Table 2.

PRIME Methods and Software.

Project 1	Method Acronym	Method Title	Summary	Reference
BU/ Harvard	BKMR-CMA	Bayesian Kernel Machine Regression-Causal Mediation Analysis	Performs a causal mediation analysis when exposure within the mediation framework is a mixture. Estimates a multivariate exposure response surface in a model for the mediator given exposure, and another for the outcome given the mediator and the outcome, both using BKMR.	[4]
BU/ Harvard	BMIM	Bayesian Multiple Index Model	Unifies exposure index models with the response surface method BKMR, allowing a spectrum of intermediate models of multiple indices. Models non-linear, non-additive relationships between indices and an outcome. Special cases are a single exposure index and a response surface of all exposures.	[5]
BU/ Harvard	DAG analysis	Use of causal methods for determining which exposures to include in a model	Applies directed acyclic graphs (DAGs) to determine inclusion of exposure variables. In some circumstances, including an exposure variable can increase bias. Determines causal relationships between exposures (or groups of exposures) and a health outcome.	[6]
Columbia	BN2MF	Bayesian Non-parametric non-negative Matrix Factorization	Matrix factorization that provides non-negative (and more interpretable) solutions for factors and loadings and uncertainty estimates for the estimated parameters. Used for exposure pattern identification, similar to PCP.	[7]
Columbia	PCP	Principal Component Pursuit	Unsupervised robust exposure pattern identification. Decomposes exposure matrix into a low-rank matrix (consistent patterns) and a sparse matrix (unique exposure events). Robust exposure pattern identification.	[8]
Duke	BAG	Bag of DAGs	A computationally efficient method to construct a class of non-stationary spatiotemporal processes in point-referenced geostatistical models. Accounts for uncertainty in directions of association over space and time by considering a mixture of direct acyclic graphs (DAGs)	[9]
Duke	BMC	Bayesian Matrix Completion for hypothesis testing	Bayesian inference about chemical activity on mean and variance of dose-response measurements accounting for sparsity of data. Used to characterize chemical activity and its uncertainty.	[10]
Duke	BS3FA	Bayesian partially supervised sparse and smooth factor analysis	Bayesian inference on how chemical structure relates to variation in dose-response measurements. Addresses how to jointly model structural variability in molecular features of a chemical and its dose-response profile.	[11]
Duke	FIN	Factor analysis for interactions	Bayesian factor analysis for inference on interactions. Estimates interactions between highly correlated chemical exposures and effect on health outcomes.	[12]
Duke	GIF-SIS	Generalized infinite factor model	Shrinkage prior to the loadings matrix of infinite factor models that incorporate meta covariates to inform the sparsity structure and has desirable shrinkage properties. Addresses how to incorporate a priori known structure among variables when fitting a member of the broad class of factorization models.	[13]
Duke	GL-GPs	Graph Laplacian based Gaussian Process	Gaussian process model with a covariance function that respects the geometry of highly restricted or nonlinear domains. Develops a covariance function for nonparametric regression that respects the intrinsic geometry of the domain without sacrificing computational tractability.	[14]
Duke	GriPS	Computational improvements for Bayesian multivariate regression models based on latent meshed gaussian processes	Computational improvements for Bayesian multivariate regression models based on latent Meshed Gaussian Processes. Addresses how to efficiently solve the big-n problem for GPs when the number of outcomes is large.	[15]
Duke	MixSelect	Identifying main effects and interactions among exposures using Gaussian processes	Identifies main effects and interactions among exposures using Gaussian processes. Addresses how to model potentially non-linear effects and high-order interactions of chemical exposures on health outcomes.	[16]
Duke	MrGap	Manifold Reconstruction via Gaussian Process	Local covariance Gaussian process model for estimating a manifold in high dimensional space from noisy data. Conducts inference on a low-dimensional, nonlinear manifold in high dimensional space when data are subject to measurement error.	[17]
Duke	PFA	Perturbed factor analysis	Factor analysis that captures common structure among groups of related observations. Distinguishes shared and group-specific covariance structure and expresses shared structure via a set of shared factors.	[18]
Duke	MatchAlign	Resolving rotational ambiguity in matrix sampling	Efficiently resolving rotational ambiguity in Bayesian matrix sampling with matching. Does inference on unidentifiable random matrices.	[19]
Duke	SPAMTREE	Spatial Multivariate Trees	Bayesian multivariate regression methods for big data using sparse treed Gaussian processes. Jointly models several imbalanced variables flexibly and scalably via GPs	[20]
MSSM/ Harvard	ACR	Acceptable Concentration Range model	New class of nonlinear statistical models for human data that incorporates and evaluates regulatory guideline values into analyses of health effects of exposure to chemical mixtures. Allows for human data to suggest points of departure for comparison to in vivo estimates from single chemicals.	[21]
MSSM/ Harvard	Mult DLAG	Multiple exposure distributed lag models with variable selection	A method to identify the presence of time-dependent interactions (interactions among chemical exposures experienced during different exposure windows) in a critical windows analysis. Identifies critical windows of exposure to multiple chemicals, and whether exposures experienced at different developmental windows interact with one another on a health outcome.	[22]
MSSM/ Harvard	BKMR-DLM	Bayesian Kernel Machine Regression-Distributed Lag Model	Develops distributed lag models for assessing critical windows of exposure associated with a mixture. The model simultaneously estimates a time-weighted combination of each exposure and estimates a multivariate exposures-response surface of these time-weighted exposures using BKMR.	[23]
MSSM/ Harvard	CVEK	Cross-validated kernel ensemble	Performs tests of interaction between two sets of exposures (i.e., two mixtures) while placing minimal assumptions on the main effects of each mixture. Asks whether one mixture (e.g., a collection of nutrients) modifies the effect of another (e.g., a metal mixture) as a whole.	[24]
MSSM/ Harvard	Bayes Tree Pairs	Bayesian Regression Tree Pairs	Estimates critical windows of susceptibility to an environmental mixture. Uses an additive ensemble of tree pairs to estimate main effects and interactions between time-resolved predictors with variable selection.	[25]
MSSM/ Harvard	DLMtree	Bayesian Treed Distributed Lab Models	Distributed lag linear and non-linear models. Method to improve the precision of critical window identification compared to methods that use spline or penalized spline basis functions. Interest focuses on identifying critical windows of exposure using data on a single exposure measured over time.	[26]
MSSM/ Harvard	Het-DLM	Heterogeneous distributed lag models	Methods for precision children’s environmental health—that is, methods to identify subject characteristics (child sex, maternal age, etc.) that modify distributed lag effects of exposure. Addresses which subjects exhibit the strongest associations with an exposure measured over multiple developmental windows, and whether the critical windows of exposure vary among subgroups.	[27]
MSSM/ Harvard	LWQS	Lagged Weighted Quantile Sum (WQS) regression	Uses a reverse distributed lag model for assessing critical windows of exposure associated with a mixture when the exposure temporal pattern differs across subjects. Can also incorporate strata-specific associations. Useful for identifying time-varying associations of a mixture effect and later life health/developmental outcomes.	[28]
MSSM/ Harvard	NLinteraction	Bayesian semiparametric regression with sparsity inducing priors	Estimates effects of environmental mixtures to allow for interactions of any order. Provides variable importance measures for both main effects and interactions among exposures within a mixture, while making minimal assumptions on the forms of those effects.	[29]
MSSM/ Harvard	RH-WQS	Repeated holdout Weighted Quantile Sum (WQS) regression	Generalizes WQS regression to include repeated holdout random data splits. Estimates a mixture effect using an empirically estimated weighted index.	[30]
MSSM/ Harvard	SGP-MPI	Scalable Gaussian Process regression via Median Posterior Inference	Takes a split-and-conquer strategy to fitting BKMR to big data. Yields summaries of the multivariate exposure-response surface, as well as variable importance measures of each individual exposure.	[31]
ND/Rice	BDS	Bayesian Data Synthesis	A Bayesian framework used to simulate fully synthetic datasets of mixed data types. The dataset may be comprised of mixed categorical, binary, count, and continuous datatypes. Can handle missing data and has customized metrics for attributing risk disclosure and other privacy concerns.	[32]
ND/Rice	BSSVI	Bayesian subset selection and variable importance for interpretable prediction and classification	Used to collect and summarize all near-optimal subset models to provide a complete predictive picture. Useful in the presence of correlated covariates, weak signals, and/or small sample sizes, where different subsets may be indistinguishable in their predictive accuracy.	[33]
ND/Rice	BVSM	Bayesian variable selection for understanding mixtures in environmental exposures	Variable selection via sparse summaries of a linear regression model. Given a Bayesian regression model with social and environmental covariates, addresses which variables matter most for predicting educational outcomes.	[34]
ND/Rice	FOTP	Fast, optimal, and targeted predictions using parameterized decision analysis	Computes targeted summaries and prediction for specific decision tasks. Given a target (or functional) of interest and a Bayesian model, constructs accurate, simple, and efficient predictions of future values or functionals of future values. Model summaries can be customized for each functionality.	[35]
ND/Rice	SCC	Spatiotemporal case-crossover	Presents a strategy for the case-crossover study design in a spatial-temporal setting. Incorporates a temporal case-crossover and a geometrically aware spatial random effect based on the Hausdorff distance.	[36]
ND/Rice	SiBAR	State Informed Background Removal	Computational technique to quantify ‘background’ versus ‘source influenced’ contributions to air pollutant time series. Addresses whether a hidden Markov model can be used and what the ‘background’ levels of pollutants are measured across an urban area.	[37]
UI Chicago	MVNimpute	Imputation of multivariate data by normal model	Implements multiple imputation to the data when there are missing and/or censored values.	[38]
UI Chicago	SPORM	Semi-Parametric Odds Ratio Model	Flexible semiparametric model for estimating complex relationship among multiple variables. Associations are modeled by odds ratio functions.	[14,39]
UI Chicago	TEV	Estimation and inference on the explained variation parameter	Estimates the explained variation of an outcome by a set of mixture pollutants.	[40,41]

¹ Listed in alphabetical order, by institution. Project details available at NIH RePORTER: https://reporter.nih.gov/, accessed on 21 December 2021. Institutions: Columbia University Mailman School of Public Health, University of Illinois Chicago, Icahn School of Medicine at Mount Sinai, Harvard T.H. Chan School of Public Health, University of Notre Dame, Rice University, Boston University School of Public Health, Duke University.