© 2003 prentice-hall, inc.chap 11-1 analysis of variance & post-anova analysis ie 340/440...
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© 2003 Prentice-Hall, Inc. Chap 11-1
Analysis of Variance&
Post-ANOVA ANALYSIS
IE 340/440
PROCESS IMPROVEMENT THROUGH PLANNED EXPERIMENTATION
Dr. Xueping LiUniversity of Tennessee
© 2003 Prentice-Hall, Inc. Chap 11-2
What If There Are More Than Two Factor Levels?
The t-test does not directly apply There are lots of practical situations where there
are either more than two levels of interest, or there are several factors of simultaneous interest
The analysis of variance (ANOVA) is the appropriate analysis “engine” for these types of experiments – Chapter 3, textbook
The ANOVA was developed by Fisher in the early 1920s, and initially applied to agricultural experiments
Used extensively today for industrial experiments
© 2003 Prentice-Hall, Inc. Chap 11-3
Figure 3.1 (p. 61)A single-wafer plasma etching tool.
© 2003 Prentice-Hall, Inc. Chap 11-4
Table 3.1 (p. 62)Etch Rate Data (in Å/min) from the Plasma Etching Experiment)
© 2003 Prentice-Hall, Inc. Chap 11-5
The Analysis of Variance (Sec. 3-3, pg. 65)
In general, there will be a levels of the factor, or a treatments, and n replicates of the experiment, run in random order…a completely randomized design (CRD)
N = an total runs We consider the fixed effects case…the random effects
case will be discussed later Objective is to test hypotheses about the equality of the a
treatment means
© 2003 Prentice-Hall, Inc. Chap 11-6
Models for the Data
There are several ways to write a model for the data:
is called the effects model
Let , then
is called the means model
Regression models can also be employed
ij i ij
i i
ij i ij
y
y
© 2003 Prentice-Hall, Inc. Chap 11-7
The Analysis of Variance
The name “analysis of variance” stems from a partitioning of the total variability in the response variable into components that are consistent with a model for the experiment
The basic single-factor ANOVA model is
2
1,2,...,,
1, 2,...,
an overall mean, treatment effect,
experimental error, (0, )
ij i ij
i
ij
i ay
j n
ith
NID
© 2003 Prentice-Hall, Inc. Chap 11-8
The Analysis of Variance
Total variability is measured by the total sum of squares:
The basic ANOVA partitioning is:
2..
1 1
( )a n
T iji j
SS y y
2 2.. . .. .
1 1 1 1
2 2. .. .
1 1 1
( ) [( ) ( )]
( ) ( )
a n a n
ij i ij ii j i j
a a n
i ij ii i j
T Treatments E
y y y y y y
n y y y y
SS SS SS
© 2003 Prentice-Hall, Inc. Chap 11-9
The Analysis of Variance
A large value of SSTreatments reflects large differences in treatment means
A small value of SSTreatments likely indicates no differences in treatment means
Formal statistical hypotheses are:
T Treatments ESS SS SS
0 1 2
1
:
: At least one mean is different aH
H
Chap 11-10
The Analysis of Variance
While sums of squares cannot be directly compared to test the hypothesis of equal means, mean squares can be compared.
A mean square is a sum of squares divided by its degrees of freedom:
If the treatment means are equal, the treatment and error mean squares will be (theoretically) equal. If treatment means differ, the treatment mean square will be larger than the error mean square.
1 1 ( 1)
,1 ( 1)
Total Treatments Error
Treatments ETreatments E
df df df
an a a n
SS SSMS MS
a a n
© 2003 Prentice-Hall, Inc. Chap 11-11
The Analysis of Variance is Summarized in a Table
Computing…see text, pp 70 – 73 The reference distribution for F0 is the Fa-1, a(n-1) distribution Reject the null hypothesis (equal treatment means) if
0 , 1, ( 1)a a nF F
© 2003 Prentice-Hall, Inc. Chap 11-12
Features of One-Way ANOVA
F Statistic The F Statistic is the Ratio of the Among
Estimate of Variance and the Within Estimate of Variance The ratio must always be positive df1 = a -1 will typically be small df2 = N - c will typically be large
The Ratio Should Be Close to 1 if the Null is True
© 2003 Prentice-Hall, Inc. Chap 11-13
Features of One-Way ANOVA
F Statistic
If the Null Hypothesis is False The numerator should be greater than the
denominator The ratio should be larger than 1
(continued)
© 2003 Prentice-Hall, Inc. Chap 11-14
The Reference Distribution:
© 2003 Prentice-Hall, Inc. Chap 11-15
Table 3.1 (p. 62)Etch Rate Data (in Å/min) from the Plasma Etching Experiment)
© 2003 Prentice-Hall, Inc. Chap 11-16
Table 3.4 (p. 71)ANOVA for the Plasma Etching Experiment
© 2003 Prentice-Hall, Inc. Chap 11-17
Table 3.5 (p. 72)Coded Etch Rate Data for Example 3.2
Coding the observations
More about manual calculation p.70-71
Chap 11-18
Graphical View of the ResultsDESIGN-EXPERT Plot
Strength X = A: Cotton Weight %
Design Points
A: Cotton Weight %
Str
en
gth
One Factor Plot
15 20 25 30 35
7
11.5
16
20.5
25
22
22
22 22
22 22
22
© 2003 Prentice-Hall, Inc. Chap 11-19
Model Adequacy Checking in the ANOVA
Text reference, Section 3-4, pg. 76
Checking assumptions is important Normality Constant variance Independence Have we fit the right model? Later we will talk about what to do if
some of these assumptions are violated
© 2003 Prentice-Hall, Inc. Chap 11-20
Model Adequacy Checking in the ANOVA
Examination of residuals (see text, Sec. 3-4, pg. 76)
Design-Expert generates the residuals
Residual plots are very useful
Normal probability plot of residuals
.
ˆij ij ij
ij i
e y y
y y
© 2003 Prentice-Hall, Inc. Chap 11-21
Table 3.6 (p. 76)Etch Rate Data and Residuals from Example 3.1a.
© 2003 Prentice-Hall, Inc. Chap 11-22
Figure 3.4 (p. 77)Normal probability plot of residuals for Example 3-1.
© 2003 Prentice-Hall, Inc. Chap 11-23
Figure 3.5 (p. 78)Plot of residuals versus run order or time.
© 2003 Prentice-Hall, Inc. Chap 11-24
Figure 3.6 (p. 79)Plot of residuals versus fitted values.
© 2003 Prentice-Hall, Inc. Chap 11-25
Other Important Residual PlotsDESIGN-EXPERT PlotStrength
22
22
22
22
22
22
22
Predicted
Re
sid
ua
ls
Residuals vs. Predicted
-3.8
-1.55
0.7
2.95
5.2
9.80 12.75 15.70 18.65 21.60
DESIGN-EXPERT PlotStrength
Run Number
Re
sid
ua
ls
Residuals vs. Run
-3.8
-1.55
0.7
2.95
5.2
1 4 7 10 13 16 19 22 25
© 2003 Prentice-Hall, Inc. Chap 11-26
Post-ANOVA Comparison of Means
The analysis of variance tests the hypothesis of equal treatment means
Assume that residual analysis is satisfactory If that hypothesis is rejected, we don’t know which
specific means are different Determining which specific means differ following an
ANOVA is called the multiple comparisons problem There are lots of ways to do this…see text, Section 3-5,
pg. 86 We will use pairwise t-tests on means…sometimes called
Fisher’s Least Significant Difference (or Fisher’s LSD) Method
© 2003 Prentice-Hall, Inc. Chap 11-27
Tukey’s Test
H0: Mu_i = Mu_j ; H1: Mu_i <> Mu_j T statistic
Whether Where
f is the DF of MSE a is the number of groups
n
MSfaqT E),(
Tyy ji ..
© 2003 Prentice-Hall, Inc. Chap 11-28
The Tukey-Kramer Procedure
Tells which Population Means are Significantly Different E.g., 1 = 2 3
2 groups whose means may be significantly different
Post Hoc (A Posteriori) Procedure Done after rejection of equal means in ANOVA
Pairwise Comparisons Compare absolute mean differences with
critical range
X
f(X)
1 = 2
3
© 2003 Prentice-Hall, Inc. Chap 11-29
The Tukey-Kramer Procedure: Example
1. Compute absolute mean differences:Machine1 Machine2 Machine3
25.40 23.40 20.0026.31 21.80 22.2024.10 23.50 19.7523.74 22.75 20.6025.10 21.60 20.40
1 2
1 3
2 3
24.93 22.61 2.32
24.93 20.59 4.34
22.61 20.59 2.02
X X
X X
X X
2. Compute critical range:
3. All of the absolute mean differences are greater than the critical range. There is a significant difference between each pair of means at the 5% level of significance.
( , )'
1 1Critical Range 1.618
2U c n cj j
MSWQ
n n
© 2003 Prentice-Hall, Inc. Chap 11-30
Fisher’s LSD
H0: Mu_i = Mu_j Least Significant Difference
Whether Where
Eji
aN MSnn
tLSD )11
(,2/
n
MStLSD E
aN
2,2/
LSDyy ji ..
© 2003 Prentice-Hall, Inc. Chap 11-31
Design-Expert Output
Treatment Means (Adjusted, If Necessary)Estimated StandardMean Error
1-15 9.80 1.27 2-20 15.40 1.27 3-25 17.60 1.27 4-30 21.60 1.27 5-35 10.80 1.27
Mean Standard t for H0Treatment Difference DF Error Coeff=0 Prob > |t| 1 vs 2 -5.60 1 1.80 -3.12 0.0054 1 vs 3 -7.80 1 1.80 -4.34 0.0003 1 vs 4 -11.80 1 1.80 -6.57 < 0.0001 1 vs 5 -1.00 1 1.80 -0.56 0.5838 2 vs 3 -2.20 1 1.80 -1.23 0.2347 2 vs 4 -6.20 1 1.80 -3.45 0.0025 2 vs 5 4.60 1 1.80 2.56 0.0186 3 vs 4 -4.00 1 1.80 -2.23 0.0375 3 vs 5 6.80 1 1.80 3.79 0.0012 4 vs 5 10.80 1 1.80 6.01 < 0.0001
© 2003 Prentice-Hall, Inc. Chap 11-32
Figure 3.12 (p. 99)Design-Expert computer output for Example 3-1.
© 2003 Prentice-Hall, Inc. Chap 11-33
Figure 3.13 (p. 100)Minitab computer output for Example 3-1.
© 2003 Prentice-Hall, Inc. Chap 11-34
© 2003 Prentice-Hall, Inc. Chap 11-35
Graphical Comparison of MeansText, pg. 89
© 2003 Prentice-Hall, Inc. Chap 11-36
For the Case of Quantitative Factors, a Regression Model is often
Useful
Response:Strength ANOVA for Response Surface Cubic ModelAnalysis of variance table [Partial sum of squares]
Sum of Mean FSource Squares DF Square Value Prob > FModel 441.81 3 147.27 15.85 < 0.0001A 90.84 1 90.84 9.78 0.0051A2 343.21 1 343.21 36.93 < 0.0001A3 64.98 1 64.98 6.99 0.0152Residual 195.15 21 9.29Lack of Fit 33.95 1 33.95 4.21 0.0535Pure Error 161.20 20 8.06Cor Total 636.96 24
Coefficient Standard 95% CI 95% CIFactor Estimate DF Error Low High VIF Intercept 19.47 1 0.95 17.49 21.44 A-Cotton % 8.10 1 2.59 2.71 13.49 9.03 A2 -8.86 1 1.46 -11.89 -5.83 1.00 A3 -7.60 1 2.87 -13.58 -1.62 9.03
Chap 11-37
The Regression Model
Final Equation in Terms of Actual Factors:
Strength = +62.61143-9.01143* Cotton Weight % +0.48143 * Cotton Weight %^2 -7.60000E-003 * Cotton Weight %^3
This is an empirical model of the experimental results
DESIGN-EXPERT Plot
Strength
X = A: Cotton Weight %
Design Points
15.00 20.00 25.00 30.00 35.00
7
11.5
16
20.5
25
A: Cotton Weight %
Str
en
gth
One Factor Plot
22
22
22 22
22 22
22
© 2003 Prentice-Hall, Inc. Chap 11-38
Figure 3.7 (p. 83)Plot of residuals versus ŷij for Example 3-5.
© 2003 Prentice-Hall, Inc. Chap 11-39
Table 3.9 (p. 83)Variance-Stabilizing Transformations
© 2003 Prentice-Hall, Inc. Chap 11-40
Figure 3.8 (p. 84)Plot of log Si versus logfor the peak discharge data from Example 3.5.
.iy
© 2003 Prentice-Hall, Inc. Chap 11-41
Figure 3.12 (p. 99)Design-Expert computer output for Example 3-1.
© 2003 Prentice-Hall, Inc. Chap 11-42
Figure 3.13 (p. 100)Minitab computer output for Example 3-1.
© 2003 Prentice-Hall, Inc. Chap 11-43
Display on page 103 y
© 2003 Prentice-Hall, Inc. Chap 11-44
Example 3-1 p70 EX3-1 p112
© 2003 Prentice-Hall, Inc. Chap 11-45
© 2003 Prentice-Hall, Inc. Chap 11-46
One-Way ANOVA F Test Example
As production manager, you want to see if 3 filling machines have different mean filling times. You assign 15 similarly trained & experienced workers, 5 per machine, to the machines. At the .05 significance level, is there a difference in mean filling times?
Machine1 Machine2 Machine325.40 23.40 20.0026.31 21.80 22.2024.10 23.50 19.7523.74 22.75 20.6025.10 21.60 20.40
© 2003 Prentice-Hall, Inc. Chap 11-47
One-Way ANOVA Example: Scatter Diagram
27
26
25
24
23
22
21
20
19
••
•••
•••••
••••
•
Time in SecondsMachine1 Machine2 Machine325.40 23.40 20.0026.31 21.80 22.2024.10 23.50 19.7523.74 22.75 20.6025.10 21.60 20.40
1 2
3
24.93 22.61
20.59 22.71
X X
X X
1X
2X
3X
X
© 2003 Prentice-Hall, Inc. Chap 11-48
One-Way ANOVA Example Computations
Machine1 Machine2 Machine3
25.40 23.40 20.0026.31 21.80 22.2024.10 23.50 19.7523.74 22.75 20.6025.10 21.60 20.40
2 2 25 24.93 22.71 22.61 22.71 20.59 22.71
47.164
4.2592 3.112 3.682 11.0532
/( -1) 47.16 / 2 23.5820
/( - ) 11.0532 /12 .9211
SSA
SSW
MSA SSA c
MSW SSW n c
1
2
3
24.93
22.61
20.59
22.71
X
X
X
X
5
3
15
jn
c
n
© 2003 Prentice-Hall, Inc. Chap 11-49
Summary Table
Source of
Variation
Degrees of
Freedom
Sum ofSquares
Mean Squares
(Variance)
FStatistic
Among(Factor)
3-1=2 47.1640 23.5820MSA/MSW
=25.60
Within(Error)
15-3=12
11.0532 .9211
Total15-
1=1458.2172
© 2003 Prentice-Hall, Inc. Chap 11-50
One-Way ANOVA Example Solution
F0 3.89
H0: 1 = 2 = 3
H1: Not All Equal = .05df1= 2 df2 = 12
Critical Value(s):
Test Statistic:
Decision:
Conclusion:
Reject at = 0.05.
There is evidence that atleast one i differs fromthe rest.
= 0.05
FMSA
MSW
23 5820
921125 6
.
..
© 2003 Prentice-Hall, Inc. Chap 11-51
Solution in Excel
Use Tools | Data Analysis | ANOVA: Single Factor
Excel Worksheet that Performs the One-Factor ANOVA of the Example
Microsoft Excel Worksheet
© 2003 Prentice-Hall, Inc. Chap 11-52
The Tukey-Kramer Procedure
Tells which Population Means are Significantly Different E.g., 1 = 2 3
2 groups whose means may be significantly different
Post Hoc (A Posteriori) Procedure Done after rejection of equal means in ANOVA
Pairwise Comparisons Compare absolute mean differences with
critical range
X
f(X)
1 = 2
3
© 2003 Prentice-Hall, Inc. Chap 11-53
The Tukey-Kramer Procedure: Example
1. Compute absolute mean differences:Machine1 Machine2 Machine3
25.40 23.40 20.0026.31 21.80 22.2024.10 23.50 19.7523.74 22.75 20.6025.10 21.60 20.40
1 2
1 3
2 3
24.93 22.61 2.32
24.93 20.59 4.34
22.61 20.59 2.02
X X
X X
X X
2. Compute critical range:
3. All of the absolute mean differences are greater than the critical range. There is a significant difference between each pair of means at the 5% level of significance.
( , )'
1 1Critical Range 1.618
2U c n cj j
MSWQ
n n
© 2003 Prentice-Hall, Inc. Chap 11-54
Solution in PHStat
Use PHStat | c-Sample Tests | Tukey-Kramer Procedure …
Excel Worksheet that Performs the Tukey-Kramer Procedure for the Previous Example
Microsoft Excel Worksheet
© 2003 Prentice-Hall, Inc. Chap 11-55
Levene’s Test for Homogeneity of Variance
The Null Hypothesis The c population variances are all equal
The Alternative Hypothesis Not all the c population variances are equal
2 2 20 1 2: cH
21 : Not all are equal ( 1, 2, , )jH j c
© 2003 Prentice-Hall, Inc. Chap 11-56
Levene’s Test for Homogeneity of Variance:
Procedure
1. For each observation in each group, obtain the absolute value of the difference between each observation and the median of the group.
2. Perform a one-way analysis of variance on these absolute differences.
© 2003 Prentice-Hall, Inc. Chap 11-57
Levene’s Test for Homogeneity of Variances:
Example
As production manager, you want to see if 3 filling machines have different variance in filling times. You assign 15 similarly trained & experienced workers, 5 per machine, to the machines. At the .05 significance level, is there a difference in the variance in filling times?
Machine1 Machine2 Machine325.40 23.40 20.0026.31 21.80 22.2024.10 23.50 19.7523.74 22.75 20.6025.10 21.60 20.40
© 2003 Prentice-Hall, Inc. Chap 11-58
Levene’s Test: Absolute Difference from the
Median
Machine1 Machine2 Machine3 Machine1 Machine2 Machine325.4 23.4 20 0.3 0.65 0.4
26.31 21.8 22.2 1.21 0.95 1.824.1 23.5 19.75 1 0.75 0.65
23.74 22.75 20.6 1.36 0 0.225.1 21.6 20.4 0 1.15 0
median 25.1 22.75 20.4
abs(Time - median(Time))Time
© 2003 Prentice-Hall, Inc. Chap 11-59
Summary Table
SUMMARYGroups Count Sum Average Variance
Machine1 5 3.87 0.774 0.35208Machine2 5 3.5 0.7 0.19Machine3 5 3.05 0.61 0.5005
ANOVASource of Variation SS df MS F P-value F crit
Between Groups 0.067453 2 0.033727 0.097048 0.908218 3.88529Within Groups 4.17032 12 0.347527
Total 4.237773 14
© 2003 Prentice-Hall, Inc. Chap 11-60
Levene’s Test Example:Solution
F0 3.89
H0:
H1: Not All Equal = .05df1= 2 df2 = 12
Critical Value(s):
Test Statistic:
Decision:
Conclusion:
Do not reject at = 0.05.
There is no evidence that at least one differs from the rest.
= 0.05
2 2 21 2 3
0.03370.0970
0.3475
MSAF
MSW
2j
© 2003 Prentice-Hall, Inc. Chap 11-61
Randomized Blocked Design Items are Divided into Blocks
Individual items in different samples are matched, or repeated measurements are taken
Reduced within group variation (i.e., remove the effect of block before testing)
Response of Each Treatment Group is Obtained
Assumptions Same as completely randomized design No interaction effect between treatments and
blocks
© 2003 Prentice-Hall, Inc. Chap 11-62
Randomized Blocked Design(Example)
Factor (Training Method)
Factor Levels
(Groups)
BlockedExperiment
Units
Dependent Variable
(Response)
21 hrs 17 hrs 31 hrs
27 hrs 25 hrs 28 hrs
29 hrs 20 hrs 22 hrs
© 2003 Prentice-Hall, Inc. Chap 11-63
Randomized Block Design(Partition of Total Variation)
Variation Due to Group
SSA
Variation Due to Group
SSA
Variation Among Blocks SSBL
Variation Among Blocks SSBL
Variation Among All
Observations
SST
Variation Among All
Observations
SSTCommonly referred to as: Sum of Squares Error Sum of Squares
Unexplained
Commonly referred to as:
Sum of Squares Among
Among Groups Variation
=
+
+Variation Due
to Random Sampling
SSW
Variation Due to Random Sampling
SSW
Commonly referred to as:
Sum of Squares Among Block
© 2003 Prentice-Hall, Inc. Chap 11-64
Total Variation
2
1
the number of blocks
the number of groups or levels
the total number of observations
the value in the -th block for the -th treatment level
the mean of all val
c r
ijj i
ij
i
SST X X
r
c
n n rc
X i j
X
ues in block
the mean of all values for treatment level
1
j
i
X j
df n
© 2003 Prentice-Hall, Inc. Chap 11-65
Among-Group Variation
2
1
1 (treatment group means)
1
1
c
jj
r
iji
j
SSA r X X
XX
rdf c
SSAMSA
c
© 2003 Prentice-Hall, Inc. Chap 11-66
Among-Block Variation
2
1
1 (block means)
1
1
r
ii
c
ijj
i
SSBL c X X
X
Xc
df r
SSBLMSBL
r
© 2003 Prentice-Hall, Inc. Chap 11-67
Random Error
2
1 1
1 1
1 1
c r
ij i jj i
SSE X X X X
df r c
SSEMSE
r c
© 2003 Prentice-Hall, Inc. Chap 11-68
Randomized Block F Test for Differences in c Means
No treatment effect
Test Statistic
Degrees of Freedom
0 1 2: cH
1 : Not all are equaljH
MSAF
MSE
1 1df c 2 1 1df r c
0
UF F
Reject
© 2003 Prentice-Hall, Inc. Chap 11-69
Summary Table
Source of
Variation
Degrees of
Freedom
Sum ofSquare
s
Mean Squares
FStatisti
c
AmongGroup
c – 1 SSAMSA =
SSA/(c – 1)MSA/MSE
AmongBlock
r – 1 SSBLMSBL =
SSBL/(r – 1)MSBL/MSE
Error (r – 1) c – 1)
SSEMSE =
SSE/[(r – 1)(c– 1)]
Total rc – 1 SST
© 2003 Prentice-Hall, Inc. Chap 11-70
Randomized Block Design: Example
As production manager, you want to see if 3 filling machines have different mean filling times. You assign 15 workers with varied experience into 5 groups of 3 based on similarity of their experience, and assigned each group of 3 workers with similar experience to the machines. At the .05 significance level, is there a difference in mean filling times?
Machine1 Machine2 Machine325.40 23.40 20.0026.31 21.80 22.2024.10 23.50 19.7523.74 22.75 20.6025.10 21.60 20.40
© 2003 Prentice-Hall, Inc. Chap 11-71
Randomized Block Design Example Computation
Machine1 Machine2 Machine3
25.40 23.40 20.0026.31 21.80 22.2024.10 23.50 19.7523.74 22.75 20.6025.10 21.60 20.40
2 2 25 24.93 22.71 22.61 22.71 20.59 22.71
47.164
8.4025
/( -1) 47.16 / 2 23.5820
/ ( -1) 1 8.4025 / 8 1.0503
SSA
SSE
MSA SSA c
MSE SSE r c
1
2
3
24.93
22.61
20.59
22.71
X
X
X
X
5
3
15
r
c
n
© 2003 Prentice-Hall, Inc. Chap 11-72
Randomized Block Design Example: Summary Table
Source of
Variation
Degrees of
Freedom
Sum ofSquare
s
Mean Square
s
FStatistic
AmongGroup
2SSA=
47.164MSA = 23.582
23.582/1.0503=22.452
AmongBlock
4SSBL=2.6507
MSBL =.6627
.6627/1.0503=.6039
Error SSE=8.4025
MSE = 1.0503
Total 14SST=
58.2172
© 2003 Prentice-Hall, Inc. Chap 11-73
Randomized Block Design Example: Solution
F0 4.46
H0: 1 = 2 = 3
H1: Not All Equal = .05df1= 2 df2 = 8
Critical Value(s):
Test Statistic:
Decision:
Conclusion:
Reject at = 0.05.
There is evidence that atleast one i differs fromthe rest.
= 0.05
FMSA
MSE
23 582
1.050322.45
.
© 2003 Prentice-Hall, Inc. Chap 11-74
Randomized Block Designin Excel
Tools | Data Analysis | ANOVA: Two Factor Without Replication
Example Solution in Excel Spreadsheet
Microsoft Excel Worksheet
© 2003 Prentice-Hall, Inc. Chap 11-75
The Tukey-Kramer Procedure
Similar to the Tukey-Kramer Procedure for the Completely Randomized Design Case
Critical Range
( , 1 1 )Critical Range U c r c
MSEQ
r
© 2003 Prentice-Hall, Inc. Chap 11-76
The Tukey-Kramer Procedure: Example
1. Compute absolute mean differences:Machine1 Machine2 Machine3
25.40 23.40 20.0026.31 21.80 22.2024.10 23.50 19.7523.74 22.75 20.6025.10 21.60 20.40
1 2
1 3
2 3
24.93 22.61 2.32
24.93 20.59 4.34
22.61 20.59 2.02
X X
X X
X X
2. Compute critical range:
3. All of the absolute mean differences are greater. There is a significance difference between each pair of means at 5% level of significance.
( , 1 1 )
1.0503Critical Range 4.04 1.8516
5U c r c
MSEQ
r
© 2003 Prentice-Hall, Inc. Chap 11-77
The Tukey-Kramer Procedurein PHStat
PHStat | c-Sample Tests | Tukey-Kramer Procedure …
Example in Excel Spreadsheet
Microsoft Excel Worksheet
© 2003 Prentice-Hall, Inc. Chap 11-78
Two-Way ANOVA
Examines the Effect of: Two factors on the dependent variable
E.g., Percent carbonation and line speed on soft drink bottling process
Interaction between the different levels of these two factors
E.g., Does the effect of one particular percentage of carbonation depend on which level the line speed is set?
© 2003 Prentice-Hall, Inc. Chap 11-79
Two-Way ANOVA
Assumptions Normality
Populations are normally distributed
Homogeneity of Variance Populations have equal variances
Independence of Errors Independent random samples are drawn
(continued)
© 2003 Prentice-Hall, Inc. Chap 11-80
SSE
Two-Way ANOVA Total Variation Partitioning
Variation Due to Factor A
Variation Due to Factor A
Variation Due to Random SamplingVariation Due to
Random Sampling
Variation Due to Interaction
Variation Due to Interaction
SSA
SSABSST
Variation Due to Factor B
Variation Due to Factor B
SSB
Total VariationTotal Variation
d.f.= n-1
d.f.= r-1
=
+
+ d.f.= c-1
+ d.f.= (r-1)(c-1)
d.f.= rc(n’-1)
© 2003 Prentice-Hall, Inc. Chap 11-81
Two-Way ANOVA Total Variation Partitioning
'
the number of levels of factor A
the number of levels of factor B
the number of values (replications) for each cell
the total number of observations in the experiment
the value of the -th oijk
r
c
n
n
X k
bservation for level of
factor A and level of factor B
i
j
© 2003 Prentice-Hall, Inc. Chap 11-82
Total Variation
' '
' 2
1 1 1
1 1 1 1 1 1
'
Sum of Squares Total
= total variation among all
observations around the grand mean
the overall or grand mean
r c n
ijki j k
r c n r c n
ijk ijki j k i j k
SST X X
X X
Xrcn n
© 2003 Prentice-Hall, Inc. Chap 11-83
Factor A Variation
2'
1
r
i
i
SSA cn X X
Sum of Squares Due to Factor A = the difference among the various levels of factor A and the grand mean
© 2003 Prentice-Hall, Inc. Chap 11-84
Factor B Variation
2'
1
c
j
j
SSB rn X X
Sum of Squares Due to Factor B = the difference among the various levels of factor B and the grand mean
© 2003 Prentice-Hall, Inc. Chap 11-85
Interaction Variation
2'
1 1
r c
ij i j
i j
SSAB n X X X X
Sum of Squares Due to Interaction between A and B = the effect of the combinations of factor A and factor B
© 2003 Prentice-Hall, Inc. Chap 11-86
Random Error
Sum of Squares Error = the differences among the observations within
each cell and the corresponding cell means
' 2
1 1 1
r c n
ijijki j k
SSE X X
© 2003 Prentice-Hall, Inc. Chap 11-87
Two-Way ANOVA:The F Test Statistic
F Test for Factor B Main Effect
F Test for Interaction Effect
H0: 1 .= 2 . = ••• = r .
H1: Not all i . are equal
H0: ij = 0 (for all i and j)
H1: ij 0
H0: 1 = . 2 = ••• = c
H1: Not all . j are equal
Reject if F > FU
Reject if F > FU
Reject if F > FU
1
MSA SSAF MSA
MSE r
F Test for Factor A Main Effect
1
MSB SSBF MSB
MSE c
1 1
MSAB SSABF MSAB
MSE r c
© 2003 Prentice-Hall, Inc. Chap 11-88
Two-Way ANOVASummary Table
Source ofVariation
Degrees of Freedom
Sum ofSquare
s
Mean Squares
FStatisti
c
Factor A(Row)
r – 1 SSAMSA =
SSA/(r – 1)MSA/MSE
Factor B(Column)
c – 1 SSBMSB =
SSB/(c – 1)MSB/MSE
AB(Interaction)
(r – 1)(c – 1) SSABMSAB =
SSAB/ [(r – 1)(c – 1)]MSAB/MSE
Error rc n’ – 1) SSEMSE =
SSE/[rc n’ – 1)]
Total rc n’ – 1 SST
© 2003 Prentice-Hall, Inc. Chap 11-89
Features of Two-Way ANOVA
F Test
Degrees of Freedom Always Add Up rcn’-1=rc(n’-1)+(c-1)+(r-1)+(c-1)(r-1) Total=Error+Column+Row+Interaction
The Denominator of the F Test is Always the Same but the Numerator is Different
The Sums of Squares Always Add Up Total=Error+Column+Row+Interaction
© 2003 Prentice-Hall, Inc. Chap 11-90
Kruskal-Wallis Rank Test for c Medians
Extension of Wilcoxon Rank Sum Test Tests the equality of more than 2 (c)
population medians
Distribution-Free Test Procedure Used to Analyze Completely Randomized
Experimental Designs Use 2 Distribution to Approximate if Each
Sample Group Size nj > 5 df = c – 1
© 2003 Prentice-Hall, Inc. Chap 11-91
Kruskal-Wallis Rank Test
Assumptions Independent random samples are drawn Continuous dependent variable Data may be ranked both within and among
samples Populations have same variability Populations have same shape
Robust with Regard to Last 2 Conditions Use F test in completely randomized
designs and when the more stringent assumptions hold
© 2003 Prentice-Hall, Inc. Chap 11-92
Kruskal-Wallis Rank Test Procedure
Obtain Ranks In event of tie, each of the tied values gets
their average rank Add the Ranks for Data from Each of the
c Groups Square to obtain Tj
2
2
1
123( 1)
( 1)
cj
j j
TH n
n n n
1 2 cn n n n
© 2003 Prentice-Hall, Inc. Chap 11-93
Kruskal-Wallis Rank Test Procedure
Compute Test Statistic
# of observation in j –th sample H may be approximated by chi-square
distribution with df = c –1 when each nj >5
(continued)
2
1
123( 1)
( 1)
cj
j j
TH n
n n n
1 2 cn n n n
jn
© 2003 Prentice-Hall, Inc. Chap 11-94
Kruskal-Wallis Rank Test Procedure
Critical Value for a Given Upper tail
Decision Rule Reject H0: M1 = M2 = ••• = Mc if test statistic
H > Otherwise, do not reject H0
(continued)
2U
2U
© 2003 Prentice-Hall, Inc. Chap 11-95
Machine1 Machine2 Machine325.40 23.40 20.0026.31 21.80 22.2024.10 23.50 19.7523.74 22.75 20.6025.10 21.60 20.40
Kruskal-Wallis Rank Test: Example
As production manager, you want to see if 3 filling machines have different median filling times. You assign 15 similarly trained & experienced workers, 5 per machine, to the machines. At the .05 significance level, is there a difference in median filling times?
© 2003 Prentice-Hall, Inc. Chap 11-96
Machine1 Machine2 Machine3 14 9 2 15 6 7 12 10 1 11 8 4 13 5 3
Example Solution: Step 1 Obtaining a Ranking
Raw Data Ranks
65 38 17
Machine1 Machine2 Machine325.40 23.40 20.0026.31 21.80 22.2024.10 23.50 19.7523.74 22.75 20.6025.10 21.60 20.40
© 2003 Prentice-Hall, Inc. Chap 11-97
Example Solution: Step 2 Test Statistic Computation
212
3( 1)( 1) 1
2 2 212 65 38 173(15 1)
15(15 1) 5 5 5
11.58
Tc jH n
n n nj j
© 2003 Prentice-Hall, Inc. Chap 11-98
Kruskal-Wallis Test Example Solution
H0: M1 = M2 = M3
H1: Not all equal = .05df = c - 1 = 3 - 1 = 2Critical Value(s): Reject at
Test Statistic:
Decision:
Conclusion:There is evidence that population medians are not all equal.
= .05
= .05.
H = 11.58
0 5.991
© 2003 Prentice-Hall, Inc. Chap 11-99
Kruskal-Wallis Test in PHStat
PHStat | c-Sample Tests | Kruskal-Wallis Rank Sum Test …
Example Solution in Excel Spreadsheet
Microsoft Excel Worksheet
© 2003 Prentice-Hall, Inc. Chap 11-100
Friedman Rank Test for Differences in c Medians
Tests the equality of more than 2 (c) population medians
Distribution-Free Test Procedure Used to Analyze Randomized Block
Experimental Designs Use 2 Distribution to Approximate if the
Number of Blocks r > 5 df = c – 1
© 2003 Prentice-Hall, Inc. Chap 11-101
Friedman Rank Test
Assumptions The r blocks are independent The random variable is continuous The data constitute at least an ordinal scale
of measurement No interaction between the r blocks and the
c treatment levels The c populations have the same variability The c populations have the same shape
© 2003 Prentice-Hall, Inc. Chap 11-102
Friedman Rank Test:Procedure
Replace the c observations by their ranks in each of the r blocks; assign average rank for ties
Test statistic:
R.j2 is the square of the rank total for group j
FR can be approximated by a chi-square distribution with (c –1) degrees of freedom
The rejection region is in the right tail
2
1
123 1
1
c
R jj
F R r crc c
© 2003 Prentice-Hall, Inc. Chap 11-103
Friedman Rank Test: Example
As production manager, you want to see if 3 filling machines have different median filling times. You assign 15 workers with varied experience into 5 groups of 3 based on similarity of their experience, and assigned each group of 3 workers with similar experience to the machines. At the .05 significance level, is there a difference in median filling times?
Machine1 Machine2 Machine325.40 23.40 20.0026.31 21.80 22.2024.10 23.50 19.7523.74 22.75 20.6025.10 21.60 20.40
© 2003 Prentice-Hall, Inc. Chap 11-104
Timing RankMachine 1 Machine 2 Machine 3 Machine 1 Machine 2 Machine 3
25.4 23.4 20 3 2 126.31 21.8 22.2 3 1 2
24.1 23.5 19.75 3 2 123.74 22.75 20.6 3 2 1
25.1 21.6 20.4 3 2 115 9 6
225 81 36
Friedman Rank Test: Computation Table
2. jR. jR
2
1
123 1
1
12 342 3 5 4 8.4
5 3 4
c
R jj
F R r crc c
© 2003 Prentice-Hall, Inc. Chap 11-105
Friedman Rank Test Example Solution
H0: M1 = M2 = M3
H1: Not all equal = .05df = c - 1 = 3 - 1 = 2Critical Value: Reject at
Test Statistic:
Decision:
Conclusion:There is evidence that population medians are not all equal.
= .05
= .05
FR = 8.4
0 5.991
© 2003 Prentice-Hall, Inc. Chap 11-106
Chapter Summary
Described the Completely Randomized Design: One-Way Analysis of Variance ANOVA Assumptions F Test for Difference in c Means The Tukey-Kramer Procedure Levene’s Test for Homogeneity of Variance
Discussed the Randomized Block Design F Test for the Difference in c Means The Tukey Procedure
© 2003 Prentice-Hall, Inc. Chap 11-107
Chapter Summary
Described the Factorial Design: Two-Way Analysis of Variance Examine effects of factors and interaction
Discussed Kruskal-Wallis Rank Test for Differences in c Medians
Illustrated Friedman Rank Test for Differences in c Medians
(continued)