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A Latent Trajectory Analysis for Multivariate Outcomes with Mixed-Scale: Application to Alzheimer's Disease Neuroimaging Initiative

This paper proposes a novel latent trajectory model utilizing the expectation-maximization algorithm to analyze multivariate longitudinal outcomes with mixed scales, enabling the identification of dynamic patient subgroups to support individualized treatment planning, as demonstrated through an application to the Alzheimer's Disease Neuroimaging Initiative (ADNI).

Original authors: Lindsay R. Salvati, Jungwun Lee

Published 2026-08-10

📖 7 min read🧠 Deep dive

CC BY 4.0

Original authors: Lindsay R. Salvati, Jungwun Lee

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to understand how a complex machine, like a car engine, changes over time. Usually, mechanics look at one part at a time: "Is the oil pressure dropping?" or "Is the temperature rising?" But in the real world, everything is connected. The oil pressure might drop because the temperature rose, and both might happen only in cars that have a specific type of fuel filter. If you only look at the oil, you miss the whole story. This is the challenge scientists face when studying Alzheimer's disease. They have a mountain of data: some numbers that can be measured on a scale (like memory test scores or body weight), and some categories that are just "yes" or "no" (like having high blood pressure or a specific genetic marker).

For a long time, statisticians had to choose: either look at all the numbers together or look at all the yes/no answers together, but rarely both at the same time. It's like trying to understand a symphony by listening only to the violins or only to the drums, but never the whole orchestra. The problem gets even trickier because people don't all follow the same script. Some people with Alzheimer's decline slowly, others rapidly, and some stay stable for years. Traditional math tools often assume everyone is just a slightly different version of the "average" person, which hides these important differences. This paper introduces a new way to listen to the whole orchestra at once, even when the instruments are playing different types of music.

The New Tool: A Statistical "Swiss Army Knife"

The authors of this paper, Lindsay R. Salvati and Jungwun Lee, have built a new statistical tool called the Mixed Latent Class Profile Model (Mixed-LCPM). Think of this model as a super-smart detective that doesn't just look at clues; it groups people into secret teams based on how their clues change over time.

In the world of Alzheimer's research, scientists have data from the Alzheimer's Disease Neuroimaging Initiative (ADNI). This is a massive collection of information from 919 people, tracked over 48 months (about 4 years). Every few months, these people took memory tests (continuous numbers) and had their blood pressure, body mass index (BMI), and dementia status checked (categorical yes/no or high/low).

The old way of analyzing this data was like trying to sort a mixed bag of red and blue marbles by only looking at the red ones, then separately looking at the blue ones. You might miss that the red marbles that are heavy always come with the blue marbles that are shiny. The new Mixed-LCPM model looks at the red and blue marbles simultaneously. It asks: "Who are the people whose memory scores drop while their blood pressure stays high? Who are the people whose memory stays steady even though they are overweight?"

How the Detective Works: The Two-Layer System

The model uses a clever two-step process to find these hidden groups, which the authors call latent classes and latent profiles.

The Snapshot (Latent Classes): Imagine taking a photo of everyone at a single moment in time. The model looks at that photo and groups people into "classes" based on what they look like right then. For example, one class might be "High Memory, Low Blood Pressure," while another is "Low Memory, High Blood Pressure."
The Movie (Latent Profiles): Now, imagine watching a movie of those same people over six years. The model watches how people move from one "class" to another. Do they stay in the "High Memory" group forever? Do they jump from "Low Memory" to "High Memory" (unlikely, but possible in the math)? Or do they slowly drift from "Healthy" to "Impaired"?

By combining these two steps, the model finds trajectories. It doesn't just say "Person A is sick." It says, "Person A belongs to a group that started healthy, stayed healthy for a while, and then suddenly got worse in a specific way."

What They Found: Five Groups, Six Stories

When the authors applied this tool to the ADNI data, they discovered that the patients weren't just a blur of "getting worse." They fell into five distinct groups (latent classes) based on their mix of memory scores and health risks:

Group 1: High memory scores, low risk of high blood pressure or obesity. (The "Healthy" group).
Group 2: High memory scores, but high risk of high blood pressure and obesity. (The "Healthy but Unhealthy" group).
Group 3: Mild memory problems, high risk of high blood pressure and obesity.
Group 4: Mild memory problems, but low risk of high blood pressure and obesity.
Group 5: Severe memory problems across the board, with high health risks. (The "Significant Impairment" group).

But the real magic happened when they looked at the trajectories (the stories). They found six different paths people took over time:

Steady & Healthy: People who stayed in Group 1 the whole time.
Steady but Unhealthy: People who stayed in Group 2 (good memory, bad health) the whole time.
Steady Mild & Unhealthy: People who stayed in Group 3.
Steady Mild & Healthy: People who stayed in Group 4.
The Decliners: A small group (about 4.5% of the people) who started in Group 4 (mild impairment, healthy) but, by the end of the study, had almost certainly moved to Group 5 (severe impairment).
The Steady Impaired: People who stayed in the impaired groups.

Why This Matters: Seeing the Invisible

The paper proves that if you only looked at the memory scores (the numbers), you would have missed a huge difference. For instance, the "Steady & Healthy" group and the "Steady but Unhealthy" group both had great memory scores. If you only looked at memory, they would look like the same group. But the new model saw that one group had high blood pressure and the other didn't.

Similarly, two groups with mild memory problems looked the same on memory tests, but one was very overweight and the other was not. By ignoring the "yes/no" health data, researchers would have missed these crucial differences.

The authors also checked how their new tool worked using simulations. They created fake data with known answers and ran their model on it. The results showed that the model was very good at finding the right groups, especially when they had a lot of data (like 1,000 people). It correctly identified the patterns about 95% of the time, which is a very strong result for this kind of complex math.

The Limits and the Future

The authors are careful to say that this is a statistical discovery, not a medical cure. They found that factors like age, gender, education, and the number of APOE4 alleles (a genetic marker) were linked to which "story" a person was likely to be in. For example, older people were slightly less likely to be in the "Steady but Unhealthy" group compared to the "Steady & Healthy" group.

However, the paper notes a few limitations. The math is tricky; sometimes the computer gets stuck in a "local maximum," which is like a hiker getting stuck in a small valley and thinking it's the top of the mountain. To fix this, the authors ran the model 100 times with different starting points to make sure they found the true highest peak. They also noted that their model assumes missing data (like a missed doctor's appointment) is random, which might not always be true in real life.

The Takeaway

This paper doesn't tell us how to stop Alzheimer's yet. Instead, it gives scientists a better pair of glasses. Before, they might have seen a crowd of people all looking "a bit sick." Now, with the Mixed-LCPM, they can see that the crowd is actually made up of distinct teams, each with their own unique history and future path. This clarity could help doctors in the future pick the right patients for the right clinical trials, ensuring that when they test a new drug, they are testing it on a group of people who are truly similar, rather than a messy mix of different stories.

Technical Summary: A Latent Trajectory Analysis for Multivariate Outcomes with Mixed-Scale

Problem Statement
Alzheimer's disease (AD) is a progressive neurodegenerative disorder characterized by heterogeneous clinical manifestations. Large-scale initiatives like the Alzheimer's Disease Neuroimaging Initiative (ADNI) generate comprehensive longitudinal datasets containing mixed-type variables: continuous measures (e.g., cognitive scores, biomarkers) and categorical items (e.g., diagnostic status, comorbidities). Analyzing these variables concurrently is essential for understanding disease evolution but presents significant statistical challenges.

Traditional approaches, such as mixed-effects models, often assume a single population mean trajectory, potentially obscuring meaningful heterogeneity in disease progression. While Growth Mixture Modeling (GMM) identifies latent subgroups, standard frameworks are typically limited to single outcomes or rely on parametric assumptions regarding mean vectors and variance-covariance matrices. If these parametric forms are misspecified, conclusions may be misleading. Conversely, non-parametric approaches like Latent Class Profile Models (LCPM) and Latent Transition Models (LTM) offer flexibility but are restricted to categorical outcomes, failing to accommodate continuous measurements critical for capturing the nuance of AD progression. There is a gap in methodology for modeling joint trajectories of both continuous and categorical outcomes simultaneously without imposing rigid parametric trajectory forms.

Methodology
To address this gap, the authors propose the Mixed Latent Class Profile Model (Mixed-LCPM), an extension of the standard LCPM framework.

Model Structure: The model utilizes a two-level latent structure:
1. Latent Classes ( $C_t$ ): At each time point $t$ , a vector of categorical latent variables represents subject-level response patterns. These classes summarize the joint distribution of observed outcomes (both continuous $Y_t$ and categorical $Z_t$ ).
2. Latent Profiles ( $U$ ): A higher-level categorical latent variable summarizes the sequential patterns of class memberships over time into $S$ representative trajectory profiles.
Distributional Assumptions:
- Continuous Outcomes: Modeled as following a multivariate normal distribution with class-specific means ( $\mu_c$ ) and covariance matrices ( $\Sigma_c$ ).
- Categorical Outcomes: Modeled using multinomial distributions with class-specific probabilities ( $\pi_{pk|c}$ ).
- Conditional Independence: The model assumes that outcomes at a specific time point are conditionally independent given the latent class membership at that time. Furthermore, latent class memberships across time are conditionally independent given the latent profile membership.
Parameterization:
- Measurement Parameters: Include $\mu$ and $\Sigma$ for continuous outcomes, and $\pi$ for categorical outcomes. These are assumed to be time-invariant to ensure consistent interpretation of latent classes across time periods.
- Structural Parameters: Include $\eta$ (transition probabilities between profiles and classes) and $\gamma$ (prevalence of profiles).
- Covariates: Associations between subject-level covariates ( $x$ ) and latent profile membership are modeled via multinomial logistic regression.
Estimation: Parameters are estimated using Maximum Likelihood (ML) via the Expectation-Maximization (EM) algorithm.
- E-step: Utilizes the forward-backward algorithm to efficiently compute the posterior probabilities of latent class sequences and profile memberships, addressing the computational complexity of high-dimensional latent sequences.
- M-step: Updates parameters to maximize the expected complete-data log-likelihood. Closed-form solutions are used for normal and multinomial parameters, while the Newton-Raphson algorithm is employed for regression coefficients ( $\beta$ ).
- Inference: Standard errors are derived from the inverse of the Hessian matrix of the observed data likelihood. Model selection (determining the number of classes $K$ and profiles $S$ ) is performed using the Bayesian Information Criterion (BIC).

Key Contributions

Methodological Extension: The paper extends the LCPM framework to handle multivariate outcomes with mixed scales (continuous and categorical), overcoming the limitation of existing non-parametric trajectory models that only support categorical data.
Non-Parametric Flexibility: Unlike parametric GMMs, the Mixed-LCPM does not assume a specific functional form (e.g., linear, quadratic) for the trajectory. Instead, trajectories are characterized by the sequential patterns of latent class memberships, allowing for more flexible and data-driven identification of longitudinal trends.
Joint Modeling: The approach simultaneously models the associations between variable types at specific time points and the correlations arising from repeated measurements, providing a unified framework for mixed-scale longitudinal data.
Implementation: The authors provide an R implementation of the proposed method and validate it through extensive simulation studies.

Results

Simulation Studies: The authors conducted simulations across four scenarios varying class separation (strong vs. weak) and profile balance (equal vs. unequal) with sample sizes of $N=250, 500, 1000$ $N = 250, 500, 1000$ .
- The estimator demonstrated robust performance with minimal standardized bias ($|Sbias| < 0.1$) and coverage probabilities near the nominal 0.95 level for $N=1000$ .
- Coverage probabilities were sensitive to sample size, dropping below 0.90 for $N=250$ , which is consistent with the asymptotic nature of mixture modeling standard errors.
ADNI Application: The method was applied to ADNI data ( $N=919$ $N = 919$ ) involving six time points (Baseline to Month 48).
- Latent Classes: Five distinct classes were identified, differentiating patients by cognitive status (Normal, Mild Impairment, Significant Impairment) and vascular risk profiles (Healthy vs. Unhealthy regarding Blood Pressure and BMI).
- Latent Trajectories: Six distinct trajectory profiles were identified. Notably, the model distinguished between groups with similar cognitive trajectories but different vascular risk profiles (e.g., "Steady Cognition and Steady Good Health" vs. "Steady Cognition and Steady Bad Health").
- Covariate Associations: Significant associations were found between baseline covariates (Age, Gender, Education, APOE4 alleles) and trajectory membership. For instance, older age was associated with lower odds of belonging to the "Normal and Unhealthy" profile compared to the "Normal and Healthy" reference group.

Significance and Claims
The paper claims that the proposed Mixed-LCPM offers a robust, non-parametric alternative for analyzing complex longitudinal data where outcomes are a mix of continuous and categorical variables. By identifying distinct subgroups based on joint trajectories, the method reveals heterogeneity in disease progression that univariate or single-type analyses might obscure.

The authors emphasize that while the method successfully identifies representative patterns of multivariate outcomes and their associations with covariates, the results presented from the ADNI cohort are intended to illustrate the statistical approach rather than to inform immediate clinical decision-making. They acknowledge limitations, including the reliance on the Missing at Random (MAR) assumption for handling missing data and the computational cost of model selection via grid search. Future directions suggested include extending the model to clustered data and addressing Missing Not at Random (MNAR) scenarios through joint modeling frameworks.