Toxicological investigation of environmental chemical mixtures is evolving, with attention focused on defined mixtures, involving a limited number of chemicals, and complex environmental mixtures, involving a large number of chemicals and, typically, an unidentified fraction. Dosing regimens are being developed to evaluate the toxicity of complex mixtures using approaches consistent with human environmental exposures that are typically much lower than those used in traditional toxicology studies. Consequently, newer studies are designed to evaluate the toxicity of complex mixtures (1) with the inclusion of low, environmentally relevant dose levels; (2) with the relative proportions of component chemicals similar to those measured in environmental samples; and, (3) with approaches that maintain the chemically unidentified components of the mixture [1,2]. This type of study design differs from many traditional defined-mixture studies, conducted mostly on binary mixtures at relatively high dose levels where adverse effects are more readily detected if present, but whose relevance to human health risks associated with low environmental exposures is often unclear. This newer type of study design has been proposed specifically for application to drinking water disinfection by-products (DBPs) [3–5].

Disinfection of drinking water for microbial contamination provides an essential public health benefit in reduction of water-borne disease. However, oxidizing disinfectants react with materials in the source water resulting in the formation of a wide variety of DBPs. DBP mixtures are highly complex, containing numerous chemicals not routinely measured and many that are unknown; approximately 50% of the total organic halide compounds formed when water is disinfected remains unidentified [6–8]. Some epidemiologic studies report adverse developmental effects associated with exposure to DBPs, including low birth weight at term, small for gestational age [9,10], and stillbirths [11,12]. Epidemiologic evidence is mixed regarding associations of DBPs with spontaneous abortion [13,14]. Toxicity bioassays have been conducted on approximately 35 individual DBPs and a limited number of DBP mixtures [15–19]; some of the tested DBPs and DBP mixtures were shown to be reproductive or developmental toxicants in experimental animals.

Because concerns identified from epidemiologic studies on whole DBP mixtures cannot be readily addressed by investigating either individual DBPs or simple, defined DBP mixtures, scientists from four of the national laboratories and centers of the U.S. EPA’s Office of Research and Development have developed and, along with extramural partners, undertaken a research project (the Four Lab Study) that integrates toxicological and chemical evaluation of environmentally realistic complex mixtures of DBPs [4,5,20]. For complex DBP mixtures formed by chlorination, improved understanding of in vivo reproductive/developmental toxicology is a priority; consequently, the Four Lab study included a multigenerational reproductive/developmental toxicity rodent bioassay. Preparation for this experiment included: conducting a phased series of experiments involving water concentration, toxicology, and chemistry [8,21–25]; development of new risk assessment methodology [26]; and conducting developmental toxicity screens on sodium, sulfate, and concentrated DBPs [27].

The resulting database of toxicological and analytical chemistry data on the whole DBP mixture provides important information for health risk assessment of DBPs [26]. Risk assessment investigations include the analysis of associations between positive assay results (e.g., delays in attaining puberty) and dose level of specific DBPs (e.g., dichloroacetic acid) or groups of DBPs (e.g., trihalomethanes) identified within the whole mixture. Using existing dose-response data on individual chemicals, statistical models can then be used to test whether the observed toxicity of the whole mixture can be attributed to known DBPs, their joint toxic action, or to the unidentified fraction of the DBP mixture [26]. These distinctions are important for informed decisions with regard to controlling the levels of specific DBPs or groups of DBPs in finished drinking water, resulting in the production of clean, safe water. To allow for expanded use of the results, statistical and toxicological criteria have been developed to determine the “sufficient similarity” of DBP mixture composition and toxicity potential among finished drinking waters [28]. Using such criteria, it may be possible to extrapolate Four Lab Study results on the chlorinated concentrates to evaluate the potential for health effects from different DBP mixtures that could arise from: (1) treatment with other types of disinfectants (e.g., chloramination), (2) differences among source waters (e.g., differences in NOM) or (3) differences in treatment practices across treatment plants or over time within the same treatment plant.

Experimental constraints considered in the design of the multigenerational bioassay included: the number of dams that could be accommodated at one time (maximum of 100); the extent that water could be concentrated while retaining palatability and conserving organics (a concentration factor of 136× for total organic carbon was achieved for use in the multigenerational bioassay [29]); and, the quantity of concentrate that would be produced (~1,500 liters). Designing a meaningful study given these constraints required careful attention to statistical power.

In the multigenerational bioassay, timed-pregnant Sprague-Dawley rats comprising the parental (P₀) generation would be assigned randomly to either a control group or a treatment group which would consume chlorinated water concentrate. Each P₀ dam was to deliver a litter (F₁ generation). An issue addressed in the present work was whether to breed one or two females from each F₁ litter to a non-sibling F₁ male from the same exposure group to produce the F₂ generation. Priority study endpoints were prenatal loss (number of uterine implantation sites minus number of live pups at birth, divided by implantation sites) and pup birth weight. In comparison with epidemiologic endpoints of concern, prenatal loss is analogous to spontaneous abortion, whereas reduced pup weight is analogous to small for gestational age and term low birth weight.

U.S. EPA testing guidelines for reproductive toxicity call for 20 pregnant females per group as the standard protocol for single chemical bioassays [30]. The guidelines also state “the highest dose level should induce toxicity in the parental (P) animals and intermediate dose levels should produce minimal observable toxic effects. The lowest dose level should not produce any evidence of toxicity” [30]. Under these guidelines, the requirement for toxicity in the high and middle dose levels ensures adequate statistical power to detect effects, but not necessarily at the low dose. In contrast, in the multigenerational bioassay whose design was the goal of this paper, pregnant animals were to be exposed to much lower (i.e., environmentally relevant) DBP concentrations than the guidelines suggest. Thus, study design, particularly estimation of statistical power, was essential to optimize sample sizes within the experimental constraints, so that effects, if present, would be detected, increasing the utility of the experimental results for risk assessment. This article describes the development and application of the methodology used for calculating statistical power for non-independent observations for this bioassay with chlorinated water concentrates in the Four Lab Study, taking into account the multigenerational bioassay design as well as constraints on sample size, concentration factor, and sample volume.

For analyzing pup weight, male and female pups were considered separately as well as combined; the results presented in this paper are for the combined male and female pups. The litter was treated as the experimental unit, with each pup within a litter representing a repeated measurement. A linear model was assumed for the data, so that:

where y_ijk is the weight of the k^th live pup in the j^th litter of the i^th group (where i = 1 for the control group and i = 2 for the treatment group):

and:

In this model, the pup weights, y_ijk, were assumed to follow a Gaussian distribution with mean β_i and common variance-covariance matrix ∑ within litters. A constant correlation was assumed between the weights of all pups from the same litter (i.e., a compound symmetric correlation structure), while weights of pups from different litters were assumed to be uncorrelated (independent).

An additional key endpoint, prenatal loss, was also examined with respect to power. For prenatal loss, litter again represented the experimental unit, with each implantation site representing a repeated measurement. The endpoint was then coded as:

where i equals 1 (control) or 2 (treatment).

If π_i denotes the mean prenatal loss probability among dams in the i^th group, then under a linear model:

where x_1j and x_2j are defined as above. Alternatively, when a linear logistic model was assumed for the data, then:

where x_ij and x_2j are defined as above.

The linear and linear logistic models differ based upon the alternative hypotheses being tested. In both the linear and linear logistic models for prenatal loss a binomial error distribution was assumed for the model. In the linear model, the alternative hypothesis H_α:β₁ ≠ β₂ represents the difference between the prenatal loss proportions of the control and treatment groups (e.g., the prenatal loss for the treatment group is 7.1 percentage points greater than the prenatal loss for the control group). In contrast, under the linear logistic model, the alternative hypothesis H_α:β₁ ≠ β₂ represents the difference between the log odds of the control and treatment groups, which for small π_i is approximately equal to the ratio of the prenatal loss proportions of the treatment and control groups (e.g., the prenatal loss for the treatment group is approximately twice the prenatal loss for the control group). Therefore, with respect to the biological interpretation of the endpoint, the specific question of interest, i.e., an absolute increase over control vs. a proportional increase relative to control, is important to the appropriate model specification. The results for the two tests should be asymptotically equivalent under the null hypothesis, but sensitivity may differ under different alternative hypotheses. Results for both models are presented here.

For both the pup weight data and the prenatal loss data, an explicit relationship exists between the over-dispersion parameter, which represents the proportion of the observed variability that is due to the correlation among the observations, and the intra-litter correlation. For both distributions, the over-dispersion parameter, ψ, can be expressed as a function of the intra-litter correlation, ρ:

where n_ij is the number of pups in the j^th litter of the i^th group. From this equation it follows that ψ_ij = 1 if and only if ρ = 0 or n_ij = 1. For each scenario examined, the intra-litter correlation ρ was specified based on the pilot data, and the dispersion parameter ψ was calculated according to the above equation. By doing this, the estimates for ρ given ψ closely matched those observed from modeling the Narotsky et al. [27] data.

This article describes the methodology used for calculating statistical power for non-independent observations in a two-block design for the multigenerational reproductive/developmental toxicity rodent bioassay in the Four Lab Study. It takes into account the multigenerational bioassay design as well as constraints on sample size, water concentrate volume and concentration factor.

Designing this bioassay under these constraints necessitated thoughtful consideration of statistical power. Determining power and sample size for multiple block designs is complicated, because it must account for the interaction of groups with blocks. Though the effect of the block is not of inherent interest, it might influence the group (treatment) effect if the DBP levels within the concentrate change during the course of the study. Using developmental toxicity screening data [27], power calculations were made for prenatal loss and pup weight. These endpoints were considered of primary importance because they correspond to effects reported in positive epidemiologic studies [9,10,13,14].

While several rodent developmental toxicity investigations have been conducted using exposures to concentrated tap waters [15,27,34,35], only Uriu-Hare et al. [35] reported power calculations. They calculated the sample size necessary to provide 80% power to detect a 50% increase in the number of dams with at least one resorption. The authors subsequently found a significant increase in the incidence of dams with one or more resorptions among those dams drinking unconcentrated tap water compared to dams drinking deionized water. The power calculations developed for this multigenerational bioassay are more detailed than those of Uriu-Hare et al. [35]. Other DBP toxicology studies to date show a lack of adverse developmental effects or only marginal or subtle effects on rodent development, however, without estimates of power, these results are difficult to interpret [15,24,34,35].

Based on the results of the power analyses and the physical constraints of the study, the Four Lab team selected a two-block design for the multigenerational bioassay, assigning 40 and 60 timed-pregnant rats to the control and treatment groups in each block, respectively. The two-block design achieved greater than the desired 80% power at a significance level of 0.05 with respect to pup weight for all the scenarios examined; more than half of the scenarios for the two-block design achieved 100% power. This design ensured that at least one sensitive endpoint (i.e., pup weight) had maximum power to detect an effect at the low-response region of the dose-response curve. Fetal or pup weight is known to be a sensitive developmental endpoint in animal studies that is often observed at doses below those causing other developmental effects [36]. This endpoint is also relevant to human health, as low birth weight children have been shown to be at increased risk for developing several chronic sequelae [37,38].

For this research, detecting a prenatal loss effect, if present, also was desirable. The two-block design, that optimizes the power for pup weight loss, also provides the most power for prenatal loss from among the possible designs, given the study constraints. This analysis shows that the two-block design provides a modest amount of power (i.e., at most, 57–58% using the linear model) to detect differences in prenatal loss. Analyses showed that the simplest design, consisting of a single block, would have insufficient power. A second possible single-block design, which bred two females per F₁ litter, was eliminated from consideration due to the uncertainty surrounding the inter-litter correlation, and the need for larger quantities of water concentrate. In general, the power associated with the two-block design was approximately twice that of the single-block designs considered. The somewhat large discrepancy between the power for pup weight and prenatal loss was expected and is inevitable in a reproductive toxicity study. This is because, relative to their respective means, the variance for prenatal loss is generally much larger than for pup weight (e.g., Table 1). Thus, based on this analysis, the two-block design was considered the best design option for the Four Lab multigenerational bioassay.

The constraints imposed by conducting toxicological investigations with highly complex environmental mixtures in an environmentally relevant medium at environmentally relevant dose levels are not unique to the Four Lab Study. The methodology described here may be applied to appropriately design other toxicology studies with environmentally realistic complex mixtures, as similar constraints likely will be encountered.

This work highlights the importance of considering statistical power in the design of bioassays that evaluate health effects of chemical mixtures in the low-response region of the dose-response curve. Such biostatistical analyses provide meaningful quantitative insights into the trade-offs inherent in the design of studies conducted in the low-response region and provide a clear and logical rationale for choice of study design. These analyses and insights lead to toxicological studies in the low-response region that provide meaningful results and allow for appropriate interpretation of experiments when no observable adverse effect is detected. The conduct of such toxicological studies is critical for improved dose-response assessments of complex chemical mixtures, because they increase understanding of the potential human health effects from exposure to chemical mixtures near environmental exposure levels, which are of increased relevance to human health risk assessment [39].