Introduction to Linear Mixed Models Tom Greene. Examples of Hierarchical or Clustered Data Design Level Clusters: Observational studies relating health

Introduction to Linear Mixed Models

Tom Greene

Examples of Hierarchical or Clustered Data

Design Level Clusters:•Observational studies relating health outcomes to patient and center level predictor variables for patients (level 1 units) nested within clinics (level 2 units)•Cluster randomized trials (RCTs) in which centers are randomized to different treatments (centers are level 2 units, patients are level 1 units)•Multicenter RCTs in which patients are randomized to treatments within each center, but it is thought that the treatment effect may vary between centers (centers are level 2 units, patients are level 1 units)


Design Level Clusters:•Longitudinal studies relating repeated measurements to predictor variables (patients are level 2 units, measurements at different times are level 1 units)•Complex survey designs in which primary sampling units (such as counties) are first sampled (level 2 units), and then households are sampled within the primary sampling units (level 1 units)


Naturally Occurring Clusters:•Analyses of family members (level 1 units) nested within families (level 2 units)•Analyses of eyes, ears, lungs, etc. (level 1 units), nested within patients (level 2 units)•Analyses of individual littermates (level 1 units) nested within litters (level 2 units)


Can have 3 or more levels of clustering:•Longitudinal observations (level 1 units) from patients (level 2 units) nested within centers (level 3 units) of a multicenter RCT•Complex surveys: Families (level 1) nested within census tracts (level 2) nested within counties (level 3)

Applications of Mixed Effects Models• Account for correlated data when computing

p-values or confidence intervals• Estimate level 2 quantities while borrowing

information from the full data set (shrinkage)• Estimate the variance (or standard deviation)

of level 2 quantities while accounting for “noise” due to sampling error

Applications of Mixed Effects Models• Characterize the pattern of correlations (or

covariances) of measurements over time or space

• Determine the optimal “weighting” to estimate treatment effects when data are unbalanced

• Providing approximately unbiased analyses of longitudinal data when data are missing at random

Most Basic Example: 1-Way ANOVA

Most Basic Example: 1-Way ANOVAMixed Effects Formulation:Primary goals are usually a) to estimate the overall mean, applying inferences to a broader population of “groups” (really level 2 units) from which the study groups are viewed as a random sample, b) to estimate the individual group means while incorporating information from the other groups, and c) to estimate the variance of the distribution of the group means in the population from which the sampled groups were drawn.

. . . . .µ1 µ2 µg

• Hypothetical super-population from which groups were drawn

• This population has an infinite # of µi .

• We’d like to know the mean and variance of this distribution

g Sampled groups

Most Basic Example: 1-Way ANOVA

Example 1• Research Objective: Estimate 6-month mean weight loss in

overweight diabetics resulting from 1-1 coaching program• Randomly assign subjects to 8 different coaches who have

been certified in the program (6 subjects per coach) • Yij = Observed weight loss for the jth patient assigned to the

ith coach• β0 = overall mean weight loss across the super-population of

coaches• β0 + bi = mean weight loss under the individual coaches

(without sampling error)• εij = patient variation in weight loss• Model: Yij = β0 + bi + εij, i = 1,2, .., 8; j = 1,2, …, 6

Weight Change Data: Change in Kg

Analysis Variable : YCoach #(group) N Obs Mean Std Dev

1 6 3.15 8.842 6 -7.20 2.583 6 -5.06 4.804 6 -19.61 5.765 6 -10.83 4.006 6 -9.41 5.897 6 -7.99 5.768 6 -9.26 4.24

PROC MIXED STATEMENTSStatement Function PROC MIXED Invokes linear mixed model procedure

MODEL Specifies response variable and fixed effect predictor variables (e.g., model for E(Yij))

RANDOM Specifies model for random effects (the bi)

REPEATED Designates correlation (or covariance) structure in the residuals (the εij)

CLASS Specifies class variables

ESTIMATE Designates linear functions of fixed and/or random effects for estimation

CONTRAST Designates general linear hypotheses in fixed and/or random effects

• PROC MIXED specifies standard fixed effects ANOVA/Regression with uncorrelated residuals if REPEATED and RANDOM statements are omitted.

Next timewith Rich

** Fixed Effects Analysis:proc mixed data=ydat; class group; model y=group / solution ddfm = kr noint; estimate ‘Overall Mean' group 1 1 1 1 1 1 1 1 /divisor=8; run;

Solution for Fixed Effects

GROUP EstimateStandard

Error DF t Value Pr > |t|1 3.1535 2.2496 40 1.40 0.16872 -7.2040 2.2496 40 -3.20 0.00273 -5.0556 2.2496 40 -2.25 0.03024 -19.6076 2.2496 40 -8.72 <.00015 -10.8269 2.2496 40 -4.81 <.00016 -9.4063 2.2496 40 -4.18 0.00027 -7.9894 2.2496 40 -3.55 0.00108 -9.2603 2.2496 40 -4.12 0.0002

Estimates

Label EstimateStandard

Error DF t Value Pr > |t|Overall Mean -8.2746 0.7954 40 -10.40 <.0001

ResponseCoach #

Kenward-Roger degrees of freedom(matches usual df for fixed effects models)

With intercept omitted, fixed effects estimates correspond to group means

What quantity does overall mean ± SE refer to?

** Mixed Effects Analysis;proc mixed data=ydat ; class group; model y=/solution ddfm = kr; random intercept/subject=group solution; run;

Covariance Parameter EstimatesCov Parm Subject EstimateIntercept GROUP 34.8526Residual 30.3639


Effect EstimateStandard

Error DF t Value Pr > |t|Intercept -8.2746 2.2336 7 -3.70 0.0076

Solution for Random Effects

Effect GROUP EstimateStd Err

Pred DF t Value Pr > |t|Intercept 1 9.9791 2.9326 14.8 3.40 0.0040Intercept 2 0.9348 2.9326 14.8 0.32 0.7544Intercept 3 2.8109 2.9326 14.8 0.96 0.3533Intercept 4 -9.8961 2.9326 14.8 -3.37 0.0043Intercept 5 -2.2287 2.9326 14.8 -0.76 0.4592Intercept 6 -0.9882 2.9326 14.8 -0.34 0.7409Intercept 7 0.2490 2.9326 14.8 0.08 0.9335Intercept 8 -0.8607 2.9326 14.8 -0.29 0.7732

Only fixed effect is overall mean β0

Random effects corresponding to each coach

= 34.85/(30.36+34.85)

is intra-class correlation

= 0.534

Comparison of Fixed Effects Estimates and BLUPs

Example 2• Research Objective: Compare effects of two 1-1 coaching

methods on 6-month mean weight loss in overweight diabetics

• Randomly assign subjects to 8 coaches who have been certified in both methods (6 subjects per coach)

• Randomly assign 8 coaches to 2 methods (4 coaches per method)

• Cluster randomized trial • Yij = Weight loss for the jth patient assigned to the ith coach• Xi = indicator for assignment of ith coach to method B. • Model: Yij = β0 + β1 Xi + bi + εij, i = 1,2, .., 8; j = 1,2, …, 6

Fixed effects terms

Random effects


Analysis Variable : Y

Treatment

Coach #

(GROUP) N Obs Mean Std DevA 1 6 5.15 8.84A 2 6 -5.20 2.58A 3 6 -3.06 4.80A 4 6 -17.61 5.76B 5 6 -12.83 4.00B 6 6 -11.41 5.89B 7 6 -9.99 5.76B 8 6 -11.26 4.24

** Fixed Effects Model Accounting for Group;proc mixed data=ydat;class group; model y=group / solution ddfm = kr noint cl; estimate ‘Method' group -1 -1 -1 -1 1 1 1 1 /divisor=4 cl;


GROUP EstimateStandard

Error DF Lower Upper1 5.1535 2.2496 40 0.6069 9.70012 -5.2040 2.2496 40 -9.7506 -0.65743 -3.0556 2.2496 40 -7.6022 1.49104 -17.6076 2.2496 40 -22.1541 -13.06105 -12.8269 2.2496 40 -17.3735 -8.28036 -11.4063 2.2496 40 -15.9529 -6.85977 -9.9894 2.2496 40 -14.5360 -5.44288 -11.2603 2.2496 40 -15.8069 -6.7137

Type 3 Tests of Fixed Effects

EffectNum

DFDen

DF F Value Pr > FGROUP 8 40 22.09 <.0001

Estimates


Error DF t Value Pr > |t| Alpha Lower UpperMethod -6.1923 1.5907 40 -3.89 0.0004 0.05 -9.4072 -2.9774

Produces fixed effects inferencefor Method effect

What quantity do the estimate and SE refer to?

** Fixed Effects Model Ignoring Group;proc mixed data=ydat;model y=method/ solution ddfm = kr cl;



Error DF t Value Pr > |t| Alpha Lower UpperIntercept -5.1784 1.6020 46 -3.23 0.0023 0.05 -8.4030 -1.9538method -6.1923 2.2655 46 -2.73 0.0089 0.05 -10.7526 -1.6320

Covariance Parameter EstimatesCov Parm EstimateResidual 61.5922

Does this standard error correspond to a meaningful quantity?

** Fixed Effects Model Ignoring Group;proc mixed data=ydat;model y=method/ solution ddfm = kr cl;



Error DF t Value Pr > |t| Alpha Lower UpperIntercept -5.1784 1.6020 46 -3.23 0.0023 0.05 -8.4030 -1.9538Method -6.1923 2.2655 46 -2.73 0.0089 0.05 -10.7526 -1.6320

Covariance Parameter EstimatesCov Parm EstimateResidual 61.5922

Does this standard error correspond to a meaningful quantity? NO!

Muddles together σb2 and σ2

** Mixed Effects Model with Treatment as Fixed Effect** and Group as Random Effect;proc mixed data=ydat;model y=method/ solution ddfm = kr;random group / solution;

Covariance Parameter EstimatesCov Parm Subject EstimateIntercept GROUP 39.9027Residual 30.3639

Solution for Fixed Effects (from mixed model)


Error DF t ValuePr > |

t|Intercept -5.1784 3.3527 6 -1.54 0.1734method -6.1923 4.7415 6 -1.31 0.2394

Solution for Random EffectsGROUP Estimate Std Err Pred DF t Value Pr > |t|

1 9.1690 3.6975 7.82 2.48 0.03882 -0.02273 3.6975 7.82 -0.01 0.99523 1.8839 3.6975 7.82 0.51 0.62444 -11.0302 3.6975 7.82 -2.98 0.01805 -1.2923 3.6975 7.82 -0.35 0.73596 -0.03155 3.6975 7.82 -0.01 0.99347 1.2258 3.6975 7.82 0.33 0.74898 0.09801 3.6975 7.82 0.03 0.9795

Comparison of Treatment Effect Estimates

Fixed Effects Model Ignoring Group


Error DF t Value Pr > |t|Method -6.1923 2.2655 46 -2.73 0.0089

Solution for Fixed Effects (from mixed model)



Fixed Effects Model with Group as Fixed Effect



Example 3• Research Objective: Compare effects of two 1-1 coaching methods

on 6-month mean weight loss in overweight diabetics• Randomly assign subjects to 8 coaches who have been certified in

both methods (6 subjects per coach) • Randomly assign each coach’s 6 subjects to method A or method B

(3 subjects per method for each coach)• Standard stratified randomized trial, with coaches as strata• Yij = Weight loss for the jth patient assigned to the ith coach• Xij = indicator for assignment of jth pt for the ith coach to method B.

• Model 1: Yij = β0 + β1 Xij + bi + εij, i = 1,2, .., 8; j = 1,2, …, 6– Treatment effect assumed constant for all coaches

• Model 2: Yij = β0 + β1Xij + Xijb1i + (1-Xij)b2i + bi + εij, i = 1,2, .., 8; j = 1,2, …, 6

– Treatment effect assumed to vary between coaches

Analysis Variable : YCoach #

(GROUP) Method N Mean Std Dev1 A 3 -11.19 5.44

B 3 0.55 4.632 A 3 -11.03 3.33

B 3 -14.79 8.093 A 3 -27.43 5.62

B 3 -10.87 10.824 A 3 -19.95 5.33

B 3 -28.31 2.345 A 3 -22.45 3.70

B 3 -11.26 6.376 A 3 -11.31 2.37

B 3 -17.45 3.087 A 3 -8.76 0.46

B 3 -3.07 5.748 A 3 -24.21 5.96

B 3 -13.08 8.52


** Standard Fixed Effect Model for Randomized Block Design;proc mixed data=ydat ; class group; model y= Method group/solution ddfm = kr;


Effect GROUP EstimateStandard

Error DF t Value Pr > |t|Intercept -21.0240 3.0957 39 -6.79 <.0001Method 4.7588 2.0638 39 2.31 0.0265GROUP 1 13.3267 4.1276 39 3.23 0.0025GROUP 2 5.7325 4.1276 39 1.39 0.1728GROUP 3 -0.5060 4.1276 39 -0.12 0.9031GROUP 4 -5.4842 4.1276 39 -1.33 0.1917GROUP 5 1.7910 4.1276 39 0.43 0.6667GROUP 6 4.2672 4.1276 39 1.03 0.3076GROUP 7 12.7287 4.1276 39 3.08 0.0037GROUP 8 0 . . . .

Reference group since Intercept included in model

** Standard Mixed Effect Model for Randomized Block Design;** Without Treatment x Group Interaction; proc mixed data=ydat ; class group; model y= Method/solution ddfm = kr; random group/ solution;



Error DF t Value Pr > |t|Intercept -17.0421 2.5249 10 -6.75 <.0001

Method 4.7588 2.0638 39 2.31 0.0265


GROUP EstimateStd Err

Pred DF t Value Pr > |t|1 7.4710 3.3484 15.1 2.23 0.04132 1.3995 3.3484 15.1 0.42 0.68193 -3.5881 3.3484 15.1 -1.07 0.30084 -7.5681 3.3484 15.1 -2.26 0.03905 -1.7517 3.3484 15.1 -0.52 0.60856 0.2280 3.3484 15.1 0.07 0.94667 6.9929 3.3484 15.1 2.09 0.05418 -3.1836 3.3484 15.1 -0.95 0.3567

Covariance Parameter Estimates

Cov Parm EstimateGROUP 33.9648Residual 51.1100

Estimate, SE, and DF are identical to those of fixed effects model. Hence, making “coach” a random effect does not influence the results

** Mixed Effect Model for Randomized Block Design;** With Treatment x Group Interaction; proc mixed data=ydat ; class group Cmethod; model y= Method/solution ddfm = kr; random group CMethod*group/ solution;

Output truncated



Error DF t Value Pr > |t|Intercept -17.0421 2.8536 12.8 -5.97 <.0001Method 4.7588 3.3660 7 1.41 0.2003


Effect GROUPGroup x Method Estimate

Std Err Pred DF

t Value Pr > |t|

GROUP 1 4.3603 4.8802 3.57 0.89 0.4277GROUP 2 0.8168 4.8802 3.57 0.17 0.8761GROUP 3 -2.0942 4.8802 3.57 -0.43 0.6924GROUP 4 -4.4170 4.8802 3.57 -0.91 0.4223GROUP 5 -1.0223 4.8802 3.57 -0.21 0.8455GROUP 6 0.1331 4.8802 3.57 0.03 0.9797GROUP 7 4.0813 4.8802 3.57 0.84 0.4553GROUP 8 -1.8580 4.8802 3.57 -0.38 0.7249GROUP*Cmethod 1 0 1.1347 5.2238 13.4 0.22 0.8313GROUP*Cmethod 1 1 6.4475 5.2238 13.4 1.23 0.2383GROUP*Cmethod 2 0 3.9507 5.2238 13.4 0.76 0.4625GROUP*CMethod 2 1 -2.5303 5.2238 13.4 -0.48 0.6359

Covariance Parameter EstimatesCov Parm EstimateGROUP 19.8231GROUP*Cmethod 34.4704Residual 32.5490

Example 4: Evaluate effect of ethylene glycol (EG) dose on fetal weight in mice

• EG administered at dose levels 0, 750, 1500, or 3000 mg/kg/day to 94 pregnant mice (dams) beginning just after implantation

• 94 litters included 1028 live fetuses, litter sizes ranges from 1 to 16

• Yij = fetal weight of the jth fetus from the ith litter

• Dose transformed as: Xi = Sqrt(Dose/750)

From Fitzmaurice, on the Web

Example 4: Evaluate effect of ethylene glycol (EG) dose on fetal weight in mice

• Mixed model:Yij = β0 + β1 Xi + bi + εij

• β0 and β1 are fixed effect regression coefficients, as in a standard linear regression of birth weight on X = sqrt(dose/750)

• bi is a random effect to account for clustering by litter, and assumed to vary independently across litters with bi ~ N(0,σb

2).• The εij are random errors, assumed to vary independently

across fetus within and between litters, with εij ~ N(0, σ2).• The amount of clustering or correlation among the fetal

weights within a litter is modeled by variation in the bi.

Model statement specifies fixed effects part of the mixed model: β0 + β1 Xi

Random statement specifies the random effects part of the mixed model: bi

PROC MIXED code

SAS Output:

Intra-class correlation

0.007256/(0.007256 + 0.005565)= 0.57

SAS Output:

The REML estimate of the regression parameter for (transformed) doseindicates that the mean fetal weight decreases with increasing dose.

What happens if we stupidly ignore the litter effect and run a standard regression analysis (PROC REG or PROC GLM)?

Mixed effect result was – 0.134 ± 0.0124

Example 5• Research Objective: Compare effects of two coaching

methods on mean weight loss over a 6 month period in overweight diabetics.

• Randomly assign 48 subjects to 2 different weight loss programs (24 per group)

• Standard 2-group randomized trial• Yij = Weight loss at time j for the ith patient, j = 0, 2, 4,

and 6 months • Nesting of repeated measurements within patients

Example 5

(b0i,b1i) ~ MVN(0,D)Unstructured covariance matrix to allow correlation between random Intercept and slope

• 1-Stage model formulation: Yij = β00 + β01 Xi + β10 tj + β11 Xi tj + b0i + tj b1i + εij

εij are i.i.d. N(0,σ2)

Can be relaxed (Rich will discuss)

• Xi = indicator for assignment to Method B

Illustration of 1st Stage of the 2 stage Model for analogous reaction time vs. days of sleep deprivation study

Analysis Variable : YMethod Time N Mean Std Dev

A 0 24 99.94 6.282 24 97.20 7.844 24 95.39 9.686 24 92.58 11.43

B 0 24 100.87 5.062 24 95.80 6.054 24 93.56 7.336 24 89.72 9.49


** Standard Random Intercept & Slope Model;data ydat; set ydat; timec=time;proc mixed data=ydat; class id; model y= Method timec Method*timec/solution ddfm = kr; random intercept timec/type = un subject=id ;


Cov Parm Subject EstimateStandard

Error Z Value Pr ZUN(1,1) ID 17.7925 6.3732 2.79 0.0026UN(2,1) ID 1.3181 1.3369 0.99 0.3241UN(2,2) ID 1.7033 0.5427 3.14 0.0008Residual 16.6901 2.4090 6.93 <.0001



Error DF t Value Pr > |t|Intercept 99.8594 1.1082 46 90.11 <.0001Method 0.4839 1.5673 46 0.31 0.7589timec -1.1943 0.3252 46 -3.67 0.0006Method*timec -0.5913 0.4599 46 -1.29 0.2049

*** Likelihood ratio test for linear vs. quadratic model; *** Must use method = ml;proc mixed data=ydat method=ml; class id; model y= Method timec Method*timec/solution ddfm = kr; random intercept timec/type = un subject=id ;proc mixed data=ydat method=ml ; class group id; model y= Method timec timec*timec Method*timec Method*timec*timec/ solution ddfm = kr; random intercept timec/type = un subject=id ;

Fit Statistics-2 Log Likelihood 1227.5AIC (smaller is better) 1243.5AICC (smaller is better) 1244.3BIC (smaller is better) 1258.5



Error Pr > |t|Intercept 99.8594 1.0849 <.0001Method 0.4839 1.5343 0.7538timec -1.1943 0.3183 0.0005Method*timec -0.5913 0.4502 0.1952

Fit Statistics-2 Log Likelihood 1227.0AIC (smaller is better) 1247.0AICC (smaller is better) 1248.2BIC (smaller is better) 1265.7



Error Pr > |t|Intercept 99.8417 1.1618 <.0001Method 0.8099 1.6431 0.6238timec -1.1677 0.7002 0.0977timec*timec -0.00443 0.1039 0.9661Method*timec -1.0804 0.9902 0.2772Method*timec*timec 0.08151 0.1470 0.5805

** General Longitudinal Model Estimating Separate Means for Each Visit;proc mixed data=ydat; class id time; model y= time Method*time/solution ddfm = kr noint; repeated time/subject=id type=un; estimate 'Month 2 Treatment Effect' Method*time -1 1 0 0; estimate 'Month 6 Treatment Effect' Method*time -1 0 0 1; estimate 'Mean Fup Treatment Effect' Method*time -3 1 1 1/divisor=3; estimate 'Treatment Effect on Slp per 6 mo' Method*time -3 -1 1 3/divisor=3;


Cov Parm Subject EstimateUN(1,1) ID 32.4980UN(2,1) ID 20.7188UN(2,2) ID 48.9954UN(3,1) ID 24.3278UN(3,2) ID 38.2395UN(3,3) ID 73.7292UN(4,1) ID 22.7135UN(4,2) ID 51.9496UN(4,3) ID 70.8862UN(4,4) ID 110.36

Solution for Fixed EffectsEffect Time Estimate SE DF t Value P|Time 0 99.9395 1.1637 46 85.88 <.0001Time 2 97.1954 1.4288 46 68.03 <.0001Time 4 95.3934 1.7527 46 54.43 <.0001Time 6 92.5784 2.1444 46 43.17 <.0001Method*Time 0 0.9347 1.6456 46 0.57 0.5728Method*Time 2 -1.3993 2.0206 46 -0.69 0.4921Method*Time 4 -1.8331 2.4787 46 -0.74 0.4633Method*Time 6 -2.8629 3.0327 46 -0.94 0.3501

EstimatesLabel Estimate SE DF t Value PMonth 2 Treatment Effect -2.3340 1.8270 46 -1.28 0.2078Month 6 Treatment Effect -3.7976 2.8495 46 -1.33 0.1892Mean Fup Treatment Effect -2.9665 2.0211 46 -1.47 0.1490Treatment Effect on Slp per 6 mo -3.9423 3.0658 46 -1.29 0.2049

Basic Linear Mixed Model Formulation (Laird & Ware 1982) Yij = Xij1β1 + Xij2 β2 + … + Xijp βp

+ Zij1b1i + Zij2b2i + … + Zijqbqi + εij(b1i, b2i, … bqi) ~ MVN with E(bri) = 0, r = 1, 2, … q,

Cov(bri,bsi) = Drs, r=1,2, … q; s=1,2,… q

(εi1, εi2,…, εin )~ MVN with E(εir) = 0, r = 1, 2, … ni, Cov(εir, εis) = Σrs, r=1,2, … ni; s=1,2,…, ni

(εi1, εi2,…, εin ) and (b1i, b2i, … bqi) are independent between different i, and are independent of each other.

i

i

Basic Linear Mixed Model Formulation (Laird & Ware 1982)

Yi = Xi β + Zi bi + εini x 1 ni x p ni x q ni x 1

p x 1 q x 1

bi ~ MVN(0,D)p x p

εi ~ MVN(0,Σi)ni x ni

b1, b2, …. bg, ε1, ε2,…, εg are independent

Yi = response for subject i

Xi, Zi = measured covariates for subject i β = fixed effects

bi = random effects for subject i

εi = residuals for subject i

Marginal Model & Estimation Procedure

Yi = Xi β + Zi bi + εi, bi ~ MVN(0,D), εi ~ MVN(0,Σi)

Yi ~ MVN(Xi β, Zi D Zit + Σi).

The linear mixed model



Optimum Weighting of Data(if the model is valid and data are MAR)

2 4 6 8 10

Years of eGFR Follow-up From 3 Months After Randomization

-60

-40

-20

0

eGF

R S

lop

e (m

l/m

in/1

.73m

2/y

r)

GFR Slope vs. Total Follow-up Time in the AASK Study

Mixed models give more weight tothese patients when computing a group mean slope

Consequences of Missing Data• Because a likelihood based approach is used, results of

correctly specified mixed models remain valid if data are missing at random (so missingness is allowed to depend on covariates included in the model, and nonmissing outcome values)

• However, results may be biased if data are missing not at random (informative missingness).

• The use of differential weighting can exacerbate this problem.• Informative censoring due to termination of follow-up due to

competing risks can be addressed by using joint mixed models incorporating both the longitudinal outcome and the time-to-event outcome defining the competing risk

Example: GFR trajectories in the MDRD Study

Schluchter, Greene, Beck, Stat Med 2001

GFR Slope vs. Total Follow-up Time in the MDRD Study

Open circles indicate pts terminatingfollow-up prior to scheduled EOS.

Estimated Mean GFR Slope by Different Methods

Violations of Normality

• Two types of violations:– Non-normal ԑij

– Non-normal bi

• Two types of inference:– For fixed effects: Central limit theorem type phenomena

protect inferences with non-normal ԑij if either the ni or g are large. If the bi are non-normal need large g.

– For random effects: Results are quite sensitive to deviations from normality – large g does not help.

Two Most Common Misconceptions

• Inclusion of a factor (such as center) as a random effect does NOT control for confounding associated with that factor !!!!– E(Yij) = Xi β ignores the random effect terms

• Inclusion of center as a random effect in an RCT does not extend the inference space for the treatment effect unless a treatment x center random effect is included.

References

• Fitzmaurice G, Laird N, Ware J. Applied Longitudinal Analysis. Wiley, 2004.

• Verbeke G & Molenberghs G. Linear Mixed Models for Longitudinal Data. Springer 2000.

• Littell R, Milliken G, Stroup W, Wolfinger R, Schabenberger O, SAS for Mixed Models 2nd Ed, SAS 2006.

• Singer J, Willett. Applied Longitudinal Data Analysis, Modeling Change and Event Occurrence. Oxford Press, 2003.

Next Time (Nov 3)

Rich Holubkov on Correlation Structures

Documents

Introduction to Linear Mixed Models Tom Greene. Examples of Hierarchical or Clustered Data Design Level Clusters: Observational studies relating health