Mixed models with correlated measurement errors Rasmus Waagepetersen October 9, 2018
Example from Department of Health Technology 25 subjects where exposed to electric pulses of 11 different durations using two different electrodes (3=pin, 2=patch). The durations were applied in random order. In total 550 measurements of response to pulse exposure. Fixed effects in the model: electrode, Pulseform (duration), order of 22 measurements for each subject. Random effects: one random effect for each subject-electrode combination (50 random effects).
Mixed model with random intercepts Model: y ijk = µ ij + U ij + ɛ ijk where i = 1,..., 25 (subject), j = 2, 3 (electrode), and k = 1,..., 11 measurement within subject-electrode combination. µ ij fixed effect part of the model depending on electrode, Pulseform and order of measurement. U ij s and ɛ ij s independent random variables.
Using lmer fit=lmer(transfpt~electrodeid*pulseform+ electrodeid*orderfixed+(1 electrsubid),data=perception Random effects: Groups Name Variance Std.Dev. electrsubid (Intercept) 0.03479 0.1865 Residual 0.01317 0.1148 Number of obs: 550, groups: electrsubid, 50 Large subject-electrode variance 0.03479. Noise variance 0.01317
Effect of Pulseform (duration) duration effect 1.2 1.0 0.8 0.6 0.4 0.2 0.0 0 20 40 60 80 100 dur Blue: pin electrode (code 3). Black: patch electrode (code 2).
Serial correlation in measurement error? Maybe error ɛ ijk not independent of previous error ɛ ij(k 1) since measurements carried out in a sequence for each subject? For each subject-electrode combination ij plot residual r ijk (resid(fit)) against previous residual r ij(k 1) for k = 2,..., 11. 0.6 0.4 0.2 0.0 0.2 0.4 0.6 0.6 0.4 0.2 0.0 0.2 0.4 0.6 resi1 resi2
Correlation cor.test(resi1,resi2) Pearson s product-moment correlation data: resi1 and resi2 t = 8.5284, df = 498, p-value < 2.2e-16 alternative hypothesis: true correlation is not equal to 0 95 percent confidence interval: 0.2779966 0.4311777 sample estimates: cor 0.3569848
Mixed model with correlated errors Analysis of residuals r ijk (which are estimates of errors ɛ ijk ) suggests that ɛ ijk are correlated (not independent). Recall general mixed model formulation: Y = X β + ZU + ɛ where ɛ normal with mean zero and covariance Σ. So far Σ = σ 2 I meaning noise terms uncorrelated and all with same variance σ 2 Extension: Σ not diagonal meaning Cov[ɛ i, ɛ i ] 0. Many possibilities for Σ - we will focus on autoregressive covariance structure that is useful for serially correlated error terms.
Basic model for serial correlation: autoregressive Consider sequence of noise terms: ɛ 1, ɛ 2,..., ɛ m. Model for variance/covariance: Cov(ɛ i, ɛ j ) = σ 2 ρ i j Corr(ɛ i, ɛ j ) = ρ i j Thus Varɛ i = σ 2 and ρ is correlation between two consecutive noise terms e.g. ɛ i and ɛ i+1 ρ = Corr(ɛ i, ɛ i+1 )
Interpretation of autoregressive model Covariance structure arises from following autoregressive model: ɛ i+1 = ρɛ i + ν i+1 (1) where ɛ 1 N(0, σ 2 ), and ν l N(0, ω) ω = σ 2 (1 ρ 2 ) l = 2,..., m ɛ 1, ν 2,..., ν m assumed to be independent.
Implementation Not possible in lmer :( However lme (from package nlme) can do the trick (thanks, Mads): fit=lme(transfpt~factor(electrodeid)*pulseform+factor(electrodeid)*orderfixed, random=~1 electrsubid,data=perception,correlation=corar1()) lme predecessor of lmer - both have pros and cons - but here lme has the upper hand. SPSS: specification using Repeated. Here we can select repeated variable: order of observations within subject subject variable: noise terms for different subjects assumed to independent covariance structure for noise terms within subject E.g. for perception data we may have 11 serially correlated errors for each subject-electrode combination but errors are uncorrelated between different subject-electrode combinations.
Repeated in SPSS OrderFixed: from 1-11 within each electrode-subject combination Covariance structure: AR(1)
Estimates of variance parameters With uncorrelated errors: τ 2 = 0.035 σ 2 = 0.013 With autoregressive errors: τ 2 = 0.030 σ 2 = 0.018 ρ = 0.626 Variance parameters not so different but quite big estimated correlation for errors. Qualitatively same conclusions regarding significance of fixed effects (F -tests). Bigger standard errors of parameter estimates for model with correlated noise - makes sense!
Model assessment - residuals Histogram of resi resi Frequency 0.4 0.0 0.2 0.4 0.6 0 50 100 150 200 3 2 1 0 1 2 3 0.4 0.2 0.0 0.2 0.4 Normal Q Q Plot Theoretical Quantiles Sample Quantiles 2 3 0.4 0.2 0.0 0.2 0.4 a.2 e.2 i.2 b.3 f.3 j.3 0.4 0.2 0.0 0.2 0.4 3 2 1 0 1 2 3 0.4 0.2 0.0 0.2 0.4 pin Theoretical Quantiles Sample Quantiles 3 2 1 0 1 2 3 0.2 0.1 0.0 0.1 0.2 patch Theoretical Quantiles Sample Quantiles
Much larger residual variance for pin electrode than for patch electrode. Still room for improvement of model! SAS PROC MIXED has group option to obtain variance heterogeneity over groups. Seems not available in SPSS Separate analyses for two electrodes?
Exercises 1. Given the model (1) verify that ɛ i+1 N(0, σ 2 ) and Corr(ɛ i, ɛ i+1 ) = ρ. 2. Fit model with autoregressive errors to the Orthodont dataset.