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Statistics for Managers Using Microsoft Excel, 2/e © 1999 Prentice-Hall, Inc.
Chapter 11 Student Lecture Notes 11-1
© 2004 Prentice-Hall, Inc. Chap 11-1
Basic Business Statistics(9th Edition)
Chapter 11Analysis of Variance
© 2004 Prentice-Hall, Inc. Chap 11-2
The Completely Randomized Design:One-Way Analysis of Variance
ANOVA AssumptionsF Test for Differences in More than Two MeansThe Tukey-Kramer ProcedureLevene’s Test for Homogeneity of Variance
The Randomized Block DesignF Test for the Difference in More than Two MeansThe Tukey Procedure
Chapter Topics
© 2004 Prentice-Hall, Inc. Chap 11-3
Chapter Topics
The Factorial Design: Two-Way Analysis of Variance
Examine Effects of Factors and Interaction
Kruskal-Wallis Rank Test: Nonparametric Analysis for the Completely Randomized Design
Friedman Rank Test: Nonparametric Analysis for the Randomized Block Design
(continued)
Statistics for Managers Using Microsoft Excel, 2/e © 1999 Prentice-Hall, Inc.
Chapter 11 Student Lecture Notes 11-2
© 2004 Prentice-Hall, Inc. Chap 11-4
General Experimental Setting
Investigator Controls One or More Independent Variables
Called treatment variables or factorsEach treatment factor contains two or more groups (or levels)
Observe Effects on Dependent VariableResponse to groups (or levels) of independent variable
Experimental Design: The Plan Used to Test Hypothesis
© 2004 Prentice-Hall, Inc. Chap 11-5
Completely Randomized Design
Experimental Units (Subjects) are Assigned Randomly to Groups
Subjects are assumed to be homogeneous
Only One Factor or Independent VariableWith 2 or more groups (or levels)
Analyzed by One-Way Analysis of Variance (ANOVA)
© 2004 Prentice-Hall, Inc. Chap 11-6
22 hrs20 hrs29 hrs
28 hrs25 hrs27 hrs
31 hrs17 hrs21 hrsDependent
Variable(Response)
Randomly Assigned
Units
Factor Levels(Groups)
Factor (Training Method)
Completely Randomized Design Example
Statistics for Managers Using Microsoft Excel, 2/e © 1999 Prentice-Hall, Inc.
Chapter 11 Student Lecture Notes 11-3
© 2004 Prentice-Hall, Inc. Chap 11-7
One-Way Analysis of VarianceF Test
Evaluate the Difference Among the Mean Responses of 2 or More (c ) Populations
E.g., Several types of tires, oven temperature settings
AssumptionsSamples are randomly and independently drawn
This condition must be metPopulations are normally distributed
F Test is robust to moderate departure from normality
Populations have equal variancesLess sensitive to this requirement when samples are of equal size from each population
© 2004 Prentice-Hall, Inc. Chap 11-8
Why ANOVA?Could Compare the Means One Pair at a Time Using t Test for Difference of MeansEach t Test Contains Type I ErrorThe Total Type I Error with k Pairs of Means is 1- (1 - α) k
E.g., If there are 5 means and use α = .05 Must perform 10 comparisonsType I Error is 1 – (.95) 10 = .4040% of the time you will reject the null hypothesis of equal means in favor of the alternative when the null hypothesis is true!
© 2004 Prentice-Hall, Inc. Chap 11-9
Hypotheses of One-Way ANOVA
All population means are equal
No treatment effect (no variation in means among groups)
At least one population mean is different (others may be the same!)
There is a treatment effect
Does not mean that all population means are different
0 1 2: cH µ µ µ= = =L
1 : Not all are the samejH µ
Statistics for Managers Using Microsoft Excel, 2/e © 1999 Prentice-Hall, Inc.
Chapter 11 Student Lecture Notes 11-4
© 2004 Prentice-Hall, Inc. Chap 11-10
One-Way ANOVA (No Treatment Effect)
The Null Hypothesis is True
0 1 2: cH µ µ µ= = =L
1 : Not all are the samejH µ
1 2 3µ µ µ= =
© 2004 Prentice-Hall, Inc. Chap 11-11
One-Way ANOVA (Treatment Effect Present)
The Null Hypothesis is
NOT True
0 1 2: cH µ µ µ= = =L
1 : Not all are the samejH µ
1 2 3µ µ µ= ≠1 2 3µ µ µ≠ ≠
© 2004 Prentice-Hall, Inc. Chap 11-12
One-Way ANOVA(Partition of Total Variation)
Variation Due to Group SSA
Variation Due to Random Sampling SSW
Total Variation SST
Commonly referred to as:Within Group VariationSum of Squares WithinSum of Squares ErrorSum of Squares Unexplained
Commonly referred to as:Among Group Variation Sum of Squares AmongSum of Squares BetweenSum of Squares ModelSum of Squares ExplainedSum of Squares Treatment
= +
Statistics for Managers Using Microsoft Excel, 2/e © 1999 Prentice-Hall, Inc.
Chapter 11 Student Lecture Notes 11-5
© 2004 Prentice-Hall, Inc. Chap 11-13
Total Variation2
1 1
1 1
( )
: the -th observation in group : the number of observations in group
: the total number of observations in all groups: the number of groups
the over
j
j
nc
ijj i
ij
j
nc
ijj i
SST X X
X i jn jnc
XX
n
= =
= =
= −
=
∑∑
∑∑all or grand mean
© 2004 Prentice-Hall, Inc. Chap 11-14
Total Variation(continued)
( ) ( ) ( )2 2 2
11 21 cn cSST X X X X X X= − + − + + −L
X
Response, X
Group 1 Group 2 Group 3
© 2004 Prentice-Hall, Inc. Chap 11-15
2
1
( )c
j jj
SSA n X X=
= −∑
Among-Group Variation
Variation Due to Differences Among Groups
1SSAMSAc
=−
jµ 'jµ
: The sample mean of group
: The overall or grand mean
jX j
X
Statistics for Managers Using Microsoft Excel, 2/e © 1999 Prentice-Hall, Inc.
Chapter 11 Student Lecture Notes 11-6
© 2004 Prentice-Hall, Inc. Chap 11-16
Among-Group Variation(continued)
( ) ( ) ( )2 2 2
1 1 2 2 c cSSA n X X n X X n X X= − + − + + −L
X1X
2X 3X
Response, X
Group 1 Group 2 Group 3
© 2004 Prentice-Hall, Inc. Chap 11-17
Summing the variation within each group and then adding over all groups
: The sample mean of group : The -th observation in group
j
ij
X jX i j
Within-Group Variation
2
1 1
( )jnc
ij jj i
SSW X X= =
= −∑ ∑ SSWMSWn c
=−
jµ
© 2004 Prentice-Hall, Inc. Chap 11-18
Within-Group Variation(continued)
( ) ( ) ( )22 211 1 21 1 cn c cSSW X X X X X X= − + − + + −L
1X
2X 3XX
Response, X
Group 1 Group 2 Group 3
Statistics for Managers Using Microsoft Excel, 2/e © 1999 Prentice-Hall, Inc.
Chapter 11 Student Lecture Notes 11-7
© 2004 Prentice-Hall, Inc. Chap 11-19
Within-Group Variation(continued)
2 2 21 1 2 2
1 2
( 1) ( 1) ( 1)( 1) ( 1) ( 1)
c c
c
SSWMSWn c
n S n S n Sn n n
=−
− + − +•••+ −=
− + − +•••+ −
For c = 2, this is the pooled-variance in the t test.
•If more than 2 groups, use F Test.
•For 2 groups, use t test. F Test more limited.
jµ
© 2004 Prentice-Hall, Inc. Chap 11-20
One-Way ANOVAF Test Statistic
Test Statistic
MSA is mean squares amongMSW is mean squares within
Degrees of Freedom
MSAFMSW
=
1 1df c= −2df n c= −
© 2004 Prentice-Hall, Inc. Chap 11-21
One-Way ANOVA Summary Table
MSA/MSW
FStatistic
SST =SSA + SSWn – 1Total
MSW =SSW/(n – c )
SSWn – cWithin(Error)
MSA = SSA/(c – 1 )SSAc – 1Among
(Factor)
Mean Squares
(Variance)
Sum ofSquares
Degrees of
Freedom
Source ofVariation
Statistics for Managers Using Microsoft Excel, 2/e © 1999 Prentice-Hall, Inc.
Chapter 11 Student Lecture Notes 11-8
© 2004 Prentice-Hall, Inc. Chap 11-22
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 positivedf1 = c -1 will typically be smalldf2 = n - c will typically be large
The Ratio Should Be Close to 1 if the Null is True
© 2004 Prentice-Hall, Inc. Chap 11-23
Features of One-Way ANOVA F Statistic
If the Null Hypothesis is FalseThe numerator should be greater than the denominatorThe ratio should be larger than 1
(continued)
© 2004 Prentice-Hall, Inc. Chap 11-24
One-Way ANOVA F TestExample
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
Statistics for Managers Using Microsoft Excel, 2/e © 1999 Prentice-Hall, Inc.
Chapter 11 Student Lecture Notes 11-9
© 2004 Prentice-Hall, Inc. Chap 11-25
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
© 2004 Prentice-Hall, Inc. Chap 11-26
One-Way ANOVA Example Computations
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
( ) ( ) ( )2 2 25 24.93 22.71 22.61 22.71 20.59 22.71
47.1644.2592 3.112 3.682 11.0532
/( -1) 47.1640 / 2 23.5820 /( - ) 11.0532 /12 .9211
SSA
SSWMSA SSA cMSW SSW n c
⎡ ⎤= − + − + −⎣ ⎦== + + == = == = =
1
2
3
24.9322.6120.59
22.71
XXX
X
=
=
=
=
5
315
jn
cn
=
==
© 2004 Prentice-Hall, Inc. Chap 11-27
Summary Table
MSA/MSW=25.60
FStatistic
58.217215-1=14Total
.921111.053215-3=12Within(Error)
23.582047.16403-1=2Among(Factor)
Mean Squares
(Variance)
Sum ofSquares
Degrees of
Freedom
Source ofVariation
Statistics for Managers Using Microsoft Excel, 2/e © 1999 Prentice-Hall, Inc.
Chapter 11 Student Lecture Notes 11-10
© 2004 Prentice-Hall, Inc. Chap 11-28
One-Way ANOVA Example Solution
F0 3.89
H0: µ1 = µ2 = µ3
H1: Not All Equalα = .05df1= 2 df2 = 12
Critical Value(s):
Test Statistic:
Decision:
Conclusion:Reject at α = 0.05.
There is evidence that atleast one µ j differs fromthe rest.
α = 0.05
FMSAMSW
= = =23 5820
921125 6
..
.
© 2004 Prentice-Hall, Inc. Chap 11-29
Solution in Excel
Use Tools | Data Analysis | ANOVA: Single Factor Excel Worksheet that Performs the One-Factor ANOVA of the Example
Microsoft Excel Worksheet
© 2004 Prentice-Hall, Inc. Chap 11-30
The Tukey-Kramer Procedure
Tells which Population Means are Significantly Different
E.g., µ1 = µ2 ≠ µ32 groups whose means may be significantly different
Post Hoc (A Posteriori) ProcedureDone after rejection of equal means in ANOVA
Pairwise ComparisonsCompare absolute mean differences with critical range
X
f(X)
µ 1 = µ 2 µ 3
Statistics for Managers Using Microsoft Excel, 2/e © 1999 Prentice-Hall, Inc.
Chapter 11 Student Lecture Notes 11-11
© 2004 Prentice-Hall, Inc. Chap 11-31
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 1.6182U c n c
j j
MSWQn n−
⎛ ⎞= + =⎜ ⎟⎜ ⎟
⎝ ⎠
© 2004 Prentice-Hall, Inc. Chap 11-32
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
© 2004 Prentice-Hall, Inc. Chap 11-33
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 σ σ σ= = =L
21 : Not all are equal ( 1, 2, , )jH j cσ = L
Statistics for Managers Using Microsoft Excel, 2/e © 1999 Prentice-Hall, Inc.
Chapter 11 Student Lecture Notes 11-12
© 2004 Prentice-Hall, Inc. Chap 11-34
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.
© 2004 Prentice-Hall, Inc. Chap 11-35
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
© 2004 Prentice-Hall, Inc. Chap 11-36
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
Statistics for Managers Using Microsoft Excel, 2/e © 1999 Prentice-Hall, Inc.
Chapter 11 Student Lecture Notes 11-13
© 2004 Prentice-Hall, Inc. Chap 11-37
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
© 2004 Prentice-Hall, Inc. Chap 11-38
Levene’s Test Example:Solution
F0 3.89
H0: H1: Not All Equalα = .05df1= 2 df2 = 12
Critical Value(s):
Test Statistic:
Decision:
Conclusion:Do not reject at α = 0.05.
There is no evidence that at least one differs from the rest.
α = 0.05
2 2 21 2 3σ σ σ= =
0.0337 0.09700.3475
MSAFMSW
= = =
2jσ
© 2004 Prentice-Hall, Inc. Chap 11-39
Randomized Blocked DesignItems are Divided into Blocks
Individual items in different samples are matched, or repeated measurements are takenReduced within group variation (i.e., remove the effect of block before testing)
Response of Each Treatment Group is ObtainedAssumptions
Same as completely randomized designNo interaction effect between treatments and blocks
Statistics for Managers Using Microsoft Excel, 2/e © 1999 Prentice-Hall, Inc.
Chapter 11 Student Lecture Notes 11-14
© 2004 Prentice-Hall, Inc. Chap 11-40
Randomized Blocked Design(Example)
22 hrs20 hrs29 hrs
28 hrs25 hrs27 hrs
31 hrs17 hrs21 hrsDependent
Variable(Response)
BlockedExperiment
Units
Factor Levels(Groups)
Factor (Training Method)
© 2004 Prentice-Hall, Inc. Chap 11-41
Randomized Block Design(Partition of Total Variation)
Variation Due to Group
SSA
Variation Among BlocksSSBL
Variation Among All
Observations SST
Commonly referred to as:Sum of Squares ErrorSum of Squares Unexplained
Commonly referred to as:Sum of Squares AmongAmong Groups Variation
=
+
+Variation Due
to Random Sampling
SSW
Commonly referred to as:Sum of Squares Among Blocks
© 2004 Prentice-Hall, Inc. Chap 11-42
Total Variation
( )
( )
2
1 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 va
c r
ijj i
ij
i
SST X X
rcn n rcX i j
X
= =
•
= −
==
= =
=
=
∑∑
lues in block the mean of all values for treatment level
1j
iX jdf n
• =
= −
Statistics for Managers Using Microsoft Excel, 2/e © 1999 Prentice-Hall, Inc.
Chapter 11 Student Lecture Notes 11-15
© 2004 Prentice-Hall, Inc. Chap 11-43
Among-Group Variation
( )2
1
1 (treatment group means)
1
1
c
jj
r
iji
j
SSA r X X
XX
rdf c
SSAMSAc
•=
=•
= −
=
= −
=−
∑
∑
© 2004 Prentice-Hall, Inc. Chap 11-44
Among-Block Variation
( )2
1
1 (block means)
1
1
r
ii
c
ijj
i
SSBL c X X
XX
cdf r
SSBLMSBLr
•=
=•
= −
=
= −
=−
∑
∑
© 2004 Prentice-Hall, Inc. Chap 11-45
Random Error
( )( )( )
( )( )
2
1 1
1 1
1 1
c r
ij i jj i
SSE X X X X
df r cSSEMSE
r c
• •= =
= − − +
= − −
=− −
∑∑
Statistics for Managers Using Microsoft Excel, 2/e © 1999 Prentice-Hall, Inc.
Chapter 11 Student Lecture Notes 11-16
© 2004 Prentice-Hall, Inc. Chap 11-46
Randomized Block F Test for Differences in c Means
No treatment effect
Test Statistic
Degrees of Freedom
0 1 2: cH µ µ µ• • •= = ••• =
1 : Not all are equaljH µ•
MSAFMSE
=
1 1df c= −( )( )2 1 1df r c= − −
0
α
UF F
Reject
© 2004 Prentice-Hall, Inc. Chap 11-47
Summary Table
SSTrc – 1Total
MSE = SSE/[(r – 1)•(c– 1)]
SSE(r – 1) •(c – 1)Error
MSBL/MSE
MSA/MSE
FStatistic
MSBL =SSBL/(r – 1)
SSBLr – 1AmongBlocks
MSA = SSA/(c – 1)SSAc – 1Among
Groups
Mean Squares
Sum ofSquares
Degrees of Freedom
Source ofVariation
© 2004 Prentice-Hall, Inc. Chap 11-48
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
Statistics for Managers Using Microsoft Excel, 2/e © 1999 Prentice-Hall, Inc.
Chapter 11 Student Lecture Notes 11-17
© 2004 Prentice-Hall, Inc. Chap 11-49
Randomized Block Design Example Computation
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
( ) ( ) ( )
( )
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
SSEMSA SSA c
MSE SSE r c
⎡ ⎤= − + − + −⎣ ⎦=== = =
= − = =⎡ ⎤⎣ ⎦
1
2
3
24.9322.6120.59
22.71
XXX
X
•
•
•
=
=
=
=
5315
rcn
===
© 2004 Prentice-Hall, Inc. Chap 11-50
Randomized Block Design Example: Summary Table
SST=58.217214Total
MSE = 1.0503
SSE=8.40258Error
.6627/1.0503=.6039
23.582/1.0503=22.452
FStatistic
MSBL =.6627
SSBL=2.65074Among
Blocks
MSA = 23.582
SSA=47.1642Among
Groups
Mean Squares
Sum ofSquares
Degrees of Freedom
Source ofVariation
© 2004 Prentice-Hall, Inc. Chap 11-51
Randomized Block Design Example: Solution
F0 4.46
H0: µ1 = µ2 = µ3
H1: Not All Equalα = .05df1= 2 df2 = 8
Critical Value(s):
Test Statistic:
Decision:
Conclusion:Reject at α = 0.05.
There is evidence that atleast one µ j differs fromthe rest.
α = 0.05
FMSAMSE
= = =23 5821.0503
22.45.
Statistics for Managers Using Microsoft Excel, 2/e © 1999 Prentice-Hall, Inc.
Chapter 11 Student Lecture Notes 11-18
© 2004 Prentice-Hall, Inc. Chap 11-52
Randomized Block Designin Excel
Tools | Data Analysis | ANOVA: Two Factor Without ReplicationExample Solution in Excel Spreadsheet
Microsoft Excel Worksheet
© 2004 Prentice-Hall, Inc. Chap 11-53
The Tukey Procedure
Similar to the Tukey Procedure for the Completely Randomized Design Case
Critical Range
( )( )( , 1 1 )Critical Range U c r cMSEQ
r− −=
© 2004 Prentice-Hall, Inc. Chap 11-54
The Tukey Procedure: Example1. Compute absolute mean
differences: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
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 cMSEQ
r− −= = =
Statistics for Managers Using Microsoft Excel, 2/e © 1999 Prentice-Hall, Inc.
Chapter 11 Student Lecture Notes 11-19
© 2004 Prentice-Hall, Inc. Chap 11-55
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?
© 2004 Prentice-Hall, Inc. Chap 11-56
Two-Way ANOVA
AssumptionsNormality
Populations are normally distributed
Homogeneity of Variance
Populations have equal variances
Independence of Errors
Independent random samples are drawn
(continued)
© 2004 Prentice-Hall, Inc. Chap 11-57
SSE
Two-Way ANOVA Total Variation Partitioning
Variation Due to Factor A
Variation Due to Random Sampling
Variation Due to Interaction
SSA
SSABSST
Variation Due to Factor B
SSB
Total Variation
d.f.= n-1
d.f.= r-1
=
+
+d.f.= c-1
+d.f.= (r-1)(c-1)
d.f.= rc(n’-1)
Statistics for Managers Using Microsoft Excel, 2/e © 1999 Prentice-Hall, Inc.
Chapter 11 Student Lecture Notes 11-20
© 2004 Prentice-Hall, Inc. Chap 11-58
Two-Way ANOVA Total Variation Partitioning
'
the number of levels of factor Athe number of levels of factor Bthe number of values (replications) for each cellthe total number of observations in the experiment
the value of the -th oijk
rcnnX k
==
=== bservation for level of
factor A and level of factor Bi
j
© 2004 Prentice-Hall, Inc. Chap 11-59
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 XX
rcn n
= = =
= = = = = =
= −
= =
=
∑∑∑
∑∑∑ ∑∑∑
© 2004 Prentice-Hall, Inc. Chap 11-60
Factor A Variation
( )2'
1
r
ii
SSA cn X X••
=
= −∑
Sum of Squares Due to Factor A = the difference among the means of the various levels of factor A and the grand mean
Statistics for Managers Using Microsoft Excel, 2/e © 1999 Prentice-Hall, Inc.
Chapter 11 Student Lecture Notes 11-21
© 2004 Prentice-Hall, Inc. Chap 11-61
Factor B Variation
( )2'
1
c
jj
SSB rn X X• •
=
= −∑
Sum of Squares Due to Factor B = the difference among the means of the various levels of factor B and the grand mean
© 2004 Prentice-Hall, Inc. Chap 11-62
Interaction Variation
( )2'
1 1
r c
ij i ji 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
© 2004 Prentice-Hall, Inc. Chap 11-63
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 •
= = =
= −∑∑∑
Statistics for Managers Using Microsoft Excel, 2/e © 1999 Prentice-Hall, Inc.
Chapter 11 Student Lecture Notes 11-22
© 2004 Prentice-Hall, Inc. Chap 11-64
Two-Way ANOVA:The F Test Statistic
F Test for Factor B Main Effect
F Test for Interaction Effect
H0: µ1 ..= µ2 .. = ••• = µr ..
H1: Not all µi .. are equal
H0: ΑΒij = 0 (for all i and j)
H1: ΑΒij ≠ 0
H0: µ.1. = µ.2. = ••• = µ.c.
H1: Not all µ.j. are equal
Reject if F > FU
Reject if F > FU
Reject if F > FU
1
MSA SSAF MSAMSE r
= =−
F Test for Factor A Main Effect
1
MSB SSBF MSBMSE c
= =−
( )( )
1 1MSAB SSABF MSABMSE r c
= =− −
© 2004 Prentice-Hall, Inc. Chap 11-65
Two-Way ANOVASummary Table
SSTr•c •n’ – 1Total
MSE = SSE/[r•c •(n’ – 1)]
SSEr•c •(n’ – 1)Error
MSAB/MSE
MSB/MSE
MSA/MSE
FStatistic
MSAB =SSAB/ [(r – 1)(c – 1)]SSAB(r – 1)(c – 1)AB
(Interaction)
MSB =SSB/(c – 1)
SSBc – 1Factor B(Column)
MSA = SSA/(r – 1)SSAr – 1Factor A
(Row)
Mean Squares
Sum ofSquares
Degrees of Freedom
Source ofVariation
© 2004 Prentice-Hall, Inc. Chap 11-66
Features of Two-Way ANOVA F Test
Degrees of Freedom Always Add Uprcn’-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 UpTotal=Error+Column+Row+Interaction
Statistics for Managers Using Microsoft Excel, 2/e © 1999 Prentice-Hall, Inc.
Chapter 11 Student Lecture Notes 11-23
© 2004 Prentice-Hall, Inc. Chap 11-67
Kruskal-Wallis Rank Test
Used to Analyze Completely Randomized Experimental Designs Extension of Wilcoxon Rank Sum Test
Tests the equality of more than 2 population medians
Distribution-Free Test ProcedureUse χ2 Distribution to Approximate if Each Sample Group Size nj > 5
df = c – 1
© 2004 Prentice-Hall, Inc. Chap 11-68
Kruskal-Wallis Rank Test
AssumptionsIndependent random samples are drawnContinuous dependent variableData may be ranked both within and among samplesPopulations have same variabilityPopulations have same shape
Robust with Regard to Last 2 ConditionsUse F test in completely randomized designs and when the more stringent assumptions hold
© 2004 Prentice-Hall, Inc. Chap 11-69
Kruskal-Wallis Rank Test Procedure
Obtain RanksIn 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 Tj2
2
1
12 3( 1)( 1)
cj
j j
TH n
n n n=
⎡ ⎤= − +⎢ ⎥
+⎢ ⎥⎣ ⎦∑
1 2 cn n n n= + + +L
Statistics for Managers Using Microsoft Excel, 2/e © 1999 Prentice-Hall, Inc.
Chapter 11 Student Lecture Notes 11-24
© 2004 Prentice-Hall, Inc. Chap 11-70
Kruskal-Wallis Rank Test Procedure
Compute Test Statistic
# of observation in the j –th sample ; ( j = 1, 2, …, c )
sum of the ranks assigned to the j –thsample
square of H may be approximated by chi-square distribution with df = c –1 when each nj >5
(continued)
2
1
12 3( 1)( 1)
cj
j j
TH n
n n n=
⎡ ⎤= − +⎢ ⎥
+⎢ ⎥⎣ ⎦∑
1 2 cn n n n= + + +L
jn =
jT =
2jT = jT
© 2004 Prentice-Hall, Inc. Chap 11-71
Kruskal-Wallis Rank Test Procedure
Critical Value for a Given αUpper tail
Decision RuleReject H0: M1 = M2 = ••• = Mc if test statistic H >Otherwise, do not reject H0
(continued)
2χU
2χU
© 2004 Prentice-Hall, Inc. Chap 11-72
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?
Statistics for Managers Using Microsoft Excel, 2/e © 1999 Prentice-Hall, Inc.
Chapter 11 Student Lecture Notes 11-25
© 2004 Prentice-Hall, Inc. Chap 11-73
Machine1 Machine2 Machine314 9 215 6 712 10 111 8 413 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
© 2004 Prentice-Hall, Inc. Chap 11-74
Example Solution: Step 2 Test Statistic Computation
212 3( 1)
( 1) 1
2 2 212 65 38 17 3(15 1)15(15 1) 5 5 5
11.58
Tc jH nn n nj j
⎡ ⎤⎢ ⎥= − +∑⎢ ⎥+ =⎢ ⎥⎣ ⎦
⎡ ⎤⎛ ⎞⎢ ⎥⎜ ⎟= + + − +
⎜ ⎟+⎢ ⎥⎝ ⎠⎣ ⎦=
© 2004 Prentice-Hall, Inc. Chap 11-75
Kruskal-Wallis Test Example Solution
H0: M1 = M2 = M3
H1: Not all equalα = .05df = c - 1 = 3 - 1 = 2Critical Value(s): Reject at
Test Statistic:
Decision:
Conclusion:There is evidence that population medians are not all equal.
α = .05
α = .05.
H = 11.58
0 5.991
Statistics for Managers Using Microsoft Excel, 2/e © 1999 Prentice-Hall, Inc.
Chapter 11 Student Lecture Notes 11-26
© 2004 Prentice-Hall, Inc. Chap 11-76
Kruskal-Wallis Test in PHStat
PHStat | c-Sample Tests | Kruskal-Wallis Rank Sum Test …Example Solution in Excel Spreadsheet
Microsoft Excel Worksheet
© 2004 Prentice-Hall, Inc. Chap 11-77
Friedman Rank Test
Used to Analyze Randomized Block Experimental DesignsTests the equality of more than 2 population medians
Distribution-Free Test Procedure
Use χ2 Distribution to Approximate if the Number of Blocks r > 5
df = c – 1
© 2004 Prentice-Hall, Inc. Chap 11-78
Friedman Rank Test
AssumptionsThe r blocks are independentThe random variable is continuousThe data constitute at least an ordinal scale of measurementNo interaction between the r blocks and the ctreatment levelsThe c populations have the same variabilityThe c populations have the same shape
Statistics for Managers Using Microsoft Excel, 2/e © 1999 Prentice-Hall, Inc.
Chapter 11 Student Lecture Notes 11-27
© 2004 Prentice-Hall, Inc. Chap 11-79
Friedman Rank Test:Procedure
Replace the c observations by their ranks in each of the r blocks; assign average rank for tiesTest 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 freedomThe rejection region is in the right tail
( ) ( )2
1
12 3 11
c
R jj
F R r crc c ⋅
=
= − ++ ∑
© 2004 Prentice-Hall, Inc. Chap 11-80
Friedman Rank Test: ExampleAs 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
© 2004 Prentice-Hall, Inc. Chap 11-81
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 1 224.1 23.5 19.75 3 2 1
23.74 22.75 20.6 3 2 125.1 21.6 20.4 3 2 1
15 9 6225 81 36
Friedman Rank Test: Computation Table
2. jR. jR
( ) ( )
( ) ( ) ( ) ( ) ( ) ( )
2
1
12 3 11
12 342 3 5 4 8.45 3 4
c
R jj
F R r crc c ⋅
=
= − ++
= − =
∑
Statistics for Managers Using Microsoft Excel, 2/e © 1999 Prentice-Hall, Inc.
Chapter 11 Student Lecture Notes 11-28
© 2004 Prentice-Hall, Inc. Chap 11-82
Friedman Rank Test Example Solution
H0: M1 = M2 = M3
H1: Not all equalα = .05df = c - 1 = 3 - 1 = 2Critical Value: Reject at
Test Statistic:
Decision:
Conclusion:There is evidence that population medians are not all equal.
α = .05
α = .05
FR = 8.4
0 5.991
© 2004 Prentice-Hall, Inc. Chap 11-83
Chapter Summary
Described the Completely Randomized Design: One-Way Analysis of Variance
ANOVA AssumptionsF Test for Differences in More than Two MeansThe Tukey-Kramer ProcedureLevene’s Test for Homogeneity of Variance
Discussed the Randomized Block DesignF Test for the Difference in More than Two MeansThe Tukey Procedure
© 2004 Prentice-Hall, Inc. Chap 11-84
Chapter Summary
Described the Factorial Design: Two-Way Analysis of Variance
Examine effects of factors and interaction
Discussed Kruskal-Wallis Rank Test: Nonparametric Analysis for the Completely Randomized Design
Illustrated Friedman Rank Test: Nonparametric Analysis for the Randomized Block Design
(continued)
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