Supplement for Kelley & Lai ( 2011 )

Using MBESS to implement the Methods Discussed The methods discussed and developed in Kelley and Lai (2011a) are implemented with the freely available MBESS (Kelley, 2007a, 2007b; Kelley & Lai, 2011b) R (R Development Core Team, 2011) package.1 First, it is helpful to know that a confidence interval for ε can be formed with the function ci.rmsea(). To form a 95% confidence interval for εwhen ε̂ = .05, ν = 25, and N = 450, ci.rmsea() can be called as follows: > ci.rmsea(rmsea=.05, df=25, N=450, conf.level=.95), which yields a 95% confidence interval of [.0269, .0720]. Notice that only summary statistics are necessary, thus allowing researchers to obtain a confidence interval for ε given only summary statistics generally reported in applications of SEM. Note that this task can also be done indirectly with SAS’s CNONCT function (2004) and Steiger’s Noncentral Distribution Calculator Steiger (2007), among others. The AIPE sample size method for the RMSEA we developed can be implemented with the ss.aipe.rmsea() function. To replicate the empirical example we discussed in Kelley and Lai (2011a), in the scenario where the researcher set ε∗ to .04 for the model in Holahan, Moos, Holahan, and Brennan (1997) and desires to obtain a 95% confidence interval that is no wider than 0.035, ss.aipe.rmsea() can be used as follows: > ss.aipe.rmsea(RMSEA=.04, df=30, width=0.035, conf.level=.95), which yields a necessary sample size 643, the same as that in our previous example. As explained in Kelley and Lai (2011a), any sample size planning methods need to be viewed as approximations, since they necessarily require some information that is not fully known about population parameters as input. In the present article, the N calculated is conditional on ε∗, ω, and ν. To verify how well the AIPE implied N leads to a confidence interval width compared to the desired width ω, a Monte Carlo simulation study can be performed. First, the researcher needs to provide a model M(·) and a covariance matrix Σ; there should be some discrepancy between M(·) and Σ, so that ε is larger than 0. There are two common ways to create the discrepancy. One way is to use the model-implied covariance matrix M(θ) as Σ, and then misspecify the original model by adding or removing paths. Let the misspecified model be M∗(·), then ε can be obtained by fitting M∗(·) to Σ. We used this method to create misspecified models in our Monte Carlo simulation study discussed previously in the text. Another way is to use the Cudeck-Browne procedure (Cudeck & Browne,