# mudiff.alc.equalvar: Bayesian sample size determination for differences in normal... In SampleSizeMeans: Sample size calculations for normal means

## Description

The function `mudiff.alc.equalvar` returns the required sample sizes to reach a given posterior credible interval length on average for a fixed coverage probability for the difference between two normal means, when variances are equal.

## Usage

 `1` ```mudiff.alc.equalvar(len, alpha, beta, n01, n02, level = 0.95, equal = TRUE) ```

## Arguments

`len`

The desired average length of the posterior credible interval for the difference between the two unknown means

`alpha`

First prior parameter of the Gamma density for the common precision (reciprocal of the variance)

`beta`

Second prior parameter of the Gamma density for the common precision (reciprocal of the variance)

`n01`

Prior sample size equivalent for the mean for the first population

`n02`

Prior sample size equivalent for the mean for the second population

`level`

The desired fixed coverage probability of the posterior credible interval (e.g., 0.95)

`equal`

logical. Whether or not the final group sizes (n1, n2) are forced to be equal:

 when equal = TRUE, final sample sizes n1 = n2; when equal = FALSE, final sample sizes (n1, n2) minimize the expected posterior variance given a total of n1+n2 observations

## Details

Assume that a sample from each of two populations will be collected in order to estimate the difference between two independent normal means. Assume that the precisions of the two normal sampling distributions are unknown but equal, with prior information in the form of a Gamma(alpha, beta) density. Assume that the means are unknown, but have prior information equivalent to (n01, n02) previous observations, respectively. The function `mudiff.alc.equalvar` returns the required sample sizes to attain the desired average length len for the posterior credible interval of fixed coverage probability level for the difference between the two unknown means.

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 sizes (n1, n2) for each group given the inputs to the function.

## Note

The sample sizes returned by this function are exact.

## 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.

`mudiff.acc.equalvar`, `mudiff.modwoc.equalvar`, `mudiff.acc`, `mudiff.alc`, `mudiff.modwoc`, `mudiff.varknown`, `mudiff.mblacc.equalvar`, `mudiff.mblalc.equalvar`, `mudiff.mblmodwoc.equalvar`, `mudiff.mblacc`, `mudiff.mblalc`, `mudiff.mblmodwoc`, `mudiff.mbl.varknown`, `mudiff.freq`, `mu.acc`, `mu.alc`, `mu.modwoc`, `mu.varknown`, `mu.mblacc`, `mu.mblalc`, `mu.mblmodwoc`, `mu.mbl.varknown`, `mu.freq`
 `1` ```mudiff.alc.equalvar(len=0.2, alpha=2, beta=2, n01=10, n02=50) ```