Fitting Nonlinear Latent Growth Curve Models With Individually Varying Time Points

Abstract

Individual growth trajectories of psychological phenomena are often theorized to be nonlinear. Additionally, individuals’ measurement schedules might be unique. In a structural equation framework, latent growth curve model (LGM) applications typically have either (a) modeled nonlinearity assuming some degree of balance in measurement schedules, or (b) accommodated truly individually varying time points, assuming linear growth. This article describes how to fit 4 popular nonlinear LGMs (polynomial, shape-factor, piecewise, and structured latent curve) with truly individually varying time points, via a definition variable approach. The extension is straightforward for certain nonlinear LGMs (e.g., polynomial and structured latent curve) but in the case of shape-factor LGMs requires a reexpression of the model, and in the case of piecewise LGMs requires introduction of a general framework for imparting piecewise structure, along with tools for its automation. All 4 nonlinear LGMs with individually varying time scores are demonstrated using an empirical example on infant weight, and software syntax is provided. The discussion highlights some advantages of modeling nonlinear growth within structural equation versus multilevel frameworks, when time scores individually vary.

Keywords:

• definition variables
• latent curve models
• nonlinear growth

ACKNOWLEDGMENTS

The author thanks Kristopher J. Preacher and Daniel J. Bauer for helpful comments.

Notes

¹ Some models discussed later (namely, structured latent curve [SLC] models) also allow certain parameters to enter the model nonlinearly. This kind of nonlinearity is distinguished from whether a model accommodates a nonlinear relationship between the construct and time (our focus here).

² If there were a limited number of different measurement schedules, a modified specification was used that placed individuals with the same schedule in a group and fit the LGM simultaneously to all groups (e.g., Duncan & Duncan, 1994; McArdle & Hamagami, 1992; Muthén, Kaplan, & Hollis, 1987).

³ For an example person in Cohort 2, definition (a) implies the loading matrix on the left for a fitted quadratic LGM, and definition (b) implies the matrix on the right. .

⁴ Regarding degree of individual variation, a simulation was conducted (generating parameters = 4.65; = –.47; = 1.4; = .35; = –.13; ; = .5) with normally distributed time scores generated to have some variation at Levels 1 and 2: . Two alternative amounts of individual variation in time were considered; either ( and ), or ( and , the latter implying considerably overlapping windows. t = 0, 1, 2, 3, 4, 5, or 6. There was trivial to no bias (≤ 1.2% relative bias) in any model parameter across 100 samples of N = 1,000.

⁵ This is an arbitrary decision. If the alternative decision had been made (i.e., when an intercept falls at the boundary between two phases, assign it to the later—not earlier—phase), the exact same lambda matrix would result, so long as the decision was made consistently.

⁶ These persons could have missing data; missing data are not immediately relevant to this illustration.

⁷ The %PIECEWISE macro outputs the transformed time scores in long format. The Mplus syntax provided in the online Appendix transforms them to wide format before fitting piecewise LGMs.

⁸ Loadings for any pieces with higher order (e.g., quadratic, cubic) slopes can be specified as squares or cubes of linear time scores in the data, for instance using nonlinear constraints (see example code).

⁹ That is, repeated measures are assumed by SEM software to be simple weighted sums of growth coefficients × predictors, plus error. In more complex functional forms, some [growth coefficients × predictor] terms might be in an exponent or logarithm, under a radical, or inside a trigonometric term.

¹⁰ Note that MLM also uses a conditional likelihood, and will listwise delete observations, not cases, with missing time scores (but will do so for Case II or Case III data).

¹¹ One extreme case (18 kg at 1 year) who had small to moderate influence on estimates was deleted.

¹² Normally, we could also conduct an LRT between a piecewise linear and piecewise quadratic LGM, but a very small Time 7 residual variance in the latter model had to be constrained to 0 to prevent a nonpositive definite solution. This additional constraint made the models nonnested.

¹³ Relatedly, it is useful to visualize that making additional observations of existing subjects necessitates adding rows to lambda (as in the balanced case shape-factor model also), corresponding with new measurement windows.