# propdiff.mblalc: 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.mblalc` returns the required sample sizes to reach a given posterior credible interval length on average for a fixed coverage probability using a mixed Bayesian/likelihood approach for the difference between two binomial proportions.

## Usage

 `1` ```propdiff.mblalc(len, c1, d1, c2, d2, level = 0.95, m = 10000, mcs = 3) ```

## Arguments

 `len` The desired average 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) `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 average length. Usually 10000 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 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.mblalc` 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 proportions.

This function uses a Mixed Bayesian/Likelihood (MBL) approach. MBL approaches use the prior information to derive the predictive distribution of the data, but use 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 are calculated via Monte Carlo simulations, and therefore may vary from one call to the next.

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