Walter W. Piegorsch

Abstracts of Earlier Articles

Links to Publishers' Listings for more recent publications are available here.

DATA MINING POTENCY ESTIMATORS FROM TOXICOLOGICAL DATABASES


By

Walter W. Piegorsch, Susan J. Simmons, and Errol Zeiger

Key Words and Phrases: Ames test, bioinformatics, feature extraction, knowledge discovery in databases, mutagenic potency, Salmonella assay.

Methods are discussed for quantifying the risk associated with exposure to bio-environmental hazards. Attention is directed at problems in dose-response/concentration-response modeling when potency estimation is a primary goal. Both discrete and continuous endpoints are considered. Possible potency measures include observed-effect levels, effective doses/effective concentrations, rates of change, and added risk/extra risk measures. Estimation of and inferences on effective doses/effective concentrations are highlighted. [Handbook of Statistics Vol. 18: Bioenvironmental and Public Health Statistics, P.K. Sen and C.R. Rao, eds., New York: North-Holland/Elsevier (2000), pp. 441-463.]

An Introduction to Binary Response Regression and Associated Trend Analyses

WALTER W. PIEGORSCH

University of South Carolina, Columbia, SC 29208

An introduction and tutorial is presented for regression and trend analyses when data are observed in the form of proportions. The basic binomial probability model is assumed. Emphasis is placed on quantal response analyses, useful when a concomitant predictor variable or other form of ordered measure is recorded with each proportion observation. These include regression methods and trend tests, residual analysis, and inverse regression/effective stimulus estimation. [Journal of Quality Technology 30, 269-281 (1998).]

Some comments on potency measures in mutagenicity research

by Barry H. Margolin, Byung Soo Kim, Melissa G. Smith, Bethel A. Fetterman, Walter W. Piegorsch and Errol Zeiger

Statistical models for genetic susceptibility in toxicological and epidemiological investigations

by Walter W. Piegorsch

Biometrical methods for testing dose effects of environmental stimuli in laboratory studies

Walter W. Piegorsch

ABSTRACT

In the areas of toxicology and environmental biology, the damaging effects of environmental chemicals or other stimuli on biological systems are often studied in controlled laboratory experiments. These usually involve animal and microbial systems. Data from such experiments are analyzed via various statistical approaches, depending on the nature of the endpoint and aspects of the particular assay under study. Major endpoints of interest include carcinogenicity, mutagenicity, and teratogenicity. When the environmental stimulus is administered over a range of doses, it is of interest to estimate and/or test features of the dose-response. In some instances, variance heterogeneity and overdispersion are present, and adjustments to the statistical methods are required. Herein, such methods for assessing dose response for the major endpoints noted above are discussed, with emphasis directed at testing for an increasing dose-response. [Environmetrics 4, 483-505 (1993).]

KEY WORDS: Discrete data; Dose response; Overdispersion; Quantal response; Significance test; Trend test.

Relative rates of mutagenic translesion synthesis on the leading and lagging strands during replication of UV-irradiated DNA in a human cell extract

David C. Thomas,1 Dinh C. Nguyen,1 Walter W. Piegorsch,2 and Thomas A. Kunkel*,1

Laboratory of Molecular Genetics and Statistics and Biomathematics Branch,
National Institute of Environmental Health Sciences, Research Triangle Park, North Carolina 27709

* To whom correspondences should be sent.
1 Laboratory of Molecular Genetics.
2 Statistics and Biomathematics Branch.

Minimum Mean-Square Error Quadrature

Walter W. Piegorsch
Statistics and Biomathematics Branch, National Institute of Environmental Health Sciences, Research Triangle Park, North Carolina 27709
and
A. John Bailer
Department of Mathematics and Statistics, Miami University, Oxford, Ohio 45056

Minimum mean squared error linear estimators of the area under a curve are considered for cases when the observations are observed with error. The underlying functional form giving rise to the observations is left unspecified, leading to use of quadrature estimators for the true area. The optimal estimator is calculated as a shrinkage of some preliminary estimator (based on, e.g., the trapezoidal rule). Applications to selected exponential functions demonstrate that savings in mean squared error varies with the levels of underlying variance. For cases where variance at each time point is large, the proposed rule can bring about savings in mean squared error of as much as 30%. For experiments with small underlying variance at each time point, squared bias is of greater importance than variance in contributing to mean squared error, and the value of higher-order quadrature routines that focus on minimizing approximation error is noted. [Journal of Statistical Computation and Simulation 46, 217-234 (1993).]

Statistical Methods for Assessing Environmental Effects on Human Genetic Disorders

Walter W. Piegorsch1 and Jack A. Taylor2

ABSTRACT


Methods are presented for assessing interactions and other effects between genetic and environmental factors for human disease or cancer susceptibility. Statistical estimation and testing approaches are based on a simple multinomial sampling model for the case-control sampling scenario. It is noted that logistic regression methods can facilitate computation of likelihood-based statistics in this setting. Additional models for collapsibility over genotypes within the genetic factor are considered. Monte Carlo comparisons show that the method appears to retain nominal significance levels at total sample sizes above 100. At smaller sample sizes, a goodness-of-fit statistic is suggested for testing the interactive effect between the genetic and environmental factors. [Environmetrics 3, 369-384 (1992).]

Key Words: Case-control study, ecogenetics, epidemiology, gene-environment interaction, genetic susceptibility, logistic regression, maximum likelihood, multinomial sampling, multiplicative null model, odds ratio, synergy.

1,2Division of Biometry and Risk Assessment, U.S. National Institute of Environmental Health Sciences, Research Triangle Park, NC 27709 U.S.A.

NON-PARAMETRIC METHODS TO ASSESS NON-MONOTONE DOSE
RESPONSE: APPLICATIONS TO GENETIC TOXICOLOGY

Walter W. Piegorsch
Statistics and Biomathematics Branch
U.S. National Institute of Environmental Health Sciences
Research Triangle Park, NC 27709, USA

ABSTRACT
Rank-based statistical methods are described for analysis of data arising from various bioassays for genotoxic damage in micro-organisms. The alternative hypothesis is a specific form of unimodal departure known as an "umbrella pattern," where the dose-response first increases, then decreases. Motivation for this alternative structure is taken from a genotoxicity assay that assesses chromosome loss (a form of "aneuploidy") in yeast. Monte Carlo evaluations are employed to illustrate the small-sample operating characteristics of the umbrella response methods. These methods are generally applicable to any toxicity assay that exhibits a downturn in dose response. Examples are presented, illustrating applications to data from the aneuploidy assay, and from a mutagenicity assay in bacteria. [Order Statistics and Nonparametrics: Theory and Applications, P.K. Sen and I.A. Salama, eds. Amsterdam: North-Holland, 419-430 (1992).]

AMS Subject Classification: Primary 62G10, Secondary 62E99
Key Words and Phrases: Mutagenesis; recursive estimation and testing; test for trend; umbrella alternatives.

Assessing overdispersion and dose response in the male dominant lethal assay

Ann-Marie Lockharta, Walter W. Piegorschb and Jack B. Bishopb
aComputer Sciences Corporation, Research Triangle Park, NC USA, and b National Institute of Environmental Health Sciences,
Research Triangle Park, NC USA

Keywords: Statistical methods; Germ cell mutagenesis; Heritable disease; Litter effect; Mouse; Extra-binomial variability; Underdispersion; Dose-response analysis

Summary

In 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 extra-binomial or extra-Poisson departures from variability predicted under these simple models. Extra-variability in the fetal response may originate from parental contributions. These can lead to over- or under-dispersion seen as, e.g., extra-binomial variability in the proportion response. Utilizing a large control database, we investigated the relative impact of extra variability from male or female contributions on the endpoints of interest. Male-related effects did not seem to contribute to overdispersion in our database; female-related 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 distribution-free statistical methods such as a nonparametric permutation test or rank-based tests for trend can be recommended for use. [Mutation Research 272, 35-58 (1992).]

Complementary log regression for generalized linear models

WALTER W. PIEGORSCH*

Use and implementation of the complementary log regression model are discussed, integrating various applications of the model under the form of a generalized linear model. Some motivation is drawn from cases where an underlying random variable is reduced to a dichotomous form. Estimation and testing are facilitated by recognizing the complementary log as a specific link function within a generalized linear framework. Testing for goodness-of-link via efficient scores is also discussed. [American Statistician 46, 94-99 (1992).]

KEY WORDS: Binomial model; Extended link family; Data Truncation; Goodness-of-link testing; Logistic regression; Non-linear regression.

* Walter W. Piegorsch was a Mathematical Statistician in the Statistics and Biomathematics Branch, National Institute of Environmental Health Sciences, Research Triangle Park, NC 27709 at the time of preparation of this article. He thanks Norman Kaplan, Barry H. Margolin, Clarice R. Weinberg and two anonymous reviewers for their helpful suggestions during the preparation of this manuscript, and Arnold R. Brody and Lilla H. Hill for providing selected data.

Acrylamide: Dermal exposure produces genetic damage in male mouse germ cells

Gustavo E. Gutierrez-Espeleta,* Lori A. Hughes,* Walter W. Piegorsch,+ Michael D. Shelby,+ and Walderico M. Generoso*
*Biology Division, Oak Ridge National Laboratory, Oak Ridge, TN 37788; and
+National Institute of Environmental Health Sciences, Research Triangle Park, NC 27709

Acrylamide: Dermal exposure produces genetic damage in male mouse germ cells. (1992) Fundamental and Applied Toxicology 18, 189-192.

Acrylamide 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 chromotography, 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 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 live-born 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.

Developmental responses of zygotes exposed to similar mutagens

W.M. Generosoa, A.G. Shourbajia, W.W. Piegorschb and J.B. Bishopb
aBiology Division, Oak Ridge National Laboratory, Oak Ridge, TN 37788 and
bNational Institute of Environmental Health Sciences,
Research Triangle Park, NC 27709 (U.S.A.)

Keywords: Foetal anomalies; Zygotes; Methyl methanesulfonate; Dimethyl sulphate; Diethyl sulphate

Summary

Exposures 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 withinduced 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 effects 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 zygote-derived anomalies in mice. Effects of zygote exposure to the compounds methyl methanesulfonate (MMS), dimethyl sulfate (DMS), and diethyl sulfate (DES), which have similar DNA-binding 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 site-specific alkylation product was identifiable as the critical target for these zygote-derived anomalies. We speculate that the developmental anomalies arose as a result of altered programming of gene expression during embryogenesis. [Mutation Research 250, 439-446 (1991).]

Measuring intra-assay agreement for the Ames Salmonella assay

Walter W. Piegorsch and Errol Zeiger

National Institute of Environmental Health Sciences
Research Triangle Park, NC 27709, USA

ABSTRACT

Statistical approaches are discussed for determination of the level of agreement among data exhibiting qualitative outcomes. Application is made to a database of results from the Ames Salmonella mutagenicity assay involving 239 chemicals from the U.S. National Toxicology Program. Two forms of average concordance are presented for descriptive measures of the percentage agreement. The first is conservative in nature, allowing for concordance only if all outcomes for a particular subject chemical agree. The second is a more informative pairwise measure, comparing the number of paired concordances to the total number of possible pairs. Inference is based on an additional measure, known as the Kappa coefficient. This value measures qualitative association after correcting for any agreement arising by chance alone. The results for the Salmonella assay show significant positive agreement throughout the database, both as a whole and for pertinent sub-divisions of the data. [Statistical Methods in Toxicology, L. Hothorn, ed. Lecture Notes in Medical Informatics, Vol. 43, Heidelberg: Springer-Verlag, 35-41 (1991).]

KEY WORDS: Ames assay; concordance; genotoxicology; Kappa coefficient; measures of agreement; measures of association; mutagenicity.

Multiple comparisons for analyzing dichotomous response data

Walter W. Piegorsch
Statistics and Biomathematics Branch
National Institute of Environmental Health Sciences
Research Triangle Park, NC 27709, U.S.A.

SUMMARY

Dichotomous response models are common in many experimental settings. Statistical parameters of interest are typically the probabilities, pi, that an experimental unit will respond at various treatment levels indexed by i. Herein, simultaneous procedures are considered for multiple comparisons among these probabilities, with attention directed at construction of simultaneous confidence intervals for various functions of the pi. The inferences are based on the asymptotic normality of the maximum likelihood estimator of pi. Specific applications include all pairwise comparisons and comparisons with a fixed (control) treatment. Monte Carlo evaluations are undertaken to examine the small-sample properties of the various procedures. It is seen that use of the usual estimates of variance leads to less-than-nominal empirical coverage for most sample sizes examined. For very large samples, nominal coverage is achieved. A reformulation of the pairwise comparisons using a form of inverted score test is shown to exhibit generally nominal empirical coverage, and is recommended for use with small-to-moderate sample sizes [Biometrics 47, 45-52 (1991).]

Key Words: Binomial distribution; Comparisons with a control; Confidence intervals; Monte Carlo evaluations; Pairwise comparisons; Quantal response; Simultaneous inference.

Estimating integrals using quadrature methods with an application in pharmacokinetics

A. John Bailer1,2 and Walter W. Piegorsch2

1Department of Mathematics and Statistics, Miami University,
Oxford, Ohio 45056, U.S.A.

and

2Statistics and Biomathematics Branch,
National Institute of Environmental Health Sciences,
Research Triangle Park, NC 27709, U.S.A.

SUMMARY

The 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 Newton-Cotes numerical quadrature procedures is examined from the perspective of minimizing mean-squared error (MSE). The MSEs are examined for a variety of functions commonly encountered in pharmacokinetics. It is seen that the simplest Newton-Cotes procedure, the trapezoidal rule, frequently provides minimum MSE for a variety of concentration-time shapes and under a variety of response variance conditions. A biopharmaceutical example is presented to illustrate these considerations. [Biometrics 46, 1201-1211 (1990).]

Key words: Area under curve; Exponential disposition models; Mean-squared error, Newton-Cotes quadrature, Numerical integration.

Fisher's contributions to genetics and heredity, with special emphasis on the Gregor Mendel controversy

Walter W. Piegorsch
Statistics and Biomathematics Branch
National Institute of Environmental Health Sciences
Research Triangle Park, NC 27709, U.S.A.

SUMMARY

R.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. [Biometrics 46, 915-924 (1990).]

Key words: Chi-Square Test, Evolution, Goodness of fit, History of science, Natural selection, P-values.

Maximum likelihood estimation for the negative binomial dispersion parameter

Walter W. Piegorsch
Statistics and Biomathematics Branch
National Institute of Environmental Health Sciences
Research Triangle Park, NC 27709, U.S.A.

SUMMARY

A follow-up investigation to that given by Clark and Perry (1989 Biometrics 45, 309-316) is presented, giving details for maximum likelihood estimation for the dispersion parameter from a negative binomial distribution. [Biometrics 46, 863-867 (1990).]

Key words: Generalized linear model; Monte Carlo evaluation; Negative binomial distribution.

One-sided significance tests for generalized linear models under dichotomous response

Walter W. Piegorsch
Statistics and Biomathematics Branch
National Institute of Environmental Health Sciences
Research Triangle Park, NC 27709, U.S.A.

SUMMARY

Dichotomous 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 one-sided departures from some null effect. In these cases, procedures can be developed for testing the null effect against a one-sided alternative. These include Bonferroni-type adjustments of univariate Wald tests, and likelihood ratio tests that employ inequality-constrained multivariate theory. This article examines such tests of significance. Monte Carlo evaluations are undertaken to examine the small-sample 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 one-sided tests against a control or standard are also considered. [Biometrics 46, 309-316 (1990).]

Key words: Bonferroni adjustments; Chi-bar-squared statistic; Complementary-log regression; Dunnett's test; Linear inequality constraints; Logistic regression; Probit regression; Simultaneous inference.

Durand's rules for approximate integration

 WALTER W. PIEGORSCH

National Institute of Environmental Health Sciences,
Research Triangle Park, North Carolina 27709

    In the late 19th century the engineer William F. Durand (1859-1958) derived a series of approximation rules for use in numerical integration. These algorithms were based primarily on trapezoidal and parabolic interpolating formulas (which correspond to the two simplest formulas of the Newton-Cotes family of quadrature rules). This discussion reviews Durand's derivations, with special attention given to the applied, practical perspective that lay behind development of his new numerical routines. it is proposed that Durand's emphasis was toward practical application of theoretical results. This led him to emphasize simplicity and applicability in his new routines for approximate integration and to consider heightened accuracy as only a secondary concern. [Historia Mathematica 16, 324-333 (1989).]

    AMS subject classifications: Primary, 01A55, 65-03, 65D30; Secondary, 65C60.
    KEY WORDS: numerical integration, quadrature, trapezoidal rule, Simpson's rule.

Optimal design allocations for estimating area under curves for studies employing destructive sampling

Walter W. Piegorsch1 and A. John Bailer2

Optimal allocations of experimental resources for the estimation of integrals is considered for experiments that use destructive sampling. Given a set of sampling times, a minimum mean square error rule is given for the allotment of fixed experimental resources to the independent variable. The results are seen to be functionally dependent upon the pattern f underlying variability assumed in the model and upon the quadrature rule used to estimate the integral. Extensions to other optimality criteria, including a minimum mean absolute deviation criterion, and to cases involving multiple treatment groups, are also noted. [Journal of Pharmacokinetics and Biopharmaceutics 17, 493-507 (1989).]

KEY WORDS: mean squared error; mean absolute deviation; nonlinear experimental design; numerical quadrature; trapezoidal rule.

1Statistics and Biomathematics Branch, National Institute of Environmental Health Sciences, Research Triangle Park, NC 27709.
2Department of Mathematics and Statistics, Miami University, Oxford, Ohio 45056.

THE EARLY USE OF MATRIX DIAGONAL INCREMENTS IN STATISTICAL PROBLEMS

WALTER W. PIEGORSCH1,2 and GEORGE CASELLA1

Abstract. The early motivation for and development of diagonal increments to ease matrix inversion in least squares (LS) problems is discussed. It is noted that this diagonal incrementation evolved from three major directions: modification of existing methodology in nonlinear LS, utilization of additional information in linear regression, and improvement of the numerical condition of a matrix. The interplay among these factors and the advent of ridge regression are considered in an historical and comparative framework. [SIAM Review 31, 428-434 (1989).] Erratum: Inverting a sum of matrices. SIAM Review 32, 470 (1990).

Key words. matrix inversion, matrix ill-conditioning, nonlinear least squares

AMS(MOS) subject classifications. 65F10, 65C60

1 Biometrics Unit, Cornell University, Ithaca, NY 14853.
2 Dept. of Statistics, Univ. of So. Carolina, Columbia, SC 29208.

Quantitative approaches for assessing chromosome loss in Saccharomyces cerevisiae: general methods for analyzing downturns in dose response

Walter W. Piegorscha, Friedrich K. Zimmermannb, Seymour Fogelc, Stephen G. Whittakerc and Michael A. Resnicka
aNational Institute of Environmental Health Sciences, Research Triangle Park, NC, USA
bInstitut fuer Mikrobiologie, Schnittspahnstrasse 10, D-6100 Darmstadt, FRG,
cUniversity of California, Berkeley, CA 94720 USA

Keywords: Aneuploidy; Chromosome loss; Control reproducibility trials; Umbrella alternatives; Nonparametric statistical tests

Summary

Statistical 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, "umbrella-patterned" 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 two-stage sampling rules to first gauge the dose window of peak response, then test if the response deviates significantly from untreated levels. [Mutation Research 224, 11-29 (1989).]

Quantitative methods for assessing a synergistic or potentiated genotoxic response

Walter W. Piegorsch1 and Barry H. Margolin2
1Statistics and Biomathematics Branch, National Institute of Environmental Health Sciences, Research Triangle Park, NC, 27709
and 2Department of Biostatistics, University of North Carolina, Chapel Hill, NC 27514 (U.S.A.)

Keywords: Interaction; Binary data; Fluctuation assay; Chromosome aberrations; Factorial experiments; Simple independent action; Simple similar action

Summary

The 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 (yes-no). 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 data-analytic strategies are described for detecting synergy and potentiation in factorially designed experiments. This approach is illustrated with a series of analyses of various genotoxicity data sets. [Mutation Research 216, 1-8 (1989).]

QUANTIFICATION OF TOXIC RESPONSE AND THE DEVELOPMENT OF THE MEDIAN EFFECTIVE DOSE (ED50) - AN HISTORICAL PERSPECTIVE

WALTER W. PIEGORSCH

Division of Biometry and Risk Assessment
National Institute of Environmental Health Sciences
Research Triangle Park, North Carolina
    The development of the widely-used median effective (or lethal) dose as a summary measure in quantifying toxic response is reviewed. Attention is directed at those mathematical properties noted by the originator of the media effective dose, the English physiologist John William Trevan, that make the measure a useful summary statistic for toxicological studies. Consideration is also given to the development of precursor measures to the median effective dose, such as minimal effective dose. [Toxicology and Industrial Health 5, 55-62 (1989).]
Key words: history of science, median lethal dose, sigmoid response curve.
Abbreviations: ED50, median effective dose; LD50, median lethal dose.

Confidence bands for logistic regression with restricted predictor variables

Walter W. Piegorsch
Statistics and Biomathematics Branch
National Institute of Environmental Health Sciences
Research Triangle Park, NC 27709, U.S.A.

and

George Casella
Biometrics Unit
Cornell University
Ithaca, NY 14853, U.S.A.

SUMMARY

Confidence bands are constructed for the logistic response function when there is an interval restriction on each of the predictor variables. The construction involves application of a general fitting procedure using Scheffé's S-method. Specific details are given for the case of one predictor variable, along with details for a fixed-width alternative to the S-method bands. For the one-predictor case, Monte Carlo results suggest that both bands are conservative if sample sizes are as low as N= 25. By N= 200, the S-method's coverage probabilities attain their nominal levels, while the fixed-width bands remain conservative. The procedures are illustrated with data from a genetic toxicology experiment. [Biometrics 44, 739-750 (1988).]

Key words: Binomial distribution; Monte Carlo evaluations; Quantal response; Restricted predictor variables, Simultaneous inference.

Exploring relationships between mutagenic and carcinogenic potencies

Walter W. Piegorsch and David G. Hoel

Division of Biometry and Risk Assessment, National Institute of Environmental Health Sciences,
Research Triangle Park, NC 27709 (U.S.A.)

Keywords: Bioassays, Chemical class analysis, Correlation, Predictivity, TD50

Summary

Salmonella 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 non-zero 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 recently-studied compounds are seen to trend lower than those of compounds studied 10 or more years ago. [Mutation Research 196, 161-175 (1988).]

Exploring simple independent action in multifactor tables of proportions

Walter W. Piegorsch, Clarice R. Weinberg, and Barry H. Margolin
Division of Biometry and Risk Assessment
National Institute of Environmental Health Sciences
Research Triangle Park, NC 27709, U.S.A.

SUMMARY

The problem of assessing synergistic or antagonistic departure from simple independent action in multifactor tables of proportions is discussed. A generalized linear model is employed in which additivity corresponds to simple independent action. Data-analytic strategies are proposed for exploring departures from simple independent action in various extensions of the 2x2 table of proportions. This methodology is illustrated with a series of models fitted to cellular differentiation and murine toxicity data. [Biometrics 44, 595-603 (1988).]

Key Words: Dichotomous response; Exploratory data analysis; GLIM; Interaction; Maximum likelihood; Synergy.

Model robustness for simultaneous confidence bands

WALTER W. PIEGORSCH*

 Confidence bands for the simple linear model are examined to assess their degree of robustness to departures from the model. All calculations are made under an interval constraint on the range of interest for the predictor variable. The true model is taken to be a quadratic polynomial, and departure from the linear case is considered in terms of increasing magnitude of a curvature parameter. Proposed measures of robustness include the actual coverage over the predictor variable axis when the band fails to cover the true quadratic model. The different band functions considered include hyperbolic bands constructed from Scheffé's S method, linear-segmented bands, and fixed-width (minimax) and minimax-regret bands. In terms of preserving coverage probability under quadratic misspecification, the fixed-width and linear-segment bands perform best, the former being preferred when the constraint on the predictor variable is small. When coverage is lost over some portion of the constraint interval, the fixed-width bands are also shown to preserve the greatest percentage of covered (predictor) axis values. The favorable performance of the fixed-width bands may be due to their generally wider, more rigid shape, relative to more curvilinear competitors. This may allow the bands to retain more extreme quadratic misspecification than other bands. [Journal of the American Statistical Association 82, 879-885 (1987).]

KEY WORDS: Coverage probability; Mean axis coverage; Quadratic regression; Simple linear regression.

* Walter W. Piegorsch was a Mathematical Statistician in the Biometry and Risk Assessment Program, National Institute of Environmental Health Sciences, Research Triangle Park, NC 27709 at the time of preparation of this article. The author thanks Barry Margolin for helping to develop the problem and giving continuing encouragement and Kenneth Risko and a referee for their many helpful comments.

Performance of likelihood-based interval estimates for two-parameter exponential samples subject to type I censoring

Walter W. Piegorsch

Biometry and Risk Assessment Program
National Institute of Environmental Health Sciences
Research Triangle Park, NC 27709
    Inferences on the parameters in a two-parameter exponential lifetime model when the data are Type I censored are studied, using asymptotic approximations or conditional constructions around the observed number of lifetimes. The asymptotic methods are simple to implement when constructing confidence regions on the scale and guarantee time parameters of the model. Their small-sample properties have not been previously explored, however. This study provides Monte Carlo results to evaluate these properties. For univariate inference on the scale parameter, convergence of the coverage probabilities to nominal levels is slow until the sample size reaches 25. For the guarantee time parameter, two asymptotically equivalent procedures behave similarly, an F distribution-based method performing slightly better for smaller sample sizes. In addition, a simultaneous confidence region and confidence bands on the survivor function are constructed from the univariate intervals. Their performance mimics that of the single intervals on the scale parameter, suggesting caution in application when the sample size is small. [Technometrics 29, 41-49 (1987).]

    KEY WORDS: Confidence region; Maximum likelihood estimation; Monte Carlo evaluation; Simultaneous confidence bounds.

Testing for synergistic effects for simultaneous exposures with stratified dichotomous response

WALTER W. PIEGORSCH and CLARICE R. WEINBERG

Biometry and Risk Assessment Program, National Institute of Environmental Health Sciences, Research Triangle Park, North Carolina 27709, U.S.A.

This paper compares small sample performances of test statistics for assessing the degree of interaction between two exposure with respect to the occurrence of a time-stratified dichotomous outcome. Particular attention is focused towards cancer bioassays. A null model of simple, independent, joint action is adopted. Sizes and powers are examined via Monte Carlo simulations for the various test statistics proposed under a variety of parameter configurations. [Journal of Statistical Computation and Simulation 26, 1-19 (1986).]

KEY WORDS: Interaction, simple independent action, simulation.

Testing for simple independent action between two factors for dichotomous response data

Walter W. Piegorsch, Clarice R. Weinberg, and Joseph K. Haseman

Biometry and Risk Assessment Program,
National Institute of Environmental Health Sciences,
Research Triangle Park, North Carolina 27709, U.S.A.

SUMMARY

This paper compares three test statistics for testing simple independent action between two dichotomous factors with respect to the occurrence of a dichotomous outcome. Sizes and powers are examined for the statistics proposed, under a variety of model parameterizations. The results suggest that a test based on the ratio of the nonresponse probability estimates [considered originally by Wahrendorf, Zentgraf, and Brown (1981, Biometrics 37, 45-54)] has proper size and acceptable power, and is recommended for use in this setting. [Biometrics 42, 413-419 (1986).]

Key words: Interaction; Maximum likelihood; Size of test, Synergy.

Confidence bands for polynomial regression with fixed intercepts

Walter W. Piegorsch

Biometry and Risk Assessment Program
National Institute of Environmental Health Sciences
Research Triangle Park, NC 27709
    Confidence bands are considered for linear and quadratic regression when the regression equation is forced through a fixed point. In the linear case, it is noted that the construction of confidence bands reduces to the construction of a confidence interval on the slope parameter. The corresponding bands are straight lines whose widths are independent of the size of the set over which the band is constructed. The construction for the quadratic case involves more complex arguments; these are developed, and tables are presented for band use in both the homoscedastic and heteroscedastic settings. An example is included. [Technometrics 28, 241-246 (1986).]

    KEY WORDS: Simultaneous inference; Regression through the origin; Quadratic regression; Weighted least squares.

Average width optimality for confidence bands in simple linear regression

WALTER W. PIEGORSCH*

 The problem of constructing optimal confidence bands for a simple linear regression over the whole real line is considered. Optimality is defined as minimization of the average width of the bands weighted by a normalized function. This weight function is presented as an indicator of experimental interest in the varying width of the band. A comparisons between the commonly seen hyperbolic bands and segmented-line bands is presented. [Journal of the American Statistical Association 80, 692-697 (1985).]

KEY WORDS: Simultaneous confidence intervals; Prior weight functions; Constrained minimization.

* Walter W. Piegorsch was a Mathematical Statistician in the Biometry and Risk Assessment Program, National Institute of Environmental Health Sciences, Research Triangle Park, NC 27709 at the time of appearance of this article. This research was initiated while the author was a graduate fellow in the Biometrics Unit, Cornell University, and was supported in part by grants from Sigma Xi, the Scientific Research Society. Thanks are due George Casella, for his aid and assistance during the preparation of the manuscript, L.D. Brown, the associate editor, and the referees for their many helpful comments.

Admissible and optimal confidence bands in simple linear regression1

By WALTER W. PIEGORSCH

Cornell University

A framework is presented for deciding among functional forms when constructing confidence bands in simple linear regression. Using the concept of tautness, definitions of admissibility and completeness are developed. These lead to characterization of a minimal complete class of band forms. A type of average width optimality within this class is briefly discussed. [Annals of Statistics 13, 801-810 (1985).]

1 Research supported, in part, by grants from Sigma Xi, the Scientific Research Society. Additional computing support provided by the National Institute of Environmental Health Sciences, Research Triangle Park, NC. This is paper no. BU-835-M in the Biometrics Unit Series, Cornell University, Ithaca, New York.
AMS subject classifications. Primary 62C15, secondary 62F25.
Key words and phrases. Simultaneous inference, complete classes.

The existence of the first negative moment

WALTER W. PIEGORSCH and GEORGE CASELLA*

The question of the existence of negative moments of a continuous probability density function is explored. A sufficient condition for the existence of the first negative moment is given. The condition is easy to verify, as it involves limits rather than integrals. An example is given, however, that shows that this simple condition is not necessary for the existence of the first negative moment. The delicacy of the characterization of existence is explored further with some results concerning the existence of moments surrounding the first negative moment. [American Statistician 39, 60-62 (1985).]

KEY WORDS: Inverse moment; Sufficient conditions; Limit conditions.

* Walter W. Piegorsch was a Mathematical Statistician in the Biometry and Risk Assessment Program, National Institute of Environmental Health Sciences, Research Triangle Park, NC 27709 and George Casella was Associate Professor, Biometrics Unit, Cornell University, Ithaca, New York 14853 at the time of appearance of this article. The work was performed while both authors were at Cornell University. The second author's research was supported by National Science Foundation Grant MCS81-02541. The authors wish to thank the associate editor and referees for their helpful comments.


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