Longitudinal Data Analysis

Overview

Definitions

Longitudinal data = data collected from the same individuals or units repeatedly over time.

Data Structure

Long Format

One row per subject-time combination.
usually preferred for mixed-effects models, GEE, time-varying Cox models, and most longitudinal analyses

Wide Format

One row per subject, with repeated measurements stored in separate columns.
can be useful for descriptive summaries, paired comparisons, MANOVA, or some machine learning workflows
but: less flexible for irregular follow-up times

Characteristics

Repeated observations are usually correlated
Individuals may differ in their baseline level and in their rate of change over time
Measurement times may be regular or irregular
Missing data are common
Dropout may be informative, especially in health studies
The number of observations per subject may vary
Time can be modeled as continuous, categorical, nonlinear, or event-based

Common Research Questions

Change Over Time: Does a biomarker change with age or follow-up time?
Group Differences in Trajectories: Do different groups change differently over time? This is often tested using a time x group interaction.
Baseline Predictor of Future Change: Does a baseline variable predict later trajectory?
Time-Varying Exposure or Biomarker: Does the updated value of a biomarker predict future risk?
Trajectory Subgroups: Are there distinct groups of people with different longitudinal patterns?
Longitudinal Biomarker and Event Outcome: Is a biomarker trajectory associated with time-to-event outcomes?

Modeling Strategies

Modeling Methods

Method	When to use	Pros	Cons	Health-data example
Change-score analysis	Compare baseline-to-follow-up change	Intuitive; easy to report	Noisy; sensitive to measurement error; limited to few time points	Compare 6-month weight loss between interventions
ANCOVA (ANOVA & Post-hoc Tests#ANCOVA)	Compare one follow-up outcome between groups, adjusting for baseline	Simple; interpretable; often more efficient than change scores	Usually limited to one follow-up; does not model trajectories	Compare 12-month blood pressure by treatment, adjusting for baseline
Repeated-measures ANOVA (ANOVA & Post-hoc Tests)	Compare fixed, common time points in a balanced design	Simple; familiar	Needs near-complete, regularly timed data; assumes sphericity; inflexible with missingness	Compare mean HbA1c at baseline, 3, 6, and 12 months
MANOVA (ANOVA & Post-hoc Tests#MANOVA)	Treat repeated time points as correlated outcomes	Avoids sphericity; models cross-time correlation	Often needs complete data; poor fit for many or irregular time points	Compare treatment groups on blood pressure at three visits
Mixed-effects models (Linear mixed models)	Model repeated or nested outcomes, including irregular visits and incomplete data	Models individual trajectories; handles unbalanced data; supports random effects	Requires covariance/random-effect choices; inference depends on model and missingness assumptions	Estimate patient-specific eGFR decline over time
Generalized estimating equations (GEE) (Generalized Linear Models#Relationship to GEE)	Estimate population-average effects for correlated continuous, binary, or count outcomes	Robust population-average inference; flexible outcome distributions	Does not model individual trajectories; dropout can bias results	Estimate the average treatment effect on repeated hypertension status
Structural equation models (SEM) (Structural Equation Models)	Model complex relations among observed and latent variables	Handles latent constructs; tests direct and indirect paths	Specification-sensitive; complex; often needs large samples	Model links among latent frailty, inflammation, and disability
Latent growth curve models (Structural Equation Models#Latent Growth Curve Models)	Model average and individual change within an SEM framework	Estimates growth factors; supports latent variables	Needs adequate waves and sample size; specification-sensitive	Model cognitive decline using repeated test scores
Transition models	Model the current outcome conditional on prior outcomes	Captures short-term dependence and state changes	Interpretation is history-dependent; less suited to long-term trajectories	Model transitions among healthy, prediabetes, and diabetes states
Time-varying Cox models (Survival Analysis#Cox (Proportional Hazards) Model)	Relate updated exposures or biomarkers to time-to-event outcomes	Uses changing covariates; handles censoring	Assumes proportional hazards unless extended; time-dependent confounding may bias estimates	Relate updated blood pressure to stroke risk
Joint models (a model for trajectory + a model for survival)	Analyze a longitudinal marker and related event time together	Links marker trajectory to event risk; accounts for informative dropout	Complex; computationally intensive; specification-sensitive	Link PSA trajectory to prostate-cancer recurrence
Latent class trajectory models	Identify subgroups with distinct longitudinal patterns	Finds clinically meaningful trajectory groups; allows heterogeneous trends	Class number can be unstable; assignments are probabilistic	Identify distinct BMI trajectories from childhood to adulthood

Missing Data Handling

if Missing at Random (Handling Missing Data#^656f1e): mixed-effects models are valid under missing at random assumptions if the model is correctly specified
if Missing Not at Random (Handling Missing Data#^b7a197):common and problematic in health studies, can try
- mixed-effects models
- multiple imputation
- inverse probability weighting
- sensitivity analysis
- joint modeling of longitudinal and dropout/event processes

Time Data Handling

Design of time definition

Time is a key design choice in longitudinal analysis. Possible definitions:

time since baseline
age
calendar time
time since diagnosis
time before event
treatment phase
visit number

Model of time

Time can be modeled as:

continuous
categorical
piecewise linear
spline-based
nonlinear
random slope

Key Model Interpretation

Fixed Effect of Time

Average change in the outcome per unit time.

Group Effect

Difference between groups at the reference time point, often baseline.

Time x Group Interaction

Difference in rate of change between groups.
This is often the most important term when comparing trajectories.

Random Intercept

Allows each subject to have their own baseline level.

Random Slope

Allows each subject to have their own rate of change.

Within-Person Effect & Between-Person Effect

Within-person and between-person effects can differ and should not always be interpreted as the same thing.

Within-Person Effect:

Association between changes within the same person over time.\Example:

When a person's blood pressure increases above their own average, does their risk increase?

Between-Person Effect

Association between differences across people.
Example:

Do people with higher average blood pressure have higher risk than others?

Practical Workflow

Understand the study design.
Define the time scale.
Visualize individual trajectories with spaghetti plot.
Plot group-level mean/smoothed trajectories.
Check missingness patterns.
Decide whether the goal is population-average inference, subject-specific inference, prediction, or trajectory discovery.
Choose the model accordingly.
Fit a simple model first.
Add nonlinear time, interactions, or random slopes if needed.
Check assumptions and model fit.
Interpret time effects and time x group interactions carefully.
Conduct sensitivity analyses for missing data and dropout.

Health Data Examples

Repeated Blood Biomarkers

Question:

Does inflammatory biomarker level increase faster among people who later develop cardiovascular disease?

Possible methods:

mixed-effects model
time-varying Cox model
joint model

Cognitive Decline

Question:

Do APOE-e4 carriers show faster cognitive decline?

Possible methods:

mixed-effects model
latent growth curve model
joint model with dementia onset

Disease Progression

Question:

What are the common trajectories of kidney function decline?

Possible methods:

mixed-effects model
latent class trajectory model
joint model with kidney failure outcome

Treatment Follow-Up

Question:

Does a treatment reduce blood pressure over repeated follow-up visits?

Possible methods:

ANCOVA for simple baseline-follow-up design
mixed-effects model for multiple visits
GEE for population-average treatment effec