Latent Growth Model

Latent Growth Model for Repeated Health Outcomes

In a latent growth model the repeated scores of each patient, for example an EQ-5D index recorded at several follow-up visits, are treated as indicators of a few unobserved growth factors, and the timing of each visit is written into the factor loadings. The means of those factors give the average trajectory of the cohort, and their variances and covariance say how far patients depart from it. Health economists meet these models when trial or registry data show how health-related quality of life recovers or declines after an event, where an average gain may hide patients who get worse. As a special case of the structural equation model, it can test whether a proposed shape of change fits the observed means and covariances, and it can link growth to predictors or to later outcomes. Its estimates depend on choices about time coding, residuals and missing data, which need to be stated.

Repeated measures as indicators of growth factors

Preacher describes latent growth curve modelling as one application of structural equation modelling to change, in which repeated measures are indicators of latent variables representing aspects of change, typically an intercept and a linear slope. The structure is that of a confirmatory factor analysis whose loadings are fixed in advance to encode the hypothesised trajectory, so the factors are defined by time.

$$\mathbf{y}_i = \boldsymbol{\Lambda}\boldsymbol{\eta}_i + \boldsymbol{\varepsilon}_i, \qquad \boldsymbol{\eta}_i = \boldsymbol{\alpha} + \boldsymbol{\zeta}_i$$

where $\mathbf{y}_i$ is the vector of $T$ repeated measures for patient $i$, $\boldsymbol{\Lambda}$ is the $T \times k$ loading matrix whose first column is all ones and whose second column, in a linear model, holds the measurement times, $\boldsymbol{\eta}_i$ holds the $k$ growth factors of patient $i$, $\boldsymbol{\varepsilon}_i$ holds the residuals with covariance matrix $\boldsymbol{\Theta}$, $\boldsymbol{\alpha}$ is the vector of growth factor means, and $\boldsymbol{\zeta}_i$ holds the deviations of patient $i$ from those means, with covariance matrix $\boldsymbol{\Psi}$. McNeish and Matta give this form, with a term for time-invariant covariates added to the second equation.

The model therefore implies a mean vector and a variance-covariance matrix for the observed scores.

$$\boldsymbol{\mu} = \boldsymbol{\Lambda}\boldsymbol{\alpha}, \qquad \boldsymbol{\Sigma} = \boldsymbol{\Lambda}\boldsymbol{\Psi}\boldsymbol{\Lambda}^{\top} + \boldsymbol{\Theta}$$

where $\boldsymbol{\mu}$ is the implied mean of each repeated measure, $\boldsymbol{\Sigma}$ is their implied covariance matrix, and the other symbols are as above. For a linear model with time code $\lambda_t$ at occasion $t$, each diagonal element of $\boldsymbol{\Sigma}$ is a variance.

$$\operatorname{Var}(y_{it}) = \psi_{II} + 2\lambda_t\psi_{IS} + \lambda_t^{2}\psi_{SS} + \theta_t$$

where $\psi_{II}$ is the intercept variance, $\psi_{SS}$ the slope variance, $\psi_{IS}$ their covariance and $\theta_t$ the residual variance at occasion $t$. Estimation chooses $\boldsymbol{\alpha}$, $\boldsymbol{\Psi}$ and $\boldsymbol{\Theta}$ so that $\boldsymbol{\mu}$ and $\boldsymbol{\Sigma}$ come as close as possible to the sample means and covariances, so the growth factors are latent variables, not lines fitted patient by patient.

Coding time and choosing the trajectory shape

The time codes in $\boldsymbol{\Lambda}$ give every growth parameter its meaning. Preacher separates the origin, where the intercept is interpreted, from the scale, the unit of the slope, and asks that both be justified by theory and context rather than chosen arbitrarily. Coding visits at 0, 6, 12 and 24 months as 0, 0.5, 1 and 2 makes the slope a change per year and the intercept the expected score at the first visit.

Curran, Obeidat and Losardo describe the usual shapes. An intercept-only model is flat, polynomial terms add curvature, and a piecewise model joins linear segments. Momenyan and colleagues fitted a piecewise model with one intercept and two slope factors to EQ-5D-5L data from 1,480 Canadian road trauma survivors, assessed at baseline (pre-injury) and at 2, 4, 6 and 12 months. Scores fell by 2 months and then rose without returning to baseline by 12 months.

A latent basis model frees some slope loadings so that the data set the shape. McNeish and Matta regard it as an easy option when growth is nonlinear in an unknown way, while Preacher warns that freed loadings make the analysis exploratory rather than a test of a stated trajectory.

Identification, estimation and model fit

Preacher states that time-balanced data need at least $k + 1$ repeated measures for $k$ growth factors, so a linear model needs three occasions and a cubic model five. With $T$ occasions the data supply $T$ means and $T(T+1)/2$ variances and covariances, and the degrees of freedom are what remains after the growth factor means, $\boldsymbol{\Psi}$ and the residual variances are estimated.

Estimation is usually by maximum likelihood estimation, which Curran and colleagues note assumes continuous, normally distributed repeated measures, with other estimators available for non-normal or ordinal outcomes. In the myocardial infarction cohort reported by Dondo and colleagues, the upper quartile of the baseline EQ-5D index was already 1.00, a ceiling effect a normal model does not represent. McNeish and Matta add that full maximum likelihood underestimates random-effect variances and standard errors with fewer than about 50 people.

Global fit sets the structural equation version apart. Preacher notes that the chi-square test reacts to sample size and trivial misfit, recommends several fit indices from different classes, with a 90% confidence interval for RMSEA, and warns that the incremental indices printed by most software use a null model that is wrong for growth models. McNeish and Matta report that SRMR rarely detects misfit in the mean structure, the part that describes average change. Preacher treats negative residual variances (Heywood cases) and growth factor correlations outside minus 1 to 1 as signs of misspecification or estimation error.

Residuals are commonly held equal across occasions or freely estimated, and Grimm and Widaman note that this choice can have sizeable effects on the estimated intercept and slope variances and covariances, the parameters that describe heterogeneity between patients.

Covariates, groups and parallel processes

Curran and colleagues describe time-invariant covariates, such as treatment arm, which predict the growth factors, and time-varying covariates, which predict the repeated measures after allowing for the growth factors. Multiple-group models let variances differ as well, and their example is an intervention that reduces variability in the treated group.

Growth factors can also predict later outcomes, and two outcomes can grow together in a multivariate model with correlated intercepts and slopes. McNeish and Matta judge such embedding straightforward in the latent curve framework, and note that its full information likelihood keeps patients with missing predictors, whereas most mixed-model software deletes them.

Worked example with EQ-5D scores over two years

All numbers in this example are illustrative. A cohort starting a new treatment completes the EQ-5D-5L at 0, 6, 12 and 24 months, coded as 0, 0.5, 1 and 2 years. A linear model with equal residual variances gives $\alpha_I = 0.62$, $\alpha_S = 0.04$ per year, $\psi_{II} = 0.040$, $\psi_{SS} = 0.0016$, $\psi_{IS} = -0.0024$ and a residual variance of 0.010 at every occasion.

1. Implied mean trajectory. The means are 0.62 at baseline, 0.62 + 0.04 × 0.5 = 0.64 at 6 months, 0.62 + 0.04 × 1 = 0.66 at 12 months and 0.62 + 0.04 × 2 = 0.70 at 24 months.

2. Implied variances. At baseline the variance is 0.040 + 0.010 = 0.050, of which 0.040 / 0.050 = 0.80 comes from the growth factors. At 24 months the positive terms are 0.040 + 4 × 0.0016 + 0.010 = 0.0564, and the covariance term 2 × 2 × 0.0024 = 0.0096 enters with a negative sign, so the variance is 0.0564 less 0.0096, or 0.0468.

3. Heterogeneity in change. The slope standard deviation is the square root of 0.0016, or 0.04 per year. If slopes are normally distributed, the share of patients whose latent trajectory declines is $\Phi(-0.04/0.04) = \Phi(-1) = 0.159$, about one in six, although the mean improves. The intercept and slope correlate at minus 0.0024 / (0.20 × 0.04) = minus 0.30, so patients who start higher tend to improve less.

4. Moving the time origin. Re-coding time so that zero falls at 24 months makes the intercept the level at two years. Its variance becomes 0.040 + 4 × 0.0016 = 0.0464 less 4 × 0.0024 = 0.0096, or 0.0368. Its covariance with the slope becomes 2 × 0.0016 = 0.0032 less 0.0024, or 0.0008, a correlation of 0.0008 / (0.1918 × 0.04) = 0.10.

5. Degrees of freedom. Four occasions supply 4 means and 4 × 5 / 2 = 10 variances and covariances, 14 moments in all. The model estimates 2 means, 3 elements of $\boldsymbol{\Psi}$ and 1 residual variance, 6 parameters, which leaves 8 degrees of freedom. Freeing the residual variance at each occasion uses 9 parameters and leaves 5.

6. Area under the mean curve. Over two years the area under the mean trajectory is 2 × 0.62 = 1.24 from the intercept plus 0.04 × 4 / 2 = 0.08 from the slope, 1.32 quality-adjusted life years per patient alive and responding throughout, against 1.24 if scores had stayed at baseline. The 0.08 difference is undiscounted, conditional on survival and not a treatment effect, because there is no comparator.

The average gain is modest and individual change varies widely. For an economic evaluation the one in six with declining trajectories, and whether baseline covariates identify them, may matter more than the mean slope.

Latent growth, mixed-effects growth and trajectory classes

A growth curve model can be fitted as a multilevel model or within the structural equation framework. Curran and colleagues state that the two are numerically identical in many situations, that the multilevel model extends more easily to higher levels of nesting, and that the structural equation approach suits latent variables that remove measurement error. McNeish and Matta show that mathematical equivalence is not always practical equivalence. In their summary the mixed-effects model suits small samples, individually varying visit times and three-level data, and the latent curve approach suits global fit testing, growth embedded in larger models and unconstrained time-varying covariates.

A single latent growth model assumes one set of growth factor means for everyone. For unobserved subgroups with different trajectories, Curran and colleagues point to growth mixture, latent class growth and group-based trajectory models. The growth mixture model and the group-based trajectory model combine growth modelling with the unobserved classes of latent class analysis, and movement between discrete unobserved states belongs to latent transition analysis.

Use in economic evaluation and health technology assessment

The applied studies cited here use latent growth models to describe trajectories and their predictors. Dondo and colleagues used them for EQ-5D-3L change over 12 months in 9,474 patients with myocardial infarction admitted to 77 hospitals in England, scored in hospital and at 1, 6 and 12 months. They report a baseline mean of 0.76 and a mean increase of 0.003 per month.

Preacher warns that attrition is typically not missing at random and can bias intercept and slope means and variances, while direct maximum likelihood and multiple imputation both assume data missing at random or completely at random, as Curran and colleagues note. A mean trajectory among survivors describes quality of life given survival, so QALYs also need a survival model. The area under the curve beyond the last visit is extrapolation, since Preacher notes that no fit can confirm the trend at times not observed. A slope difference between arms supports a causal reading only through the study design, and Preacher advises sparing causal language.

Limitations and common misreadings

Common errors come from reading growth parameters apart from the model and time coding that define them. The parameters summarise a fitted structure and are not direct measurements of individual patients.

1. Good fit read as proof of the trajectory. Preacher notes that growth models are never literally true and that a model with zero degrees of freedom fits any data perfectly.
2. An intercept and slope correlation read as a fixed property of patients. It depends on where time zero is placed, as step 4 shows.
3. A mean gain read as everyone improving. The slope variance can imply a sizeable share of patients who decline.

Sources

• Curran PJ, Obeidat K, Losardo D. Twelve frequently asked questions about growth curve modeling. Journal of Cognition and Development. 2010;11(2):121-136. https://doi.org/10.1080/15248371003699969
• Dondo TB, Munyombwe T, Hurdus B, Aktaa S, Hall M, Soloveva A, et al. Association of baseline and changes in health-related quality of life with mortality following myocardial infarction: multicentre longitudinal linked cohort study. European Heart Journal. Quality of Care and Clinical Outcomes. 2025;11(6):730-738. https://doi.org/10.1093/ehjqcco/qcae036
• Grimm KJ, Widaman KF. Residual structures in latent growth curve modeling. Structural Equation Modeling. 2010;17(3):424-442. https://doi.org/10.1080/10705511.2010.489006
• McNeish D, Matta T. Differentiating between mixed-effects and latent-curve approaches to growth modeling. Behavior Research Methods. 2018;50(4):1398-1414. https://doi.org/10.3758/s13428-017-0976-5
• Momenyan S, Chan H, Erdelyi S, Pei LX, Jae L, Taylor JA, et al. Longitudinal modeling of health-related quality of life trajectories over 12 months following road trauma. PLoS One. 2025;20(11):e0336144. https://doi.org/10.1371/journal.pone.0336144
• Preacher KJ. Latent growth curve models. In: Hancock GR, Mueller RO, eds. The Reviewer's Guide to Quantitative Methods in the Social Sciences. New York: Routledge; 2010:185-198. https://quantpsy.org/pubs/preacher_2010.pdf