BACprior.CV {BACprior}

A Cross-Validation Procedure for the Choice of the Omega Value in the Bayesian Adjustment for Confounding Algorithm.

Description

The BACprior.CV function proposes two cross-validation procedures to select BAC's omega value in an attempt to minimize the mean squared error (MSE) of the exposure effect estimator. The data are split into half V times. Each time, the exposure effect is estimated with omega = infinity on one half of the data. On the other half of the data, the exposure effect is estimated for each of the selected omega values. A criterion to be minimized is then computed. Details can be found in Lefebvre et al. (2014), which is inspired by a procedure proposed by Brookhart and van der Laan (2006). The BACprior.CV function uses the BACprior.lm function to estimate the exposure effect.

Usage

BACprior.CV(Y, X, U, 
omega = c(1, 1.1, 1.3, 1.6, 2, 5, 10, 30, 50, 100, Inf),
maxmodels = 150, cutoff = 0.0001, V = 100, criterion = "CVm")

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 cross-validation 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.

V

The number of times the data are split into half. Larger numbers reduce Monte Carlo error, but require more computation time.

criterion

The criterion based on MSE to be computed and minimized. The possible values are “CVm” and “CV” and the default is “CVm”. Both criteria are detailed in Lefebvre et al. (2014).

Details

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

BACprior.CV 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 selected criterion.

Criterion.Value

The criterion values for the selected omega values.

BACprior.CV also returns a plot of the criterion values 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.boot, 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.CV(Y, X, U, maxmodels = 150, criterion = "CVm"); 

# $best
# [1] 1
# $Criterion
#  [1] 0.0008764926 0.0008817157 0.0008916412 0.0009056528 0.0009233560
#  0.0010425601 0.0012070083 0.0015884799 0.0017678894 0.0019616864 0.0022413220
# 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.