Bayesian Matrix Completion for Hypothesis Testing
Bora Jin, David B. Dunson, Julia E. Rager, David Reif, Stephanie M. Engel, Amy H. Herring
BBayesian Matrix Completionfor Hypothesis Testing
Bora Jin , David B. Dunson , Julia E. Rager , David Reif , Stephanie M.Engel , and Amy H. Herring Department of Statistical Science, Duke University Department of Environmental Sciences and Engineering, University ofNorth Carolina at Chapel Hill Department of Biological Sciences, North Carolina State University Department of Epidemiology, University of North Carolina at Chapel Hill * email: [email protected] 15, 2020 Abstract
The United States Environmental Protection Agency (EPA) screens thousandsof chemicals primarily to differentiate those that are active vs inactive for differenttypes of biological endpoints. However, it is not feasible to test all possible com-binations of chemicals, assay endpoints, and concentrations, resulting in a majorityof missing combinations. Our goal is to derive posterior probabilities of activity foreach chemical by assay endpoint combination. Therefore, we are faced with a taskof matrix completion in the context of hypothesis testing for sparse functional data.We propose a Bayesian hierarchical framework, which borrows information acrossdifferent chemicals and assay endpoints. Our model predicts bioactivity profiles ofwhether the dose-response curve is constant or not, using low-dimensional latent at-tributes of chemicals and of assay endpoints. This framework facilitates out-of-sample a r X i v : . [ s t a t . A P ] O c t rediction of bioactivity potential for new chemicals not yet tested, while capturingheteroscedastic residuals. We demonstrate the performance via extensive simulationstudies and an application to data from the EPA’s ToxCast/Tox21 program. Our ap-proach allows more realistic and stable estimation of potential toxicity as shown fortwo disease outcomes: neurodevelopmental disorders and obesity. Code is availableon Github. Keywords:
Bayesian hierarchical; Bioactivity profiles; Dose-response modeling; Heteroscedas-ticity; Latent factor models; Multiple testing; ToxCast/Tox212
Introduction
The United States Environmental Protection Agency (EPA) has been charged with main-taining a chemical inventory and regulating hazardous chemicals since the Toxic SubstancesControl Act in 1976. As massive numbers of new chemicals have been added to the existinglist of untested chemicals, the EPA has struggled to keep up with screening chemicals tomeasure their potential toxicity and hazards to human health. The traditional animal or in vivo testing paradigms are infeasible due to financial and time constraints (Dix et al.,2007; Judson et al., 2010); in addition, it is desirable to minimize animals used in anytesting procedure. Therefore, a high-throughput screening (HTS) mechanism based on invitro assays and a large number of chemicals has been developed, enabling the EPA tooperate the relatively low-cost and rapid Toxicity Forecaster (ToxCast) and Toxicology inthe 21st Century (Tox21). These programs are designed to identify chemicals that likelyinduce toxicity in humans and prioritize them for further testing (Judson et al., 2010).The ToxCast/Tox21 program tests thousands of chemicals against numerous high-throughput assay endpoints. Although the HTS mechanism has provided a relatively cheapand quick way to conduct millions of tests, it is still only possible to test a small minority ofall (chemical, assay endpoint) combinations. This leads to many non-tested combinations,which are shown as empty cells in Figures 1 & 2. Figure 1 illustrates the ToxCast/Tox21data structure relevant to obesity; many cells have zero observations, and the number ofobservations largely fluctuates across cells. Here, an observation is the result from an exper-iment where a chemical is applied to an assay endpoint at a certain dose. Figure 2 providesadditional details on a few cells from Figure 1, including the observed measurements. Somecells have few observations, while others have multiple replicates at a finer grid of doses.These data can be arranged as matrix-structured sparse functional data, with the rows of3 ssay endpoint C he m i c a l Number of Observations501−870101−50011−1002−1010
Figure 1:
Heat map of the number of observations in ToxCast/Tox21 data for obesity, based on30 chemicals (rows) and 271 assay endpoints (columns). the matrix corresponding to different chemicals, the columns to different assay endpoints,and the cells containing dose-response measurements.Traditional matrix completion focuses on the problem of filling in the missing elementsof a large matrix based on observations on a small proportion of the cells. Typically, theobserved cells contain a scalar that is assumed to be measured without error. A classicapplication is recommender systems (Koren, 2009; Rendle et al., 2009) in which rows areusers, columns are products, and the observations are user ratings of products, thoughmatrix completion applications are broad (Candes et al., 2015; Lu et al., 2018; Nguyenet al., 2019). We are faced with a latent binary matrix completion problem with eachcell containing a binary indicator of whether a particular chemical is active for a specificassay endpoint. This is a type of matrix-structured multiple hypothesis testing problemfor assessing dose response relationships.There have been many Bayesian approaches to multiple hypothesis testing (Scott andBerger, 2006; Thomas et al., 2009; Scott and Berger, 2010; Li and Zhang, 2010; Scheel et al.,4 ll ll lll l lllll llll ll ll llllll ll ll l ll lll lll llll lll lllllllllll ll ll l l ll ll ll l llll llll llll ll llll l ll lll lllll llll lll llll ll ll l llllll ll ll l l l ll ll ll l ll ll lll l lllll lllllll lllllllll llllllll lllllll lllll llll l lll llllll l l ll ll ll ll l l lll llll l l llll ll ll lll ll l ll lll ll llll l lll ll ll ll ll l ll ll ll lll lll llll ll ll l l ll ll ll l ll lll l ll l l ll ll lll ll l llll llllllll lllll l l l l l l ll ll l l l l llll lll ll l ll l lll l l lll llll l l llll ll l l lll ll lll l lll l ll ll llll lll lll ll ll l l ll l l l l ll lll l ll lll l ll llll l lll ll lll ll l l l l lll ll l l ll llll ll ll l ll lll l lll ll llll ll ll l ll ll l l ll lll l lll lll l lll ll lll l ll ll ll ll l ll lll lll lll lll ll lllll ll l l l ll ll l lll llll ll l lll llll llll ll ll l lllll l lll l ll l ll ll lll lll llll ll ll l l ll lll l ll l l ll l ll lll l l ll lllll l l l llll lll l ll lll llll lll lllll lll lll llll l lll lllllll l llll l l l l lll llll l llll llll l l l ll l llll l l lll ll l lll llll lll l ll lll l ll l l lll ll llll ll ll l l l llll l l lll llll ll llll l ll ll l ll lll l lll ll lll l l l ll llll ll l l ll l lll ll ll l l ll llll lllll l ll ll l ll l l lll l llll l l l l lll lll l l l ll lll ll l ll l ll ll l ll lll llll llll l l ll ll l ll l l lll l lll ll ll ll lll lll ll ll ll ll l l ll lll l l ll l l llll llll ll ll lll l l l llll ll llll ll ll llll ll l l l l ll ll l lllll l lll lll ll ll l l l ll l l l lll lll ll l ll lll l ll llll lll l l l l lll l l l l lll ll ll ll l llll ll l ll ll ll lll lll lll llll l l l lll ll lll l l l ll lllllll l l lll llll l l llllll l lll ll lll l l lll ll llllll l lll ll ll l llll ll lll ll l lll l lll l ll ll l ll llll llll lllll ll ll l ll llll llll llll l lll ll ll l ll lll l lll llll l llll l ll llll l l lll l ll lll ll ll l llll ll ll l ll l lll llll l ll ll ll ll ll ll ll ll lll lll lll l llll lllll lll ll lll l ll ll ll ll ll ll lll l l ll ll l lll lll l ll ll ll l l ll ll ll l ll l llll ll ll l lll lll l lll lll lll ll ll ll l l ll l lll ll l ll ll lllll ll llll ll l llll ll ll llll ll lll lll l l lll l ll l l lllll ll l lll ll l ll llll ll lll lll lll lllll l l llll l ll ll ll llll l llll l ll ll ll l ll l lll l ll ll l ll l lll llll l ll l l lll l ll l lll l l lll ll l l l l ll l l lll ll l lll l ll lll l ll llllll l ll l llll ll l ll l lllll ll l ll ll l ll ll l llll llll l l ll llll l l lllll l ll l ll l ll l l lllll l ll llll lll l ll ll lll ll l l ll ll l l lll l lll llll ll l l l ll l ll l ll lllll l lllll lllll ll l llll ll lll lll lll lll l lllll lll ll lllll ll l lll ll ll l lll l l lllll l ll l ll ll l l lll l lll lll lll ll l llll llllll lll llll lll ll l l ll l ll ll l lll l llll l l l lll l l ll l llll l lll lllll l l ll l l l lll l llll l l l l lllllll l l lll llll l llll l ll l l lll lllllllllll l l lllll lll l l l ll l lllll ll llll l l l lllll lll ll l ll lll ll l l l llll l lll llll l ll l l llllll l ll lll llll ll lll l l ll l ll ll lll lll l ll l l lll ll ll l l llll l l l ll l l ll ll lll lllll l l ll ll lll l lll ll lll l ll l ll ll lll lll ll lll lll l l l ll lll l l l lll ll ll ll l ll ll l llll l l ll l l l ll lll l l l lll ll ll lll lll l l ll ll lll ll l l l ll ll lll l l ll l lll lll lll ll lll l ll ll l l l lll l l ll l ll ll l l l l ll l l l ll ll lll ll l ll l ll lllll l l ll ll ll l ll l l lll lll ll llllll lll ll l ll l ll l llll l l l l lll lll ll lll lll ll ll ll lllll ll l l l ll l l l ll ll l l ll llll llll ll l lll l ll llll l ll l l ll ll lllll l l l l lll l l lll l l lll lll l l lll llll l llll ll ll ll ll ll l l ll ll ll l llll l lll lllll llll l llll llll ll ll l lll l ll l l ll llll l lll lll l ll lll l llll ll lllll l ll lll l ll l l ll l lll l ll l llll ll l l l lllll lll ll ll llll lllll lll l l l ll lll llll l ll ll llll ll l ll l l l ll llll llll lll
ATG_ERE_CIS_up CEETOX_H295R_OHPROG_up NVS_ADME_hCYP19A1_Activator OT_AR_ARSRC1_0960 TOX21_AR_LUC_MDAKB2_Antagonist B i s pheno l A C h l o r p y r i f o s C y pe r − m e t h r i n D i bu t y l ph t ha l a t e D i e t h y l ph t ha l a t ep , p ' − DD E T r i c l o s an −2 −1 0 1 2 −2 −1 0 1 2 −2 −1 0 1 −2 −1 0 1 2 −3 −2 −1 0 1 2−4004080120−4004080−2002040−20−10010−20−10010050100150200−50050100 Dose (log m M) R e s pon s e Figure 2:
Detailed illustration of the ToxCast/Tox21 data structure. Sample data for 7 chemicals(rows) and 5 assay endpoints (columns). Each cell contains a set of test results of a single chemicalagainst one assay endpoint, which is functional data on a dose-response curve. a priori . For example, variable selection caseshave hypotheses H j : γ j = 0 and H j : γ j = 1 in which γ j is an indicator of whetherthe j th variable is included for j = 1 , . . . , p , and π = P r ( γ j = 1) ∼ Beta ( a, b ) is a globalparameter controlling model size. A variety of more elaborate non-exchangeable priors havebeen proposed for γ = ( γ , . . . , γ p ) (cid:48) , designed to include “prior covariates” Z j informing P r ( γ j = 1) (Thomas et al., 2009) and known structure among covariates represented byan undirected graph (Li and Zhang, 2010). 5here has also been some consideration of matrix-structured multiple testing. Scheelet al. (2013) impose a generalized linear model on the prior probability of no insuranceclaims in municipality i on day j . In relation to dose-response curves, Wilson et al. (2014)test for dose effects on the mean using a generalized linear mixed effects model. The meaneffect indicator γ ij for a (chemical i , assay endpoint j ) pair follows a Bernoulli distributionwith π ij = P r ( γ ij = 1). Then π ij is further structured with an assay endpoint random effect,chemical-level fixed effect and a probit link: π ij = Φ( α j + αx i ) where x is the chemical-levelcovariate. However, it is nontrivial to find informative chemical-level covariates x i in thiscontext. In their ToxCast/Tox21 application, Wilson et al. (2014) found their covariate,chemical solubility, was not significant in explaining γ ij , resulting in a simplified modelwith only random effects for assay endpoints.In order to account for mechanistic similarities among chemicals and/or assay endpointsas well as to tackle sparsity of the data, we require a more sophisticated hierarchy thatborrows information across both rows and columns of the matrix. Tansey et al. (2019)propose hierarchical functional matrix factorization methods to infer dose-response curves,approximating the row and the column space using low-dimensional latent attributes. How-ever, their model lacks a formal testing framework and assumes a matrix data structurewhere all cells have the same number of replicates at the same number of unique doses.In the ToxCast/Tox21 data, the number of unique doses differs within a column, and thenumber of replicates at each dose varies within a cell. Still, we can adapt low rank approx-imations addressing matrix completion problems (Mnih and Salakhutdinov, 2008; Korenet al., 2009; Purushotham et al., 2012; Tansey et al., 2019) to a multiple hypothesis testingframework. We construct π ij with a latent factor model, assuming that low-dimensionallatent attributes account for associations relevant to the mean effect among chemicals oramong assay endpoints. 6ther important characteristics of the ToxCast/Tox21 data are irregular dose-responseshapes and heteroscedasticity. Many previous studies placed monotone increasing shaperestrictions on dose-response curves (Neelon and Dunson, 2004; Ritz, 2010; Wilson et al.,2014) and did not consider heteroscedasticity. Our approach is strongly motivated byevidence that disruption in centrality or dispersion of intricately-controlled biological path-ways observed in vitro can lead to in vivo toxicity and ultimately connect to detrimentalhealth effects (Klaren et al., 2019; Knapen et al., 2020). We define any changes in meanand variance of dose-response curves as “activity” and simultaneously model both. Theseconsiderations provide a more holistic perspective on active chemicals than previous re-search.We propose a new hierarchical Bayesian matrix completion (BMC) approach for hy-pothesis testing, which is particularly useful to tackle sparsity of the ToxCast/Tox21 data.A posterior summary matrix of γ ij naturally prioritizes chemicals, putting forward thoselikely to be active for further exhaustive screening. Our BMC approach is designed to cap-ture any non-constant shapes and heteroscedastic changes of dose-response curves, whilefacilitating out-of-sample prediction of bioactivity for new chemicals not yet assayed.The remainder of the paper is organized as follows. Section 2 further explains moti-vating aspects to develop a model tailored for the ToxCast/Tox21 application. Section3 summarizes the ToxCast/Tox21 data of relevance to neurodevelopmental disorders andobesity. Then, the BMC approach is described by subparts throughout section 4. Wecompare the performance of BMC with existing methods on simulated data sets and showthe application results in section 5, highlighting chemicals that pose greater risks for thetwo diseases. Potential areas of future research are discussed in section 6.7 Motivating Aspects and Relevant Literature
Hierarchical Structures
A simple approach for matrix-structured data would be to consider each cell indepen-dently. The EPA has developed an R package “tcpl” (Filer et al., 2017) to facilitate inde-pendent dose-response modeling of the ToxCast data. This R package provides three de-fault models: a constant model at zero, a three-parameter Hill model, and a five-parametergain-loss model for each (chemical, assay endpoint) combination separately. Unfortunately,independently inferring dose-response relationships does not have predictive power: it can-not predict activity for cells having no data. Further, it is likely to have low power andhigh variance in estimation due to the intrinsic sparsity of the ToxCast/Tox21 data shownin Figure 1 & 2. In the ToxCast/Tox21 data, the median number of unique doses testedfor each pair is 8 (Figure S1), and about 30% of them are without replicates. Therefore,hierarchical methods for borrowing information are crucial.
Splines without Shape Restrictions
In estimating dose-response curves, researchers have often forced parametric or mono-tone restrictions on shapes of the curves to increase interpretability. The EPA’s defaultmodels currently available through the tcpl package heavily depend on parametric assump-tions and are restricted to positive responses to reduce the parameter space, requiring aninverse transformation to fit negative responses. Wilson et al. (2014) model dose-responsefunctions by piecewise log-linear splines with constrained parameters to ensure responsesare monotone and non-decreasing. In the ToxCast/Tox21 data, it appears difficult tostandardize shapes of the dose-response curves (Figure 2). In addition, we observe someexamples of decreasing trends between the assay endpoint TOX21 ERa LUC BG1 Agonistand multiple chemicals. Figure 3 shows two such examples of toxic chemicals with de-8 l l ll lll l l l ll ll ll l l lll ll ll ll lll lll l l l llll l ll l lll l ll l ll ll l ll l l l ll l lll ll l l l ll l ll ll ll l lll ll ll l l ll l l ll ll l l ll l llll l l l l l l ll ll l l l l lll l l l l ll ll l −202 −3 −2 −1 0 1 2
Dose (log m M) R e s pon s e Di−n−octyl phthalate l llll lll lll l l l ll l ll ll ll l lll l lll l ll l ll ll l lll l lll l l l l l l ll llll l ll lll ll l lll l ll l l ll l ll ll l l l l l ll l l lll ll l ll ll lll l lll l l l l ll l l l lll l ll l l l l l ll l ll l −6−4−202 −3 −2 −1 0 1 2
Dose (log m M) R e s pon s e Figure 3:
Scatter plots of two chemicals on the TOX21 ERa LUC BG1 Agonist assay endpoint.The solid lines and gray shaded areas represent the average dose-response curves and 95% confi-dence intervals fitted via locally estimated scatterplot smoothing (LOESS). creasing trends. Thus, we propose a non-restricted spline model robust to any shapes ofdose-response curves, given that both upturns and downturns in dose-response functionsare suggestive of potential toxicity.
Heteroscedastic Variances
The ToxCast/Tox21 data have innate heteroscedasticity. Such heteroscedasticity is in-evitable because dose effects are variable by nature, with variability often amplified athigh doses. Differences in the ability of assays to absorb chemical doses further inflatethis variability. That is, a certain chemical can exhibit clearer effects on some assays andless obvious effects on others testing the same endpoint. Wilson et al. (2014) attemptedto reduce such heteroscedasticity by log transforming the data. However, data transfor-mations may be hard to justify theoretically (Leslie et al., 2007) and may be insufficientpractically. In genetics, multiple studies have been conducted to detect genetic loci thataffect heteroscedastic errors of quantitative traits of interest (Par´e et al., 2010; R¨onneg˚ardand Valdar, 2012; Yang et al., 2012). It is widely appreciated that analyzing differences invariance could reveal a previously unknown genetic influence and alternative biological rel-9vance. Although detection of heteroscedastic variances is routinely considered in geneticanalysis (Corty and Valdar, 2018), it has not been of main interest in chemical toxicityanalysis. Without data transformations, we consider heteroscedasticity as another sourceof information. We compute posterior probabilities of the variance effect for observed cells.
This paper uses data from the ToxCast/Tox21 project (invitroDBv3.2, released on March2019), available at https://epa.figshare.com/articles/ToxCast/Tox21 Database invitroDB for Mac Users/6062620 . We focus on a subset of the ToxCast/Tox21 data thatcontain assay endpoints relevant to neurodevelopmental disorders and obesity, along withchemicals tested over those assay endpoints. As a result of selection criteria for chemicalsand assay endpoints described in the Supplemental Material, 30 chemicals evaluated across131 assay endpoints are studied for neurodevelopmental disorders. These create in total3930 cells, from which 2024 cells (51.5%) are missing. For obesity, we use the same 30chemicals evaluated across 271 assay endpoints. Among the total 8130 cells, 3274 cells(40.3%) contain no data.
We specify motivations for each part of the model including the prior specification.
Primary interest lies in differentiating active and inactive chemicals. First, we conductmultiple hypothesis testing of whether the dose-response curve is constant or not across10hemicals and assay endpoints. We introduce latent binary indicators { γ ij } , with γ ij = 1denoting that the average dose-response curve is not constant for the (chemical i , assayendpoint j ) pair. Let vector γ i = ( γ i , . . . , γ iJ ) T represent chemical i ’s mean effect profile across the J assay endpoints for i = 1 , . . . , m . We assume that chemicals and assayendpoints explain dose effects on the mean via low rank latent features, for which weexploit a sparse Bayesian factor model (Bhattacharya and Dunson, 2011). Since each γ ij takes { , } values, we impose a generalized factor model using a probit link: P r ( γ ij = 1) = π ij = Φ( λ Ti η j ) . (1)A data-augmented form rewrites (1) as γ ij = ( z ij >
0) where z ij ∼ N ( λ Ti η j , . (2)In this factor model, λ il represents the coefficient of the l th latent pathway for the i thchemical to have the mean effect, and η lj represents that for the j th assay endpoint to havethe mean effect for l = 1 , . . . , q and q (cid:28) min( m, J ) . The inequality is reasonable, assumingthat not every assay endpoint (or chemical) forms an idiosyncratic latent pathway for themean effect. The ToxCast/Tox21 application lets either λ i be treated as factor loadingsand η j as latent factors or vice versa, depending on researchers’ interests. Provided thatone is interested in latent covariance structure among chemicals with regards to the meaneffect, a standard factor model puts a multivariate standard normal prior on latent factors η j ∼ N q ( , I ) . Integrating out η j from (2) yields z j ∼ N m ( , ΛΛ T + I ) where Λ has λ Ti asits i th row. This factor model provides a low dimensional representation of the underlyingcovariance structure of chemicals. We employ a multiplicative gamma process shrinkage11rior on factor loadings as in Bhattacharya and Dunson (2011): λ il ∼ N (0 , φ − il τ − l ) , φ il ∼ Gamma ( ν/ , ν/ , τ l = l (cid:89) h =1 ζ h , l = 1 , . . . , q, (3) ζ ∼ Gamma ( a , , ζ h ∼ Gamma ( a , , h ≥ . (4)This prior choice is supported by Judson et al. (2010) who elucidated relationships be-tween chemicals and published pathways. The authors discovered that chemicals are ac-tivating various human genes and pathways, but the number of activated pathways varieswidely across chemicals. The multiplicative gamma process shrinkage prior tends to shrinkcolumns of a loading matrix towards zero through the τ l ’s. At the same time, it is possibleto strongly shrink only a subset of elements in a certain column through local shrinkageparameters φ il ’s, retaining sparse signals.As alternatives to the above model, we considered adding another parameter controllingthe overall proportion of γ ij ’s equal to one. This can be accomplished through adding anintercept and letting P r ( γ ij = 1) = Φ( ξ + λ Ti η j ) or including an unknown mean at thelatent variable level as λ il ∼ N ( ξ, φ − il τ − l ). We found that, in both these cases, adding anunknown ξ parameter did not improve results, and indeed can lead to worse performance.This is likely due to the fact that the extra parameter is effectively redundant, leading toan over-parameterized model. Hence, we set ξ = 0 as in (1) in all the analyses we reportin this paper. Splines without Shape Restrictions
Let x ijk be a test dose (in log base 10 scale in micromolar ( µM )) of the k th measure-ment for a (chemical i , assay endpoint j ) pair, and let y ijk be the corresponding response.12onsider the model y ijk = γ ij f ij ( x ijk ) + (cid:15) ∗ ijk , where the error distribution is (cid:15) ∗ ijk ∼ N (0 , σ ∗ ijk ) for i = 1 , . . . , m , j = 1 , . . . , J , and k =1 , . . . , K ij . Non-constant dose-response curves are estimated when γ ij = 1. We model thedose-response function f ij using cubic B-splines, which is equivalent to estimating β ij in( f ij ( x ij ) , . . . , f ij ( x ijK ij )) T = X ij β ij with the B-spline basis matrix X ij of size ( K ij × p ). Wenormalize responses and center columns of the B-spline basis matrix by ( i, j ) pairs prior toany analyses in order to exclude the intercept. The prior distributions of spline coefficientsand their hyperparameters are as follows: β ij ind. ∼ N p (0 , Σ j ) , Σ − j iid ∼ W ish p ( a, R − ) , a, R fixedwhere Ω ∼ W ish p ( m, A ) is a Wishart distribution in p -dimensions with E (Ω) = mA .We suggest the following default choices for our application. As Figure 2 suggests, dose-response functions share more similarities within an assay endpoint than between differentassay endpoints. This suggests a formulation in which spline coefficients of different chem-icals have a common prior covariance matrix for the same assay endpoint. For assayendpoint-specific covariance matrices, R is determined as the empirical covariance of theordinary least squares estimates for chemical-assay endpoint pairs. The degrees of freedomparameter a is chosen to be p + 2 so that Σ j is loosely centered around R . Heteroscedastic Variances
Figure S2 illustrates that ranges of responses may vary substantially by assay end-points. This suggests modeling errors with assay endpoint-specific variances. Moreover,we are motivated to capture heteroscedasticity to explain another dimension of chem-ical activity. We use a log-linear model on σ ∗ ijk so that log σ ∗ ijk = δ j + x ijk δ ij and13 ∗ ijk = exp( δ j /
2) exp( x ijk δ ij / δ j /
2) with σ j givesthe final model equation y ijk = γ ij f ij ( x ijk ) + exp( x ijk δ ij / (cid:15) ijk , (cid:15) ijk ∼ N (0 , σ j ) . (5)The assay endpoint-specific variances have an inverse-Gamma distribution a priori :1 /σ j iid ∼ Gamma (cid:18) ν , ν σ (cid:19) , ν , σ fixed.In our application, we suggest fixing the hyperparameters ν at 1 and σ at the samplevariance of the response variable to have the prior distribution weakly centered around thevanilla estimate from data. With respect to the heteroscedastic noise, let t ij = ( δ ij (cid:54) = 0)so that t indicates activity causing variance changes of responses. Prior and hyperpriordistributions of heteroscedasticity parameters are given similarly to Leslie et al. (2007): t ij iid ∼ Bernoulli ( π t ) π t ∼ U nif (0 , δ ij | t ij = 1 iid ∼ N (0 , v δ ) , P r ( δ ij = 0 | t ij = 0) = 1 , v δ fixed.In our case, we found that ensuring a large enough value for v δ that appropriately coversthe data range improves estimation of δ ij and t ij . Provided that conditional standarddeviations of responses given doses can be proxies for exp( x ijk δ ij /
2) in (5), a range of δ ij ’sis obtained. The variance parameter v δ of δ ij is then determined as the square of the rangedivided by 4, which makes ± δ ij cover its sample range.We finally fix v δ at the maximum of the above value and the sample variance of the responsevariable. Combined with σ , this allows the prior distributions of two variance parts - theassay endpoint-specific and the heteroscedastic variance - to place enough probability onthe observed variability from data. 14n conclusion, (5) is the final model where γ ij is an indicator specifying whether the i th chemical activates the j th assay endpoint in the mean, f ij is a dose-response function,the exponential term allows for heteroscedastic noise, and error is modeled with normaldistributions having assay endpoint-specific variances. Our posterior samples are obtained using a partially collapsed Gibbs sampler and Metropolis-Hastings steps. Most of the parameters have conjugate posterior distributions which leadto a straightforward Gibbs sampler. Details are provided in the Supplemental Materialwith code at https://github.com/jinbora0720/BMC . Simulation studies were conducted to evaluate the performance of BMC in learning thelatent correlation structure among chemicals, predicting the mean effect probabilities, andestimating the parameters. Two broad scenarios of simulations were examined correspond-ing to data simulated from BMC (Simulation 1) or an alternative (Simulation 2). Forpredictive performance, BMC was compared to three variations in the prior structure of γ ij . Instead of a latent factor model, we assume simpler structures a priori as follows: P r ( γ ij = 1) = π ∀ i, j, and π ∼ Beta (1 , P r ( γ ij = 1) = π i ∀ j, and π i iid ∼ Beta (1 , ∀ i ; (7) P r ( γ ij = 1) = π j ∀ i, and π j iid ∼ Beta (1 , ∀ j. (8)15e call models with (6), (7), (8) BMC , BMC i , and BMC j , respectively. BMC i assumesthat each chemical has its own intrinsic mean effect probability, while BMC j assumes thateach assay endpoint has its own mean effect probability. For estimation performance,the proposed model is compared to the zero-inflated piecewise log-logistic model (ZIPLL)(Wilson et al., 2014) and tcpl (Filer et al., 2017). The ZIPLL code at https://github.com/AnderWilson/ZIPLL utilizes a Bayesian hierarchical approach whose testing frameworkfor the mean effect adopts (6). Since the code does not allow missing pairs in the data, weonly use ZIPLL for estimation and not prediction. The tcpl models are currently used byEPA and treat dose-response curves independently.In Simulation 1 in which BMC is the true data generating process, mimicking theToxCast/Tox21 application, the number of chemicals m was set to 30, and the number ofassay endpoints J to 150. We generate 50 data sets, and in each set we hold out 135 pairs atrandom, which are 3% of the total cells in the data matrix. The profiles of the mean effectfor chemical-assay endpoint pairs were sampled assuming a factor model, which induceda correlation structure among chemicals (Figure 4). For pairs having dose effects on themean, dose-response functions were given as one of the three categories: mostly increasingand decreasing at higher doses; monotonically increasing; and decreasing. Figure 7 presentsexamples of dose-response functions of each category. Heteroscedasticity is expected at onefifth of chemical-assay endpoint combinations. More specific settings of Simulation 1 aredescribed in the Supplemental Material.Figure 4 illustrates that the latent correlations relevant to the mean effect among chem-icals are accurately estimated. Two factors ( q = 2) generated the truth, and the samplerran with a guess of three more factors. The multiplicative gamma process shrinkage priorhelped recover the true number of factors q = 2 by shrinking factor loadings of redundantfactors to zero (Figure 5). Figure 6 displays an example of mean effect profiles. The truth16 hemical C he m i c a l Estimate
Chemical C he m i c a l −1.0−0.50.00.51.0 Correlation
Truth
Figure 4:
Heat map of the estimated and true correlation matrix among chemicals with respectto the mean effect. The results are from Simulation 1.
Factor C he m i c a l −202Loading Estimated L Factor C he m i c a l −505Loading True L Figure 5:
The estimated and true entries of loading matrix Λ from Simulation 1. The estimatedΛ is rotated for better visualization. is adequately captured via the estimated and predicted probabilities. Results from a 5 × C he m i c a l Estimate
Assay endpoint C he m i c a l g ij Truth
Figure 6:
Heat map of estimated and true profiles of the mean effect from Simulation 1. Figurepresents the results from a 5 × γ ij . Cells with outer lines ((3,3) and (2,5) elements) are hold-out pairsfor which γ ij ’s are predicted. It also produces tighter 95% credible intervals for the average dose-response curves thancompetitors. The competitors, ZIPLL and tcpl models, do not seem robust enough tovarious dose-response curves. In particular, ZIPLL estimates a decreasing trend as constant,which is evident in Figure 7 C . For generally increasing curves ( A, B ), the ZIPLL and tcplmodels sometimes miss the true dose-response functions, which becomes more noticeablewhen heteroscedasticity exists. These results suggest that in some cases, BMC can lead tomore precise inferences on values estimated through dose-response curves, such as Emax(greatest attainable response) or AC50 (chemical dose producing half maximal response inan assay endpoint).Table 1 summarizes simulation results when the data generating process is BMC. Onlythe results from BMC j are presented because it showed slightly improved performance overthe other two. Note that Area Under the ROC Curves (AUCs) from tcpl in Table 1 & S1were computed slightly differently than those from other methods. BMC, three variations,18 Dose (log m M) R e s pon s e Truth: (Pr(Mean Effect), Pr(Var Effect)) = ( 1 , 1 ) BMC: ( 1 , 1 ) ZIPLL: ( 0.982 , NA ) tcpl: ( 1 , NA ) A −2024 0.5 1.0 1.5 2.0 Dose (log m M) R e s pon s e Truth: (Pr(Mean Effect), Pr(Var Effect)) = ( 1 , 1 ) BMC: ( 1 , 1 ) ZIPLL: ( 0.89 , NA ) tcpl: ( 1 , NA ) B −2−101 0.5 1.0 1.5 2.0 Dose (log m M) R e s pon s e Truth: (Pr(Mean Effect), Pr(Var Effect)) = ( 1 , 0 ) BMC: ( 1 , 0.099 ) ZIPLL: ( 0 , NA ) tcpl: ( 1 , NA ) C Model BMC tcpl ZIPLL
Figure 7:
Dose-response curves (solid line) with fitted mean functions by BMC (long dash line),ZIPLL (dot-dashed line), and tcpl (dashed line) in Simulation 1. The true curve is mostly in-creasing and decreasing at higher dose in A , monotonically increasing in B , and monotonicallydecreasing in C . Darker gray areas around estimated functions represent 95% credible intervalsfor the average dose-response curves computed by BMC and ZIPLL, and confidence intervals bytcpl. Lighter gray areas illustrate 95% predictive intervals from BMC for data points. and ZIPLL all produce probability of active responses, which can be any value between 0and 1. In order to evaluate the accuracy of estimates compared to the true γ ij ∈ { , } values, ROC curves and the corresponding AUCs are computed by changing thresholdsbetween 0 and 1. On the other hand, EPA provides a binary hit-call variable for the meaneffect through ToxCast/Tox21 based on invitroDBv2. We hereafter refer to this variableas EPA’s hit-call. The EPA’s hit-call identifies a pair as active if the fitted Hill or gain-loss model have lower Akaike information criterion than a constant model, and both theestimated and observed maximum responses exceed a cutoff chosen for the assay endpoint.This classification of whether each pair is active or not is directly comparable to the true γ ij without changing thresholds. In simulations, assay endpoint-specific cutoffs are set to19able 1: Summary of results from Simulation 1. Root Mean Squared Error (RMSE), Area Underthe ROC curve (AUC) results for probability of the mean effect, and AUC for probability ofthe variance effect are presented. The displayed values are the mean (standard error) across 50simulations.
BMC BMC j ZIPLL tcplRMSE 0.402 (0.018) 0.397 (0.017) 0.820 (0.034) 0.645 (0.017)In-sample AUC for γ ij γ ij t ij j ) have lower RMSEs and higher AUCs compared totcpl or ZIPLL. Poor performance of ZIPLL in these simulations is partially due to thefacts that monotone increasing shape restrictions fail to fit decreasing trends and thatZIPLL does not allow for different σ j ’s. BMC outperforming tcpl may be due to theborrowing of information across chemicals and assay endpoints. Moreover, the originalBMC model produced in- and out-of-sample AUCs that are uniformly better than thosefrom BMC j . Hence, when the factor model provides a realistic characterization of thedependence structure across assay endpoints and chemicals, it is not suggested to use asimplified model for multiple testing. Less structure in γ ij results in lower out-of-sampleAUCs for γ ij (BMC : 0 .
504 (0 . i : 0 .
486 (0 . does not utilizechemical- or assay endpoint-specific information in prediction.Another benefit of BMC is the capability of modeling heteroscedasticity. The AUCs20or t ij in Table 1 exhibit highly accurate estimation performance for the probability ofheteroscedastic variances. Figure 7 illustrates that BMC closely recovers the true curveseven in the existence of heteroscedasticity ( A and B ) and that it nicely differentiates variancechanges and mean changes - for instance, the estimated probability of the variance effectis around 0.1, while the probability of the mean effect is 1 in C . Figure 8 shows residualsversus fitted values for the heteroscedastic pairs in Figure 7. The ZIPLL does not considerheteroscedasticity in the model and consequently results in heteroscedastic residuals. Incontrast, BMC is able to properly account for heteroscedasticity, and residuals do notshow any patterns against fitted values. Note that in Figure 8 residuals from ZIPLLare obtained by subtracting the fitted values from observations, while those from BMCare the posterior mean of “normalized” residuals whose value at s th iteration is ( y ijk − γ ( s ) ij f ( s ) ij ( x ijk )) / exp( x ijk δ ( s ) ij / −2−101 −0.5 0.0 0.5 1.0 Fitted value R e s i dua l A −1.0−0.50.00.51.0 −1.0 −0.5 0.0 0.5 1.0 Fitted value R e s i dua l B Model BMC ZIPLL
Figure 8:
Residuals versus fitted values using BMC and ZIPLL in Simulation 1. The (normalized)residuals and fitted values in A and B are computed using observations and fitted lines from A and B in Figure 7, respectively. Simulation 2 generates data from an alternative model, ZIPLL. Despite misalignmentin data structure assumed by BMC and by ZIPLL, BMC performs similarly to ZIPLL and21utperforms tcpl with respect to RMSE and AUC. The high in-sample AUC for γ ij fromBMC (0.982) suggests its stable estimation performance even with relatively small numberof chemicals and assay endpoints. We provide a full discussion of Simulation 2 results inthe Supplemental Material. This section presents results from the ToxCast/Tox21 data analysis with a focus on neu-rodevelopmental disorders and obesity. We ran the sampler for 40,000 iterations from which30,000 were discarded as burn-in, and every 10th sample was saved for the next 10,000 it-erations. This long burn-in is to be conservative; trace plots and effective sample sizes forMCMC samples indicated good mixing and apparent convergence after 15,000 iterations. −2−10123 −1 0 1
Dose (log m M) R e s pon s e EPA hit−call = 1 Pr(Mean Effect) = 1 Pr(Var Effect) = 0.015
Bisphenol A,NVS_TR_hDAT −2024 −3 −2 −1 0 1 2
Dose (log m M) R e s pon s e EPA hit−call = 1 Pr(Mean Effect) = 0.256 Pr(Var Effect) = 1 p,p'−DDE,TOX21_ERR_Antagonist
Figure 9:
Results for select chemical-assay endpoint pairs estimated by BMC to be active, havingdose effects on the mean (left) or variance (right).
Figures 9 - 11 show estimated dose-response curves from BMC as dashed lines with95% credible intervals as shaded areas with darker gray. The lighter gray shaded areas22llustrate 95% predictive intervals for the data points drawn as dots. “Pr(Mean Effect)” isthe mean effect probability for a (chemical i , assay endpoint j ) pair, which is computed asthe posterior mean of γ ij . Similarly, “Pr(Var Effect)” means the variance effect probabilitywhose value is the posterior mean of t ij . The posterior mean of π t is 0.12, meaning thatabout 12% of the observed cells have heteroscedastic variances.Figure 9 shows that BMC is able to differentiate dose effects on the mean from thoseon the variance of dose-response curves. Recall that the EPA’s hit-call is an indicationof mean changes. In the left panel, BMC and the EPA agree that mean changes exist,which is supported by an increasing trend. In the right panel, the EPA’s hit-call claimsthat the average dose-response is not constant. However, BMC estimates that the meancurve is constant at zero with probability 0.75, but with there being clear evidence ofheteroscedasticity. Therefore, Figure 9 suggests that (1) the EPA’s hit-call for the meaneffect might be misled by heteroscedastic variances; and (2) BMC can separate mean andvariance effects (at least in some cases). −2−10123 −1 0 1 2 Dose (log m M) R e s pon s e EPA hit−call = 0 Pr(Mean Effect) = 1 Pr(Var Effect) = 0.137
Dibutyl phthalate,ATG_PPARg_TRANS_up −202 −2 −1 0 1 2
Dose (log m M) R e s pon s e EPA hit−call = 0 Pr(Mean Effect) = 1 Pr(Var Effect) = 1
Bisphenol A,ATG_PPARg_TRANS_up
Figure 10:
Results for select chemical-assay endpoint pairs estimated by BMC to be active,having dose effects on the mean. γ receptors are potential obesogens because PPAR γ is a master regulator in formulating fatcells (Evans et al., 2004). Therefore, it is not unexpected for BPA and Dibutyl phthalateto be active for the assay endpoint, ATG PPAR γ TRANS up. Several pieces of evidencereinforce the validity of results from BMC over the EPA’s hit-call. −202 −3 −2 −1 0 1 2
Dose (log m M) R e s pon s e EPA hit−call = 0 Pr(Mean Effect) = 0.15 Pr(Var Effect) = 1
Diisobutyl phthalate,TOX21_CAR_Antagonist −2024 −3 −2 −1 0 1 2
Dose (log m M) R e s pon s e EPA hit−call = 0 Pr(Mean Effect) = 0.037 Pr(Var Effect) = 1
Di(2−ethylhexyl) phthalate,TOX21_Aromatase_Inhibition
Figure 11:
Results for select chemical-assay endpoint pairs estimated by BMC to be active,having dose effects on the variance.
Figure 11 shows cases where the EPA’s hit-call can have low power because it missessignals manifest in the variance instead of the mean. Given that phthalates are related toobesity, we expect disruptive patterns on assay endpoints presenting toxicity of Diisobutyl24
Dose (log m M) R e s pon s e Predicted Pr(Mean Effect) = 0.831 p,p'−DDE,BSK_SAg_CD38_down
01 0.0 0.5 1.0 1.5
Dose (log m M) R e s pon s e Predicted Pr(Mean Effect) = 0.722 p,p'−DDE,BSK_3C_IL8_down
Figure 12:
Results for select chemical-assay endpoint pairs with seemingly identical measurementsbut predicted by BMC to have different probabilities of the mean effect. phthatlate and Di(2-ethylhexyl) phthatlate. However, the EPA’s hit-call suggests thatthese phthalates are not active at the doses tested. This may be true in terms of meanchanges, but variances seem clearly heteroscedastic.Figures 12 and S3 show predicted results for hold-out pairs. Note that we do notpredict observations or the average dose-response curves. BMC only predicts the meaneffect probability, which is often the primary focus of many studies and helps researchersto prioritize chemicals for further testing.Figure 12 displays advantages of BMC’s probabilistic approach to evaluate dose effectson the mean. The left and right panels exhibit seemingly identical dose-response resultsof the same chemical and different assay endpoints. These pairs have different mean ef-fect probabilities that reflect different assay endpoint effects. The chemical p,p’-DDE ispredicted to cause mean changes in dose-response curves with average probability of 0.808across assay endpoints. Across chemicals, the assay endpoint BSK SAg CD38 down (left)is more likely to have the mean effect with the average posterior probability 0.874 than25SK 3C IL8 down (right) with 0.694. Hence, the predicted probabilities for the mean ef-fect can be thought to be pulled towards the probability of each assay endpoint from thechemical’s probability. This implies that BMC appropriately addresses chemical and assayendpoint effects in the mean effect probabilities through the latent factor model. On theother hand, the EPA’s hit-call is 1 for both cases. Their deterministic approach mightnot always be informative when researchers attempt to arrange chemicals by evidence oftoxicity. When the researchers are more informed by the probabilities, however, they caneasily prioritize chemicals - even choosing among those with the same hit-call.We observed seemingly active pairs with mean changes can have a wide range of meaneffect probabilities. (Refer to Figure S3 for some relatively active pairs.) In fact, we foundthat in the ToxCast/Tox21 application, 95% highest density intervals for the estimatedand predicted γ ij ’s are (0.205, 1) and (0.510, 0.863), respectively. These suggest the lackof conclusive evidence of inactivity in most cases, while the EPA’s hit-call forces chemical-assay endpoint pairs to be classified as active or inactive. The EPA’s hit-call may tend toflag too many pairs as inactive.As discussed so far, it is valuable to assess toxicity of chemicals based on their activityprobability, which could be computed as P r ( γ ij = 1 ∪ t ij = 1) = 1 − P r ( γ ij = 0 ∩ t ij = 0).Figure 13 shows chemicals by the order of average activity probability over obesity-relatedassay endpoints. Top chemicals that are most likely to disrupt biological processes asso-ciated with obesity include p,p’-DDE, Dichlorodiphenyltrichloroethane (DDT), Triclosan,BPA, 2,4,5-Trichlorophenol, Chlorpyrifos, and Benzyl butyl phthalate. The rankings ofchemicals by BMC and by the EPA’s hit-call show subtle differences.Chemicals are similarly ranked by the average probability over assay endpoints relatedto neurodevelopmental disorders (Figure S4). Top chemicals that are most likely to disruptneurodevelopmental processes include Triclosan, DDT, 2,4,5-Trichlorophenol, p,p’-DDE,26 .00.20.40.60.8 D i e t h y l ph t ha l a t e D i m e t h y l pho s pha t e1 , − D i c h l o r oben z ene M onobu t y l ph t ha l a t e O , O − D i e t h y l d i t h i opho s pha t e2 , − D i c h l o r opheno l D e l t a m e t h r i n2 , ' , , ' , , ' − H e x a c h l o r ob i phen y l , − D i c h l o r opheno l , , − T r i c h l o r opheno l D i − n − o c t y l ph t ha l a t e M onoben zy l ph t ha l a t e3 , , − T r i c h l o r o − − p y r i d i no l D i ( − e t h y l he xy l ) ph t ha l a t e C y pe r m e t h r i n P e r m e t h r i n D ii s obu t y l ph t ha l a t e M a l a t h i on M E H P F enp r opa t h r i n D i bu t y l ph t ha l a t e C y f l u t h r i n2 − H y d r o xy − − m e t ho xy ben z ophenone B en zy l bu t y l ph t ha l a t e C h l o r p y r i f o s , , − T r i c h l o r opheno l B i s pheno l A T r i c l o s an D i c h l o r od i phen y l t r i c h l o r oe t hanep , p ' − DD E Chemical A c t i v e p r obab ili t y BMCEPA
Obesity
Figure 13:
Chemical ranks by the average activity probability from BMC (dots) and the averagehit-call from EPA (triangles) over obesity-related assay endpoints.
Fenpropathrin, 2-Hydroxy-4-methoxybenzophenone, and BPA. Between the two sets ofthe most active chemicals associated with neurodevelopmental disorders and obesity, fivechemicals - Triclosan, BPA, 2,4,5-Trichlorophenol, DDT, and p,p’-DDE - overlap, and wecall them the top 5 chemicals. These bioactivity rankings are based on the data that arecurrently available. As data expand, it will be informative to revisit such rankings.Figure 14 provides a list of assay endpoints of relevance to neurodevelopmental disorders,which are highly likely ( > .
9) to be “activated” by the top 5 chemicals. The list includesboth agonist and antagonist assays, and thus the “activated” probability encompassesagonist and antagonist directions in the mean effect. The assay endpoints in the list areexpected to have important implications in disease progression, from which thirty-one assay27 .9000.9250.9500.9751.000
BSK _h D F C G F _ PA I BSK _ K F C T _u PA _up T O X A R _L UC _ M D AKB A n t agon i s t _ S pe c i f i c i t y T O X G R _ B L A _ A gon i s t _ r a t i o BSK _h D F C G F _ PA I w n N VS _ NR _h P RN VS _ I C _ r N a C h_ s i t e2 N VS _ T R _h D A T N VS _ E N Z _ r M A O B C _ A c t i v a t o r O T _ E R _ E R a E R a_0480 BSK _3 C _u PA R _up BSK _3 C _u PA R _do w n N VS _ NR _ r A R T O X SS H _3 T G L I A gon i s t C EE T O X _ H R _ O H P R OG _dn C EE T O X _ H R _ O H P R OG _up N VS _ G P CR _ r V A T G _ P a x C I S _dn A T G _ P a x C I S _up N VS _ G P CR _p5 H T C A T G _ E R a_ T R A N S _up N VS _ G P CR _h5 H T N VS _ NR _h A RN VS _ T R _h N E T A T G _ M R E _ C I S _dn T O X G R _ B L A _ A n t agon i s t _ r a t i o A T G _ E R E _ C I S _up A T G _ M R E _ C I S _up T O X SS H _3 T G L I A n t agon i s t T O X E R a_L UC _ B G A n t agon i s t A C EA _ A R _an t agon i s t _80h r N VS _L G I C _ r G l u N M D A _ A gon i s t N VS _ T R _ r N E T O T _ A R _ A R S RC O T _ A R _ A R S RC O T _ E R _ E R a E R a_1440 O T _ E R a_ E R E G F P _0120 O T _ E R a_ E R E G F P _0480 T O X A R _ B L A _ A gon i s t _ r a t i o T O X A R _ B L A _ A n t agon i s t _ r a t i o T O X A R _L UC _ M D AKB A gon i s t _ C oun t e r sc r een T O X A R _L UC _ M D AKB A n t agon i s t T O X E R a_ B L A _ A gon i s t _ r a t i o T O X E R a_L UC _ B G A gon i s t Assay endpoint A c t i v a t ed p r obab ili t y Involved in both diseases FALSE TRUE
Neurodevelopmental Disorders
Figure 14:
Ranks of assay endpoints associated with neurodevelopmental disorders in termsof probabilities to be activated by the top 5 chemicals. Only a subset of assay endpoints haveactivation probabilities higher than 0.9. The assay endpoints with dots are marked uniquely forneurodevelopmental disorders, while those with triangles are marked for two disease classes. endpoints show impacts on both disease classes. The same list for the obesity-related assayendpoints is provided in the Supplemental Material (Figure S5).To study sensitivity of rankings to the choice of chemicals, we expanded our analysis to326 chemicals. They consist of the original 30 chemicals and those screened in Phase I ofthe ToxCast that have been exclusively used in other toxicity studies including Martin et al.(2010) and Wilson et al. (2014). Within this larger collection, relative positions of the 30chemicals remained intact with a few exceptions. BPA and Triclosan were positioned lowerin the larger set, while Cyfluthrin and MEHP were positioned higher. One of explanationsfor these shifts is an altered correlation structure among chemicals. The Phase I chemicals28re mostly pesticides, and the four chemicals might have different relationship with thosefrom what they had with the 30 chemicals in terms of the mean effect.
We have proposed a Bayesian multiple testing approach for inference on activity of chemi-cals in settings involving multiple chemicals and assay endpoints and possible heteroscedas-ticity. Our BMC approach can be applied directly in other settings involving a similarmatrix-structured experimental design. For example, this is common in pharmaceuticalstudies assessing drug activity - studies will look for evidence of activity for different healthoutcomes. Also, in microbial genetics, similar designs are conducted but for different typesof bacteria and environmental conditions.The ultimate goal of many analyses using in vitro data is to make inferences on hu-man health and inform protective regulations. Accordingly, chemicals and assay endpointsstudied in the ToxCast/Tox21 application are carefully selected: the chemicals are also mea-sured in human epidemiology studies, and the assay endpoints cover a variety of speciesand several types of tissue targets. It will be interesting to follow up on the top rankingchemicals for neurodevelopmental disorders and obesity outcomes identified in our analysesto further elucidate their role in human disease risk.When extending in vitro results to in vivo toxicity, doses need to be carefully considered.All the results presented in the paper should be interpreted in terms of tested doses, so wedo not conclude a chemical with a high probability of inactivity is inactive at higher dosesthan those tested. Simultaneously, it is recommended to ensure that the doses tested invitro can physiologically occur in animals/humans. This recommendation is reinforced byKlaren et al. (2019) in which in vivo toxicity prediction using in vitro assays performs much29etter with toxicokinetic modeling. Therefore, future research linking in vitro data and invivo implications could be greatly assisted by assuring dose applicability in animals/humansas well as widening the range of tested doses.
We are grateful for the financial support of the National Institute of Environmental HealthSciences through grants R01ES027498 and R01ES028804. The authors would like to thankBrett Winters for identifying assay endpoints relevant to neurodevelopmental disorders andobesity, Evan Poworoznek for sharing computer code, Kelly Moran for helpful comments,and Matthew Wheeler for help in processing ToxCast/Tox21 data.
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S1 Figures
Number of unique doses C oun t o f c he m i c a l − a ss a y endpo i n t pa i r s Figure S1:
Histogram of the number of unique doses tested for each chemical-assay endpointpair in the ToxCast/Tox21 data related to neurodevelopmental disorders. The vertical solid lineis the median. Half the pairs have the number of tested doses less than or equal to 8. llllllll llllllll llllllllllllllllllllllllllllllllllllll llllllll llllllllllllllll llllllllllllllll llllllllllllllllllllllll llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll llllllllllllllllllllllllllllllll llllllllllllllllllllll llllllllllllllllllllllll llllllll llllllllllllllll llllllllllllllllllllllllllllllllllllllllllllllll llllllll llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll 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lllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll 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−1001020
Assay endpoint R e s pon s e Figure S2:
Scatter plot of the responses normalized by chemical-assay endpoint pairs. Points oneach vertical line represent responses from one assay endpoint. This figure is based on a subsetof assay endpoints. Dose (log m M) R e s pon s e Predicted Pr(Mean Effect) = 0.668
Dichlorodiphenyltrichloroethane,ATG_IR1_CIS_dn −2−101 −1 0 1 2
Dose (log m M) R e s pon s e Predicted Pr(Mean Effect) = 0.85
Benzyl butyl phthalate,ATG_PPARg_TRANS_dn −2−1012 −3 −2 −1 0 1 2
Dose (log m M) R e s pon s e Predicted Pr(Mean Effect) = 0.78
Bisphenol A,TOX21_ERb_BLA_Antagonist_ratio
036 −3 −2 −1 0 1 2
Dose (log m M) R e s pon s e Predicted Pr(Mean Effect) = 0.641
Di(2−ethylhexyl) phthalate,TOX21_FXR_BLA_antagonist_ratio
Figure S3:
Results for select chemical-assay endpoint pairs predicted by BMC to likely have doseeffects on the mean. .00.20.40.60.8 D i e t h y l ph t ha l a t e D i m e t h y l pho s pha t e M onobu t y l ph t ha l a t e1 , − D i c h l o r oben z ene2 , − D i c h l o r opheno l , ' , , ' , , ' − H e x a c h l o r ob i phen y l O , O − D i e t h y l d i t h i opho s pha t e D i − n − o c t y l ph t ha l a t e2 , − D i c h l o r opheno l D e l t a m e t h r i n D i bu t y l ph t ha l a t e M onoben zy l ph t ha l a t e M E H P , , − T r i c h l o r opheno l , , − T r i c h l o r o − − p y r i d i no l D ii s obu t y l ph t ha l a t e C y f l u t h r i n C y pe r m e t h r i n P e r m e t h r i n D i ( − e t h y l he xy l ) ph t ha l a t e M a l a t h i on B en zy l bu t y l ph t ha l a t e C h l o r p y r i f o s B i s pheno l A − H y d r o xy − − m e t ho xy ben z ophenone F enp r opa t h r i np , p ' − DD E , , − T r i c h l o r opheno l D i c h l o r od i phen y l t r i c h l o r oe t hane T r i c l o s an Chemical A c t i v e p r obab ili t y BMCEPA
Neurodevelopmental Disorders
Figure S4:
Chemical ranks by the average active probability from BMC (dots) and the averagehit-call from EPA (triangles) over assay endpoints related to neurodevelopmental disorders. . . . . . BSK _h D F C G F _ I L8_up A T G _ M yc _ C I S _up BSK _3 C _ M C P w n N VS _ E N Z _h ES BSK _ BE C _ PA I w n BSK _ K F C T _ I C A M BSK _ K F C T _ T G F b1_do w n A T G _ PB R E M _ C I S _up T O X A h R _L UC _ A gon i s t N VS _ T R _h D A T BSK _h D F C G F _ I L8_do w n N VS _ T R _ r V M A T BSK _L PS _ I L8_do w n BSK _ BE C _ I P w n BSK _ K F C T _ I C A M w n BSK _ SA g_ M C P w n BSK _3 C _ M C P A T G _ PB R E M _ C I S _dn BSK _ BE C _ I P BSK _ K F C T _ I P w n A T G _ C _ EBP _ C I S _dn A T G _ C _ EBP _ C I S _up N VS _ G P CR _h5 H T BSK _L PS _ M C P w n T O X HR E _ B L A _ A gon i s t _ r a t i o BSK _ K F C T _ MM P BSK _h D F C G F _ PA I w n BSK _L PS _ T N F a_do w n BSK _3 C _ I C A M BSK _L PS _ M C P BSK _ K F C T _ MM P w n BSK _ SA g_ I L8_do w n A T G _ S R EBP _ C I S _dn A T G _ PX R _ T R A N S _up BSK _h D F C G F _ PA I BSK _3 C _ I C A M w n A T G _ A h r _ C I S _up A T G _ PX R _ T R A N S _dn A T G _ PPA R g_ T R A N S _dn N VS _ NR _ r A R A T G _ S R EBP _ C I S _up A T G _ A h r _ C I S _dn O T _ E R _ E R a E R a_0480 N VS _ G P CR _h O p i a t e_ D N VS _ G P CR _p5 H T C N VS _ E N Z _h S I R T A c t i v a t o r A T G _ PPA R g_ T R A N S _up BSK _ K F C T _ I P BSK _4 H _ M C P w n T O X PPA R g_ B L A _ A gon i s t _ r a t i o BSK _h D F C G F _ I P w n N VS _ E N Z _ r M A O B C N VS _ NR _h A R N VS _ T R _ r N E T BSK _h D F C G F _ I P BSK _ SA g_ I L8_up A T G _ P a x C I S _dn BSK _4 H _ M C P N VS _ E N Z _ r M A O B C _ A c t i v a t o r N VS _ G P CR _ r V BSK _ SA g_ CD N VS _ T R _h N E T BSK _L PS _ T N F a_up BSK _ SA g_ CD w n A T G _ E R a_ T R A N S _up T O X PPA R g_ B L A _an t agon i s t _ r a t i o A T G _ P a x C I S _up O T _ E R _ E R a E R a_1440 O T _ PPA R g_ PPA R g S RC T O X G R _ B L A _ A n t agon i s t _ r a t i o T O X E R b_ B L A _ A gon i s t _ r a t i o A T G _ E R E _ C I S _up A C EA _ A R _an t agon i s t _80h r A T G _ NR F A R E _ C I S _dn A T G _ NR F A R E _ C I S _up A T G _ PX R E _ C I S _dn A T G _ PX R E _ C I S _up A T G _ V DR E _ C I S _dn A T G _ V DR E _ C I S _up N VS _ E N Z _h S I R T N VS _ G P CR _ r O p i a t e_ N on S e l e c t i v e N a N VS _L G I C _ r G l u N M D A _ A gon i s t N VS _ M P _h PB R N VS _ M P _ r PB R O T _ A R _ A R S RC O T _ A R _ A R S RC O T _ E R _ E R b E R b_0480 O T _ E R _ E R b E R b_1440 O T _ E R a_ E R E G F P _0120 O T _ E R a_ E R E G F P _0480 O T _ F X R _ F X R S RC O T _ F X R _ F X R S RC O T _ PPA R g_ PPA R g S RC T O X A R _ B L A _ A gon i s t _ r a t i o T O X A R _ B L A _ A n t agon i s t _ r a t i o T O X A R _L UC _ M D AKB A gon i s t _ C oun t e r sc r een T O X A R _L UC _ M D AKB A n t agon i s t T O X A R E _ B L A _agon i s t _ r a t i o T O X C A R _ A gon i s t T O X C A R _ A n t agon i s t T O X E R a_ B L A _ A gon i s t _ r a t i o T O X E R a_L UC _ B G A gon i s t T O X E R a_L UC _ B G A n t agon i s t T O X E R b_ B L A _ A n t agon i s t _ r a t i o T O X E RR _ A gon i s t T O X E RR _ A n t agon i s t T O X F X R _ B L A _an t agon i s t _ r a t i o T O X P G C _ E RR _ A gon i s t T O X P G C _ E RR _ A n t agon i s t T O X R O R g_L UC _ CH O _ A n t agon i s t A ss a y endpo i n t Activated probability I n v o l v ed i n bo t h d i s ea s e s F A L SE T RU E O be s i t y Figure S5:
Ranks of obesity-related assay endpoints in terms of probabilities to be activated bythe top 5 chemicals. Only a subset of assay endpoints are presented with the activated probabilitieshigher than 0.9. The assay endpoints with dots are marked uniquely for obesity, while those withtriangles are marked for both diseases. This section explains our selection criteria for chemicals and assay endpoints related to neu-rodevelopmental disorders and obesity in the ToxCast/Tox21 data. The exact procedure tofind assay endpoints of interest is as follows: First, molecules are identified that have knownassociations with each disease through the Comparative Toxicogenomics Database (CTD)(Davis et al., 2019) and Ingenuity ® PathwayAnalysis (IPA) Knowledgebase (QIAGEN Inc., ).Here, the known associations include: molecules are biomarkers of the disease; are knownto play a role in the disease etiology; and are therapeutic targets for treatment of the dis-ease. The databases CTD and IPA maintain curated and published associations betweenmolecules and diseases. It is noteworthy that they originated from a variety of speciesand tissue targets. In later steps, the databases are compared to the ToxCast/Tox21 data,whose assay endpoints were derived from different species. Moreover, the assay endpointsin the ToxCast/Tox21 program were tested across several types of tissue targets that maydifferently mediate the relationship of even the same molecular target and the same assayendpoint. Therefore, it is important to ensure that a wide variety of species and tissue tar-gets are well represented in both ToxCast/Tox21 and the databases from which moleculartargets are identified. Second, we filter molecular targets in ToxCast/Tox21 that over-lap with the identified molecules from CTD and IPA. Third, we choose assay endpointsthat those overlapping molecular targets are screened over. As a result of these steps, 132and 352 assay endpoints were identified as relevant to neurodevelopmental disorders andobesity, respectively, which were further filtered based on chemical coverage, as detailedbelow.A partial list of chemicals is considered due to a particular interest in human data.We featured a set of overlapping chemicals measured in ToxCast/Tox21 and an existingobservational study of environmental risk factors for neurodevelopmental disorders andobesity. In doing so, we believe future application of the ToxCast/Tox21 results to humanswill be more viable. A total of 48 chemicals were selected, 30 of which were tested within theabove mentioned list of assay endpoints. Due to the reduced list of chemicals, the numberof assay endpoints has diminished as well. Following recommended practice (Judson et al.,2016), we retained only the doses lower than a cytotoxicity point for each chemical, whichremoved two percent of the data. We employed the cytotoxicity median values stored ina variable “cyto pt um” in the tcpl package (Filer et al., 2017). Consequently, our finaldata involve 30 chemicals and 131 and 271 assay endpoints related to neurodevelopmentaldisorders and obesity, respectively. 5
For all simulations, we used eight unique doses { } in log µM chosen based on the frequency of appearance in the Tox-Cast/Tox21 data. B-spline knots are set at the minimum value, three quartiles, and themaximum of the doses.In Simulation 1, we generated 50 data sets with K ij = 24 at each combination ofchemical and assay endpoint, which represents 3 replicates at each of the eight doses.Elements of η were drawn independently from the standard normal distribution. Elementsin the m × q matrix Λ were sampled as in (3) and (4) with ν = 3, a = 2 .
1, and a = 3 . /σ j ∼ Gamma ( , × . ). Heteroscedasticity is expected at one fifth of chemical-assayendpoint combinations by setting π t = 0 .
2. For heteroscedastic pairs, d ij was sampled from N (2 , . ), which gives roughly exp( x ijk ) (cid:15) ijk for the error term. For BMC and ZIPLL,20,000 samples were drawn, of which 1,000 samples were saved and analyzed. First 10,000samples were discarded as burn-in, and every 10th sample was retained for the next 10,000samples. Trace plots and effective sample sizes for MCMC samples suggested convergenceand good mixing.In Simulation 2 where misalignment exists between the true data generating processand BMC, we generated data under the ZIPLL model. The number of chemicals m was setto 15, and the number of assay endpoints J to 15. The bottom right 3 × γ ij . We generated 50 data sets with K ij = 8so that each chemical-assay endpoint pair has one observation at each of the eight doses.No replicates at such scarce doses make it impractical to evaluate heteroscedasticity, whichis consequently not considered in Simulation 2. The model y ijk = γ ij f ij ( x ijk ) + (cid:15) ijk , (cid:15) ijk ∼ N (0 , . )was considered where around half the pairs were randomly assigned to have the mean effectwith γ ij ∼ Bernoulli (0 . f ij ( x ijk ) = t ij − t ij − b ij { w ij (log x ijk − log a ij ) } where t ij ∼ U nif (0 , b ij = 0, a ij = max ( x ijk ), and w ij ∼ U nif (1 , . The RMSEand AUC results are summarized in Table S1. Only the results from BMC are presentedbecause ZIPLL adopts BMC in its γ ij testing framework.6able S1: Summary of results from Simulation 2. The RMSEs and AUC results of the meaneffect probabilities are presented. The displayed values are the mean (standard error) across 50simulated data sets.
BMC BMC ZIPLL tcplRMSE 0.094 (0.002) 0.094 (0.002) 0.088 (0.002) 0.282 (0.017)In-sample AUC for γ ij γ ij γ ij even when ZIPLL is the true datagenerating process. In Simulation 2, BMC and ZIPLL outperform tcpl models. SmallerRMSEs and higher AUCs from BMC and ZIPLL compared to those from tcpl suggestincreased robustness of spline methods than parametric ones for dose-response functions.The improved metrics also indicate benefits of hierarchical methods over the independentcurve fitting that ignores correlations between chemicals or assay endpoints. In particular,the achievement of the high in-sample AUC for γ ij from BMC is encouraging despite therelatively small number of chemicals and assay endpoints. The disadvantage of these small m and J , however, is apparent in poor predictive AUCs of BMC and BMC . The out-of-sample AUC for γ ij from BMC is only slightly better than random guessing (AUC =0.5), while the models with simpler structures on γ ij are not as good as random guessing(BMC i : 0 .
475 (0 . j : 0 . . S4 Posterior Computation
Under the prior specification in section 4, posterior samples are obtained by iterating thefollowing partially collapsed MCMC sampler.
Heteroscedasticity
1. Update t ij and δ ij simultaneously using the Metropolis-Hastings algorithm. Propose t pij as follows: for each j , choose random number of elements and random indices toupdate. For those selected ( i, j ) pairs, flip zero and one. Given the proposed t pij , propose δ pij using t-distribution with 4 degrees of freedom centered at the current δ cij .7ccept ( t pij , δ pij ) with probability min { , r } where r = (cid:81) K ij k =1 N ( y ijk − γ ij ( x Bijk ) T β ij ; 0 , exp( x ijk δ pij / σ j ) (cid:81) K ij k =1 N ( y ijk − γ ij ( x Bijk ) T β ij ; 0 , exp( x ijk δ cij / σ j ) × Bernoulli ( t pij ; π t ) { N ( δ pij | t pij = 1; 0 , v δ ) ( t pij = 1) + 1 × ( t pij = 0) } Bernoulli ( t cij ; π t ) { N ( δ cij | t cij = 1; 0 , v δ ) ( t cij = 1) + 1 × ( t cij = 0) } and ( x Bijk ) T is the k th row of the B-spline basis matrix X ij .2. Update π t from ( π t | t ij ∀ i, j ) ∼ Beta (cid:32) n t , J (cid:88) j =1 m j − n t (cid:33) , where n t = (cid:80) i (cid:80) j ( t ij = 1) and m j = (cid:80) i ( K ij > Functional Mean
Once Steps 1&2 are completed in every iteration, the data (
X, Y ) need to be refor-mulated: y ijk is replaced by y ijk / exp( x ijk δ ij / x Bijk by x Bijk / exp( x ijk δ ij / λ i , η j ) as in (Bhattacharya and Dunson, 2011). Priors, hyperparametersspecification and posterior distributions are fully explained in (Bhattacharya andDunson, 2011) and (Durante, 2017). We shall not repeat the sampling algorithms for λ i and η j here. With the sampled ( λ i , η j ), update π ij using (1).4. Update z ij from ( z ij | γ ij = 1 , λ i , η j ) ∼ T N (0 , ∞ ) ( λ Ti η j , , ( z ij | γ ij = 0 , λ i , η j ) ∼ T N ( −∞ , ( λ Ti η j , T N ( a,b ) ( µ, σ ) denotes a normal distribution truncated to the interval ( a, b )with mean µ , variance σ . 8. Update γ ij from the conditional Bernoulli distribution with β ij marginalized out.Using the following probabilities, P r ( γ ij = 1 | y ij , X ij , σ j , Σ j , π ij ) ∝ π ij | Σ j X Tij X ij /σ j + I p | − / × exp (cid:18) σ j y Tij X ij (cid:0) X Tij X ij /σ j + Σ − j (cid:1) − X Tij y ij (cid:19) , (S1) P r ( γ ij = 0 | y ij , X ij , σ j , Σ j , π ij ) ∝ (1 − π ij ) , (S2)( γ ij | y ij , X ij , σ j , Σ j , π ij ) ∼ Bernoulli (cid:18) ( S S
1) + ( S (cid:19) where y ij = [ y ij, , . . . , y ij,K ij ] T .6. Update β ij from the conditional normal distribution only if γ ij = 1( β ij | γ ij = 1 , y ij , X ij , σ j , Σ j ) ∼ N p (cid:16)(cid:0) Σ − j + X Tij X ij /σ j (cid:1) − X Tij y ij /σ j , (cid:0) Σ − j + X Tij X ij /σ j (cid:1) − (cid:17) .
7. Update Σ j from (cid:0) Σ − j | β j , . . . , β m j ,j (cid:1) ∼ W ish a + m j , (cid:32) R + m j (cid:88) i =1 β ij β Tij (cid:33) − . Assay endpoint-specific variance
8. Update σ j from(1 /σ j | y ij , γ ij , X ij , β ij ∀ i = 1 , . . . , m j ) ∼ Gamma (cid:32) ν + (cid:80) m j i =1 K ij , ν σ + (cid:80) m j i =1 (cid:80) K ij k =1 ( y ijk − γ ij ( x Bijk ) T β ij ) (cid:33) ..