Growth Mixture Model

Growth Mixture Model for Patient Trajectories of Health and Cost

A growth mixture model fits several growth curves at once to repeated measurements, such as EQ-5D utility scores after surgery or yearly health care costs, assuming that each patient belongs to one of a few unobserved classes. Each class has its own mean path over time, and patients within a class still vary around that path through random effects, which is what separates the method from a group-based trajectory model. Health economists meet it in studies of quality of life after hip fracture surgery and when choosing a trajectory model for health care costs. The classes depend heavily on distributional and covariance assumptions, so a fitted solution describes the data under a model and does not by itself show that distinct kinds of patient exist.

One growth curve extended to several unobserved classes

A growth curve model describes repeated measures with one average trajectory and person-specific deviations from it. Ram and Grimm present the growth mixture model as a multiple-group growth model in which the grouping variable is latent, so membership is estimated from each person's observed trajectory. Herle and colleagues describe it as several mixed effects models, one per class, each with its own mean and shape, with the classes weighted as in a finite mixture model. In Muthén and Shedden's formulation, latent trajectory classes of heavy drinking predicted later alcohol dependence.

Class-specific growth and the mixture likelihood

Within each class the outcome follows a mixed effects growth model with its own mean intercept and slope. Herle and colleagues give the linear case, written here with each person's own measurement times $t_{ij}$.

$$y_{ij} \mid c = (\beta_{0c} + u_{0ic}) + (\beta_{1c} + u_{1ic})\, t_{ij} + \varepsilon_{ijc}$$

where $y_{ij}$ is the outcome for person $i$ at occasion $j$, $t_{ij}$ is its time, $c$ is the class ($c = 1, \ldots, C$), $\beta_{0c}$ and $\beta_{1c}$ are the class mean intercept and slope, $u_{0ic}$ and $u_{1ic}$ are person-specific deviations, jointly normal with mean zero and covariance matrix $\Omega_c$, and $\varepsilon_{ijc}$ is a residual with variance $\sigma_c^2$.

Because membership is unknown, each person's measurements follow a weighted sum of the class distributions.

$$f(\mathbf{y}_i) = \sum_{c=1}^{C} \pi_c \, \phi\left(\mathbf{y}_i ;\, \mathbf{X}_i \boldsymbol{\beta}_c ,\, \mathbf{Z}_i \Omega_c \mathbf{Z}_i^{\top} + \sigma_c^2 \mathbf{I}\right)$$

where $\mathbf{y}_i$ is person $i$'s vector of measurements, $\pi_c$ is the population share of class $c$ (the shares sum to one), $\phi$ is the multivariate normal density, $\mathbf{X}_i$ and $\mathbf{Z}_i$ hold the time terms for the class means and the random effects, $\boldsymbol{\beta}_c$ holds the class mean coefficients and $\mathbf{I}$ is an identity matrix. The random effects make one patient's measurements correlated within a class.

Bayes' theorem then gives each person a posterior probability for each class.

$$\Pr(c \mid \mathbf{y}_i) = \frac{\pi_c \, f_c(\mathbf{y}_i)}{\sum_{k=1}^{C} \pi_k \, f_k(\mathbf{y}_i)}$$

where $f_c(\mathbf{y}_i)$ is the class $c$ normal density from the previous equation and $k$ runs over all classes. Herle and colleagues note their use for modal assignment, to the most probable class, or as weights when estimating class frequencies.

Estimation is by maximum likelihood estimation through the EM algorithm. Herle and colleagues expect several local maxima in models this complex and recommend multiple starting points. Ram and Grimm note that missing outcomes are handled under a missing at random assumption, so patients with incomplete follow-up stay in the analysis.

Specification choices that move the classes

Which parameters may differ between classes changes how many classes appear. Ram and Grimm let classes differ in mean change, then also in growth factor variances and covariances, then in the shape of change, and fit one more class than expected so that it can be rejected.

Herle and colleagues report that a covariance structure that is too simple may need more classes to absorb the variability, and they recommend class-specific $\Omega_c$ and time-specific residual variances when choosing the number of classes, as far as study size allows. Fixing every $\Omega_c$ at zero gives latent class growth analysis, also called the group-based trajectory model, in which members of a class share one trajectory. They add that when data truly come from a mixture with random effects, latent class growth analysis needs more classes to reproduce the main features.

The outcome distribution is the second choice. Bauer and Curran showed that several classes can be estimated, and appear optimal, when non-normal repeated measures come from a single group. Health outcomes are rarely normal. Mihaylova and colleagues review cost methods that must cope with skewness, excess zeros, multimodality and heavy right tails, and Hernández Alava, Wailoo and Ara describe EQ-5D values as right bounded at 1 with a substantial gap below, left bounded and multimodal. A normal growth mixture fitted to raw costs or utilities can return classes that only trace the shape of the distribution.

Choosing the number of classes

Ram and Grimm note that the chi-square fit indices of structural equation modelling are unavailable for mixtures. The Akaike information criterion, the Bayesian information criterion and a sample-size adjusted BIC are compared instead, lower being better.

A likelihood ratio test of $C$ against $C-1$ classes cannot use the usual chi-square distribution, so corrected and bootstrap versions are used, valid only between models that differ in the number of classes alone. Herle and colleagues report that the bootstrap test does well in simulations but is computationally intensive and weak in small samples, and that criteria can disagree. Entropy summarises classification quality on a scale of 0 to 1, and Ram and Grimm read values above 0.80 as confident classification, while Herle and colleagues caution that this reading requires the model to be correct. They also list avoiding very small classes, for example under 5% of the study population.

Ram and Grimm's example shows why fit statistics are not decisive. With nine cortisol measurements from 34 older adults, a three-class model had the lowest information criteria, yet some variance estimates were negative, one class held an implied 3.99 people and two classes had very similar paths, so they chose two classes.

Worked example with two illustrative EQ-5D recovery classes

All numbers in this example are illustrative. A cohort has EQ-5D utilities at baseline and 12 months after an operation, and in a fitted two-class model the recovering class (share 0.70) has mean utility 0.60 and then 0.80, and the declining class (share 0.30) has means 0.50 and 0.35. Each class has a random intercept with variance 0.01 and a residual variance of 0.01.

1. Build the within-class covariance. Each measurement has variance 0.01 + 0.01 = 0.02, the shared intercept gives a covariance of 0.01, and the determinant is 0.0004 minus 0.0001 = 0.0003.

2. Measure one patient's distance from each class. A patient records 0.52 and 0.62. The deviations $d_1$ and $d_2$ are minus 0.08 and minus 0.18 from the recovering class and 0.02 and 0.27 from the declining class. The squared distance is $Q = (0.02 d_1^2-0.02 d_1 d_2+0.02 d_2^2)/0.0003$, which gives (0.000128 minus 0.000288 + 0.000648) / 0.0003 = 1.627 for the recovering class and (0.000008 minus 0.000108 + 0.001458) / 0.0003 = 4.527 for the declining class.

3. Apply Bayes' theorem. With a shared covariance each density is proportional to $\exp(-Q/2)$, giving 0.4434 and 0.1040. The weighted terms are 0.70 × 0.4434 = 0.3104 and 0.30 × 0.1040 = 0.0312, summing to 0.3416, so the posterior probability of the recovering class is 0.3104 / 0.3416 = 0.909.

4. Remove the random intercept. With the same means but independent errors of variance 0.02, as in a group-based model, the distances are (0.0064 + 0.0324) / 0.02 = 1.94 and (0.0004 + 0.0729) / 0.02 = 3.665, the values of $\exp(-Q/2)$ 0.3791 and 0.1600 and the weighted terms 0.2654 and 0.0480. The posterior for the recovering class falls to 0.2654 / 0.3134 = 0.847. The random intercept reads the low start partly as a lasting personal offset, so the rise of 0.10 counts for more.

5. Convert class paths to QALYs. With straight lines between the points, the area under the recovering path is (0.60 + 0.80) / 2 = 0.700 QALYs and under the declining path (0.50 + 0.35) / 2 = 0.425 QALYs, a gap of 0.275. The mixture mean is 0.70 × 0.700 + 0.30 × 0.425 = 0.6175, the same as the area under a single mean path of 0.57 and 0.665.

6. Interpretation. The mixture leaves the mean QALY unchanged in this linear case, but it shows that 30% of patients follow a path worth 0.275 fewer QALYs over the year. That matters for targeting, provided membership can be predicted before treatment.

Trajectory classes in outcome and cost studies

Nguyen and colleagues fitted growth mixture models to EQ-5D-3L scores of 382 older patients at admission, 6 months and 1 year after hip fracture surgery. The four-class model had the lowest BIC and AIC, entropy of 0.86 and average posterior probabilities above 0.80 in every class. Its classes held 60.47%, 21.99%, 12.83% and 4.71% of patients, the last under the 5% example given by Herle and colleagues. Osteosarcopenia and a lower pre-fracture utility predicted the class with a steep, persistent decline, but no baseline covariate significantly predicted the recovered class. Only patients with complete EQ-5D-3L data at all three time points were analysed.

For costs, Trenaman and colleagues studied 224,285 high-need high-cost patients in British Columbia and chose a group-based model. They note that the growth mixture model's random effects cost more computation but capture differences among individuals in the same group, detail not needed for their aim of groups that tailored policies could target.

What growth mixture classes can support in a decision

Classes found from outcomes after treatment starts are not baseline subgroups. A subgroup analysis or targeting rule needs membership predicted from characteristics known at the decision, and the hip fracture study had a class that baseline data could not predict. Herle and colleagues compare one-step joint estimation with a three-step approach that corrects for classification error, and report similar results when entropy exceeds 0.80.

In a cost-effectiveness analysis, class-specific utilities or costs should travel with the posterior probabilities and the uncertainty in class shares, not with modal labels treated as observed. Class utilities must still be valid health state utility values for the model's population, and a fitted polynomial says nothing beyond the last measurement.

Neighbouring longitudinal methods and where each applies

Neighbouring models differ from the growth mixture model in one assumption. Herle and colleagues advise fitting the most general mixed effects model first and comparing mixture solutions before concluding that trajectory types exist.

Limitations and common misreadings

Most errors arise when classes are asked to mean more than the model can deliver. Others are computational.

1. Reading classes as patient types. Bauer and Curran found that within-class estimates from such solutions are largely uninterpretable and that predictive relationships can be hidden or spurious.
2. Choosing the number of classes on fit alone. Ram and Grimm set aside the solution with the lowest information criteria.
3. Trusting one run. EM can stop at a local maximum, so solutions from multiple starting values should agree.
4. Analysing complete cases only. Dropping patients who died or left follow-up understates the declining classes if their paths were worse, and missingness that depends on unobserved health is missing not at random.

Sources

• Bauer DJ, Curran PJ. Distributional assumptions of growth mixture models: implications for overextraction of latent trajectory classes. Psychological Methods. 2003;8(3):338-363. https://doi.org/10.1037/1082-989X.8.3.338
• Herle M, Micali N, Abdulkadir M, et al. Identifying typical trajectories in longitudinal data: modelling strategies and interpretations. European Journal of Epidemiology. 2020;35(3):205-222. https://doi.org/10.1007/s10654-020-00615-6
• Hernández Alava M, Wailoo AJ, Ara R. Tails from the peak district: adjusted limited dependent variable mixture models of EQ-5D questionnaire health state utility values. Value in Health. 2012;15(3):550-561. https://doi.org/10.1016/j.jval.2011.12.014
• Mihaylova B, Briggs A, O'Hagan A, Thompson SG. Review of statistical methods for analysing healthcare resources and costs. Health Economics. 2011;20(8):897-916. https://doi.org/10.1002/hec.1653
• Muthén B, Shedden K. Finite mixture modeling with mixture outcomes using the EM algorithm. Biometrics. 1999;55(2):463-469. https://doi.org/10.1111/j.0006-341X.1999.00463.x
• Nguyen TTB, Tran QS, Nguyen TT, et al. Quality of life trajectories following geriatric hip fracture surgery: a growth mixture modelling approach. Journal of Orthopaedic Surgery and Research. 2026;21:117. https://doi.org/10.1186/s13018-026-06664-z
• Ram N, Grimm KJ. Growth mixture modeling: a method for identifying differences in longitudinal change among unobserved groups. International Journal of Behavioral Development. 2009;33(6):565-576. https://doi.org/10.1177/0165025409343765
• Trenaman L, Guh D, McGrail K, et al. Longitudinal trajectories of healthcare costs among high-need high-cost patients: a population-based retrospective cohort study in British Columbia, Canada. BMJ Open. 2025;15(4):e089693. https://doi.org/10.1136/bmjopen-2024-089693