BACprior.boot {BACprior}

A Bootstrap Procedure to Guide the Choice of Omega in the Bayesian Adjustment for Confounding Algorithm.

Description

The BACprior.boot function proposes a bootstrap procedure to select BAC's omega value in an attempt to minimize the mean squared error (MSE) of the exposure effect estimator. A number B of bootstrap samples are taken from the original sample. Then, the MSE is estimated for each selected omega value, considering the exposure estimate with omega = infinity from the original sample as the true value. The BACprior.boot function uses the BACprior.lm function to estimate the exposure effect.

Usage

BACprior.boot(Y, X, U, 
omega = c(1, 1.1, 1.3, 1.6, 2, 5, 10, 30, 50, 100, Inf),
maxmodels = 150, cutoff = 0.0001, B = 100)

Arguments

Y

A vector of observed values for the continuous outcome.

X

A vector of observed values for the continuous exposure.

U

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

omega

A vector of omega values for which the bootstrap procedure is performed. The default is c(1, 1.1, 1.3, 1.6, 2, 5, 10, 30, 50, 100, Inf).

maxmodels

The maximum number of outcome and exposure models of each size to be considered. Larger numbers improves the approximation, but can greatly increase the computational burden. The default is 150.

cutoff

Minimum posterior probability needed for an outcome model to be considered in the weighted average of the posterior mean and standard deviation of the exposure effect. Smaller values of cutoff improves the approximation, but add computational complexity. The default is 0.0001.

B

The number of bootstrap samples to be taken. Larger numbers reduce Monte Carlo error, but require more computation time.

Details

Since BACprior.boot uses the BACprior.lm function to estimate the exposure effect, users should refer to the BACprior.lm documentation for details of implementation.

BACprior.boot assumes there are no missing values. The objects X, Y and U should be processed beforehand so that every case is complete. The na.omit function which removes cases with missing data or an imputation package might be helpful.

Value

Best

The omega value, among the omega values given in input, which minimizes the estimated MSE.

MSE

The estimated MSE for each of the selected omega values.

BACprior.boot also returns a plot of the estimated MSEs according to the selected omega values.

Author(s)

Denis Talbot, Genevieve Lefebvre, Juli Atherton.

References

Brookhart, M.A., van der Laan, M.J. (2006). A semiparametric model selection criterion with applications to the marginal structural model, Computational Statistics & Data Analysis, 50, 475-498.

Lefebvre, G., Atherton, J., Talbot, D. (2014). The effect of the prior distribution in the Bayesian Adjustment for Confounding algorithm, Computational Statistics & Data Analysis, 70, 227-240.

See Also

BACprior.lm, BACprior.CV, na.omit

Examples

# Required package to simulate from a multivariate normal distribution.
require(mvtnorm);


# Simulate data
# n = 500 observations with 5 covariates.
# (U1, U2, U4) is multivariate normal with mean vector 0,
# variances of 1 and 0 pairwise correlations.
# U3 and U5 are causal effects of U2 and U4, respectively.
# X is a causal effect of U1, U2 and U4.
# Y is a causal effect of U3, U4, U5 and X.

set.seed(3417817);
n = 500;
U = rmvnorm(n = n, mean = rep(0, 5), sigma = diag(1, nrow = 5) + matrix(0, nrow = 5, ncol = 5));
U[,3] = U[,2] + rnorm(n);
U[,5] = U[,4] + rnorm(n);
X = U[,1] + U[,2] + U[,4] + rnorm(n);
Y = U[,3] + 0.1*U[,4] + U[,5] + 0.1*X + rnorm(n);

# Remove  ``#'' to run example 
# BACprior.boot(Y, X, U, maxmodels = 150); 
# $best
# [1] 1
# $MSE
#  [1] 0.001467631 0.001480494 0.001505006 0.001539194 0.001580756
# 0.001803000 0.002017034 0.002375198 0.002516998 0.002662188 0.002865611
# Best omega value would be 1
BACprior.lm(Y, X, U);
# $results
#       omega Posterior mean Standard deviation
#  [1,]   1.0      0.1089228         0.02951582
#  [2,]   1.1      0.1087689         0.02971457
#  [3,]   1.3      0.1084802         0.03008991
#  [4,]   1.6      0.1080900         0.03060449
#  [5,]   2.0      0.1076376         0.03121568
#  [6,]   5.0      0.1057020         0.03426854
#  [7,]  10.0      0.1046804         0.03696670
#  [8,]  30.0      0.1044711         0.04124805
#  [9,]  50.0      0.1047315         0.04291842
# [10,] 100.0      0.1051211         0.04462874
# [11,]   Inf      0.1058021         0.04703111

# Posterior mean doesn't change much with omega,
# but posterior standard deviation greatly increases.
# This supports the choice of omega = 1.