Mixed Models for Longitudinal Data: An Applied Introduction Don Hedeker Department of Public Health Sciences Biological Sciences Division University of Chicago [email protected]Hedeker, D. (2004). An introduction to growth modeling. In D. Kaplan (Ed.), Quantitative Method- ology for the Social Sciences. Thousand Oaks CA: Sage. Hedeker, D. & Gibbons, R.D. (2006). Longitudinal Data Analysis, chapters 4 & 5. Wiley. This work was supported by National Institute of Mental Health Contract N44MH32056. 1
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Mixed Models for Longitudinal Data: An Applied Introduction · 2019-03-29 · Mixed Models for Longitudinal Data: An Applied Introduction Don Hedeker Department of Public Health Sciences
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Mixed Models for Longitudinal Data:An Applied Introduction
Time might be years or months, and could differ for each subject
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The conditional variance-covariance matrix is now of the form:
• Σyi = ZiΣυZ′i + σ2Ini
For example, with r = 2, n = 3, and Z ′i =
1 1 10 1 2
the conditional variance-covariance Σyi = σ2Ini+
σ2υ0
σ2υ0
+ συ0υ1 σ2υ0
+ 2συ0υ1
σ2υ0
+ συ0υ1 σ2υ0
+ 2συ0υ1 + σ2υ1
σ2υ0
+ 3συ0υ1 + 2σ2υ1
σ2υ0
+ 2συ0υ1 σ2υ0
+ 3συ0υ1 + 2σ2υ1
σ2υ0
+ 4συ0υ1 + 4σ2υ1
• variances and covariances change across time
More general models allow autocorrelated errors, εi ∼ N (0, σ2Ωi),where Ω might represent AR or MA process
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Example: Drug Plasma Levels and Clinical Response
Riesby and associates (Riesby et al., 1977) examined therelationship between Imipramine (IMI) and Desipramine (DMI)plasma levels and clinical response in 66 depressed inpatients(37 endogenous and 29 non-endogenous)
i = 1 . . . 66 patientsj = 1 . . . ni observations (max = 6) for patient i
b0i = week 0 HD level for patient ib1i = weekly change in HD for patient i
Between-subjects models
b0i = β0 + υ0ib1i = β1 + υ1i
β0 = average week 0 HD levelβ1 = average HD weekly improvementυ0i = individual deviation from average interceptυ1i = individual deviation from average improvement
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Under “File” click on “Open Existing Model Setup”
Open C:\SuperMixEn Examples\Workshop\Continuous\reisby.mum(or C:\SuperMixEn Student Examples\Workshop\Continuous\reisby.mum)
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Empirical Bayes Estimates of Random EffectsSelect “Analysis” > “View Level-2 Bayes Results”
ID, random effect number, estimate, variance, name
b0i = week 0 HD level for patient ib1i = weekly change in HD for patient i
Between-subjects models
b0i = β0 + β2Dxi + υ0ib1i = β1 + β3Dxi + υ1i
β0 = average week 0 HD level for NE patients (Dxi = 0)β1 = average HD weekly improvement for NE patients (Dxi = 0)β2 = average week 0 HD difference for E patientsβ3 = average HD weekly improvement difference for endogenous patientsυ0i = individual deviation from average interceptυ1i = individual deviation from average improvement
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Under “File” click on “Open Existing Model Setup”
Open C:\SuperMixEn Examples\Workshop\Continuous\reisby2.mum(or C:\SuperMixEn Student Examples\Workshop\Continuous\reisby2.mum)
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⇒ χ22 = 4.11, p ns, compared to model with β2 = β3 = 0
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Select “File” > “Model-based Graphs” > “Trends”(sorry, but “Trends” is not include in the student edition)
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⇒ Endogenous group by time interaction is non-significant; groupsare about 2 points different at all timepoints
b0i = week 2 HD level for patient i with both ln IMI and lnDMI = 0b1i = weekly change in HD for patient ib2i = change in HD due to ln IMIb3i = change in HD due to lnDMI
Between-subjects models
b0i = β0 + υ0ib1i = β1 + υ1ib2i = β2b3i = β3
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β0 = average week 2 HD level for subjects with ln drug values of 0β1 = average HD weekly improvementβ2 = average HD difference for unit change in ln IMIβ3 = average HD difference for unit change in lnDMIυ0i = individual intercept deviation from modelυ1i = individual slope deviation from model
Here, week 2 is the actual study week (i.e., one week after the drug washoutperiod), which is coded as 0 in this analysis of the last four study timepoints