====== Algorithm Implementation of Power for Model 14 Conditional Processes ======

Power analysis is conducted for the model below:

{{:manual:modmed14.png?600|}}

===== Mediation =====

X is the predictor and Y is the dependent variable . Variable X’s effect on a second variable Y is said to be mediated by a third variable M if X causally influences M and M in turn causally influences Y, and then M is the mediator in the model.

===== Moderation =====

A variable’s effect on another is moderated if its size depends on a third variable—a moderator. W is the moderator in the model.
The indirect process model:

$$
M=i_{M}+a_{1}X
$$

The conditional direct model:

$$
Y=i_{Y}+c'X+b_{1}M+c_{1}W+c_{2}XW
$$

XW is the product of X and W. If W moderates X’s effect on Y, it is said that X and W interact in their influence on Y. Equation above can be written in an equivalent form as

$$
Y=i_{Y}+(c'+c_{2}W)X+b_{1}M+c_{1}W
$$

In the model, Conditional direct effect of X on Y is $(c'+c_{1}W)$, indirect effect of X on Y is $a_{1}b_{1}$

In the power analysis, we use Monte Carlo simulation by assuming all variables are normally distributed. We require the input of the correlations between X,W,M,XW,Y. Based on the correlation matrix, $R$ sets of data are randomly generated from a multivariate normal distribution. For each set of data, least squares regression is used to get the parameter estimates and the associated test of significance.

Power for Conditional Indirect Effect (Joint Significance - Recommended)is computed by the following formulas:

$$
\begin{align*}
\mathrm{JSa}&=\frac{\text{times of }a_{1}\text{ and }c_{2}\text{ passes the test}}{\text{ times of the simulation}} 
\end{align*}
$$

===== How to use =====

We developed a web app based on the wp.modmed.m14 function in the WebPower package. The input of the web app include the following:

  * Sample Size
  * Significance Level(alpha): the default value is 0.05.
  * Number of Simulations
  * Regression coefficient of mediator (m) on predictor (x)	
  * Regression coefficient of outcome (y) on predictor (x)	
  * Regression coefficient of outcome (y) on mediator (m)	
  * Regression coefficient of mediator (y) on moderator (w)	
  * Regression coefficient of mediator (y) on the product (mw)	
  * Variance of predictor (x)	
  * Variance of moderator (w)	
  * Variance of error in the first regression equation	
  * Variance of error in the second regression equation	
  * Covariance between predictor (x) and moderator (w)	
  * Moderator value	
  * Method

===== References =====
  * Chris Aberson (2021). pwr2ppl: Power Analyses for Common Designs (Power to the People). R package version 0.2.0. https://CRAN.R-project.org/package=pwr2ppl.
  * Aberson, C. L. (2019). Applied power analysis for the behavioral sciences. Routledge.