# Statistical Power Analysis for Structural Equation Modeling / Mediation based on Monte Carlo Simulation: bootstrap method Structural equation modeling (SEM) is a multivariate technique used to analyze relationships among observed and latent variables. It can be viewed as a combination of factor analysis and multivariate regression analysis. A mediation model can be viewed as a SEM model. Funtions wp.sem.chisq and wp.sem.rmsea provide anlytical solutions of power analysis for SEM. Function wp.mediation provides anlytical solutions of power analysis for a simple mediatoin model. This function provides a solution based on Monte Carlo simulation (see Zhang, 2014 ) and a bootstrap method for testing the indirect /mediation effects. The solution is extended from the general framework for power analysis for complex mediation models using Monte Carlo simulation in Mplus (Muthén & Muthén, 2011) proposed by Thoemmes et al. (2010) . We extended the framework in three ways. First, we proposes a general method to conduct power analysis for mediation models based on the bootstrap method. The method is still based on Monte Carlo simulation but uses the bootstrap method to test mediation effects. Second, the method allows the specification of nonnormal data in the Monte Carlo simulation and can thereby reflect more closely practical data collection. Third, the function wp.mc.sem.boot of a free, open-source R package, WebPower, is developed to ease power anlysis for mediation models using the proposed method. ```r wp.mc.sem.boot(model, indirect = NULL, nobs = 100, nrep = 1000, nboot = 1000, alpha = 0.95, skewness = NULL, kurtosis = NULL, ovnames = NULL, se = "default", estimator = "default", parallel = "no", ncore = Sys.getenv("NUMBER_OF_PROCESSORS"), cl = NULL, ...) ``` ## Arguments - `model`: Model specified using lavaan syntax. More about model specification can be found in Rosseel (2012) . - `indirect`: Indirect effect difined using lavaan syntax. - `nobs`: Sample size. It is 100 by default. - `nrep`: Number of replications for the Monte Carlo simulation. It is 1000 by default. - `nboot`: Number of replications for the bootstrap to test the specified parameter (e.g., mediation). It is 1000 by default. - `alpha`: significance level chosed for the test. It equals 0.05 by default. - `skewness`: A sequence of skewnesses of the observed variables. It is not required. - `kurtosis`: A sequence of kurtosises of the observed variables. It is not required. - `ovnames`: Names of the observed variables in the model. It is not required. - `se`: The method for calculatating the standard errors. Its default method "default" is regular standard errors. More about methods specification standard errors calculatationcan be found in Rosseel (2012) . - `estimator`: Estimator. It is Maxmum likelihood estimator by default. More about estimator specification can be found in Rosseel (2012) . - `parallel`: Parallel computing (`"no"` or `"parallel"` or `"snow"`). It is "no" by default, which means it will not use parallel computing. The option "parallel" is to use multiple cores in a computer for parallel computing. It is used with the number of cores (ncore ). The option "snow" is to use clusters for parallel computing. It is used with the number of clusters (cl ). - `ncore`: Number of processors used for parallel computing. By default, ncore = Sys.getenv('NUMBER_OF_PROCESSORS'). - `cl`: Number of clusters. It is NULL by default. When it is NULL, the program will detect the number of clusters automatically. - `...`: Extra arguments. It is not required. ## Returns An object of the power analysis. The power for all parameters in the model as well as the indirect effects if specified. ## Examples ```r set.seed(20220722) #To specify the model demo =" y ~ cp*x + start(0)*x + b*m + start(0.39)* m m ~ a*x + start(0.39)*x x ~~ start(1)*x m ~~ start(1)*m y ~~ start(1)*y " #To specify the indirect effects mediation = " ab := a*b abc:= a*b + cp " #Power for mediation based on MC method when bootstrap method is used to test the effects: # change nrep and nboot to at least 1,000 in real analysis mediation.boot = wp.mc.sem.boot(model=demo, indirect=mediation, nobs=100, nrep=10, nboot=10, skewness=c(0, 0, 1.3), kurtosis=c(0,0,10), ovnames=c("x","m","y")) #To print the power analysis results summary(mediation.boot) #Example: Power for Simple Mediation Analysis ex1model <- " math ~ c*ME + start(0)*ME + b*HE + start(0.39)*HE HE ~ a*ME + start(0.39)*ME " indirect <- "ab:=a*b" # change nrep and nboot to at least 1,000 in real analysis boot.normal <- wp.mc.sem.boot(ex1model,indirect, 50, nrep=10, nboot=10) summary(boot.normal) boot.non.normal <- wp.mc.sem.boot(ex1model,indirect, 100, nrep=10, nboot=10, skewness=c(-0.3, -0.7, 1.3), kurtosis=c(1.5, 0, 5), ovnames=c('ME','HE','math')) summary(boot.non.normal) #Example: Multiple Group Mediation Analysis (Moderated Mediation) ex3model <- " y ~ start(c(0.283, 0.283))*x + c(c1,c2)*x + start(c(0.36, 0.14))*m + c(b1,b2)*m m ~ start(c(0.721, 0.721))*x + c(a1,a2)*x m =~ c(1,1)*m1 + start(c(0.8, 0.8))*m2 + start(c(0.8, 0.8))*m3 x ~~ start(c(0.25, 0.25))*x y ~~ start(c(0.81, 0.95))*y m ~~ start(c(0.87, 0.87))*m m1 ~~ start(c(0.36, 0.36))*m1 m2 ~~ start(c(0.36, 0.36))*m2 m3 ~~ start(c(0.36, 0.36))*m3 " # med1 and med2 are the mediation effect for group1 and group2, respectively. indirect <- " med1 := a1*b1 med2 := a2*b2 diffmed := a1*b1 - a2*b2 " # change nrep and nboot to at least 1,000 in real analysis bootstrap <- wp.mc.sem.boot(ex3model, indirect, nobs=c(400,200), nrep=10, nboot=10) summary(bootstrap) #Example: A Longitudinal Mediation Model ex4model <- " x2 ~ start(.9)*x1 + x*x1 x3 ~ start(.9)*x2 + x*x2 m2 ~ start(.3)*x1 + a*x1 + start(.3)*m1 + m*m1 m3 ~ start(.3)*x2 + a*x2 + start(.3)*m2 + m*m2 y2 ~ start(.3)*m1 + b*m1 + start(.7)*y1 + y*y1 y3 ~ start(.3)*m2 + b*m2 + start(.7)*y2 + y*y2 + start(0)*x1 + c*x1 x1 ~~ start(.37)*m1 x1 ~~ start(.27)*y1 y1 ~~ start(.2278)*m1 x2 ~~ start(.19)*x2 x3 ~~ start(.19)*x3 m2 ~~ start(.7534)*m2 m3 ~~ start(.7534)*m3 y2 ~~ start(.3243)*y2 y3 ~~ start(.3243)*y3 " indirect <- "ab := a*b" # change nrep and nboot to at least 1,000 in real analysis set.seed(10) bootstrap <- wp.mc.sem.boot(ex4model, indirect, nobs=500, nrep=10, nboot=10) summary(bootstrap) ``` ## References Rosseel, Y. (2012). Lavaan: An R package for structural equation modeling and more. Version 0.5–12 (BETA). Ghent, Belgium: Ghent University. Thoemmes, F., MacKinnon, D. P., & Reiser, M. R. (2010). Power analysis for complex mediational designs using Monte Carlo methods. Structural Equation Modeling, 17(3), 510-534. Zhang, Z. (2014). Monte Carlo based statistical power analysis for mediation models: Methods and software. Behavior research methods, 46(4), 1184-1198. Zhang, Z., & Yuan, K.-H. (2018). Practical Statistical Power Analysis Using Webpower and R (Eds). Granger, IN: ISDSA Press.

wp.mc.sem.boot function