Sounds good, right?<\/p>\n
Bayesian priors allow cross-loadings and residual covariances of SEM’s to vary a small degree (i.e., replace exact zeros with approximate zeros from informative, small-variance priors) and be evaluated (see\u00a0Asparouhov, Muth\u00e9n, & Morin, 2015;\u00a0Muth\u00e9n, & Asparouhov 2012). \u00a0The researcher can thereby discover whether 0 cross-loadings likely exist in the “true” population model, given their data, and refine their model accordingly.<\/p>\n
I recently had the opportunity to apply this technique to a bi-factor model for a new scale that had previously only been subjected to\u00a0a traditional CFA. \u00a0The model had strong theoretical support, but fit indices and some loadings were not supportive of the general factor as theorized. \u00a0These conditions provide a\u00a0perfect opportunity to use\u00a0Bayesian CFA (BCFA) to refine the model\u00a0for cross-validation.<\/p>\n
MPlus 7 was used to specify and test a model where cross-loadings were assigned\u00a0normally distributed priors with\u00a00 means and variances of .01. \u00a0The technique\u00a0is not difficult for those familiar with CFA in MPlus. \u00a0Working from a typical CFA, the researcher need only:<\/p>\n
ANALYSIS: ESTIMATOR = BAYES; !Uses two independent MCMC chains\r\n PROCESSORS = 2; !To speed up computations if you have 2 processors\r\n FBITERATIONS=15000; !Minimum number of Markov Chain iterations<\/pre>\n
f1 BY y1-y3;\r\nf2 BY y4-y10;\r\n\r\nto\r\n\r\nf1 BY y1-y3\r\n y4-y10; !Cross-loadings\r\nf2 BY y4-y10\r\n y1-y3; !Cross-loadings<\/pre>\n
f1 BY y1-y3\r\n y4-y10 (f1xlam1-f1xlam7); !Cross-loadings (with assigned labels)\r\nf2 BY y4-y10\r\n y1-y3(f2xlam1-f2xlam3); !Cross-loadings (with assigned labels)<\/pre>\n
MODEL PRIORS:\r\n f1xlam1-f2xlam3~N(0,0.01); !You can list across factors if ordered<\/pre>\nNaturally, identification and metric setting needs to be addressed\u00a0first (for example, MPlus sets the first loading of a factor to one by default, but you may want to free that and instead set the variance of each factor to 1 instead). \u00a0Yet, hopefully this illustrates\u00a0the ease at which cross-loadings can be assigned Bayesian priors. \u00a0For those interested in assigning\u00a0residual covariances prior distributions the concept is very similar. \u00a0A full example is given in the Asparouhov, Muth\u00e9n, and Morin (2015) article.<\/p>\n
Now for the beauty (i.e., application) of BCFA. \u00a0Cross-loadings were assigned a normal prior distribution with mean 0 and variance of 0.01.\u00a0 Were the actual\u00a0loadings 0 given our data? \u00a0The prior and posterior distributions for each loading can be viewed\u00a0to verify this (choose\u00a0“Plot” > “View Plots” in MPlus) or the confidence intervals for the cross-loadings can be examined in the MPlus output. \u00a0Below\u00a0is an example of a cross-loading for an item I analyzed:<\/p>\n
First, examine the prior distribution (made up of 15,000 iterations):
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