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

## Description

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

## Usage

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

## Arguments

 `n` sample size. `power` power for testing b_2=0 for the linear regression y_i=b0+b1 x_i + b2 m_i + ε_i, ε_i\sim N(0, σ_e^2). `sigma.m` standard deviation of the mediator. `sigma.e` standard deviation of the random error term in the linear regression y_i=b0+b1 x_i + b2 m_i + ε_i, ε_i\sim N(0, σ_e^2). `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 linear regressions:

y_i=b_0+b_1 x_i + b_2 m_i + ε_i, ε_i\sim N(0, σ^2_{e})

Vittinghoff et al. (2009) showed that for the above linear 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

y_i=b_0+b_1 x_i + b_2 m_i + ε_i, ε_i\sim N(0, σ^2_{e}).

The reduced model is

y_i=b_0+b_1 x_i + ε_i, ε_i\sim N(0, σ^2_{e}).

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`, `ssMediation.VSMc`

## Examples

 ```1 2 3 4``` ``` # example in section 3 (page 544) of Vittinghoff et al. (2009). # minimum effect is =0.1 minEffect.VSMc(n = 863, power = 0.8, sigma.m = 1, sigma.e = 1, corr.xm = 0.3, alpha = 0.05, verbose = TRUE) ```

### Example output

``` 0.0999804
```

powerMediation documentation built on March 24, 2021, 1:06 a.m.