Modeling Repeated Measures in Clinical Research: Fixed and Random Effects Explained

Mayta
1 hour ago
4 min read

Introduction

In clinical research, data are frequently collected from the same individuals at multiple time points. These repeated measures are vital for understanding how outcomes evolve in response to interventions, natural disease progression, or other temporal factors. However, repeated observations violate the assumption of independence that underpins standard regression techniques. Specialized modeling strategies are thus required to capture the within-subject correlation and nested data structure. This article introduces the logic of modeling repeated measures, explains multilevel structures, and outlines when and how to use fixed versus random effects.

Understanding Repeated Measures and Time Dependency

Repeated measures refer to data where each subject contributes multiple outcome values across time or condition. The value of the outcome variable (Y) at each time point is influenced by several factors:

Individual characteristics: baseline traits or idiosyncratic variations
Group membership: such as intervention or control group
Time: the specific time at which the outcome is measured
Interaction between group and time: capturing whether the effect of treatment changes over time

Because time is central to the structure of repeated measures, it must be explicitly included as a predictor in any regression model intended to analyze such data.

Multilevel Models and Their Terminology

Repeated measures generate hierarchically structured data, often described using a multilevel or mixed-effects framework. The nesting of measurements (level 1) within individuals (level 2) creates dependency, which must be modeled appropriately.

Key Terminological Clarifications

Fixed Effect Model: Assumes the same parameter applies to all individuals or clusters.
Random Effect Model: Allows parameters to vary across individuals or clusters.
Mixed-Effects (Multilevel) Model: Incorporates both fixed and random effects.

Why the Terminology Confusion?

The phrase “mixed model” may refer to different configurations across disciplines. The term multilevel model is widely accepted in biostatistics and epidemiology to describe models that explicitly account for hierarchical structure.

Fixed and Random Intercepts and Slopes

1. Fixed Intercept, Fixed Slope

This basic model assumes:

All individuals within a group start at the same baseline value of the outcome (fixed intercept).
The trajectory or change in outcome over time is identical across individuals (fixed slope).

Example: In a study comparing two rehabilitation programs, both groups may show a linear decrease in pain score over 6 months. A fixed model would assume the same rate of improvement for all individuals within each group.

2. Random Intercept, Fixed Slope

This model permits each individual to start at a unique baseline, but assumes the rate of change is constant across individuals.

Example: In a nutritional trial, patients might begin with varying body mass indices (BMIs), but all lose weight at a similar rate following dietary counseling.

3. Random Intercept, Random Slope

Here, both the starting point and the trajectory over time vary by individual. This is the most flexible and often realistic scenario in clinical settings.

Example: In a follow-up of hypertensive patients, not only might the baseline blood pressure differ across individuals, but the response to medication over time may also vary depending on genetic or lifestyle factors.

Modeling Strategies for Repeated Measures

Depending on the complexity of the data and study design, several modeling strategies are available:

Ordinary Least Squares Regression (Naïve Approach)

Assumes all observations are independent. This is inappropriate for repeated measures, as it ignores intra-subject correlation and underestimates standard errors.

Clustered Robust Standard Errors

Applies a correction to standard errors to account for clustering but retains the standard regression framework. Useful for exploratory analysis but does not fully model the data structure.

Generalized Estimating Equations (GEE)

Estimates population-averaged (marginal) effects.
Requires specification of a working correlation structure (e.g., independent, exchangeable, autoregressive).
GEE is robust to misspecification of correlation but less efficient if misspecified.

Mixed-Effects Models (Multilevel Models)

Estimates subject-specific effects.
Incorporates random intercepts and/or random slopes to model within-subject variation explicitly.
Allows flexible modeling of complex longitudinal structures.

Model Specification Examples in Practice

Fixed Intercept & Fixed Slope:

reg size group month

Allowing for Group-Time Interaction:

gen month_group = month * group
reg size group month month_group

Random Intercept Only:

mixed size group month || patient:

Random Intercept & Random Slope:

mixed size group month month_group || patient: month, cov(uns)

Each progression adds complexity and fidelity to the model, better capturing the nature of individual variation over time.

Interpreting Clinical Trajectories

The effect of an intervention may be constant over time or change across follow-up periods. For instance, an early chemotherapy group may initially improve more rapidly than a concurrent therapy group, but later both converge. This interaction is tested by adding a product term between group and time.

Visual interpretation—such as plotting predicted mean outcomes by group and time—is crucial. These plots can reveal:

Non-parallel trends indicating interaction
Convergence/divergence over time
Non-linear changes not captured by linear terms

Conclusion

Repeated measures analysis is indispensable in longitudinal clinical studies. Selecting the appropriate model—whether GEE for marginal estimates or mixed models for individual trajectories—depends on the research question, data structure, and intended interpretation. Understanding the interplay between fixed and random effects, and between group-time interactions, is essential for valid inference and clinical insight.

If you'd like, I can also generate GEE or mixed-effects Stata code for your specific dataset or guide you through model selection decisions based on your study's structure.