# mu.mblacc: Bayesian sample size determination for estimating a single... In SampleSizeMeans: Sample size calculations for normal means

## Description

The function `mu.mblacc` returns the required sample size to reach a given coverage probability on average - using a mixed Bayesian/likelihood approach - for a posterior credible interval of fixed length for a normal mean.

## Usage

 `1` ```mu.mblacc(len, alpha, beta, level = 0.95, m = 10000, mcs = 3) ```

## Arguments

 `len` The desired fixed length of the posterior credible interval for the mean `alpha` First prior parameter of the Gamma density for the precision (reciprocal of the variance) `beta` Second prior parameter of the Gamma density for the precision (reciprocal of the variance) `level` The desired average coverage probability of the posterior credible interval (e.g., 0.95) `m` The number of points simulated from the preposterior distribution of the data. For each point, the probability coverage of the highest posterior density interval of fixed length len is estimated, in order to approximate the average coverage probability. Usually 10000 is sufficient, but one can increase this number at the expense of program running time. `mcs` The Maximum number of Consecutive Steps allowed in the same direction in the march towards the optimal sample size, before the result for the next upper/lower bound is cross-checked. In our experience, mcs = 3 is a good choice.

## Details

Assume that a sample will be collected in order to estimate the mean of a normally distributed random variable. Assume that the precision (reciprocal of the variance) of this random variable is unknown, but has prior information in the form of a Gamma(alpha, beta) density. The function `mu.mblacc` returns the required sample size to attain the desired average coverage probability level for the posterior credible interval of fixed length len for the unknown mean.

This function uses a Mixed Bayesian/Likelihood (MBL) approach. MBL approaches use the prior information to derive the predictive distribution of the data, but use only the likelihood function for final inferences. This approach is intended to satisfy investigators who recognize that prior information is important for planning purposes but prefer to base final inferences only on the data.

## Value

The required sample size given the inputs to the function.

## Note

The sample size is calculated via Monte Carlo simulations, and therefore may vary from one call to the next.

## Author(s)

Lawrence Joseph lawrence.joseph@mcgill.ca and Patrick Belisle

## References

Joseph L, Belisle P.
Bayesian sample size determination for Normal means and differences between Normal means
The Statistician 1997;46(2):209-226.

`mu.mblalc`, `mu.mblmodwoc`, `mu.mbl.varknown`, `mu.acc`, `mu.alc`, `mu.modwoc`, `mu.varknown`, `mu.freq`, `mudiff.mblacc`, `mudiff.mblalc`, `mudiff.mblmodwoc`, `mudiff.mblacc.equalvar`, `mudiff.mblalc.equalvar`, `mudiff.mblmodwoc.equalvar`, `mudiff.mbl.varknown`, `mudiff.acc`, `mudiff.alc`, `mudiff.modwoc`, `mudiff.acc.equalvar`, `mudiff.alc.equalvar`, `mudiff.modwoc.equalvar`, `mudiff.varknown`, `mudiff.freq`
 `1` ```mu.mblacc(len=0.2, alpha=2, beta=2) ```