NBCEE {BCEE}

Naive BCEE Implementation

Description

N-BCEE implementation of the BCEE algorithm. This function supports exposures that can be modeled with generalized linear models (e.g., binary, continuous or Poisson), but only continuous outcomes.

Usage

NBCEE(X, Y, U, omega, niter = 5000, nburn = 500, nthin = 10,
 maxmodelX = NA, maxmodelY = NA, family.X = "gaussian")

Arguments

X

A vector of observed values for the exposure.

Y

A vector of observed values for the continuous outcome.

U

A matrix of observed values for the M potential confounding covariates, where each column contains observed values for a potential confounding factor. A recommended implementation is to only consider pre-exposure covariates.

omega

The value of the hyperparameter omega in the BCEE's outcome model prior distribution. A recommended implementation is to take omega = sqrt(n)*c, where n is the sample size and c is a user-supplied constant value. Simulation studies suggest that values of c between 100 and 1000 yield good results.

niter

The number of post burn-in iterations in the Markov chain Monte Carlo model composition (MC^3) algorithm (Madigan et al. 1995), prior to applying thinning. The default is 5000.

nburn

The number of burn-in iterations (prior to applying thinning). The default is 500.

nthin

The thinning of the chain. The default is 10.

maxmodelX

The maximum number of exposure models the algorithm can explore. See maxmodelY and the note below.

maxmodelY

The maximum number of distinct outcome models that the algorithm can explore. Choosing a smaller value can shorten computing time. However, choosing a value that is too small will cause the algorithm to crash. The default is NA; the maximum number of outcome models that can be explored is then set to the minimum of niter + nburn and 2^M.

family.X

A description of the error distribution and link function to be used in the model. This can be a character string naming a family function, a family function or the result of a call to a family function. (See family for details of family functions.) The default is "gaussian"

Details

NBCEE assumes there are no missing values in the objects X, Y and U. The na.omit function which removes cases with missing data or an imputation package might be helpful.

Value

betas

A vector containing the sampled values for the exposure effect.

models.X

A Boolean matrix identifying the sampled exposure models. See models.Y.

models.Y

A Boolean matrix identifying the sampled outcome models. Each row corresponds to a sampled outcome model. Within each row, the mth element equals 1 if and only if the mth potential confounding covariate is included in the sampled outcome model (and 0 otherwise).

Note

Variability of the exposure effect estimator is generally underestimated by the N-BCEE implementation of BCEE. The A-BCEE, which also happens to be faster, is thus preferred. Another option is to use N-BCEE with nonparametric bootstrap (B-BCEE) to correctly estimate variability.

The difference in computing time between A-BCEE and N-BCEE is mostly explainable by the method used to compute the posterior distribution of the exposure model. In A-BCEE, this posterior distribution is calculated as a first step using bic.glm. In N-BCEE, the posterior distribution of the exposure model is computed inside the MC^3 algorithm.

Author(s)

Denis Talbot, Genevieve Lefebvre, Juli Atherton.

References

Madigan, D., York, J., Allard, D. (1995) Bayesian graphical models for discrete data, International Statistical Review, 63, 215-232.

Talbot, D., Lefebvre, G., Atherton, J. (2015) The Bayesian causal effect estimation algorithm, Journal of Causal Inference, 3(2), 207-236.

See Also

na.omit, ABCEE.

Examples

# In this example, U1 and U2 are potential confounding covariates
# generated as independent N(0,1).
# X is generated as a function of both U1 and U2 with a N(0,1) error.
# Y is generated as a function of X and U1 with a N(0,1) error.
# Variable U1 is the only confounder.
# The causal effect of X on Y equals 1. 
# The exposure effect estimator (beta hat) in the outcome model  
# including U1 and U2 or including U1 only is unbiased.
# The sample size is n = 200.

# Generating the data
set.seed(418949); 
U1 = rnorm(200); 
U2 = rnorm(200);
X = 0.5*U1 + 1*U2 + rnorm(200);
Y = 1*X + 0.5*U1 + rnorm(200);

# Using NBCEE to estimate the causal exposure effect
n = 200;
omega.c = 500;
results = NBCEE(X,Y,cbind(U1,U2), omega = omega.c*sqrt(n),
 niter = 1000, nthin = 5, nburn = 20);

# The posterior mean of the exposure effect:
mean(results$betas);
# The posterior standard deviation of the exposure effect:
sd(results$betas);
# The posterior probability of inclusion of each covariate in the exposure model:
colMeans(results$models.X);
# The posterior distribution of the exposure model:
table(apply(results$models.X, 1, paste0, collapse = ""));
# The posterior probability of inclusion of each covariate in the outcome model:
colMeans(results$models.Y);
# The posterior distribution of the outcome model:
table(apply(results$models.Y, 1, paste0, collapse = ""));