This article introduces a new SAS procedure written by the authors that analyzes longitudinal data (developmental trajectories) by fitting a mixture model. The TRAJ procedure fits semiparametric (discrete) mixtures of censored normal, Poisson, zero-inflated Poisson, and Bernoulli distributions to longitudinal data. Applications to psychometric scale data, offense counts, and a dichotomous prevalence measure in violence research are illustrated. In addition, the use of the Bayesian information criterion to address the problem of model selection, including the estimation of the number of components in the mixture, is demonstrated.
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References
1.
Bryk, Anthony S.
and Stephen W. Raudenbush. 1987. “Application of Hierarchical Linear Models to Assessing Change.”Psychology Bulletin101:147-158.
2.
Bryk, Anthony S.
and Stephen W. Raudenbush. 1992. Hierarchical Linear Models for Social and Behavioral Research: Application and Data Analysis Methods. Newbury Park, CA: Sage.
3.
Clogg, Clifford C.
1995. “Latent Class Models.” In Handbook of Statistical Modeling for the Social and Behavioral Sciences, edited by Gerhard Arminger, Clifford C. Clogg, and Michael E. Sobel. New York: Plenum.
4.
Dennis, John E.
, David M. Gay, and Roy E. Welsch. 1981. “An Adaptive Nonlinear Least-Squares Algorithm.”ACM Transactions on Mathematical Software7:348-383.
5.
Dennis, John E.
and Howell W. Mei. 1979. “Two New Unconstrained Optimization Algorithms Which Use Function and Gradient Values.”Journal of Optimization Theory and Applications28:453-483.
6.
D'Unger, Amy V.
, Kenneth C. Land, Patricia L. McCall, and Daniel S. Nagin. 1998. “How Many Latent Classes of Delinquent/Criminal Careers? Results From Mixed Poisson Regression Analyses of the London, Philadelphia, and Racine Cohorts Studies.”American Journal of Sociology103:1593-1630.
7.
Farrington, David P.
and Donald J. West. 1990. “The Cambridge Study in Delinquent Development: A Prospective Longitudinal Study of 411 Males.” In Criminality: Personality, Behavior, and Life History, edited by Hans-Jürgen Kerner and G. Kaiser. New York: Springer-Verlag.
8.
Fergusson, David M.
, Michael T. Lynskey, and L. John Horwood. 1996. “Factors Associated With Continuity and Change in Disruptive Behavior Patterns During Childhood and Adolescence.”Journal of Abnormal Child Psychology24:533-553.
9.
Fraley, Chris
and Adrian E. Raftery. 1998. “How Many Clusters? Which Clustering Method? Answers Via Model-Based Cluster Analysis.”Computer Journal41:578-588.
10.
Ghosh, Jayanta K.
and Pranab K. Sen. 1985. “On the Asymptotic Performance of the Log Likelihood Ratio Statistic for the Mixture Model and Related Results.” In Proceedings of the Berkeley Conference in Honor of Jerzy Neyman and Jack Kiefer, vol. 3, edited by Lucien M. LeCam and Richard A. Olshen. Monterey, CA: Wadsworth.
Jeffreys, Harold
. 1961. Theory of Probability. 3d ed.London: Oxford University Press.
13.
Kass, Robert E.
and Adrian E. Raftery. 1995. “Bayes Factors.”Journal of the American Statistical Association90:773-795.
14.
Kass, Robert E.
and Larry Wasserman. 1995. “A Reference Bayesian Test for Nested Hypotheses and Its Relationship to the Schwarz Criterion.”Journal of the American Statistical Association90:928-934.
15.
Keribin, Christine
. 1997. “Consistent Estimation of the Order of Mixture Models.” Working Paper No. 61. Laboratorie Analyse et Probabilité, Université d'Évry-Val d'Essonne, Évry, France.
16.
Lambert, Diane
. 1992. “Zero-Inflated Poisson Regressions, With an Application in Manufacturing.”Technometrics34:1-13.
17.
Land, Kenneth C.
, Patricia McCall, and Daniel S. Nagin. 1996. “A Comparison of Poisson, Negative Binomial, and Semiparametric Mixed Poisson Regression Models With Empirical Applications to Criminal Careers Data.”Sociological Methods & Research24:387-440.
18.
Laub, John H.
, Daniel S. Nagin, and Robert J. Sampson. 1998. “Good Marriages and Trajectories of Change in Criminal Offending.”American Sociological Review63:225-238.
19.
Loeber, Rolf
and Marc LeBlanc. 1990. “Toward a Developmental Criminology.” In Crime and Justice: An Annual Review of Research, vol. 12, edited by Michael Tonry and Norval Morris. Chicago: University of Chicago Press.
20.
Meredith, William
and John Tisak. 1990. “Latent Curve Analysis.”Psychometrika55(1):107-122.
21.
Moffitt, Terrie E.1993. “Adolescence-Limited and Life-Course Persistent Antisocial Behavior: A Developmental Taxonomy.”Psychological Review100:674-701.
22.
Muthen, Bengt O.1989. “Latent Variable Modeling in Heterogeneous Populations.”Psychometrika54(4):557-585.
23.
Nagin, Daniel S.1999. “Analyzing Developmental Trajectories: A Semi-Parametric, Group-Based Approach.”Psychological Methods4:139-177.
24.
Nagin, Daniel S.
, David P. Farrington, and Terrie E. Moffitt. 1995. “Life-Course Trajectories of Different Types of Offenders.”Criminology33:111-139.
25.
Nagin, Daniel S.
and Kenneth C. Land. 1993. “Age, Criminal Careers, and Population Heterogeneity: Specific Estimation of a Nonparametric, Mixed Poisson Model.”Criminology31:327-362.
26.
Nagin, Daniel S.
and Richard E. Tremblay. 1999. “Trajectories of Boys' Physical Aggression, Opposition, and Hyperactivity on the Path to Physically Violent and Non Violent Juvenile Delinquency.”Child Development70:1181-1196.
27.
Patterson, Gerald R.
1996. “Some Characteristics of a Developmental Theory for Early-Onset Delinquency.” In Frontiers of Developmental Psychopathology, edited by Mark F. Lenzenweger and Jeffrey J. Haugaard. Oxford, UK: Oxford University Press.
28.
Patterson, Gerald R.
, Barbara D. DeBaryshe, and E. Ramsey. 1989. “A Developmental Perspective on Antisocial Behavior.”American Psychologist44:329-335.
29.
Patterson, Gerald R.
, Marion S. Forgatch, Karen L. Yoerger, and Mike Stoolmiller. 1998. “Variables That Initiate and Maintain an Early-Onset Trajectory for Juvenile Offending.”Development and Psychopathology10:531-547.
30.
Patterson, Gerald R.
and Karen L. Yoerger. 1997. “A Developmental Model for Late-Onset Delinquency.” In Motivation and Delinquency, edited by D. Wayne Osgood. Lincoln: University of Nebraska Press.
31.
Raftery, Adrian E.
1995. “Bayesian Model Selection in Social Research (With Discussion).” In Sociological Methodology, edited by Peter V. Marsden. Cambridge, MA: Blackwell.
32.
Roeder, Kathryn
, Kevin G. Lynch, and Daniel S. Nagin. 1999. “Modeling Uncertainty in Latent Class Membership: A Case Study in Criminology.”Journal of the American Statistical Association94:766-776.
33.
Sampson, Robert J.
and John H. Laub. 1993. Crime in the Making: Pathways and Turning Points Through Life. Cambridge, MA: Harvard University Press.
34.
Schwarz, Gideon1978. “Estimating the Dimension of a Model.”Annals of Statistics6:461-464.
35.
Willett, John B.
and Aline G. Sayer. 1994. “Using Covariance Structure Analysis to Detect Correlates and Predictors of Individual Change Over Time.”Psychological Bulletin116(2):363-381.
