**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 data** data a1; infile 'c:\...\CH25PR15.TXT'; input mpg driver car; proc print data=a1; run;

**The data** Obs mpg driver car 1 25.3 1 1 2 25.2 1 1 3 28.9 1 2 4 30.0 1 2 5 24.8 1 3 6 25.1 1 3 7 28.4 1 4 8 27.9 1 4 9 27.1 1 5 10 26.6 1 5

**Prepare the data for a plot** 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 data** title1 'Plot of the data'; symbol1 v=circle i=none c=black; proc gplot data=a1; plot mpg*dc/frame; run;

**Find the means** proc means data=a1; output out=a2 mean=avmpg; var mpg; by driver car;

**Plot the means** title1 '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 Revisited** • 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 + i + j + ()ij + εijk • Σi =0 (unknown constants) • j ~ N(0, σ2) • ()ij ~ N(0, σ2) • εij ~ N(0, σ2) • σY2 = σ2 + σ2 + σ2

**Mixed effects model(restricted)** • Yijk = μ + i + j + ()ij + εijk • Σi =0 (unknown constants) • Σ(b)ij =0 for all j • εij ~ N(0, σ2) • σY2 = σ2 + ((a-1)/a)σ2 + σ2

**Parameters** • There are a+3 parameters in this model • a fixed effects means • σ2 • σ2 • 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 + bnΣi2 /(a-1)+ nσ2 • E(MSB) = σ2 + anσ2 + nσ2 • E(MSAB) = σ2 + nσ2 • 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 + bnΣi2 /(a-1)+ nσ2 • E(MSB) = σ2 + anσ2 • E(MSAB) = σ2 + nσ2 • 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 Diff here

**Hypotheses (unrestricted)** • H0A: σ2 = 0; H1A: σ2 ≠ 0 • H0A is tested by F = MSA/MSAB with df a-1 and (a-1)(b-1) • H0B: σ2 = 0; H1B : σ2 ≠ 0 • H0B is tested by F = MSB/MSAB with df b-1 and (a-1)(b-1) • H0AB : σ2 = 0; H1AB : σ2 ≠0 • H0AB is tested by F = MSAB/MSE with df (a-1)(b-1) and ab(n-1)

**Hypotheses (restricted)** • H0A: σ2 = 0; H1A: σ2 ≠ 0 • H0A is tested by F = MSA/MSAB with df a-1 and (a-1)(b-1) • H0B: σ2 = 0; H1B : σ2 ≠ 0 • H0B is tested by F = MSB/MSE with df b-1 and ab(n-1) • H0AB : σ2 = 0; H1AB : σ2 ≠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

**Using Proc Mixed** proc 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

**SAS Output**

**SAS Output**

**SAS Output**

**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, e.g. 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