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

## Description

The function `mu.modwoc` calculates 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.modwoc(len, alpha, beta, n0, level = 0.95, worst.level = 0.95) ```

## Arguments

 `len` The desired 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) `n0` Prior sample size equivalent for the mean `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

## 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. Assume that the mean is unknown, but has prior information equivalent to n0 previous observations. The function `mu.modwoc` 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 fully Bayesian approach to sample size determination. Therefore, the desired coverages and lengths are only realized if the prior distributions input to the function are used for final inferences. Researchers preferring to use the data only for final inferences are encouraged to use the Mixed Bayesian/Likelihood version of the function.

## Value

The required sample size given the inputs to the function.

## Note

The sample size returned by this function is exact.

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.acc`, `mu.alc`, `mu.varknown`, `mu.mblacc`, `mu.mblalc`, `mu.mblmodwoc`, `mu.mbl.varknown`, `mu.freq`, `mudiff.acc`, `mudiff.alc`, `mudiff.modwoc`, `mudiff.acc.equalvar`, `mudiff.alc.equalvar`, `mudiff.modwoc.equalvar`, `mudiff.varknown`, `mudiff.mblacc`, `mudiff.mblalc`, `mudiff.mblmodwoc`, `mudiff.mblacc.equalvar`, `mudiff.mblalc.equalvar`, `mudiff.mblmodwoc.equalvar`, `mudiff.mbl.varknown`, `mudiff.freq`
 `1` ```mu.modwoc(len=0.2, alpha=2, beta=2, n0=10) ```