1. Alexandra Kuznetsova Biostatistician, Leo Pharma
Mixed models
Alexandra Kuznetsova
Biostatistician, Leo Pharma

2. Efficacy data Simulated data in an ADaM-like format
30 subjects with psoriasis skin disease were randomized to 3 treatment arms in 1:1:1. Treatments had a concentration of an active drug 1%, 2%, 3%.
Effectiveness of the treatments was measured in terms of thickness and redness of the skin
drug 1%
Drug application
Redness assessments
drug 2%
Thickness measurements
drug 3%
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8

3. Efficacy data
Thickness measured at Day 1 (baseline), Day 4, Day 6, Day 8
Redness measured at Day 1, Day2, Day3, Day 4, Day 5, Day 6, Day 7, Day 8
Objective: compare treatments in terms of thickness, redness at Day 8

4. Efficacy data, redness
Profile plots

5. Modelling, mixed model with one random subject effect
Main treatment effect
Fixed effects
Main day effect
Day * treatment interaction effect
Random effect
Random subject effect

6. Modelling, mixed model with one random subject effect
Variance-covariance matrix of Redness:

7. Modelling, MMRM, spatial gaussian correlation
Variance-covariance matrix of Redness:

8. Mixed models, R lme4 package nlme package Fast, can handle large data
can handle multiple crossed effects
User-friendly
JSS (
nlme package
Can handle variance covariance structures (MMRM)
Well documented (Pinheiro and Bates, 2010)
Difficult syntax
Cannot handle multiple crossed random effects

9. lme4 package, CRAN

10. Other R- packages that depend on lme4

11. Useful packages that depend on lme4
emmeans for calculating least squares means, differences of least- squares means e t.c.
pbkrtest for calculating Kenward-Roger’s adjusted F-statistics and denominator degrees of freedom
lmerTest for calculating Satterthwaites degrees of freedom, Type Ⅲ hypothesis tests, step-wise selection …

12. Mixed model, lme4

13. Modelling, mixed model with one random subject effect, lme4
Fixed effects
Random effect
NOTE: by default the first level in fixed effect is set to 0 (in SAS the last one)
NOTE: check classes of variables before fitting a model!
Estimates of the coefficients, REML, variance estimates
F-tests for fixed effects

14. Modelling, mixed model with one random subject effect, lme4

15. nlme package, CRAN

16. Modelling, MMRM, nlme
Fixed effects
NOTE: ADY must be covariate!

17. Modelling, MMRM, nlme

18. MMRM. Treatment differences at Day 8

19. MMRM, unstructured covariance

20. Exercises, MMRM
Use the same data dat_mmrm.csv. Subset the data for response variable thickness
Plot the data
Fit a model with a random scalar effect either using nlme package and gls function or lme4 package.
Extend the model by adding correlation structure.
Try different correlation structures. Use ?corClasses
Make diagnostic plots
Formulate a conclusion about difference between treatments at Day 8.