1. ENGR 610 Applied Statistics Fall 2007 - Week 9
Marshall University
CITE
Jack Smith

2. Overview for Today Review Design of Experiments, Ch 10
One-Factor Experiments
Randomized Block Experiments
Go over homework problems: 10.27, 10.28
Design of Experiments, Ch 11
Two-Factor Factorial Designs
Factorial Designs Involving Three or More Factors
Fractional Factorial Design
The Taguchi Approach
Homework assignment

3. Design of Experiments R.A. Fisher (Rothamsted Ag Exp Station)
Study effects of multiple factors simultaneously
Randomization
Homogeneous blocking
One-Way ANOVA (Analysis of Variance)
One factor with different levels of “treatment”
Partitioning of variation - within and among treatment groups
Generalization of two-sample t Test
Two-Way ANOVA
One factor against randomized blocks (paired treatments)
Generalization of two-sample paired t Test

4. One-Way ANOVA ANOVA = Analysis of Variance
However, goal is to discern differences in means
One-Way ANOVA = One factor, multiple treatments (levels)
Randomly assign treatment groups
Partition total variation (sum of squares)
SST = SSA + SSW
SSA = variation among treatment groups
SSW = variation within treatment groups (across all groups)
Compare mean squares (variances): MS = SS / df
Perform F Test on MSA / MSW
H0: all treatment group means are equal
H1: at least one group mean is different

5. Partitioning of Total Variation
Within-group variation
Among-group variation
(Grand mean)
(Group mean)
c = number of treatment groups
n = total number of observations
nj = observations for group j
Xij = i-th observation for group j

6. Mean Squares (Variances)
Total mean square (variance)
MST = SST / (n-1)
Within-group mean square
MSW = SSW / (n-c)
Among-group mean square
MSA = SSA / (c-1)

7. F Test F = MSA / MSW Reject H0 if F > FU(,c-1,n-c) [or p<]
FU from Table A.7
One-Way ANOVA Summary
Source
Degrees of Freedom (df)
Sum of Squares (SS)
Mean Square (MS) (Variance) F
p-value
Among groups
c-1
SSA
MSA = SSA/(c-1)
MSA/MSW
Within groups
n-c
SSW
MSW = SSW/(n-c)
Total
n-1
SST

8. Tukey-Kramer Comparison of Means
Critical Studentized range (Q) test
qU(,c,n-c) from Table A.9
Perform on each of the c(c-1)/2 pairs of group means
Analogous to t test using pooled variance for comparing two sample means with equal variances

9. One-Way ANOVA Assumptions and Limitations
Assumptions for F test
Random and independent (unbiased) assignments
Normal distribution of experimental error
Homogeneity of variance within and across group (essential for pooling assumed in MSW)
Limitations of One-Factor Design
Inefficient use of experiments
Can not isolate interactions among factors

10. Randomized Block Model
Matched or repeated measurements assigned to a block, with random assignment to treatment groups
Minimize within-block variation to maximize treatment effect
Further partition within-group variation
SSW = SSBL + SSE
SSBL = Among-block variation
SSE = Random variation (experimental error)
Total variation: SST = SSA + SSBL + SSE
Separate F tests for treatment and block effects
Two-way ANOVA, treatment groups vs blocks, but the focus is only on treatment effects

11. Partitioning of Total Variation
Among-group variation
Among-block variation
(Grand mean)
(Group mean)
(Block mean)

12. Partitioning, cont’d Random error c = number of treatment groups
r = number of blocks
n = total number of observations (rc)
Xij = i-th block observation for group j

13. Mean Squares (Variances)
Total mean square (variance)
MST = SST / (rc-1)
Among-group mean square
MSA = SSA / (c-1)
Among-block mean square
MSBL = SSBL / (r-1)
Mean square error
MSE = SSE / (r-1)(c-1)

14. F Test for Treatment Effects
F = MSA / MSE
Reject H0 if F > FU(,c-1,(r-1)(c-1))
FU from Table A.7
Two-Way ANOVA Summary
Source
Degrees of Freedom (df)
Sum of Squares (SS)
Mean Square (MS) (Variance) F
p-value
Among groups
c-1
SSA
MSA = SSA/(c-1)
MSA/MSE
Among blocks
r-1
SSBL
MSBL = SSBL/(r-1)
MSBL/MSE
Error
(r-1)(c-1)
SSE
MSE = SSE/(r-1)(c-1)
Total
rc-1
SST

15. F Test for Block Effects
F = MSBL / MSE
Reject H0 if F > FU(,r-1,(r-1)(c-1))
FU from Table A.7
Assumes no interaction between treatments and blocks
Used only to examine effectiveness of blocking in reducing experimental error
Compute relative efficiency (RE) to estimate leveraging effect of blocking on precision

16. Estimated Relative Efficiency
Estimates the number of observations in each treatment group needed to obtain the same precision for comparison of treatment group means as with randomized block design.
nj (without blocking)  RE*r (with blocking)

17. Tukey-Kramer Comparison of Means
Critical Studentized range (Q) test
qU(,c,(r-1)(c-1)) from Table A.9
Where group sizes (number of blocks, r) are equal
Perform on each of the c(c-1)/2 pairs of group means
Analogous to paired t test for the comparison of two-sample means (or one-sample test on differences)

18. Factorial Designs Two or more factors simultaneously
Includes interaction terms
Typically
2-level: high(+), low(-)
3-level: high(+), center(0), low(-)
Replicates
Needed for random error estimate

19. Partitioning for Two-Factor ANOVA (with Replication)
Total variation
Factor A variation
Factor B variation
(Grand mean)
(Mean for i-th level
of factor A)
(Mean for j-th level
of factor B)

20. Partitioning, cont’d Variation due to interaction of A and B
Random error
(Mean for replications of i-j combination)
r = number of levels for factor A
c = number of levels for factor B
n’ = number of replications for each
n = total number of observations (rcn’)
Xijk = k-th observation for
i-th level of factor A and
j-th level of factor B

21. Mean Squares (Variances)
Total mean square
MST = SST / (rcn’-1)
Factor A mean square
MSA = SSA / (r-1)
Factor B mean square
MSB = SSB / (c-1)
A-B interaction mean square
MSAB = SSAB / (r-1)(c-1)
Mean square error
MSE = SSE / rc(n’-1)

22. F Tests for Effects Factor A effect Factor B effect
F = MSA / MSE
Reject H0 if F > FU(,r-1,rc(n’-1))
Factor B effect
F = MSB / MSE
Reject H0 if F > FU(,c-1,rc(n’-1))
A-B interaction effect
F = MSAB / MSE
Reject H0 if F > FU(,(r-1)(c-1),rc(n’-1))

23. Two-Way ANOVA (with Repetition) Summary Table
Source
Degrees of Freedom (df)
Sum of Squares (SS)
Mean Square (MS) (Variance) F
p-value A
r-1
SSA
MSA = SSA/(r-1)
MSA/MSE B
c-1
SSB
MSB = SSB/(c-1)
MSB/MSE AB
(r-1)(c-1)
SSAB
MSAB = SSAB/(r-1)(c-1)
MSAB/MSE
Error
rc(n’-1)
SSE
MSE = SSE/rc(n’-1)
Total
rcn’-1
SST

24. Tukey-Kramer Comparisons
Critical range (Q) test for levels of factor A
qU(,r,rc(n’-1)) from Table A.9
Perform on each of the r(r-1)/2 pairs of levels
Critical range (Q) test for levels of factor B
qU(,c,rc(n’-1)) from Table A.9
Perform on each of the c(c-1)/2 pairs of levels

25. Main Effects and Interaction Effects
No interaction
Interaction
Crossing Effect

26. Three-Way ANOVA (with Repetition) Summary Table
Source
Degrees of Freedom (df)
Sum of Squares (SS)
Mean Square (MS) (Variance) F
p-value A
i-1
SSA
MSA = SSA/(i-1)
MSA/MSE B
j-1
SSB
MSB = SSB/(j-1)
MSB/MSE C
k-1
SSC
MSC = SSC/(k-1)
MSC/MSE AB
(i-1)(j-1)
SSAB
MSAB = SSAB/(i-1)(j-1)
MSAB/MSE BC
(j-1)(k-1)
SSBC
MSBC = SSBC/(j-1)(k-1)
MSBC/MSE AC
(i-1)(k-1)
SSAC
MSAC = SSAC/(i-1)(k-1)
MSAC/MSE
ABC
(i-1)(j-1)(k-1)
SSABC
MSABC = SSABC/(i-1)(j-1)(k-1)
MSABC/MSE
Error
ijk(n’-1)
SSE
MSE = SSE/ijk(n’-1)
Total
Ijkn’-1
SST

27. Main and Interaction Effects
For a k-factor design
Number of main effects
Number of 2-way interaction effects
Number of 3-way interaction effects
See text (p 529) for sample plots

28. 3-Factor 2-Level Design Notation
ABC
(1) = a-lo, b-lo, c-lo
a = a-hi, b-lo, c-lo
b = a-lo, b-hi, c-lo
c = a-lo, b-lo, c-hi
ab = a-hi, b-hi, c-lo
bc = a-lo, b-hi, c-hi
ac = a-hi, b-lo, c-hi
abc = a-hi, b-hi, c-hi

29. Contrasts and Estimated Effects
A = (1/4n’)[a + ab + ac + abc - (1) - b - c - bc]
B = (1/4n’)[b + ab + bc + abc - (1) - a - c - ac]
C = (1/4n’)[c + ac + bc + abc - (1) - a - b - ab]
AB = (1/4n’)[abc - bc + ab - b - ac + c - a + (1)]
BC = (1/4n’)[(1) - a + b - ab - c + ac - bc + abc]
AC = (1/4n’)[(1) + a - b - ab - c - ac + bc + abc]
ABC = (1/4n’)[abc - bc - ac + c - ab + b + a - (1)]
Effect = (1/n’2k-1)Contrast
SS = (1/n’2k)(Contrast)2
Sum over replications
k = number of factors
n’ = number of replicates

30. 3-Factor 2-Level Contrast Table
Notation A B C AB AC BC
ABC
(1) - + a b c ab ac bc
abc

31. Using Normal Probability Plots
Cumulative percentage for i-th ordered effect
pi = (Ri - 0.5)/(2k - 1)
Ri = ordered rank of I-th effect
k = number of factors
Plot on normal probability paper, or use PHStat
Note deviations from zero and from the nearly straight vertical line for normal random variation
See example in text (p 535)

32. Fractional Factorial Design
Choose a defining contrast
Typically highest interaction term
Halves the number of combinations
But introduces confounding interactions
Aliasing
Resolution III, IV, V designs
based on types of confounding interactions remaining in design
==>
==>

33. Taguchi Approach Parameter design Quadratic Loss Function
Loss = k(Yi-T)2
Partition into design parameters (inner array) and noise factors
Use Signal-to-Noise (S/N) ratios to meet target or minimize/maximize response
Use of orthogonal arrays

34. Homework Work through Appendix 11.1 Work through Problems
Review for Exam #2
Chapters 8-11
Take-home
“Given out” end of class Oct 25
Due beginning of class Nov 1