Statistical Analysis

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Statistical Analysis

• Professor Lynne Stokes
• Department of Statistical Science
• Lecture 19
• Analysis of Designs with Random Factor Levels

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Fermentation Process ExperimentMGH Ex 10.17
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Fermentation Process Experiment
Proc GLM dataFerment class batch process
model response batch process random batch /
test lsmeans process / stderr pdiff run
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Fermentation Process Experiment
The GLM Procedure Dependent Variable Response
Sum of Source
DF Squares Mean
Square F Value Pr gt F Model
7 389.0000000 55.5714286 1.73
0.1934 Error 12
386.0000000 32.1666667 Corrected Total
19 775.0000000 R-Square
Coeff Var Root MSE Response Mean
0.501935 6.958977 5.671567
81.50000 Source DF Type
I SS Mean Square F Value Pr gt F Batch
4 324.0000000 81.0000000
2.52 0.0965 Process 3
65.0000000 21.6666667 0.67
0.5846 Source DF Type III
SS Mean Square F Value Pr gt F Batch
4 324.0000000 81.0000000
2.52 0.0965 Process 3
65.0000000 21.6666667 0.67 0.5846
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Fermentation Process Experiment
Mason, Gunst, Hess Exercise 10.17
The GLM Procedure Source
Type III Expected Mean Square
Batch Var(Error) 4
Var(Batch) Process Var(Error)
Q(Process)
The Random Statement Produces This Output
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Fermentation Process Experiment
Mason, Gunst, Hess Exercise 10.17
7 The GLM
Procedure Tests of Hypotheses for
Mixed Model Analysis of Variance Dependent
Variable Response Source DF
Type III SS Mean Square F Value Pr gt
F Batch 4 324.000000
81.000000 2.52 0.0965 Process
3 65.000000 21.666667
0.67 0.5846 Error MS(Error) 12
386.000000 32.166667
The Random Statement Produces This Output
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Fermentation Process Experiment
Mason, Gunst, Hess Exercise 10.17
8 Least
Squares Means Response
Standard LSMEAN
Process LSMEAN Error Pr gt
t Number F1 79.0000000
2.5364017 lt.0001 1 F2
82.0000000 2.5364017 lt.0001
2 F3 84.0000000
2.5364017 lt.0001 3 F4
81.0000000 2.5364017 lt.0001
4 Least Squares Means
for effect Process Pr gt t
for H0 LSMean(i)LSMean(j)
Dependent Variable Response i/j
1 2 3
4 1 0.4193
0.1886 0.5874 2
0.4193 0.5874 0.7852
3 0.1886 0.5874
0.4193 4 0.5874
0.7852 0.4193
LSMEANS Standard Errors Only use the Fixed
Effects Computing Formulas These are
Incorrect (See Proc Mixed Results)
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Estimation of Variance Components Method of
Moments
Equate mean squares to their expected mean
squares and solve
Random Main Effects Model
Method of Moments
F Test MSA / MSE
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Fermentation Process Experiment
Mason, Gunst, Hess Exercise 10.17
The GLM Procedure Source
Type III Expected Mean Square
Batch Var(Error) 4
Var(Batch) Process Var(Error)
Q(Process)
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Estimation of Variance Components Method of
Moments
Equate mean squares to their expected mean
squares and solve
Three-Factor Random Effects Model
Method of Moments
F Test MSABC / MSE
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Estimation of Variance Components
Three-Factor Random Effects Model
F Test MSAB / MSABC
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Estimation of Variance Components
Three-Factor Random Effects Model
F Test No Exact Test
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Estimation of Variance Components
Confidence Intervals
b
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Estimation of Variance Components
Confidence Intervals
b
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Estimation of Variance Components
Confidence Intervals
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Testing Variance Components
Three-Factor Random Effects Model
F Test No Exact Test
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Satterthwaites Approximate F Statistic
Assumptions
MS1, MS2, ... , MSk are Pairwise Independent
ANOVA Mean Squares
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Satterthwaites Approximate F Statistic
Approximation
L wL c2(nL)
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Satterthwaites Approximate F Statistic
Solution
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Satterthwaites Approximate F Statistic
Application
Under Ho
Select to be Independent of L
Regardless of Ho
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Satterthwaites Approximate F Statistic
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Satterthwaites Approximate F Statistic
Approximation 1
M1 F Can Be Negative
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Satterthwaites Approximate F Statistic
Approximation 2
M2 F Are Positive
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Random Effects Testing
Three-Factor Random Effects Model
Source Mean Square Expected Mean
Square A MSA se2 rsabc2 crsab2
brsag2 bcrsa2 AB MSAB se2 rsabc2 crsab2
ABC MSABC se rsabc2 Error MSE se2

• Effects Not Necessarily Tested Against Error
• Test Main Effects Even if Interactions are
  Significant
• May Not be an Exact Test (Mixed Effects Models)

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Random Effects Testing
Three-Factor Random Effects Model
Source Mean Square Expected Mean
Square A MSA se2 rsabc2 crsab2
brsag2 bcrsa2 AB MSAB se2 rsabc2 crsab2
ABC MSABC se rsabc2 Error MSE se2
Proc GLM Random ... / Test Produces
Satterthwaite Approximate Test Statistics Fixed
Effects Standard Errors May be Incorrect
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Restricted Maximum Likelihood
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Proc Mixed
Proc Mixed dataFerment Cl class batch
process model response process random
batch lsmeans process / adjusttukey
pdiff run
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Fermentation Process Experiment
Mason, Gunst, Hess Exercise 10.17
8 The Mixed
Procedure Covariance
Parameter Estimates Cov Parm
Estimate Alpha Lower Upper
Batch 12.2083 0.05 2.8125
2023.05 Residual 32.1667
0.05 16.5405 87.6518
Balanced Design Same as Method of Moments
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Fermentation Process Experiment
Type 3 Tests of Fixed Effects
Num Den Effect
DF DF F Value Pr gt F
Process 3 12 0.67
0.5846 Least Squares
Means
Standard Effect Process Estimate
Error DF t Value Pr gt t Process
F1 79.0000 2.9791 12
26.52 lt.0001 Process F2
82.0000 2.9791 12 27.53
lt.0001 Process F3 84.0000
2.9791 12 28.20 lt.0001 Process
F4 81.0000 2.9791 12
27.19 lt.0001
Correct Standard Errors
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Fermentation Process Experiment
Differences of Least Squares Means
Standard Effect
Process _Process Estimate Error DF
t Value Pr gt t Process F1 F2
-3.0000 3.5870 12 -0.84
0.4193 Process F1 F3 -5.0000
3.5870 12 -1.39 0.1886
Differences of Least Squares Means
Effect Process _Process Adjustment
Adj P Process F1 F2
Tukey-Kramer 0.8363 Process
F1 F3 Tukey-Kramer 0.5261
Other Pairwise Comparisons on the Next Output Page
Note Standard Error of a Difference is Smaller
than Random Effects Cancel in yi1
yi2 (Pairwise Balance Needed)
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Randomized Complete Block Designs
Factorial Structure with Main Effect for Blocks
Nothing New
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Latin Square Designs

• Control Two Sources of Variability
• Restrictions
• Factor of Interest and Two blocking Factors Each
  at k Levels
• No Interactions Among the Experimental and
  Blocking Factors

Experiment Size Latin Square n k2 Complete
Factorial n k3 r
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Analysis of Latin Square Designs
ith Row Block Effect
Error Variation From All Sources Except Blocks
Factor Main Effects
jth Column Block Effect
kth Factor Level Effect
Main Effects Analysis of Variance Model
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Balanced Incomplete Block Designs
Used when blocks contain fewer experimental
units than the number of unique factor-level
combinations

• b blocks
• f factor-level combinations
• k lt f experimental units per block

No interactions with the design factor(s)
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Asphalt-Pavement Rating Study
Purpose Assess the Deterioration of Highway
Pavement Response Rating 0 No pavement
remaining 100 excellent condition Design
Factor 16 District Engineers
(Random) Blocking Factor 16 Road Segments
(Random)
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Asphalt-Pavement Rating Study

• b 16 Road Segments (Blocks)
• f 16 Engineers (Factor-Level Combinations)
• k 6 Engineers/Road Segment

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Asphalt-Pavement Rating Study
Design Engineer
R o a d S e g m e n t
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Analysis of Variance with Unbalanced Data
Error Sums of Squares
Models 1 2 are Hierarchical Model 2 has a
Subset of Model 1 Terms
SSE2 SSE1
Reduction in Error Sums of Squares
R(M1 M2) SSE2 - SSE1
df n2 - n1
Testing Effects
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Balanced Incomplete Block Design
Model 1
Model 2
Block Effect R(M1 M2) SSE2 - SSE1
Model 3
Factor Effect R(M1 M3) SSE3 - SSE1
SAS PROC GLM Type I Sums of Squares Two Model Fits
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Asphalt-Pavement Rating Study
Sum
of Source DF Squares
Mean Square F Value Pr gt F Model
30 13422.12500 447.40417
7.10 lt.0001 Error 65
4098.83333 63.05897 Corrected Total
95 17520.95833 R-Square
Coeff Var Root MSE Rating Mean
0.766061 12.93405 7.940968
61.39583 Source DF
Type I SS Mean Square F Value Pr gt F Road
15 11786.95833
785.79722 12.46 lt.0001 Engineer
15 1635.16667 109.01111 1.73
0.0668 Source DF Type
III SS Mean Square F Value Pr gt F Road
15 11005.16667
733.67778 11.63 lt.0001 Engineer
15 1635.16667 109.01111 1.73
0.0668
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Asphalt-Pavement Rating Study
Sum
of Source DF Squares
Mean Square F Value Pr gt F Model
30 13422.12500 447.40417
7.10 lt.0001 Error 65
4098.83333 63.05897 Corrected Total
95 17520.95833 R-Square
Coeff Var Root MSE Rating Mean
0.766061 12.93405 7.940968
61.39583 Source DF
Type I SS Mean Square F Value Pr gt
F Engineer 15 2416.95833
161.13056 2.56 0.0047 Road
15 11005.16667 733.67778
11.63 lt.0001 Source DF
Type III SS Mean Square F Value Pr gt
F Engineer 15 1635.16667
109.01111 1.73 0.0668 Road
15 11005.16667 733.67778
11.63 lt.000
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Asphalt-Pavement Rating Study
Asphalt-Paving Rating Study Mason, Gunst,
Hess Table 10.4
The GLM Procedure Source Type
III Expected Mean Square Engineer
Var(Error) 5.3333 Var(Engineer) Road
Var(Error) 5.3333 Var(Road)
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Asphalt-Pavement Rating Study
The Mixed Procedure Convergence criteria
met Covariance Parameter Estimates
Cov Parm Estimate Alpha Lower
Upper Road 121.96 0.05
63.5064 323.11 Engineer
7.9899 0.05 2.2767 222.91
Residual 63.4641 0.05 46.1892
92.6823
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Allergic Reaction StudyRandomized Complete
Block Design
Not Additive
MGH Table 10.6
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Balanced Incomplete Block Designs
Multiple Comparisons Use Adjusted Factor-Level
Averages
Average of r Block Averages Containing
Factor-Level i
MGH Exhibit 10.5