This glossary collects the key concepts, people, and ideas you will meet in this lesson. Use it as a reference while you work through the material, or as a review before assessments. Type in the search box to filter entries.

Introduction and Overview

What Are Repeated Measures?

Repeated measures data arise when multiple measurements are taken over time on the same subjects (Wikipedia, 2025). This is one of the most common data structures in health sciences research—think of clinical trials where patients are measured at baseline and multiple follow-up visits, or cohort studies that track health outcomes over years (Diggle, Heagerty, Liang, & Zeger, 2002).

Longitudinal studies (which collect repeated measures) differ fundamentally from cross-sectional studies, which measure each subject only once. The key advantage of longitudinal designs is their ability to assess within-subject change over time, making them more powerful for detecting the effects of within-subject predictors.

Key Terminology

Missing Data and Drop-Outs

Missing data is very common in repeated measures studies. Subjects may miss individual visits (intermittent missingness) or drop out permanently (monotone missingness). The pattern and mechanism of missingness can substantially affect the validity of the analysis (Little, 1995). Methods that can handle unbalanced data (such as mixed models and GEE) are therefore particularly valuable for repeated measures, and multiple imputation under MAR is a widely used complementary tool (Sterne et al., 2009).

Descriptive Approaches

Introduction and Overview

Before mixed models, what was the field doing? Section 1 gave us the descriptive picture of repeated-measures data. This section walks through the methods that dominated longitudinal analysis for decades — split-plot ANOVA (univariate, with strong assumptions like compound symmetry/sphericity), MANOVA (multivariate, weaker assumptions but lower power and intolerant of missing data), and summary-statistic approaches that collapse each subject’s trajectory to a single value. Knowing these methods is more than historical: they still appear in legacy literature and teaching, and seeing where they fail motivates the modern approaches in Sections 3 and 4.

Simple Approaches to Repeated Measures

Before turning to complex mixed models, it is worth understanding the simpler methods that have traditionally been used for repeated measures data. These methods either reduce the data to avoid modelling correlations altogether, or make strong assumptions about the correlation structure.

Covariance and Correlation Matrices

library(lme4);  library(lmerTest);  library(nlme);  library(geepack)
dat <- read.csv("phaa_repeated.csv", stringsAsFactors = FALSE)
dat$id  <- factor(dat$id)
dat$arm <- factor(dat$arm, levels = c("control","intervention"))

# 1. Random-intercept mixed model on the continuous outcome
m_lmm <- lmer(sbp_mmhg ~ visit * arm + age + female + (1 | id),
              data = dat)
summary(m_lmm)

# 2. Add an AR(1) within-subject correlation structure with nlme
m_lme <- lme(sbp_mmhg ~ visit * arm + age + female,
             random      = ~ 1 | id,
             correlation = corAR1(form = ~ visit | id),
             data        = dat, na.action = na.omit)
summary(m_lme)

# 3. Compare correlation structures with AIC
m_cs <- update(m_lme, correlation = corCompSymm(form = ~ visit | id))
m_un <- update(m_lme, correlation = corSymm(form    = ~ 1     | id))
AIC(m_lme, m_cs, m_un)

# 4. Same trial, binary outcome (adherence) -- GLMM and GEE
m_bin <- glmer(adherent ~ visit * arm + age + female + (1 | id),
               data    = dat, family = binomial,
               control = glmerControl(optimizer = "bobyqa"))
exp(fixef(m_bin))               # subject-specific ORs

Introduction and Overview

Where the modern toolkit takes over. Section 2 showed why the classical methods strain when measurement spacing is irregular, missingness is informative, or the within-subject correlation pattern is more complex than “all pairs equally correlated.” Mixed models with explicit residual correlation structures — compound symmetry, AR(1), Toeplitz, unstructured — are the response. They let us model the temporal dependence directly rather than assume it away, handle unbalanced or missing-at-random data gracefully, and combine random effects with structured residuals to capture both subject-level heterogeneity and within-subject autocorrelation. This section is the heart of the lesson.

Beyond Random Intercepts

A random intercept model assumes compound symmetry—that all pairs of measurements on the same subject are equally correlated (Wikipedia, 2025). For most repeated measures data, this assumption is violated because of autocorrelation. We need to extend the mixed model to include explicit correlation structures for the error term ε — the framework formalised in the classic Laird & Ware random-effects model for longitudinal data (Laird & Ware, 1982; Fitzmaurice, Laird, & Ware, 2011).

Key Correlation Structures

Combining Random Effects with Correlation Structures

An important practical consideration is how random effects interact with error correlation structures. Some combinations are redundant and cannot be separately identified:

• Random intercepts + compound symmetry errors = redundant — both produce the same correlation structure
• Random intercepts + AR(1) errors = useful — produces a structure where correlations decay but do not reach zero
• Unstructured errors + random effects = pointless — the unstructured covariance already captures everything

Covariance pattern models use no random effects at all, relying entirely on the structured covariance of the errors to capture within-subject correlation.

Model Selection

For nested correlation structures (e.g., AR(1) is nested within Toeplitz), use likelihood ratio tests. For non-nested structures (e.g., AR(1) vs. compound symmetry), use AIC or similar information criteria. Models should be compared with the same fixed effects and random effects structure.

Introduction and Overview

Pulling the threads together. Section 3 framed within-subject correlation as something to be modelled directly. This section closes the loop on three remaining concerns. First, trend models with random slopes: rather than (or in addition to) structuring the residuals, we let each subject have their own trajectory over time — a particularly intuitive approach when the question is about individual change. Second, discrete longitudinal outcomes: extending the GLMM machinery from Lesson 11 to repeated binary or count measurements. Third, generalised estimating equations (GEE): the marginal alternative to mixed models, which prioritises population-average effects and is robust to mis-specification of the working correlation. Together with Section 3, this section gives you a complete repeated-measures toolbox.

Trend Models with Random Slopes

An alternative to modelling the error correlation directly is to include random slopes for time. This allows each subject to have their own rate of change (growth or decline) over time, with the population-average trend captured by the fixed effect of time.

The variation in individual trajectories naturally induces autocorrelation—subjects who start high and decline slowly will have correlated measurements. This can be sufficient to capture the temporal structure in many datasets, especially when the primary interest is in individual trajectories.

The time variable can be parameterized in different ways: linear (for constant rates of change), polynomial (for curved trajectories), or log-transformed (for rapid early change that levels off).

Discrete Repeated Measures Data

Extending mixed models to discrete outcomes (binary, count) with correlation structures is much harder than for continuous outcomes. The fundamental challenge is that in GLMs, the error term and the linear predictor operate on different scales—the link function transforms the relationship, making it difficult to add correlation structures to the error term in a meaningful way.

Transition Models

One approach for discrete repeated measures is the transition model, which includes the previous outcome as a predictor. This captures autocorrelation informally through dependence on the prior outcome.

Here, γ is the log odds ratio comparing those with versus without the previous event. A positive γ means that having the event at the previous time point increases the odds of having it at the current time point.

Generalised Estimating Equations (GEE)

GEE is a population-averaged (marginal) approach that does not require specifying random effects (Liang & Zeger, 1986). Instead, it specifies a “working” correlation structure and uses robust (sandwich) standard errors that provide valid inference even if the working correlation is misspecified.

Bringing It All Together

Repeated-measures data are the most common form of clustered data in health research, and this lesson built a layered toolbox for analysing them. Section 1 defined the structure — multiple measurements on the same subjects, with the new ingredient of time ordering — and showed why simple descriptive views (spaghetti plots, mean profiles, empirical correlation matrices) are the right starting point for any longitudinal analysis. Section 2 walked through the classical methods that long dominated the field: separate-time-point analyses, summary-statistic reductions, RM-ANOVA with its compound-symmetry assumption and Huynh–Feldt correction, and MANOVA's distribution-free but missingness-intolerant approach.

Section 3 introduced the modern workhorse: linear mixed models with explicit residual correlation structures. Compound symmetry, AR(1), ARMA(1,1), Toeplitz, and unstructured matrices each encode a different temporal dependence pattern, and the choice among them — guided by the empirical correlation matrix, AIC, and likelihood-ratio tests — is one of the most consequential decisions in a longitudinal analysis. Combined with random intercepts and slopes, these structures capture both subject-level heterogeneity and within-subject autocorrelation, and they handle the unbalanced and irregularly spaced designs that defeat the classical methods.

Section 4 extended the framework in three directions: random-slope trend models that turn individual trajectories into modelled objects, GLMM-based and transition models for discrete repeated outcomes, and generalised estimating equations as a marginal alternative whose working correlation can be mis-specified without invalidating fixed-effect inference. Together, the four sections give you the methods and the decision logic you need for the longitudinal data you will encounter in your research career — and they close out the HSCI 230 → 341 → 410 series with the most general analytic tools you have met so far.

This final assessment covers all material from this lesson. You must answer all 15 questions correctly (100%) and complete the final reflection to finish the lesson.