Threshold models for combination data from reproductive and developmental experiments

Pamela F. Schwartz, Chris Gennings, Vernon M. Chinchilli

Research output: Contribution to journalArticlepeer-review

40 Scopus citations


In risk assessment, thresholds are generally believed to exist for toxic reproductive and developmental agents. Therefore, dose combinations exist below the threshold where the response is not distinguishable from background and above the threshold where a dose-response trend results. Because data from combination experiments are more reflective of human exposure to environmental agents, threshold models that incorporate the effect of interactions among these agents are developed. For s agents acting in combination, the threshold is an s-dimensional surface. More specifically, for a combination of two agents, the threshold is a two-dimensional contour of dose combinations, and for a single agent, the threshold is represented by a single dose value, not necessarily a dose level used in the experiment. In toxicological experiments designed to investigate the reproductive or developmental effects of a chemical on laboratory animals, litter effects typically are present, yielding overdispersion. Instead of assuming a particular distributional form for the binary responses, quasi-likelihood methods often are used where parameters are introduced to measure the overdispersion. This article describes threshold models for combination reproductive and developmental experiments and develops parameter estimation techniques using quasi-likelihood methods. Once parameters are estimated, a hypothesis test for an overall dose-response relationship is performed using a quasi-likelihood ratio statistic. Next, a confidence interval about the threshold parameter is constructed using a quasi-likelihood ratio statistic, and finally, a confidence region is constructed about the threshold surface. Illustrative examples are presented that use single-agent reproductive data from the National Toxicology Program and three-agent combination developmental data from the Environmental Protection Agency and ManTech Environmental Technology, Inc.

Original languageEnglish
Pages (from-to)862-870
Number of pages9
JournalJournal of the American Statistical Association
Issue number431
StatePublished - Sep 1995
Externally publishedYes


  • Confidence regions
  • Noncarcinogenic
  • Overdispersion
  • Quasi-likelihood


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