Latent Growth Curve Analysis
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Overview
There are several statistical approaches that can be used to analyze trajectories of change in a variable or variables over time. The simplest approach is to describe the change in a variable from one time point to a second time point – the change score. However, in longitudinal studies involving more than two assessments over time, more sophisticated techniques are required to describe not only the trajectories of change, but also to determine interrelationships of variables and predictors.
Description
Latent growth curve analysis (LGCA) is a powerful technique that is based on structural equation modeling. Another approach, which will not be directly discussed here, is multilevel modeling, which employs the statistical techniques of general linear regression and specifies fixed and random effects. LGCA, on the other hand, considers change over time in terms of an underlying, latent, unobserved process. The two approaches are similar and in most cases they yield identical estimates. However, LGCA is more flexible than multilevel modeling in some regards and can explore questions that are not possible with multilevel modeling. For example, multilevel modeling treats time scores differently. While time scores are considered data in MLM they can be parameters in LGCA and can be estimated. With LGCA, in terms of trajectories of change, the direction and functional form can be described. Also, the intercept of a curve, or its “initial level,” can be examined if it is of interest to the research question. But perhaps most importantly, instead of a group statistic that provides the mean curve or intercept of the sample, LGCA can represent unique curves for each individual or groups of individuals, represented as deviations from the average function, in addition to testing hypothesis about trajectories of interest to the researcher. The loss of information that results from averaging unique trajectories is sometimes referred to as “aggregation bias.” If individuals are postulated to be able to have positive changes as well as negative changes in a variable over time, then a procedure that aggregates or averages trajectories is not appropriate to describe changes, since an individual’s trajectory may cancel or mask the effect of others’ trajectories.
One of the strengths of LGCA is its flexibility. One can analyze and model a range of parameters of interest, starting from a single growth trajectory in a single variable (characterized by an intercept and slope) and progressing to more complex models. In fact, several models can be considered special or restricted cases of LGC models, which are possible when particular assumptions, are met: for example, repeated measures analysis of variance and factor analysis. LGCA allows one to partition the variance in growth and analyze different predictors. One can locate clusters of cases that have unique growth curves. It is also possible to incorporate timespecific measurement error, unlike in traditional regression approaches, which allows for heteroscedasticity and ensures reliability in results by enabling one to estimate parameters separately from measurement error. LGCA also allows for covariance among the variance for slope and the variance for the intercept, unlike traditional regression approaches. Finally, multivariate LGCA models allow one to model and test longitudinal associations among several outcome variables measured repeatedly over time. One of the weaknesses of LGCA is that it uses the language and techniques of structural equation models (SEM), which require expertise and advanced training to design and analyze. Also, missing observations and unequally spaced observations over time require special treatment in models.
In order to understand the analytic approach of LGCA, some background in SEM is required. SEMs represent a general modeling framework that enables the testing of associational patterns between observed and latent variables. They allow tests of the relationships among variables through tests of the variances and covariances among the variables. SEMs have been used for various purposes, including causal modeling, path analysis, factor analysis, and regression models. Many SEMs can be written as path diagrams. For example, the simple linear regression equation Y=aX + e can be represented in the following path diagram. In path diagrams, arrows connect independent variables to dependent variables, with the arrow pointing towards the dependent variable and with the weighting coefficient noted above the arrow bar. The variances of independent variables and error terms (or residuals) is also represented and connected to the appropriate variable with curved lines without arrowheads. Observed variables are surrounded by a box, and latent or unobserved variables are surrounded by an oval.
One describes the hypothesized interrelationships among variables by using such a path diagram. The underlying rules of SEM allow calculation of the variances and covariances of the variables using observed data. Then, formal testing of the variances and covariances is performed to see if the model fits the data. In order to program LGCA models, in general, at least 3 repeated measures of a variable over time are required to model trajectories (with only 2 measurements at different time points, the best that can be estimated is a straight line). Also, a general rule of thumb is that at least 300500 cases are preferable for adequate power in analyses.
Programming and analyzing LGC models often follows a particular sequence. One potential sequence for the analysis is as follows: 1) a twofactor (intercept, slope), linear growth trajectory model; 2) exploratory models assessing for quadratic and cubic trends in the trajectory; 3) analysis of potential predictors of the intercept and slope; 4) model fit assessments; and 5) calculation or reporting of the probability of being a member of a particular group or trajectory. Specification and reporting of a path diagram is also usually a part of this sequence of analysis and illustrates the hypothesized relationships among factors in an accessible form. Interestingly, time is not treated as an explanatory variable as in traditional regression techniques. Rather, in LGCA the factor loadings for the variable with repeated measures are constrained to represent the postulated time trend. In this sense, each time point is treated as a separate variable, and LGCA is called a “multivariate” approach as a result. There is a variety of software programs that are designed to build and analyze structural equation models, including Mplus, Amos, EQS, LISREL, and SAS. There are different estimation techniques that can be used in SEMs in general, and LGCA in particular. One estimation procedure, which is optimal for missing data, is full information maximum likelihood (FIML) estimation, which uses all available data and does not require listwise deletion of variables in the case of missing values.
Finally, an essential step in LGCA analysis is to assess model fit. As it is important to assess the probability of rejecting the hypothesized latent growth curve model, when the model fit is not good, or in other terms not correct in the population. Since most LGM don’t fit exactly in the population, employing a high statistical power would lead to rejection of very good fitting models. Therefore, when fitting models, one shouldn’t rely heavily on the likelihood ratio test and should utilize other methods that may involve several indices including chisquare, normed chisquare index, the comparative fit index, TuckerLew Index, and Root Mean Square Approximation (RMSEA). Note that these fit indices are distinct from those used in traditional regression, such as AIC and BIC. Different models can be compared using these fit statistics, but it is important to remember that a goodfitting model does not imply that it illustrates a causal connection among the postulated factors.
Readings
Textbooks & Chapters

Li, Fuzhong. Latent curve analysis: a manual for research data analysts. Oregon Research Institute, Eugene, OR. Available here Accessed April 2013. (Provides a brief conceptual introduction followed by detailed programming approaches and code for LGCA using several different programs.)

Willet JB, Bub K. Structural Equation Modeling: latent growth curve analysis in: Encyclopedia of Statistics in Behavioral Science, ed: Everitt BS and Howell DC. John Wiley & Sons, Chichester, 2005: 19121922. (An exposition of the use of LGCA couched in terms of structural equation modeling, with an example application.)

Latent growth curve modeling. In: Preacher KJ, editor. Los Angeles:: SAGE; 2008. Available here and here (This book provides a strong introduction to latent growth curve models. The authors describe succinctly how latent growth curve models and multilevel modeling are related and how SEM represents latent growth curve models)
Methodological Articles

Andruff H, Carraro N, Thompson A, Gaudreau P. Latent class growth modeling: a tutorial. Tut Quant Meth Psych 2009;5(1):1124. (An excellent introduction to LGCA with detailed explanation of the programming of an analysis using SAS)

Hertzog C, Nesselroade JR. Assessing psychological change in adulthood: an overview of methodological issues. Psych and Aging 2003;18(4):639657. (A comprehensive review of issues in measuring and explaining change over time. Provides much detail about the conceptual statistical basis of LGCA and multilevel modeling, and compares and contrasts the use of both.)

Llabre M, Spitzer S, Siegel S, et al. Applying latent growth curve modeling to the investigation of individual differences in cardiovascular recovery from stress. Psychosom Med2004;66:2941.(An illustration of the use of LGCA with a good, brief conceptual introduction to LGCA. An appendix also has code for the analysis itself.)

Stoel RD, van Den Wittenboer G, Hox J. Analyzing longitudinal data using multilevel regression and latent growth curve analysis. Metodologia de las Ciencias del Comportamiento2003. (A clear and well organized comparison of LGCA and multilevel modeling. The paper illustrates differences between MLM and LGCA when certain assumptions are violated and also shows examples where LGCA is preferred over MLM and viceversa.)

MacCallum RC, Kim C, Malarkey WB, KiecoltGlaser JK. Studying Multivariate Change Using Multilevel Models and Latent Curve Models. Multivariate behavioral research. 1997;32(3):215. (Comparisons have been drawn between multilevel modeling and latent growth curve models. This paper presents an overview of both, their relationship and instances where use of one would be superior to the use of the other. Simply, MacCallum et al, provide a (very long), yet informative review of MLM and LGCA in univariate change, they also show how they apply to multivariate change, present an example and then compare the two approaches.)
Application Articles

Brunet J, Sabiston CM, Chaiton M, et al. The association between past and current physical activity and depressive symptoms in young adults: a 10year prospective study. Ann Epi2013;23:2530. (An example of the use of LGCA to describe the longitudinal associations between adolescent moderatevigorous physical activity and depressive symptoms in young adulthood. Illustrates the overlap between LGCA and multilevel modeling approaches to analyzing change over time.)

Griffin MJ, Wardell JD, Read JP. Recent sexual victimization and drinking behavior in newly matriculated college students: a latent growth analysis. Psych Add Behav 2013: Advance online publication. Doi:10.1037/a0031831. (An example of the use of LGCA to describe associations among timing of sexual victimization and timing of drinking behavior. Illustrates well the stepwise approach to model building and interpretation.)

Heine C, Browning C, Cowlishaw S, Kendig H. Trajectories of older adults’ hearing difficulties: examining the influence of health behaviors and social activity over 10 years. Geriatr Gerontol Int 2013: online doi: 10.1111/ggi.12030. (An example of the use of LGCA to describe the trajectories of hearing decline in relation to age and lifestyle factors such as nutrition, smoking and social activity. Incorporates a brief summary of the use of LGCA and a lucid explanation of the results and the way to interpret LGCA models.)

Zahodne LB, Devanand DP, Stern Y. Coupled cognitive and functional change in Alzheimer’s Disease and the influence of depressive symptoms. J Alzh Dis 2013;34:85160.(An example of the use of a multivariate LCGA model to examine longitudinal associations among cognition, function, and depression in Alzheimer’s Disease patients followed every 6 months over 5.5 years.)

Muthen, Bengt O. “Analysis of longitudinal data using latent variable models with varying parameters.” (2011).

Jackson, Joshua J., et al. “Can an old dog learn (and want to experience) new tricks? Cognitive training increases openness to experience in older adults. “Psychology and aging 27.2 (2012): 286 (This paper was one of the first studies describing change in personality trait (openness) as a result to exposure to an intervention aimed at improving cognitive functioning. In the analysis, the researchers utilized secondorder LGM because they wanted to adjust for measurement error resulting from repeated measures.)
Courses

A workshop on growth curve modeling is available through EPIC

Introduction to Structural Equation Modeling, Online course Available Here