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Keywords:

  • Calibration;
  • Cytogenetic dosimetry;
  • Dependent Dirichlet process;
  • Dose–response curve;
  • Markov chain Monte Carlo;
  • Nonparametric mixture models

Summary

  1. Top of page
  2. Summary
  3. 1. Introduction
  4. 2. Methods
  5. 3. Posterior Inference
  6. 4. Data Illustrations
  7. 5. Discussion
  8. 6. Supplementary Materials
  9. Acknowledgments
  10. References
  11. Supporting Information

We develop a Bayesian nonparametric mixture modeling framework for quantal bioassay settings. The approach is built upon modeling dose-dependent response distributions. We adopt a structured nonparametric prior mixture model, which induces a monotonicity restriction for the dose–response curve. Particular emphasis is placed on the key risk assessment goal of calibration for the dose level that corresponds to a specified response. The proposed methodology yields flexible inference for the dose–response relationship as well as for other inferential objectives, as illustrated with two data sets from the literature.

1. Introduction

  1. Top of page
  2. Summary
  3. 1. Introduction
  4. 2. Methods
  5. 3. Posterior Inference
  6. 4. Data Illustrations
  7. 5. Discussion
  8. 6. Supplementary Materials
  9. Acknowledgments
  10. References
  11. Supporting Information

The quantal bioassay dose–response setting has been valuable in many fields, from ecology to medicine. Though quantitative measurement of a response is preferred, there are several experiments for which the response can only be recorded on a binary scale, either occurring or not occurring. The dose–response relationship is based on observed data from experimental animal, human clinical, or cell studies. In these types of settings, as the level of exposure increases, the responses generally become more severe. However, within a population, a wide range of responses may be encountered as some individuals are more susceptible and others resistant.

Regarding the quantal response data structure, at each of a typically small number of dose levels, a number of subjects are exposed to the substance and the subset that have the response of interest is recorded. The dose–response curve, denoted by inline image, is defined as the probability of positive response as a function of dose x. One of the standard assumptions in the analysis of quantal bioassay problems is that inline image is a non-decreasing function and can therefore be modeled as a distribution function. Under this assumption, the distribution corresponding to inline image is referred to as the tolerance distribution. In parametric modeling, inline image is assumed to be a member of a parametric family of distribution functions. Standard parametric models, such as probit or logit models, are simple to implement but do not have the ability to capture complex curves, including skewness and multimodality.

To extend the inferential scope of standard models, finite mixtures for the tolerance distribution have been considered in the literature. Lwin and Martin (1989) present a location-scale mixture model, which is developed based on the idea of a finite number of unobserved, underlying subpopulations. Geweke and Keane (1999) propose mixtures of probit models to more accurately describe the shape of the dose–response curve, while Basu and Mukhopadyay (2000) use a finite scale mixture of normal distribution functions. In addition to theoretical advantages, Bayesian nonparametric mixture priors, as developed in this article, offer practical benefits relative to finite parametric mixtures with respect to Markov chain Monte Carlo (MCMC) posterior simulation as well as prior specification.

The quantal bioassay setting corresponds to one of the earliest applications for Bayesian nonparametrics. One of the more common approaches involves assigning a Dirichlet process (DP) prior (Ferguson, 1973) to the distribution function inline image (e.g., Antoniak, 1974; Bhattacharya, 1981; Disch, 1981; Kuo, 1983, 1988; Gelfand and Kuo, 1991; Mukhopadhyay, 2000). Also along these lines is the work in Muliere and Walker (1997) where Pólya tree priors are used to determine the maximum tolerated dose. A monotone nonparametric regression framework is presented in Bornkamp and Ickstadt (2009), where the monotone function is modeled as a mixture of shifted and scaled two-sided power distribution functions.

An alternative approach is to model the binary responses with latent continuous variables, and assume mixtures of probit or logit functions for the dose–response curve. MacEachern (1998) assumes a DP prior directly on the distribution of the latent variables. Presented in terms of multivariate probit regression, Jara, García-Zattera, and Lesaffre (2007) mix on the intercept of the linear mean and the covariance matrix of the distribution of the subject specific latent variable distribution.  Casanova, Iglesias, and Bolfarine (2010) place a scale normal DP mixture prior on the distribution of the latent variables.

We also represent the model through latent continuous responses, but build the inference for the dose–response curve from nonparametric mixture modeling for dose-dependent response distributions. We adopt a structured dependent DP prior for the collection of dose-dependent mixing distributions, which begets a monotonicity restriction and produces smooth realizations of the dose–response curve. We place particular emphasis on inference for calibration, where we seek to estimate the dose level associated with a given vector of responses. This objective is especially important in cytogenetic dosimetry, a particular area of dose–response modeling concerned with the relationship between exposure to radiation and some measure of genetic aberration. In these studies, only a portion of the dose–response curve is observed, rendering extrapolation beyond observed dose levels a primary inferential target.

The outline of the article is as follows. We develop the model in Section 2, including the hierarchical formulation with latent continuous responses, and a method for prior specification. The approach to posterior simulation and inference for risk assessment is provided in Section 3. In Section 4, we illustrate the modeling approach with a standard data set from the literature as well as with the cytogenetic dosimetry application. Finally, Section 5 concludes with discussion of possible extensions of the methodology.

2. Methods

  1. Top of page
  2. Summary
  3. 1. Introduction
  4. 2. Methods
  5. 3. Posterior Inference
  6. 4. Data Illustrations
  7. 5. Discussion
  8. 6. Supplementary Materials
  9. Acknowledgments
  10. References
  11. Supporting Information

2.1. The Modeling Approach

The starting point of our methodology is nonparametric mixture modeling for the collection of response distributions, which are indexed by dose level inline image, where, typically, inline image, as dose is commonly recorded on a logarithmic scale. With replicated binary responses at a number of observed dose values, the natural mixture structure involves mixing of Bernoulli distributions,

  • display math(1)

where inline image is the standard normal distribution function, and thus inline image. Given the mixture setting, the choice of the link function for the kernel probability is driven mainly by convenience in model implementation where the probit link offers advantages. We aim to develop flexible nonparametric modeling for the response distributions inline image, and for the implied dose–response relationship, inline image, inline image, retaining however the traditional assumption of monotonicity for the dose–response curve. To this end, the model needs to be completed with an appropriate nonparametric prior for the collection of mixing distributions, inline image inline image.

Given the nature of quantal bioassay studies, we expect response distributions associated with nearby dose levels to be more similar than those which are far apart. Under our mixture model setting, this situation gives rise to the need for a prior which relates the mixing distributions inline image across dose level x to varying degrees. A powerful option for this modeling problem is the dependent Dirichlet process (DDP) prior (MacEachern, 2000). The DDP prior is motivated by the (almost sure) discrete representation of DP realizations (Sethuraman, 1994), where, in full generality, both the stick-breaking weights and the atoms may evolve with x. Because the general DDP prior is complicated to implement and requires large data sets to sufficiently learn about its hyperparameters, simpler versions are typically employed in applications. Under minimal conditions, Barrientos et al. (2012) have established full support (according to weak convergence) for both of the simplified DDP prior specifications discussed next.

In particular, a “single-p” DDP prior involves a countable mixture of realizations from a stochastic process over inline image, with weights matching those from the standard DP, that is, inline image inline image. Here, the inline image inline image are i.i.d. realizations from a base stochastic process, inline image, over inline image, and the weights arise from stick-breaking: inline image inline image, inline image inline image for inline image, with the inline image i.i.d. from a Betainline image distribution (independently of the inline image). Under a single-p DDP prior for the mixing distributions in (1), the dose–response curve, inline image inline image inline image, inline image. This model does not enforce monotonic dose–response relationships, although an increasing trend can be incorporated in prior expectation. This structure can be achieved through a Gaussian process for inline image with constant variance, inline image, and linear mean function, inline image + inline image, with inline image. Then, the inline image distribution for inline image, induced by the Gaussian process inline image at inline image, is stochastically ordered, and thus inline image inline image is increasing in x being the expectation of the increasing function inline image with respect to inline image (depending on the context, we use Ninline image for either the normal distribution or density with mean m and variance inline image).

Contrarily, we may consider a DDP prior structure where only the weights evolve with x, that is, inline image inline image, with the inline image i.i.d. from a base distribution inline image on inline image, independently of the stochastic mechanism that generates the inline image inline image. This “single’inline image” DDP formulation presents a formidable complication with regard to the main inferential objectives for bioassay experiments. A monotonically increasing dose–response relationship, at least in prior expectation, is imperative in terms of prediction at unobserved dose levels to anchor the inference with an increasing trend. In the case of the single’inline image DDP prior, there is no means to force such a trend for the dose–response curve, inline image inline image. More specifically, using the monotone convergence theorem and letting inline image inline image (where a finite mean is assumed for inline image), we obtain inline image inline image inline image, for any inline image. Thus, the prior expectation of the dose–response curve is constant in x, rendering interpolation and extrapolation inference practically useless.

Hence, the single-p DDP prior emerges as the preferred choice for dose–response inference built from modeling dependent response distributions. The general version of the single-p DDP prior can be used for settings where one seeks the extra flexibility of non-monotonic dose–response functions with the increasing trend only in prior expectation. We have studied such DDP mixture modeling for developmental toxicity experiments, which involve clustered categorical responses and multiple dose–response curves for distinct endpoints (Fronczyk and KottasFronczyk and Kottasin press, Kottas and Fronczyk, 2013). For these more involved experiments, the capacity of the DDP mixture model to uncover non-monotonic dose–response relationships is a practically relevant feature. However, for the simpler quantal bioassay setting, we seek a more structured nonparametric prior to incorporate the standard monotonicity assumption for the dose–response curve with prior probability one rather than only in prior expectation.

To this end, we explore the linear-DDP prior formulation, which simplifies the single-p DDP stochastic process realizations inline image to linear functions of x, that is, inline image inline image + inline image. Hence, the base stochastic process inline image is replaced with a distribution inline image that generates the component specific intercept and slope parameters. In particular, we assume the inline image are i.i.d. from inline image, and independently, the inline image are i.i.d. from inline image. We refer to DeIorio et al. (2009) for survival regression with linear-DDP priors, and DeIorio et al. (2004) for the closely related ANOVA-DDP model.

The essential observation from the construction of the linear-DDP prior is the correspondence of inline image and inline image inline image, where inline image DPinline image with inline image inline image. In particular, the atoms inline image, sampled originally from inline image, and the weights, inline image, in the stick-breaking construction of G induce the atoms inline image inline image in conjunction with the same weights, inline image, in the stick-breaking construction of inline image, for all x. Hence, the linear-DDP mixture model, inline image, where inline image inline image, can be equivalently formulated as a DP mixture model

  • display math

The corresponding dose–response curve is now defined by

  • display math(2)

Hence, under the more structured linear-DDP mixture model, the restriction inline image, implemented with a distribution inline image supported by inline image, yields a sufficient condition for dose–response curve realizations to be increasing with prior probability one.

For most bioassay experiments, it is natural to view the observed binary response as a proxy for a latent continuous response variable. This procedure offers a tractable way to estimate the parameters of the proposed DP mixture model. In particular, with the restriction inline image, we can reparameterize the Bernoulli kernel, and, consequently, the underlying latent response distribution, to a location-scale mixture of normal distributions. Specifically, let inline image inline image and inline image inline image, and denote by z the latent inline image-valued response associated with the binary outcome y. Then, we obtain

  • display math

where the final probability arises under a location-scale normal DP mixture for the latent tolerance distribution, inline image, where inline image with inline image inline image an appropriate distribution on inline image. Because the inline image parameterization expedites prior specification, we work with the equivalent location-scale normal DP mixture formulation (suppressing the inline image notation in the following).

2.2. Hierarchical Model Formulation

The general structure for the observations comprises data = inline image, where inline image are the binary responses from the inline image subjects at dose level inline image. Moreover, we denote by inline image the latent (unobserved) continuous response corresponding to inline image. Inclusion of the latent responses facilitates posterior simulation; see Section 3. Hereinafter, we use the notation inline image to indicate the relationship between inline image and inline image, where inline image when inline image, and inline image otherwise.

Using the location-scale parameterization for the normal DP mixture tolerance distribution developed in Section 2.1, the mixture model for the data are given by

  • display math

with inline image and inline image defined through independent Ninline image and inv-gammainline image components, where the inverse gamma distribution has mean inline image provided inline image. Here, c is fixed and inline image inline image denotes the parameters of the DP base distribution which are assigned hyperpriors.

As a computational tool, we use a DP truncation approximation replacing G with inline image inline image. Truncation from the outset provides closed form full conditionals for posterior simulation via Gibbs sampling and a straightforward means for obtaining inference for all objectives. Here, the inline image, given inline image, are i.i.d. inline image, and the weights inline image arise from a truncated version of the stick-breaking construction: inline image, inline image inline image, inline image, and inline image inline image, with the inline image i.i.d., given inline image, from Betainline image. The truncation level can be chosen using distributional properties for the tail probability of the stick-breaking weights, inline image inline image. More specifically, inline image inline image and inline image inline image (Ishwaran and Zarepour, 2000). These expressions can be averaged over the prior for inline image to estimate inline image and inline image. Given a tolerance level for the approximation, the former expression yields the corresponding value L. For both data examples of Section 4, we used a gammainline image prior for inline image. Consequently, we set the truncation level to inline image, which, after averaging over the prior for inline image, results in inline image, and inline image.

To represent the mixture component with which each data point inline image is associated, configuration variables inline image are introduced. Then, the hierarchical model for the data, augmented with the latent continuous responses, can be written as follows:

  • display math(3)

where the prior distribution for inline image inline image is given by inline image inline image. Here, inline image inline image, inline image inline image, and inline image inline image. In addition to the gamma prior for inline image, we place normal inline image and inline image priors on inline image and inline image, respectively, and an exponential prior on inline image with mean inline image.

2.3. Prior Specification

Regarding the DP base distribution parameters, inline image inline image, empirical evidence suggests robustness to their hyperprior choice. In general, we recommend setting the shape parameter, c, of the inverse gamma distribution for the inline image, and of the inverse gamma prior for inline image to values that yield relatively dispersed distributions, but with finite variance. Then, working with the expressions for the marginal mean, inline image, and variance, inline image, of the latent responses under a single component of the mixture model, we are able to define the hyperpriors through only the range of the dose levels. In particular, inline image inline image inline image inline image inline image inline image, and thus inline image can be specified by the midrange of the doses. Analogously, inline image inline image + inline image inline image + inline image + inline image inline image + inline image + inline image. Therefore, with c and inline image specified, the variance components, inline image, inline image, and inline image, can be determined through a proxy for the marginal variance obtained by dividing the range of the dose levels by 4 (or 6) and squaring.

The DP precision parameter inline image controls the number of distinct components, inline image, in mixture model (3), that is, the number of distinct inline image associated with the inline image inline image binary outcomes. In particular, for moderately large inline image, inline image inline image, and such expressions can be used to guide prior choice for inline image. For the data examples of Section 4.1 and 4.2, the gammainline image prior for inline image implies inline image 10 and 16, respectively.

3. Posterior Inference

  1. Top of page
  2. Summary
  3. 1. Introduction
  4. 2. Methods
  5. 3. Posterior Inference
  6. 4. Data Illustrations
  7. 5. Discussion
  8. 6. Supplementary Materials
  9. Acknowledgments
  10. References
  11. Supporting Information

3.1. MCMC Posterior Simulation

We use blocked Gibbs sampling (Ishwaran and James, 2001) for simulation from the posterior distribution of the DP mixture model in (3). Posterior samples for the model parameters are used in inference for risk assessment as discussed in Section 3.2.

Each latent response inline image is imputed based on the value of inline image. If inline image (inline image), inline image is drawn from a Ninline image distribution truncated below (above) at inline image.

We denote the inline image distinct values of inline image by inline image, and let inline image inline image, for inline image, and inline image inline image, for inline image. Then, if l does not correspond to a distinct component, the inline image are sampled, given inline image, from inline image. For the active components, that is, for inline image, inline image arises from a normal distribution with mean inline image and variance inline image. Similarly, inline image is drawn from an inverse gamma distribution with shape parameter inline image and scale parameter inline image + inline image.

Each subject-specific configuration variable inline image is drawn from a discrete distribution with probabilities proportional to inline image, for inline image. The updates for the vector of weights, inline image, and the precision parameter, inline image, are the same with a generic DP mixture model (e.g., Ishwaran and Zarepour, 2000).

image

Figure 1. Trypanosome data. Left panel: posterior mean estimate (solid line) and 90% uncertainty bands (dashed lines) for the dose–response curve. Middle panel: inversion estimates inline image for inline image. Right panel: posterior densities for the calibrated dose level, given inline image and inline image, under a inline image prior (dotted line), a uniform prior on inline image (dashed–dotted line), and a uniform prior across the full dose range (solid line).

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The conditionally conjugate priors used for the parameters of inline image lead to standard updates for the components of inline image. Specifically, the posterior full conditional for inline image is normal with mean inline image and variance inline image, and for inline image it is given by an inverse gamma distribution with shape parameter inline image and scale parameter inline image. Finally, inline image has a gamma posterior full conditional with shape parameter inline image and rate parameter inline image.

3.2. Inference for Risk Assessment

The most important inference under the quantal bioassay setting is for the dose–response relationship. Expressed in terms of the tolerance distribution, and under the DP truncation approximation, the dose–response curve in (2) becomes

  • display math

Section 2.1 elucidates the monotonicity property for the dose–response curve in terms of the inline image restriction, which carries over to the tolerance distribution formulation.

Another inferential objective is inversion, where interest lies in estimation of the dose-level, inline image, corresponding to a specified probability, q. Posterior draws for inline image are obtained by numerically inverting the posterior realization of the dose–response curve at each iteration of the MCMC algorithm.

An important feature of the linear-DDP mixture modeling framework is that it lends itself to inference for calibration, which is a key risk assessment inferential objective for certain types of experiments. Here, we consider a specified vector of responses, inline image inline image and seek to estimate the dose level, inline image, which is associated with this new vector of responses. This inference can be obtained by augmenting the hierarchical model with the components associated with inline image and expanding the parameter vector to include the unknown dose inline image. Because the DDP prior is defined over the uncountable space inline image, it induces a proper hierarchical model formulation for any inline image. Specifically, the full hierarchical model, inline image, for calibrating dose inline image is given by:

  • display math

where the prior for the inline image and for inline image, and the hyperpriors for inline image and inline image, are equivalent to model (3). In Section 4, we experiment with both normal and uniform priors, inline image, for inline image, obtaining empirical evidence that a plausible prior range for the calibrated dose level suffices for robust posterior inference.

Regarding posterior simulation, conditional on inline image, the main MCMC updates are as in Section 3.1 now extended to inline image. The full conditional for inline image is proportional to inline image, where inline image 1 (0) if inline image (inline image). Hence, draws from the full conditional of inline image arise from a truncated version of its prior distribution, truncated below at inline image and above at inline image.

4. Data Illustrations

  1. Top of page
  2. Summary
  3. 1. Introduction
  4. 2. Methods
  5. 3. Posterior Inference
  6. 4. Data Illustrations
  7. 5. Discussion
  8. 6. Supplementary Materials
  9. Acknowledgments
  10. References
  11. Supporting Information

The performance of the linear-DDP mixture model was evaluated using two synthetic data sets generated from tolerance distributions that exhibit skewness and multimodality. Inference results were also contrasted with the single-p DDP prior setting, discussed in Section 2.1, which supports increasing dose–response curves only in prior expectation. In both cases, the linear-DDP model recovers successfully the true dose–response curve with narrower uncertainty bands than the general DDP prior model. Details from this simulation study are provided in Web Appendix A.

Here, we illustrate the methodology with two data sets, one studied throughout the literature and the other an application to cytogenetic dosimetry.

Table 1. Trypanosome data reported in Ashford and Walker (1972)
Log concentration4.74.84.95.05.15.25.35.4
Exposed5549605553535150
Dead08181822374750

4.1. Trypanosome Data

Ashford and Walker (1972) first analyzed data reporting the death rate of protozoan trypanosome found in Table 1. Trypanosomes are parasites which do not grow readily in artificial culture (when they do grow, they change from the blood form to the insect form) and no reliable quantitative genetic characters have been found nor have accurate methods been available to assess characters such as drug resistance. Here, we look at results from the trypanosome sensitivity to the log concentrations of pure neutral acriflavine (Walker, 1966).

Following the prior specification approach discussed in Section 2.3, we implemented model (3) setting inline image and placing a Ninline image prior on inline image, an inv-gammainline image prior on inline image, and an exponential prior with mean inline image on inline image.

The posterior mean and 90% uncertainty bands for the dose–response curve are shown in the left panel of Figure 1. The posterior mean estimate follows closely the path of the observed data, and the interval bands include the majority of the points. The extremes of the observed proportions are at 0 and 1, which are known to be burdensome for most parametric models. In general, the linear-DDP mixture model captures well the non-standard, bimodal shape for the tolerance distribution.

The middle panel of Figure 1 includes the posterior estimates for dose levels, inline image, for inversion probabilities inline image inline image, and inline image. The spread of the densities depends on the width of the probability intervals around the dose–response curve at the given response probability. The posterior range for inline image is comparable to that obtained by Mukhopadhyay (2000), using a DP prior for the dose–response function.

Finally, we consider inference for the calibrated dose level, inline image, which corresponds to a new response vector comprising 26 positive responses from 52 subjects, that is, inline image and inline image, in the notation of Section 3.2. This new response vector resembles the observed counts at dose 5.1, thus offering a useful setting for study of sensitivity to the prior specification for inline image. The right panel of Figure 1 plots the posterior density for inline image under three different priors: a normal prior with mean 5.1 and standard deviation 0.07, a uniform prior on inline image, and a uniform prior over the entire dose range. The general uniform prior results in the largest spread and the spread decreases as the uniform support shrinks or switches to the normal distribution. The specific normal prior leads to a relatively small amount of learning in the resulting posterior density for the calibrated dose level. However, it is encouraging that the normal prior and the uniform prior on the relatively wide range inline image yield similar posterior densities.

4.2. Cytogenetic Dosimetry Application

Cytogenetic dosimetry is a biological tool for dose assessment in cases of radiological accidents and suspected overexposures. The main focus of these studies is to determine the relationship between the dosage of exposure to radiation and some measure of genetic aberration. The typical experiment considers samples of cell cultures exposed to a range of levels of an agent with some measure of cell disability recorded as the response. The two main inferential objectives are prediction of response at unobserved exposure levels, and inference for unknown dose levels given observed responses.

For an illustration, we consider part of a larger data set where blood samples from individuals were exposed to inline imageCo radiation at doses of 0, 20, 50, 100, 200, 300, 400, and 500 cGy (centograms). The resulting cultures were tested for binucleated cells with a recorded number of micronuclei (MN). The full data set, found in Madruga et al. (1996), groups the outcomes into none, one, or two or more MN; see Table 2 for the subset we work with here from healthy, older subjects. This particular ordinal classification is used because when many MN are present, they can be difficult to count exactly. We collapse to two classification groups, no MN and one or more MN, in order to apply the linear-DDP Bernoulli mixture model. In particular, model (3) is fitted setting inline image and using a Ninline image prior for inline image, an inv-gammainline image prior for inline image, and an exponential prior with mean 500 for inline image.

Table 2. Cytogenetic dosimetry data
inline imageCo radiationinline imageMN1 MN0 MN
  1. inline imageThe cell counts for the binucleated cells from healthy, older subjects, as reported in Madruga et al. (1996).

0 cGy231920
20 cGy841989
50 cGy1456933
100 cGy32114939
200 cGy67176794
300 cGy59209683
400 cGy107256742
500 cGy143327771

Plotted in the top panels of Figure 2 are posterior estimates of the dose–response curve, and the inversion dose levels for inline image, and inline image. The 90% probability bands of the dose–response curve are much tighter than those found in the trypanosome data, as here the number of subjects per dose level is roughly 20 times larger. The data are only observed to about 40% of the full curve, and therefore the intervals get much wider beyond the largest experimental dose level. The inversion inference is interesting in that at the smaller levels of q (0.05 and 0.15), the estimates of the corresponding dose level have relatively narrow densities, whereas the larger values (0.25 and 0.3) result in densities with heavier tails and substantially larger spread. This coincides with the uncertainty bands around the dose–response curve in these areas.

To illustrate inference for calibration, we consider two vectors of new responses which are equivalent to the observed vectors at 400 and 500 cGy. In each case, we used a relatively dispersed normal prior for inline image centered at the corresponding observed dose level with variance inline image. Figure 2 provides results from the model fitted to the full data set in Table 2, but also to the reduced data with the respective observations at 400 and 500 cGy excluded. In the former case (Figure 2, bottom left panel), the calibrated dose level posterior density captures fairly well the value of 400 cGy, and interestingly, with a relatively small increase in uncertainty when the corresponding data vector is not included in the model fit. For the 500 cGy dose level, excluding the respective observations yields a challenging extrapolation test for the model with data available up to dose 400 cGy and with <35% of the dose–response curve observed. The value of 500 cGy is contained in the support of the resulting calibrated dose distribution, but the corresponding density (Figure 2, bottom right panel) shows a more noticeable effect from the extrapolation with a larger increase in uncertainty.

image

Figure 2. Cytogenetic dosimetry data. Top left panel: posterior mean estimate (solid line) and 90% uncertainty bands (dashed lines) for the dose–response curve. Top right panel: inversion estimates inline image for inline image. Bottom left panel: posterior densities for the calibrated dose level given the observed vector at dose 400 cGy including the data vector (dotted line) and leaving the vector out (solid line). Bottom right panel: posterior densities for the calibrated dose level given the observed vector at dose 500 cGy including the data vector (dotted line) and leaving the vector out (solid line).

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5. Discussion

  1. Top of page
  2. Summary
  3. 1. Introduction
  4. 2. Methods
  5. 3. Posterior Inference
  6. 4. Data Illustrations
  7. 5. Discussion
  8. 6. Supplementary Materials
  9. Acknowledgments
  10. References
  11. Supporting Information

We have presented a Bayesian nonparametric approach to the analysis of quantal data from bioassay studies. The modeling approach is built from nonparametric mixing driven by a simplified version of a DDP prior, the linear-DDP prior. The implied dose–response curve is increasing, the inversion inferential objective becomes straightforward to address, and the calibration objective can be implemented without immense computational complexity. We have provided illustration with a commonly used data example which bears evidence of multiple modes and skewness in the dose–response curve. Further demonstration through portions of a cytogenetic dosimetry data set reveals the practicality of the model.

Note that, under the single-p DDP mixture modeling approach discussed in Section 2.1, if the collection of mixing distributions inline image is stochastically ordered in x, then inline image inline image is increasing in x. This provides a general sufficient condition to create mixture models that support monotonic dose–response curves through stochastic processes for the DDP atoms inline image with increasing sample paths. Of methodological interest are extensions of the linear-DDP prior based on more general specifications for the DDP atoms, using structured Gaussian processes with increasing realizations. This extension is not likely to offer significant practical utility to the standard quantal bioassay setting where the entire dose–response curve is typically observed. However, it has potential for more flexible calibration inference in cytogenetic dosimetry where extrapolation for the dose–response relationship is necessary.

The cytogenetic dosimetry application provides the motivation for another extension of the methodology to modeling for dose–response studies with outcomes comprising an ordinal classification, as in the original experiment considered in Section 4.2. In this context, fully nonparametric mixture modeling is particularly attractive as it enables different shapes for the distinct dose–response curves, and a more practical modeling approach for a moderate to large number of ordinal categories than the model in Kottas, Branco, and Gelfand (2002) based on stochastically ordered DP priors.

6. Supplementary Materials

  1. Top of page
  2. Summary
  3. 1. Introduction
  4. 2. Methods
  5. 3. Posterior Inference
  6. 4. Data Illustrations
  7. 5. Discussion
  8. 6. Supplementary Materials
  9. Acknowledgments
  10. References
  11. Supporting Information

Web Appendix A referenced in Section 4, and the Fortran 90 code for implementing the MCMC algorithm presented in Section 3.1 are available with this paper at the Biometrics website on Wiley Online Library.

Acknowledgments

  1. Top of page
  2. Summary
  3. 1. Introduction
  4. 2. Methods
  5. 3. Posterior Inference
  6. 4. Data Illustrations
  7. 5. Discussion
  8. 6. Supplementary Materials
  9. Acknowledgments
  10. References
  11. Supporting Information

This research is part of the Ph.D. dissertation of Kassandra Fronczyk, completed at University of California, Santa Cruz, and was supported in part by the National Science Foundation under award DMS 1310438. The authors wish to thank the Associate Editor and two reviewers for their comments and feedback.

References

  1. Top of page
  2. Summary
  3. 1. Introduction
  4. 2. Methods
  5. 3. Posterior Inference
  6. 4. Data Illustrations
  7. 5. Discussion
  8. 6. Supplementary Materials
  9. Acknowledgments
  10. References
  11. Supporting Information
  • Antoniak, C. (1974). Mixtures of Dirichlet processes with applications to Bayesian nonparametric problems. The Annals of Statistics 2, 11521174.
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Supporting Information

  1. Top of page
  2. Summary
  3. 1. Introduction
  4. 2. Methods
  5. 3. Posterior Inference
  6. 4. Data Illustrations
  7. 5. Discussion
  8. 6. Supplementary Materials
  9. Acknowledgments
  10. References
  11. Supporting Information

All Supplemental Data may be found in the online version of this article.

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biom12120-sm-0001-SupData-S1.pdf405KSupplementary Materials.
biom12120-sm-0002-SupData-S2.zip22KSupplementary Materials Code.

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