Joint Modeling of Longitudinal Item Response Data and Survival - PowerPoint PPT Presentation

Joint Modeling of Longitudinal Item Response Data and Survival Jean-Paul Fox University of Twente Department of Research Methodology, Measurement and Data Analysis Faculty of Behavioural Sciences Enschede, Netherlands J.-P. Fox Bayesian Item Response Modeling

Overview Introduction Cross-classified Response Data J.-P. Fox Bayesian Item Response Modeling

Overview Introduction Cross-classified Response Data Longitudinal Response Data Mini-Mental State Examination (MMSE) J.-P. Fox Bayesian Item Response Modeling

Overview Introduction Cross-classified Response Data Longitudinal Response Data Mini-Mental State Examination (MMSE) Survival Analysis Joint Modeling of Latent Developmental Trajectories and Survival Joint (Response, Survival) Analysis Outcomes J.-P. Fox Bayesian Item Response Modeling

Overview Introduction Cross-classified Response Data Longitudinal Response Data Mini-Mental State Examination (MMSE) Survival Analysis Joint Modeling of Latent Developmental Trajectories and Survival Joint (Response, Survival) Analysis Outcomes Discussion J.-P. Fox Bayesian Item Response Modeling

Overview Introduction Cross-classified Response Data Longitudinal Response Data Mini-Mental State Examination (MMSE) Survival Analysis Joint Modeling of Latent Developmental Trajectories and Survival Joint (Response, Survival) Analysis Outcomes Discussion J.-P. Fox Bayesian Item Response Modeling

Responses to Test Items Collection of responses on tests, i = 1 , . . . , N persons who answer k = 1 , . . . , K items, resulting in N × K 0/1 responses Y :   1 0 1 . . . Y 1 K 0 1 1 . . . Y 2 K   Y =  . .  ... . .   . .   Y N 1 0 1 . . . Y NK J.-P. Fox Bayesian Item Response Modeling

Responses to Test Items Collection of responses on tests, i = 1 , . . . , N persons who answer k = 1 , . . . , K items, resulting in N × K 0/1 responses Y :   1 0 1 . . . Y 1 K 0 1 1 . . . Y 2 K   Y =  . .  ... . .   . .   Y N 1 0 1 . . . Y NK • Develop a model to say something about the structure of this data set J.-P. Fox Bayesian Item Response Modeling

Responses to Test Items Collection of responses on tests, i = 1 , . . . , N persons who answer k = 1 , . . . , K items, resulting in N × K 0/1 responses Y :   1 0 1 . . . Y 1 K 0 1 1 . . . Y 2 K   Y =  . .  ... . .   . .   Y N 1 0 1 . . . Y NK • Develop a model to say something about the structure of this data set • Structure: person and item effects. J.-P. Fox Bayesian Item Response Modeling

Stage 1: Modeling Success Probabilities P ( Y ik = 1 | θ i , ξ k ) F ( a k θ i − b k ) = µ θ , σ 2 � � θ i ∼ N θ J.-P. Fox Bayesian Item Response Modeling

Stage 1: Modeling Success Probabilities P ( Y ik = 1 | θ i , ξ k ) F ( a k θ i − b k ) = µ θ , σ 2 � � θ i ∼ N θ • Response observations k are nested within persons, random person effects (latent variable) J.-P. Fox Bayesian Item Response Modeling

Stage 1: Modeling Success Probabilities P ( Y ik = 1 | θ i , ξ k ) F ( a k θ i − b k ) = µ θ , σ 2 � � θ i ∼ N θ • Response observations k are nested within persons, random person effects (latent variable) • Response observations k are nested within items, fixed/random item effects. J.-P. Fox Bayesian Item Response Modeling

Two-Parameter Item Response Model y ik m q q i a k s q b k Item k Individual i J.-P. Fox Bayesian Item Response Modeling

Likelihood-Model Collection of N × K responses, N persons and K items J.-P. Fox Bayesian Item Response Modeling

Likelihood-Model Collection of N × K responses, N persons and K items � exp( d ( a k θ i − b k )) Logistic Model P ( Y ik = 1 | θ i , a k , b k ) = 1+exp( d ( a k θ i − b k )) Φ( a k θ i − b k ) Probit Model J.-P. Fox Bayesian Item Response Modeling

Likelihood-Model Collection of N × K responses, N persons and K items � exp( d ( a k θ i − b k )) Logistic Model P ( Y ik = 1 | θ i , a k , b k ) = 1+exp( d ( a k θ i − b k )) Φ( a k θ i − b k ) Probit Model �� � F ( η ik ) y ik (1 − F ( η ik )) 1 − y ik � p ( y | θ , a , b ) = i k where η ik = a k θ i − b k J.-P. Fox Bayesian Item Response Modeling

Population Model for Item Parameters Stage 2: Prior for Item Parameters ( a k , b k ) t ∼ N ( µ ξ , Σ ξ ) I A k ( a k ) , where the set A k = { a k ∈ R , a k > 0 } J.-P. Fox Bayesian Item Response Modeling

Population Model for Item Parameters Stage 2: Prior for Item Parameters ( a k , b k ) t ∼ N ( µ ξ , Σ ξ ) I A k ( a k ) , where the set A k = { a k ∈ R , a k > 0 } Stage 3: Hyper prior ∼ IW ( ν, Σ 0 ) Σ ξ µ ξ | Σ ξ ∼ N ( µ 0 , Σ ξ /K 0 ) . J.-P. Fox Bayesian Item Response Modeling

Population Model for Person Parameter Stage 2: Prior for Person Parameters θ i ∼ N ( µ θ , σ 2 θ ) . Respondents are sampled independently and identically distributed. J.-P. Fox Bayesian Item Response Modeling

Population Model for Person Parameter Stage 2: Prior for Person Parameters θ i ∼ N ( µ θ , σ 2 θ ) . Respondents are sampled independently and identically distributed. Stage 3: Hyper prior σ 2 θ ∼ IG ( g 1 , g 2 ) µ θ | σ 2 θ ∼ N ( µ 0 , σ 2 θ /n 0 ) . J.-P. Fox Bayesian Item Response Modeling

Longitudinal Item Response Data • Discrete response data Y ijk : (subject i , measurement occasion j , item k ) J.-P. Fox Bayesian Item Response Modeling

Longitudinal Item Response Data • Discrete response data Y ijk : (subject i , measurement occasion j , item k ) • Several measurement occasions j = 1 , . . . , n i , several points in time. J.-P. Fox Bayesian Item Response Modeling

Longitudinal Item Response Data • Discrete response data Y ijk : (subject i , measurement occasion j , item k ) • Several measurement occasions j = 1 , . . . , n i , several points in time. • Latent Growth Modeling J.-P. Fox Bayesian Item Response Modeling

Latent Growth Modeling • Model Latent Developmental Trajectories: θ ij = β 0 i + β 1 i Time ij + e ij β 0 i = γ 00 + u 0 i β 1 i = γ 10 + u 1 i J.-P. Fox Bayesian Item Response Modeling

Latent Growth Modeling • Model Latent Developmental Trajectories: θ ij = β 0 i + β 1 i Time ij + e ij β 0 i = γ 00 + u 0 i β 1 i = γ 10 + u 1 i 1. Subjects not measured on the same time points across time (include all data) J.-P. Fox Bayesian Item Response Modeling

Latent Growth Modeling • Model Latent Developmental Trajectories: θ ij = β 0 i + β 1 i Time ij + e ij β 0 i = γ 00 + u 0 i β 1 i = γ 10 + u 1 i 1. Subjects not measured on the same time points across time (include all data) 2. Number of observations per subject may vary J.-P. Fox Bayesian Item Response Modeling

Latent Growth Modeling • Model Latent Developmental Trajectories: θ ij = β 0 i + β 1 i Time ij + e ij β 0 i = γ 00 + u 0 i β 1 i = γ 10 + u 1 i 1. Subjects not measured on the same time points across time (include all data) 2. Number of observations per subject may vary 3. Follow-up times not uniform across subjects (time a continuous variable, individualized schedule) J.-P. Fox Bayesian Item Response Modeling

Latent Growth Modeling • Model Latent Developmental Trajectories: θ ij = β 0 i + β 1 i Time ij + e ij β 0 i = γ 00 + u 0 i β 1 i = γ 10 + u 1 i 1. Subjects not measured on the same time points across time (include all data) 2. Number of observations per subject may vary 3. Follow-up times not uniform across subjects (time a continuous variable, individualized schedule) 4. Handle time-invariant and time-varying covariates J.-P. Fox Bayesian Item Response Modeling

Latent Growth Modeling • Model Latent Developmental Trajectories: θ ij = β 0 i + β 1 i Time ij + e ij β 0 i = γ 00 + u 0 i β 1 i = γ 10 + u 1 i 1. Subjects not measured on the same time points across time (include all data) 2. Number of observations per subject may vary 3. Follow-up times not uniform across subjects (time a continuous variable, individualized schedule) 4. Handle time-invariant and time-varying covariates 5. Estimate subject-specific change across time (average change) J.-P. Fox Bayesian Item Response Modeling

Latent Growth Modeling • Specify curvilinear individual change, e.g., polynomial individual change of any order J.-P. Fox Bayesian Item Response Modeling

Latent Growth Modeling • Specify curvilinear individual change, e.g., polynomial individual change of any order • Model the covariance structure of the level-1 measurement errors explicitly. J.-P. Fox Bayesian Item Response Modeling

Latent Growth Modeling • Specify curvilinear individual change, e.g., polynomial individual change of any order • Model the covariance structure of the level-1 measurement errors explicitly. • Model change in several domains simultaneously. J.-P. Fox Bayesian Item Response Modeling

Overview Introduction Cross-classified Response Data Longitudinal Response Data Mini-Mental State Examination (MMSE) Survival Analysis Joint Modeling of Latent Developmental Trajectories and Survival Joint (Response, Survival) Analysis Outcomes Discussion J.-P. Fox Bayesian Item Response Modeling