mixed models with variance...

Introduction Extending the basic model Simulation Analytical Explanation Summary Extra Slides

Mixed models with variance heterogeneity

David Afshartous, Ph.D.

Miller School of Medicine, Division of Clinical PharmacolgoyUniversity of Miami

Joint with Geert Verbeke, Ph.D., Katholieke Universiteit Leuven; Richard A.Preston, M.D., University of Miami

Vanderbilt University School of MedicineSeptember 30, 2009


Outline

1 Introduction

2 Extending the basic model

3 Simulation

4 Analytical Explanation

5 Summary

6 Extra Slides


Background

Mixed-effects models very popular across wide range ofapplications, e.g., longitudinal or repeated measures dataarising from clinical trials; hierarchical or clustered dataarising from observational studies (Brown & Prescott 2006;Verbeke & Molenberghs 2000)

Motivations: better estimates of uncertainty; borrowingstrength; modeling variability between and within groups;irregularly spaced time points, missing dataFixed effects describe the mean structure while randomeffects account for the association structure in the data andlead to estimates of subject-specific profiles.


Background

Mixed-effects models very popular across wide range ofapplications, e.g., longitudinal or repeated measures dataarising from clinical trials; hierarchical or clustered dataarising from observational studies (Brown & Prescott 2006;Verbeke & Molenberghs 2000)Motivations: better estimates of uncertainty; borrowingstrength; modeling variability between and within groups;irregularly spaced time points, missing data

Fixed effects describe the mean structure while randomeffects account for the association structure in the data andlead to estimates of subject-specific profiles.


Background

Mixed-effects models very popular across wide range ofapplications, e.g., longitudinal or repeated measures dataarising from clinical trials; hierarchical or clustered dataarising from observational studies (Brown & Prescott 2006;Verbeke & Molenberghs 2000)Motivations: better estimates of uncertainty; borrowingstrength; modeling variability between and within groups;irregularly spaced time points, missing dataFixed effects describe the mean structure while randomeffects account for the association structure in the data andlead to estimates of subject-specific profiles.


Stochastic Components of Linear Mixed Model

Time

Response Average evolution

Subject 1

Subject 2


Notation

Yi = Xiβ +Zibi + εi , (1)

Yi is an ni -vector of observations for the i th subject, Xi isan ni ×p vector of covariatesβ is a p-vector of fixed effects, Zi is an ni ×q design matrixfor the random effectsbi : q-vector of random effectsεi : ni vector of within-subject residual errorbi ∼ N(0,D) and εi ∼ N(0,σ2I) with bi independent of εi .


Notation


Yi is an ni -vector of observations for the i th subject, Xi isan ni ×p vector of covariates

β is a p-vector of fixed effects, Zi is an ni ×q design matrixfor the random effectsbi : q-vector of random effectsεi : ni vector of within-subject residual errorbi ∼ N(0,D) and εi ∼ N(0,σ2I) with bi independent of εi .


Notation


Yi is an ni -vector of observations for the i th subject, Xi isan ni ×p vector of covariatesβ is a p-vector of fixed effects, Zi is an ni ×q design matrixfor the random effects

bi : q-vector of random effectsεi : ni vector of within-subject residual errorbi ∼ N(0,D) and εi ∼ N(0,σ2I) with bi independent of εi .


Notation


Yi is an ni -vector of observations for the i th subject, Xi isan ni ×p vector of covariatesβ is a p-vector of fixed effects, Zi is an ni ×q design matrixfor the random effectsbi : q-vector of random effectsεi : ni vector of within-subject residual error

bi ∼ N(0,D) and εi ∼ N(0,σ2I) with bi independent of εi .


Notation


Yi is an ni -vector of observations for the i th subject, Xi isan ni ×p vector of covariatesβ is a p-vector of fixed effects, Zi is an ni ×q design matrixfor the random effectsbi : q-vector of random effectsεi : ni vector of within-subject residual errorbi ∼ N(0,D) and εi ∼ N(0,σ2I) with bi independent of εi .


Differential variability for population subgroups

Standard mixed model does not account for differentialvariability across population subgroups

Treatment growth curves may exhibit higher variability if thetreatment worked very well on some subjects and poorlyon others; lower variability if treatment depresses outcomevariable to some common lower boundQuestion: Stratify random effect variance and/or residualerror variance?



Standard mixed model does not account for differentialvariability across population subgroupsTreatment growth curves may exhibit higher variability if thetreatment worked very well on some subjects and poorlyon others; lower variability if treatment depresses outcomevariable to some common lower bound

Question: Stratify random effect variance and/or residualerror variance?



Standard mixed model does not account for differentialvariability across population subgroupsTreatment growth curves may exhibit higher variability if thetreatment worked very well on some subjects and poorlyon others; lower variability if treatment depresses outcomevariable to some common lower boundQuestion: Stratify random effect variance and/or residualerror variance?


Example from potassium handling clinical trial (N=19)


Literature

Gelman & Hill (2007), Brown & Prescott (2006), Snijders &Bosker (1999) discuss differential treatment groupvariability

Littell et al. (2006) warn that failure to account forheterogeneity can lead to inefficient and possiblymisleading inferences for fixed effectsVerbeke & Lesaffre 1996,1997; Magder & Zeger 1996;Butler & Louis 1992; mixture model, variance constantacross subgroupsMarhshall & Baron (2000), Brant et al. (2003) “fullyheteroscedastic” modelDunson (2009), Bayesian nonparametric infinite mixturemodel


Literature

Gelman & Hill (2007), Brown & Prescott (2006), Snijders &Bosker (1999) discuss differential treatment groupvariabilityLittell et al. (2006) warn that failure to account forheterogeneity can lead to inefficient and possiblymisleading inferences for fixed effects

Verbeke & Lesaffre 1996,1997; Magder & Zeger 1996;Butler & Louis 1992; mixture model, variance constantacross subgroupsMarhshall & Baron (2000), Brant et al. (2003) “fullyheteroscedastic” modelDunson (2009), Bayesian nonparametric infinite mixturemodel


Literature

Gelman & Hill (2007), Brown & Prescott (2006), Snijders &Bosker (1999) discuss differential treatment groupvariabilityLittell et al. (2006) warn that failure to account forheterogeneity can lead to inefficient and possiblymisleading inferences for fixed effectsVerbeke & Lesaffre 1996,1997; Magder & Zeger 1996;Butler & Louis 1992; mixture model, variance constantacross subgroups

Marhshall & Baron (2000), Brant et al. (2003) “fullyheteroscedastic” modelDunson (2009), Bayesian nonparametric infinite mixturemodel


Literature

Gelman & Hill (2007), Brown & Prescott (2006), Snijders &Bosker (1999) discuss differential treatment groupvariabilityLittell et al. (2006) warn that failure to account forheterogeneity can lead to inefficient and possiblymisleading inferences for fixed effectsVerbeke & Lesaffre 1996,1997; Magder & Zeger 1996;Butler & Louis 1992; mixture model, variance constantacross subgroupsMarhshall & Baron (2000), Brant et al. (2003) “fullyheteroscedastic” modelDunson (2009), Bayesian nonparametric infinite mixturemodel


Outline

1 Introduction


3 Simulation


5 Summary

6 Extra Slides


Stratified random effects variance

Yi = Xiβ +Zibi + εi

Var(bi) = Dk , k = 1,2

Intuition: if one instead estimates common random effectsvariance, estimate will lie between stratified estimates

Common random effect variance will be downward biasedfor the high variance group and upward biased for the lowvariance groupWhy is bias in the random effect variance estimatesimportant?


Stratified random effects variance


Var(bi) = Dk , k = 1,2Intuition: if one instead estimates common random effectsvariance, estimate will lie between stratified estimates

Common random effect variance will be downward biasedfor the high variance group and upward biased for the lowvariance groupWhy is bias in the random effect variance estimatesimportant?


Stratified random effects variance - cont

Why is bias in random effect variance estimate important?

Variance components are often of substantive interest intheir own rightVariance components affect shrinkage, i.e., pulling subjectprofile towards average evolutionDownward bias in random effect variance estimate⇒ increased shrinkage, i.e., subject profile will be pulledmore towards average profile.Effect on random effects estimates (predictions)themselves. Rankings of random effects.



Why is bias in random effect variance estimate important?Variance components are often of substantive interest intheir own rightVariance components affect shrinkage, i.e., pulling subjectprofile towards average evolution

Downward bias in random effect variance estimate⇒ increased shrinkage, i.e., subject profile will be pulledmore towards average profile.Effect on random effects estimates (predictions)themselves. Rankings of random effects.



Why is bias in random effect variance estimate important?Variance components are often of substantive interest intheir own rightVariance components affect shrinkage, i.e., pulling subjectprofile towards average evolutionDownward bias in random effect variance estimate⇒ increased shrinkage, i.e., subject profile will be pulledmore towards average profile.Effect on random effects estimates (predictions)themselves. Rankings of random effects.


Stratified within-subject residual error variance


Var(εi) = σ2k I, k = 1,2

Similar intuition as for random effects variance

However, opposite impact since residual error affectspooling in opposite mannerDownward bias in residual error variance estimate⇒ decreased shrinkage, i.e., subject profile will be pulledless towards average profile.




Var(εi) = σ2k I, k = 1,2


However, opposite impact since residual error affectspooling in opposite manner

Downward bias in residual error variance estimate⇒ decreased shrinkage, i.e., subject profile will be pulledless towards average profile.




Var(εi) = σ2k I, k = 1,2


However, opposite impact since residual error affectspooling in opposite mannerDownward bias in residual error variance estimate⇒ decreased shrinkage, i.e., subject profile will be pulledless towards average profile.


Outline

1 Introduction


3 Simulation


5 Summary

6 Extra Slides


Simulation Design

Balanced parallel group design, random intercept modelPlacebo group response γ0 = 4.4, treatment effect γ1 = 4.4

Sim I: random effects variance stratifiedSim II: residual error variance stratifiedSim III: both stratified1000 simulations; lme/lmer R packagesmodels: unstratified, RE-strat, resid-strat, both-strat

Residual error Random effects Sample sizesSim I: RE strat σy = .6,1.2,2.2,3.2,4.2 σd1 = 2.2, σd2 = 4.2 N = 10,20,40,80;ni = 4,8,16Sim II: Resid strat σy1 = 2.2, σy2 = 4.2 σd = .6,1.2,2.2,3.2,4.2 N = 10,20,40,80;ni = 4,8,16Sim III: Both strat A: (σy1 = 3.2, σy2 = 4.2) A: (σd1 = 2.2, σd2 = 4.2) N = 10,20,40,80;ni = 4,8,16

B: (σy1 = 4.2, σy2 = 6.2) B: (σd1 = 2.2, σd2 = 4.2) N = 10,20,40,80;ni = 4,8,16C: (σy1 = 0.6, σy2 = 1.2) C: (σd1 = 2.2, σd2 = 4.2) N = 10,20,40,80;ni = 4,8,16


Simulation Design

Balanced parallel group design, random intercept modelPlacebo group response γ0 = 4.4, treatment effect γ1 = 4.4Sim I: random effects variance stratifiedSim II: residual error variance stratifiedSim III: both stratified

1000 simulations; lme/lmer R packagesmodels: unstratified, RE-strat, resid-strat, both-strat




Simulation Design

Balanced parallel group design, random intercept modelPlacebo group response γ0 = 4.4, treatment effect γ1 = 4.4Sim I: random effects variance stratifiedSim II: residual error variance stratifiedSim III: both stratified1000 simulations; lme/lmer R packagesmodels: unstratified, RE-strat, resid-strat, both-strat




Sim I results: variance components

Table: N = 10, ni = 4; residual error variance

unstratified random effects strat residual error strat both stratσy σ̂y σ̂y σ̂y1 σ̂y2 σ̂y1 σ̂y20.6 0.6 0.6 0.6 0.6 0.6 0.61.2 1.2 1.2 1.2 1.2 1.1 1.12.2 2.2 2.2 2.1 2.1 2.1 2.13.2 3.2 3.1 3.1 3.2 3.1 3.24.2 4.1 4.2 4.1 4.2 4.1 4.1

Table: N = 10, ni = 4; random effect variance

unstratified random effects strat residual error strat both stratσy σ̂d σ̂d1 (2.2) σ̂d2 (4.2) σ̂d σ̂d1 (2.2) σ̂d2(4.2)0.6 3.2 2.0 4.0 3.2 1.9 3.81.2 3.2 2.0 4.0 3.2 2.0 3.82.2 3.2 2.0 3.8 3.2 1.9 3.93.2 3.1 1.0 3.8 3.1 1.9 3.84.2 3.1 1.8 3.8 3.0 1.8 3.6


Sim I results: fixed effect standard errors

N = 10,ni = 4 unstratified RE stratified resid strat both strat empirical SEσy γ̂0 γ̂1 γ̂0 γ̂1 γ̂0 γ̂1 γ̂0 γ̂1 γ̂0 γ̂10.6 1.45 2.05 0.91 2.04 1.45 2.05 0.90 2.01 0.82 1.841.2 1.47 2.08 0.94 2.07 1.47 2.08 0.95 2.04 0.85 1.862.2 1.53 2.16 1.03 2.16 1.50 2.12 1.03 2.15 0.93 1.903.2 1.58 2.24 1.17 2.25 1.58 2.24 1.17 2.29 1.02 2.014.2 1.69 2.39 1.33 2.42 1.70 2.42 1.32 2.46 1.14 2.17


N = 40,ni = 4 unstratified RE stratified resid strat both strat empirical SEσy γ̂0 γ̂1 γ̂0 γ̂1 γ̂0 γ̂1 γ̂0 γ̂1 γ̂0 γ̂10.6 0.74 1.05 0.49 1.05 0.74 1.05 0.48 1.06 0.48 1.031.2 0.75 1.06 0.50 1.07 0.75 1.07 0.50 1.07 0.49 1.042.2 0.78 1.10 0.54 1.10 0 .78 1.10 0.54 1.10 0.53 1.083.2 0.82 1.16 0.60 1.16 0.82 1.16 0.60 1.16 0.59 1.144.2 0.87 1.24 0.67 1.24 0.86 1.23 0.67 1.24 0.65 1.21


Sim I results: fixed effect standard errors, cont.



Sim I results: fixed effect standard errors, cont.

What if Placebo group is high variability group?

N = 10, ni = 4 unstratifed RE-stratified empirical SEσy γ̂0 γ̂1 γ̂0 γ̂1 γ̂0 γ̂10.6 1.43 2.03 1.74 2.02 1.56 1.821.2 1.46 2.07 1.77 2.06 1.58 1.852.2 1.51 2.15 1.83 2.15 1.63 1.923.2 1.59 2.26 1.91 2.27 1.70 2.024.2 1.71 2.42 1.98 2.44 1.76 2.16

N = 40, ni = 4 unstratified RE stratified empirical SEσy γ̂0 γ̂1 γ̂0 γ̂1 γ̂0 γ̂10.6 0.75 1.06 0.93 1.05 0.91 1.031.2 0.75 1.07 0.94 1.07 0.92 1.042.2 0.78 1.11 0.96 1.11 0.94 1.083.2 0.83 1.18 1.01 1.18 0.98 1.154.2 0.88 1.24 1.03 1.24 1.01 1.21


Sim II results: variance components

Table: N = 10, ni = 4; residual error variance

unstratified random effects strat residual error strat both stratσd σ̂y σ̂y σ̂y1(2.2) σ̂y2(4.2) σ̂y1(2.2) σ̂y2(4.2)

0.6 3.2 3.1 2.1 4.0 2.0 3.91.2 3.2 3.2 2.1 4.1 2.1 3.92.2 3.2 3.2 2.1 4.1 2.1 4.03.2 3.3 3.3 2.1 4.1 2.2 3.94.2 3.3 3.3 2.1 4.1 2.3 3.9

Table: N = 10, ni = 4; random effect variance

unstratified random effects strat residual error strat both stratσd σ̂d σ̂d1 σ̂d2 σ̂d σ̂d1 σ̂d20.6 0.5 0.5 0.5 0.5 0.5 0.71.2 1.0 1.0 1.0 1.0 1.0 1.12.2 2.0 2.0 1.9 2.0 2.0 1.83.2 3.0 3.0 2.9 3.0 3.0 2.74.2 4.0 3.9 3.9 4.0 4.0 3.8


Sim II results: fixed effect standard errors

N = 10,ni = 4 unstratified RE stratified resid strat both strat empirical SEσd γ̂0 γ̂1 γ̂0 γ̂1 γ̂0 γ̂1 γ̂0 γ̂1 γ̂0 γ̂10.6 0.56 0.80 0.57 0.82 0.57 1.14 0.53 1.19 0.46 0.681.2 0.71 1.01 0.70 1.02 0.71 1.30 0.70 1.34 0.60 0.892.2 1.06 1.50 1.04 1.50 1.04 1.69 1.04 1.71 0.93 1.343.2 1.46 2.06 1.43 2.06 1.44 2.20 1.43 2.21 1.28 1.854.2 1.88 2.66 1.82 2.66 1.87 2.78 1.82 2.78 1.62 2.38




Sim II results: fixed effect standard errors, cont.



Outline

1 Introduction


3 Simulation


5 Summary

6 Extra Slides


Standard error calculation

Var(

γ̂0γ̂1

)=

(N

∑i=1

X ′i WiXi

)−1

Wi = Vi−1,Vi = ZiDZ ′i +σ

2y

calculate for the correct modelassess impact of model mis-specification


Standard error calculation - Sim I

Placebo Group, suppose ni = 2:

X ′i WiXi =

(1 10 0

)(σ2

d1 +σ2y σ2

d1σ2

d1 σ2d1 +σ2

y

)−1(1 10 0

)=

2K1

(σ2

y 00 0

)Treatment Group, suppose ni = 2:

X ′i WiXi =

(1 11 1

)(σ2

d2 +σ2y σ2

d2σ2

d2 σ2d2 +σ2

y

)−1(1 11 1

)=

2K2

(σ2

y σ2y

σ2y σ2

y

)Ki = 2σ2

diσ2y +σ4

y



(N

∑i=1

X ′i WiXi

)−1

=

(N/2

∑i=1

X ′i WiXi +N

∑i=N/2+1

X ′i WiXi

)−1

=

[NK1

(σ2

y 00 0

)+

NK1

(σ2

y σ2y

σ2y σ2

y

)]−1

=1N

K1σ2

y−K1

σ2y

−K1σ2

y

K2σ2

y+ K2

σ2y

⇒ Var(γ̂0) = 1

N (σ2y +2σ2

d1)

⇒ Var(γ̂1) = 2N (σ2

y +σ2d1 +σ2

d2)



General result:

Var(γ̂0) = 2N (

σ2y

ni+σ2

d1)

What happens with unstratified model? resid-strat?

Upward bias in random effect variance⇒ increased SE

What happens with both-strat model?

Split data explanation



General result:

Var(γ̂1) = 2N (2σ2

yni

+σ2d1 +σ2

d2)

What happens with unstratified model? resid-strat?

Any bias cancels out due to summation




Standard error calculation - Sim II

General result:

Var(γ̂0) = 2N (

σ2y1ni

+σ2d )

What happens with unstratified model? RE-stratified model?

Largely unaffected




Standard error calculation - Sim II

General result:

Var(γ̂1) = 2N (

σ2y1ni

+σ2

y1ni

+2σ2d )

What happens with unstratified model? RE-stratified model?

Largely unaffected as resid error estimates unaffected




Outline

1 Introduction


3 Simulation


5 Summary

6 Extra Slides


Key Points

Heterogeneity in random effects and/or residual error variance

What happens when model is misspecified?Effect on variance components estimatesEffect on shrinkage and random effects estimatesEffect on fixed effects SE

Future directions

multivariate mixed model

One dependent variable, two groups, N subjectsTwo variables, one group, N subjects


Key Points

Heterogeneity in random effects and/or residual error variance

What happens when model is misspecified?Effect on variance components estimatesEffect on shrinkage and random effects estimatesEffect on fixed effects SE

Future directions

multivariate mixed modelOne dependent variable, two groups, N subjectsTwo variables, one group, N subjects


References

Bates, D. (2008). http://www.r-project.org/. The lme4 Package.

Brown, R. and Prescott, H. (2006). Applied Mixed Models in Medicine, New York: Wiley & Sons, 2nd Edition.

Bulter, S. and Louis, T. (1992). Random effects models with nonparametric priors Statistics in Medicine, 11:1981–2000.Brant, L.J., Sheng, C.H., Morrell, C.H., Verbeke, G., Lesaffre, E., Carter, H.B. (2003). Screening for prostatecancer using random-effects models Journal of the Royal Statistical Society, Series A, 166(1): 51–62.

Gelman, A. and Hill, J. (2007). Data Analysis Using Regression and Multilevel/Hierarchical Models, NewYork: Cambridge.

Littell, R., Milliken, G., Stroup, W., Wolfinger, R., and Schabenberge, O. (2006). SAS for Mixed Models,Cary, NC: SAS Institute, Inc., 2nd ed.

Magder, L. and Zeger, S. (1996). A Smooth Nonparametric Estimate of a Mixing Distribution Using Mixturesof Gaussians Journal of the American Statistical Association, 91: 1141–1151.

Marshall, G. and Baron, A.E. (2000). Linear discriminant models for unbalanced longitudinal data Statisticsin Medicine, 19: 1969–1981.

Preston, R.A., Afshartous, D. and Alonso, A. (2007). Effects of Selective versus NonselectiveCyclooxygenase Inhibition on Dynamic Renal Potassium Excretion: A Randomized Trial. ClinicalPharmacology and Therapeutics, 84(2): 208-211.

Snijders, T.A.B and Bosker, R.J. (1999). Multilevel Analysis, London: SAGE.

Verbeke, G. and Molenberghs, G. (2000). Linear Mixed Models for Longitudinal Data, New York: Springer.

Verbeke, G. and Lesaffre, E. (1996). A linear mixed model with heterogeneity in the random effectspopulation. Journal of the American Statistical Association, 91: 217-221.

Verbeke, G. and Lesaffre, E. (1997). The effect of misspecifying the random effects distribution in linearmixed models for longitudinal data. Computational Statistics and Data Analysis, 23: 541-556.

Zeger, S. and Liang, K. (1986). Longitudinal data analysis using generalized linear models, Biometrika, 73:13-22.


Outline

1 Introduction


3 Simulation


5 Summary

6 Extra Slides


Potassium Handling Study

6 repeated measures of potassium excretion on placebofollowed by 6 repeated measures on treatment; Yij is the j thmeasurement on the i th individual, tij represents time

5 random intercept models, different variance assumptions:M1: Common random effects varianceM2: Random treatment effectM3: Stratified random effect varianceM4: Stratified random effect and residual error varianceM5: Only stratified residual error varianceestimation via restricted maximum likelihood (REML);lmer R package (Bates 2008)


Model Equations

M1:

Yij =

{(γ0 +bi)+β1tij +β2t2

ij + εij , if placebo(γ0 +bi)+ γ1 +(β1 +β3)tij +(β2 +β4)t2

ij + εij , if drug

bi ∼ N(0,σ2d ),εij ∼ N(0,σ2

y ).

M2:

Yij =

{(γ0 +b0i)+β1tij +β2t2

ij + εij , if placebo(γ0 +b0i)+(γ1 +b1i)+(β1 +β3)tij +(β2 +β4)t2

ij + εij , if drug

b0i ∼ N(0,σ2d̃ ),b1i ∼ N(0,σ2

T ).

M3:

Yij =

{(γ0 +bi1)+β1tij +β2t2

ij + εij , if placebo(γ0 +bi2)+ γ1 +(β1 +β3)tij +(β2 +β4)t2

ij + εij , if drug

}bik ∼ N(0,σ2

dk ),k = 1,2.


Estimation results

M1 M2 M3 M4 M5log-likelihood -1398.44 -1397.31 -1397.31 -1375.97 -1377.60Fixed Effects:γ0 (intercept) 67.3 (25.6) 67.3 (26.6) 67.3 (26.6) 67.3 (31.5) 67.3 (32.4)γ1 (drug) 25.6 (35.0) 25.6 (35.9) 25.6 (35.9) 25.6 (35.9) 25.6 (34.9)β1 (linear) 117.0 (23.3) 117.0 (22.8) 117.0 (22.8) 117.0 (28.9) 117.0 (29.4)β2 (quadratic) -20.0 (4.4) -20.0 (4.3) -20.0 (4.3) -20.0 (5.6) -20.0 (5.6)β3 (linear×Drug) -78.1 (33.0) -78.1 (32.3) -78.1 (32.3) -78.1 (32.3) -78.1 (32.9)β3(quadratic×Drug) 14.9 (6.3) 14.9 (6.2) 14.9 (6.2) 14.9 (6.2) 14.9 (6.3)Variance Components:√

Var(bi ) (intercept) 28.77 36.97√Var(b0i ) (intercept all) 47.18√Var(b1i ) (intercept drug) 46.18

corr(b0i ,b1i ) -0.86√Var(bi1) (intercept placebo) 47.19 29.09√Var(bi2) (intercept drug) 24.20 44.35

corr(bi1 ,bi2) 0.29 0.265√Var(εij ) (error all) 119.32 116.93 116.93√Var(εij1) (error placebo) 148.21 150.49√Var(εij2) (error drug) 73.34 75.28

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