Outline Rasch model Mixture models Rasch mixture models Illustration: Simulated data Application: Verbal aggression Mixtures of Rasch Models with R Package psychomix Hannah Frick, Carolin Strobl, Friedrich Leisch, Achim Zeileis http://eeecon.uibk.ac.at/~frick/ Introduction Rasch Model Latent traits measured through probabilistic models for item Probability for person i to solve item j : response data. P ( Y ij = y ij | θ i , β j ) = exp { y ij ( θ i − β j ) } 1 + exp { θ i − β j } . Here, Rasch model for binary items. Crucial assumption of measurement invariance: All items measure y ij : Response by person i to item j . the latent trait in the same way for all subjects. θ i : Ability of person i . Check for heterogeneity in (groups of) subjects, either based on β j : Difficulty of item j . observed covariates or unobserved latent classes. Mixtures of Rasch models to address heterogeneity in latent By construction: classes. No covariates, all information is captured by ability and difficulty. Both parameters θ and β are on the same scale: If β 1 > β 2 , then item 1 is more difficult than item 2 for all subjects. Central assumption of measurement invariance needs to be checked for both manifest and latent subject groups.
Rasch Model: Estimation Mixture Models Joint estimation of θ and β is inconsistent. Assumption: Data stems from different classes but class membership is unknown. Conditional ML estimation: Use factorization of the full likelihood on basis of the scores r i = � m j = 1 y ij : Modeling tool: Mixture models. Mixture model = � weight × component. L ( θ, β ) = f ( y | θ, β ) Components represent the latent classes. They are densities or = h ( y | r , θ, β ) g ( r | θ, β ) (regression) models. = h ( y | r , β ) g ( r | θ, β ) . Weights are a priori probabilities for the components/classes, treated either as parameters or modeled through concomitant Estimate β from maximization of h ( y | r , β ) . variables. Also maximizes L ( θ, β ) if g ( r |· ) is assumed to be independent of θ and β ; but potentially depending on auxiliary parameters δ : g ( r | δ ) . Rasch Mixture Models: Framework Rasch Mixture Models: Score Probabilities Full mixture: Original proposition by Rost (1990): Discrete distribution with parameters (probabilities) g ( r ) = Ψ r . Weights: Either (non-parametric) prior probabilities π k or weights π ( k | x , α ) based on concomitant variables x , e.g., a Number of parameters necessary is potentially very high: (number of items - 1) × (number of components). multinomial logit model. Components: Conditional likelihood for item parameters and More parsimonious: Assume parametric model on score specification of score probabilities probabilities, e.g., using mean and variance parameters. General approach: Conditional logit model encompassing the n K original saturated parameterization and a mean/variance � � f ( y | π, α, β, δ ) = π ( k | x i , α ) h ( y i | r i , β k ) g ( r i | δ k ) . parameterization (with only two parameters per component) as i = 1 k = 1 special cases Estimation of all parameters via ML through the EM algorithm. exp { z ⊤ r δ } g ( r | δ ) = . � m − 1 j = 1 exp { z ⊤ j δ }
Rasch Mixture Models: Score Probabilities Rasch Mixture Models: Score Models Motivation: When checking for measurement invariance, items are of Mean/variance: interest, not the scores. Parsimonious: 2 parameters per class. Mixture might catch on to latent score groups, even when no Idea: Use differential item functioning (DIF) is present. g ( r ) = constant Saturated: Equivalent to: Score distribution is the same over all components. Non-identified if no DIF present, as a mixture of multinomial models is itself a multinomial model. Interpretation: Possibly too many parameters to detect moderate DIF . Score distribution is irrelevant to the mixture. Constant: Consequently, the mixture is only influenced by latent classes Mixture only influenced by latent groups in items (i.e., DIF), yet regarding the item parameters. parsimonious. Differences in the score distribution (if any) do not influence the Potentially less accurate if latent groups are present in both scores mixture, neither if coincident with differences in the item and items – and the groups coincide. parameters nor if w.r.t. other classes. Trade accuracy for robustness. Software Illustration: No DIF Available in R in package psychomix at Data generating process: http://CRAN.R-project.org/package=psychomix m = 20 items, n = 100 subjects. Based on package flexmix (Grün and Leisch, 2008) for flexible No DIF: all item difficulties β = 0. estimation of mixture models. Differences in scores through 2 different abilities: {− 1 . 8 , 1 . 8 } . Based on package psychotools for estimation of Rasch models. Frick et al. (2011), provides implementation details and hands-on 0 5 10 15 20 practical guidance. See also vignette("raschmix", package 1 2 = "psychomix") . 60 Percent of Total 40 20 0 0 5 10 15 20 scores
Illustration: No DIF Illustration: No DIF meanvar score model constant score model Centered item difficulty parameters ● ● ● ● 0.5 ● AIC ● AIC ● ● ● ● ● ● 0.30 ● ● ● ● 2080 BIC BIC 2140 ● ● ● ● 0.0 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.20 ● ● Density ● 2040 ● 2100 −1.0 0.10 ● 2000 ● Comp. 1 ● Comp. 2 ● 0.00 2060 −2.0 ● ● ● ● ● 1 2 1 2 1 5 10 15 20 0 5 10 15 20 number of components number of components Items scores Figure: Mixture Rasch model with 1 to 2 classes and a meanvar (left) and a Figure: Estimated item parameters (left) and score probabilities with empirical constant (right) specification of the score model. score distribution (right) of the 2-class Rasch mixture model with a meanvar score specification. Illustration: Moderate DIF Illustration: Moderate DIF Data generating process: saturated score model constant score model m = 20 items, n = 1000 subjects. 28000 2 items with DIF: β = ( − 1 . 2 , 1 . 2 ) and β = ( 1 . 2 , − 1 . 2 ) , ● ● AIC ● AIC all other items with β = 0. ● BIC BIC All abilities θ = 0. 29250 27850 ● ● 27700 29150 1.5 1.5 1.5 ● ● ● ● ● ● ● ● Item difficulty Item difficulty Item difficulty 0.5 0.5 0.5 1 2 1 2 ● ● ●●●●●● ●●●●●● ●●●●●●●● ●●●●●●●● ●●●● ●●●● ●●●●●●●●●●●●● ●●●●●●●●●●●●● ●●● ●●● ●●●●● ●●●●● ●●●●●● ●●●●●● ●●● ●●● ●●● ●●● ● ● ● ● ● ● ● ● −0.5 −0.5 −0.5 number of components number of components ● ● ● ● −1.5 −1.5 −1.5 1 5 10 15 20 1 5 10 15 20 1 5 10 15 20 Figure: Mixture Rasch model with 1 to 2 classes and a saturated (left) and a Items Items Items constant (right) specification of the score model. Figure: Estimated item difficulties for whole sample and in both subsamples.
Illustration: Moderate DIF Application: Verbal Aggression Data Behavioral study of psychology students: 243 women and 73 men. Description of frustrating situations: ● ● Comp. 1 3 S1: A bus fails to stop for me. ● Comp. 2 S2: I miss a train because a clerk gave me faulty information. Item difficulty 2 Behavioral mode: Want or do. 1 ● Verbally aggressive response: Curse, scold, or shout. ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0 ● ● ● ● ● ● ● ● ● ● ● 12 resulting items: S1WantCurse, S1DoCurse, S1WantScold, . . . , ● −1 ● S2WantShout, S2DoShout ● Covariates: Gender and an anger score. 1 5 10 15 20 Items Figure: Estimated item difficulties in a 2-class Rasch mixture model with a constant score model. Verbal Aggression: Analysis Verbal aggression: Item profiles Fit model: R> set.seed(1) Centered item difficulty parameters R> mix <- raschmix(resp2 ~ 1, data = va12, k = 1:4, Comp. 1 Comp. 2 Comp. 3 + scores = "constant", nrep = 5) 4 ● R> mixC <- raschmix(resp2 ~ gender + anger, data = va12, ● + k = 2:4, scores = "constant", nrep = 5) ● ● ● 2 ● ● ● ● ● ● ● Select model: ● ● ● 0 ● ● ● ● ● ● ● ● ● ● ● R> rbind(mix = BIC(mix), mixC = c(NA, BIC(mixC))) ● ● ● ● ● −2 ● ● ● ● 1 2 3 4 ● mix 3881.065 3854.193 3847.796 3865.268 1 2 3 4 5 6 7 8 9 10 11 12 1 2 3 4 5 6 7 8 9 10 11 12 1 2 3 4 5 6 7 8 9 10 11 12 mixC NA 3861.127 3850.129 3867.554 Item R> va12_mix <- getModel(mixC, which = "3") Figure: Item difficulty profiles for the 3-component Rasch mixture model. Plot item profiles and effects of concomitant variables: Items 1–6: Situation S1 (bus). Items 7–12: Situation S2 (train). Order: want/do curse, want/do scold, want/do shout. R> xyplot(va12_mix) R> effectsplot(va12_mix)
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