bama: Bayesian Mediation Analysis

Description

bama is a Bayesian inference method that uses continuous shrinkage priors for high-dimensional Bayesian mediation analysis, developed by Song et al (2019, 2020). bama provides estimates for the regression coefficients as well as the posterior inclusion probability for ranking mediators.

Usage

bama(
  Y,
  A,
  M,
  C1,
  C2,
  method,
  burnin,
  ndraws,
  weights = NULL,
  inits = NULL,
  control = list(k = 2, lm0 = 1e-04, lm1 = 1, lma1 = 1, l = 1, lambda0 = 0.04, lambda1 =
    0.2, lambda2 = 0.2, phi0 = 0.01, phi1 = 0.01, a0 = 0.01 * ncol(M), a1 = 0.05 *
    ncol(M), a2 = 0.05 * ncol(M), a3 = 0.89 * ncol(M)),
  seed = NULL
)

Value

If method = "BSLMM", then bama returns a object of type "bama" with 12 elements:

beta.m: ndraws x p matrix containing outcome model mediator coefficients.
r1: ndraws x p matrix indicating whether or not each beta.m belongs to the larger normal component (1) or smaller normal component (0).
alpha.a: ndraws x p matrix containing the mediator model exposure coefficients.
r3: ndraws x p matrix indicating whether or not each alpha.a belongs to the larger normal component (1) or smaller normal component (0).
beta.a: Vector of length ndraws containing the beta.a coefficient.
pi.m: Vector of length ndraws containing the proportion of non zero beta.m coefficients.
pi.a: Vector of length ndraws containing the proportion of non zero alpha.a coefficients.
sigma.m0: Vector of length ndraws containing the standard deviation of the smaller normal component for mediator-outcome coefficients (beta.m).
sigma.m1: Vector of length ndraws containing standard deviation of the larger normal component for mediator-outcome coefficients (beta.m).
sigma.ma0: Vector of length ndraws containing standard deviation of the smaller normal component for exposure-mediator coefficients (alpha.a).
sigma.ma1: Vector of length ndraws containing standard deviation of the larger normal component for exposure-mediator coefficients (alpha.a).
call: The R call that generated the output.

NOTE: GMM not currently supported. Instead, use method = 'PTG' If method = "GMM", then bama returns a object of type "bama" with:

beta.m: ndraws x p matrix containing outcome model mediator coefficients.
alpha.a: ndraws x p matrix containing the mediator model exposure coefficients.
betam_member: ndraws x p matrix of 1's and 0's where item = 1 only if beta.m is non-zero.
alphaa_member: ndraws x p matrix of 1's and 0's where item = 1 only if alpha.a is non-zero.
alpha.c: ndraws x (q2 + p) matrix containing alpha_c coefficients. Since alpha.c is a matrix of dimension q2 x p, the draws are indexed as alpha_c(w, j) = w * p + j
beta.c: ndraws x q1 matrix containing beta_c coefficients. Since beta.c is a matrix of dimension q1 x p
beta.a: Vector of length ndraws containing the beta.a coefficient.
sigma.sq.a: Vector of length ndraws variance of beta.a effect
sigma.sq.e: Vector of length ndraws variance of outcome model error
sigma.sq.g: Vector of length ndraws variance of mediator model error

If method = "PTG", then bama returns a object of type "bama" with:

beta.m: ndraws x p matrix containing outcome model mediator coefficients.
alpha.a: ndraws x p matrix containing the mediator model exposure coefficients.
alpha.c: ndraws x (q2 + p) matrix containing alpha_c coefficients. Since alpha.c is a matrix of dimension q2 x p, the draws are indexed as alpha_c(w, j) = w * p + j
beta.c: ndraws x q1 matrix containing beta_c coefficients. Since beta.c is a matrix of dimension q1 x p
betam_member: ndraws x p matrix of 1's and 0's where item = 1 only if beta.m is non-zero.
alphaa_member: ndraws x p matrix of 1's and 0's where item = 1 only if alpha.a is non-zero.
beta.a: Vector of length ndraws containing the beta.a coefficient.
sigma.sq.a: Vector of length ndraws variance of beta.a effect
sigma.sq.e: Vector of length ndraws variance of outcome model error
sigma.sq.g: Vector of length ndraws variance of mediator model error

Arguments

Y

Length n numeric outcome vector

A

Length n numeric exposure vector

M

n x p numeric matrix of mediators of Y and A

C1

n x nc1 numeric matrix of extra covariates to include in the outcome model

C2

n x nc2 numeric matrix of extra covariates to include in the mediator model

method

String indicating which method to use. Options are

"BSLMM" - mixture of two normal components; Song et al. 2019
"PTG" - product threshold Gaussian prior; Song et al. 2020
"GMM" - NOTE: GMM not currently supported. Instead, use method = 'PTG'. four-component Gaussian mixture prior; Song et al. 2020

burnin

number of iterations to run the MCMC before sampling

ndraws

number of draws to take from MCMC (includes burnin draws)

weights

Length n numeric vector of weights

inits

list of initial values for the Gibbs sampler. Options are

beta.m - Length p numeric vector of initial beta.m in the outcome model. See details for equation
alpha.a - Length p numeric vector of initial alpha.a in the mediator model. See details for equation

control

list of Gibbs algorithm control options. These include prior and hyper-prior parameters. Options vary by method selection. If method = "BSLMM"

k - Shape parameter prior for inverse gamma
lm0 - Scale parameter prior for inverse gamma for the small normal components
lm1 - Scale parameter prior for inverse gamma for the large normal components of beta_m
lma1 - Scale parameter prior for inverse gamma for the large normal component of alpha_a
l - Scale parameter prior for the other inverse gamma distributions

If method = "PTG"

lambda0 - threshold parameter for product of alpha.a and beta.m effect
lambda1 - threshold parameter for beta.m effect
lambda2 - threshold parameter for alpha.a effect
ha - inverse gamma shape prior for sigma.sq.a
la - inverse gamma scale prior for sigma.sq.a
h1 - inverse gamma shape prior for sigma.sq.e
l1 - inverse gamma scale prior for sigma.sq.e
h2 - inverse gamma shape prior for sigma.sq.g
l2 - inverse gamma scale prior for sigma.sq.g
km - inverse gamma shape prior for tau.sq.b
lm - inverse gamma scale prior for tau.sq.b
kma - inverse gamma shape prior for tau.sq.a
lma - inverse gamma scale prior for tau.sq.a

If method = "GMM". NOTE: GMM not currently supported. Instead, use method = 'PTG'.

phi0 - prior beta.m variance
phi1 - prior alpha.a variance
a0 - prior count of non-zero beta.m and alpha.a effects
a1 - prior count of non-zero beta.m and zero alpha.a effects
a2 - prior count of zero beta.m and non-zero alpha.a effects
a3 - prior count of zero beta.m and zero alpha.a effects
ha - inverse gamma shape prior for sigma.sq.a
la - inverse gamma scale prior for sigma.sq.a
h1 - inverse gamma shape prior for sigma.sq.e
l1 - inverse gamma scale prior for sigma.sq.e
h2 - inverse gamma shape prior for sigma.sq.g
l2 - inverse gamma scale prior for sigma.sq.g

seed

numeric seed for GIBBS sampler

Details

bama uses two regression models for the two conditional relationships, $Y | A, M, C$ and $M | A, C$. For the outcome model, bama uses $$Y = M \beta_M + A * \beta_A + C* \beta_C + \epsilon_Y$$ For the mediator model, bama uses the model $$M = A * \alpha_A + C * \alpha_C + \epsilon_M$$

For high dimensional tractability, bama employs continuous Bayesian shrinkage priors to select mediators and makes the two following assumptions: First, it assumes that all the potential mediators contribute small effects in mediating the exposure-outcome relationship. Second, it assumes that only a small proportion of mediators exhibit large effects ("active" mediators). bama uses a Metropolis-Hastings within Gibbs MCMC to generate posterior samples from the model.

NOTE: GMM not currently supported. Instead, use method = 'PTG'.

References

Song, Y, Zhou, X, Zhang, M, et al. Bayesian shrinkage estimation of high dimensional causal mediation effects in omics studies. Biometrics. 2019; 1-11. tools:::Rd_expr_doi("10.1111/biom.13189")

Song, Yanyi, Xiang Zhou, Jian Kang, Max T. Aung, Min Zhang, Wei Zhao, Belinda L. Needham et al. "Bayesian Sparse Mediation Analysis with Targeted Penalization of Natural Indirect Effects." arXiv preprint arXiv:2008.06366 (2020).

Examples

Run this code

library(bama)

Y <- bama.data$y
A <- bama.data$a

# grab the mediators from the example data.frame
M <- as.matrix(bama.data[, paste0("m", 1:100)], nrow(bama.data))

# We just include the intercept term in this example as we have no covariates
C1 <- matrix(1, 1000, 1)
C2 <- matrix(1, 1000, 1)
beta.m  <- rep(0, 100)
alpha.a <- rep(0, 100)

out <- bama(Y = Y, A = A, M = M, C1 = C1, C2 = C2, method = "BSLMM", seed = 1234,
            burnin = 100, ndraws = 110, weights = NULL, inits = NULL, 
            control = list(k = 2, lm0 = 1e-04, lm1 = 1, lma1 = 1, l = 1))

# The package includes a function to summarise output from 'bama'
summary <- summary(out)
head(summary)
# \donttest{

# Product Threshold Gaussian 
ptgmod = bama(Y = Y, A = A, M = M, C1 = C1, C2 = C2, method = "PTG", seed = 1234,
              burnin = 100, ndraws = 110, weights = NULL, inits = NULL, 
              control = list(lambda0 = 0.04, lambda1 = 0.2, lambda2 = 0.2))

mean(ptgmod$beta.a)
apply(ptgmod$beta.m, 2, mean)
apply(ptgmod$alpha.a, 2, mean)
apply(ptgmod$betam_member, 2, mean)
apply(ptgmod$alphaa_member, 2, mean)

# }

Run the code above in your browser using DataLab