Joint Modeling of Feedback-Use and Time Data Advances in Bayesian Item Response Modeling Jean-Paul Fox University of Twente Department of Research Methodology, Measurement and Data Analysis Faculty of Behavioural Sciences Enschede, Netherlands J.-P. Fox Advances in Bayesian Item Response Modeling
Outline Overview 1 Introduction Feedback Behavior Study Bayesian Response Modeling J.-P. Fox Advances in Bayesian Item Response Modeling
Outline Overview 1 Introduction Feedback Behavior Study Bayesian Response Modeling 2 Complex Multivariate Count Data Multivariate Zero-Inflated Poisson Modeling Results Feedback Behavior Study: Use (Latent) Predictors Results J.-P. Fox Advances in Bayesian Item Response Modeling
Outline Overview 1 Introduction Feedback Behavior Study Bayesian Response Modeling 2 Complex Multivariate Count Data Multivariate Zero-Inflated Poisson Modeling Results Feedback Behavior Study: Use (Latent) Predictors Results 3 Discussion J.-P. Fox Advances in Bayesian Item Response Modeling
Introduction Feedback Behavior Study Overview 1 Introduction Feedback Behavior Study Bayesian Response Modeling 2 Complex Multivariate Count Data Multivariate Zero-Inflated Poisson Modeling Results Feedback Behavior Study: Use (Latent) Predictors Results 3 Discussion J.-P. Fox Advances in Bayesian Item Response Modeling
Introduction Feedback Behavior Study Formative Computer-Based Assessment ◮ Two-stage testing: Ability - feedback use ◮ Observe response times (speed) and feedback times (reading) ◮ Dutch study: Differential use of feedback in test assessment J.-P. Fox Advances in Bayesian Item Response Modeling
Introduction Feedback Behavior Study Formative Computer-Based Assessment ◮ Two-stage testing: Ability - feedback use ◮ Observe response times (speed) and feedback times (reading) ◮ Dutch study: Differential use of feedback in test assessment J.-P. Fox Advances in Bayesian Item Response Modeling
Introduction Feedback Behavior Study Formative Computer-Based Assessment ◮ Two-stage testing: Ability - feedback use ◮ Observe response times (speed) and feedback times (reading) ◮ Dutch study: Differential use of feedback in test assessment J.-P. Fox Advances in Bayesian Item Response Modeling
Introduction Feedback Behavior Study Bayesian Modeling of Multivariate Count Data A Bayesian Modeling Approach: ◮ Hierarchical Structured Data, uncertainty/sampling error at different levels ◮ Use Powerful Simulation Techniques ◮ Use Prior Knowledge J.-P. Fox Advances in Bayesian Item Response Modeling
Introduction Feedback Behavior Study Bayesian Modeling of Multivariate Count Data A Bayesian Modeling Approach: ◮ Hierarchical Structured Data, uncertainty/sampling error at different levels ◮ Use Powerful Simulation Techniques ◮ Use Prior Knowledge J.-P. Fox Advances in Bayesian Item Response Modeling
Introduction Feedback Behavior Study Bayesian Modeling of Multivariate Count Data A Bayesian Modeling Approach: ◮ Hierarchical Structured Data, uncertainty/sampling error at different levels ◮ Use Powerful Simulation Techniques ◮ Use Prior Knowledge J.-P. Fox Advances in Bayesian Item Response Modeling
9 7 40 30 20 10 0 40 30 20 10 0 Number of Subjects Feedback Use 11 Feedback Time (Seconds) 5 Percentage Subjects 3 1 250 200 150 100 50 0 10 8 6 4 2 0 50 Complex Multivariate Count Data Feedback-Use and Feedback-Time Data J.-P. Fox Advances in Bayesian Item Response Modeling
100 Nine Pages Five Pages 0 20 40 60 80 100 Six Pages Seven Pages Eight Pages 0 80 10 20 30 40 Ten Pages 0 20 40 60 80 Feedback Time | Feedback Use 60 Eleven Pages 40 0 20 40 60 80 100 0 10 20 30 Zero Pages 40 One Page 0 10 20 30 40 Two Pages Three Pages Four Pages 0 20 100 Complex Multivariate Count Data Feedback-Use and Feedback-Time Data J.-P. Fox Advances in Bayesian Item Response Modeling
Complex Multivariate Count Data Modeling Multivariate Count Data Count Data No. Pages Total Times 2 7 Subjects 0 0 . . . . . . y f y t i i J.-P. Fox Advances in Bayesian Item Response Modeling
Complex Multivariate Count Data Modeling Multivariate Count Data Count Data No. Pages Total Times 2 7 Subjects 0 0 . . . . . . y f y t i i Summary Statistics Mean SD % Zeros Mean | No Zeros Feedback Use 2.35 5.35 .43 4.11 Feedback Times 2.75 6.19 .43 9.35 J.-P. Fox Advances in Bayesian Item Response Modeling
Complex Multivariate Count Data Multivariate Zero-Inflated Poisson Modeling Overview 1 Introduction Feedback Behavior Study Bayesian Response Modeling 2 Complex Multivariate Count Data Multivariate Zero-Inflated Poisson Modeling Results Feedback Behavior Study: Use (Latent) Predictors Results 3 Discussion J.-P. Fox Advances in Bayesian Item Response Modeling
Complex Multivariate Count Data Multivariate Zero-Inflated Poisson Modeling Feedback-Use No. Pages The idea is to model feedback use (yes or no), feedback pages (count pages), feedback times (count seconds) Mixture of Observed Feedback Pages � 0 , with probability 1 − φ i Y f ∼ � � λ ( f ) i with probability φ i , Poisson , i J.-P. Fox Advances in Bayesian Item Response Modeling
Complex Multivariate Count Data Multivariate Zero-Inflated Poisson Modeling Feedback-Use No. Pages The idea is to model feedback use (yes or no), feedback pages (count pages), feedback times (count seconds) Mixture of Observed Feedback Pages � 0 , with probability 1 − φ i Y f ∼ � � λ ( f ) i with probability φ i , Poisson , i Model Feedback Count Data � i = 0 | λ i = λ ( f ) � Y f (1 − φ i ) + φ i e − λ i = P i e − λ i λ j � � Y f i = j | λ i = λ ( f ) i = P φ i , i j ! J.-P. Fox Advances in Bayesian Item Response Modeling
Complex Multivariate Count Data Multivariate Zero-Inflated Poisson Modeling Feedback Times Mixture of Observed Feedback Times � with probability 1 − φ i 0 , T f ∼ � � λ ( t ) i with probability φ i , Poisson , i J.-P. Fox Advances in Bayesian Item Response Modeling
Complex Multivariate Count Data Multivariate Zero-Inflated Poisson Modeling Feedback Times Mixture of Observed Feedback Times � with probability 1 − φ i 0 , T f ∼ � � λ ( t ) i with probability φ i , Poisson , i Model Feedback Time Count Data � � i = 0 | λ i = λ ( t ) T f (1 − φ i ) + φ i e − λ i P = i e − λ i λ j � i = j | λ i = λ ( t ) � T f i P = φ i , i j ! J.-P. Fox Advances in Bayesian Item Response Modeling
Complex Multivariate Count Data Multivariate Zero-Inflated Poisson Modeling Feedback Use Identify (non-)users of feedback pages using explanatory subject information Observed Feedback Use � � Y f i = 0 , T f with probability (1 − φ i ) P 0 , i = 0 Z i | λ ( t ) i , λ ( f ) ∼ i � � �� Y f i = 0 , T f 1 , with probability φ i 1 − P i = 0 J.-P. Fox Advances in Bayesian Item Response Modeling
Complex Multivariate Count Data Multivariate Zero-Inflated Poisson Modeling Feedback Use Identify (non-)users of feedback pages using explanatory subject information Observed Feedback Use � � Y f i = 0 , T f with probability (1 − φ i ) P 0 , i = 0 Z i | λ ( t ) i , λ ( f ) ∼ i � � �� Y f i = 0 , T f 1 , with probability φ i 1 − P i = 0 Feedback Use x t � � exp i α φ i = P ( Z i = 1) = 1 + exp ( x t i α ) J.-P. Fox Advances in Bayesian Item Response Modeling
Complex Multivariate Count Data Multivariate Zero-Inflated Poisson Modeling Population Model Subjects Respondents are sampled independently and identically distributed. Stage 2: Prior Expected Counts log λ ( f ) x t = i β f i log λ ( t ) x t = i β t i J.-P. Fox Advances in Bayesian Item Response Modeling
Complex Multivariate Count Data Multivariate Zero-Inflated Poisson Modeling Population Model Subjects Respondents are sampled independently and identically distributed. Stage 2: Prior Expected Counts log λ ( f ) x t = i β f i log λ ( t ) x t = i β t i Stage 2: Multivariate Prior Expected Counts � � log λ ( f ) i ∼ N ( x β , Σ λ ) log λ ( t ) i J.-P. Fox Advances in Bayesian Item Response Modeling
Complex Multivariate Count Data Multivariate Zero-Inflated Poisson Modeling Population Results Joint Model (No Predictors) Component Parameter Mean HPD Feedback Use (Bernoulli part) Use Feedback Intercept, α 0 .30 (.13,.45) 1 − φ No Feedback .43 (.38,.46) Feedback Behavior (Poisson part) No. Pages Intercept, µ 1 3.06 (2.69,3.46) Time Intercept, µ 2 7.09 (6.35,7.92) Correlation,Σ 12 .20 (.13,.27) – HPD: 95% Highest Posterior Density interval J.-P. Fox Advances in Bayesian Item Response Modeling
Complex Multivariate Count Data Feedback Behavior Study: Use (Latent) Predictors Overview 1 Introduction Feedback Behavior Study Bayesian Response Modeling 2 Complex Multivariate Count Data Multivariate Zero-Inflated Poisson Modeling Results Feedback Behavior Study: Use (Latent) Predictors Results 3 Discussion J.-P. Fox Advances in Bayesian Item Response Modeling
Complex Multivariate Count Data Feedback Behavior Study: Use (Latent) Predictors Ability-Speed Model Collection of Responses and Response Times, N persons and K items J.-P. Fox Advances in Bayesian Item Response Modeling
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