Latent Growth Curve Models

19.1 The Model, From the Path Diagram Up

The latent growth curve model, or latent curve model, represents each person’s trajectory through two latent factors. An intercept factor that every repeated measure loads on with a fixed loading of one, so that it represents the common level shared across occasions, and a slope factor whose loadings are fixed to the time codes, zero at the first occasion, one at the second, and so on, so that it represents the linear change per unit of time. Figure 19.1 draws the model for five waves. A person’s score at a given occasion is the intercept factor plus the slope factor times that occasion’s time code plus a residual, which is algebraically the person-specific line of Chapter 14 written as a factor model. What is genuinely new to the structural equation reader is the mean structure. Ordinary factor analysis models only the covariances among the indicators and leaves their means aside; a growth model must model the means, because the average trajectory, the mean starting level and the mean rate of change, is the primary object of study. The factor means, the mean of the intercept factor and the mean of the slope factor, carry the average trajectory, while the factor variances carry the individual differences around it and the factor covariance carries their association. This is the conceptual hurdle for those trained on covariance-only factor analysis, and it repays patience: a growth model is a model of means and covariances together.

Note. The intercept factor loads on every occasion with a fixed loading of one; the slope factor loads with the time codes (\(0,1,2,3,4\)). The factor means (not shown) carry the average trajectory, the factor variances carry the individual differences, and the factor covariance carries their association. Residuals on each indicator are omitted for clarity. The model is the person-specific line of Chapter 14 written as a factor model.

Seeing the model this way reveals its deepest feature: a growth model is a set of constraints on the moments. The factor structure implies a specific mean vector and covariance matrix for the observed repeated measures, in which the means grow linearly and the variances fan out or in according to the intercept and slope variances and their covariance. Fitting the model tests whether the observed means and covariances match that implied structure, and Figure 19.2 shows the match on the running data: the model-implied means grow linearly through the observed means, and the implied variances track the observed fanning. This implied-moment view is what powers the global fit testing that distinguishes the structural equation approach, because a discrepancy between the observed and implied moments is exactly what the fit indices quantify. In lavaan the model is written either through the convenience function growth, which sets the intercept and slope loadings and the factor means automatically, or by hand, and this chapter teaches the manual specification first so that the convenience is understood rather than trusted.

Note. Observed moments (red points) against the implied moments (blue lines) of the linear latent curve model. The model reproduces both the linearly growing means and the fanning variances. A growth model constrains the mean vector and the covariance matrix jointly, and the fit indices quantify the discrepancy between observed and implied.

19.2 Identification and Data Demands

Because a growth model is a set of constraints on the moments, its identification is a matter of counting knowns against unknowns. The knowns are the observed moments: with \(T\) waves there are \(T\) means and \(T(T+1)/2\) variances and covariances. The unknowns are the model parameters: the two factor means, the two factor variances, their covariance, and the residual variances. Table 19.1 works the arithmetic. A linear model with a mean structure is identified with three waves, but only just, leaving no degrees of freedom to test the growth form, so three waves can fit a line but cannot adjudicate whether the line is right. Four waves are the honest minimum for form adjudication, and more are better for nonlinear shapes. A distinct problem is empirical underidentification: when the slope variance is genuinely near zero, the data carry little information about it, and estimation can produce a negative variance estimate, a Heywood case. This is the boundary problem of Chapter 13 in structural-equation clothing, and the response is the same, to diagnose whether the negative estimate reflects a true near-zero variance, sampling noise, or misspecification, and to respond by constraining, simplifying, or, as Chapter 17 showed, moving to Bayesian estimation that keeps the variance in bounds, rather than by panicking or by blindly fixing the estimate to zero.

19.3 The Equivalence Set Piece

The claim that the multilevel and latent-curve frameworks fit the same model is best demonstrated, not asserted. Fitting the unconditional linear growth model of Chapter 14 to the same data both ways, as a mixed model in lme4 and as a latent curve model in lavaan, returns identical estimates, and Figure 19.3 plots them against each other: every parameter, the intercept and slope means, their variances and covariance, and the residual variance, lies exactly on the line of equality, matching to three decimal places. Table 19.2 is the bilingual dictionary that translates each parameter’s name between the frameworks. The equivalence holds under specific conditions, which are worth stating because their violation is where the frameworks diverge: the time codes must be the same for every person, and the residual structure must match, typically a single residual variance across waves. When those conditions hold the two are the same model estimated by the same likelihood, and the choice between them is one of convenience.

Note. Each parameter of the unconditional linear growth model, estimated as a mixed model (horizontal axis) and as a latent curve model (vertical axis), plotted against the line of equality. Every parameter matches to three decimal places. The two frameworks are dialects of one model when the time codes and residual structure are matched.

The equivalence has boundaries, and touring them shows what each framework offers. The multilevel framework handles individually varying measurement times natively, because time enters as a numeric predictor that can differ across persons, whereas the standard latent-curve model fixes the time codes in the loadings and so assumes a common schedule; accommodating varying times in the structural equation framework requires definition variables that let each person’s loadings equal their own measurement times, available in some programs and a genuine limitation of others (Mehta & Neale, 2005; Mehta & West, 2000). The structural equation framework, in turn, supplies global fit machinery that the multilevel framework lacks, and this is not idle, because it detects misspecification of the growth form that a multilevel model, which does not test the implied covariance structure against the observed, can miss. The structural equation framework also embeds the growth factors in larger structural systems, letting a slope factor predict a distal outcome or mediate an effect, which the multilevel framework cannot do directly. Both handle structured residuals, with different ergonomics, and both default to full-information maximum likelihood under missingness. Table 19.3 is the decision guide, and the book’s position is bilingualism: the frameworks are dialects, not denominations (Curran, 2003; Bauer, 2003; McNeish & Matta, 2018).

19.4 Fit Evaluation for Growth Models

Because a growth model constrains the means and the covariances jointly, its fit indices carry information the multilevel framework does not surface, but they must be read with care. A poor comparative fit index or a large root mean square error of approximation signals that the implied moments do not match the observed, and the first diagnostic question is where: misfit in a growth model is often in the mean structure, an average trajectory of the wrong shape, rather than in the covariances. The forensic tool is to inspect the residual moments, the differences between observed and implied means and covariances, to localize the discrepancy. Figure 19.4 shows the diagnosis on a decelerating outcome fitted with a linear model: the linear model’s implied means bow systematically away from the observed means, over-predicting in the middle and under-predicting at the ends, the fingerprint of a wrong growth form, while a model that estimates the shape reproduces the means. The benchmarks for the fit indices carry the same cautions rehearsed in Chapter 18, that they are rules of thumb from particular conditions, and they are sensitive here to the sample size and to the magnitude of the slope variance. Nested models for the growth form, linear against a freed-loading shape against a quadratic, are compared by the chi-square difference and the information criteria. Table 19.4 is the protocol, and a recurring lesson is that good global fit does not certify the correct growth form, only that the chosen form is not grossly contradicted (Wu et al., 2009).

Note. Observed means (grey points) of a decelerating outcome against the implied means of a linear model (red) and a freed-loading model (blue). The linear model’s implied means bow away from the observed, the signature of a misspecified growth form, which resides in the mean structure; the freed-loading model reproduces the means. Locate growth-model misfit by inspecting the residual moments.

19.5 Conditional Models and Consequences of Growth

A conditional growth model adds covariates that predict the growth factors, and it is the structural-equation counterpart of the cross-level interactions of Chapter 14, the same estimand in different clothing. A time-invariant covariate regressed on the intercept and slope factors gives its effect on the starting level and on the rate of change, and Figure 19.5 shows the predicted trajectories at several covariate values, diverging because the covariate raises the slope. On the running data the covariate’s effect on the slope factor recovers its true value. Time-varying covariates enter as occasion-specific predictors of the repeated measures, carrying the same warnings rehearsed in Chapter 14, that a covariate trending with time can absorb the growth it should sit alongside. The distinctive structural-equation extension is to place the growth factors as predictors of a distal outcome, the slope-as-predictor model in which the rate of change forecasts a later result. This is powerful and hazardous: the slope factor is estimated with error, so its reliability bounds what it can predict, and the time window matters, because predicting an outcome from change measured over an interval that overlaps the outcome courts circularity. These cautions are the price of the structural-equation framework’s reach.

Note. Predicted trajectories at three values of a time-invariant covariate that is regressed on the intercept and slope factors. A higher covariate value raises both the starting level and the rate of change, so the trajectories diverge. This is the latent-curve counterpart of the cross-level interaction of Chapter 14.

19.6 Multiple-Group Growth

Fitting a growth model in multiple groups compares trajectories across, for instance, the arms of a trial, and it makes the treatment effect a difference in the slope-factor mean. Figure 19.6 shows the arm-specific fitted trajectories on the therapy trial of Chapter 14, now analyzed as a multiple-group latent curve model with full-information maximum likelihood for the dropout, and the treatment arm’s steeper decline, a slope of about one point per week faster than the control arm, is the treatment effect. The analysis proceeds by a constraint-testing sequence, from a configural model with the growth form free in each group, to a model constraining the growth-factor means equal, and the test of that constraint is the test of the treatment effect, which here rejects equality decisively. Table 19.5 lays out the sequence. This multiple-group parameterization is exactly equivalent to the arm-by-time interaction of the Chapter 14 conditional model, the treatment effect appearing as an interaction in one framework and as a group difference in slope means in the other, and demonstrating that equivalence again is part of the bilingual discipline. When the construct is measured by multiple items, the comparison rests on the measurement invariance of Chapter 18, and the full chain, invariant measurement supporting comparable growth factors, is the second-order growth model developed in Chapter 20.

Note. Fitted trajectories for the two arms of the therapy trial, from a multiple-group latent curve model with full-information maximum likelihood. The treatment arm declines faster, and the test constraining the growth-factor means equal across arms rejects, establishing the treatment effect. This parameterization is equivalent to the arm-by-time interaction of Chapter 14.

19.7 Parallel-Process Growth

A parallel-process model fits growth in two domains at once and estimates the associations among their growth factors, including the slope-slope covariance that asks whether change in one domain accompanies change in the other. Figure 19.7 shows the estimated growth slopes of two outcomes plotted against each other, with a positive association: people whose first outcome grows faster tend to have a faster-growing second outcome. The interpretive caution is essential and easily forgotten. A slope-slope correlation is a statement of correlated change, that two processes move together across people, and it is not evidence that one process drives the other. Establishing which process leads requires a model of the temporal dynamics, the lead-lag structure that a static association cannot reveal, and that is precisely what the latent change score and cross-lagged panel models of Chapters 20 and 21, and the dynamic models of the intensive-longitudinal part, are built to provide. The parallel-process model describes the coupling; it does not explain it, and reporting it as though it did is a common overreach.

Note. Estimated growth slopes of two outcomes across persons, with their positive association. The slope-slope correlation describes correlated change, that the two processes move together, and is not evidence that one drives the other. Establishing direction requires the dynamic models of the chapters to come.

19.8 Latent-Basis Growth

When the shape of change is unknown, the structural equation framework can estimate it rather than impose it, through a latent-basis or freed-loading model. The first and last slope loadings are fixed, to zero and one, so that the slope factor represents the total change from the first occasion to the last, and the interior loadings are freed and estimated, so that each interior loading is the proportion of the total change that has occurred by that wave. Figure 19.8 shows the result on a decelerating outcome: the freed loadings recover the true nonlinear shape, and they differ visibly from the equal spacing a linear model would impose. On the running data the latent-basis model fits better than the linear model and comparably to a quadratic, at fewer parameters. The freed-loading approach has real power, giving an interpretable proportion-of-change reading and adapting to any monotone shape, but it carries traps. It capitalizes on chance, fitting the sample’s particular shape, so the estimated basis should be cross-validated or theoretically motivated; and comparing a latent basis across groups requires constraining the loadings, because an unconstrained basis that differs across groups conflates a difference in shape with a difference in the measurement of change. Used with these cautions, the latent basis is often preferable to a polynomial, which imposes a rigid functional form and extrapolates poorly.

Note. The estimated freed loadings (blue), the true shape (grey), and the equal spacing a linear model imposes (red), as proportions of the total change by wave. The freed loadings recover the true nonlinear shape, which the linear model cannot. Each interior loading reads as the proportion of total change achieved by that wave.

19.9 Reporting a Latent Growth Analysis

A latent growth analysis is reported so that its constraints and choices are legible, and Table 19.6 is the checklist. The report states the time coding and its origin, because every intercept-related quantity depends on it exactly as in Chapter 14; names the growth form and the basis for choosing it; gives the factor means, variances, and covariance with the average trajectory and the individual differences interpreted; reports the global fit with the residual-moment diagnosis where fit is imperfect; describes the missing-data handling; and, for conditional, multiple-group, or parallel-process extensions, reports the relevant coefficients, constraint tests, or factor associations with their cautions. Where the frameworks are equivalent, noting the correspondence with the multilevel parameterization aids a bilingual readership.

19.10 Running Latent Growth Models in R

The lavaan package fits the model through the convenience function growth, which sets the intercept and slope loadings and estimates the factor means, or through manual syntax that makes every constraint explicit.

library(lavaan)
lgm <- '
  I =~ 1*y0 + 1*y1 + 1*y2 + 1*y3 + 1*y4          # intercept: loadings all 1
  S =~ 0*y0 + 1*y1 + 2*y2 + 3*y3 + 4*y4          # slope: loadings = time codes
  y0 ~~ r*y0; y1 ~~ r*y1; y2 ~~ r*y2             # equal residuals (to match the mixed model)
  y3 ~~ r*y3; y4 ~~ r*y4
  I ~ 1; S ~ 1                                    # the mean structure: factor means '
fit <- growth(lgm, data = wide)                   # or sem(lgm, ..., meanstructure = TRUE)
summary(fit, fit.measures = TRUE, standardized = TRUE)

Conditional, multiple-group, parallel-process, and latent-basis models are small extensions of this syntax: a covariate is regressed on the factors (I   x; S   x); groups are compared with the group and group.equal arguments; a parallel process adds a second pair of factors; and a latent basis frees the interior loadings (S =  0*y0 + NA*y1 + NA*y2 + NA*y3 + 1*y4).

growth(lgm, data = wide, group = "arm", missing = "fiml")   # multiple-group, FIML
anova(fit_configural, fit_equalmeans)                        # test the group effect
# equivalence check: the same model as a mixed model
library(lme4); lmer(y ~ wave + (wave | id), data = long)

The complete analysis, including the parameter-by-parameter equivalence with lme4, the moment inspection, the conditional and multiple-group and parallel-process models, and the latent-basis shape recovery, is the shipped script ch19_analysis_V01.R, with figures drawn by ch19_figures_V01.R and the dataset generated by gen_growth_wide_V01.R.

19.11 Common Misconceptions

Several beliefs about latent growth models mislead. The first is that structural-equation growth modeling is more advanced or more correct than multilevel growth modeling; the two are the same model under matched assumptions, and they differ only in ergonomics (Figure 19.3). The second is that a poor comparative fit index kills the growth model; misfit should first be localized, because it often lies in the mean structure and signals the wrong growth form rather than a fatal flaw. The third is that a nonzero slope variance proves individual differences in change exist; the variance is estimated with error and can hit a boundary, so its precision, and the power to detect it, must be weighed. The fourth is that a parallel-process slope correlation shows one domain drives the other; it shows correlated change, and direction requires a dynamic model. The fifth is that good fit certifies the growth form; good fit is necessary but not sufficient, and a well-fitting linear model can still be the wrong shape at unobserved times.

Where to Go Next

This chapter and Chapter 14 together complete the growth-modeling core in both dialects. Chapter 20 extends the latent-curve framework in two directions: the second-order growth model that fits the trajectory to the invariant latent factors of Chapter 18, cashing the measurement check that a composite score silently assumes, and the latent change score model that reparameterizes growth as a sequence of change scores and so begins to model the dynamics of change rather than only its shape. Chapter 21 places growth factors inside the cross-lagged panel systems that ask which process leads, resolving the directional question the parallel-process model can only raise. Chapter 22 asks whether a single growth model conceals a mixture of distinct trajectory classes. Throughout, the bilingual habit this chapter builds, seeing every growth model as both a mixed model and a set of constraints on the moments, is the working competence of longitudinal structural equation modeling.

Exercises

1. 19.1 Implied moments by hand. For a four-wave linear model with given factor means, variances, covariance, and residual variance, compute the implied mean vector and covariance matrix, and verify them against a lavaan fit.
2. 19.2 Reproduce and break the equivalence. Fit the unconditional linear model as a mixed model and a latent curve model and confirm the estimates match; then add an autoregressive residual structure and explain why the equivalence no longer holds.
3. 19.3 Misfit forensics. Fit a linear model to a dataset with a planted quadratic shape, and use the residual moments to locate and diagnose the misfit.
4. 19.4 Multiple-group analysis. Fit a multiple-group growth model with the constraint sequence, test the group effect, and write the results, noting the equivalence with the interaction parameterization.
5. 19.5 Latent basis. Fit a freed-loading model, interpret the loadings as proportions of total change, and compare it to a quadratic by fit and parsimony.
6. 19.6 Bilingual translation. Translate a published multilevel growth table into latent-curve syntax and confirm the two produce the same estimates.

References