Nonparametric Bayesian analysis of developmental toxicity ...thanos/ISBA2012-talk.pdfOutline 1 Introduction 2 The basic modeling framework 3 Modeling for multicategory classiﬁcation

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


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


Introduction









Introduction









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


Introduction













Introduction













Introduction













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).


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).


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


Introduction










Introduction










Introduction










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+}



Model formulation









Model formulation









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




GX (·) =∞∑

`=1

ω`δη`X (·)



∏`−1r=1(1−ζr )



)





GX (·) =∞∑

`=1

ω`δη`X (·)



∏`−1r=1(1−ζr )



)





GX (·) =∞∑

`=1

ω`δη`X (·)



∏`−1r=1(1−ζr )



)





GX (·) =∞∑

`=1

ω`δη`X (·)



∏`−1r=1(1−ζr )



)




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




f (z | m;GX ) =

∫Bin

(z ;m,

exp(θ)

1 + exp(θ)














f (z | m;GX ) =

∫Bin

(z ;m,

exp(θ)

1 + exp(θ)














f (z | m;GX ) =

∫Bin

(z ;m,

exp(θ)

1 + exp(θ)














f (z | m;GX ) =

∫Bin

(z ;m,

exp(θ)

1 + exp(θ)













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



































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

ωèxp(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!)



Dose-response curve



D(x) =

∫exp(θ)


∞∑`=1

ωèxp(η`(x))

1 + exp(η`(x)), x ∈ X




Dose-response curve



D(x) =

∫exp(θ)


∞∑`=1

ωèxp(η`(x))

1 + exp(η`(x)), x ∈ X




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



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


0 2 4 6 8 10 12

0.0

0.2

0.4

0.6

0.8

1.0

dose 45 mg/kg


0 2 4 6 8 10 12

0.0

0.2

0.4

0.6

0.8

1.0

new dose 50 mg/kg


0 2 4 6 8 10 12

0.0

0.2

0.4

0.6

0.8

1.0

dose 60 mg/kg


0 2 4 6 8 10 12

0.0

0.2

0.4

0.6

0.8

1.0

dose 75 mg/kg


0 2 4 6 8 10 120.0

0.2

0.4

0.6

0.8

1.0

dose 90 mg/kg


0 2 4 6 8 10 12

0.0

0.2

0.4

0.6

0.8

1.0

new dose 100 mg/kg


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).



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).



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


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)



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




f (R, y | m;GX ) =



→ GX =∑∞







f (R, y | m;GX ) =



→ GX =∑∞







f (R, y | m;GX ) =



→ GX =∑∞






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)





Pr(R∗ = 1;Gx) =




Pr(y∗ = 1 | R∗ = 0;Gx) =


, x ∈ X








Pr(R∗ = 1;Gx) =




Pr(y∗ = 1 | R∗ = 0;Gx) =


, x ∈ X






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)


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







DDP mixture model:

f (R, y∗,u∗ | m;GX ) =


m−R∏k=1


×m−R∏k=1









DDP mixture model:

f (R, y∗,u∗ | m;GX ) =


m−R∏k=1


×m−R∏k=1





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






E(u∗ | R∗ = 0;Gx) =


, x ∈ X


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






E(u∗ | R∗ = 0;Gx) =


, x ∈ X


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



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





Pr(R∗ = 1 or y∗ = 1 or u∗ < U;Gx) =1−










Pr(R∗ = 1 or y∗ = 1 or u∗ < U;Gx) =1−








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



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)


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


Conclusions

Summary





Conclusions

Summary





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


Conclusions

MANY THANKS !!!


Documents

Nonparametric Bayesian analysis of developmental toxicity ...thanos/ISBA2012-talk.pdfOutline 1 Introduction 2 The basic modeling framework 3 Modeling for multicategory classiﬁcation