Quantitative risk assessments for physical, chemical, biological, occupational, or environmental agents rely on scientific studies to support their conclusions. These studies often include relatively few observations, and, as a result, models used to characterize the risk may include large amounts of uncertainty. The motivation, development and assessment of new methods for risk assessment is facilitated by the availability of a set of experimental studies that span a range of dose-response patterns that are observed in practice. We describe construction of such an historical database focusing on quantal data in chemical risk assessment, and we employ this database to develop priors in Bayesian analyses. The database is assembled from a variety of existing toxicological data sources and contains 733 separate quantal dose-response data sets. As an illustration of the database’s use, prior distributions for individual model parameters in Bayesian dose-response analysis are constructed. Results indicate that including prior information based on curated historical data in quantitative risk assessments may help stabilize eventual point estimates, producing dose-response functions that are more stable and precisely estimated. These in turn produce potency estimates that share the same benefit. We are confident that quantitative risk analysts will find many other applications and issues to explore using this database.

Public and environmental health officials studying the impacts of adverse physical, chemical, biological, occupational, or environmental agents rely on scientific studies to assess risk. Statistical modeling of data from these studies can be used to generate quantitative estimates of the risks associated with exposure to such agents. As statisticians, data scientists, and toxicologists develop new techniques to address these risk-analytic questions, it is often of interest to examine performance of the new methods on a ‘sandbox’ or ‘test bed’ of pertinent data sets containing experimental results that span a range of possible dose-response patterns. We describe the development and application of such a database here.

Modeling and analysis for quantal dose-response data has a long history of
study and continues to be an active area of research in quantitative risk assessment
^{(1–3)}. ^{(4)} although the database is designed for users to apply any
statistical programming language/package, or even code directly, at their
discretion.

To populate and curate our Quantal Risk Assessment Database (QRAD), we
extracted quantal-response data from multiple sources. To begin, we accessed the
Integrated Risk Information System (IRIS) data warehouse developed by the U.S.
Environmental Protection Agency (EPA, see ^{(5)} for constructing a smaller quantal database. For
simplicity, we limited the selection to rat or mouse tumorigenicity and only if
the animals’ exposure to the substances was oral (diet, drinking water,
or gavage) or thru inhalation, although we did allow for differing response
endpoints, such as lung and liver cancer, from the same experiment. The IRIS
data provided an initial group of 90 individual quantal data sets for study.

Next, we followed Wignall et al. ^{(6)} who included quantal dose-responses over a variety of
endpoints and chemical sources. These were originally sourced from IRIS, but
also included quantal data from the U.S. EPA’s Provisional Peer Reviewed
Toxicity Values (PPRTV), the EPA Office of Pesticide Programs (OPP), and its
Health Effects Assessment Summary Tables (HEAST); from the State of California
EPA (CalEPA); and from the U.S. Centers for Disease Control and
Prevention’s Agency for Toxic Substances and Disease Registry (ATSDR). In
particular, the data sets sourced from Wignall et al. contained IRIS entries
above and beyond the original 90 quantal data sets from Nitcheva et al. Any
duplicate entries were removed, leading to a total of 144 IRIS quantal data sets
for our use. Further, many of the CalEPA studies were themselves duplicates of
the IRIS studies, and these were also removed.

In all, 733 unique quantal data sets for 333 different chemicals were
extracted from these five data sources. The 333 unique chemicals in this
database have all been the subject of previous risk assessments. Among them,
only 1.6% of the experiments have 7 or more dose groups (the modal number of
dose groups including control is 4, the maximum is 16) and over 80% of the
experiments had between 3 and 5 dose groups. Also, more than 50% of the database
is derived from IRIS (371 entries) with CalEPA contributing 235 entries. Only 3
entries were from ATSDR. Our QRAD can be found online at

Directions for accessing QRAD can be found in §2.2, below.
Approaches for constructing summary descriptions of the database are described
in §2.3 and in Sections

Within the database, each data set provides the following information:

a unique identifier code (

each agent’s

the source from which we compiled the data set
(

the chemical’s Chemical Abstracts Service Registry Number
(

the

doses scaled to the unit interval, i.e.,

the number of independent subjects,

the number of observed responses,

the

the particular toxicological or carcinogenic outcome recorded to
produce the values in

if available, the publication or other product
(

Our collected QRAD construct is fashioned as an R data frame and is also
available as a comma delimited table. In what follows we access the database
from R. The core file can be read into R using the

Alternatively, one may wish to load this database from the website. To do this, enter the command

It is simple to list the variables included in QRAD (cf. the list
above); e.g., the command

It is also relatively straightforward to learn more about these data.
For example, query which organ systems were tested via

Additionally, if the database is saved as a ^{®} for
additional analysis. Information about the variables in the SAS data set
‘^{(7)}:

After downloading the database, one might wish to start exploring its
features. We often employ the ^{(8)} to build data frames with
particular summary information. In the code below, we process the data by ID to
calculate the number of dose groups in each ID and to save this as a new data
frame.

After constructing this new data frame with counts of the number of dose
groups associated with each ID, we can generate relative frequencies for the
numbers of dose groups in each experiment. In the sample code below, the

The output (edited for presentation) is:

We discern in QRAD that the modal experiment size is 4 dose groups (40%
of the studies) and approximately 80% of the studies have between 3 and 5 dose
groups. This is consistent with past reports on numbers of dose levels in
toxicological assays: Nitcheva et al. ^{(5)} found that that the most common number of doses used
among the 91 rodent carcinogenicity studies they studied (most of which are
contained in our database) was three; next most common was four. Similarly, Muri
et al. ^{(9)} reported that the
most prevalent study design they found among 20 pesticide risk analyses again
employed only four doses, and did so almost twice as often as the next
most-common design (which used five doses). Within this context, one surprise in
our database is the 1% of studies possessing 10 or more dose groups: this
appears to be a relatively uncommon occurrence in studies used in support of
toxicological risk assessments.

A simple graphical display for the numbers of dose groups can be
generated using the ^{(10)}. Sample code is

which produces the histogram in

A supplemental document illustrates a few other, basic commands for manipulating QRAD, including ways to subset it and to apply a function to all data sets in the database.

We illustrate a more-advanced use of QRAD by addressing an intriguing
problem in dose-response analytics: how to specify prior distributions on
unknown parameters in a Bayesian quantal-response analysis. We assume the
responses _{i} are independent binomial variates,
_{i} ~ Bin.(_{i},
_{i})), where
_{i}) is the underlying
function describing the response probability P[_{i} =
1|_{i}] at dose _{i} (i
= 1,..., m). The dose-response function
_{0},
_{1},

For the possible dose-response function we employed the following
quantal-response models, available in the U.S. EPA’s popular BMDS system
^{(11)}:

Bayesian analysis of quantal dose-response data has a long and rich
history—see, e.g., Ramsay ^{(12)}; Tsutakawa ^{(13)}; Messig and Strawderman ^{(14)}; Kuo and Cohen ^{(15)}; or Novelo et al. ^{(16)}. As these sources suggest, Bayesian
quantal-response modeling often expands the inferential capacity of the
analysis, offering the analyst heightened flexibility, greater connection to the
data, and in particular more efficient use of information that can be gleaned
from previous studies. Indeed, its application here has the added benefit of
enhancing the limited information in the data when only a few doses are studied.
As we saw in

To cement the notation, consider use of the quantal-linear model
_{1}(_{1} parameter, say,
g(_{1},). Writing the data vector as
_{1} …
_{m}]^{T}, the likelihood function under
the binomial model is _{1} as
p(_{1}) and the prior for
_{1} is
p(_{1}|_{1})ƒ(_{1});
the posterior for _{1},
including those we propose below, the posterior
p(_{1}|^{(17, §6.3)}: simulate a (long) Markov chain of
_{1} values whose stationary distribution
is designed to converge to
p(_{1}|_{1} as an
approximation of a random sample from the posterior
p(_{1}|_{1}) can be taken from
appropriate components of the transformed chain of
g(_{1}) values.

Specification of prior(s) on the parameters for such Bayesian
dose-response analyses can be a thorny issue ^{(18–20)}. When previous experimental results or domain-specific
expert knowledge are available from which to elicit the prior, such knowledge
should be the first recourse ^{(21)}. Otherwise, however, determination or specification of
prior components in a Bayesian (or any hierarchical) model is often a difficult
exercise. Here, we query QRAD to identify potential prior forms for use in a
Bayesian quantal-response analysis. Our approach has the flavor of a
^{(22)}, where information in a large knowledgebase such as
QRAD is employed to expand our understanding of the quantal-response issue(s)
under study and to advance the consequent data analysis.

To develop the priors from QRAD we first filtered out those experiments
providing only minimal information on the shape of their dose-response
relationships, namely, experiments with fewer than 4 dose groups or fewer than
100 observations. The requirement for 100 or more observations ensured that
sufficient dose-response information was available to accurately represent the
underlying biological processes, while the requirement of four or more dose
groups was used to help prevent the three-parameter models from overfitting.
This resulted in 269 data sets, which we employed to develop historically-based
priors for the various parameters (i.e., the _{j} ≥ 0 (j = 1,2) for all models,
although we left _{0} unconstrained in Models
(

Our strategy adopted a relatively straightforward approach: using the
filtered database, Models (^{2}) with mean exp{μ +
½σ^{2}} and variance {exp(σ^{2})
– 1}exp{2μ + σ^{2}} for parameters restricted to a
positive support and displaying a clear positive skew, (ii) a normal
distribution N(μ,σ^{2}) mean μ and variance
σ^{2} for parameters varying over the entire real line and
where the empirical pattern was more general, and (iii) a beta distribution
Beta(

To produce specific prior recommendations for each model/parameter
combination, we kept the effort simple and appealed to moment-type estimation
schemes. For (i) the LN(μ,σ^{2}) prior we took a
logarithmic transformation of the 269 estimated values for that parameter and
then set μ equal to its log-mean and σ^{2} equal to its
log-variance. Cases where the original parameter estimate was
zero—corresponding to a ‘boundary’ occurrence for that
parameter—were excluded, as were cases where the BMDS software ran the
point estimate up to its arbitrary maximum of 18, which we viewed as indicators
of aberrant response for that particular model.

For the special case of the

For (ii) the N(μ,σ^{2}) prior, we found the sample
mean and sample variance of the 269 estimated values for the given parameter and
set them equal to μ and σ^{2}, respectively.

For (iii) the Beta(

Our results appear in _{j}
coefficients (_{j}
parameters—the lognormal prior forces
P[_{j} = 0] = 0. In effect, this assumes
that the associated dose response is real. To incorporate the possibility that
the dose response is flat, a more-complex prior needs to be constructed, e.g., a
mixture of a continuous density for _{j}
> 0 and a non-zero mass at P[_{j} = 0]
or some other form of spike-and-slab prior ^{(23)}.

It is also useful to note that the lognormal priors all correspond to
parameters whose parent model places restrictions on their range. _{1} parameter from the probit
dose-response model (

In those cases from _{0} parameter in Models (

We list in ^{2}/2} are close to or above 1.0; smallest is the log-probit
_{1} parameter with prior expectation
1.108. This despite the fact that in no cases were the underlying parameter
estimates constrained to be greater than or equal to 1.0, as is sometimes
recommended for many of these models. Simply from an empirical perspective this
gives evidence that shape parameters for many of the models in ^{(2, 24)} to down-weight use of supralinear
curves—i.e., curves where the shape parameter is less than 1—with
dose-response data such as found in QRAD ^{(25)}.

The empirical distributions as well as the histograms in _{1} for the log-logistic (_{1}
for the log-probit (^{(26)}
these parameters are constrained to be at or smaller than 18, and options are
often available to also force them larger than or equal to 1. Yet, for a
majority of occurrences in QRAD—say, 75% of the empirical values for each
parameter, between its 12.5% and 87.5% percentiles—these shape parameters
appear to lie in a much smaller range of values, with many values less than 1
and few above roughly 4; see

Alert readers will no doubt have noticed that the various empirical
priors recommended for the background probability parameter
^{(16, 27, §7.4)} for risk
analysts seeking such a database.

We should also mention that the number of significant digits displayed
for the various recommended priors in

As a formal example, we analyze renal tubule hyperplasia data in male
Sprague-Dawley rats exposed to 3-monochloropropane-1,2-diol, a chemical food
contaminant and suspected mammalian carcinogen ^{(28)}. At exposure doses of _{1} = 0,
_{2} = 25, _{3} = 100, and
_{4} = 400 ppm the following proportions of affected
animals were observed: _{1}/_{1}
= 1/50, _{2}/_{2} = 11/50,
_{3}/_{3} = 21/50, and
_{4}/_{4} = 36/50
respectively. These data do not appear in QRAD; however, they have been the subject
of previous risk assessments by the European Food Safety Authority (EFSA)
^{(29)} and by the Food and
Agricultural Organization of the United Nations (FAO) in concert with the World
Health Organization (WHO) ^{(30)}.
Intriguingly, across both studies carcinogenic potency estimates varied by an order
of magnitude. One source of the disparities seen in such previous risk assessments
with these data was the different modeling assumptions applied by previous analysts,
using the Weibull (

We conducted a Bayesian analysis similar to the EFSA risk assessment
^{(29)} but applied
(independent) informative priors developed from QRAD. For comparison purposes, we
also applied independent, vague, uniform priors similar to those suggested by Shao
and Shapiro ^{(31)}. In particular,
with the Weibull and Gamma models we assumed _{0} ~ U(–18, 18),
_{1} ~ U(0, 18), and

We applied all these various hierarchical models to the
3-monochloropropane-1,2-diol data using ^{(32)}. To
conduct the MCMC operations for each model fit we sampled 25,000 realizations of the
pertinent Markov chain(s), with the first 1,000 samples discarded as burn-in. As a
measure of carcinogenic potency/risk, we calculated the benchmark dose
(^{(27, §4.3)}. For reporting purposes, we
followed standard practice and found the lower 95% limit on the BMD (a 95%

It is worth noting that the use of models with restrictions such as

We have described the construction and exploration of a test bed knowledgebase of quantal dose-response data, the Quantal Risk Assessment Database (QRAD), useful for coordinated study of new or existing data-analytic methods in quantitative risk analysis. The database contains 733 separate quantal dose-response data sets, representing 333 unique chemicals and exhibiting a panoply of patterns and shapes.

Our initial analyses with QRAD illustrate the database’s flexibility
for addressing different scientific questions of interest to the environmental risk
assessment community. Obviously, however, much more could be accomplished; e.g., one
might explore QRAD to help define a core range of dose-response configurations
typically encountered in toxicological dose-response experimentation. Indeed, our
example in §3 with specification of prior distributions for the model
parameters in Bayesian dose-response analysis is only one such potential
illustration, and even there we only scratch the surface. A deeper analysis would
consider other distributional forms for the various priors, along with comparisons
of the resulting posterior inferences. For the many positive parameters in _{0} parameters, alternative priors include the
extreme value (Gumbel), logistic—not to be confused with the logistic
response model (^{(33)} or Thomopoulos ^{(34)} for details on any of these distributional forms.)

In the end, our hope is that availability of this database will enable risk analysts to answer questions about the shape and type of models used in quantal dose-response analysis. For instance, it is well-known that the probit and logistic models produce very similar fits over large portions of the dose-response curve, and some may question the need for both models in risk assessment practice. As QRAD contains over 700 dose-response data sets—all having been employed for risk assessment purposes—the effect of removing the probit or the logistic model could be explored by analyzing their model fits across the entire database. Such an analysis would add a transparency to any future decisions based on these models, and may allow for the creation of general dose-response modeling guidance based upon data used in risk assessment.

Thanks are due Drs. Christine Whittaker and D. Dean Billheimer for their helpful input during the preparation of this material and Drs. Woodrow Setzer and Kan Shao for useful comments on an earlier version of the manuscript. This research was supported in part by grant #R03-ES027394 from the U.S. National Institutes of Health and grant #CCF-1740858 from the National Science Foundation. John Bailer was supported by NIOSH funding via an Intergovernmental Personnel Act appointment. The contents herein are solely the responsibility of the authors and do not necessarily reflect the official views of any Federal agency or external company.

Competing Financial Interests

The authors declare they have no actual or potential competing financial interests.

Supplemental Material

Supplemental material is provide that contains sample computer code for manipulating the new database and exploring its contents, along with graphical summaries of selected parametric models fitted via data extracted from the database.

Numbers of dose groups per data set in the quantal-response database

Empirical distribution (histogram) of shape parameter
β_{1} for the probit dose-response model (_{1} ~ LN(0.473, 0.638), from

Empirical distribution (histogram) of parameter β_{0} for
the log-logistic dose-response model (_{0} ~ N(0.203, 4.061), from

Empirical distribution (histogram) relating to shape parameter α
for the Weibull dose-response model (

Summary statistics and recommended empirical prior distributions for
Models (^{2}) indicates a normal
distribution with mean μ and variance σ^{2},
LN(μ,σ^{2}) indicates a log-normal distribution with
mean

sample statistics | |||||||
---|---|---|---|---|---|---|---|

Model | parameter | minimum | median | maximum | mean | variance | recommended prior |

Quantal linear ( | 0 | 0.041 | 1 | 0.109 | 0.025 | ||

_{1} | 0 | 0.583 | 17.770 | −0.470 | 1.688 | _{1} ~
LN(-0.470, 1.688) | |

Multi-stage ( | 0 | 0.041 | 0.947 | 0.112 | 0.025 | ||

_{1} | 0 | 0.241 | 11.394 | −1.006 | 4.949 | _{1} ~
LN(-1.006, 4.949) | |

_{2} | 0 | 0.063 | 15.874 | −1.016 | 5.013 | _{2} ~
LN(-1.016, 5.013) | |

Weibull ( | 0 | 0.034 | 0.947 | 0.095 | 0.022 | ||

0.20 | 1.051 | 17.619 | −0.243 | 2.467 | |||

0 | 0.706 | 11.388 | −0.464 | 1.535 | β ~ LN(-0.464, 1.535) | ||

Gamma ( | 0 | 0.034 | 1 | 0.097 | 0.023 | ||

0.20 | 1.098 | 17.799 | −0.153 | 2.637 | |||

0 | 0.924 | 16.119 | −0.587 | 5.659 | β ~ LN(-0.587, 5.659) | ||

Logistic ( | _{0} | −16.491 | −2.446 | 2.889 | −2.526 | 3.733 | _{0} ~
N(-2.526, 3.733) |

_{1} | 0 | 2.734 | 16.734 | 1.018 | 0.603 | _{1} ~
LN(1.018, 0.603) | |

Log-Logistic ( | 0 | 0.031 | 1 | 0.097 | 0.023 | ||

_{0} | −4.789 | 0.095 | 7.353 | 0.203 | 4.061 | _{0} ~
N(0.203, 4.061) | |

_{1} | 0.200 | 1.314 | 17.760 | 0.274 | 0.960 | _{1} ~
LN(0.274, 0.960) | |

Probit ( | _{0} | −11.844 | −1.431 | 1.619 | −1.660 | 3.264 | _{0} ~
N(-1.66, 3.264) |

_{1} | 0 | 1.537 | 13.519 | 0.473 | 0.638 | _{1} ~
LN(0.473, 0.638) | |

Log-Probit ( | 0 | 0.050 | 0.947 | 0.104 | 0.022 | ||

_{0} | −17.400 | 0.047 | 11.176 | −0.514 | 10.337 | _{0} ~
N(-0.514,10.337) | |

_{1} | 0.200 | 0.914 | 5.926 | −0.186 | 0.579 | _{1} ~
LN(-0.186, 0.579) |

Reported mean and variance based on sample values after logarithmic
transformation; values for

Summary statistics for various parameters in selected lognormal models
from

Model (no.) | parameter | mean, μ | variance, σ^{2} | expected value | percentiles: | 12.5% | 87.5% |
---|---|---|---|---|---|---|---|

Quantal-linear ( | _{1} | −0.470 | 1.688 | 1.453 | 0.130 | 2.902 | |

Multi-stage ( | _{1} | −1.006 | 4.949 | 4.341 | 0.0 | 2.052 | |

_{2} | −1.016 | 5.013 | 4.440 | 0.0 | 1.303 | ||

Weibull ( | −0.243^{†} | 2.467^{†} | 2.693^{†} | 0.370 | 2.594 | ||

−0.464 | 1.535 | 1.355 | 0.130 | 2.659 | |||

Gamma ( | −0.153^{†} | 2.637^{†} | 3.206^{†} | 0.315 | 3.510 | ||

−0.587 | 5.659 | 9.414 | 0.039 | 5.854 | |||

Logistic ( | _{1} | 1.018 | 0.603 | 3.744 | 1.100 | 6.620 | |

Log-Logistic ( | _{1} | 0.274 | 0.960 | 2.125 | 0.448 | 3.777 | |

Probit ( | _{1} | 0.473 | 0.638 | 2.206 | 0.589 | 3.687 | |

Log-Probit ( | _{1} | −0.186 | 0.579 | 1.108 | 0.311 | 1.948 |

For ^{2}),
E[^{2}}.

Values for

Summary characteristics from MCMC-based posterior distributions of the
Benchmark Dose (BMD) at BMR = 0.10 for four dose-response models referenced in
Section 3. The benchmark dose lower bound (BMDL) is the lower 5% percentile from
the generated MCMC posterior. Results are stratified by type of priors used in
the analysis: (i) Informative Priors from

Model (no.) | Mean | Sth. Dev. | IQR | BMDL | Mean/BMDL |
---|---|---|---|---|---|

Informative Priors | |||||

Weibull ( | 0.68 | 0.49 | 0.58 | 0.11 | 6.18 |

Gamma ( | 0.55 | 0.45 | 0.53 | 0.06 | 9.17 |

Log-Logistic ( | 0.94 | 0.55 | 0.67 | 0.23 | 4.09 |

Log-Probit ( | 1.05 | 0.56 | 0.68 | 0.32 | 3.28 |

Flat Priors | |||||

Weibull ( | 0.99 | 0.68 | 0.77 | 0.20 | 4.95 |

Gamma ( | 1.09 | 0.83 | 0.90 | 0.20 | 5.45 |

Log-Logistic ( | 1.19 | 0.73 | 0.83 | 0.31 | 3.84 |

Log-Probit ( | 1.35 | 1.79 | 0.83 | 0.35 | 3.86 |

Abbreviations: IQR = lower limit. interquartile range; BMDL = (95%) benchmark dose