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Step 4 · Complex designs · Topic 16 of 17 · 10 min

Mixed model with a random intercept

How mixed models analyze incomplete repeated measurements, and which assumptions to justify.

In short#

  • Model: yᵢⱼ = xᵢⱼβ + uᵢ + εᵢⱼ, where uᵢ ~ N(0, σᵤ²) and εᵢⱼ ~ N(0, σ²). A random intercept gives each experimental unit its own baseline shift and models dependence among its repeated measurements.
  • A mixed model can use available outcomes under a correctly specified model and a missing-at-random (MAR) assumption. Under MAR, missingness can depend on observed data, but not on the missing outcome after conditioning on those data. No values are imputed.
  • REML estimates variance components using restricted likelihood. Use ML for likelihood-ratio comparisons of models with different fixed effects.
  • Satterthwaite degrees of freedom are approximate. Asymptotic p values can be unreliable with few experimental units. Boundary variance estimates make p values and intervals unavailable (N/A).

In plain words#

Imagine the same experimental unit measured several times. Its values may be related because it has its own baseline level. A random intercept models a shift for each unit and accounts for this dependence.

With repeated-measures ANOVA, a missing condition usually means excluding all measurements for that unit. A mixed model can use its other available measurements when the missingness assumption is justified.

This flexibility needs justification. Report the estimation method, degrees-of-freedom method and assumption about the missingness mechanism.

In Linza#

  1. Open the practice project: eight experimental units were measured under three repeated conditions, with two measurements missing. The “Repeated measures” table already uses a mixed model with a random intercept. Experimental-unit IDs link the available rows.
  2. The practice uses REML with Satterthwaite inference. Check warnings and the included observation count. The model uses available measurements, but this does not establish MAR (missing at random): after conditioning on observed data, missingness does not depend on missing values. Justify the assumption and check the model.
  3. Random slopes and arbitrary covariance structures are not supported yet.
Try it in Linza Opens in a new tab with synthetic data. Work in the original tab remains unchanged.

Common mistakes#

  • Choosing a mixed model only because data are missing, without justifying the missingness assumption.
  • Relying on asymptotic p values with few experimental units.

Check yourself#

Why can a mixed model use incomplete repeated data?

Optional. Your answer is not saved.

Sources#

  • Laird N.M., Ware J.H. (1982) Random-effects models for longitudinal data. Biometrics, 38(4). Search for publication Opens in a new tab
  • Kuznetsova A., Brockhoff P.B., Christensen R.H.B. (2017) lmerTest package: tests in linear mixed effects models. Journal of Statistical Software, 82(13). Open publication Opens in a new tab
  • Luke S.G. (2017) Evaluating significance in linear mixed-effects models in R. Behavior Research Methods, 49. Search for publication Opens in a new tab

References are for further reading. A method’s suitability depends on the study design and its assumptions.