1. Experimental Statistics - week 8
Chapter 17:
Models with Random Effects

2. Models with Random Effects
Fixed-Effects Models
-- the models we’ve studied to this point
-- factor levels have been specifically selected
- investigator is interested in testing effects of these specific levels on the response variable
Examples:
-- CAR data
- interested in performance of these 5 gasolines
-- Pilot Plant data
- interested in the specific temperatures (160o and o) and catalysts (C1 and C2)

3. Random-Effect Factor
-- the factor has a large number of possible levels
-- the levels used in the analysis are a random sample from the population of all possible levels
- investigator wants to draw conclusions about the population from which these levels were chosen
(not the specific levels themselves)

4. Fixed Effects vs Random Effects
This determination affects
- the model
- the hypothesis tested
- the conclusions drawn
- the F-tests involved (sometimes)

5. 1-Factor Random Effects Model
Assumptions:

6. Hypotheses: Ho says (considering the variability of the yij’s) :
Ha: sa
Ho says (considering the variability of the yij’s) :
- the component of the variance due to “Factor” has zero variance
-- i.e. no factor “level-to-level” variation
- all of the variability observed is just unexplained subject-to-subject variation
-- at least none is explained by variation due to the factor

7. DATA one;
INPUT operator output;
DATALINES; ;
PROC GLM;
CLASS operator;
MODEL output=operator;
RANDOM operator;
TITLE ‘Operator Data: One Factor Random Effects Model';
RUN;
These are data from an experiment studying the effect of four operators (chosen randomly) on the output of a particular machine.
t =
n =

8. One Factor Random effects Model
The GLM Procedure
Dependent Variable: output
Sum of
Source DF Squares Mean Square F Value Pr > F
Model
Error
Corrected Total
R-Square Coeff Var Root MSE output Mean
Source DF Type I SS Mean Square F Value Pr > F
operator
Source Type III Expected Mean Square
operator Var(Error) + 4 Var(operator)

9. Conclusion:
We reject Ho : sa2 = 0 (p = .0002)
and we conclude that there is
variability due to operator
Note:
Multiple comparisons are not used in random effects analyses
-- we are interested in whether there is variability due to operator
- not interested in which operators performed better, etc. (they were randomly chosen)

10. RECALL: 1-Factor (Fixed-Effects) ANOVA Table
(page 389)
Rationale for F-test and critical region:
estimates
estimates
+ constant ×
- if no factor effects, we expect F ≈ 1;
- if factor effects, we expect F > 1

11. Expected Mean Squares for 1-Factor ANOVA’s (p.979)
  EMS
Source SS df MS Fixed Effects Random Effects
Treatments SST t MST
Error SSE t(n - 1) MSE
  Total TSS tn - 1
Rationale for Test Statistic and Critical Region is the Same: Fixed or Random

12. DATA one;
INPUT operator output;
DATALINES; ;
PROC GLM;
CLASS operator;
MODEL output=operator;
RANDOM operator;
TITLE ‘Operator Data: One Factor Random Effects Model';
RUN;
These are data from an experiment studying the effect of four operators (chosen randomly) on the output of a particular machine.

13. One Factor Random effects Model
The GLM Procedure
Dependent Variable: output
Sum of
Source DF Squares Mean Square F Value Pr > F
Model
Error
Corrected Total
R-Square Coeff Var Root MSE output Mean
Source DF Type I SS Mean Square F Value Pr > F
operator
Source Type III Expected Mean Square
operator Var(Error) + 4 Var(operator)

14. Estimating Variance Components
Solving for sa2 we get:
so, we estimate sa2 by
Also,

15. For OPERATOR Data,

16. RECALL: 2-Factor Fixed-Effects Model
where

17. Expected Mean Squares for 2-Factor ANOVA with Fixed Effects:
Expected MS
F-test A
MSA/MSE B
MSB/MSE AB
MSAB/MSE
Error

18. 2-Factor Random Effects Model
Assumptions:
Sum-of-Squares obtained as in Fixed-Effects case

19. Expected Mean Squares for 2-Factor ANOVA with Random Effects:
Expected MS A B AB
Error

20. To Test:
we use F =
we use F =
we use F =
Note: Test each of these 3 hypotheses (no matter whether Ho:sab2 = 0 is rejected)

21. 2-Factor Random Effects ANOVA Table
Source SS df MS F
Main Effects
A SSA a - 1
B SSB b- 1
Interaction
AB SSAB (a - 1)(b- 1)
Error SSE ab(n - 1)
  Total TSS abn - 1

22. Estimating Variance Components 2-Factor Random Effects Model
(note error on page 986)

23. Filtration Process: Response - % material lost through filtration
DATA one;
INPUT operator filter loss;
DATALINES; . ;
PROC GLM;
CLASS operator filter;
MODEL loss=operator filter operator*filter;
TITLE ‘2-Factor Random Effects Model';
RANDOM operator filter operator*filter/test;
RUN;
Filtration Process:
Response - % material lost through filtration
A – Operator (randomly selected) (a = )
B – Filter (randomly selected) (b = )
n =
Operator
Filter

24. SAS Random-Effects Output
(Filtration Data)
2-Factor Random Effects Model
General Linear Models Procedure
Dependent Variable: LOSS
Sum of Mean
Source DF Squares Square F Value Pr > F
Model
Error
Corrected Total
R-Square C.V Root MSE LOSS Mean
Source DF Type III SS Mean Square F Value Pr > F
OPERATOR
FILTER
OPERATOR*FILTER
Source Type III Expected Mean Square
OPERATOR Var(Error) + 3 Var(OPERATOR*FILTER) + 9 Var(OPERATOR)
FILTER Var(Error) + 3 Var(OPERATOR*FILTER) + 12 Var(FILTER)
OPERATOR*FILTER Var(Error) + 3 Var(OPERATOR*FILTER)

25. SAS Random-Effects Output – continued
“../test” option
Tests of Hypotheses for Random Model Analysis of Variance
Dependent Variable: LOSS
Source: OPERATOR
Error: MS(OPERATOR*FILTER)
Denominator Denominator
DF Type III MS DF MS F Value Pr > F
Source: FILTER
Source: OPERATOR*FILTER
Error: MS(Error)

26. Filtration Problem Results and Conclusions

27. Variable 1: Active Ingredient (in mg/mL) at End of Storage Period
Table Factor ANOVA - Ex 15.41, page mg/mL Data
The GLM Procedure
Dependent Variable: mgml
Sum of
Source DF Squares Mean Square F Value Pr > F
Model <.0001
Error
Corrected Total
R-Square Coeff Var Root MSE mgml Mean
Source DF Type III SS Mean Square F Value Pr > F
time <.0001
lab <.0001
time*lab

28. Table 3. Calculations for LSD comparisons of mg/mL Cell Means
T3L1 T1L1 T1L2 T6L1 T3L2 T9L1 T6L2 T9L2
Comparison Actual Difference (lsd = .086)
T3L1 vs T9L
T3L1 vs T6L
T3L1 vs T9L
T3L1 vs T3L
T3l1 vs T6L
T3L1 vs T1L
T3L1 vs T1L X
T1L1 vs T9L
T1L1 vs T6L
T1L1 vs T9L
T1L1 vs T3L
T1L1 vs T6L X
T1L2 vs T9L
T1L2 vs T6L
T1L2 vs T9L
T1L2 vs T3L
T6L1 vs T9L
T6L1 vs T6L
T6L1 vs T9L
T6L1 vs T3L
T3L2 vs T9L
T3L2 vs T6L
T3L2 vs T9L
T9L1 vs T9L X

29. T3L1 T1L1 T1L2 T6L1 T3L2 T9L1 T6L2 T9L2

30. 2-Factor Mixed Effects Model
random
fixed
Assumptions:
Sum-of-Squares obtained as before

31. Expected Mean Squares for 2-Factor ANOVA with Mixed Effects:
Expected MS A
(fixed)
(random) B AB
Error

33. Expected Mean Squares for 2-Factor ANOVA with Mixed Effects:
SAS Expected MS
Book’s Expected MS A
(fixed)
(random) B AB
Error

34. Mixed-Effects Model To Test:
use F =
SAS uses F =
use F =
Again: Test each of these 3 hypotheses as in random-effects case.

35. 2-Factor Mixed-Effects ANOVA Table (using SAS Expected MS)
Source SS df MS F
Main Effects
A SSA a - 1
B SSB b- 1
Interaction
AB SSAB (a - 1)(b- 1)
Error SSE ab(n - 1)
  Total TSS abn - 1

36. Estimating Variance Components 2-Factor Mixed-Effects Model
(based on SAS Expected MS)
Note: A is a fixed effect

37. Response – fatigue of mechanical part
(F)ull
Military
Inspect.
(R)educed
Military
Inspect.
Product Inspection
(C)ommercial
Response – fatigue of mechanical part
A – type of inspection (a = )
B – inspector (randomly selected) (b = )
n =
Inspector

38. Mixed-Effects Data DATA one; INPUT insp$ level$ fatigue; DATALINES;.
2 C 5.68
3 C 6.21
3 C 5.66
3 C 5.36
3 C 5.90
3 C 6.12 ;
PROC GLM;
CLASS insp level;
MODEL fatigue= level insp level*insp;
TITLE 'Mixed-Effects Model';
RANDOM insp level*insp/test;
RUN;
PROC MEANS mean var;
CLASS level;
VAR fatigue;

39. SAS Mixed-Effects Output
Mixed-Effects Model
The GLM Procedure
Dependent Variable: fatigue
Sum of
Source DF Squares Mean Square F Value Pr > F
Model
Error
Corrected Total
R-Square Coeff Var Root MSE fatigue Mean
Source DF Type III SS Mean Square F Value Pr > F
level
insp
insp*level

40. SAS Mixed-Effects Output - Continued
Mixed-Effects Model
The GLM Procedure
Source Type III Expected Mean Square
level Var(Error) + 5 Var(insp*level) + Q(level)
insp Var(Error) + 5 Var(insp*level) + 15 Var(insp)
insp*level Var(Error) + 5 Var(insp*level)
Tests of Hypotheses for Mixed Model Analysis of Variance
Dependent Variable: fatigue
Source DF Type III SS Mean Square F Value Pr > F
level
insp
Error
Error: MS(insp*level)
insp*level
Error: MS(Error)

41. Multiple Comparisons for Fixed Effect (Inspection Level)
-- Use MSAB in place of MSE
where ▪ N denotes the # of observations involved in the computation of a marginal mean
▪ v denotes the df associated with AB interaction

42. SAS Mixed-Effects Output –
Output from PROC Means
The MEANS Procedure
Analysis Variable : fatigue N
level Obs Mean Variance
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43. Mixed-Effects Example Results and Conclusions: