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

## Description

The function `mudiff.modwoc.equalvar` calculates conservative sample sizes, in the sense that the desired posterior credible interval coverage and length for the difference between two normal means are guaranteed over a given proportion of data sets that can arise according to the prior information, when variances are equal.

## Usage

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

## Arguments

`len`

The desired total 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)

`worst.level`

The probability that the length of the posterior credible interval of fixed coverage probability level will be at most len

`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.modwoc.equalvar` returns the required sample sizes to attain the desired length len for the posterior credible interval of fixed coverage probability level for the difference between the two unknown means. 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 sizes (n1, n2) for each group given the inputs to the function.

## Note

The sample sizes returned by this function are 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 sizes returned.

## Author(s)

Lawrence Joseph [email protected] 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.alc.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.modwoc.equalvar(len=0.2, alpha=2, beta=2, n01=10, n02=50) ```