Experimental%20Statistics%20%20%20%20%20%20-%20week%208

1
Experimental Statistics - week 8
Chapter 17 Models with Random Effects
2
Models with Random Effects
Fixed-Effects Models -- the models weve
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
180o) 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 sa2 0 Ha sa2 0
Ho says (considering the variability of the
yijs) - 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 1
175.4 1 171.7 1 173.0 1 170.5 2 168.5 2 162.7 2
165.0 2 164.1 3 170.1 3 173.4 3 175.7 3 170.7 4
175.2 4 175.7 4 180.1 4 183.7 PROC GLM CLASS
operator MODEL outputoperator 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
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One Factor Random effects Model
The GLM
Procedure Dependent Variable output
Sum of
Source DF Squares Mean
Square F Value Pr gt F Model
3 371.8718750 123.9572917 14.91
0.0002 Error 12
99.7925000 8.3160417 Corrected Total
15 471.6643750 R-Square
Coeff Var Root MSE output Mean
0.788425 1.674472 2.883755
172.2188 Source DF Type I
SS Mean Square F Value Pr gt F
operator 3 371.8718750
123.9572917 14.91 0.0002
The GLM Procedure
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 gt 1
11
Expected Mean Squares for 1-Factor ANOVAs
(p.979)  
EMS Source SS
df MS Fixed Effects Random
Effects   Treatments SST
t - 1 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 1
175.4 1 171.7 1 173.0 1 170.5 2 168.5 2 162.7 2
165.0 2 164.1 3 170.1 3 173.4 3 175.7 3 170.7 4
175.2 4 175.7 4 180.1 4 183.7 PROC GLM CLASS
operator MODEL outputoperator 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 gt F Model
3 371.8718750 123.9572917 14.91
0.0002 Error 12
99.7925000 8.3160417 Corrected Total
15 471.6643750 R-Square
Coeff Var Root MSE output Mean
0.788425 1.674472 2.883755
172.2188 Source DF Type I
SS Mean Square F Value Pr gt F
operator 3 371.8718750
123.9572917 14.91 0.0002
The GLM Procedure
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 Hosab2 0 is rejected)
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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
DATA one INPUT operator filter
loss DATALINES 1 1 16.2 1 1 16.8 1 1
17.1 1 2 16.6 1 2 16.9 1 2 16.8 . .
. 4 1 14.9 4 2 15.4 4 2 14.6 4 2 15.9 4
3 16.1 4 3 15.4 4 3 15.6 PROC GLM CLASS
operator filter MODEL lossoperator filter
operatorfilter TITLE 2-Factor Random Effects
Model' RANDOM operator filter
operatorfilter/test RUN
Filtration Process Response - material lost
through filtration A Operator (randomly
selected) (a ) B Filter (randomly
selected) (b )
n
Operator
1 2 3 4 16.2 15.9 15.6
14.9 1 16.8 15.1 15.9 15.2 17.1 14.5
16.1 14.9 16.6 16.0 16.1 15.4 2 16.9
16.3 16.0 14.6 16.8 16.5 17.2 15.9
16.7 16.5 16.4 16.1 3 16.9 16.9 17.4
15.4 17.1 16.8 16.9 15.6
Filter
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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 gt F Model
11 16.60888889 1.50989899
8.16 0.0001 Error 24
4.44000000 0.18500000 Corrected Total
35 21.04888889 R-Square
C.V. Root MSE LOSS
Mean 0.789062 2.664175
0.4301163 16.144444 Source
DF Type III SS Mean Square
F Value Pr gt F OPERATOR 3
10.31777778 3.43925926 18.59
0.0001 FILTER 2
4.63388889 2.31694444 12.52
0.0002 OPERATORFILTER 6
1.65722222 0.27620370 1.49 0.2229
Source Type III Expected Mean
Square OPERATOR Var(Error) 3
Var(OPERATORFILTER) 9 Var(OPERATOR) FILTER
Var(Error) 3 Var(OPERATORFILTER) 12
Var(FILTER) OPERATORFILTER Var(Error) 3
Var(OPERATORFILTER)
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SAS Random-Effects Output continued ../test
option
Tests of Hypotheses for Random Model Analysis of
Variance Dependent Variable LOSS Source
OPERATOR Error MS(OPERATORFILTER)
Denominator Denominator
DF Type III MS DF MS
F Value Pr gt F 3 3.4392592593
6 0.2762037037 12.4519
0.0055 Source FILTER Error MS(OPERATORFILTER)
Denominator
Denominator DF Type III MS
DF MS F Value Pr gt F
2 2.3169444444 6 0.2762037037
8.3885 0.0183 Source OPERATORFILTER Erro
r MS(Error)
Denominator Denominator DF Type III
MS DF MS F Value
Pr gt F 6 0.2762037037 24
0.185 1.4930 0.2229
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Filtration Problem Results and Conclusions
27
Variable 1 Active Ingredient (in mg/mL) at End
of Storage Period
Table 1. 2-Factor ANOVA - Ex
15.41, page 935 -- mg/mL Data
The GLM
Procedure Dependent Variable mgml
Sum of
Source DF Squares
Mean Square F Value Pr gt F Model
7 0.46740000
0.06677143 27.30 lt.0001 Error
16 0.03913333
0.00244583 Corrected Total 23
0.50653333 R-Square
Coeff Var Root MSE mgml Mean
0.922743 0.165090 0.049455
29.95667 Source
DF Type III SS Mean Square F Value
Pr gt F time 3
0.29376667 0.09792222 40.04 lt.0001
lab 1
0.09126667 0.09126667 37.32 lt.0001
timelab 3
0.08236667 0.02745556 11.23 0.0003
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Table 3. Calculations for LSD comparisons of
mg/mL Cell Means T3L1 T1L1 T1L2 T6L1
T3L2 T9L1 T6L2 T9L2 30.17 30.09
30.08 30.01 29.90 29.81 29.80
29.80 Comparison Actual Difference
(lsd .086) T3L1 vs T9L2 .37 T3L1 vs
T6L2 .37 T3L1 vs T9L1 .36
T3L1 vs T3L2 .27 T3l1 vs T6L1
.16 T3L1 vs T1L2 .09 T3L1 vs T1L1
.38 X T1L1 vs T9L2 .29 T1L1 vs
T6L2 .29 T1L1 vs T9L2 .28
T1L1 vs T3L2 .19 T1L1 vs T6L1
.08 X T1L2 vs T9L2 .28 T1L2 vs T6L2
.28 T1L2 vs T9L2 .27 T1L2 vs
T3L2 .18 T6L1 vs T9L2 .21
T6L1 vs T6L2 .21 T6L1 vs T9L2
.20 T6L1 vs T3L2 .17 T3L2 vs T9L2
.10 T3L2 vs T6L2 .10 T3L2 vs
T9L2 .09 T9L1 vs T9L2 .01 X
T3L1 T1L1 T1L2 T6L1 T3L2 T9L1
T6L2 T9L2 30.17 30.09 30.08
30.01 29.90 29.81 29.80 29.80
------------- ----------------------
------- ------------------------
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T3L1 T1L1 T1L2 T6L1 T3L2
T9L1 T6L2 T9L2 30.17 30.09
30.08 30.01 29.90 29.81 29.80 29.80
------------- -------------------
--- ------- ------------------------
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2-Factor Mixed Effects Model
random
fixed
Assumptions
Sum-of-Squares obtained as before
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Expected Mean Squares for 2-Factor ANOVA with
Mixed Effects
Expected MS
A
(fixed)
B
(random)
AB
Error
32
(No Transcript)
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Expected Mean Squares for 2-Factor ANOVA with
Mixed Effects
SAS Expected MS
Books Expected MS
A
(fixed)
B
(random)
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.
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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
(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 )
7.50 7.08 6.15 7.42
6.17 5.52 1 5.85 5.65
5.48 5.89 5.30 5.48 5.35
5.02 5.98 7.58 7.68
6.17 6.52 5.86 6.20 2 6.54
5.28 5.44 5.64 5.38
5.75 5.12 4.87 5.68
7.70 7.19 6.21 6.82
6.19 5.66 3 6.42 5.85
5.36 5.39 5.35 5.90 5.35
5.01 6.12
n
Inspector
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Mixed-Effects Data
DATA one INPUT insp level fatigue DATALINES 1
F 7.50 1 F 7.42 1 F 5.85 1 F
5.89 . . . 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
levelinsp TITLE 'Mixed-Effects Model'
RANDOM insp levelinsp/test RUN PROC MEANS
mean var CLASS level VAR fatigue RUN
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SAS Mixed-Effects Output

Mixed-Effects Model
The GLM
Procedure Dependent Variable fatigue
Sum of
Source DF Squares
Mean Square F Value Pr gt F Model
8 2.70711111
0.33838889 0.53 0.8282 Error
36 23.11448000
0.64206889 Corrected Total 44
25.82159111 R-Square
Coeff Var Root MSE fatigue Mean
0.104839 13.35141 0.801292
6.001556 Source
DF Type III SS Mean Square F Value
Pr gt F level 2
2.58739111 1.29369556 2.01 0.1481
insp 2
0.02523111 0.01261556 0.02 0.9806
insplevel 4
0.09448889 0.02362222 0.04 0.9973
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SAS Mixed-Effects Output - Continued

Mixed-Effects Model
The GLM Procedure Source
Type III Expected Mean Square
level Var(Error) 5
Var(insplevel) Q(level) insp
Var(Error) 5 Var(insplevel) 15
Var(insp) insplevel
Var(Error) 5 Var(insplevel)
Mixed-Effects Model
The GLM
Procedure Tests of Hypotheses
for Mixed Model Analysis of Variance
Dependent Variable fatigue Source
DF Type III SS Mean Square F
Value Pr gt F level 2
2.587391 1.293696 54.77
0.0012 insp 2
0.025231 0.012616 0.53 0.6229
Error 4 0.094489
0.023622 Error MS(insplevel)
Source DF Type III SS
Mean Square F Value Pr gt F insplevel
4 0.094489 0.023622
0.04 0.9973 Error MS(Error)
36 23.114480 0.642069
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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
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SAS Mixed-Effects Output Output from PROC
Means
The MEANS Procedure
Analysis Variable fatigue N
level Obs Mean Variance
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C 15 5.8066667
0.0981810 F 15 6.3393333
0.8208638 R 15 5.8586667
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43
Mixed-Effects Example Results and Conclusions