Mplus 5.21 (Muthén & Muthén, 2009) was used for all analyses. For structural equation modeling results used in ITT analyses, multiple indices of model fit are provided. The use of multiple fit indices to assess structural equation model fit has become common in recent decades (Fan & Sivo, 2005). A good-fitting structural equation model should contain at least two of the following: a nonsignificant chi-square test, although this can be violated in large samples (Bentler & Bonnett, 1980); comparative fit index (CFI) close to 1.0 (Hu & Bentler, 1999); Tucker-Lewis index close to 1.0 (Bentler & Bonnet, 1980); and root mean square error approximation (RMSEA) less than .08 (MacCallum, Browne, & Sugawara,1996). FCU effects were determined by including the dummy-coded intervention status variable (1 = intervention) as a predictor of problem behavior. To assess whether increasingly strict definitions of engagement increased the preventative effect of the FCU on early-child disruptive behavior, CACE analysis with a growth mixture framework was used. CACE enables a distinction to be made between those participants who actually receive the intervention and those who do not
