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## Overview

A trajectory describes the course of a measured variable over age or time. Investigators in epidemiology and other fields are often interested not only in the trajectory of variables over time, but also in how covariates may affect their shape. Traditionally, hierarchical modeling and latent curve analysis have been used to measure these relationships, but in recent years, a group-based approach known as group-based trajectory modeling has increased in popularity as an alternative. Whereas hierarchical modeling and latent curve analysis estimate the population average trajectory and use covariates to explain variability about this average, group-based trajectory modeling assumes that the population is composed of distinct groups, each with a different underlying trajectory. It identifies groups of individuals following similar progressions of some phenomenon over time and estimates the effects of covariates not only on trajectory shape, but also group membership. According to the assumptions of group-based trajectory modeling, time stable covariates influence group membership and time dependent covariates explain variation about the average trajectory within each group. Another benefit of trajectory analysis is that it assumes that the outcomes at different points are also outcomes of the initial treatment by adjusting for group behavior.

Group-based trajectory modeling may be a preferable option to hierarchical and latent curve modeling when handling non-monotonic trajectories and trajectories that do not vary regularly in the population. This method was originally developed for use in the fields of criminology, psychology, and sociology, but has since been applied to a much wider range of fields, including epidemiology.

## Description

**Group-based trajectory model estimation**

The parameters of group-based trajectory models are generated by maximum likelihood estimation. The equation describing the likelihood of an individual’s observed repeated measures is composed of two elements – the probability of group membership and the probability of the observed data given group membership. The probability of group membership is modeled with a generalized logit model. Time stable covariates are added to this model to explain group membership. Additional models must also be specified for the average trajectories observed in each group. These models are specified to reflect the distribution of the observed data, the shape of the group trajectory, and any time dependent covariates. Group-based trajectory modeling can accommodate a number of different data distributions, including Poisson, zero-inflated Poisson (ZIP), normal, censored normal, and binary.

SAS is the primary package used for group-based trajectory modeling. The SAS procedure developed to estimate group-based trajectory models is known as Proc Traj. While Proc Traj is not included in the standard SAS package, it can be easily downloaded from the following website: www.andrew.cmu.edu/~bjones. In addition to the Proc Traj download, this website also contains examples of the application of Proc Traj with sample code. Other software programs that can replicate or approximate Proc Traj include R, MPlus, Stata, and Latent Gold.

## Readings

### Textbooks & Chapters

For a detailed explanation of the statistical and theoretical underpinnings of group-based trajectory modeling and examples of its application:

Nagin, DS. Group-Based Modeling of Development Over the Life Course. Cambridge, MA: Harvard University Press. 2005

### Methodological Articles

For a discussion of the statistical underpinnings of group-based trajectory modeling:

Nagin DS. Analyzing developmental trajectories: a semiparametric group-based approach. Psychological Methods 1999;4:137-157.

Roeder K, Lynch K, Nagin DS. Modeling uncertainty in latent class membership: a case study in criminology. Journal of the American Statistical Association 1999;94:766-776.

For a comparison of group-based trajectory models with generalized linear mixed models and latent growth curve models:

Charnigo, R., et al. (2011). “Joint modeling of longitudinal data in multiple behavioral change.” Eval Health Prof 34(2): 181-200.

For a discussion of the SAS procedure developed for group-based trajectory modeling:

Jones BJ, Nagin DS, Roeder K. A SAS procedure based on mixture models for estimating developmental trajectories. Sociological Methods and Research 2001;29:374-393.

Jones BL, Nagin DS. Advances in group-based trajectory modeling and an SAS procedure for estimating them. Sociological Methods and Research 2007;35:542-571.

Knight KE. Using SAS® Proc Traj to Improve Estimates of Assortative Mating for Problem Behavior. WUSS 2009.

For a discussion of controversies in group-based trajectory modeling:

Nagin DS, Tremblay RE. Developmental trajectory groups: fact or a useful statistical fiction? Criminology 2005;43:873-902.

### Application Articles

Mayo, N. E., et al. (2014). “Necessary and sufficient causes of participation post-stroke: practical and philosophical perspectives.” Qual Life Res 23(1): 39-47.

http://www.ncbi.nlm.nih.gov/pubmed/23754685

Pines, H. A., et al. (2014). “Sexual Risk Trajectories Among MSM in the United States: Implications for Pre-exposure Prophylaxis Delivery.” J Acquir Immune Defic Syndr 65(5): 579-586. http://www.ncbi.nlm.nih.gov/pubmed/24378726

Wang, B., et al. (2014). “The impact of youth, family, peer and neighborhood risk factors on developmental trajectories of risk involvement from early through middle adolescence.” Soc Sci Med 106: 43-52. http://www.ncbi.nlm.nih.gov/pubmed/24530616

Zimmer, Z., et al. (2014). “Examining late-life functional limitation trajectories and their associations with underlying onset, recovery, and mortality.” J Gerontol B Psychol Sci Soc Sci 69(2): 275-286. http://www.ncbi.nlm.nih.gov/pubmed/24531526

## Websites

SAS: Download Proc Traj and view examples of its application:

http://www.andrew.cmu.edu/user/bjones/index.htm

R package for latent class mixed models to implement group-based trajectory models:

http://cran.r-project.org/web/packages/lcmm/index.html

Stata: Implementation and example of GBTM:

http://ssrc.indiana.edu/seminars/wimdocs/2013-03-29_nagin_trajectory_stata-plugin-info.pdf

Powerpoint presentation of examples of trajectory modeling for binary longitudinal data:

http://www.stats.uwo.ca/faculty/bellhouse/stat%209945b/Presentation_MaggieChu.pdf