Bayesian Sensitivity Analysis of a Nonlinear Dynamic Factor Analysis Model with Nonparametric Prior and Possible Nonignorable Missingness

Many psychological concepts are unobserved and usually represented as latent factors apprehended through multiple observed indicators. When multiple-subject multivariate time series data are available, dynamic factor analysis models with random effects offer one way of modeling patterns of within- and between-person variations by combining factor analysis and time series analysis at the factor level. Using the Dirichlet process (DP) as a nonparametric prior for individual-specific time series parameters further allows the distributional forms of these parameters to deviate from commonly imposed (e.g., normal or other symmetric) functional forms, arising as a result of these parameters’ restricted ranges. Given the complexity of such models, a thorough sensitivity analysis is critical but computationally prohibitive. We propose a Bayesian local influence method that allows for simultaneous sensitivity analysis of multiple modeling components within a single fitting of the model of choice. Five illustrations and an empirical example are provided to demonstrate the utility of the proposed approach in facilitating the detection of outlying cases and common sources of misspecification in dynamic factor analysis models, as well as identification of modeling components that are sensitive to changes in the DP prior specification.

[1]  A. Troxel,et al.  AN INDEX OF LOCAL SENSITIVITY TO NONIGNORABILITY , 2004 .

[2]  J G Ibrahim,et al.  Monte Carlo EM for Missing Covariates in Parametric Regression Models , 1999, Biometrics.

[3]  W. Wong,et al.  The calculation of posterior distributions by data augmentation , 1987 .

[4]  Peter C. M. Molenaar,et al.  A dynamic factor model for the analysis of multivariate time series , 1985 .

[5]  Marco Lippi,et al.  The Generalized Dynamic Factor Model , 2002 .

[6]  D. Pfeffermann,et al.  Small area estimation , 2011 .

[7]  H. Ishwaran,et al.  Markov chain Monte Carlo in approximate Dirichlet and beta two-parameter process hierarchical models , 2000 .

[8]  D. Rubin,et al.  Inference from Iterative Simulation Using Multiple Sequences , 1992 .

[9]  Joseph G Ibrahim,et al.  Local Influence for Generalized Linear Models with Missing Covariates , 2009, Biometrics.

[10]  A. Linde.  Local influence on posterior distributions under multiplicative modes of perturbation , 2007 .

[11]  R. Millar,et al.  Assessment of locally influential observations in Bayesian models , 2007 .

[12]  Bengt Muthén,et al.  Bayesian structural equation modeling: a more flexible representation of substantive theory. , 2012, Psychological methods.

[13]  Martyn Plummer,et al.  JAGS: A program for analysis of Bayesian graphical models using Gibbs sampling , 2003 .

[14]  D. Rubin,et al.  Statistical Analysis with Missing Data , 1988 .

[15]  Ming-Hui Chen,et al.  Monte Carlo Estimation of Bayesian Credible and HPD Intervals , 1999 .

[16]  J. Ibrahim,et al.  Perturbation selection and influence measures in local influence analysis , 2007, 0803.2986.

[17]  P. Gustafson.  Local sensitivity of posterior expectations , 1996 .

[18]  Sy-Miin Chow,et al.  Dynamic Structure of Emotions Among Individuals with Parkinson's Disease , 2004 .

[19]  Sik-Yum Lee,et al.  Structural equation modelling: A Bayesian approach. , 2007 .

[20]  P. Molenaar,et al.  Dynamic Factor Analysis of Psychophysiological Signals , 1994 .

[21]  E. Diener,et al.  The personality structure of affect. , 1995 .

[22]  C. Antoniak.  Mixtures of Dirichlet Processes with Applications to Bayesian Nonparametric Problems , 1974 .

[23]  Emilio Ferrer,et al.  Modeling affective processes in dyadic relations via dynamic factor analysis. , 2003, Emotion.

[24]  Stuart R. Lipsitz,et al.  Marginal models for the analysis of longitudinal measurements with nonignorable non-monotone missing data , 1998 .

[25]  Shinto Eguchi,et al.  Local model uncertainty and incomplete‐data bias (with discussion) , 2005 .

[26]  A. Rotnitzky,et al.  A note on the bias of estimators with missing data. , 1994, Biometrics.

[27]  Snigdhansu Chatterjee,et al.  Structural Equation Modeling, A Bayesian Approach , 2008, Technometrics.

[28]  Ian T. Jolliffe,et al.  Introduction to Multiple Time Series Analysis , 1993 .

[29]  N. Nicolson,et al.  The role of stressful events in the relationship between positive and negative affects: evidence from field and experimental studies. , 2000, Journal of personality.

[30]  Roderick J. A. Little,et al.  Statistical Analysis with Missing Data: Little/Statistical Analysis with Missing Data , 2002 .

[31]  Gilbert Swinnen,et al.  Biased latent variable mean comparisons due to measurement noninvariance. A simulation study , 2018 .

[32]  J. Ibrahim,et al.  Bayesian influence analysis: a geometric approach. , 2011, Biometrika.

[33]  John R. Nesselroade,et al.  Perceived control and academic performance: A comparison of high- and low-performing children on within-person change patterns , 2002 .

[34]  Dipak K. Dey,et al.  On local sensitivity measures in Bayesian analysis , 1996 .

[35]  G Molenberghs,et al.  Sensitivity Analysis for Nonrandom Dropout: A Local Influence Approach , 2001, Biometrics.

[36]  John Geweke,et al.  Maximum Likelihood "Confirmatory" Factor Analysis of Economic Time Series , 1981 .

[37]  P. Whittle,et al.  Latent Variables in Socio‐Economic Models , 1978 .

[38]  Shun-ichi Amari,et al.  Differential-geometrical methods in statistics , 1985 .

[39]  Nian-Sheng Tang,et al.  ANALYSIS OF NONLINEAR STRUCTURAL EQUATION MODELS WITH NONIGNORABLE MISSING COVARIATES AND ORDERED CATEGORICAL DATA , 2006 .

[40]  David B. Dunson,et al.  Bayesian Nonparametrics: Nonparametric Bayes applications to biostatistics , 2010 .

[41]  P. Gustafson.  Local Sensitivity of Inferences to Prior Marginals , 1996 .

[42]  S. Walker,et al.  Coherent psychometric modelling with Bayesian nonparametrics. , 2009, The British journal of mathematical and statistical psychology.

[43]  Helmut Ltkepohl,et al.  New Introduction to Multiple Time Series Analysis , 2007 .

[44]  Helmut Lütkepohl,et al.  Introduction to multiple time series analysis , 1991 .

[45]  D. Kaplan,et al.  A study of the power associated with testing factor mean differences under violations of factorial invariance , 1995 .

[46]  Steven N. MacEachern,et al.  Nonparametric Bayesian modelling for item response , 2008 .

[47]  J G Ibrahim,et al.  Maximum Likelihood Methods for Cure Rate Models with Missing Covariates , 2001, Biometrics.

[48]  Lutz Kilian,et al.  NEW INTRODUCTION TO MULTIPLE TIME SERIES ANALYSIS, by Helmut Lütkepohl, Springer, 2005 , 2006, Econometric Theory.

[49]  J. Sethuraman.  A CONSTRUCTIVE DEFINITION OF DIRICHLET PRIORS , 1991 .

[50]  Xin-Yuan Song,et al.  Semiparametric Bayesian analysis of structural equation models with fixed covariates , 2008, Statistics in medicine.

[51]  A. Rotnitzky,et al.  Missing Data in Longitudinal Studies: Strategies for Bayesian Modeling and Sensitivity Analysis by DANIELS, M. J. and HOGAN, J. W , 2009 .

[52]  Geert Molenberghs,et al.  Kernel weighted influence measures , 2005, Comput. Stat. Data Anal..

[53]  Lancelot F. James,et al.  Gibbs Sampling Methods for Stick-Breaking Priors , 2001 .

[54]  T. Ferguson.  A Bayesian Analysis of Some Nonparametric Problems , 1973 .

[55]  Hongtu Zhu,et al.  Bayesian estimation of semiparametric nonlinear dynamic factor analysis models using the Dirichlet process prior. , 2011, The British journal of mathematical and statistical psychology.

[56]  Sy-Miin Chow,et al.  Bayesian Factor Analysis as a Variable-Selection Problem: Alternative Priors and Consequences , 2016, Multivariate behavioral research.

[57]  Hongtu Zhu,et al.  Bayesian Sensitivity Analysis of Statistical Models with Missing Data. , 2014, Statistica Sinica.

[58]  D. Reiss,et al.  Sensitivity Analysis of Multiple Informant Models When Data Are Not Missing at Random , 2013, Structural equation modeling : a multidisciplinary journal.

[59]  Hongtu Zhu,et al.  Bayesian Influence Measures for Joint Models for Longitudinal and Survival Data , 2012, Biometrics.

[60]  Geert Molenberghs,et al.  The nature of sensitivity in monotone missing not at random models , 2006, Comput. Stat. Data Anal..

[61]  Geert Molenberghs,et al.  A local influence approach to sensitivity analysis of incomplete longitudinal ordinal data , 2001 .

[62]  David A. Sbarra,et al.  Predicting the Onset of Emotional Recovery Following Nonmarital Relationship Dissolution: Survival Analyses of Sadness and Anger , 2006, Personality & social psychology bulletin.

[63]  T. Griffiths,et al.  Modeling individual differences using Dirichlet processes , 2006 .

[64]  L. Tierney,et al.  Approximate methods for assessing influence and sensitivity in Bayesian analysis , 1989 .

[65]  R. Engle,et al.  A One-Factor Multivariate Time Series Model of Metropolitan Wage Rates , 1981 .

[66]  Asim Ansari,et al.  Semiparametric Thurstonian Models for Recurrent Choices: A Bayesian Analysis , 2006 .

[67]  James O. Berger,et al.  An overview of robust Bayesian analysis , 1994 .

[68]  R. Weiss.  An approach to Bayesian sensitivity analysis , 1996 .

[69]  M. Lavine.  Sensitivity in Bayesian Statistics: The Prior and the Likelihood , 1991 .

[70]  Bengt Muthén,et al.  Bayesian SEM : A more flexible representation of substantive theory , 2010 .