Overview MAXENT-Modeling: A framework for Discrete MAXENT-Models and RMs IRT-Modeling? ◮ Conceptual foundation and relationships of the MAXENT-framework ◮ The canonical MAXENT-distribution is structurally similar to Dipl.-Psych. Georg Hosoya Rasch’s modeling framework (Rasch, 1961) FU-Berlin ◮ Example: PCM Cluster Languages of Emotion ◮ Model generation: Is it possible to apply the MAXENT-modeling framework to solve practical problems 10. Februar 2012 with IRT? Dipl.-Psych. Georg Hosoya MAXENT-Modeling: A framework for IRT-Modeling? Dipl.-Psych. Georg Hosoya MAXENT-Modeling: A framework for IRT-Modeling? Overview MAXENT-Models and RMs A RM for the assessment of intraindividual variability ◮ Are interindividual differences in intraindividual variability measurable? ◮ Definition of a model for multivariate, discrete time series Principle of Maximum-Entropy (ambulatory assessment) Reasoning under uncertainty: given a set of constraints, choose the ◮ Model properties, transition matrix, CRFs, difference of logits one distribution (model) that a.) is in congruence with the available of adjacent categories for two persons information (constraints) and b.) has maximum information entropy. (see e.g. Jaynes, 1957a, 1957b and 2003, chap. 11). Application to ambulatory assessment data ◮ Estimation of model parameters via MCMC ◮ Heuristic assessment of model fit via standardized residuals ◮ Andrich-Reliability Dipl.-Psych. Georg Hosoya MAXENT-Modeling: A framework for IRT-Modeling? Dipl.-Psych. Georg Hosoya MAXENT-Modeling: A framework for IRT-Modeling?
MAXENT-Models and RMs MAXENT-Models and RMs Expectation of f k ( x ) under the model Solution for the discrete case (details see Jaynes, 2003, chap. 11) � f k � = − ∂ log Z (3) ∂λ k m 1 (Exponential family → sufficient statistics [Expectations should fit � p ( X = x ) = Z ( λ 1 , . . . , λ m ) · exp − λ j f j ( x ) (1) the acutual data F k ]) j =1 Variance of f k ( x ) Partition function: n m − � f k � 2 = − ∂ 2 log Z f 2 � � � � (4) . Z ( λ 1 , . . . , λ m ) = exp − λ j f j ( x ) (2) k ∂λ 2 k i =1 j =1 Useful for assessing test-information and model-fit (standardized residuals) Dipl.-Psych. Georg Hosoya MAXENT-Modeling: A framework for IRT-Modeling? Dipl.-Psych. Georg Hosoya MAXENT-Modeling: A framework for IRT-Modeling? MAXENT-Models and RMs MAXENT-Models and RMs Compatible modeling frameworks: Rasch’s (1961) and Jayne’s framework (1957) ◮ Undirected graphical models (fusion of probability theory and graph theory) exp[ φ x θ v + ψ x α i + χ x θ v α i + ω x ] P ( X vi = x ) = (5) ◮ Markov random fields � m l =0 exp[ φ l θ v + ψ l α i + χ l θ v α i + ω l ] ◮ Boltzmann distribution / Gibbs distribution � � − � m exp j =1 λ j f j ( x ) ◮ Logistic Regression / Perzeptron p ( X = x ) = (6) � � ◮ Multinomial Logit Regression � n − � m i =1 exp j =1 λ j f j ( x ) ◮ Bayes Networks ◮ HMMs Structurally equivalent, only the notation differs. ◮ ... Dipl.-Psych. Georg Hosoya MAXENT-Modeling: A framework for IRT-Modeling? Dipl.-Psych. Georg Hosoya MAXENT-Modeling: A framework for IRT-Modeling?
MAXENT-Models and RMs MAXENT-Models and RMs Expectation of x vi under the model � m l =1 l · exp { θ v · l + β il } ∂ logZ = (9) Example PCM (Masters, 1982) � m ∂θ v l =1 exp { θ v · l + β il } = � x vi � (10) exp { θ v x vi + β ix } p ( X vi = x vi ) = l =1 exp { θ v l + β il } , (7) � m Variance of x vi under the model m � Z = exp { θ v l + β il } . (8) l =1 l 2 · exp { θ v l + β il } l =1 � m ∂ 2 logZ = − (11) � m ∂θ 2 l =1 exp { θ v l − β il } v � 2 �� m l =1 l · exp { θ v l + β il } � m l =1 exp { θ v l + β il } − � x vi � 2 . x 2 � � = (12) vi Dipl.-Psych. Georg Hosoya MAXENT-Modeling: A framework for IRT-Modeling? Dipl.-Psych. Georg Hosoya MAXENT-Modeling: A framework for IRT-Modeling? MAXENT-Models and RMs A RM for the assessment of intraindividual variability Intermediate conlcusions Model based on absolute sucessive difference ◮ RMs and MAXENT-Models are structurally very similar, if not equivalent � � p ( X vi [ t ] = x vit | x vi [ t − 1] ) = exp | x vi [ t ] − x vi [ t − 1] | · η v − β ix ◮ MAXENT-Models fit into the broader framework of � m � � l =1 exp | l − x vi [ t − 1] | · η v − β il undirected graphical models ◮ The MAXENT-framework gives a reason rooted in information theory for the form of a certain model ◮ x vit : response of a person v on item i at timepoint t ◮ Probabilistic psychometrics is not an ivory tower, very broad ◮ x vi [ t − 1] : response of a person v on item i at timepoint t − 1 (mostly unexplored) connections to other disciplines via the ◮ η v : variability parameter of Person v modeling technique ◮ β ix : easiness of category l of item i Next step: Exploration of the framework for the formulation of new ◮ sum zero norming over item-specific category parameters probabilistic IRT-models for research questions in psychometrics Dipl.-Psych. Georg Hosoya MAXENT-Modeling: A framework for IRT-Modeling? Dipl.-Psych. Georg Hosoya MAXENT-Modeling: A framework for IRT-Modeling?
A RM for the assessment of intraindividual variability A RM for the assessment of intraindividual variability Parameters (one item, 4 response categories) Parameters (one item, 4 response categories) η v = 0, β = [ − 0 . 5 , 0 . 5 , 0 . 5 , − 0 . 5] η v = − 1 . 5, β = [0 , 0 , 0 , 0] Transition matrix Relatively stable transition matrix x vi [ t ] =1 x vi [ t ] =2 x vi [ t ] =3 x vi [ t ] =4 x vi [ t ] =1 x vi [ t ] =2 x vi [ t ] =3 x vi [ t ] =4 x vi [ t − 1] =1 .13 .37 .37 .13 x vi [ t − 1] =1 .57 .35 .08 .00 x vi [ t − 1] =2 .13 .37 .37 .13 x vi [ t − 1] =2 .06 .76 .17 .01 x vi [ t − 1] =3 .13 .37 .37 .13 x vi [ t − 1] =3 .01 .17 .76 .06 x vi [ t − 1] =4 .13 .37 .37 .13 x vi [ t − 1] =4 .00 .08 .35 .57 Independence of x vit from x vi [ t − 1] . Dipl.-Psych. Georg Hosoya MAXENT-Modeling: A framework for IRT-Modeling? Dipl.-Psych. Georg Hosoya MAXENT-Modeling: A framework for IRT-Modeling? A RM for the assessment of intraindividual variability A RM for the assessment of intraindividual variability 4 conditional CRFs per Item Difference of logits for adjacent category probabilities for two persons � p ( x vit +1 | x vi [ t − 1] ) � p ( x vit +1 | x vi [ t − 1] ) � ′ � log − log p ( x vit | x vi [ t − 1] ) p ( x vit | x vi [ t − 1] ) = η v − η ′ (13) v [ f ( x vit + 1) − f ( x vit )] f ( x vit ) = | x vit − x vi [ t − 1] | (14) f ( x vit + 1) = | ( x vit + 1) − x vi [ t − 1] | (15) Dipl.-Psych. Georg Hosoya MAXENT-Modeling: A framework for IRT-Modeling? Dipl.-Psych. Georg Hosoya MAXENT-Modeling: A framework for IRT-Modeling?
Application to ambulatory assessment data Application to ambulatory assessment data Dataset: Crayen et al. 2012 Subscale: pleasant-unpleasent mood ◮ 165 students at Freie Universit¨ at Berlin ◮ unwell-well ◮ study of mood regulation ◮ bad-good ◮ 2 weeks of ambulatory assessment with handheld device (7-8 ◮ unhappy-happy signals/day) ◮ discontent-content ◮ relevant here: MDBF short-form (Steyer, Schwenkmezger, ◮ 4-point rating scale Nostitz & Eid, 1997) ◮ n=64760 Dipl.-Psych. Georg Hosoya MAXENT-Modeling: A framework for IRT-Modeling? Dipl.-Psych. Georg Hosoya MAXENT-Modeling: A framework for IRT-Modeling? Application to ambulatory assessment data Application to ambulatory assessment data MCMC-results Posterior distributions of category- and person parameters Dipl.-Psych. Georg Hosoya MAXENT-Modeling: A framework for IRT-Modeling? Dipl.-Psych. Georg Hosoya MAXENT-Modeling: A framework for IRT-Modeling?
Application to ambulatory assessment data Application to ambulatory assessment data Paremeters of latent score distribution Reliability Andrich: Rel = var (ˆ η ) − var ( e ) = 0 . 225 − 0 . 011 = 0 . 95 (16) var (ˆ η ) 0 . 225 Based on latent score distribution: Rel = var (ˆ var ( η ) = 0 . 225 η ) 0 . 238 = 0 . 95 (17) Dipl.-Psych. Georg Hosoya MAXENT-Modeling: A framework for IRT-Modeling? Dipl.-Psych. Georg Hosoya MAXENT-Modeling: A framework for IRT-Modeling? Application to ambulatory assessment data Application to ambulatory assessment data Standardized residuals Item Outfit χ 2 Item Outfit df p well 1.06 17217.63 16190 0.00 good 1.00 16239.95 16190 0.39 contempt 0.99 16068.96 16190 0.75 happy 0.88 14232.33 16190 1.00 Dipl.-Psych. Georg Hosoya MAXENT-Modeling: A framework for IRT-Modeling? Dipl.-Psych. Georg Hosoya MAXENT-Modeling: A framework for IRT-Modeling?
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