Growth Curve Model — longitudinal trajectories with random slope on time

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Models per-subject TRAJECTORIES of a continuous outcome over time using a linear mixed-effects framework with random intercept + random slope on time: y_it = β₀ + β₁ t + u_0i + u_1i t + ε_it.

Reports the average trajectory (fixed effects), between-subject variation in baseline intercepts σ_u₀ and growth rates σ_u₁, the correlation between them ρ_u₀u₁, and (optionally) differences in trajectories between groups via time × group interactions. Standard tool in developmental psychology, education research, and clinical trial trajectory analyses.

Worked example

How do individuals change over time, and do their trajectories differ?

A latent growth-curve model estimates the average intercept and slope over repeated occasions plus individual variation around them.

Result

Scores rose on average (slope = 1.4 per occasion, p < .001) with significant individual differences in slopes (variance p = .008).

How you'd report it (APA)

A growth-curve model showed a significant average increase (slope = 1.4, p < .001) with reliable individual differences in trajectories.

When to use it

  • Repeated measures per subject (longitudinal trajectories)
    Depression ~ week + (week | patient) in 100 patients × 8 weekly measures.
  • Non-linear (polynomial) trajectories
    Task performance ~ t + t² + (t | subject) — practice effects level off; quadratic captures the asymptote, random linear slope captures inter-individual learning rate.

When NOT to — use instead

Hypotheses

H₀: β_time = 0 (no average change over time) AND σ²_slope = 0 (no between-subject variation in growth rate).
Hₐ: β_time ≠ 0 and/or σ²_slope > 0.

Parameter tested: fixed-effect trajectory coefficients + random-effect covariance Ψ

Assumptions (and what to do if they fail)

σ²_slope > 0 — between-subject variation in rate of change. Low σ²_slope means everyone grows at nearly the same rate; a random-intercept LMM would suffice.medium

Check: LRT p-value.

If violated: If LRT is non-significant, the random slope is not adding explanatory power.

Level-1 residuals (ε_ij) are approximately normal.medium

Check: Shapiro-Wilk p-value.

If violated: Non-normal residuals affect small-sample CIs.

Random intercepts and random slopes are approximately normally distributed across subjects.medium

Check: Shapiro-Wilk on BLUP intercepts + BLUP slopes.

If violated: Non-normal BLUPs can indicate outlying subjects or an omitted cluster-level predictor.

No individual subject contributes a wildly extreme slope (|BLUP_slope| > 3·SD(slope)).low

Check: n of outlier subjects.

If violated: Extreme individual trajectories may pull the fixed-effect time slope.

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