D.measure {RobustBayesianCopas}

D Measure for Quantifying Publication Bias

Description

This function computes Bai's D measure for quantifying publication bias based on the robust Bayesian Copas (RBC) selection model. Let \pi_{rbc}(\theta | y) be the posterior distribution for \theta under the full RBC (bias-corrected) model, and let \pi_{\rho=0} (\theta | y) be the posterior distribution for \theta under the non-bias corrected model (when \rho is fixed at \rho=0). The D measure is the Hellinger distance H between the bias-corrected and non-bias-corrected posteriors.

D = H(\pi_{rbc}(\theta | y), \pi_{\rho=0} (\theta | y)).

D is always between 0 and 1, with D \approx 0 indicating negligible publication bias and D \approx 1 indicating a very high magnitude of publication bias.

The random effects distributions for the bias-corrected and non-bias-corrected models should be the same in order for the D measure to be valid. The posterior densities for \pi_{rbc}(\theta | y) and \pi_{\rho=0} (\theta | y) are approximated using MCMC samples. Numerical integration is used to compute the Hellinger distance between them.

Usage

D.measure(samples.RBCmodel, samples.nobiasmodel)

Arguments

samples.RBCmodel

A vector of the MCMC samples from the RBC model. These can be obtained from the output of the function RobustBayesianCopas.

samples.nobiasmodel

A vector of the MCMC samples from the non-bias-corrected model. These can be obtained from the output of the function BayesNonBiasCorrected. In order for the D measure to be valid, the random effects distribution used in BayesNonBiasCorrected should be the same as the random effects distribution used in RobustBayesianCopas.

Value

The function returns Bai's D measure, a value between 0 and 1. D \approx 0 means negligible publication bias, and D \approx 1 means a very high magnitude of publication bias.

References

Bai, R., Lin, L., Boland, M. R., and Chen, Y. (2020). "A robust Bayesian Copas selection model for quantifying and correcting publication bias." arXiv preprint arXiv:2005.02930.

Examples

######################################
# Example on the Barlow2014 data set #
######################################
data(Barlow2014)
attach(Barlow2014)
# Observed treatment effect
y.obs = Barlow2014[,1]
# Observed standard error
s.obs = Barlow2014[,2]

#############################################
# Compute D measure using posterior samples #
#############################################
# Fit RBC (bias-corrected) model with t-distributed random effects.
# NOTE: Use default burn-in (burn=10000) and post-burn-in samples (nmc=10000)
RBC.mod = RobustBayesianCopas(y=y.obs, s=s.obs, re.dist="StudentsT", burn=500, nmc=500)

# Fit non-bias-corrected model with t-distributed random effects.
# NOTE: Use default burn-in (burn=10000) and post-burn-in samples (nmc=10000)
RBCNoBias.mod = BayesNonBiasCorrected(y=y.obs, s=s.obs, re.dist="StudentsT", burn=500, nmc=500)

# Compute the D measure based on posterior samples for theta
D = D.measure(RBC.mod$theta.samples, RBCNoBias.mod$theta.samples)




############################################
# Example on the antidepressants data set. #
# This is from Section 6.2 of the paper    #
# by Bai et al. (2020).                    #
############################################

# Set seed, so we can reproduce the exact same results as in the paper.
seed = 123
set.seed(seed)

# Load the full data
data(antidepressants)
attach(antidepressants)

# Extract the 50 published studies
published.data = antidepressants[which(antidepressants$Published==1),]

# Observed treatment effect
y.obs = published.data$Standardized_effect_size

# Observed standard error
s.obs = published.data$Standardized_SE


################################
# Compute the D measure for    #
# quantifying publication bias #
################################

# Fit RBC model with different random effects distributions and compare them using DIC.

# Normal
RBC.mod.normal = RobustBayesianCopas(y=y.obs, s=s.obs, re.dist="normal", seed=seed) 
RBC.mod.normal$DIC # DIC=369.3126
# Student's t
RBC.mod.StudentsT = RobustBayesianCopas(y=y.obs, s=s.obs, re.dist="StudentsT", seed=seed)
RBC.mod.StudentsT$DIC # DIC=515.5151
# Laplace
RBC.mod.laplace = RobustBayesianCopas(y=y.obs, s=s.obs, re.dist="Laplace", seed=seed)
RBC.mod.laplace$DIC # DIC=475.5845
# Slash
RBC.mod.slash = RobustBayesianCopas(y=y.obs, s=s.obs, re.dist="slash", seed=seed)
RBC.mod.slash$DIC # DIC=381.2705

# The model with normal random effects gave the lowest DIC, so we use the model
# with normal study-specific effects for our final analysis.

RBC.mod.normal$rho.hat # rho.hat=0.804
# Plot posterior for rho, which suggests strong publication bias.
hist(RBC.mod.normal$rho.samples) 

# Fit non-biased-corrected model. Make sure that we specify the random effects as normal.
RBCNoBias.mod = BayesNonBiasCorrected(y=y.obs, s=s.obs, re.dist="normal", seed=seed)

# Compute D measure using posterior samples for theta
D.RBC = D.measure(RBC.mod.normal$theta.samples, RBCNoBias.mod$theta.samples) # D=0.95