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A Dynamic Factor Model for Multivariate Counting Process Data

Fangyi Chen, Hok Kan Ling, Zhiliang Ying

Abstract

We propose a dynamic multiplicative factor model for process data, which arise from complex problem-solving items, an emerging testing mode in large-scale educational assessment. The proposed model can be viewed as an extension of the classical frailty models developed in survival analysis for multivariate recurrent event times, but with two important distinctions: (i) the factor (frailty) is of primary interest; (ii) covariates are internal and embedded in the factor. It allows us to explore low dimensional structure with meaningful interpretation. We show that the proposed model is identifiable and that the maximum likelihood estimators are consistent and asymptotically normal. Furthermore, to obtain a parsimonious model and to improve interpretation of parameters therein, variable selection and estimation for both fixed and random effects are developed through suitable penalisation. The computation is carried out by a stochastic EM combined with the Metropolis algorithm and the coordinate descent algorithm. Simulation studies demonstrate that the proposed approach provides an effective recovery of the true structure. The proposed method is applied to analysing the log-file of an item from the Programme for the International Assessment of Adult Competencies (PIAAC), where meaningful relationships are discovered.

A Dynamic Factor Model for Multivariate Counting Process Data

Abstract

We propose a dynamic multiplicative factor model for process data, which arise from complex problem-solving items, an emerging testing mode in large-scale educational assessment. The proposed model can be viewed as an extension of the classical frailty models developed in survival analysis for multivariate recurrent event times, but with two important distinctions: (i) the factor (frailty) is of primary interest; (ii) covariates are internal and embedded in the factor. It allows us to explore low dimensional structure with meaningful interpretation. We show that the proposed model is identifiable and that the maximum likelihood estimators are consistent and asymptotically normal. Furthermore, to obtain a parsimonious model and to improve interpretation of parameters therein, variable selection and estimation for both fixed and random effects are developed through suitable penalisation. The computation is carried out by a stochastic EM combined with the Metropolis algorithm and the coordinate descent algorithm. Simulation studies demonstrate that the proposed approach provides an effective recovery of the true structure. The proposed method is applied to analysing the log-file of an item from the Programme for the International Assessment of Adult Competencies (PIAAC), where meaningful relationships are discovered.

Paper Structure

This paper contains 35 sections, 25 theorems, 444 equations, 2 figures, 13 tables.

Table of Contents

  1. Introduction
  2. Notation and Model Specification
  3. Main Theoretical Results
  4. Implementation
  5. Choice of regularization parameter
  6. Application to PIAAC data
  7. Simulation Study
  8. Discussion
  9. Simulation setting and results
  10. Additional simulation results
  11. Under model misspecification
  12. Scenarios with right censoring
  13. More discussions on model identifiability
  14. The necessity of adopting generic identifiability
  15. Identifiability when only external covariates are present
  16. ...and 20 more sections

Key Result

Theorem 1

Under Conditions (c)-(f), model (eq:our_model) is generically identifiable.

Figures (2)

  • Figure 1: Screenshot of the sample item provided on the OECD website.
  • Figure 2: Screenshot of the sample item provided on the OECD website.

Theorems & Definitions (51)

  • Definition 1: Generic Identifiability
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem S.2.1
  • proof
  • Theorem S.2.2
  • proof
  • Lemma S.2.1
  • ...and 41 more