Analysis of proportional data in reproductive and developmental toxicity studies: Comparison of sensitivities of logit transformation, arcsine square root transformation, and nonparametric analysis.

BACKGROUND: In developmental and reproductive toxicity studies, analysis of litter-based binary endpoints (e.g., incidence of malformed fetuses) is complex in that littermates often are not entirely independent of one another. It is well established that the litter, not the individual fetus, is the proper independent experimental unit in statistical analysis. Accordingly, analysis is often based on the proportion affected per litter and the litter proportions are analyzed as continuous data. Because these proportional data generally do not meet assumptions of symmetry or normality, data are typically analyzed by nonparametric methods, arcsine square root transformation, or logit transformation. METHODS: We conducted power calculations to compare different approaches (nonparametric, arcsine square root-transformed, logit-transformed, untransformed) for analyzing litter-based proportional data. A reproductive toxicity study with a control and one treated group provided data for two endpoints: prenatal loss, and fertility by in utero insemination (IUI). Type 1 error and power were estimated by 10,000 simulations based on two-sample one-tailed t tests with varying numbers of litters per group. To further compare the different approaches, we conducted additional analyses with shifted mean proportions to produce illustrative scenarios. RESULTS: Analyses based on logit-transformed proportions had greater power than those based on untransformed or arcsine square root-transformed proportions, or nonparametric procedures. CONCLUSION: The logit transformation is preferred to the other approaches considered when making inferences concerning litter-based proportional endpoints, particularly with skewed distributions. The improved performance of the logit transformation becomes increasingly pronounced as the response proportions are increasingly close to the boundaries of the parameter space.

Evaluating the similarity of complex drinking-water disinfection by-product mixtures: overview of the issues.

Humans are exposed daily to complex mixtures of environmental chemical contaminants, which arise as releases from sources such as engineering procedures, degradation processes, and emissions from mobile or stationary sources. When dose-response data are available for the actual environmental mixture to which individuals are exposed (i.e., the mixture of concern), these data provide the best information for dose-response assessment of the mixture. When suitable data on the mixture itself are not available, surrogate data might be used from a sufficiently similar mixture or a group of similar mixtures. Consequently, the determination of whether the mixture of concern is "sufficiently similar" to a tested mixture or a group of tested mixtures is central to the use of whole mixture methods. This article provides an overview for a series of companion articles whose purpose is to develop a set of biostatistical, chemical, and toxicological criteria and approaches for evaluating the similarity of drinking-water disinfection by-product (DBPs) complex mixtures. Together, the five articles in this series serve as a case study whose techniques will be relevant to assessing similarity for other classes of complex mixtures of environmental chemicals. Schenck et al. (2009) describe the chemistry and mutagenicity of a set of DBP mixtures concentrated from five different drinking-water treatment plants. Bull et al. (2009a, 2009b) describe how the variables that impact the formation of DBP affect the chemical composition and, subsequently, the expected toxicity of the mixture. Feder et al. (2009a, 2009b) evaluate the similarity of DBP mixture concentrates by applying two biostatistical approaches, principal components analysis, and a nonparametric "bootstrap" analysis. Important factors for determining sufficient similarity of DBP mixtures found in this research include disinfectant used; source water characteristics, including the concentrations of bromide and total organic carbon; concentrations and proportions of individual DBPs with known toxicity data on the same endpoint; magnitude of the unidentified fraction of total organic halides; similar toxicity outcomes for whole mixture testing (e.g., mutagenicity); and summary chemical measures such as total trihalomethanes, total haloacetic acids, total haloacetonitriles, and the levels of bromide incorporation in the DBP classes.

Evaluating sufficient similarity for disinfection by-product (DBP) mixtures: multivariate statistical procedures.

For evaluation of the adverse health effects associated with exposures to complex chemical mixtures in the environment, the U.S. Environmental Protection Agency (EPA) (2000) states, "if no data are available on the mixture of concern, but health effects data are available on a similar mixture ... a decision must be made whether the mixture on which health effects are available is 'sufficiently' similar to the mixture of concern to permit a risk assessment." This article provides a detailed discussion of statistical considerations for evaluation of the similarity of mixtures. Multivariate statistical procedures are suggested to determine whether individual samples of drinking-water disinfection by-products (DBPs) vary significantly from a group of samples that are considered to be similar. The application of principal components analysis to (1) reduce the dimensionality of the vectors of water samples and (2) permit visualization and statistical comparisons in lower dimensional space is suggested. Formal analysis of variance tests of homogeneity are illustrated. These multivariate statistical procedures are applied to a data set describing samples from multiple water treatment plants. Essential data required for carrying out sensitive analyses include (1) identification and measurement of toxicologically sensitive process input and output characteristics, and (2) estimates of variability within the data to construct statistically efficient estimates and tests.

Evaluating sufficient similarity for drinking-water disinfection by-product (DBP) mixtures with bootstrap hypothesis test procedures.

In chemical mixtures risk assessment, the use of dose-response data developed for one mixture to estimate risk posed by a second mixture depends on whether the two mixtures are sufficiently similar. While evaluations of similarity may be made using qualitative judgments, this article uses nonparametric statistical methods based on the "bootstrap" resampling technique to address the question of similarity among mixtures of chemical disinfectant by-products (DBP) in drinking water. The bootstrap resampling technique is a general-purpose, computer-intensive approach to statistical inference that substitutes empirical sampling for theoretically based parametric mathematical modeling. Nonparametric, bootstrap-based inference involves fewer assumptions than parametric normal theory based inference. The bootstrap procedure is appropriate, at least in an asymptotic sense, whether or not the parametric, distributional assumptions hold, even approximately. The statistical analysis procedures in this article are initially illustrated with data from 5 water treatment plants (Schenck et al., 2009), and then extended using data developed from a study of 35 drinking-water utilities (U.S. EPA/AMWA, 1989), which permits inclusion of a greater number of water constituents and increased structure in the statistical models.