Approximate Bayesian computation (ABC) coupled with Bayesian model   averaging method for estimating mean and standard deviation
  Background: We proposed approximate Bayesian computation with single distribution selection (ABC-SD) for estimating mean and standard deviation from other reported summary statistics. The ABC-SD generates pseudo data from a single parametric distribution thought to be the true distribution of underlying study data. This single distribution is either an educated guess, or it is selected via model selection using posterior probability criterion for testing two or more candidate distributions. Further analysis indicated that when model selection is used, posterior model probabilities are sensitive to the prior distribution(s) for parameter(s) and dependable on the type of reported summary statistics. Method: We propose ABC with Bayesian model averaging (ABC-BMA) methodology to estimate mean and standard deviation based on various sets of other summary statistics reported in published studies. We conduct a Monte Carlo simulation study to compare the new proposed ABC-BMA method with our previous ABC-SD method. Results: In the estimation of standard deviation, ABC-BMA has smaller average relative errors (AREs) than that of ABC-SD for normal, lognormal, beta, and exponential distributions. For Weibull distribution, ARE of ABC-BMA is larger than that of ABC-SD but <0.05 in small sample sizes and moves toward zero as sample size increases. When underlying distribution is highly skewed and available summary statistics are only quartiles and sample size, ABC-BMA is recommended but it should be used with caution. Comparison of mean estimation between ABC-BMA and ABC-SD shows similar patterns of results as for standard deviation estimation. Conclusion: ABC-BMA is easy to implement and it performs even better than our previous ABC-SD method for estimation of mean and standard deviation. 
Deukwoo Kwon and Isildinha M. Reis
stat.ME