Group-Based Trajectory Model

Group-Based Trajectory Model for Cost, Adherence and Outcome Paths

A group-based trajectory model takes a repeated measure, such as a patient's quarterly health care cost or monthly medication supply, and estimates a few typical paths through time together with the share of the population on each path. Each person then receives a probability of belonging to every path, so a cohort can be described as a stable low cost group, a rising cost group and a persistently high cost group. Health economists apply the method to claims and trial data to find patterns that an average curve hides, and the number, size and shape of the groups depend on modelling choices that have to be reported and checked. The method is also known as latent class growth analysis.

Discrete paths as an approximation to individual variation

Nagin and Land set out the method in 1993 to study criminal careers. In Nagin's 1999 account the method identifies groups instead of assuming them, estimates the proportion of the population following each group, relates membership to individual characteristics and uses membership probabilities to profile group members.

The model is a finite mixture model applied to longitudinal data. Elmer, Jones and Nagin stress that in most applications the population is not literally made up of a finite number of groups. The mixture approximates an unknown continuous distribution of trajectories, so a "high cost group" is a region of that distribution, not a fixed type of patient.

The mixture likelihood and posterior membership

Each person's repeated measurements are treated as drawn from one of a fixed number of unobserved groups. Following Elmer, Jones and Nagin, the likelihood for one person is a weighted sum over groups.

$$P(Y_i) = \sum_{j=1}^{J} \pi_j \, P(Y_i \mid j; \theta_j)$$

where $Y_i = (y_{i1}, \ldots, y_{iT})$ is person $i$'s sequence of measurements over $T$ occasions, $J$ is the number of groups, $\pi_j$ is the probability that a randomly chosen person belongs to group $j$, and $P(Y_i \mid j; \theta_j)$ is the distribution of the sequence given membership of group $j$, with parameter vector $\theta_j$.

Within a group, measurements on the same person are assumed independent, so the group term is a product over occasions, and the expected outcome is usually a polynomial in time with coefficients specific to each group.

$$P(Y_i \mid j; \theta_j) = \prod_{t=1}^{T} p(y_{it} \mid j; \theta_j), \qquad \mu_{jt} = \beta_{0j} + \beta_{1j} t + \beta_{2j} t^2 + \beta_{3j} t^3$$

where $p(y_{it} \mid j; \theta_j)$ is the probability or density of the measurement at occasion $t$ for a member of group $j$, $\mu_{jt}$ is that group's expected outcome at time $t$, and $\beta_{0j}$ to $\beta_{3j}$ are its intercept and linear, quadratic and cubic coefficients. The independence holds only within a group, so outcomes stay correlated across the population.

The outcome distribution must be specified. The TRAJ procedure for SAS described by Jones, Nagin and Roeder fits censored normal, Poisson, zero-inflated Poisson and Bernoulli distributions. Elmer and colleagues later added the beta distribution for outcomes on a 0 to 1 scale. Fitting uses maximum likelihood estimation. Complete data are not required, and in the basic model missing values are assumed missing at random, with an extension for dropout caused by an external event such as death.

Posterior probabilities for each person

Once the model is fitted, Bayes' theorem turns the estimated shares and group likelihoods into each person's posterior probability of membership. Analysts commonly assign each person to the group with the highest posterior probability.

$$\Pr(j \mid Y_i) = \frac{\pi_j \, P(Y_i \mid j; \theta_j)}{\sum_{l=1}^{J} \pi_l \, P(Y_i \mid l; \theta_l)}$$

where $\Pr(j \mid Y_i)$ is the probability that person $i$ belongs to group $j$ given the observed sequence, and the denominator sums the weighted likelihood over all $J$ groups. Elmer and colleagues specify $\pi_j$ as a multinomial logistic function, and Jones, Nagin and Roeder let baseline risk factors enter it, so characteristics such as age or comorbidity can shift the chance of membership of each group.

Choosing the number of groups and the shape of each path

No test statistic identifies the number of components in a finite mixture, as Elmer and colleagues note, so selection combines information criteria with judgement about which groups are substantively distinct. Jones, Nagin and Roeder demonstrated the Bayesian information criterion for this purpose.

$$\text{BIC} = -2 \ln \hat{L} + k \ln n$$

where $\hat{L}$ is the maximised likelihood, $k$ is the number of estimated parameters and $n$ is the sample size. A smaller value is preferred. The TRAJ procedure reports BIC as $\ln \hat{L} - 0.5 k \ln n$, which is this quantity divided by minus 2, so in TRAJ output the largest value is preferred.

Mésidor and colleagues describe the usual sequence. Models with an increasing number of groups are fitted with cubic terms, and higher-order terms not statistically significant at 5% are removed. The average posterior probability (AvePP) of a group is the mean posterior probability among people assigned to it, and values above 0.7 in every group are treated as adequate. Nguena Nguefack and colleagues add that each group should hold more than 5% of the sample and that estimated shares should match the proportions assigned. Lennon and colleagues treat odds of correct classification (OCC) above 5.0 as acceptable.

$$\text{OCC}_j = \frac{\text{AvePP}_j / (1 - \text{AvePP}_j)}{\hat{\pi}_j / (1 - \hat{\pi}_j)}$$

where $\text{AvePP}_j$ is the average posterior probability for group $j$ and $\hat{\pi}_j$ is its estimated share, and the ratio compares correct assignment with random assignment in proportion to group size. Mésidor and colleagues regard OCC as a reformulation of AvePP and found that relative entropy and mismatch detected spurious groups better than AvePP did.

Worked example with three annual cost paths

All numbers in this example are illustrative. A health board records three years of annual cost, in thousands of GBP, for adults with a long-term condition and selects three groups on BIC. Cost is treated as normal with a standard deviation of 1.0 in every group and year, although real analyses more often model log cost with a censored normal.

1. Read the fitted paths. The stable low group (share 0.60) costs 1.0 in each year. The rising group (share 0.25) costs 1.0, 2.5 and 4.0. The persistently high group (share 0.15) costs 5.0 in each year.

2. Compute one patient's group likelihoods. A patient's observed costs are 1.4, 2.2 and 3.1. With a common standard deviation, each group likelihood is proportional to $\exp(-S/2)$, where $S$ is the sum of squared deviations from the group path, and the constant cancels in step 3. For the stable low group the deviations are 0.4, 1.2 and 2.1, so $S$ = 0.16 + 1.44 + 4.41 = 6.01, giving 0.04954. For the rising group they are 0.4, minus 0.3 and minus 0.9, so $S$ = 0.16 + 0.09 + 0.81 = 1.06, giving 0.58860. For the high group they are minus 3.6, minus 2.8 and minus 1.9, so $S$ = 12.96 + 7.84 + 3.61 = 24.41, giving 0.00000501.

3. Apply Bayes' theorem. The weighted terms are 0.60 × 0.04954 = 0.02972, 0.25 × 0.58860 = 0.14715 and 0.15 × 0.00000501 = 0.00000075, which sum to 0.17688. The posterior probabilities are 0.02972 / 0.17688 = 0.168 for stable low and 0.14715 / 0.17688 = 0.832 for rising, with about 0.000004 for high. The patient is assigned to the rising group with a one in six chance of being wrongly placed.

4. Check classification. Suppose people assigned to the rising group have an AvePP of 0.85. The odds of correct assignment are 0.85 / 0.15 = 5.667 and the odds under random assignment are 0.25 / 0.75 = 0.333, so the OCC is 17.0, above 5.0.

5. Interpret the cost burden. Three-year costs are 3.0, 7.5 and 15.0 thousand GBP per person in the three groups. The group contributions are 0.60 × 3.0 = 1.800, 0.25 × 7.5 = 1.875 and 0.15 × 15.0 = 2.250, which sum to a population mean of 5.925 thousand GBP. The high group holds 15% of people and 2.250 / 5.925 = 0.38, or 38%, of cost. Weighting the paths by this patient's posterior probabilities gives 0.168 × 3.0 + 0.832 × 7.5 = 6.74 thousand GBP, below the 7.5 implied by hard assignment, so carrying the probabilities forward keeps classification uncertainty.

Applications to claims, adherence and trial data

Han and colleagues analysed claims data on 309,876 commercially insured adults with type 2 diabetes in Texas between 2016 and 2019. A censored normal model of log quarterly cost with cubic time terms gave the smallest BIC with five groups. Two groups (10.1% and 8.6% of patients) had rising costs and three (17.1%, 31.5% and 32.7%) had stable low, medium and high costs. The two rising groups differed mainly because patients enrolled with the insurer at different times, so the authors treated them as one pattern.

For medication adherence, Franklin and colleagues found among 264,789 statin initiators that a six-group model summarised observed monthly adherence better than the proportion of days covered, with a C statistic of 0.938 against 0.881.

Harvey and colleagues modelled self-efficacy, EQ-5D values, perceived stress and burnout together over two years in the PROMISE randomised trial of remote person-centred care for common mental disorders. Four groups had average posterior probabilities above 0.9, and costs were reported by group and in a cost-consequence analysis. No statistically significant difference between trial arms in group allocation was found.

What trajectory groups can support in a decision

Groups built from outcomes measured after randomisation are not baseline subgroups. Harvey and colleagues contrast them with conventional subgroup analysis, often based on predetermined characteristics such as age and sex that the intervention cannot change. Comparing trial arms within a trajectory group conditions on a post-randomisation outcome, so a group-specific treatment effect needs separate causal justification. In observational data, Nagin and Odgers describe combining trajectory groups with propensity score matching as one route to causal inference.

A decision model that uses trajectory groups as cohorts or cost profiles has to predict membership of a new patient from baseline characteristics. Treating modal assignment as known removes classification uncertainty, as step 5 shows, and extending a fitted polynomial beyond observed follow-up is an extrapolation the data do not support.

Group-based trajectories, growth mixtures and growth curves

Nguena Nguefack and colleagues describe the group-based model as a simpler specification of the growth mixture model. A growth mixture model estimates within-class variation through random effects on the growth parameters, while the group-based model assumes none and so estimates fewer parameters. A growth curve model has one mean path with continuous individual variation around it. Latent class analysis classifies people from responses at one time point. The review warns that these terms are often used interchangeably, so the label alone does not show whether within-group random effects were estimated.

Spurious groups and other common errors

The main risk is artefactual groups. In six simulated scenarios, Mésidor and colleagues found the correct number of trajectories in only two. The method found trajectories in data with no subgroups and split people on one continuum into parallel bands, and AvePP missed most of these solutions. They attribute the parallel bands mainly to the assumption of homogeneity within groups. A non-normal continuous outcome fitted with normal components also tends to inflate the number of groups.

1. Expecting groups to replicate. Nagin and Odgers caution that the number, size and shape of groups need not stay constant across populations, and describe groups splitting apart as follow-up lengthens.
2. Selecting on BIC alone. Lennon and colleagues note that BIC can keep falling as groups are added, reflecting overfitting, and combine it with classification diagnostics, individual trajectory plots and clinical plausibility.
3. Under-reporting the steps. Nguena Nguefack and colleagues find descriptions often too thin for replication and point to the GRoLTS reporting checklist for latent trajectory studies.

Sources

• Elmer J, Jones BL, Nagin DS. Using the Beta distribution in group-based trajectory models. BMC Medical Research Methodology. 2018;18:152. https://doi.org/10.1186/s12874-018-0620-9
• Franklin JM, Shrank WH, Pakes J, et al. Group-based trajectory models: a new approach to classifying and predicting long-term medication adherence. Medical Care. 2013;51(9):789-796. https://doi.org/10.1097/MLR.0b013e3182984c1f
• Han G, Spencer MS, Ahn S, et al. Group-based trajectory analysis identifies varying diabetes-related cost trajectories among type 2 diabetes patients in Texas. BMC Health Services Research. 2023;23:1116. https://doi.org/10.1186/s12913-023-10118-1
• Harvey BP, Kebede TT, Fors A, et al. Group-based trajectory modelling of multiple health outcomes in a cost-consequence framework. Journal of Patient-Reported Outcomes. 2026;10:54. https://doi.org/10.1186/s41687-026-01043-y
• Jones BL, Nagin DS, Roeder K. A SAS procedure based on mixture models for estimating developmental trajectories. Sociological Methods & Research. 2001;29(3):374-393. https://doi.org/10.1177/0049124101029003005
• Lennon H, Kelly S, Sperrin M, et al. Framework to construct and interpret latent class trajectory modelling. BMJ Open. 2018;8(7):e020683. https://doi.org/10.1136/bmjopen-2017-020683
• Mésidor M, Rousseau MC, O'Loughlin J, Sylvestre MP. Does group-based trajectory modeling estimate spurious trajectories? BMC Medical Research Methodology. 2022;22:194. https://doi.org/10.1186/s12874-022-01622-9
• Nagin DS. Analyzing developmental trajectories: a semiparametric, group-based approach. Psychological Methods. 1999;4(2):139-157. https://doi.org/10.1037/1082-989X.4.2.139
• Nagin DS, Odgers CL. Group-based trajectory modeling (nearly) two decades later. Journal of Quantitative Criminology. 2010;26(4):445-453. https://doi.org/10.1007/s10940-010-9113-7
• Nguena Nguefack HL, Pagé MG, Katz J, et al. Trajectory modelling techniques useful to epidemiological research. Clinical Epidemiology. 2020;12:1205-1222. https://doi.org/10.2147/CLEP.S265287