Definition

Longitudinal data: Repeated measurements collected from the same subjects over time.

Key goal: Understand both:

• Within-person changes (trajectories)
• Between-person differences in those changes

Example: Six Cities Study (FEV1 Data)

• Tracked annual lung function in children
• This is the Topeka, Kansas subset (girls) of the Six Cities Study
• 300 girls, 1-12 repeated measurements per child
• FEV1 = forced expiratory volume; modeled on the log scale (logfev1)
• Available: data/fev1.txt
• FEV1/Six Cities is an instructor-chosen course-wide running example (formally introduced in FLW Ch. 9, Table 9.1), not one of FLW Chapter 1’s four motivating studies (TLC lead trial, Muscatine obesity, anti-epileptic drug trial, Connecticut Child Surveys; FLW Section 1.3)

Visualizing Repeated Measures

fev <- read.table("../../data/fev1.txt", header = TRUE)

# Sample 15 children for clarity
set.seed(667)
ids <- sample(unique(fev$id), 15)
fev_sub <- fev[fev$id %in% ids, ]

ggplot(fev_sub, aes(x = age, y = logfev1, group = id, color = factor(id))) +
  geom_line(linewidth = 0.8) +
  geom_point(size = 2) +
  theme_minimal() +
  theme(legend.position = "none") +
  labs(title = "Lung Function Trajectories (15 Children)",
       x = "Age (years)", y = "log(FEV1)")

Key Observation

Each child’s trajectory is correlated: measurements don’t bounce randomly.

• Cross-sectional: one dot per child
• Longitudinal: connected trajectory per child

Cross-Sectional vs. Longitudinal

Advantages

• Efficiency: Each subject serves as their own control
• Separates aging from cohort effects: longitudinal designs isolate the within-person change over time from cohort effects that confound cross-sectional comparisons (FLW Section 1.2, body-fat/menarche example)
• Temporal ordering: exposure/measurement precedes outcome (a structural feature, not proof of causation)
• Change rates: Directly estimate slopes and trajectories
• Precision: Baseline adjustment improves power (we return to this in Ch. 5)

Tradeoffs

• Dropout: Missing data patterns (MAR/MNAR)
• Irregular timing: Unequal visit schedules
• Complex dependence: Requires covariance modeling (Ch. 7-8)

Why Correlation Matters

Ignoring within-subject correlation leads to:

• Incorrect standard errors (the point estimates stay consistent; only the SEs are wrong)
• Invalid hypothesis tests
• Overstated significance

Proper models must distinguish within- vs. between-subject effects.

See It: Naive vs. Cluster SE

The cluster bootstrap here is a pedagogical illustration of the concept. The methods FLW and this course actually use to fix correlation-induced SE problems are correctly specified covariance/mixed models (Ch. 7-8) and the robust sandwich/empirical estimator (GEE, Ch. 13), not the bootstrap.

set.seed(667)

# Simulate clustered data: 50 subjects, 8 visits each,
# strong positive within-subject correlation (shared random intercept).
# The covariate of interest is a BETWEEN-subject (time-invariant) exposure:
# one value per subject, so it is confounded with the cluster structure and
# ignoring within-subject correlation understates its SE.
n_sub <- 50; n_vis <- 8; var_b <- 3   # subject-level variance
sim <- do.call(rbind, lapply(1:n_sub, function(i) {
  b_i <- rnorm(1, 0, sqrt(var_b))      # subject random intercept
  x_i <- rnorm(1)                      # subject-level exposure (one value per subject)
  y   <- 1 + 0.5 * x_i + b_i + rnorm(n_vis)
  data.frame(id = i, x = x_i, y = y)
}))

fit <- lm(y ~ x, data = sim)
naive_se <- summary(fit)$coefficients["x", "Std. Error"]

# Cluster bootstrap: resample whole subjects (keeps within-subject correlation)
boot_b <- replicate(500, {
  ids <- sample(unique(sim$id), replace = TRUE)
  d   <- do.call(rbind, lapply(ids, function(j) sim[sim$id == j, ]))
  coef(lm(y ~ x, data = d))["x"]
})
cluster_se <- sd(boot_b)

se_tab <- data.frame(SE = c("Naive (independence)", "Cluster bootstrap"),
                     value = c(naive_se, cluster_se))

ggplot(se_tab, aes(x = SE, y = value, fill = SE)) +
  geom_col(width = 0.6) +
  geom_text(aes(label = round(value, 3)), vjust = -0.4) +
  theme_minimal() +
  theme(legend.position = "none") +
  labs(title = "Naive SE understates true uncertainty",
       subtitle = "Independence SE is the smaller bar; the cluster bootstrap respects within-subject correlation and is the honest one",
       x = NULL, y = "Standard error of slope")

See It: The Takeaway

For the subject-level exposure x, at set.seed(667):

• Naive (independence) SE: 0.105
• Cluster bootstrap SE: 0.215

The honest cluster SE (0.215) is about 2 times larger than the naive SE (0.105).

Scientific Questions

Longitudinal analysis addresses:

• Describe: Mean trajectories, individual variation
• Explain: How covariates shift trajectories
• Predict: Future outcomes for individuals/populations
• Compare: Treatment effects over time
• Account for dependence: Valid inference with correlation

Longitudinal vs. Clustered Data

Techniques overlap, but time structure adds constraints and opportunities.

Summary

• What: repeated measures on the same subjects over time.
• Why: efficient (own control), temporal ordering, direct change rates.
• Catch: within-subject correlation corrupts SEs (estimates stay consistent).
• Use: describe, explain, predict, and compare trajectories.

Canonical Resources

• Author slides: content.sph.harvard.edu/fitzmaur/ala2e/
• Datasets: content.sph.harvard.edu/fitzmaur/ala2e/datasets.html
• Sample R code: content.sph.harvard.edu/fitzmaur/ala2e/SampleR.html

For Next Time

Read Chapter 2 of Fitzmaurice, Laird & Ware (2011)

• Lecture 2 (Ch. 2): longitudinal data structures, notation, and the sources of within-subject correlation