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

## Description

The function `mu.mblmodwoc` uses a mixed Bayesian/likelihood approach to determine conservative sample sizes, in the sense that the desired posterior credible interval coverage and length for a normal mean are guaranteed over a given proportion of data sets that can arise according to the prior information.

## Usage

 `1` ```mu.mblmodwoc(len, alpha, beta, level = 0.95, worst.level = 0.95, m = 50000, mcs = 3) ```

## Arguments

 `len` The desired total 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 fixed coverage probability of the posterior credible interval (e.g., 0.95) `worst.level` The probability that the length of the posterior credible interval of fixed coverage probability level will be at most len `m` The number of points simulated from the preposterior distribution of the data. For each point, the length of the highest posterior density interval of fixed coverage probability level is estimated, in order to approximate the (100*worst.level)%-percentile of the posterior credible interval length. Usually 50000 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.mblmodwoc` returns the required sample size to attain the desired length len for the posterior credible interval of fixed coverage probability level for the unknown mean. The Modified Worst Outcome Criterion used is conservative, in the sense that the posterior credible interval length len is guaranteed over the worst.level proportion of all possible data sets that can arise according to the prior information, for a fixed coverage probability level.

This function uses a Mixed Bayesian/Likelihood (MBL) approach. MBL approaches use the prior information to derive the predictive distribution of the data, but uses 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.

It is also correct to state that the coverage probability of the posterior credible interval of fixed length len will be at least level with probability worst.level with the sample size returned.

## 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.mblacc`, `mu.mblalc`, `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.mblmodwoc(len=0.2, alpha=2, beta=2) ```