Topic 32: Two-Way Mixed Effects Model

Outline

• Two-way mixed models • Three-way mixed models

Data for two-way design• Y is the response variable• Factor A with levels i = 1 to a• Factor B with levels j = 1 to b• Yijk is the kth observation in cell (i, j)

k = 1 to nij

• Have balanced designs with n = nij

Two-way mixed model• Two-way mixed model has

– One fixed effect– One random effect

• Tests: – Again use EMS as guide– Two possible models

• Unrestricted mixed model (SAS)• Restricted mixed model (Text)

KNNL Example• KNNL Problem 25.15, p 1080• Y is fuel efficiency in miles per gallon • Factor A represents four different

drivers, a=4 levels• Factor B represents five different cars

of the same model , b=5 • Each driver drove each car twice over

the same 40-mile test course

Read and check the datadata a1; infile 'c:\...\CH25PR15.TXT';input mpg driver car;

proc print data=a1; run;

The dataObs mpg driver car1 25.3 1 12 25.2 1 13 28.9 1 24 30.0 1 25 24.8 1 36 25.1 1 37 28.4 1 48 27.9 1 49 27.1 1 510 26.6 1 5

Prepare the data for a scatterplot

data a1; set a1;if (driver eq 1)*(car eq 1) then dc='01_1A';

if (driver eq 1)*(car eq 2) then dc='02_1B';

⋮if (driver eq 4)*(car eq 5)then dc='20_4E';

Plot the datatitle1 'Plot of the data';symbol1 v=circle i=none c=black;proc gplot data=a1;

plot mpg*dc/frame;run;

Find the meansproc means data=a1;

output out=a2 mean=avmpg;var mpg;by driver car;

Plot the meanstitle1 'Plot of the means';symbol1 v='A' i=join c=black;symbol2 v='B' i=join c=black;symbol3 v='C' i=join c=black;symbol4 v='D' i=join c=black;symbol5 v='E' i=join c=black;proc gplot data=a2;

plot avmpg*driver=car/frame;run;

Example Revision• Suppose that the four drivers were not

randomly selected and there is interest in comparing the four drivers in the study

• Driver (A) is now a fixed effect• Still consider Car (B) to be a random

effect

Mixed effects model(unrestricted)

• Yijk = μ + αi + βj + (αβ)ij + εijk

• αi are unknown constants• βj ~ N(0, σβ2)• (αβ)ij ~ N(0, σαβ2)• εij ~ N(0, σ2)• σY

2 = σβ2 + σαβ2 + σ2

• Σαi =0 (common restriction)

Mixed effects model(restricted)

• Yijk = μ + αi + βj + (αβ)ij + εijk

•• εij ~ N(0, σ2)• σY

2 = σβ2 + ((a-1)/a)σαβ2 + σ2

• Common restrictions on “fixed” effects – Σαi =0– Σ(αb)ij =0 for all j

2 2j ij

1~ (0, ) and ( ) ~ (0, )aN Naβ αββ σ αβ σ−

Parameters• There are a+3 parameters in this

model– a fixed effects means– σβ

2

– σαβ2

– σ2

ANOVA table• The terms and layout of the ANOVA

table are the same as what we used for the fixed effects model

• The expected mean squares (EMS) are different and vary based on the choice of unrestricted or restricted mixed model

EMS (unrestricted)• E(MSA) = σ2 + bnΣαi

2 /(a-1)+ nσαβ2

• E(MSB) = σ2 + anσβ2 + nσαβ2

• E(MSAB) = σ2 + nσαβ2

• E(MSE) = σ2

• Estimates of the variance components can be obtained from these equations, replacing E(MS) with table value, or other methods such as ML

EMS (restricted)• E(MSA) = σ2 + bnΣαi

2 /(a-1)+ nσαβ2

• E(MSB) = σ2 + anσβ2

• E(MSAB) = σ2 + nσαβ2

• E(MSE) = σ2

• Estimates of the variance components can be obtained from these equations, replacing E(MS) with table value, or other methods such as ML

Different here

Hypotheses (unrestricted)• H0A: σα2 = 0; H1A: σα2 ≠ 0

– H0A is tested by F = MSA/MSAB with dfa-1 and (a-1)(b-1)

• H0B: σβ2 = 0; H1B : σβ2 ≠ 0– H0B is tested by F = MSB/MSAB with df

b-1 and (a-1)(b-1)• H0AB : σαβ2 = 0; H1AB : σαβ2 ≠ 0

– H0AB is tested by F = MSAB/MSE with df(a-1)(b-1) and ab(n-1)

Hypotheses (restricted)• H0A: σα2 = 0; H1A: σα2 ≠ 0

– H0A is tested by F = MSA/MSAB with dfa-1 and (a-1)(b-1)

• H0B: σβ2 = 0; H1B : σβ2 ≠ 0– H0B is tested by F = MSB/MSE with df

b-1 and ab(n-1)• H0AB : σαβ2 = 0; H1AB : σαβ2 ≠ 0

– H0AB is tested by F = MSAB/MSE with df(a-1)(b-1) and ab(n-1)

Comparison of Means• To compare fixed levels of A, std

error is

• Degrees of freedom for t tests and CIs are then (a-1)(b-1)

• This is true for both unrestricted and restricted mixed models

2MSAB / bn

Using Proc Mixedproc mixed data=a1;

class car driver;model mpg=driver;random car car*driver / vcorr;lsmeans driver / adjust=tukey;

run;SAS considers unrestricted model only…results in slightly different variance estimates. Not crucial if goal is to compare fixed effect means

SAS OutputCovariance Parameter

EstimatesCov Parm Estimatecar 2.9343car*driver 0.01406Residual 0.1757

Type 3 Tests of Fixed Effects

EffectNum

DFDen DF F Value Pr > F

driver 3 12 458.26 <.0001

SAS OutputLeast Squares Means

Effect driver EstimateStandard

Error DF t Value Pr > |t|driver 1 26.9300 0.7793 12 34.56 <.0001driver 2 34.1500 0.7793 12 43.82 <.0001driver 3 28.8500 0.7793 12 37.02 <.0001driver 4 30.2600 0.7793 12 38.83 <.0001

Proc GLM provides wrong SE here

SAS OutputDifferences of Least Squares Means

Effect driver _driver EstiateStandard

Error DF t Value Pr > |t| Adjustment Adj Pdriver 1 2 -7.2200 0.2019 12 -35.76 <.0001 Tukey-

Kramer<.0001

driver 1 3 -1.9200 0.2019 12 -9.51 <.0001 Tukey-Kramer

<.0001

driver 1 4 -3.3300 0.2019 12 -16.49 <.0001 Tukey-Kramer

<.0001

driver 2 3 5.3000 0.2019 12 26.25 <.0001 Tukey-Kramer

<.0001

driver 2 4 3.8900 0.2019 12 19.26 <.0001 Tukey-Kramer

<.0001

driver 3 4 -1.4100 0.2019 12 -6.98 <.0001 Tukey-Kramer

<.0001

Three-way models• We can have zero, one, two, or three

random effects• EMS indicate how to do tests• In some cases the situation is

complicated and we need approximations of an F test– Example: when all are random, use

MS(AB)+MS(AC)-MS(ABC) to test A

Last slide• Finish reading KNNL Chapter 25• We used program topic32.sas to

generate the output for today