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
)

Arguments

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'.

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

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. 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

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)


# 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)

Synthetic example data for bama

Description

Synthetic example data for bama

Usage

bama.data

Format

A data.frame with 1000 observations on 102 variables:

y

Numeric response variable.

a

Numeric exposure variable.

m[1-100]

Numeric mediator variables

Bayesian Mediation Analysis Controlling For False Discovery

Description

fdr.bama uses the permutation test to estimate the null PIP distribution for each mediator and determines a threshold (based off of the fdr parameter) for significance.

Usage

fdr.bama(
  Y,
  A,
  M,
  C1,
  C2,
  beta.m,
  alpha.a,
  burnin,
  ndraws,
  weights = NULL,
  npermutations = 200,
  fdr = 0.1,
  k = 2,
  lm0 = 1e-04,
  lm1 = 1,
  lma1 = 1,
  l = 1,
  mc.cores = 1,
  type = "PSOCK"
)

Arguments

Value

fdr.bama returns a object of type "fdr.bama" with 5 elements:

bama.out

Output from the bama run.

pip.null

A p x npermutations matrices containing the estimated null PIP distribution for each mediator.

threshold

The cutoff significance threshold for each PIP controlling for the false discovery rate.

fdr

The false discovery rate used to calculate threshold.

call

The R call that generated the output.

Author(s)

Alexander Rix

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. doi:10.1111/biom.13189

Examples

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)

set.seed(12345)

out <- fdr.bama(Y, A, M, C1, C2, beta.m, alpha.a, burnin = 100,
                ndraws = 120, npermutations = 10)

# The package includes a function to summarise output from 'fdr.bama'
summary(out)

Printing bama objects

Description

Print a bama object.

Usage

## S3 method for class 'bama'
print(x, ...)

Arguments

Printing bama objects

Description

Print a bama object.

Usage

## S3 method for class 'fdr.bama'
print(x, ...)

Arguments

Summarize objects of type "bama"

Description

summary.bama summarizes the 'beta.m' estimates from bama and generates an overall estimate, credible interval, and posterior inclusion probability.

Usage

## S3 method for class 'bama'
summary(object, rank = F, ci = c(0.025, 0.975), ...)

Arguments

Value

A data.frame with 4 elements. The beta.m estimates, the estimates' credible interval (which by default is 95\ inclusion probability (pip) of each 'beta.m'.

Summarize objects of type "fdr.bama"

Description

summary.fdr.bama summarizes the beta.m estimates from fdr.bama and for each mediator generates an overall estimate, credible interval, posterior inclusion probability (PIP), and PIP threshold for significance controlling for the specified false discovery rate (FDR).

Usage

## S3 method for class 'fdr.bama'
summary(
  object,
  rank = F,
  ci = c(0.025, 0.975),
  fdr = object$fdr,
  filter = T,
  ...
)

Arguments

Value

A data.frame with 4 elements. The beta.m estimates, the estimates' credible interval (which by default is 95\ inclusion probability (pip) of each 'beta.m'.