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The practical use and limitation of mixed model analysis
Jonas Ranstam
Analysis unit errorsAnalysis 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.
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 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 12 9 1 1 6.1 13 9 2 1 6.3 14 10 1 1 8.1 15 10 2 1 7.7
Placebo
Active substance
Exampel: Data from a blood pressure trial on 10 patients
Placebo Active substance
Student's t-test of the 10 patients 1st measurements
t = 1.86, df = 8, p-value = 0.10
Estimated treatment effect: 1.8 (-0.3 - 3.1)
Assumptions
1. Gaussian distribution2. Equal variances between groups3. Independent observations
Student's t-test of the 10 patients 15 measurements (including dependent observations)
t = 3.00, df = 13, p-value = 0.01
Estimated treatment effect: 1.6 (0.4 – 2.7)
Assumptions
1. Gaussian distribution2. Equal variances between groups3. Independent observations Departure from
assumption!!!
Student's t-test of the 10 patients 15 measurementsusing patients' average value
t = 1.91, df = 8, p-value = 0.09
Estimated treatment effect: 1.3 (-0.3 - 2.9)
Assumptions
1. Gaussian distribution2. Equal variances between groups3. Independent observations
Mixed model analysis of the 10 patients all 15 measurements (assuming compound symmetry)
t = 1.92, df = 8, p-value = 0.09
Estimated treatment effect: 1.3 (-0.3 - 2.7)
Assumptions
1. Gaussian distribution2. Equal variances between groups3. Compound symmetry (Equal covariances between
pairs of variables)
P-value summary
Student's t-test
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
Repeated measures data - two types of mixed models
μ = interceptb = baseline covariate effectpre = baseline valuetk = treatment effect at treatment kmj = time effect at jth visiteij = 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
A covariance pattern model
Yi = μ + b•pre + tk + mj + (tm)jk + eij
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 θθ
σ2
θ θ
θ
θ
σ2 θ
θ
θ
θ σ2
Toeplitz σ2 θ1
θ2
θ3
θ1 σ2
θ1 θ2
θ2 θ1 σ2 θ
1
θ3 θ2 θ
1 σ2
2. Random coefficients models
Yi = μ + b•pre + tk + m•timeij + eij
Mixed effect models
- continuous endpoints (linear models)
- count endpoints (Poisson models)
- binary endpoints (logistic models)
- survival endpoints (Cox models with frailty term)
Baseline Follow up
Baseline Visit 1 Visit 2 Visit 3
Time
Revised
Censored
Mixed models can be analysed using standard statistical packages
RSASS-plusSPSS*STATAEtc.
* linear models only