Markov-switching autoregressive latent variable models for longitudinal data

Transcription

1 Markov-swching autoregressive latent variable models for longudinal data Silvia Bacci Francesco Bartolucci Fulvia Pennoni Universy of Perugia (Italy) Universy of Perugia (Italy) Universy of Milano Bicocca (Italy)

2 Summary Introduction Different approaches for the treatment of longudinal data Proposed model: the Markov-swching LAR (SW-LAR) model SW-LAR model: special cases SW-LAR model: estimation Application Future developments References

3 Introduction Context: analysis of longudinal data (we refer to the case of ordinal response variables y depending on covariates x ) Problem: taking into account the effect that unobservable factors have on the occasion-specific response variables Different approaches: 1. Individual-specific random intercept model. Latent autoregressive (LAR) model (Chi and Reinsel, 1989) 3. Latent Markov (LM) regression model (Wiggins, 1973) Aim: We propose a generalization of the LAR model based on assuming a latent Markov-swching AR(1) process wh correlation coefficient depending on the regime of the chain

4 Individual-specific random intercept model The unobserved heterogeney is taken into account through individual-specific random intercepts Ordinal response variable for subject i at occasion t wh j = 1,..., l categories p( y j ui 0, x ) log u j i 0 p( y j u, x ) i 0 Random part of the intercept ui0 ~ Covariates x ' β N (0, ) t Parsimony The effect of unobservable factors is assumed to be time constant

5 LAR model The unobserved heterogeney is taken into account through the inclusion, whin subjects, of occasion-specific random effects which follow an AR(1) process y i1,..., y it are condionally independent given the latent variables u and the covariates x log p( y p( y j j u u, x, x ) ) j u x ' β Occasion-specific continuous latent variable u ui1 N (0, ) for t 1 u u for t 1 ~ i, t 1 i, t 1 N (0, 1 ~ )

6 LAR model Parsimony The effect of unobservable factors is time varying In many applications, error terms are naturally represented by continuous random variables Estimation may be problematic from the computational point of view (Heiss, 008)

7 LM regression model The unobserved heterogeney is taken into account through the inclusion of a sequence of discrete latent variables which follow a first-order Markov chain y i1,..., y it are condionally independent given the latent variables u and the covariates x log p( y p( y j j u u, x, x ) ) j u x ' β Occasion-specific discrete latent variable 1. Any latent variable u is condionally independent of u i1,..., u i, t- given u i, t-1. The latent variable u can assume k different regimes (or states)

8 LM regression model It may reach a better f than the LAR model It is easier to estimate than the LAR model It provides a classification of subjects in a reduced number of groups It may be seen as a semi-parametric version of the LAR model It is less parsimonious than the LAR model: the LM model is based on k-1 inial probabilies and k(k-1) transion probabilies, whereas the LAR model is based on only parameters for the latent process.

9 We formulate a model for longudinal data based on the assumption that the error terms follow a Markov-swching AR(1) process (Hamilton, 1989) Main characteristics: 1. The latent process is continous as in the LAR model, but the correlation coefficient is not restricted to be constant.. A set of different regimes are possible, wh each regime corresponding to a different value of the correlation coefficient 3. How a subject moves between regimes is governed by a timehomogenous latent Markov chain We expect that the resulting model has a f comparable to that of a LM model, but is more parsimonious

10 Proposed model: Markov-Swching LAR model (SW-LAR) Assumptions of LAR model are substuted by: u u i, v u for t 1, t 1 i, t 1 v v ~ N (0, 1 v ) Note that every latent variable N(0, ) as in the LAR model. u has marginal distribution

11 SW-LAR model v,..., v i1 it follow a Markov chain wh k latent states: 1. Each latent state corresponds to a correlation coefficient:,..., 1 k. The latent states are characterized by a vector of inial probabilies: λ, v 1,..., v and by a transion probabily matrix: Π k, v0, v k 1,..., v 0v

12 SW-LAR model: special cases k 1 Π I Basic LAR model: The correlation coefficient is the same for all subjects and occasions SW-LAR 1 model: The correlation coefficient may be different between subjects belonging to different latent states, but not between occasions Π 1 λ' SW-LAR model: The correlation coefficient may change between subjects and occasions, since each subject randomly moves betwen different regimes

13 SW-LAR model: estimation We maximize the log-likelihood: ( θ) log p ( y X ) i i i Manifest probabily It is based on a T-dimensional integral Sequential numerical integration method (Heiss, 008 which is strictly related to Baum et al., 1970)

14 SW-LAR model: sequential numerical integration p ( y i X i ) qit ( u, v) du v q ( u, v) p( u u, v v, y,..., y i1 it ) We compute first q ( u, v) p( y u) g( u) i1 i1 v and then for t 1 q ( u, v) p( y u) q ( u, v ) g( u u, v) du v v i, t v0 0 Each integral above is computed by a Gaussian Quadrature

15 Application Data from the Health and Retirement Study (Universy of Michigan) A set of 1000 American people who self-evaluated their health status over 8 occasions Health status is an ordinal qualative response variable: poor, fair, good, very good, excellent Time-constant covariates: gender, race, education Time- varying covariate: age We consider three models: LAR, SW-LAR 1 wh latent states, SW-LAR wh latent states Model selection crerion: BIC

16 Results: parameter estimates, maximum loglikelihood and BIC LAR SW-LAR 1 SW-LAR female non whe education age log-likelihood # parameters BIC We have two different levels of persistence of the effect of the unobservable factors on the response variables SW-LAR 1 model has only two more parameters than LAR model SW-LAR 1 model has a better f than LAR model

17 What s next? Simulation study to detect the differences among LAR, LM and SW-LAR models Implementation of a sequential numerical integration algorhm to estimate a general SW-LAR model and to obtain standard errors for the parameter estimates

18 References Baum L.E., Petrie, T., Solues, G., and Weiss, N. (1970). A maximization technique occurring in the statistical analysis of probabilistic functions of Markov chains. Annals of Mathematical Statistics, 41, Chi, E.M., and Reinsel, G.C. (1989). Models for longudinal data wh random effects and AR(1) errors. Journal of the American Statistical Association, 84, Hamilton, J.D. (1989). A new approach to the economic analysis of nonstationary time series and the business cycle. Eocnometrica, 57, Heiss, F. (008). Sequential numerical integration in nonlinear state space models for microeconometric panel data. Journal of Applied Econometrics, 3, Wiggins, L.M. (1973). Panel analysis: latent probabily models for attude and behavior processes. Amsterdam: Elsevier.