# propdiff.mblwoc: Bayesian sample size determination for the difference between... In SampleSizeProportions: Calculating sample size requirements when estimating the difference between two binomial proportions

## Description

The function `propdiff.mblwoc` uses a mixed Bayesian/likelihood approach to determine conservative sample sizes for the difference between two binomial proportions, in the sense that the desired posterior credible interval coverage and length are guaranteed over all possible data sets.

## Usage

 `1` ```propdiff.mblwoc(len, c1, d1, c2, d2, level = 0.95) ```

## Arguments

 `len` The desired length of the posterior credible interval for the difference between the two unknown proportions `c1` First prior parameter of the Beta density for the binomial proportion for the first population `d1` Second prior parameter of the Beta density for the binomial proportion for the first population `c2` First prior parameter of the Beta density for the binomial proportion for the second population `d2` Second prior parameter of the Beta density for the binomial proportion for the second population `level` The fixed coverage probability of the posterior credible interval (e.g., 0.95)

## Details

Assume that a sample from each of two populations will be collected in order to estimate the difference between two independent binomial proportions. Assume that the proportions have prior information in the form of Beta(c1, d1) and Beta(c2, d2) densities in each population, respectively. The function `propdiff.mblwoc` 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 proportions. The Worst Outcome Criterion used is conservative, in the sense that the posterior credible interval length len is guaranteed over all possible data sets that can arise, 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 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 the sample sizes returned.

## Author(s)

Lawrence Joseph lawrence.joseph@mcgill.ca, Patrick Belisle and Roxane du Berger

## References

Joseph L, du Berger R, and Belisle P.
Bayesian and mixed Bayesian/likelihood criteria for sample size determination
Statistics in Medicine 1997;16(7):769-781.

`propdiff.mblacc`, `propdiff.mblalc`, `propdiff.mblmodwoc`, `propdiff.acc`, `propdiff.alc`, `propdiff.modwoc`, `propdiff.woc`
 `1` ```propdiff.mblwoc(len=0.05, c1=3, d1=11, c2=11, d2=54) ```