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1.  Using regression mixture models with non-normal data: Examining an ordered polytomous approach 
Journal of statistical computation and simulation  2011;83(4):10.1080/00949655.2011.636363.
Mild to moderate skew in errors can substantially impact regression mixture model results; one approach for overcoming this includes transforming the outcome into an ordered categorical variable and using a polytomous regression mixture model. This is effective for retaining differential effects in the population; however, bias in parameter estimates and model fit warrant further examination of this approach at higher levels of skew. The current study used Monte Carlo simulations; three thousand observations were drawn from each of two subpopulations differing in the effect of X on Y. Five hundred simulations were performed in each of the ten scenarios varying in levels of skew in one or both classes. Model comparison criteria supported the accurate two class model, preserving the differential effects, while parameter estimates were notably biased. The appropriate number of effects can be captured with this approach but we suggest caution when interpreting the magnitude of the effects.
doi:10.1080/00949655.2011.636363
PMCID: PMC3653334  PMID: 23687397
Regression mixture models; non-normal errors; differential effects
2.  Not quite normal: Consequences of violating the assumption of normality in regression mixture models 
Regression mixture models are a new approach for finding differential effects which have only recently begun to be used in applied research. This approach comes at the cost of the assumption that error terms are normally distributed within classes. The current study uses Monte Carlo simulations to explore the effects of relatively minor violations of this assumption, the use of an ordered polytomous outcome is then examined as an alternative which makes somewhat weaker assumptions, and finally both approaches are demonstrated with an applied example looking at differences in the effects of family management on the highly skewed outcome of drug use. Results show that violating the assumption of normal errors results in systematic bias in both latent class enumeration and parameter estimates. Additional classes which reflect violations of distributional assumptions are found. Under some conditions it is possible to come to conclusions that are consistent with the effects in the population, but when errors are skewed in both classes the results typically no longer reflect even the pattern of effects in the population. The polytomous regression model performs better under all scenarios examined and comes to reasonable results with the highly skewed outcome in the applied example. We recommend that careful evaluation of model sensitivity to distributional assumptions be the norm when conducting regression mixture models.
doi:10.1080/10705511.2012.659622
PMCID: PMC3384700  PMID: 22754273
3.  Developing measures on the perceptions of the built environment for physical activity: a confirmatory analysis 
Background
Minimal validity evidence exists for scales assessing the built environment for physical activity. The purpose of this study was to assess the test-retest reliability and invariance of a three-factor model (Neighborhood Characteristics, Safety/Crime, and Access to Physical Activity Facilities) across gender, race, geographic location, and level of physical activity.
Methods
To assess measurement invariance, a random sample of 1,534 adults living in North Carolina or Mississippi completed a computer assisted telephone interview that included items examining perceptions of the neighborhood for physical activity. Construct level test-retest reliability data were collected from a purposeful sample of 106 participants who were administered the questionnaire twice, approximately two weeks apart. Fit indices, Cronbach's alpha, Mokken H and Spearman correlation coefficients (SCC) were used to evaluate configural and co/variance invarianc,e and intraclass correlation coefficients (ICC) were used to assess reliability.
Results
Construct test-retest reliability was strong (ICC 0.90 to 0.93). SCC for Neighborhood Characteristics and Crime/Safety were weak with Access (0.21 and 0.25), but strong between Crime/Safety and Neighborhood Characteristics (0.62). Acceptable fit and evidence of measurement invariance was found for gender, race (African American and White), geographic location, and level of physical activity. Fit indices consistently approached or were greater than 0.90 for goodness of fit index, normed fit index, and comparative fit index which is evidence of configural invariance. There was weak support of variance and covariance invariance for all groups that was indicative of factorial validity.
Conclusions
Support of the validity and reliability of the three-factor model across groups expands the possibilities for analysis to include latent variable modeling, and suggests these built environment constructs may be used in other settings and populations.
doi:10.1186/1479-5868-7-72
PMCID: PMC2959084  PMID: 20929571

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