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| Growth Modeling of Longitudinal Data | |
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Growth models examine the development of individuals on one or more outcome variables over time. These outcome variables can be observed variables or continuous latent variables. Observed outcome variables can be continuous, censored, binary, ordered categorical (ordinal), counts, or combinations of these variable types if more than one growth process is being modeled. In growth modeling, random effects are used to capture individual differences in development. In a latent variable modeling framework, the random effects are reconceptualized as continuous latent variables, that is, growth factors. Mplus takes a multivariate approach to growth modeling such that an outcome variable measured at four occasions, for example, gives rise to a four-variate outcome vector. In contrast, multilevel modeling typically takes a univariate approach to growth modeling where an outcome variable measured at four occasions gives rise to a single outcome for which observations at the different occasions are nested within individuals, resulting in two-level data. Due to the use of the multivariate approach, Mplus does not consider a growth model to be a two-level model as in multilevel modeling but a single-level model. With longitudinal data, the number of levels in Mplus is one less than the number of levels in conventional multilevel modeling programs. The multivariate approach allows flexible modeling of relationships between the outcomes such as correlated residuals over time and regressions among the outcomes over time. In Mplus, there are two options for handling the relationship between the outcome and time. One approach allows time scores to be parameters in the model so that the growth function can be estimated. This is the approach used in structural equation modeling. The second approach allows time to be a variable that reflects individually-varying times of observations. This variable has a random slope. This is the approach used in multilevel modeling. Random effects in the form of random slopes are also used to represent individual variation in the influence of time-varying covariates on outcomes. Mplus growth modeling allows the analysis of multiple processes, both parallel and sequential; allows regressions among growth factors and random effects; and allows the growth model to be part of a larger latent variable model. When observed outcome variables are all continuous, Mplus has seven estimator choices: maximum likelihood (ML), maximum likelihood with robust standard errors and chi-square (MLR, MLF, MLM, MLMV), generalized least squares (GLS), and weighted least squares (WLS) also referred to as ADF. When at least one outcome variable is binary or ordered categorical, Mplus has seven estimator choices: weighted least squares (WLS), robust weighted least squares (WLSM, WLSMV), maximum likelihood (ML), maximum likelihood with robust standard errors and chi-square (MLR, MLF), and unweighted least squares (ULS). When at least one outcome variable is censored, unordered categorical, or a count, Mplus has six estimator choices: weighted least squares (WLS), robust weighted least squares (WLSM, WLSMV), maximum likelihood (ML), and maximum likelihood with robust standard errors and chi-square (MLR, MLF). |
