by Albert Major
Exam-style mixed model questions (WUCLAP)
• The lecturer showed full lmer() outputs and asked things like:
• “Which random effect structure is correct for this output?”
• “Which interpretation of the fixed effect is correct?”
• Idea: you get an R output, several statements, and you must choose which one is true or false.
• Random effect structure example discussed:
• Formula fragment like: (1 + Days | Subject)
• Or sometimes written without the “1 +”, for example (Days | Subject).
• R treats these as including both a random intercept and a random slope for Days per Subject.
• Interpretation of that structure:
• Each subject has their own intercept.
• Each subject has their own slope for Days.
• These random effects model how individuals differ around the average trend.
• The lecturer advised you to:
• Fit models with different random structures.
• Use coef(model) to inspect subject-specific intercepts and slopes.
• See how changing the random effects formula changes those.
• Fixed effect of Days ≈ 10.47 (large t-value, no p-value shown but clearly significant).
• Interpretation: on average, each extra day increases reaction time by about 10.47 units, across all subjects.
• This is a fixed effect, so it describes the average trend, not individual deviations.
• Answer: True, as used in this course.
• There are other growth models (for example in structural equation modeling), but here “growth curve model” means a mixed model with time as a predictor.
• Answer: Time.
• Follow-up: in a conditional growth model, the main extra fixed factor is the primary independent variable of interest (for example treatment group).
Mixed models reminder and where growth curves fit
• General mixed model form (simplified):
• Outcome = fixed part + random part + residual error.
• Fixed part works like a standard regression.
• Random part gives random intercepts and/or slopes for clustered or repeated data.
• For time data:
• Observations are repeated within units (subjects, chickens, etc.).
• We need random effects for those units because repeated measures violate independence.
• Growth curve modeling is a special case of mixed models where:
• Time is explicitly included as a predictor.
• Outcome is measured at multiple time points per unit.
• We model change over time, possibly with curves.
What is a growth curve model in this course
• Growth curve model here means:
• A mixed effects model with time as a predictor of an outcome.
• The model can have random intercepts and random slopes for time.
• You can add polynomial terms of time (time², time³, etc.) to capture curves.
• “Growth” refers to any systematic change over time, not only physical growth.
• If you only include a linear time term, you model straight-line growth.
• If you include time², time³, you model curved trajectories over time.
When to use growth curves
• Use growth curve models when:
• You have longitudinal or repeated measures data.
• You measure the same units (people, animals, etc.) at several time points.
• You think the outcome changes over time in a systematic way.
• Examples mentioned:
• Placebo vs treatment group measured weekly.
• Cognitive performance measures taken across multiple sessions.
• Visual search performance over trials or sessions.
• Two key questions:
• How does the outcome change over time on average.
• Does that pattern differ between conditions or groups.
Unconditional vs conditional growth models
5.1 Unconditional growth model
• Simplest growth model: