Nonparametric Bayesian analysis of developmentaltoxicity experiments with clustered discrete-continuous
outcomes
Athanasios Kottas
Department of Applied Mathematics and Statistics, University of California, Santa Cruz
Joint work with Kassandra Fronczyk, Department of Biostatistics, University of
Texas MD Anderson Cancer Center & Department of Statistics, Rice University
ISBA 2012 World Meeting, Kyoto JapanJune 25–29, 2012
Athanasios Kottas (UCSC) NP Bayes in Developmental Toxicology June 29, 2012 1 / 27
Outline
1 Introduction
2 The basic modeling framework
3 Modeling for multicategory classification responses
4 Toxicity experiments with discrete-continuous outcomes
5 Conclusions
Athanasios Kottas (UCSC) NP Bayes in Developmental Toxicology June 29, 2012 2 / 27
Introduction
Developmental toxicity studies
→ Birth defects induced by toxic chemicals are investigated through developmentaltoxicity experiments
→ A number of pregnant laboratory animals (dams) are exposed to a toxin
→ Recorded from each animal are the number of resorptions and/or prenataldeaths, the number of live pups, and the number of live malformed pups; mayalso include continuous outcomes from the live pups (typically, body weight)
→ Key objective is to examine the relationship between the level of exposure to
the toxin (dose level) and the probability of response for the different endpoints:
embryolethality; malformation; low birth weight
Athanasios Kottas (UCSC) NP Bayes in Developmental Toxicology June 29, 2012 3 / 27
Introduction
Developmental toxicity studies
→ Birth defects induced by toxic chemicals are investigated through developmentaltoxicity experiments
→ A number of pregnant laboratory animals (dams) are exposed to a toxin
→ Recorded from each animal are the number of resorptions and/or prenataldeaths, the number of live pups, and the number of live malformed pups; mayalso include continuous outcomes from the live pups (typically, body weight)
→ Key objective is to examine the relationship between the level of exposure to
the toxin (dose level) and the probability of response for the different endpoints:
embryolethality; malformation; low birth weight
Athanasios Kottas (UCSC) NP Bayes in Developmental Toxicology June 29, 2012 3 / 27
Introduction
Developmental toxicity studies
→ Birth defects induced by toxic chemicals are investigated through developmentaltoxicity experiments
→ A number of pregnant laboratory animals (dams) are exposed to a toxin
→ Recorded from each animal are the number of resorptions and/or prenataldeaths, the number of live pups, and the number of live malformed pups; mayalso include continuous outcomes from the live pups (typically, body weight)
→ Key objective is to examine the relationship between the level of exposure to
the toxin (dose level) and the probability of response for the different endpoints:
embryolethality; malformation; low birth weight
Athanasios Kottas (UCSC) NP Bayes in Developmental Toxicology June 29, 2012 3 / 27
Introduction
Developmental toxicology data
Data structure for Segment II designs (exposure after implantation)
→ Data at dose levels, xi , i = 1, ...,N, including a control group (dose = 0)
→ ni dams at dose level xi
→ For the j-th dam at dose xi :
mij : number of implants
Rij : number of resorptions and prenatal deaths (Rij ≤ mij)
y∗ij = {y∗ijk : k = 1, ...,mij − Rij}: binary malformation indicators for the live
pups (yij =∑mij−Rij
k=1 y∗ijk : number of live pups with a malformation)
u∗ij = {u∗ijk : k = 1, ...,mij − Rij} the continuous outcomes (on body weight)for the live pups
Athanasios Kottas (UCSC) NP Bayes in Developmental Toxicology June 29, 2012 4 / 27
Introduction
Developmental toxicology data
Data structure for Segment II designs (exposure after implantation)
→ Data at dose levels, xi , i = 1, ...,N, including a control group (dose = 0)
→ ni dams at dose level xi
→ For the j-th dam at dose xi :
mij : number of implants
Rij : number of resorptions and prenatal deaths (Rij ≤ mij)
y∗ij = {y∗ijk : k = 1, ...,mij − Rij}: binary malformation indicators for the live
pups (yij =∑mij−Rij
k=1 y∗ijk : number of live pups with a malformation)
u∗ij = {u∗ijk : k = 1, ...,mij − Rij} the continuous outcomes (on body weight)for the live pups
Athanasios Kottas (UCSC) NP Bayes in Developmental Toxicology June 29, 2012 4 / 27
Introduction
Developmental toxicology data
Data structure for Segment II designs (exposure after implantation)
→ Data at dose levels, xi , i = 1, ...,N, including a control group (dose = 0)
→ ni dams at dose level xi
→ For the j-th dam at dose xi :
mij : number of implants
Rij : number of resorptions and prenatal deaths (Rij ≤ mij)
y∗ij = {y∗ijk : k = 1, ...,mij − Rij}: binary malformation indicators for the live
pups (yij =∑mij−Rij
k=1 y∗ijk : number of live pups with a malformation)
u∗ij = {u∗ijk : k = 1, ...,mij − Rij} the continuous outcomes (on body weight)for the live pups
Athanasios Kottas (UCSC) NP Bayes in Developmental Toxicology June 29, 2012 4 / 27
Introduction
Developmental toxicology data
Data structure for Segment II designs (exposure after implantation)
→ Data at dose levels, xi , i = 1, ...,N, including a control group (dose = 0)
→ ni dams at dose level xi
→ For the j-th dam at dose xi :
mij : number of implants
Rij : number of resorptions and prenatal deaths (Rij ≤ mij)
y∗ij = {y∗ijk : k = 1, ...,mij − Rij}: binary malformation indicators for the live
pups (yij =∑mij−Rij
k=1 y∗ijk : number of live pups with a malformation)
u∗ij = {u∗ijk : k = 1, ...,mij − Rij} the continuous outcomes (on body weight)for the live pups
Athanasios Kottas (UCSC) NP Bayes in Developmental Toxicology June 29, 2012 4 / 27
Introduction
To begin with, consider simplest data form, {(mij , zij) : i = 1, . . . , N, j = 1, . . . , ni},where zij = Rij + yij is the number of combined negative outcomes
0 20 40 60 80
0.0
0.2
0.4
0.6
0.8
1.0
dose mg/kg
73 87 97 76 44 25dams
0 50 100 150
0.0
0.2
0.4
0.6
0.8
1.0
dose mg/kg x 1000
30 26 26 17 9dams
Figure: 2,4,5-T data (left) and DEHP data (right).
Athanasios Kottas (UCSC) NP Bayes in Developmental Toxicology June 29, 2012 5 / 27
Introduction
To begin with, consider simplest data form, {(mij , zij) : i = 1, . . . , N, j = 1, . . . , ni},where zij = Rij + yij is the number of combined negative outcomes
0 20 40 60 80
0.0
0.2
0.4
0.6
0.8
1.0
dose mg/kg
73 87 97 76 44 25dams
0 50 100 1500.0
0.2
0.4
0.6
0.8
1.0
dose mg/kg x 1000
30 26 26 17 9dams
Figure: 2,4,5-T data (left) and DEHP data (right).
Athanasios Kottas (UCSC) NP Bayes in Developmental Toxicology June 29, 2012 5 / 27
Introduction
Overview and research objectives
Develop nonparametric Bayesian methodology for risk assessment in develop-mental toxicology
→ overcome limitations of parametric approaches, while retaining a fullyinferential probabilistic model setting
→ modeling framework that provides flexibility in both the response distri-bution and the dose-response relationship
Build flexible risk assessment inference tools from nonparametric modeling fordose-dependent response distributions
→ nonparametric mixture models with increasing levels of complexity in thekernel structure to account for the different data types
→ dependent DP priors for the dose-dependent mixing distributions
→ emphasis on properties of the implied dose-response relationships
Athanasios Kottas (UCSC) NP Bayes in Developmental Toxicology June 29, 2012 6 / 27
Introduction
Overview and research objectives
Develop nonparametric Bayesian methodology for risk assessment in develop-mental toxicology
→ overcome limitations of parametric approaches, while retaining a fullyinferential probabilistic model setting
→ modeling framework that provides flexibility in both the response distri-bution and the dose-response relationship
Build flexible risk assessment inference tools from nonparametric modeling fordose-dependent response distributions
→ nonparametric mixture models with increasing levels of complexity in thekernel structure to account for the different data types
→ dependent DP priors for the dose-dependent mixing distributions
→ emphasis on properties of the implied dose-response relationships
Athanasios Kottas (UCSC) NP Bayes in Developmental Toxicology June 29, 2012 6 / 27
Introduction
Overview and research objectives
Develop nonparametric Bayesian methodology for risk assessment in develop-mental toxicology
→ overcome limitations of parametric approaches, while retaining a fullyinferential probabilistic model setting
→ modeling framework that provides flexibility in both the response distri-bution and the dose-response relationship
Build flexible risk assessment inference tools from nonparametric modeling fordose-dependent response distributions
→ nonparametric mixture models with increasing levels of complexity in thekernel structure to account for the different data types
→ dependent DP priors for the dose-dependent mixing distributions
→ emphasis on properties of the implied dose-response relationships
Athanasios Kottas (UCSC) NP Bayes in Developmental Toxicology June 29, 2012 6 / 27
Introduction
Overview and research objectives
Develop nonparametric Bayesian methodology for risk assessment in develop-mental toxicology
→ overcome limitations of parametric approaches, while retaining a fullyinferential probabilistic model setting
→ modeling framework that provides flexibility in both the response distri-bution and the dose-response relationship
Build flexible risk assessment inference tools from nonparametric modeling fordose-dependent response distributions
→ nonparametric mixture models with increasing levels of complexity in thekernel structure to account for the different data types
→ dependent DP priors for the dose-dependent mixing distributions
→ emphasis on properties of the implied dose-response relationships
Athanasios Kottas (UCSC) NP Bayes in Developmental Toxicology June 29, 2012 6 / 27
The basic modeling framework
Model formulation
Again, begin with the DDP mixture model for the simplest data structure,{(mij , zij) : i = 1, . . . ,N, j = 1, . . . , ni}, where zij is the number of combinednegative outcomes on resorptions/prenatal deaths and malformations
Number of implants is a random variable, though with no information aboutthe dose-response relationship (the toxin is administered after implantation)
→ f (m) = Poisson(m;λ), m ≥ 1 (more general models can be used)
Focus on the dose-dependent conditional response distributions f (z | m)
→ for dose level x , model f (z | m) ≡ f (z | m;Gx) through a nonparametricmixture of Binomial distributions
→ DDP prior for the collection of mixing distributions {Gx : x ∈ X ⊆ R+}
Athanasios Kottas (UCSC) NP Bayes in Developmental Toxicology June 29, 2012 7 / 27
The basic modeling framework
Model formulation
Again, begin with the DDP mixture model for the simplest data structure,{(mij , zij) : i = 1, . . . ,N, j = 1, . . . , ni}, where zij is the number of combinednegative outcomes on resorptions/prenatal deaths and malformations
Number of implants is a random variable, though with no information aboutthe dose-response relationship (the toxin is administered after implantation)
→ f (m) = Poisson(m;λ), m ≥ 1 (more general models can be used)
Focus on the dose-dependent conditional response distributions f (z | m)
→ for dose level x , model f (z | m) ≡ f (z | m;Gx) through a nonparametricmixture of Binomial distributions
→ DDP prior for the collection of mixing distributions {Gx : x ∈ X ⊆ R+}
Athanasios Kottas (UCSC) NP Bayes in Developmental Toxicology June 29, 2012 7 / 27
The basic modeling framework
Model formulation
Again, begin with the DDP mixture model for the simplest data structure,{(mij , zij) : i = 1, . . . ,N, j = 1, . . . , ni}, where zij is the number of combinednegative outcomes on resorptions/prenatal deaths and malformations
Number of implants is a random variable, though with no information aboutthe dose-response relationship (the toxin is administered after implantation)
→ f (m) = Poisson(m;λ), m ≥ 1 (more general models can be used)
Focus on the dose-dependent conditional response distributions f (z | m)
→ for dose level x , model f (z | m) ≡ f (z | m;Gx) through a nonparametricmixture of Binomial distributions
→ DDP prior for the collection of mixing distributions {Gx : x ∈ X ⊆ R+}
Athanasios Kottas (UCSC) NP Bayes in Developmental Toxicology June 29, 2012 7 / 27
The basic modeling framework
DDP prior defined by extending the (almost sure) discrete representation forthe regular DP:
GX (·) =∞∑
`=1
ω`δη`X (·)
→ the η`X = {η`(x) : x ∈ X} arise i.i.d. from a base stochastic processG0X over X
→ weights generated through stick-breaking: ω1 = ζ1, ω` = ζ`
∏`−1r=1(1−ζr )
for r ≥ 2, with ζ` i.i.d. Beta(1,α) (independent of the η`X )
key feature of the DDP prior: for any finite set (x1, ..., xk) it induces a multi-variate DP prior for the set of mixing distributions (Gx1 , ...,Gxk
)
single-p DDP prior structure offers a natural choice for this application
Athanasios Kottas (UCSC) NP Bayes in Developmental Toxicology June 29, 2012 8 / 27
The basic modeling framework
DDP prior defined by extending the (almost sure) discrete representation forthe regular DP:
GX (·) =∞∑
`=1
ω`δη`X (·)
→ the η`X = {η`(x) : x ∈ X} arise i.i.d. from a base stochastic processG0X over X
→ weights generated through stick-breaking: ω1 = ζ1, ω` = ζ`
∏`−1r=1(1−ζr )
for r ≥ 2, with ζ` i.i.d. Beta(1,α) (independent of the η`X )
key feature of the DDP prior: for any finite set (x1, ..., xk) it induces a multi-variate DP prior for the set of mixing distributions (Gx1 , ...,Gxk
)
single-p DDP prior structure offers a natural choice for this application
Athanasios Kottas (UCSC) NP Bayes in Developmental Toxicology June 29, 2012 8 / 27
The basic modeling framework
DDP prior defined by extending the (almost sure) discrete representation forthe regular DP:
GX (·) =∞∑
`=1
ω`δη`X (·)
→ the η`X = {η`(x) : x ∈ X} arise i.i.d. from a base stochastic processG0X over X
→ weights generated through stick-breaking: ω1 = ζ1, ω` = ζ`
∏`−1r=1(1−ζr )
for r ≥ 2, with ζ` i.i.d. Beta(1,α) (independent of the η`X )
key feature of the DDP prior: for any finite set (x1, ..., xk) it induces a multi-variate DP prior for the set of mixing distributions (Gx1 , ...,Gxk
)
single-p DDP prior structure offers a natural choice for this application
Athanasios Kottas (UCSC) NP Bayes in Developmental Toxicology June 29, 2012 8 / 27
The basic modeling framework
DDP prior defined by extending the (almost sure) discrete representation forthe regular DP:
GX (·) =∞∑
`=1
ω`δη`X (·)
→ the η`X = {η`(x) : x ∈ X} arise i.i.d. from a base stochastic processG0X over X
→ weights generated through stick-breaking: ω1 = ζ1, ω` = ζ`
∏`−1r=1(1−ζr )
for r ≥ 2, with ζ` i.i.d. Beta(1,α) (independent of the η`X )
key feature of the DDP prior: for any finite set (x1, ..., xk) it induces a multi-variate DP prior for the set of mixing distributions (Gx1 , ...,Gxk
)
single-p DDP prior structure offers a natural choice for this application
Athanasios Kottas (UCSC) NP Bayes in Developmental Toxicology June 29, 2012 8 / 27
The basic modeling framework
DDP prior defined by extending the (almost sure) discrete representation forthe regular DP:
GX (·) =∞∑
`=1
ω`δη`X (·)
→ the η`X = {η`(x) : x ∈ X} arise i.i.d. from a base stochastic processG0X over X
→ weights generated through stick-breaking: ω1 = ζ1, ω` = ζ`
∏`−1r=1(1−ζr )
for r ≥ 2, with ζ` i.i.d. Beta(1,α) (independent of the η`X )
key feature of the DDP prior: for any finite set (x1, ..., xk) it induces a multi-variate DP prior for the set of mixing distributions (Gx1 , ...,Gxk
)
single-p DDP prior structure offers a natural choice for this application
Athanasios Kottas (UCSC) NP Bayes in Developmental Toxicology June 29, 2012 8 / 27
The basic modeling framework
DDP mixture of Binomial distributions:
f (z | m;GX ) =
∫Bin
(z ;m,
exp(θ)
1 + exp(θ)
)dGX (θ), GX ∼ DDP(α, G0X )
→ Gaussian process (GP) for G0X with:
linear mean function, E(η`(x) | β0, β1) = β0 + β1x
constant variance, Var(η`(x) | σ2) = σ2
isotropic power exponential correlation function,Corr(η`(x), η`(x
′) | φ) = exp(−φ|x − x ′|d) (with fixed d ∈ [1, 2])
→ hyperpriors for α and ψ = (β0, β1, σ2, φ)
→ MCMC posterior simulation using blocked Gibbs sampling
→ Posterior predictive inference over observed and new dose levels, using the
posterior samples from the model and GP interpolation for the DDP locations
Athanasios Kottas (UCSC) NP Bayes in Developmental Toxicology June 29, 2012 9 / 27
The basic modeling framework
DDP mixture of Binomial distributions:
f (z | m;GX ) =
∫Bin
(z ;m,
exp(θ)
1 + exp(θ)
)dGX (θ), GX ∼ DDP(α, G0X )
→ Gaussian process (GP) for G0X with:
linear mean function, E(η`(x) | β0, β1) = β0 + β1x
constant variance, Var(η`(x) | σ2) = σ2
isotropic power exponential correlation function,Corr(η`(x), η`(x
′) | φ) = exp(−φ|x − x ′|d) (with fixed d ∈ [1, 2])
→ hyperpriors for α and ψ = (β0, β1, σ2, φ)
→ MCMC posterior simulation using blocked Gibbs sampling
→ Posterior predictive inference over observed and new dose levels, using the
posterior samples from the model and GP interpolation for the DDP locations
Athanasios Kottas (UCSC) NP Bayes in Developmental Toxicology June 29, 2012 9 / 27
The basic modeling framework
DDP mixture of Binomial distributions:
f (z | m;GX ) =
∫Bin
(z ;m,
exp(θ)
1 + exp(θ)
)dGX (θ), GX ∼ DDP(α, G0X )
→ Gaussian process (GP) for G0X with:
linear mean function, E(η`(x) | β0, β1) = β0 + β1x
constant variance, Var(η`(x) | σ2) = σ2
isotropic power exponential correlation function,Corr(η`(x), η`(x
′) | φ) = exp(−φ|x − x ′|d) (with fixed d ∈ [1, 2])
→ hyperpriors for α and ψ = (β0, β1, σ2, φ)
→ MCMC posterior simulation using blocked Gibbs sampling
→ Posterior predictive inference over observed and new dose levels, using the
posterior samples from the model and GP interpolation for the DDP locations
Athanasios Kottas (UCSC) NP Bayes in Developmental Toxicology June 29, 2012 9 / 27
The basic modeling framework
DDP mixture of Binomial distributions:
f (z | m;GX ) =
∫Bin
(z ;m,
exp(θ)
1 + exp(θ)
)dGX (θ), GX ∼ DDP(α, G0X )
→ Gaussian process (GP) for G0X with:
linear mean function, E(η`(x) | β0, β1) = β0 + β1x
constant variance, Var(η`(x) | σ2) = σ2
isotropic power exponential correlation function,Corr(η`(x), η`(x
′) | φ) = exp(−φ|x − x ′|d) (with fixed d ∈ [1, 2])
→ hyperpriors for α and ψ = (β0, β1, σ2, φ)
→ MCMC posterior simulation using blocked Gibbs sampling
→ Posterior predictive inference over observed and new dose levels, using the
posterior samples from the model and GP interpolation for the DDP locations
Athanasios Kottas (UCSC) NP Bayes in Developmental Toxicology June 29, 2012 9 / 27
The basic modeling framework
DDP mixture of Binomial distributions:
f (z | m;GX ) =
∫Bin
(z ;m,
exp(θ)
1 + exp(θ)
)dGX (θ), GX ∼ DDP(α, G0X )
→ Gaussian process (GP) for G0X with:
linear mean function, E(η`(x) | β0, β1) = β0 + β1x
constant variance, Var(η`(x) | σ2) = σ2
isotropic power exponential correlation function,Corr(η`(x), η`(x
′) | φ) = exp(−φ|x − x ′|d) (with fixed d ∈ [1, 2])
→ hyperpriors for α and ψ = (β0, β1, σ2, φ)
→ MCMC posterior simulation using blocked Gibbs sampling
→ Posterior predictive inference over observed and new dose levels, using the
posterior samples from the model and GP interpolation for the DDP locations
Athanasios Kottas (UCSC) NP Bayes in Developmental Toxicology June 29, 2012 9 / 27
The basic modeling framework
Key aspects of the DDP mixture model:
→ flexible inference at each observed dose level through a nonparametric Binomialmixture (overdispersion, skewness, multimodality ...)
→ prediction at unobserved dose levels (within and outside the range of observeddoses)
→ level of dependence between Gx and Gx′ , and thus between f (z | m;Gx) andf (z | m;Gx′), is driven by the distance between x and x ′
→ in prediction for f (z | m;Gx), we learn more from dose levels x ′ nearby x thanfrom more distant dose levels
→ inference for the dose-response relationship is induced by flexible modeling forthe underlying response distributions
Athanasios Kottas (UCSC) NP Bayes in Developmental Toxicology June 29, 2012 10 / 27
The basic modeling framework
Key aspects of the DDP mixture model:
→ flexible inference at each observed dose level through a nonparametric Binomialmixture (overdispersion, skewness, multimodality ...)
→ prediction at unobserved dose levels (within and outside the range of observeddoses)
→ level of dependence between Gx and Gx′ , and thus between f (z | m;Gx) andf (z | m;Gx′), is driven by the distance between x and x ′
→ in prediction for f (z | m;Gx), we learn more from dose levels x ′ nearby x thanfrom more distant dose levels
→ inference for the dose-response relationship is induced by flexible modeling forthe underlying response distributions
Athanasios Kottas (UCSC) NP Bayes in Developmental Toxicology June 29, 2012 10 / 27
The basic modeling framework
Key aspects of the DDP mixture model:
→ flexible inference at each observed dose level through a nonparametric Binomialmixture (overdispersion, skewness, multimodality ...)
→ prediction at unobserved dose levels (within and outside the range of observeddoses)
→ level of dependence between Gx and Gx′ , and thus between f (z | m;Gx) andf (z | m;Gx′), is driven by the distance between x and x ′
→ in prediction for f (z | m;Gx), we learn more from dose levels x ′ nearby x thanfrom more distant dose levels
→ inference for the dose-response relationship is induced by flexible modeling forthe underlying response distributions
Athanasios Kottas (UCSC) NP Bayes in Developmental Toxicology June 29, 2012 10 / 27
The basic modeling framework
Key aspects of the DDP mixture model:
→ flexible inference at each observed dose level through a nonparametric Binomialmixture (overdispersion, skewness, multimodality ...)
→ prediction at unobserved dose levels (within and outside the range of observeddoses)
→ level of dependence between Gx and Gx′ , and thus between f (z | m;Gx) andf (z | m;Gx′), is driven by the distance between x and x ′
→ in prediction for f (z | m;Gx), we learn more from dose levels x ′ nearby x thanfrom more distant dose levels
→ inference for the dose-response relationship is induced by flexible modeling forthe underlying response distributions
Athanasios Kottas (UCSC) NP Bayes in Developmental Toxicology June 29, 2012 10 / 27
The basic modeling framework
Key aspects of the DDP mixture model:
→ flexible inference at each observed dose level through a nonparametric Binomialmixture (overdispersion, skewness, multimodality ...)
→ prediction at unobserved dose levels (within and outside the range of observeddoses)
→ level of dependence between Gx and Gx′ , and thus between f (z | m;Gx) andf (z | m;Gx′), is driven by the distance between x and x ′
→ in prediction for f (z | m;Gx), we learn more from dose levels x ′ nearby x thanfrom more distant dose levels
→ inference for the dose-response relationship is induced by flexible modeling forthe underlying response distributions
Athanasios Kottas (UCSC) NP Bayes in Developmental Toxicology June 29, 2012 10 / 27
The basic modeling framework
Dose-response curve
Exploit connection of the DDP Binomial mixture for the negative outcomeswithin a dam and a DDP mixture model with a product of Bernoullis kernelfor the set of binary responses for all implants corresponding to that dam
Using the equivalent mixture model formulation for the underlying binaryoutcomes, define the dose-response curve as the probability of a negativeoutcome for a generic implant expressed as a function of dose level
D(x) =
∫exp(θ)
1 + exp(θ)dGx(θ) =
∞∑`=1
ω`exp(η`(x))
1 + exp(η`(x)), x ∈ X
If β1 > 0, the prior expectation E(D(x)) is non-decreasing with x , but prior(and thus posterior) realizations for the dose-response curve are not struc-turally restricted to be non-decreasing (a model asset!)
Athanasios Kottas (UCSC) NP Bayes in Developmental Toxicology June 29, 2012 11 / 27
The basic modeling framework
Dose-response curve
Exploit connection of the DDP Binomial mixture for the negative outcomeswithin a dam and a DDP mixture model with a product of Bernoullis kernelfor the set of binary responses for all implants corresponding to that dam
Using the equivalent mixture model formulation for the underlying binaryoutcomes, define the dose-response curve as the probability of a negativeoutcome for a generic implant expressed as a function of dose level
D(x) =
∫exp(θ)
1 + exp(θ)dGx(θ) =
∞∑`=1
ω`exp(η`(x))
1 + exp(η`(x)), x ∈ X
If β1 > 0, the prior expectation E(D(x)) is non-decreasing with x , but prior(and thus posterior) realizations for the dose-response curve are not struc-turally restricted to be non-decreasing (a model asset!)
Athanasios Kottas (UCSC) NP Bayes in Developmental Toxicology June 29, 2012 11 / 27
The basic modeling framework
Dose-response curve
Exploit connection of the DDP Binomial mixture for the negative outcomeswithin a dam and a DDP mixture model with a product of Bernoullis kernelfor the set of binary responses for all implants corresponding to that dam
Using the equivalent mixture model formulation for the underlying binaryoutcomes, define the dose-response curve as the probability of a negativeoutcome for a generic implant expressed as a function of dose level
D(x) =
∫exp(θ)
1 + exp(θ)dGx(θ) =
∞∑`=1
ω`exp(η`(x))
1 + exp(η`(x)), x ∈ X
If β1 > 0, the prior expectation E(D(x)) is non-decreasing with x , but prior(and thus posterior) realizations for the dose-response curve are not struc-turally restricted to be non-decreasing (a model asset!)
Athanasios Kottas (UCSC) NP Bayes in Developmental Toxicology June 29, 2012 11 / 27
The basic modeling framework
Data examples
2,4,5-T data: data set from a developmental toxicity study regarding the effects of the
herbicide 2,4,5-trichlorophenoxiacetic (2,4,5-T) acid (Holson et al., 1991)
0 20 40 60 80
0.0
0.2
0.4
0.6
0.8
1.0
dose mg/kg
73 87 97 76 44 25dams
Athanasios Kottas (UCSC) NP Bayes in Developmental Toxicology June 29, 2012 12 / 27
The basic modeling framework
0 2 4 6 8 10 12
0.0
0.2
0.4
0.6
0.8
1.0
dose 0 mg/kg
Number of negative outcomes
0 2 4 6 8 10 12
0.0
0.2
0.4
0.6
0.8
1.0
dose 30 mg/kg
Number of negative outcomes
0 2 4 6 8 10 12
0.0
0.2
0.4
0.6
0.8
1.0
dose 45 mg/kg
Number of negative outcomes
0 2 4 6 8 10 12
0.0
0.2
0.4
0.6
0.8
1.0
new dose 50 mg/kg
Number of negative outcomes
0 2 4 6 8 10 12
0.0
0.2
0.4
0.6
0.8
1.0
dose 60 mg/kg
Number of negative outcomes
0 2 4 6 8 10 12
0.0
0.2
0.4
0.6
0.8
1.0
dose 75 mg/kg
Number of negative outcomes
0 2 4 6 8 10 120.0
0.2
0.4
0.6
0.8
1.0
dose 90 mg/kg
Number of negative outcomes
0 2 4 6 8 10 12
0.0
0.2
0.4
0.6
0.8
1.0
new dose 100 mg/kg
Number of negative outcomes
Figure: 2,4,5-T data: For the 6 observed and 2 new doses, posterior mean estimates
(denoted by “o”) and 90% uncertainty bands (red) for f (z | m = 12; Gx).
Athanasios Kottas (UCSC) NP Bayes in Developmental Toxicology June 29, 2012 13 / 27
The basic modeling framework
0 20 40 60 80 100
0.0
0.2
0.4
0.6
0.8
1.0
dose mg/kg
Bin
om
ial M
od
el
0 20 40 60 80 100
0.0
0.2
0.4
0.6
0.8
1.0
dose mg/kg
Be
ta-B
ino
mia
l M
od
el
0 20 40 60 80 100
0.0
0.2
0.4
0.6
0.8
1.0
dose mg/kg
DD
P B
ino
mia
l M
od
el
Figure: 2,4,5-T data: Posterior mean estimate and 90% uncertainty bands for the dose-
response curve under a Binomial-logistic model (left), a Beta-Binomial model (middle),
and the DDP Binomial mixture model (right).
Athanasios Kottas (UCSC) NP Bayes in Developmental Toxicology June 29, 2012 14 / 27
The basic modeling framework
DEHP data: data from an experiment that explored the effects of diethylhexalphthalate
(DEHP), a commonly used plasticizing agent (Tyl et al., 1983)
0 50 100 150
0.0
0.2
0.4
0.6
0.8
1.0
dose mg/kg x 1000
30 26 26 17 9dams
0 50 100 1500.0
0.2
0.4
0.6
0.8
1.0
dose mg/kg x 1000
Figure: Posterior mean estimate and 90% uncertainty bands for the dose-response curve
— the dip at small toxin levels may indicate a hormetic dose-response relationship
Athanasios Kottas (UCSC) NP Bayes in Developmental Toxicology June 29, 2012 15 / 27
Modeling for multicategory classification responses
Inference for embryolethality and malformation endpoints
Full version of the DEHP data
0 50 100 150
0.0
0.2
0.4
0.6
0.8
1.0
R/m
dose mg/kg x 1000
0 50 100 150
0.0
0.2
0.4
0.6
0.8
1.0
y/(m-R)
dose mg/kg x 1000
0 50 100 150
0.0
0.2
0.4
0.6
0.8
1.0
(R+y)/m
dose mg/kg x 1000
Clustered categorical responses: for the j-th dam at dose xi , Rij resorptions and prenatal
deaths (Rij ≤ mij), and yij malformations among the live pups (yij ≤ mij − Rij)
Athanasios Kottas (UCSC) NP Bayes in Developmental Toxicology June 29, 2012 16 / 27
Modeling for multicategory classification responses
DDP mixture model for endpoints of embryolethality (R) and malformationfor live pups (y)
f (R, y | m;GX ) =
∫Bin (R;m, π(γ)) Bin (y ;m − R, π(θ)) dGX (γ, θ)
→ π(v) = exp(v)/{1 + exp(v)}, v ∈ R, denotes the logistic function
→ GX =∑∞
`=1 ω`δη`X ∼ DDP(α, G0X ), where η`(x) = (γ`(x), θ`(x))
→ G0X defined through two independent GPs with linear mean functions,E(γ`(x) | ξ0, ξ1) = ξ0 + ξ1x , and E(θ`(x) | β0, β1) = β0 + β1x
Equivalent mixture model (with product Bernoulli kernels) for binary responses:R∗ non-viable fetus indicator; y∗ malformation indicator
Athanasios Kottas (UCSC) NP Bayes in Developmental Toxicology June 29, 2012 17 / 27
Modeling for multicategory classification responses
DDP mixture model for endpoints of embryolethality (R) and malformationfor live pups (y)
f (R, y | m;GX ) =
∫Bin (R;m, π(γ)) Bin (y ;m − R, π(θ)) dGX (γ, θ)
→ π(v) = exp(v)/{1 + exp(v)}, v ∈ R, denotes the logistic function
→ GX =∑∞
`=1 ω`δη`X ∼ DDP(α, G0X ), where η`(x) = (γ`(x), θ`(x))
→ G0X defined through two independent GPs with linear mean functions,E(γ`(x) | ξ0, ξ1) = ξ0 + ξ1x , and E(θ`(x) | β0, β1) = β0 + β1x
Equivalent mixture model (with product Bernoulli kernels) for binary responses:R∗ non-viable fetus indicator; y∗ malformation indicator
Athanasios Kottas (UCSC) NP Bayes in Developmental Toxicology June 29, 2012 17 / 27
Modeling for multicategory classification responses
DDP mixture model for endpoints of embryolethality (R) and malformationfor live pups (y)
f (R, y | m;GX ) =
∫Bin (R;m, π(γ)) Bin (y ;m − R, π(θ)) dGX (γ, θ)
→ π(v) = exp(v)/{1 + exp(v)}, v ∈ R, denotes the logistic function
→ GX =∑∞
`=1 ω`δη`X ∼ DDP(α, G0X ), where η`(x) = (γ`(x), θ`(x))
→ G0X defined through two independent GPs with linear mean functions,E(γ`(x) | ξ0, ξ1) = ξ0 + ξ1x , and E(θ`(x) | β0, β1) = β0 + β1x
Equivalent mixture model (with product Bernoulli kernels) for binary responses:R∗ non-viable fetus indicator; y∗ malformation indicator
Athanasios Kottas (UCSC) NP Bayes in Developmental Toxicology June 29, 2012 17 / 27
Modeling for multicategory classification responses
DDP mixture model for endpoints of embryolethality (R) and malformationfor live pups (y)
f (R, y | m;GX ) =
∫Bin (R;m, π(γ)) Bin (y ;m − R, π(θ)) dGX (γ, θ)
→ π(v) = exp(v)/{1 + exp(v)}, v ∈ R, denotes the logistic function
→ GX =∑∞
`=1 ω`δη`X ∼ DDP(α, G0X ), where η`(x) = (γ`(x), θ`(x))
→ G0X defined through two independent GPs with linear mean functions,E(γ`(x) | ξ0, ξ1) = ξ0 + ξ1x , and E(θ`(x) | β0, β1) = β0 + β1x
Equivalent mixture model (with product Bernoulli kernels) for binary responses:R∗ non-viable fetus indicator; y∗ malformation indicator
Athanasios Kottas (UCSC) NP Bayes in Developmental Toxicology June 29, 2012 17 / 27
Modeling for multicategory classification responses
Dose-response curves
Probability of embryolethality
Pr(R∗ = 1;Gx) =
∫π(γ) dGx(γ, θ), x ∈ X
(monotonic in prior expectation provided ξ1 > 0)
Probability of malformation
Pr(y∗ = 1 | R∗ = 0;Gx) =
∫{1− π(γ)}π(θ) dGx(γ, θ)∫{1− π(γ)} dGx(γ, θ)
, x ∈ X
Combined risk function
Pr(R∗ = 1 or y∗ = 1;Gx) = 1−∫{1− π(γ)}{1− π(θ)} dGx(γ, θ), x ∈ X
(monotonic in prior expectation provided ξ1 > 0 and β1 > 0)
Athanasios Kottas (UCSC) NP Bayes in Developmental Toxicology June 29, 2012 18 / 27
Modeling for multicategory classification responses
Dose-response curves
Probability of embryolethality
Pr(R∗ = 1;Gx) =
∫π(γ) dGx(γ, θ), x ∈ X
(monotonic in prior expectation provided ξ1 > 0)
Probability of malformation
Pr(y∗ = 1 | R∗ = 0;Gx) =
∫{1− π(γ)}π(θ) dGx(γ, θ)∫{1− π(γ)} dGx(γ, θ)
, x ∈ X
Combined risk function
Pr(R∗ = 1 or y∗ = 1;Gx) = 1−∫{1− π(γ)}{1− π(θ)} dGx(γ, θ), x ∈ X
(monotonic in prior expectation provided ξ1 > 0 and β1 > 0)
Athanasios Kottas (UCSC) NP Bayes in Developmental Toxicology June 29, 2012 18 / 27
Modeling for multicategory classification responses
Dose-response curves
Probability of embryolethality
Pr(R∗ = 1;Gx) =
∫π(γ) dGx(γ, θ), x ∈ X
(monotonic in prior expectation provided ξ1 > 0)
Probability of malformation
Pr(y∗ = 1 | R∗ = 0;Gx) =
∫{1− π(γ)}π(θ) dGx(γ, θ)∫{1− π(γ)} dGx(γ, θ)
, x ∈ X
Combined risk function
Pr(R∗ = 1 or y∗ = 1;Gx) = 1−∫{1− π(γ)}{1− π(θ)} dGx(γ, θ), x ∈ X
(monotonic in prior expectation provided ξ1 > 0 and β1 > 0)
Athanasios Kottas (UCSC) NP Bayes in Developmental Toxicology June 29, 2012 18 / 27
Modeling for multicategory classification responses
DEHP data, Vol. 2: Posterior mean estimates and 90% uncertainty bands for the three
dose-response curves
0 50 100 150
0.00.2
0.40.6
0.81.0
probability of non-viable fetus
dose mg/kg x 1000
Pr(R*=1;G
x)
0 50 100 150
0.00.2
0.40.6
0.81.0
probability of malformation
dose mg/kg x 1000
Pr(Y*=1|R
*=0;G
x)
0 50 100 150
0.00.2
0.40.6
0.81.0
combined risk
dose mg/kg x 1000
r(x;Gx)
Athanasios Kottas (UCSC) NP Bayes in Developmental Toxicology June 29, 2012 19 / 27
Toxicity experiments with discrete-continuous outcomes
The modeling approach
General setting with multiple categorical endpoints and continuous outcomes
For a generic dam (at dose x) with m implants and R non-viable fetuses
→ (y∗,u∗) = {(y∗k , u∗k ) : k = 1, ...,m − R}: binary malformation and con-tinuous responses (body weight) for live pups
DDP mixture model:
f (R, y∗,u∗ | m;GX ) =
∫Bin (R;m, π(γ))
m−R∏k=1
Bern (y∗k ;π(θ))
×m−R∏k=1
N (u∗k ;µ, ϕ) dGX (γ, θ, µ)
with an extended centering process G0X in the DDP prior for GX
Athanasios Kottas (UCSC) NP Bayes in Developmental Toxicology June 29, 2012 20 / 27
Toxicity experiments with discrete-continuous outcomes
The modeling approach
General setting with multiple categorical endpoints and continuous outcomes
For a generic dam (at dose x) with m implants and R non-viable fetuses
→ (y∗,u∗) = {(y∗k , u∗k ) : k = 1, ...,m − R}: binary malformation and con-tinuous responses (body weight) for live pups
DDP mixture model:
f (R, y∗,u∗ | m;GX ) =
∫Bin (R;m, π(γ))
m−R∏k=1
Bern (y∗k ;π(θ))
×m−R∏k=1
N (u∗k ;µ, ϕ) dGX (γ, θ, µ)
with an extended centering process G0X in the DDP prior for GX
Athanasios Kottas (UCSC) NP Bayes in Developmental Toxicology June 29, 2012 20 / 27
Toxicity experiments with discrete-continuous outcomes
The modeling approach
General setting with multiple categorical endpoints and continuous outcomes
For a generic dam (at dose x) with m implants and R non-viable fetuses
→ (y∗,u∗) = {(y∗k , u∗k ) : k = 1, ...,m − R}: binary malformation and con-tinuous responses (body weight) for live pups
DDP mixture model:
f (R, y∗,u∗ | m;GX ) =
∫Bin (R;m, π(γ))
m−R∏k=1
Bern (y∗k ;π(θ))
×m−R∏k=1
N (u∗k ;µ, ϕ) dGX (γ, θ, µ)
with an extended centering process G0X in the DDP prior for GX
Athanasios Kottas (UCSC) NP Bayes in Developmental Toxicology June 29, 2012 20 / 27
Toxicity experiments with discrete-continuous outcomes
Risk assessment inference
Dose-response curves for endpoints of embryolethality, Pr(R∗ = 1; Gx), andmalformation, Pr(y∗ = 1 | R∗ = 0;Gx), as before
Expected fetal weight
E(u∗ | R∗ = 0;Gx) =
∫µ{1− π(γ)} dGx(γ, θ, µ)∫{1− π(γ)} dGx(γ, θ, µ)
, x ∈ X
Probability of low fetal weight
Pr(u∗ < U | R∗ = 0;Gx) =
∫{1− π(γ)}Φ(ϕ−1/2(U − µ)) dGx(γ, θ, µ)∫
{1− π(γ)} dGx(γ, θ, µ), x ∈ X
Athanasios Kottas (UCSC) NP Bayes in Developmental Toxicology June 29, 2012 21 / 27
Toxicity experiments with discrete-continuous outcomes
Risk assessment inference
Dose-response curves for endpoints of embryolethality, Pr(R∗ = 1; Gx), andmalformation, Pr(y∗ = 1 | R∗ = 0;Gx), as before
Expected fetal weight
E(u∗ | R∗ = 0;Gx) =
∫µ{1− π(γ)} dGx(γ, θ, µ)∫{1− π(γ)} dGx(γ, θ, µ)
, x ∈ X
Probability of low fetal weight
Pr(u∗ < U | R∗ = 0;Gx) =
∫{1− π(γ)}Φ(ϕ−1/2(U − µ)) dGx(γ, θ, µ)∫
{1− π(γ)} dGx(γ, θ, µ), x ∈ X
Athanasios Kottas (UCSC) NP Bayes in Developmental Toxicology June 29, 2012 21 / 27
Toxicity experiments with discrete-continuous outcomes
Risk assessment inference
Dose-response curves for endpoints of embryolethality, Pr(R∗ = 1; Gx), andmalformation, Pr(y∗ = 1 | R∗ = 0;Gx), as before
Expected fetal weight
E(u∗ | R∗ = 0;Gx) =
∫µ{1− π(γ)} dGx(γ, θ, µ)∫{1− π(γ)} dGx(γ, θ, µ)
, x ∈ X
Probability of low fetal weight
Pr(u∗ < U | R∗ = 0;Gx) =
∫{1− π(γ)}Φ(ϕ−1/2(U − µ)) dGx(γ, θ, µ)∫
{1− π(γ)} dGx(γ, θ, µ), x ∈ X
Athanasios Kottas (UCSC) NP Bayes in Developmental Toxicology June 29, 2012 21 / 27
Toxicity experiments with discrete-continuous outcomes
Combined risk function for discrete endpoints, Pr(R∗ = 1 or y∗ = 1; Gx), asbefore
Full combined risk function
Pr(R∗ = 1 or y∗ = 1 or u∗ < U;Gx) =1−
∫{1− π(γ)}{1− π(θ)}{1− Φ(ϕ−1/2(U − µ))} dGx(γ, θ, µ), x ∈ X
Dose-dependent inference for intracluster correlations
→ between malformation responses for two live pups
→ between continuous endpoints for two live pups
→ between discrete-continuous outcomes within a live pup
Athanasios Kottas (UCSC) NP Bayes in Developmental Toxicology June 29, 2012 22 / 27
Toxicity experiments with discrete-continuous outcomes
Combined risk function for discrete endpoints, Pr(R∗ = 1 or y∗ = 1; Gx), asbefore
Full combined risk function
Pr(R∗ = 1 or y∗ = 1 or u∗ < U;Gx) =1−
∫{1− π(γ)}{1− π(θ)}{1− Φ(ϕ−1/2(U − µ))} dGx(γ, θ, µ), x ∈ X
Dose-dependent inference for intracluster correlations
→ between malformation responses for two live pups
→ between continuous endpoints for two live pups
→ between discrete-continuous outcomes within a live pup
Athanasios Kottas (UCSC) NP Bayes in Developmental Toxicology June 29, 2012 22 / 27
Toxicity experiments with discrete-continuous outcomes
Combined risk function for discrete endpoints, Pr(R∗ = 1 or y∗ = 1; Gx), asbefore
Full combined risk function
Pr(R∗ = 1 or y∗ = 1 or u∗ < U;Gx) =1−
∫{1− π(γ)}{1− π(θ)}{1− Φ(ϕ−1/2(U − µ))} dGx(γ, θ, µ), x ∈ X
Dose-dependent inference for intracluster correlations
→ between malformation responses for two live pups
→ between continuous endpoints for two live pups
→ between discrete-continuous outcomes within a live pup
Athanasios Kottas (UCSC) NP Bayes in Developmental Toxicology June 29, 2012 22 / 27
Toxicity experiments with discrete-continuous outcomes
Data illustration
EG data: data set from a study on the toxic effects of ethylene glycol (EG), an industrial
chemical widely used as an antifreeze (Price et al., 1985)
0 500 1000 1500 2000 2500 3000
0.0
0.2
0.4
0.6
0.8
1.0
R/m
dose mg/kg
0 500 1000 1500 2000 2500 3000
0.0
0.2
0.4
0.6
0.8
1.0
sum y*/(m-R)
dose mg/kg
0 500 1000 1500 2000 2500 3000
0.0
0.2
0.4
0.6
0.8
1.0
1.2
u*
dose mg/kg
Athanasios Kottas (UCSC) NP Bayes in Developmental Toxicology June 29, 2012 23 / 27
Toxicity experiments with discrete-continuous outcomes
Posterior mean estimates and 90% uncertainty bands for six dose-response curves
0 500 1000 1500 2000 2500 3000
0.0
0.2
0.4
0.6
0.8
1.0
dose mg/kg
D(x)
0 500 1000 1500 2000 2500 3000
0.0
0.2
0.4
0.6
0.8
1.0
dose mg/kgM(x)
0 500 1000 1500 2000 2500 3000
0.0
0.2
0.4
0.6
0.8
1.0
dose mg/kg
rd(x)
0 500 1000 1500 2000 2500 3000
0.5
0.6
0.7
0.8
0.9
1.0
1.1
1.2
dose mg/kg
E(u*
|R*=
0;G
x)
0 500 1000 1500 2000 2500 3000
0.0
0.2
0.4
0.6
0.8
1.0
dose mg/kg
Pr(u
* < U
|R*=
0;G
x)
0 500 1000 1500 2000 2500 3000
0.0
0.2
0.4
0.6
0.8
1.0
dose mg/kgrf(x)
Athanasios Kottas (UCSC) NP Bayes in Developmental Toxicology June 29, 2012 24 / 27
Conclusions
Summary
General modeling framework for replicated count responses, and discrete-continuous outcomes in dose-response settings — emphasis on data fromdevelopmental toxicity studies
Further elaboration to fully nonparametric modeling for experiments with pre-implantation exposure
More structured nonparametric mixture models for traditional bioassay exper-iments, including quantal responses and ordinal response classifications
Athanasios Kottas (UCSC) NP Bayes in Developmental Toxicology June 29, 2012 25 / 27
Conclusions
Summary
General modeling framework for replicated count responses, and discrete-continuous outcomes in dose-response settings — emphasis on data fromdevelopmental toxicity studies
Further elaboration to fully nonparametric modeling for experiments with pre-implantation exposure
More structured nonparametric mixture models for traditional bioassay exper-iments, including quantal responses and ordinal response classifications
Athanasios Kottas (UCSC) NP Bayes in Developmental Toxicology June 29, 2012 25 / 27
Conclusions
Summary
General modeling framework for replicated count responses, and discrete-continuous outcomes in dose-response settings — emphasis on data fromdevelopmental toxicity studies
Further elaboration to fully nonparametric modeling for experiments with pre-implantation exposure
More structured nonparametric mixture models for traditional bioassay exper-iments, including quantal responses and ordinal response classifications
Athanasios Kottas (UCSC) NP Bayes in Developmental Toxicology June 29, 2012 25 / 27
Conclusions
Papers
Fronczyk, K. and Kottas, A. (2012). “Risk assessment for toxicity experiments com-prising joint discrete-continuous outcomes: A Bayesian nonparametric approach.”Draft manuscript.
Kottas A. and Fronczyk K. (2011). “Flexible Bayesian modelling for clustered cat-egorical responses in developmental toxicology.” Book chapter for the Oxford Uni-versity Press book in tribute of Adrian Smith.
Fronczyk, K. and Kottas, A. (2010). “A Bayesian Nonparametric Modeling Frame-
work for Developmental Toxicity Studies.” AMS Technical Report, School of Engi-
neering, University of California, Santa Cruz.
Acknowledgment: Funding from NSF, Division of Environmental Biology and the
Committee on Research, UC Santa Cruz
Athanasios Kottas (UCSC) NP Bayes in Developmental Toxicology June 29, 2012 26 / 27
Conclusions
MANY THANKS !!!
Athanasios Kottas (UCSC) NP Bayes in Developmental Toxicology June 29, 2012 27 / 27