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

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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

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Item Response Profiles (IRPs)

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Pattern Differences IRPs for classes of examinees with different patterns

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

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Latent Variable Model PARALLELNON-PARALLEL C Latent Class Model C is a latent categorical variable with as many levels as # of classes C is a nominal latent variable C is a ordinal latent variable

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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

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

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

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

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

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

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

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

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

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Example 9 dichotomously scored items measuring 3 aspects of psychosocial research: 1.Confidentiality 2.Generalizability 3.Informed Consent Sample 2,259 incoming freshmen tested in low- stakes conditions prior to start of classes

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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

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Model Fit Indices ModelLL# parasBICSSA-BICLMR FM1f-12628182539525338NA 2f-12576262535225270NA 3f-12547332534825243NA LCM1c-1293892594625918NA 2c-126491925445253840.00 3c-125912925406253140.07 4c-125383925377252530.01 5c-125204925418252620.14 SPFM1f2c-126182125399253320.01 FMM1f2c-125393525348252370.00

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IRPs of 4 Class LCM 0.18 0.36 0.25 0.20 generalizability

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

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Visually Conveying Loading Information XY

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Validity Evidence for 2-class FMM Solution 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 XY

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

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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?

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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 (2 nd 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.

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