1. Topic 31: Two-way Random Effects Models

2. Outline
Two-way random effects design
Model
F tests

3. 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

4. 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

5. Read and check the data data a1; infile 'c:\...\CH25PR15.TXT';
input mpg driver car;
proc print data=a1;
run;

6. The data Obs mpg driver car 1 25.3 1 1 2 25.2 1 1 3 28.9 1 2

7. 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';

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

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

11. 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;

13. Random effects model Yijk = μij + εijk the μij are N(μ, σμ2)
the εijk are iid N(0, σ2)
μij and εijk are independent
Dependence among the Yijk can be most easily described by specifying the covariance matrix of the vector (Yijk)’

14. Random factor effects model
Yijk = μ + i + j + ()ij + εijk
i ~ N(0, σ2)
j ~ N(0, σ2)
()ij ~ N(0, σ2)
εij ~ N(0, σ2)
σY2 = σ2 + σ2 + σ2 + σ2

15. Parameters There are five parameters in this model μ σ2 σ2 σ2 σ2
The cell means are random variables, not parameters

16. ANOVA table
The terms and layout of the ANOVA table are the same as what we used for the fixed effects two-way model
The expected mean squares (EMS) are different

17. EMS and parameter estimates
E(MSA) = σ2 + bnσ2 + nσ2
E(MSB) = σ2 + anσ2 + nσ2
E(MSAB) = σ2 + nσ2
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

18. Hypotheses H0A: σ2 = 0; H1A: σ2 ≠ 0 H0B: σ2 = 0; H1B : σ2 ≠ 0
H0AB : σ2 = 0; H1AB : σ2 ≠ 0

19. Hypothesis H0A H0A: σ2 = 0; H1A: σ2 ≠ 0 E(MSA) = σ2 + bnσ2 + nσ 2
E(MSE) = σ2
E(MSAB) = σ2 + nσ 2
So H0A is tested by F = MSA/MSAB with df a-1 and (a-1)(b-1)

20. Hypothesis H0B H0B: σ2 = 0; H1B: σ2 ≠ 0 E(MSB) = σ2 + anσ2 + nσ 2
E(MSAB) = σ2 + nσ 2
So H0B is tested by F = MSB/MSAB with df b-1 and (a-1)(b-1)

21. Hypothesis H0AB H0AB: σ 2 = 0; H1AB: σ2 ≠ 0 E(MSAB) = σ2 + nσ2
E(MSE) = σ2
So H0AB is tested by F = MSAB/MSE with df (a-1)(b-1) and ab(n-1)

22. Run proc glm proc glm data=a1; class driver car; model mpg=driver car
random driver car
driver*car/
test;
run;

23. Model and error output Source DF Sum of Squares Mean Square F Value
Pr > F
Model 19
113.03
<.0001
Error 20
Corrected Total 39

24. Only the interaction test
Factor effects output
All using MSE in denominator
Source DF
Type I SS
Mean Square
F Value
Pr > F
driver 3
531.60
<.0001
car 4
134.73
driver*car 12
1.16
0.3715
Only the interaction test
is valid here

25. Random statement output
Source
Type III Expected Mean Square
driver
Var(Error) + 2 Var(driver*car) + 10 Var(driver)
car
Var(Error) + 2 Var(driver*car) + 8 Var(car)
driver*car
Var(Error) + 2 Var(driver*car)

26. Random/test output Source DF Type III SS Mean Square F Value Pr > F
driver 3
458.26
<.0001
car 4
116.14
Error 12
Error: MS(driver*car)

27. Random/test output Source DF Type III SS Mean Square F Value Pr > F
driver*car 12
1.16
0.3715
Error: MS(Error) 20

28. Proc varcomp proc varcomp data=a1; class driver car;
model mpg=driver car
driver*car;
run;

29. Output MIVQUE(0) Estimates Variance Component mpg Var(driver) 9.32244
Var(car)
Var(driver*car)
Var(Error)

30. Covariance Parameter Estimates
MIXED Output
Fit Statistics
-2 Res Log Likelihood
86.8
AIC (smaller is better)
94.8
AICC (smaller is better)
96.0
BIC (smaller is better)
93.2
Covariance Parameter Estimates
Cov Parm
Estimate
car
2.9343
driver
9.3224
car*driver
Residual
0.1757
DEFAULT: F-tests only supplied for fixed effects

31. Estimated V Correlation Matrix for Subject 1
Row
Col1
Col2
Col3
Col4
Col5
Col6
Col7
Col8
Col9
Col10
Col11
Col12
Col13
Col14
Col15 1
1.0000
0.9859
0.7490
0.2358 2 3 4 5 6

32. Covariances/Correlations

33. Using Proc Mixed proc mixed data=a1 covtest; class car driver;
model mpg=;
random car driver
car*driver/vcorr;
run;

34. Covariance Parameter Estimates
MIXED Output
Covariance Parameter Estimates
Cov Parm
Estimate
Standard Error
Z Value
Pr > Z
car
2.9343
2.0929
1.40
0.0805
driver
9.3224
7.6284
1.22
0.1108
car*driver
0.28
0.3893
Residual
0.1757
3.16
0.0008

35. Last slide Finish reading KNNL Chapter 25
We used program topic31.sas to generate the output for today