Dena Pastor James Madison University pastorda@jmu

1. Modeling item response profiles using factor models, latent class models, and latent variable hybrids Dena Pastor James Madison University pastorda@jmu.edu

2. Purposes of the Presentation • To present the model-implied item response profiles (IRPs) that correspond to latent variable models used with dichotomous item response data • To provide an example of how these models can be used in practice

3. Item Response Profiles (IRPs)

4. Pattern Differences IRPs for classes of examinees with different patterns

5. Elevation Differences IRPs for classes of examinees with the same pattern, but differences in elevation

6. Latent Variable Model C NON-PARALLEL PARALLEL Latent Class Model C is a latent categoricalvariable with as many levels as # of classes C is a nominal latent variable C is a ordinal latent variable

7. Exploratory Process • In latent class modeling a variety of models are fit to the data with differing numbers of classes • 1-class model, 2-class model, 3-class model, etc. • Use fit indices and a priori expectations to determine the number of classes to retain • Can allow latent categorical variable to be nominal and examine resulting profiles; can also constrain latent categorical variable to be ordinal

8. F Alternative Model for Parallel Profiles Do we have 3 classes, with no variability within class? Factor Model F is a latent continuousvariable OR Do we have 1 profile with systematic variability within class?

9. Different Models for Different IRPs …+ within profile variability 1 profile… LCM: 1 class Factor Model 2 parallel profiles… …+ within profile variability Latent Variable Hybrids Semi-parametric Factor Model LCM: 2 classes (C is ordinal) 2 non-parallel profiles… …+ within profile variability Factor Mixture Model LCM: 2 classes (C is nominal)

10. Number of profiles? (number of classes) 1+ 1 Nature of profile differences? Non-parallel Parallel Decisions Systematic variability within profiles? Systematic variability within profiles? Systematic variability within profiles? no yes no yes no yes Latent class model (LCM) Factor model (FM) LCM with parallel profiles Semi-parametric factor model (SPFM) LCM with non-parallel profiles Factor mixture model (FMM) Models IRPs

11. F F F Latent class model (LCM) C C C F F F C C C 2 classes 2 classes C Factor mixture model(FMM) Semi-parametric factor model (SPFM) 2 classes, w/in class factor variance = 0 2 classes, w/in class factor variance = 0 1 class: Factor Model! 1 class: Factor Model!

12. Marginal probability of getting an item correct is sum across classes of probability of getting item correct conditional on class membership F C F C Conditional probability differs across models C Factor mixture model (FMM) Semi-parametric factor model (SPFM) Latent class model (LCM)

13. Latent class model (LCM) Latent Variable Distribution IRP Path diagram C C C C is ordinal C is nominal

14. F F Semi-Parametric Factor Model (SPFM) C C Latent Variable Distribution IRP Path diagram Measurement Invariance Same measurement model parameters (thresholds, loadings) for each class Quantitative differences between classes

15. Factor Mixture Model (FMM) F F C C Latent Variable Distribution IRP Path diagram Measurement Non-Invariance Different measurement model parameters (thresholds, loadings) for each class Qualitative differences between classes

16. Example • 9 dichotomously scored items measuring 3 aspects of psychosocial research: • Confidentiality • Generalizability • Informed Consent • Sample 2,259 incoming freshmen tested in low-stakes conditions prior to start of classes

17. Exploratory Model Selection • Exploratory model selection approach to answer the question, “What type and number of latent variables are most salient for our data?” • Reasons to believe that IRPs would differ in pattern and/or elevation because students differ in: • Completion of psychosocial coursework • Effort they put forth on test

18. Model Fit Indices

19. IRPs of 4 Class LCM 0.18 0.36 0.25 0.20 generalizability

20. 2-class FMM Factor Variability Within Each Class 0.44 0.56 26. Which ethical practice is not considered by Marty? She failed to obtain informed consent from her participants She failed to randomly select participants … …

21. Visually Conveying Loading Information X Y

22. Validity Evidence for 2-class FMM Solution X Y • Students with higher SAT-V scores, who reported put forth more effort on the test, and who have completed psychosocial coursework more likely to be in Class X • Positive relationship between SAT-V, coursework completion and factor scores in that class (negative relationship with effort) • Negative relationship between number of missing responses and factor scores in Class Y

23. Correspondence Between Models C & D from LCM, Y from FMM A & B from LCM, X from FMM X & Y from FMM with intervals

24. Parting Thoughts… • These models are like potato chips… • It was so much easier to settle on a brand of chip when I had a limited number of brands to choose from • But I also like having more brands because it increases my chances of finding the brand that is right for me • With all these brands, it is possible that some are selling essentially the same chip….but which ones? • When two brands are essentially the same chip, what criteria do I use to choose between the two brands?

25. Questions? pastorda@jmu.edu Pastor, D. A., & Gagné, P. (2013). Mean and covariance structure mixture models. In G. R. Hancock & R. O. Mueller (Eds.), Structural Equation Modeling: A Second Course (2nd Ed.). Greenwich, CT: Information Age. Pastor, D. A., Lau, A. R., & Setzer, J. C. (2007, August). Modeling item response profiles using factor models, latent class models, and latent variable hybrids. Poster presented at the annual meeting of the American Psychological Association, San Francisco.