The document discusses mixed model analysis for repeated measures data. It provides an example dataset from a blood pressure trial with 10 patients measured at 1-2 timepoints under placebo or active substance. Student's t-test assuming independence violates assumptions when applied to this dataset. Mixed models that account for dependency between observations, such as compound symmetry, provide a better analysis. Two common mixed models are presented - covariance pattern models that model covariance structure and random coefficients models that model random effects. Mixed models can handle different endpoint types and be analyzed in standard statistical packages.

1. The practical use and limitation
of mixed model analysis
Jonas Ranstam

2. Analysis unit errors
Analysis unit errors are surprisingly common. Of 142
reviewed papers 42% involved such errors*.
* Bryant et al. How Many Patients? How Many Limbs?
Analysis of Patients or Limbs in the Orthopaedic
Litterature. J Bone Joint Surg Am.2006;88:41-45.

3. Exampel: Data from a blood pressure trial on 10 patients
seq pat mst trt eff
­­­ ­­­ ­­­ ­­­ ­­­
1 1 1 0 4.2
2 1 2 0 4.4
3 2 1 0 4.1
4 3 1 0 4.3 Placebo
5 4 1 0 4.5
6 4 2 0 4.8
7 5 1 0 6.7
8 6 1 1 6.1
9 7 1 1 4.9
10 8 1 1 5.4
11 8 2 1 5.7
Active substance
12 9 1 1 6.1
13 9 2 1 6.3
14 10 1 1 8.1
15 10 2 1 7.7

4. Exampel: Data from a blood pressure trial on 10 patients
Placebo Active substance

5. Student's t­test of the 10 patients 1st measurements
t = 1.86, df = 8, p­value = 0.10
Estimated treatment effect: 1.8 (­0.3 ­ 3.1)
Assumptions
1. Gaussian distribution
2. Equal variances between groups
3. Independent observations

6. Student's t­test of the 10 patients 15 measurements
(including dependent observations)
t = 3.00, df = 13, p­value = 0.01
Estimated treatment effect: 1.6 (0.4 – 2.7)
Assumptions
1. Gaussian distribution
2. Equal variances between groups
3. Independent observations Departure from
assumption!!!

7. Student's t­test of the 10 patients 15 measurements
using patients' average value
t = 1.91, df = 8, p­value = 0.09
Estimated treatment effect: 1.3 (­0.3 ­ 2.9)
Assumptions
1. Gaussian distribution
2. Equal variances between groups
3. Independent observations

8. Mixed model analysis of the 10 patients all 15
measurements (assuming compound symmetry)
t = 1.92, df = 8, p­value = 0.09
Estimated treatment effect: 1.3 (­0.3 ­ 2.7)
Assumptions
1. Gaussian distribution
2. Equal variances between groups
3. Compound symmetry (Equal covariances between
pairs of variables)

9. P­value summary
Student's t­test
of 1st measurements p = 0.10
of measurement averages p = 0.09
of dependent observations p = 0.01
Mixed model analysis
assuming compound symmetry p = 0.09

10. Repeated measures data ­ two types of mixed models
μ = intercept
b = baseline covariate effect
pre = baseline value
tk = treatment effect at treatment k
mj = time effect at jth visit
eij = residual term for the ith patient at the jth visit
1. Covariance pattern models
Yi = μ + b•pre + tk + mj + (tm)jk + eij
2. Random coefficients models
Yi = μ + b•pre + tk + m•timeij + eij

11. A covariance pattern model
Yi = μ + b•pre + tk + mj + (tm)jk + eij

12. Covariance patterns for a trial with 4 time points
General σ21 θ12 θ13 θ14
θ12 σ22 θ23 θ24
θ13 θ23 σ23 θ34
θ14 θ24 θ34 σ24
Compound symmetry σ2 θ θ θ
θ σ2 θ θ
θ θ σ2
θ
θ θ θ σ2
Toeplitz σ2 θ1 θ2 θ3
θ1 σ2 θ1 θ2
θ2 θ1 σ2 θ1
θ3 θ2 θ1 σ2

13. 2. Random coefficients models
Yi = μ + b•pre + tk + m•timeij + eij

14. Mixed effect models
- continuous endpoints (linear models)
- count endpoints (Poisson models)
- binary endpoints (logistic models)
- survival endpoints (Cox models with frailty term)

16. Baseline Follow up

18. Baseline Visit 1 Visit 2 Visit 3

20. Revised
Censored
Time

24. Mixed models can be analysed using
standard statistical packages
R
SAS
S-plus
SPSS*
STATA
Etc.
* linear models only