Statistics for CSAI 2

by Albert Major

1. Warm-up question about model interpretation

• Students saw an equation and had to choose what is “certainly true.”
• Most students selected “multiple linear regression predicting memory score with practice time and participant.”
  • This was incorrect.
• The correct interpretation:
  • The model includes a random effect for participant.
  • The syntax used (1 | participant) indicates a mixed effects model, not a standard linear regression.
• Extra note:
  • A first-order polynomial is degree 1, which matches linear regression.
  • But the presence of the random-effect notation makes this a mixed model, regardless of polynomial degree.

1. Why the model cannot run with LM

• LM only fits standard linear regression.
• Mixed models use LMER (from lme4).
• If you try to fit (1 | participant) inside LM, R will throw an error.
• This confirms that the model uses mixed-effects structure, not standard multiple regression.

1. Introduction to mixed effects models

• Goal: extend multiple regression to data that is nested or repeated.
• Nested data means:
  • Multiple observations come from the same unit.
  • Examples:
    • Many measurements from each student.
    • Many patients treated by the same therapist.
    • Several timepoints for each participant.
    • Students grouped by classrooms.
• Standard regression assumes independence of observations.
  • But repeated or grouped data violate this assumption.
• Mixed effects models let us model this dependence directly.

1. What regression assumed, and what mixed models change

   Left side (old regression):

• Y = β0 + β1·X + error