# minEffect.VSMc.poisson: Minimum detectable slope for mediator in poisson regression... In powerMediation: Power/Sample Size Calculation for Mediation Analysis

## Description

Calculate minimal detectable slope for mediator given sample size and power in poisson regression based on Vittinghoff, Sen and McCulloch's (2009) method.

## Usage

 ```1 2 3 4 5 6 7``` ```minEffect.VSMc.poisson(n, power, sigma.m, EY, corr.xm, alpha = 0.05, verbose = TRUE) ```

## Arguments

 `n` sample size. `power` power for testing b_2=0 for the poisson regression \log(E(Y_i))=b0+b1 x_i + b2 m_i. `sigma.m` standard deviation of the mediator. `EY` the marginal mean of the outcome `corr.xm` correlation between the predictor x and the mediator m. `alpha` type I error rate. `verbose` logical. `TRUE` means printing minimum absolute detectable effect; `FALSE` means not printing minimum absolute detectable effect.

## Details

The test is for testing the null hypothesis b_2=0 versus the alternative hypothesis b_2\neq 0 for the poisson regressions:

\log(E(Y_i))=b_0+b_1 x_i + b_2 m_i

Vittinghoff et al. (2009) showed that for the above poisson regression, testing the mediation effect is equivalent to testing the null hypothesis H_0: b_2=0 versus the alternative hypothesis H_a: b_2\neq 0, if the correlation `corr.xm` between the primary predictor and mediator is non-zero.

The full model is

\log(E(Y_i))=b_0+b_1 x_i + b_2 m_i

The reduced model is

\log(E(Y_i))=b_0+b_1 x_i

Vittinghoff et al. (2009) mentioned that if confounders need to be included in both the full and reduced models, the sample size/power calculation formula could be accommodated by redefining `corr.xm` as the multiple correlation of the mediator with the confounders as well as the predictor.

## Value

 `b2 ` minimum absolute detectable effect. `res.uniroot ` results of optimization to find the optimal sample size.

## Note

The test is a two-sided test. For one-sided tests, please double the significance level. For example, you can set `alpha=0.10` to obtain one-sided test at 5% significance level.

## Author(s)

Weiliang Qiu stwxq@channing.harvard.edu

## References

Vittinghoff, E. and Sen, S. and McCulloch, C.E.. Sample size calculations for evaluating mediation. Statistics In Medicine. 2009;28:541-557.

`powerMediation.VSMc.poisson`, `ssMediation.VSMc.poisson`
 ```1 2 3 4 5 6``` ``` # example in section 5 (page 546) of Vittinghoff et al. (2009). # minimum effect is = log(1.35) = 0.3001046 minEffect.VSMc.poisson(n = 1239, power = 0.7998578, sigma.m = sqrt(0.25 * (1 - 0.25)), EY = 0.5, corr.xm = 0.5, alpha = 0.05, verbose = TRUE) ```