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