeri, F. (2014). Wiley StatsRef: Statistics Reference Online. Chichester: John Wiley & Sons.
 Cutter, S. L., Emrich, C. T., Mitchell, J. T., Piegorsch, W. W., Smith, M. M., & Weber, L. (2014). Hurricane Katrina and the Forgotten Coast of Mississippi. Cambridge: Cambridge University Press.
 Piegorsch, W. W. (2012). Hurricane Katrina and the forgotten coast of Mississippi.
 Piegorsch, W. W. (2009). Genenutrient interactions in nutritional epidemiology.
 Piegorsch, W. W. (2005). Analyzing Environmental Data.
 Piegorsch, W. W. (2000). 14 Quantitative potency estimation to measure risk with bioenvironmental hazards.
 Piegorsch, W. W. (1994). 15 Environmental biometry: Assessing impacts of environmental stimuli via animal and microbial laboratory studies.
Chapters
 Piegorsch, W. W. (2015). Joint action models. In Wiley StatsRef: Statistics Reference Online. Chichester: John Wiley & Sons. doi:10.1002/9781118445112.stat07719.pub2
 Piegorsch, W. W. (2014). Binary data and quantal response. In Wiley StatsRef: Statistics Reference Online(pp doi:10.1002/9781118445112.stat07340). Chichester: John Wiley & Sons.
 Piegorsch, W. W. (2014). Dispersion parameter. In Wiley StatsRef: Statistics Reference Online(pp doi:10.1002/9781118445112.stat07375). Chichester: John Wiley & Sons.
 Piegorsch, W. W. (2014). Distribution function. In Wiley StatsRef: Statistics Reference Online(pp doi:10.1002/9781118445112.stat07524). Chichester: John Wiley & Sons.
 Piegorsch, W. W. (2014). Environmental mutagenesis, Statistics in. In Wiley StatsRef: Statistics Reference Online(pp doi:10.1002/9781118445112.stat03788). Chichester: John Wiley & Sons.
 Piegorsch, W. W. (2014). Joint action models. In Wiley StatsRef: Statistics Reference Online(pp doi:10.1002/9781118445112.stat07719). Chichester: John Wiley & Sons.
 Piegorsch, W. W. (2014). Lowdose extrapolation. In Wiley StatsRef: Statistics Reference Online(pp doi:10.1002/9781118445112.stat03804). Chichester: John Wiley & Sons.
 Piegorsch, W. W. (2014). Mutagenicity study. In Wiley StatsRef: Statistics Reference Online(pp doi:10.1002/9781118445112.stat05476). Chichester: John Wiley & Sons.
 Piegorsch, W. W. (2014). Potency estimation. In Wiley StatsRef: Statistics Reference Online(pp doi:10.1002/9781118445112.stat03811). Chichester: John Wiley & Sons.
 Piegorsch, W. W. (2014). Proportional hazards model: Introduction. In Wiley StatsRef: Statistics Reference Online(pp doi:10.1002/9781118445112.stat07448). Chichester: John Wiley & Sons.
 Piegorsch, W. W. (2014). Quantal response data. In Wiley StatsRef: Statistics Reference Online(pp doi:10.1002/9781118445112.stat07556). Chichester: John Wiley & Sons.
 Piegorsch, W. W. (2014). Random effects. In Wiley StatsRef: Statistics Reference Online(pp doi:10.1002/9781118445112.stat07558). Chichester: John Wiley & Sons.
 Piegorsch, W. W. (2014). ZStatistic. In Wiley StatsRef: Statistics Reference Online(pp doi:10.1002/9781118445112.stat07427). Chichester: John Wiley & Sons.
 Piegorsch, W. W., & Bailer, A. J. (2014). Combining information. In Wiley StatsRef: Statistics Reference Online(pp doi:10.1002/9781118445112.stat03704). Chichester: John Wiley & Sons.
Journals/Publications
 Aberasturi, D. T., Piegorsch, W. W., Bedrick, E. J., & Lussier, Y. A. (2022). Accounting for extrabinomial variability with differentially expressed genetic pathway data: a collaborative bioinformatic study. Stat, 12, Article No. e518. doi:10.1002/sta4.518
 Liu, J., Piegorsch, W. W., Cutter, S. L., McCaster, R. R., & Schissler, A. G. (2022). Adjusting statistical benchmark risk analysis to account for nonspatial autocorrelation, with application to natural hazard risk assessment. Journal of Applied Statistics, 49, 23492369. doi:10.1080/02664763.2021.1904385
 Aberasturi, D., Pouladi, N., Zaim, S. R., Kenost, C., Berghout, J., Piegorsch, W. W., & Lussier, Y. A. (2021). 'Singlesubject studies'derived analyses unveil altered biomechanisms between very small cohorts: implications for rare diseases. Bioinformatics (Oxford, England), 37(Suppl_1), i67i75.More infoIdentifying altered transcripts between very small human cohorts is particularly challenging and is compounded by the low accrual rate of human subjects in rare diseases or substratified common disorders. Yet, singlesubject studies (S3) can compare paired transcriptome samples drawn from the same patient under two conditions (e.g. treated versus pretreatment) and suggest patientspecific responsive biomechanisms based on the overrepresentation of functionally defined gene sets. These improve statistical power by: (i) reducing the total features tested and (ii) relaxing the requirement of withincohort uniformity at the transcript level. We propose InterNof1, a novel method, to identify meaningful differences between very small cohorts by using the effect size of 'singlesubjectstudy'derived responsive biological mechanisms.
 Piegorsch, W. W., McCaster, R. R., & Cutter, S. L. (2021). From terrorism to flooding: how vulnerable is your city?. Significance, 18(1), 2227. doi:10.1111/17409713.01487
 SansFuentes, M. A., & Piegorsch, W. W. (2021). Benchmark dose risk analysis with mixedfactor quantal data in environmental risk assessment. Environmetrics, 32(5), Article No. e2677.
 Wheeler, M. W., Piegorsch, W. W., & Bailer, A. J. (2019). Quantal Risk Assessment Database: A Database for Exploring Patterns in Quantal DoseResponse Data in Risk Assessment and its Application to Develop Priors for Bayesian DoseResponse Analysis. Risk analysis : an official publication of the Society for Risk Analysis, 39(3), 616629.More infoQuantitative risk assessments for physical, chemical, biological, occupational, or environmental agents rely on scientific studies to support their conclusions. These studies often include relatively few observations, and, as a result, models used to characterize the risk may include large amounts of uncertainty. The motivation, development, and assessment of new methods for risk assessment is facilitated by the availability of a set of experimental studies that span a range of doseresponse patterns that are observed in practice. We describe construction of such a historical database focusing on quantal data in chemical risk assessment, and we employ this database to develop priors in Bayesian analyses. The database is assembled from a variety of existing toxicological data sources and contains 733 separate quantal doseresponse data sets. As an illustration of the database's use, prior distributions for individual model parameters in Bayesian doseresponse analysis are constructed. Results indicate that including prior information based on curated historical data in quantitative risk assessments may help stabilize eventual point estimates, producing doseresponse functions that are more stable and precisely estimated. These in turn produce potency estimates that share the same benefit. We are confident that quantitative risk analysts will find many other applications and issues to explore using this database.
 Piegorsch, W. W. (2018). Autologistic models for benchmark risk or vulnerability assessment of urban terrorism outcomes. Journal of the Royal Statistical Society. Series A: Statistics in Society.
 Peña, E. A., Wu, W., Piegorsch, W. W., West, R. W., & An, L. (2017). Model Selection and Estimation with QuantalResponse Data in Benchmark Risk Assessment. Risk analysis : an official publication of the Society for Risk Analysis, 37(4), 716732. doi:10.1111_risa.12644More infoThis article describes several approaches for estimating the benchmark dose (BMD) in a risk assessment study with quantal doseresponse data and when there are competing model classes for the doseresponse function. Strategies involving a twostep approach, a modelaveraging approach, a focusedinference approach, and a nonparametric approach based on a PAVAbased estimator of the doseresponse function are described and compared. Attention is raised to the perils involved in data "doubledipping" and the need to adjust for the modelselection stage in the estimation procedure. Simulation results are presented comparing the performance of five model selectors and eight BMD estimators. An illustration using a real quantalresponse data set from a carcinogenecity study is provided.
 Piegorsch, W. W. (2017). Are pvalues under attack? Contribution to the discussion of 'A critical evaluation of the current "pvalue controversy" '. Biometrical journal. Biometrische Zeitschrift, 59(5), 889891.
 Schissler, A. G., Li, Q., Chen, J. L., Kenost, C., Achour, I., Billheimer, D. D., Li, H., Piegorsch, W. W., & Lussier, Y. A. (2016). Analysis of aggregated cellcell statistical distances within pathways unveils therapeuticresistance mechanisms in circulating tumor cells. Bioinformatics (Oxford, England), 32(12), i80i89.More infoAs 'omics' biotechnologies accelerate the capability to contrast a myriad of molecular measurements from a single cell, they also exacerbate current analytical limitations for detecting meaningful singlecell dysregulations. Moreover, mRNA expression alone lacks functional interpretation, limiting opportunities for translation of singlecell transcriptomic insights to precision medicine. Lastly, most singlecell RNAsequencing analytic approaches are not designed to investigate small populations of cells such as circulating tumor cells shed from solid tumors and isolated from patient blood samples.
 Fang, Q., Piegorsch, W. W., Simmons, S. J., Li, X., Chen, C., & Wang, Y. (2015). Bayesian modelaveraged benchmark dose analysis via reparameterized quantalresponse models. Biometrics, 71(4), 116875.More infoAn important objective in biomedical and environmental risk assessment is estimation of minimum exposure levels that induce a prespecified adverse response in a target population. The exposure points in such settings are typically referred to as benchmark doses (BMDs). Parametric Bayesian estimation for finding BMDs has grown in popularity, and a large variety of candidate doseresponse models is available for applying these methods. Each model can possess potentially different parametric interpretation(s), however. We present reparameterized doseresponse models that allow for explicit use of prior information on the target parameter of interest, the BMD. We also enhance our Bayesian estimation technique for BMD analysis by applying Bayesian model averaging to produce point estimates and (lower) credible bounds, overcoming associated questions of model adequacy when multimodel uncertainty is present. An example from carcinogenicity testing illustrates the calculations.
 Piegorsch, W. W. (2015). Bayesian benchmark dose analysis. Environmetrics.
 Piegorsch, W. W. (2015). Bayesian model averaging for benchmark dose estimation. Environmental and Ecological Statistics.
 Piegorsch, W. W. (2015). Nonparametric Benchmark Dose Estimation with Continuous DoseResponse Data. Scandinavian Journal of Statistics.
 Schissler, A. G., Gardeux, V., Li, Q., Achour, I., Li, H., Piegorsch, W. W., & Lussier, Y. A. (2015). Dynamic changes of RNAsequencing expression for precision medicine: Nof1pathways Mahalanobis distance within pathways of single subjects predicts breast cancer survival. Bioinformatics (Oxford, England), 31(12), i293302.More infoThe conventional approach to personalized medicine relies on molecular data analytics across multiple patients. The path to precision medicine lies with molecular data analytics that can discover interpretable singlesubject signals (Nof1). We developed a global framework, Nof1pathways, for a mechanisticanchored approach to singlesubject gene expression data analysis. We previously employed a metric that could prioritize the statistical significance of a deregulated pathway in single subjects, however, it lacked in quantitative interpretability (e.g. the equivalent to a gene expression foldchange).
 Piegorsch, W. W. (2014). A pooladjacentviolatorsalgorithm approach to detect infinite parameter estimates in oneregressor doseresponse models with asymptotes. Journal of Statistical Computation and Simulation.
 Piegorsch, W. W. (2014). Benchmark dose analysis via nonparametric regression modeling. Risk Analysis.
 Piegorsch, W. W. (2014). Environmetrics silver anniversary special issue. Environmetrics.
 Piegorsch, W. W. (2014). Model uncertainty in environmental doseresponse risk analysis. Statistics and Public Policy, 1(1), 7985.
 Piegorsch, W. W., Xiong, H., Bhattacharya, R. N., & Lin, L. (2014). Benchmark Dose Analysis via Nonparametric Regression Modeling. Risk analysis : an official publication of the Society for Risk Analysis, 34(1), 13551.More infoEstimation of benchmark doses (BMDs) in quantitative risk assessment traditionally is based upon parametric doseresponse modeling. It is a wellknown concern, however, that if the chosen parametric model is uncertain and/or misspecified, inaccurate and possibly unsafe lowdose inferences can result. We describe a nonparametric approach for estimating BMDs with quantalresponse data based on an isotonic regression method, and also study use of corresponding, nonparametric, bootstrapbased confidence limits for the BMD. We explore the confidence limits' smallsample properties via a simulation study, and illustrate the calculations with an example from cancer risk assessment. It is seen that this nonparametric approach can provide a useful alternative for BMD estimation when faced with the problem of parametric model uncertainty.
 Deutsch, R. C., & Piegorsch, W. W. (2013). Benchmark dose profiles for jointaction continuous data in quantitative risk assessment. Biometrical journal. Biometrische Zeitschrift, 55(5), 74154.More infoBenchmark analysis is a widely used tool in biomedical and environmental risk assessment. Therein, estimation of minimum exposure levels, called benchmark doses (BMDs), that induce a prespecified benchmark response (BMR) is well understood for the case of an adverse response to a single stimulus. For cases where two agents are studied in tandem, however, the benchmark approach is far less developed. This paper demonstrates how the benchmark modeling paradigm can be expanded from the singleagent setting to jointaction, twoagent studies. Focus is on continuous response outcomes. Extending the singleexposure setting, representations of risk are based on a jointaction doseresponse model involving both agents. Based on such a model, the concept of a benchmark profilea twodimensional analog of the singledose BMD at which both agents achieve the specified BMRis defined for use in quantitative risk characterization and assessment.
 Piegorsch, W. W. (2013). In memory of George Casella. Environmetrics.
 Piegorsch, W. W., An, L., Wickens, A. A., West, R. W., Peña, E. A., & Wu, W. (2013). Informationtheoretic modelaveraged benchmark dose analysis in environmental risk assessment. Environmetrics, 24(3), 143157.More infoAn important objective in environmental risk assessment is estimation of minimum exposure levels, called Benchmark Doses (BMDs), that induce a prespecified Benchmark Response (BMR) in a doseresponse experiment. In such settings, representations of the risk are traditionally based on a specified parametric model. It is a wellknown concern, however, that existing parametric estimation techniques are sensitive to the form employed for modeling the dose response. If the chosen parametric model is in fact misspecified, this can lead to inaccurate lowdose inferences. Indeed, avoiding the impact of model selection was one early motivating issue behind development of the BMD technology. Here, we apply a frequentist model averaging approach for estimating benchmark doses, based on informationtheoretic weights. We explore how the strategy can be used to build onesided lower confidence limits on the BMD, and we study the confidence limits' smallsample properties via a simulation study. An example from environmental carcinogenicity testing illustrates the calculations. It is seen that application of this informationtheoretic, model averaging methodology to benchmark analysis can improve environmental health planning and risk regulation when dealing with lowlevel exposures to hazardous agents.
 Deutsch, R. C., & Piegorsch, W. W. (2012). Benchmark dose profiles for jointaction quantal data in quantitative risk assessment. Biometrics, 68(4), 131322.More infoBenchmark analysis is a widely used tool in public health risk analysis. Therein, estimation of minimum exposure levels, called Benchmark Doses (BMDs), that induce a prespecified Benchmark Response (BMR) is well understood for the case of an adverse response to a single stimulus. For cases where two agents are studied in tandem, however, the benchmark approach is far less developed. This article demonstrates how the benchmark modeling paradigm can be expanded from the singledose setting to jointaction, twoagent studies. Focus is on response outcomes expressed as proportions. Extending the singleexposure setting, representations of risk are based on a jointaction doseresponse model involving both agents. Based on such a model, the concept of a benchmark profile (BMP)  a twodimensional analog of the singledose BMD at which both agents achieve the specified BMR  is defined for use in quantitative risk characterization and assessment. The resulting, joint, lowdose guidelines can improve public health planning and risk regulation when dealing with lowlevel exposures to combinations of hazardous agents.
 Piegorsch, W. W., Xiong, H., Bhattacharya, R. N., & Lin, L. (2012). Nonparametric estimation of benchmark doses in environmental risk assessment. Environmetrics, 23(8), 717728.More infoAn important statistical objective in environmental risk analysis is estimation of minimum exposure levels, called benchmark doses (BMDs), that induce a prespecified benchmark response in a doseresponse experiment. In such settings, representations of the risk are traditionally based on a parametric doseresponse model. It is a wellknown concern, however, that if the chosen parametric form is misspecified, inaccurate and possibly unsafe lowdose inferences can result. We apply a nonparametric approach for calculating benchmark doses, based on an isotonic regression method for doseresponse estimation with quantalresponse data (Bhattacharya and Kong, 2007). We determine the largesample properties of the estimator, develop bootstrapbased confidence limits on the BMDs, and explore the confidence limits' smallsample properties via a short simulation study. An example from cancer risk assessment illustrates the calculations.
 Shane, B. S., Zeiger, E., Piegorsch, W. W., Booth, E. D., Goodman, J. I., & Peffer, R. C. (2012). Reevaluation of the Big Blue® mouse assay of propiconazole suggests lack of mutagenicity. Environmental and molecular mutagenesis, 53(1), 19.More infoPropiconazole (PPZ) is a conazole fungicide that is not mutagenic, clastogenic, or DNA damaging in standard in vitro and in vivo genetic toxicity tests for gene mutations, chromosome aberrations, DNA damage, and cell transformation. However, it was demonstrated to be a male mouse liver carcinogen when administered in food for 24 months only at a concentration of 2,500 ppm that exceeded the maximum tolerated dose based on increased mortality, decreased body weight gain, and the presence of liver necrosis. PPZ was subsequently tested for mutagenicity in the Big Blue® transgenic mouse assay at the 2,500 ppm dose, and the result was reported as positive by Ross et al. ([2009]: Mutagenesis 24:149152). Subsets of the mutants from the control and PPZexposed groups were sequenced to determine the mutation spectra and a multivariate clustering analysis method purportedly substantiated the increase in mutant frequency with PPZ (Ross and Leavitt. [2010]: Mutagenesis 25:231234). However, as reported here, the results of the analysis of the mutation spectra using a conventional method indicated no treatmentrelated differences in the spectra. In this article, we reexamine the Big Blue® mouse findings with PPZ and conclude that the compound does not act as a mutagen in vivo.
 West, R. W., Piegorsch, W. W., Peña, E. A., An, L., Wu, W., Wickens, A. A., Xiong, H., & Chen, W. (2012). The Impact of Model Uncertainty on Benchmark Dose Estimation. Environmetrics, 23(8), 706716.More infoWe study the popular benchmark dose (BMD) approach for estimation of low exposure levels in toxicological risk assessment, focusing on doseresponse experiments with quantal data. In such settings, representations of the risk are traditionally based on a specified, parametric, doseresponse model. It is a wellknown concern, however, that uncertainty can exist in specification and selection of the model. If the chosen parametric form is in fact misspecified, this can lead to inaccurate, and possibly unsafe, lowdose inferences. We study the effects of model selection and possible misspecification on the BMD, on its corresponding lower confidence limit (BMDL), and on the associated extra risks achieved at these values, via largescale Monte Carlo simulation. It is seen that an uncomfortably high percentage of instances can occur where the true extra risk at the BMDL under a misspecified or incorrectly selected model can surpass the target BMR, exposing potential dangers of traditional strategies for model selection when calculating BMDs and BMDLs.
 Deutsch, R. C., Grego, J. M., Habing, B., & Piegorsch, W. W. (2010). Maximum likelihood estimation with binarydata regression models: smallsample and largesample features. Advances and applications in statistics, 14(2), 101116.More infoMany inferential procedures for generalized linear models rely on the asymptotic normality of the maximum likelihood estimator (MLE). Fahrmeir & Kaufmann (1985, Ann. Stat., 13, 1) present mild conditions under which the MLEs in GLiMs are asymptotically normal. Unfortunately, limited study has appeared for the special case of binomial response models beyond the familiar logit and probit links, and for more general links such as the complementary loglog link, and the less wellknown complementary log link. We verify the asymptotic normality conditions of the MLEs for these models under the assumption of a fixed number of experimental groups and present a simple set of conditions for any twice differentiable monotone link function. We also study the quality of the approximation for constructing asymptotic Wald confidence regions. Our results show that for small sample sizes with certain link functions the approximation can be problematic, especially for cases where the parameters are close to the boundary of the parameter space.
 Guttorp, P., & Piegorsch, W. W. (2010). Editorial. Environmetrics, 21(1), 12.
 Piegorsch, W., & Piegorsch, W. W. (2010). Translational benchmark risk analysis. Journal of risk research, 13(5).More infoTranslational development  in the sense of translating a mature methodology from one area of application to another, evolving area  is discussed for the use of benchmark doses in quantitative risk assessment. Illustrations are presented with traditional applications of the benchmark paradigm in biology and toxicology, and also with risk endpoints that differ from traditional toxicological archetypes. It is seen that the benchmark approach can apply to a diverse spectrum of risk management settings. This suggests a promising future for this important riskanalytic tool. Extensions of the method to a wider variety of applications represent a significant opportunity for enhancing environmental, biomedical, industrial, and socioeconomic risk assessments.
 Buckley, B. E., Piegorsch, W. W., & West, R. W. (2009). Confidence limits on onestage model parameters in benchmark risk assessment. Environmental and ecological statistics, 16(1), 5362.More infoIn modern environmental risk analysis, inferences are often desired on those low dose levels at which a fixed benchmark risk is achieved. In this paper, we study the use of confidence limits on parameters from a simple onestage model of risk historically popular in benchmark analysis with quantal data. Based on these confidence bounds, we present methods for deriving upper confidence limits on extra risk and lower bounds on the benchmark dose. The methods are seen to extend automatically to the case where simultaneous inferences are desired at multiple doses. Monte Carlo evaluations explore characteristics of the parameter estimates and the confidence limits under this setting.
 Liu, W., Hayter, A. J., & Piegorsch, W. W. (2009). Comparison of Hyperbolic and Constant Width Simultaneous Confidence Bands in Multiple Linear Regression under MVCS Criterion. Journal of multivariate analysis, 100(7), 14321439.More infoA simultaneous confidence band provides useful information on the plausible range of the unknown regression model, and different confidence bands can often be constructed for the same regression model. For a simple regression line, it is proposed in Liu and Hayter (2007) to use the area of the confidence set that corresponds to a confidence band as an optimality criterion in comparison of confidence bands; the smaller is the area of the confidence set, the better is the corresponding confidence band. This minimum area confidence set (MACS) criterion can clearly be generalized to the minimum volume confidence set (MVCS) criterion in study of confidence bands for a multiple linear regression model. In this paper the hyperbolic and constant width confidence bands for a multiple linear regression model over a particular ellipsoidal region of the predictor variables are compared under the MVCS criterion. It is observed that whether one band is better than the other depends on the magnitude of one particular angle that determines the size of the predictor variable region. When the angle and so the size of the predictor variable region is small, the constant width band is better than the hyperbolic band but only marginally. When the angle and so the size of the predictor variable region is large the hyperbolic band can be substantially better than the constant width band.
 Piegorsch, W. W. (2009). Introduction: Modern benchmark analysis for environmental risk assessment. Environmental and Ecological Statistics.
 Piegorsch, W. W., & Bailer, A. J. (2009). Combining information. Wiley interdisciplinary reviews. Computational statistics, 1(3), 354360.More infoThe combination of information from diverse sources is a common task encountered in computational statistics. A popular label for analyses involving the combination of results from independent studies is metaanalysis. The goal of the methodology is to bring together results of different studies, reanalyze the disparate results within the context of their common endpoints, synthesize where possible into a single summary endpoint, increase the sensitivity of the analysis to detect the presence of adverse effects, and provide a quantitative analysis of the phenomenon of interest based on the combined data. This entry discusses some basic methods in metaanalytic calculations, and includes commentary on how to combine or average results from multiple models applied to the same set of data.
 West, R. W., Nitcheva, D. K., & Piegorsch, W. W. (2009). Bootstrap methods for simultaneous benchmark analysis with quantal response data. Environmental and ecological statistics, 16(1), 6373.More infoA primary objective in quantitative risk assessment is the characterization of risk which is defined to be the likelihood of an adverse effect caused by an environmental toxin or chemcial agent. In modern riskbenchmark analysis, attention centers on the "benchmark dose" at which a fixed benchmark level of risk is achieved, with a lower confidence limits on this dose being of primary interest. In practice, a range of benchmark risks may be under study, so that the individual lower confidence limits on benchmark dose must be corrected for simultaneity in order to maintain a specified overall level of confidence. For the case of quantal data, simultaneous methods have been constructed that appeal to the large sample normality of parameter estimates. The suitability of these methods for use with small sample sizes will be considered. A new bootstrap technique is proposed as an alternative to the large sample methodology. This technique is evaluated via a simulation study and examples from environmental toxicology.
 Buckley, B. E., & Piegorsch, W. W. (2008). Simultaneous Confidence Bands for AbbottAdjusted Quantal Response Models. Statistical methodology, 5(3), 209219.More infoWe study use of a Schefféstyle simultaneous confidence band as applied to lowdose risk estimation with quantal response data. We consider two formulations for the doseresponse risk function, an Abbottadjusted Weibull model and an Abbottadjusted loglogistic model. Using the simultaneous construction, we derive methods for estimating upper confidence limits on predicted extra risk and, by inverting the upper bands on risk, lower bounds on the benchmark dose, or BMD, at which a specific level of 'benchmark risk' is attained. Monte Carlo evaluations explore the operating characteristics of the simultaneous limits.
 Piegorsch, W. W. (2008). Construction of exact simultaneous confidence bands for a simple linear regression model. International Statistical Review.
 Piegorsch, W. W., & Schuler, E. (2008). Communicating the risks, and the benefits, of nanotechnology. International journal of risk assessment and management, 10(12), 5769.More infoIssues surrounding the wide spectrum of (perceived) risks and possible benefits associated with the rapid advance of modern nanotechnology are deliberated. These include the current realities of nanotechnological hazards, their impact visàvis perceived nanotechrisks and perceived nanotechbenefits, and the consequent repercussions on the public and society. It is argued that both the risks and the benefits of nanoscientific advances must be properly communicated if the public is to support this emerging technology.
 Schmidtlein, M. C., Deutsch, R. C., Piegorsch, W. W., & Cutter, S. L. (2008). A sensitivity analysis of the social vulnerability index. Risk analysis : an official publication of the Society for Risk Analysis, 28(4), 1099114.More infoThe Social Vulnerability Index (SoVI), created by Cutter et al. (2003), examined the spatial patterns of social vulnerability to natural hazards at the county level in the United States in order to describe and understand the social burdens of risk. The purpose of this article is to examine the sensitivity of quantitative features underlying the SoVI approach to changes in its construction, the scale at which it is applied, the set of variables used, and to various geographic contexts. First, the SoVI was calculated for multiple aggregation levels in the State of South Carolina and with a subset of the original variables to determine the impact of scalar and variable changes on index construction. Second, to test the sensitivity of the algorithm to changes in construction, and to determine if that sensitivity was constant in various geographic contexts, census data were collected at a submetropolitan level for three study sites: Charleston, SC; Los Angeles, CA; and New Orleans, LA. Fiftyfour unique variations of the SoVI were calculated for each study area and evaluated using factorial analysis. These results were then compared across study areas to evaluate the impact of changing geographic context. While decreases in the scale of aggregation were found to result in decreases in the variance explained by principal components analysis (PCA), and in increases in the variance of the resulting index values, the subjective interpretations yielded from the SoVI remained fairly stable. The algorithm's sensitivity to certain changes in index construction differed somewhat among the study areas. Understanding the impacts of changes in index construction and scale are crucial in increasing user confidence in metrics designed to represent the extremely complex phenomenon of social vulnerability.
 Nitcheva, D. K., Piegorsch, W. W., & West, R. W. (2007). On use of the multistage doseresponse model for assessing laboratory animal carcinogenicity. Regulatory toxicology and pharmacology : RTP, 48(2), 13547.More infoWe explore how well a statistical multistage model describes doseresponse patterns in laboratory animal carcinogenicity experiments from a large database of quantal response data. The data are collected from the US EPA's publicly available IRIS data warehouse and examined statistically to determine how often higherorder values in the multistage predictor yield significant improvements in explanatory power over lowerorder values. Our results suggest that the addition of a secondorder parameter to the model only improves the fit about 20% of the time, while adding even higherorder terms apparently does not contribute to the fit at all, at least with the study designs we captured in the IRIS database. Also included is an examination of statistical tests for assessing significance of higherorder terms in a multistage doseresponse model. It is noted that bootstrap testing methodology appears to offer greater stability for performing the hypothesis tests than a morecommon, but possibly unstable, "Wald" test.
 Piegorsch, W. W. (2007). Vulnerability of U.S. cities to environmental hazards. Journal of Homeland Security and Emergency Management.
 Piegorsch, W. W., Cutter, S. L., & Hardisty, F. (2007). Benchmark analysis for quantifying urban vulnerability to terrorist incidents. Risk analysis : an official publication of the Society for Risk Analysis, 27(6), 141125.More infoWe describe a quantitative methodology to characterize the vulnerability of U.S. urban centers to terrorist attack, using a placebased vulnerability index and a database of terrorist incidents and related human casualties. Via generalized linear statistical models, we study the relationships between vulnerability and terrorist events, and find that our placebased vulnerability metric significantly describes both terrorist incidence and occurrence of human casualties from terrorist events in these urban centers. We also introduce benchmark analytic technologies from applications in toxicological risk assessment to this social risk/vulnerability paradigm, and use these to distinguish levels of high and low urban vulnerability to terrorism. It is seen that the benchmark approach translates quite flexibly from its biological roots to this social scientific archetype.
 Piegorsch, K. M., Watkins, K. W., Piegorsch, W. W., Reininger, B., Corwin, S. J., & Valois, R. F. (2006). Ergonomic decisionmaking: a conceptual framework for experienced practitioners from backgrounds in industrial engineering and physical therapy. Applied Ergonomics, 37(5), 58798.More infoErgonomists play an important role in preventing and controlling workrelated injuries and illnesses, yet little is known about the decisionmaking processes that lead to their recommendations. This study (1) generated a datagrounded conceptual framework, based on schema theory, for ergonomic decisionmaking by experienced practitioners in the USA and (2) assessed the adequacy of that framework for describing the decisionmaking of ergonomics practitioners from backgrounds in industrial engineering (IE) and physical therapy (PT). A combination of qualitative and quantitative analyses, within and across 54 decisionmaking situations derived from indepth interviews with 21 practitioners, indicated that a single framework adequately describes the decisionmaking of experienced practitioners from these backgrounds. Results indicate that demands of the practitioner environment and practitioner factors such as personality more strongly influence the decisionmaking of experienced ergonomics practitioners than does practitioner background in IE or PT.
 Piegorsch, W. W. (2006). Excess risk estimation under multistage model misspecification. Journal of Statistical Computation and Simulation.
 Piegorsch, W. W. (2006). Multiplicityadjusted inferences in risk assessment: Benchmark analysis with continuous response data. Environmental and Ecological Statistics.
 Piegorsch, W. W. (2005). Benchmark analysis: Shopping with proper confidence. Risk Analysis.
 Piegorsch, W. W. (2005). Low dose risk estimation via simultaneous statistical inferences. Journal of the Royal Statistical Society. Series C: Applied Statistics.
 Piegorsch, W. W. (2005). Multiplicityadjusted inferences in risk assessment: Benchmark analysis with quantal response data. Biometrics.
 Piegorsch, W. W. (2005). Simultaneous confidence bounds for lowdose risk assessment with nonquantal data. Journal of Biopharmaceutical Statistics.
 Piegorsch, W. W. (2004). Sample sizes for improved binomial confidence intervals. Computational Statistics and Data Analysis, 46(2), 309316.More infoAbstract: Sample size equations are reviewed for different types of confidence intervals on a binomial success probability. Based on recommendations for improved binomial confidence limits given by Brown et al. (Statist. Sci. 16 (2001) 101), the intervals expand upon or enhance the traditional Waldtype interval. Some useful sample size relations appear. © 2003 Elsevier B.V. All rights reserved.
 Piegorsch, W. W. (2004). Sample sizes for improved binomial confidence intervals. Computational Statistics and Data Analysis.
 Piegorsch, W. W. (2003). Combining environmental information via hierarchical modeling: An example using mutagenic potencies. Environmetrics.
 Piegorsch, W. W. (2003). Confidence Bands for LowDose Risk Estimation with Quantal Response Data. Biometrics.
 Piegorsch, W. W. (2003). Detection of oxidative DNA damage in isolated marine bivalve hemocytes using the comet assay and formamidopyrimidine glycosylase (Fpg). Mutation Research  Genetic Toxicology and Environmental Mutagenesis.
 Piegorsch, W. W. (2003). Empirical Bayes analysis for a hierarchical poisson generalized linear model. Journal of Statistical Planning and Inference.
 Piegorsch, W. W. (2003). Environmetrics: Preface. Environmetrics.
 Piegorsch, W. W. (2003). Exact onesided simultaneous confidence bands via Uusipaikka's method. Annals of the Institute of Statistical Mathematics.
 Piegorsch, W. W. (2003). Introduction to the special section on statistics and the environment. Statistical Science.
 Piegorsch, W. W. (2002). What shall we teach in environmental statistics?. Environmental and Ecological Statistics.
 Piegorsch, W. W. (2001). Largesample pairwise comparisons among multinomial proportions with an application to analysis of mutant spectra. Journal of Agricultural, Biological, and Environmental Statistics.
 Piegorsch, W. W. (2001). The male rat carcinogens limonene and sodium saccharin are not mutagenic to male Big Blue™ rats. Mutagenesis.
 Piegorsch, W. W. (2000). Asymmetric confidence bands for simple linear regression over bounded intervals. Computational Statistics and Data Analysis.
 Piegorsch, W. W. (2000). Estimation and testing with overdispersed proportions using the beta logistic regression model of Heckman and Willis. Biometrics.
 Piegorsch, W. W. (2000). From quantal counts to mechanisms and systems: The past, present, and future of biometrics in environmental toxicology. Biometrics.
 Piegorsch, W. W. (2000). On a likelihoodbased goodnessoffit test of the betabinomial model. Biometrics.
 Piegorsch, W. W. (2000). Parametric empirical Bayes estimation for a class of extended loglinear regression models. Environmetrics.
 Piegorsch, W. W. (2000). Statistical modeling and analyses of a basespecific Salmonella mutagenicity assay. Mutation Research  Genetic Toxicology and Environmental Mutagenesis.
 Piegorsch, W. W. (1999). Experimental evidence of subsurface feeding by the burrowing ophiuroid Amphipholis gracillima (Echinodermata). Marine Ecology Progress Series.
 Piegorsch, W. W. (1998). Introduction to binary response regression and associated trend analyses. Journal of Quality Technology.
 Piegorsch, W. W. (1998). Statistical Advances in Environmental Science. Statistical Science.
 Piegorsch, W. W. (1998). Statistical aspects for combining information and metaanalysis in environmental toxicology. Journal of Environmental Science and Health  Part C Environmental Carcinogenesis and Ecotoxicology Reviews.
 Piegorsch, W. W. (1997). Optimal statistical design for toxicokinetic studies. Statistical Methods in Medical Research.
 Piegorsch, W. W. (1997). Sources of variability in data from a positive selection lacZ transgenic mouse mutation assay: An interlaboratory study. Mutation Research  Genetic Toxicology and Environmental Mutagenesis.
 Piegorsch, W. W. (1996). Combining environmental information. I: Environmental monitoring, measurement and assessment. Environmetrics.
 Piegorsch, W. W. (1996). Empirical Bayes estimation for logistic regression and extended parametric regression models. Journal of Agricultural, Biological, and Environmental Statistics.
 Piegorsch, W. W. (1996). Lifestagespecific toxicity of sedimentassociated chlorpyrifos to a marine, infaunal copepod. Environmental Toxicology and Chemistry.
 Piegorsch, W. W. (1996). The Ames test: The twofold rule revisited. Mutation Research  Genetic Toxicology.
 Piegorsch, W. W., & Cox, L. H. (1996). Combining environmental information. II: Environmental epidemiology and toxicology. Environmetrics, 7(3), 309324.More infoAbstract: An increasingly important concern in epidemiological and toxicological studies of environmental exposures is the need to combine information from diverse sources that relate to a common endpoint. This is clearly a statistical activity, but statistical techniques for data combination are still only developmental. Herein, we illustrate some current applications of combining information in environmental epidemiology and toxicology, with emphasis on the burgeoning use of metaanalyses for environmental settings. Our goal is to inform readers about modern statistical techniques useful for combining environmental information, with emphasis on more recently developed approaches.
 Piegorsch, W. W. (1995). Discussion. Environmental and Ecological Statistics.
 Piegorsch, W. W. (1995). Study design and sample sizes for a lacI transgenic mouse mutation assay. Environmental and Molecular Mutagenesis.
 Piegorsch, W. W. (1994). Combining environmental information.. Environmental Health Perspectives.
 Piegorsch, W. W. (1994). Computer program for the analysis of mutational spectra: Application to p53 mutations. Carcinogenesis.
 Piegorsch, W. W. (1994). Nonhierarchical logistic models and caseonly designs for assessing susceptibility in populationbased casecontrol studies. Statistics in Medicine.
 Piegorsch, W. W. (1994). Some comments on potency measures in mutagenicity research. Environmental Health Perspectives.
 Piegorsch, W. W. (1994). Statistical approaches for analyzing mutational spectra: Some recommendations for categorical data. Genetics.
 Piegorsch, W. W. (1994). Statistical models for genetic susceptibility in toxicological and epidemiological investigations. Environmental Health Perspectives.
 Piegorsch, W. W., Lockhart, A. ., Margolin, B. H., Tindall, K. R., Gorelick, N. J., Short, J. M., Carr, G. J., Thompson, E. D., & Shelby, M. D. (1994). Sources of variability in data from a lacI transgenic mouse mutation assay. Environmental and Molecular Mutagenesis, 23(1), 1731.More infoPMID: 8125080;Abstract: Experimental features of a transgenic mouse mutation assay based on a lacI target transgene from Escherichia coli are considered in detail. Sources of variability in the experimental protocol that can affect the statistical nature of the observations are examined with the goal of identifying sources of excess variation in the observed mutant fractions. The sources include platetoplate (within packages), packagetopackage (within animals), and animaltoanimal (within study) variability. Data from two laboratories are evaluated, using various statistical methods to identify excess variability. Results suggest only scattered patterns of excess variability, except possibly in those cases where genomic DNA from test animals is stored for extended periods (e.g., > 90 days) after isolation from tissues. Further study is encouraged to examine the validity and implications of this time/storagerelated effect.
 Piegorsch, W. W. (1993). Biometrical methods for testing dose effects of environmental stimuli in laboratory studies. Environmetrics.
 GutierrezEspeleta, G. A., Hughes, L. A., Piegorsch, W. W., Shelby, M. D., & Generoso, W. M. (1992). Acrylamide: dermal exposure produces genetic damage in male mouse germ cells. Fundamental and applied toxicology : official journal of the Society of Toxicology, 18(2), 18992.More infoAcrylamide is used extensively in sewage and wastewater treatment plants, in the paper and pulp industry, in treatment of potable water, and in research laboratories for chromatography, electrophoresis, and electron microscopy. Dermal contact is a major route of human exposure. It has been shown that acrylamide is highly effective in breaking chromosomes of germ cells of male mice and rats when administered intraperitoneally or orally, resulting both in the early death of conceptuses and in the transmission of reciprocal translocations to liveborn progeny. It is now reported that acrylamide is absorbed through the skin of male mice, reaches the germ cells, and induces chromosomal damage. The magnitude of genetic damage appears to be proportional to the dose administered topically.
 Lockhart, A. M., Piegorsch, W. W., & Bishop, J. B. (1992). Assessing overdispersion and doseresponse in the male dominant lethal assay. Mutation research, 272(1), 3558.More infoIn dominant lethal studies the primary variables of interest are typically expressed as discrete counts or proportions (e.g., live implants, resorptions, percent pregnant). Simple statistical sampling models for discrete data such as binomial or Poisson generally do not fit this type of data because of extrabinomial or extraPoisson departures from variability predicted under these simple models. Extravariability in the fetal response may originate from parental contributions. These can lead to over or underdispersion seen as, e.g., extrabinomial variability in the proportion response. Utilizing a large control database, we investigated the relative impact of extravariability from male or female contributions on the endpoints of interest. Malerelated effects did not seem to contribute to overdispersion in our database; femalerelated effects were, however, evidenced. Various statistical methods were considered to test for significant treatment differences under these forms of sampling variability. Computer simulations were used to evaluate these methods and to determine which are most appropriate for practical use in the evaluation of dominant lethal data. Our results suggest that distributionfree statistical methods such as a nonparametric permutation test or rankbased tests for trend can be recommended for use.
 Piegorsch, W. W. (1992). Concordance of carcinogenic response between rodent species: Potency dependence and potential underestimation. Risk Analysis.
 Piegorsch, W. W. (1992). Statistical methods for assessing environmental effects on human genetic disorders. Environmetrics.
 Generoso, W. M., Shourbaji, A. G., Piegorsch, W. W., & Bishop, J. B. (1991). Developmental response of zygotes exposed to similar mutagens. Mutation research, 250(12), 439446. doi:10.1016/00275107(91)902008More infoExposure of mouse zygotes to ethylene oxide (EtO) or ethyl methanesulfonate (EMS) led to high incidences of fetal death and of certain classes of fetal malformations (Generoso et al., 1987, 1988; Rutledge and Generoso, 1989). These effects were not associated with induced chromosomal aberrations (Katoh et al., 1989) nor are they likely to be caused by gene mutations (Generoso et al., 1990). Nevertheless, the anomalies observed in these studies resemble the large class of stillbirths and sporadic defects in humans that are of unknown etiology, such as cleft palate, omphalocoel, clubfoot, hydrops and stillbirths (Czeizel, 1985; Oakley, 1986). Therefore, we continue to study the possible mechanisms relating to induction of these types of zygotederived anomalies in mice. Effects of zygote exposure to the compounds methyl methanesulfonate (MMS), dimethyl sulfate (DMS), and diethyl sulfate (DES), which have similar DNAbinding properties as EtO and EMS, were studied. DMS and DES, but not MMS, induced effects that are similar to those induced by EtO and EMS. Thus, no sitespecific alkylation product was identifiable as the critical target for these zygotederived anomalies. We speculate that the developmental anomalies arose as a result of altered programming of gene expression during embryogenesis.
 Piegorsch, W. W. (1991). Multiple comparisons for analyzing dichotomous response. Biometrics.
 Piegorsch, W. W., & Haseman, J. K. (1991). Statistical methods for analyzing developmental toxicity data. Teratogenesis, carcinogenesis, and mutagenesis, 11(3), 11533.More infoA description and review of methods for performing perlitter analyses involving extrabinomial proportion response is provided. It is stressed that the litter should be regarded as the appropriate experimental unit for quantitative analysis in studies for teratogenic or heritable mutagenic effects. Attention is directed at statistical identification of possible treatment effects, such as a positive dose response to a chemical stimulus. The methods range from distributionfree, nonparametric analyses to models involving parametric distributions such as the betabinomial density. It is seen that most current methods require computer implementation. When concern is raised over misspecification of assumptions critical to the statistical analysis, it is argued that relatively parameterfree methods are appropriate for use. These include statistical bootstrapping and rankbased analyses.
 Bailer, A. J., & Piegorsch, W. W. (1990). Estimating integrals using quadrature methods with an application in pharmacokinetics. Biometrics, 46(4), 120111.More infoThe estimation of integrals using numerical quadrature is common in many biological studies. For instance, in biopharmaceutical research the area under curves is a useful quantity in deriving pharmacokinetic parameters and in providing a surrogate measure of the total dose of a compound at a particular site. In this paper, statistical issues as separate from numerical issues are considered in choosing a quadrature rule. The class of NewtonCôtes numerical quadrature procedures is examined from the perspective of minimizing mean squared error (MSE). The MSE are examined for a variety of functions commonly encountered in pharmacokinetics. It is seen that the simplest NewtonCôtes procedure, the trapezoidal rule, frequently provides minimum MSE for a variety of concentrationtime shapes and under a variety of response variance conditions. A biopharmaceutical example is presented to illustrate these considerations.
 Piegorsch, W. W. (1990). Fisher's contributions to genetics and heredity, with special emphasis on the Gregor Mendel controversy. Biometrics, 46(4), 91524.More infoR. A. Fisher is widely respected for his contributions to both statistics and genetics. For instance, his 1930 text on The Genetical Theory of Natural Selection remains a watershed contribution in that area. Fisher's subsequent research led him to study the work of (Johann) Gregor Mendel, the 19th century monk who first developed the basic principles of heredity with experiments on garden peas. In examining Mendel's original 1865 article, Fisher noted that the conformity between Mendel's reported and proposed (theoretical) ratios of segregating individuals was unusually good, "too good" perhaps. The resulting controversy as to whether Mendel "cooked" his data for presentation has continued to the current day. This review highlights Fisher's most salient points as regards Mendel's "too good" fit, within the context of Fisher's extensive contributions to the development of genetical and evolutionary theory.
 Piegorsch, W. W. (1990). Maximum likelihood estimation for the negative binomial dispersion parameter. Biometrics, 46(3), 8637.
 Piegorsch, W. W. (1990). Onesided significance tests for generalized linear models under dichotomous response. Biometrics, 46(2), 30916.More infoDichotomous response models are common in many experimental settings. Often, concomitant explanatory variables are recorded, and a generalized linear model, such as a logit model, is fit. In some cases, interest in specific model parameters is directed only at onesided departures from some null effect. In these cases, procedures can be developed for testing the null effect against a onesided alternative. These include Bonferronitype adjustments of univariate Wald tests, and likelihood ratio tests that employ inequalityconstrained multivariate theory. This paper examines such tests of significance. Monte Carlo evaluations are undertaken to examine the smallsample properties of the various procedures. The procedures are seen to perform fairly well, generally achieving their nominal sizes at total sample sizes near 100 experimental units. Extensions to the problem of onesided tests against a control or standard are also considered.
 Whittaker, S. G., Moser, S. F., Maloney, D. H., Piegorsch, W. W., Resnick, M. A., & Fogel, S. (1990). The detection of mitotic and meiotic chromosome gain in the yeast Saccharomyces cerevisiae: effects of methyl benzimidazol2yl carbamate, methyl methanesulfonate, ethyl methanesulfonate, dimethyl sulfoxide, propionitrile and cyclophosphamide monohydrate. Mutation research, 242(3), 23158.More infoThe diploid yeast strain BR1669 was used to study induction of mitotic and meiotic chromosome gain by selected chemical agents. The test relies on a gene dosage selection system in which hyperploidy is detected by the simultaneous increase in copy number of two alleles residing on the right arm of chromosome VIII: arg48 and cup1S (Rockmill and Fogel. 1988; Whittaker et al., 1988). Methyl methanesulfonate (MMS) induced mitotic, but not meiotic, chromosome gain. Methyl benzimidazol2yl carbamate (MBC) and ethyl methanesulfonate (EMS) induced both mitotic and meiotic chromosome gain. Propionitrile, a polar aprotic solvent, induced only mitotic chromosome gain; a reliable response was only achieved by overnight incubation of treated cultures at 0 degrees C. MBC is postulated to act by binding directly to tubulin. The requirement for lowtemperature incubation suggests that propionitrile also induces aneuploidy by perturbation of microtubular dynamics. The alkylating agents MMS and EMS probably induce recombination which might in turn perturb chromosome segregation. Cyclophosphamide monohydrate and dimethyl sulfoxide (DMSO) failed to induce mitotic or meiotic chromosome gain.
 Whittaker, S. G., Zimmermann, F. K., Dicus, B., Piegorsch, W. W., Resnick, M. A., & Fogel, S. (1990). Detection of induced mitotic chromosome loss in Saccharomyces cerevisiaean interlaboratory assessment of 12 chemicals. Mutation research, 241(3), 22542.More infoInduced mitotic chromosome loss was assayed using diploid yeast strain S. cerevisiae D61.M. The test relies upon the uncovering and expression of multiple recessive markers reflecting the presumptive loss of the chromosome VII homologue carrying the corresponding wildtype alleles. An interlaboratory study was performed in which 12 chemicals were tested under code in 2 laboratories. The results generated by the Berkeley and the Darmstadt laboratories were in close agreement. The solvents benzonitrile and methyl ethyl ketone induced significantly elevated chromosome loss levels. However, a treatment regime that included overnight storage at 0 degree C was required to optimize chromosome loss induction. Hence, these agents are postulated to induce chromosome loss via perturbation of microtubular assembly. Fumaronitrile yielded inconsistent results: induction of chromosome loss and respiratory deficiency was observed in both laboratories, but the response was much more pronounced in the Darmstadt trial than that observed in Berkeley. The mammalian carcinogens, benzene, acrylonitrile, trichloroethylene, 1,1,1trichloroethane and 1,1,1,2tetrachloroethane failed to induce chromosome loss but elicited high levels of respiratory deficiency, reflecting antimitochondrial activity. Trifluralin, cyclophosphamide monohydrate, diazepam and diethylstilbestrol dipropionate failed to induce any detectable genetic effects. These data suggest that the D61.M system is a reproducible method for detecting induced chromosome loss in yeast.
 Piegorsch, W. W. (1989). Durand's rules for approximate integration. Historia Mathematica.
 Piegorsch, W. W. (1989). Early use of matrix diagonal increments in statistical problems. SIAM Review.
 Piegorsch, W. W. (1989). Optimal design allocations for estimating area under curves for studies employing destructive sampling. Journal of Pharmacokinetics and Biopharmaceutics.
 Piegorsch, W. W. (1989). Quantification of toxic response and the development of the median effective dose (ED50)  A historical perspective. Toxicology and Industrial Health.
 Piegorsch, W. W. (1989). Quantitative approaches for assessing chromosome loss in Saccharomyces cerevisiae: general methods for analyzing downturns in dose response. Mutation Research/Genetic Toxicology.
 Piegorsch, W. W., & Margolin, B. H. (1989). Quantitative methods for assessing a synergistic or potentiated genotoxic response. Mutation research, 216(1), 18.More infoThe problem of assessing chemical interactions in studies of genotoxicity is discussed. Attention is focused on assessing possible synergism or potentiation when the observed genotoxic response is binary (yesno). Different forms of enhancement are distinguished based upon different assumptions on the genotoxic activity of the experimental treatments. A generalized linear statistical model is considered that links the probability of the binary response to the doses, and dataanalytic strategies are described for detecting synergy and potentiation in factorially designed experiments. This approach is illustrated with a series of analyses of various genotoxicity datasets.
 Piegorsch, W. W., Zimmermann, F. K., Fogel, S., Whittaker, S. G., & Resnick, M. A. (1989). Quantitative approaches for assessing chromosome loss in Saccharomyces cerevisiae: general methods for analyzing downturns in dose response. Mutation research, 224(1), 1129.More infoStatistical methods are considered for analysis of data arising from a mitotic chromosome loss assay in Saccharomyces cerevisiae strain D61.M. The methods make use of reproducibility trial data from the assay (presented herein) and previous data, which suggest a unimodal, 'umbrellapatterned' dose response. Computer simulations are employed to illustrate the operating characteristics of the umbrella response methods. These methods are generally applicable to any toxicity assay that exhibits a downturn in dose response. Experimental design considerations are also discussed. These include applications of 2stage sampling rules to first gauge the dose window of peak response, then test if the response deviates significantly from untreated levels.
 Rao, G. N., Piegorsch, W. W., Crawford, D. D., Edmondson, J., & Haseman, J. K. (1989). Influence of viral infections on body weight, survival, and tumor prevalence of B6C3F1 (C57BL/6N x C3H/HeN) mice in carcinogenicity studies. Fundamental and applied toxicology : official journal of the Society of Toxicology, 13(1), 15664.More infoSendai virus (SV), mouse hepatitis virus (MHV), and pneumonia virus of mice (PVM) are common viral infections of mice. Influence of these viral infections on the prevalence of liver tumors, lung tumors, and lymphoma is of concern in chemical carcinogenicity studies. Body weight, survival, and tumor prevalence of B6C3F1 mice with and without viral infections in 33 male and 34 female untreated control groups and 32 male and 32 female low and highdose groups of 2year chemical carcinogenicity studies were evaluated. In male mice, the SV infection was associated with significantly (p less than 0.05) higher survival of control, lowdose, and highdose groups, and higher prevalence of liver tumors and lymphoma. The increases in tumor prevalence are possibly due to an increase in the survival of male mice that had SV infection. However, when interlaboratory variability and timerelated effects were taken into account, the number of significant effects was consistent with the expected falsepositive rate inherent to the statistical procedures. The MHV and PVM infections did not cause consistent changes in body weight, survival, and tumor prevalences in the control and chemical treatment groups of male mice. Viral infections did not cause consistent increases or decreases in body weight, survival, or tumor prevalence in the control and chemical treatment groups of female B6C3F1 mice.
 Whittaker, S. G., Zimmermann, F. K., Dicus, B., Piegorsch, W. W., Fogel, S., & Resnick, M. A. (1989). Detection of induced mitotic chromosome loss in Saccharomyces cerevisiaean interlaboratory study. Mutation research, 224(1), 3178.More infoThe diploid yeast strain D61.M was used to study induction of mitotic chromosome loss. The test relies upon the uncovering and expression of multiple recessive markers reflecting the presumptive loss of the chromosome VII homologue carrying the corresponding wildtype alleles. The underlying 'loss event' is probably complex since the predicted centromerelinked lethal tetrad segregations for chromosome VII are not recovered. Instead, the homologue bearing the multiple recessive markers is patently homozygous. An interlaboratory study was performed in which 16 chemicals were tested under code in 2 laboratories. The results generated by the Berkeley and Darmstadt laboratories were in close agreement. Acetonitrile, ethyl acetate, 4acetylpyridine, propionitrile and nocodazole were identified as potent inducers of mitotic chromosome loss. Acetone, dimethyl sulfoxide and 2methoxyethyl acetate either elicited weak responses or yielded ambiguous results. Water, carbon tetrachloride, 4fluoroD,Lphenylalanine, amphotericin B, griseofulvin, cadmium chloride, ethyl methanesulfonate and methylmercury(II) chloride failed to induce chromosome loss. These data suggest that the system described herein represents a reliable assay for chemically induced chromosome loss in yeast.
 Piegorsch, W. W. (1988). Confidence bands for logistic regression with restricted predictor variables. Biometrics.
 Piegorsch, W. W. (1988). Exploring relationships between mutagenic and carcinogenic potencies. Mutation Research/Reviews in Genetic Toxicology.
 Piegorsch, W. W. (1988). Exploring simple independent action in multifactor tables of proportions. Biometrics.
 Piegorsch, W. W. (1988). Respiratory tract lesions in F344/N rats and B6C3F1 mice after inhalation exposure to 1,2epoxybutane. Toxicology.
 Piegorsch, W. W., & Hoel, D. G. (1988). Exploring relationships between mutagenic and carcinogenic potencies. Mutation research, 196(2), 16175.More infoSalmonella mutagenic and rodent carcinogenic potencies are calculated for 112 compounds recently studied by the U.S. National Toxicology Program. 28 of the 112 compounds are seen to exhibit simultaneous nonzero mutagenic and carcinogenic potencies. These are combined with an earlier list of mutagenic and carcinogenic compounds (McCann et al., 1988) in order to study possible trends in the data. A significant positive correlation is exhibited between mutagenic and carcinogenic potencies in the combined data, although the observed scatter is too great for the overall result to be predictive. Classification by chemical class further indicates positive correlations near one for chemicals classified as nitroaromatic and related compounds. Patterns in mutagenic and carcinogenic potency over time are also examined. Mean potencies of recentlystudied compounds are seen to trend lower than those of compounds studied 10 or more years ago.
 Kitamura, H., Inayama, Y., Ito, T., Yabana, M., Piegorsch, W. W., & Kanisawa, M. (1987). Morphologic alteration of mouse Clara cells induced by glycerol: ultrastructural and morphometric studies. Experimental lung research, 12(4), 281302.More infoIn our studies on activation of the Clara cell by biological substances and its relationship to pulmonary carcinogenesis, we found that large doses of glycerol induced drastic morphologic changes selectively in the Clara cell among distal airway epithelial cells in ddY mice. Subcutaneous injection of glycerol (7.2 g/Kg body weight) caused cytoplasmic edema with disruption of endoplasmic reticulum membranes at 1 and 3 hours, followed by hyperplasia of smooth endoplasmic reticulum (SER) at 12 and 24 hours. Concentric lamination of SER was observed at 48 and 96 hours. Oral administration of 5% glycerol in drinking water for 2 to 8 weeks induced more conspicuous hyperplasia and hypertrophy of SER in the Clara cells. Electron microscopic morphometry revealed a 3fold increase in the profile area of SER in the Clara cells of the animals at 2 and 8 weeks. Both the profile area and the number of secretory granules increased significantly at 2 and 8 weeks, and those of mitochondria tended to increase with time of glycerol treatment. In both experiments, the mitochondria of the Clara cells exhibited marked elongation and distortion of the contour associated with appearance of prominent cristae. These results suggest that large doses of glycerol induce marked alteration in the functional activity of the mouse Clara cell.
 Piegorsch, W. W. (1987). Influence of body weight on the incidence of spontaneous tumors in rats and mice of longterm studies.. American Journal of Clinical Nutrition.
 Piegorsch, W. W. (1987). Performance of likelihoodbased interval estimates for twoparameter exponential samples subject to type I censoring.. Technometrics.
 Piegorsch, W. W. (1986). Confidence bands for polynomial regression with fixed intercepts.. Technometrics.
 Piegorsch, W. W. (1986). Testing for simple independent action between two factors for dichotomous response data.. Biometrics.
 Piegorsch, W. W. (1986). The Gregor Mendel controversy: early issues of goodnessoffit and recent issues of genetic linkage.. History of science; an annual review of literature, research and teaching.
 Piegorsch, W. W., & Gladen, B. C. (1986). Note on the use of prior interval information in constructing interval estimates for a gamma mean.. Technometrics, 28(3), 269273.More infoAbstract: Methods are presented for construction of interval estimates on the mean of a gamma distribution when there is some prior interval information as to the location of this parameter. The methods produce posterior intervals by constructing prior distributions for the mean parameter from the prior interval information. Both Bayesian and pseudoBayesian approaches for the construction of the priors are considered. These concepts are illustrated by an experiment assessing the operating characteristics of a laboratory chemical analyzer.
Presentations
 Piegorsch, W. W. (2021, April). "Collaboration is the best part of statistics". Stats+Stories™ Podcast Episode #183.
 Piegorsch, W. W. (2021, April). "The probability of the next terrorist attack". Stats+Stories™ Podcast Episode #182.
 Piegorsch, W. W. (2021, August). "Predicting drug toxicity and flood risk with data science". Arizona BIO5 Institute Science Talks, Podcast Episode #16.
 Piegorsch, W. W. (2019, May). Model uncertainty in environmental risk assessment. University of Nevada Department of Mathematics & Statistics Colloquium Series. University of Nevada, Reno, NV: University of Nevada Department of Mathematics & Statistics.More infoStatistical estimation of lowdose ‘benchmark’ points in environmental risk analysis is discussed. Focus is on the increasing recognition that model uncertainty and misspecification can drastically affect point estimators and confidence limits built from limited doseresponse data, which in turn can lead to imprecise risk assessments with uncertain, even dangerous, policy implications. Some possible remedies are mentioned, including use of parametric (frequentist) model averaging over a suite of potential doseresponse models, and nonparametric doseresponse analysis via isotonic regression. An example on formaldehyde toxicity illustrates the calculations.
 Pena, E., Wu, W., Piegorsch, W. W., West, W., & An, L. . (2013, March). Model Selection and BMD Estimation with QuantalResponse Data. The International Biometric Society (IBS)  ENAR Spring meeting. Orlando, FL: IBS Eastern North American Region.