Complex Mixtures Methods for In Vivo Toxicity Testing (1988) / Chapter Skim
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Appendix G: Empirical Modeling of the Toxicity of Mixtures
Appendix G: Empirical Modeling of the Toxicity of Mixtures
Pages 209-220
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From page 209... ...
In this setting, there is a broad class of powerful statistical tools accompanied by excellent computation software for exploratory investigation of the dose-response surface, for model fitting and diagnostic checking, and for experimental design. For toxicity data in the above format, the job of empirical 209
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An exception occurs in experimental design, because most of the standard optimal experimental design literature on the linear model focuses on construction of designs that minimize the variance of some regression parameters. In the standard linear model, variance expressions are independent ofthe value ofthe response surface, so the optimal design is relatively easy to define.
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From page 211... ...
is to approximate the relationship between toxicity and dose over the observed dose range; extrapolation on the basis of these models without additional scientific insight might be inappropnate. Where extrapolation is required, the empirical model will, at best, suggest relationships that a scientist can use to postulate more refined theones.
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17 on 18 degrees of 1 80,000 1 60,000 1 40,000 1 20,000 ~ 1 00,000 Q o 80,000 60,000 40,000 o o o 4 13 0 2 20,000 1 1 1 1 1 1 1 o 0 1 2 3 2 2 5 0 3 5 1,500 2,000 2,500 3,000 3,500 4,000 CO, ppm FIGURE G-1 Data from inhalation experiments (Levin et al., in press)
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The scaled deviance has an approximate chi-squared distribution with a number of degrees of freedom equal to the sample size minus the number of parameters in the model. The deviance for our fit is larger than one would hope the expected value for a chi-squared variate is equal to its number of degrees of freedom, so, because 24.17 is over 30% larger then the expected 18, one might worry about the adequacy of the model.
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A variety of modern statistical software packages could have been used to implement our analysis. We used the GLIM statistical package that is distributed by the Royal Statistical Society and the S statistical package,
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This section shows that additive models constitute a large class of models, including many often used to address dose-response issues. GENERALIZED ADDITIVE MODEL An additive model might not be appropriate in the raw toxicity scale, but it might hold for some alternative scale.
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In the case where the His are parametric, the generalized additive models are a special case of the generalized linear models of McCullagh and Nelder (1983)
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In the Hewlett and Plackett system, the joint action of the chemicals is classified as in Table G-2. In a Hewlett-Plackett model, a toxic event is said to occur if the administered dose exceeds some tolerance or threshold in the organism and a probability distribution for tolerances in the population is specified.
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m 1 + ~ aiZ i = 1 , (G4) where al are association constants, aizi corresponds to the effective dose of the ith component at the site of action, and hi measures the effect on the activity due
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For example, normality and constant variance are not required for estimates of the random-error component, and interactions of systematic effects can be specified to hold on an additive scale by appropriate transformations of the scale. Also, GLMs are accessible through an extensive collection of computer programs that permit various approaches to be explored and that can model joint excess risks without reliance on arbitrary biologic mechanisms.
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From page 220... ...
1986. Automatic smoothing of regression functions ingeneralizedlinear models.
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