Business Statistics, 5th ed.by Ken Black
Chapter 11
Analysis of Variance
& Design of Experiments
Discrete Distributions
PowerPoint presentations prepared by Lloyd Jaisingh, Morehead State University
Learning ObjectivesLearning Objectives
• Understand the differences between various experimental designs and when to use them.
• Compute and interpret the results of a one-way ANOVA.
• Compute and interpret the results of a random block design.
• Compute and interpret the results of a two-way ANOVA.
• Understand and interpret interaction.• Know when and how to use multiple comparison
techniques.
Introduction to Design of Experiments
Introduction to Design of Experiments
Experimental Design- a plan and a structure to test hypotheses in which the researcher controls or manipulates one or more variables.
Introduction to Design of ExperimentsIndependent Variable • Treatment variable is one that the experimenter
controls or modifies in the experiment.• Classification variable is a characteristic of the
experimental subjects that was present prior to the experiment, and is not a result of the experimenter’s manipulations or control.
• Levels or Classifications are the subcategories of the independent variable used by the researcher in the experimental design.
• Independent variables are also referred to as factors.
Introduction to Design of Experiments
• Dependent Variable - the response to the different levels of the
independent variables.• Analysis of Variance (ANOVA) – a group
of statistical techniques used to analyze experimental designs.
Three Types of Experimental Designs
Three Types of Experimental Designs
• Completely Randomized Design – subjects are assigned randomly to treatments; single independent variable.
• Randomized Block Design – includes a blocking variable; single independent variable.
• Factorial Experiments – two or more independent variables are explored at the same time; every level of each factor are studied under every level of all other factors.
Completely Randomized DesignCompletely Randomized Design
Machine Operator
Valve OpeningMeasurements
1
.
.
.
2
.
.
.
4
.
.
.
.
.
.
3
Valve Openings by Operator
1 2 3 4
6.33 6.26 6.44 6.29
6.26 6.36 6.38 6.23
6.31 6.23 6.58 6.19
6.29 6.27 6.54 6.21
6.4 6.19 6.56
6.5 6.34
6.19 6.58
6.22
Analysis of Variance: AssumptionsAnalysis of Variance: Assumptions
• Observations are drawn from normally distributed populations.
• Observations represent random samples from the populations.
• Variances of the populations are equal.
One-Way ANOVA: Procedural Overview
One-Way ANOVA: Procedural Overview
H
H
ok
a
:
:1 2 3
At least one of the means is different from the others
FMSC
MSE
If F > , reject H .
If F , do not reject H .
c o
c o
FF
One-Way ANOVA: Sums of Squares Definitions
One-Way ANOVA: Sums of Squares Definitions
valueindividual
levelor group treatmenta ofmean =
mean grand =X
leveltment given trea ain nsobservatio ofnumber
levels treatmentofnumber =
level treatmenta =
level treatmenta ofmember particular :
nn
ij
SSE + SSC = SST
squares of sumbetween + squares of sumerror = squares of sum total
XX
n
X
ij
j
j
1 1
2
1
2
1=i 1j=
2 jj
C
j
iwhere
jijjji
C
j
C
j
C
XXXXnX
Partitioning Total Sum of Squares of VariationPartitioning Total Sum of Squares of Variation
SST(Total Sum of Squares)
SSC(Treatment Sum of Squares)
SSE(Error Sum of Squares)
One-Way ANOVA: Computational Formulas
One-Way ANOVA: Computational Formulas
MSE
MSCF
SSEMSE
SSCMSC
Nn
ijSST
CNn
jijSSE
Cj
SSC
df
df
dfXX
dfXX
dfXXn
E
C
Tj
C
i
Ei
C
j
C
C
jj
j
j
1
1
1 1
2
1 1
2
1
2
where
X
: i = a particular member of a treatment level
j = a treatment level
C = number of treatment levels
= number of observations in a given treatment level
X = grand mean
column mean
= individual value
j
j
ij
n
X
One-Way ANOVA: Preliminary Calculations
One-Way ANOVA: Preliminary Calculations
1 2 3 4
6.33 6.26 6.44 6.29
6.26 6.36 6.38 6.23
6.31 6.23 6.58 6.19
6.29 6.27 6.54 6.21
6.4 6.19 6.56
6.5 6.34
6.19 6.58
6.22
Tj T1 = 31.59 T2 = 50.22 T3 = 45.42 T4 = 24.92 T = 152.15
nj n1 = 5 n2 = 8 n3 = 7 n4 = 4 N = 24
Mean 6.318000 6.277500 6.488571 6.230000 6.339583
15492.0)230.619.6()230.622.6(
)2775.636.6()2775.626.6()318.64.6(
)318.629.6()318.631.6()318.626.6()318.633.6(
23658.0)339583.623.6()339583.6488571.6(
)339583.62775.6()339583.6318.6(
22
222
2222
1 1
2
22
22
1
2
47
85[
n
jijSSE
jSSC
j
i
C
j
C
jj
XX
XXn
One-Way ANOVA: Sum of Squares Calculations
One-Way ANOVA: Sum of Squares Calculations
39150.0)339583.619.6(
)339583.622.6()339583.631.6(
)339583.626.6()339583.633.6(
2
22
22
1 1
2
n
ijSSTj
i
C
jXX
One-Way ANOVA: Sum of Squares Calculations
One-Way ANOVA: Sum of Squares Calculations
One-Way ANOVA: Mean Square and F Calculations
One-Way ANOVA: Mean Square and F Calculations
18.10007746.
078860.
007746.20
15492.
078860.3
23658.
231241
20424
3141
MSE
MSCF
SSEMSE
SSCMSC
N
CN
C
df
df
dfdfdf
E
C
T
E
C
Analysis of Variance for Valve Openings
Analysis of Variance for Valve Openings
Source of Variancedf SS MS F
Between 3 0.23658 0.07886010.18
Error 20 0.15492 0.007746Total 23 0.39150
F 20,3,05.
df1
df 2
A Portion of the F Table for = 0.05A Portion of the F Table for = 0.05
1 2 3 4 5 6 7 8 9
1161.4
5199.5
0215.7
1224.5
8230.1
6233.9
9236.7
7238.8
8240.5
4
… … … … … … … … … …
18 4.41 3.55 3.16 2.93 2.77 2.66 2.58 2.51 2.46
19 4.38 3.52 3.13 2.90 2.74 2.63 2.54 2.48 2.42
20 4.35 3.49 3.10 2.87 2.71 2.60 2.51 2.45 2.39
21 4.32 3.47 3.07 2.84 2.68 2.57 2.49 2.42 2.37
df2
One-Way ANOVA: Procedural SummaryOne-Way ANOVA: Procedural Summary
.Hreject do ,10.3 F
.Hreject ,10.3 > F
oc
oc
FF
If
If
Rejection Region
Critical Value10.3
11,9,05.F
Non rejectionRegion
20
3
2
1
others thefromdifferent is
means theof oneleast At :H
:H
a
4321o
.Hreject ,10.3 >10.18 = F Since ocF
Excel Output for the Valve Opening Example
Excel Output for the Valve Opening Example
Anova: Single Factor
SUMMARY
Groups Count Sum Average Variance
Operator 1 5 31.59 6.318 0.00277
Operator 2 8 50.22 6.2775 0.0110786
Operator 3 7 45.42 6.488571429 0.0101143
Operator 4 4 24.92 6.23 0.0018667
ANOVA
Source of Variation SS df MS F P-value F crit
Between Groups 0.236580119 3 0.07886004 10.181025 0.00028 3.09839
Within Groups 0.154915714 20 0.007745786
Total 0.391495833 23
MINITAB Output for the Valve Opening Example
MINITAB Output for the Valve Opening Example
Multiple Comparison TestsMultiple Comparison Tests
An analysis of variance (ANOVA) test is an overall test of differences among groups.
Multiple Comparison techniques are used to identify which pairs of means are significantly different given that the ANOVA test reveals overall significance.
• Tukey’s honestly significant difference (HSD) test requires equal sample sizes
• Tukey-Kramer Procedure is used when sample sizes are unequal.
Tukey’s Honestly Significant Difference (HSD) Test
Tukey’s Honestly Significant Difference (HSD) Test
HSDMSE
n
,C,N-C
,C,N-C
q
q
where: MSE = mean square error
n = sample size
= critical value of the studentized range distribution from Table A.10
Data from Demonstration Problem 11.1Data from Demonstration Problem 11.1
PLANT (Employee Age)
1 2 3
29 32 25
27 33 24
30 31 24
27 34 25
28 30 26
Group Means 28.2 32.0 24.8
nj 5 5 5
C = 3
dfE = N - C = 12 MSE = 1.63
q Values for = .01q Values for = .01
Degrees of Freedom
1
2
3
4
.
11
12
2 3 4 5
90 135 164 186
14 19 22.3 24.7
8.26 10.6 12.2 13.3
6.51 8.12 9.17 9.96
4.39 5.14 5.62 5.97
4.32 5.04 5.50 5.84
.
...
Number of Populations
. , ,.
01 3 125 04q
Tukey’s HSD Test for the Employee Age Data
Tukey’s HSD Test for the Employee Age Data
HSDMSE
nC N Cq
X
X
X
, ,.
..
. . .
. . .
. . .
5 04163
52 88
28 2 32 0 38
28 2 24 8 3 4
32 0 24 8 7 2
2
3
3
1
1
2
X
X
X
Tukey’s HSD Test for the Employee Age Data using MINITAB
Tukey’s HSD Test for the Employee Age Data using MINITAB
Intervals do notcontain 0,so significantdifferences between themeans.
Tukey-Kramer Procedure: The Case of Unequal Sample Sizes
Tukey-Kramer Procedure: The Case of Unequal Sample Sizes
HSDMSE
r sn n
,C,N-C
r
th
s
th
,C,N-C
q
n rn s
q
where: MSE = mean square error
= sample size for sample
= sample size for sample
= critical value of the studentized range distribution from Table A.10
2
1 1( )
Freighter Example: Means and Sample Sizes for the Four Operators
Freighter Example: Means and Sample Sizes for the Four Operators
Operator Sample Size Mean1 5 6.31802 8 6.27753 7 6.48864 4 6.2300
Tukey-Kramer Results for the Four OperatorsTukey-Kramer Results for the Four Operators
PairCritical Difference
|Actual Differences|
1 and 2 .1405 .0405
1 and 3 .1443 .1706*
1 and 4 .1653 .0880
2 and 3 .1275 .2111*
2 and 4 .1509 .0475
3 and 4 .1545 .2586*
*denotes significant at .05
Partitioning the Total Sum of Squares in the Randomized Block Design
Partitioning the Total Sum of Squares in the Randomized Block Design
SST(Total Sum of Squares)
SSC(Treatment
Sum of Squares)
SSE(Error Sum of Squares)
SSR(Sum of Squares
Blocks)
SSE’(Sum of Squares
Error)
A Randomized Block DesignA Randomized Block Design
Individualobservations
.
.
.
.
.
.
.
.
.
.
.
.
Single Independent Variable
BlockingVariable
.
.
.
.
.
Randomized Block Design Treatment Effects: Procedural Overview
Randomized Block Design Treatment Effects: Procedural Overview
others thefromdifferent is means theof oneleast At :H
:H
a
321o
k
FMSC
MSE
If F > , reject H .
If F , do not reject H .
c o
c o
FF
Randomized Block Design: Computational Formulas
Randomized Block Design: Computational Formulas
SSC n j C
SSR C i n
SSE ij i i C n N n C
SST ij N
MSCSSC
C
MSRSSR
n
MSESSE
N n CMSC
MSEMSR
MSE
X X df
X X df
X X X X df
X X df
F
F
j
C
C
i
n
R
i
n
j
n
E
i
n
j
n
E
treatments
blocks
2
1
2
12
11
2
11
1
1
1 1 1
1
1
1
1
( )
( )
( )
( )where: i = block group (row)
j = a treatment level (column)
C = number of treatment levels (columns)
n = number of observations in each treatment level (number of blocks - rows)
individual observation
treatment (column) mean
block (row) mean
X = grand mean
N = total number of observations
ij
j
i
X
X
X
SSC sum of squares columns (treatment)
SSR = sum of squares rows (blocking)
SSE = sum of squares error
SST = sum of squares total
Randomized Block Design: Tread-Wear Example
Randomized Block Design: Tread-Wear Example
Supplier
1
2
3
4
Slow Medium FastBlock Means ( )
3.7 4.5 3.1 3.77
3.4 3.9 2.8 3.37
3.5 4.1 3.0 3.53
3.2 3.5 2.6 3.10
5
Treatment Means( )
3.9 4.8 3.4 4.03
3.54 4.16 2.98 3.56
Speed
jX
iX
XC = 3
n = 5
N = 15
SSC n j
SSR C i
X X
X X
j
C
i
n
2
1
2 2 2
2
1
2 2 2 2 2
5
3
54 356 16 356 98 3563484
77 356 37 356 53 356 10 356 03 3561549
( )
(3. . ) (4. . ) (2. . ).
( )
(3. . ) (3. . ) (3. . ) (3. . ) (4. . ).
[
[ ]
Randomized Block Design: Sum of Squares Calculations (Part 1)
Randomized Block Design: Sum of Squares Calculations (Part 1)
Randomized Block Design: Sum of Squares Calculations (Part 2)
Randomized Block Design: Sum of Squares Calculations (Part 2)
176.5)56.34.3()56.36.2()56.34.3()56.37.3(
)(
143.0)56.303.498.24.3()56.310.398.26.2(
)56.337.354.34.3()56.377.354.37.3(
)(
2222
1 1
2
22
22
1 1
2
n
i
C
j
n
i
C
j
XX
XXXX
ijSST
ijijSSE
Randomized Block Design: Mean Square Calculations
Randomized Block Design: Mean Square Calculations
MSCSSC
C
MSRSSR
n
MSESSE
N n C
FMSC
MSE
1
3 484
21742
1
1549
40 387
1
0143
80 018
1742
0 01896 78
..
..
..
.
..
Analysis of Variance for the Tread-Wear Example
Analysis of Variance for the Tread-Wear Example
Source of VarianceSS df MS F
Treatment 3.484 2 1.74296.78
Block 1.549 4 0.38721.50
Error 0.143 8 0.018
Total 5.176 14
Randomized Block Design Treatment Effects: Procedural Summary
Randomized Block Design Treatment Effects: Procedural Summary
H
H
o
a
:
:1 2 3
At least one of the means is different from the others
78.96018.0
742.1
MSE
MSCF
F = 96.78 > = 8.65, reject H ..01,2,8 oF
Randomized Block Design Blocking Effects: Procedural Overview
Randomized Block Design Blocking Effects: Procedural Overview
H
H
o
a
:
:1 2 3 4 5
At least one of the blocking means is different from the others
5.21018.
387.
MSE
MSRF
F = 21.5 > = 7.01, reject H .F o. , ,01 4 8
Excel Output for Tread-Wear Example: Randomized Block Design
Excel Output for Tread-Wear Example: Randomized Block Design
Anova: Two-Factor Without Replication
SUMMARY Count Sum Average VarianceSupplier 1 3 11.3 3.7666667 0.4933333Supplier 2 3 10.1 3.3666667 0.3033333Supplier 3 3 10.6 3.5333333 0.3033333Supplier 4 3 9.3 3.1 0.21Supplier 5 3 12.1 4.0333333 0.5033333
Slow 5 17.7 3.54 0.073Medium 5 20.8 4.16 0.258Fast 5 14.9 2.98 0.092
ANOVASource of Variation SS df MS F P-value F critRows 1.5493333 4 0.3873333 21.719626 0.0002357 7.0060651Columns 3.484 2 1.742 97.682243 2.395E-06 8.6490672Error 0.1426667 8 0.0178333
Total 5.176 14
MINITAB Output for Tread-Wear Example: Randomized Block DesignMINITAB Output for Tread-Wear
Example: Randomized Block Design
Blocking variable Suppliers
Two-Way Factorial DesignTwo-Way Factorial Design
Cells
.
.
.
.
.
.
.
.
.
.
.
.
Column Treatment
RowTreatment
.
.
.
.
.
Two-Way ANOVA: HypothesesTwo-Way ANOVA: Hypotheses
Row Effects: H : Row Means are all equal.
H : At least one row mean is different from the others.
Columns Effects: H : Column Means are all equal.
H : At least one column mean is different from the others.
Interaction Effects: H : The interaction effects are zero.
H : There is an interaction effect.
o
a
o
a
o
a
Formulas for Computing a Two-Way ANOVA
Formulas for Computing a Two-Way ANOVA
SSR nC i R
SSC nR j C
SSI n ij i j R C
SSE ijk ij RC n
SST ijk N
MSRSSR
R
MSR
MSE
MSC
X X df
X X df
X X X X df
X X df
X X df
F
i
R
R
j
C
C
j
C
i
R
I
k
n
j
C
i
R
E
a
n
r
R
c
C
T
R
2
1
2
1
2
11
2
111
2
111
1
1
1 1
1
1
1
( )
( )
( )
( )
( )
SSC
C
MSC
MSE
MSISSI
R C
MSI
MSE
MSESSE
RC n
where
C
I
F
F
1
1 1
1
:
n = number of observations per cell
C = number of column treatments
R = number of row treatments
i = row treatment level
j = column treatment level
k = cell member
= individual observation
= cell mean
= row mean
= column mean
X = grand mean
ijk
ij
i
j
XXXX
A 2 3 Factorial Design with Interaction
A 2 3 Factorial Design with Interaction
CellMeans
C1 C2 C3
Row effects
R1
R2
Column
A 2 3 Factorial Design with Some Interaction
A 2 3 Factorial Design with Some Interaction
CellMeans
C1 C2 C3
Row effects
R1
R2
Column
A 2 3 Factorial Design with No Interaction
A 2 3 Factorial Design with No Interaction
CellMeans
C1 C2 C3
Row effects
R1
R2
Column
A 2 3 Factorial Design: Data and Measurements for CEO Dividend Example
A 2 3 Factorial Design: Data and Measurements for CEO Dividend Example
N = 24n = 4
X=2.7083
1.75 2.75 3.625
Location Where CompanyStock is Traded
How Stockholders are Informed of
DividendsNYSE AMEX OTC
Annual/Quarterly Reports
2121
2332
4343
2.5
Presentations to Analysts
2312
3324
4434
2.9167
Xj
Xi
X11=1.5
X23=3.75X22=3.0X21=2.0
X13=3.5X12=2.5
A 2 3 Factorial Design: Calculations for the CEO Dividend Example (Part 1)A 2 3 Factorial Design: Calculations for the CEO Dividend Example (Part 1)
SSR X X
SSC X X
SSI X X X X
nC i
nR j
n ij i j
i
R
j
C
j
C
i
R
2
1
2 2
2
1
2 2 2
2
11
2
4 3 2 5 2 7083 2 9167 2 7083
4 2 175 2 7083 2 75 2 7083 3 625 2 7083
4 15 2 5 175 2 7083
10418
14 0833
( )
.
( )
.
( )
( )( )[( . . ) ( . . ) ]
( )( )[( . . ) ( . . ) ( . . ) ]
[( . . . . ) ( . . . . )
( . . . . ) ( . . . . )
( . . . . ) ( . . . . ) ]
.
2 5 2 5 2 75 2 7083
35 2 5 3 625 2 7083 2 0 2 9167 175 2 7083
3 0 2 9167 2 75 2 7083 3 75 2 9167 3 625 2 7083
2
2 2
2 2
00833
A 2 3 Factorial Design: Calculations for the CEO Dividend Example (Part 2)A 2 3 Factorial Design: Calculations for the CEO Dividend Example (Part 2)
SSE X X
SST X X
ijk ij
ijk
k
n
j
C
i
R
a
n
r
R
c
C
2
111
2 2 2 2
2
111
2 2 2 2
2 15 1 15 3 375 4 3757 7500
2 2 7083 1 2 7083 3 2 7083 4 2 708322 9583
( )
( . ) ( . ) ( . ) ( . ).
( )
( . ) ( . ) ( . ) ( . ).
A 2 3 Factorial Design: Calculations for the CEO Dividend Example (Part 3)A 2 3 Factorial Design: Calculations for the CEO Dividend Example (Part 3)
MSRSSR
R
MSR
MSE
MSCSSC
C
MSC
MSE
MSISSI
R C
MSI
MSE
MSESSE
RC n
R
C
I
F
F
F
1
10418
110418
10418
0 43062 42
1
14 0833
27 0417
7 0417
0 430616 35
1 1
0 0833
20 0417
0 0417
0 4306010
1
7 7500
180 4306
..
.
..
..
.
..
..
.
..
..
Analysis of Variance for the CEO Dividend Problem
Analysis of Variance for the CEO Dividend Problem
Source of VarianceSS df MS F
Row 1.0418 1 1.0418 2.42
Column 14.0833 2 7.0417 16.35*
Interaction 0.0833 2 0.0417 0.10
Error 7.7500 18 0.4306
Total 22.9583 23
*Denotes significance at = .01.
Excel Output for the CEO Dividend Example (Part 1)
Excel Output for the CEO Dividend Example (Part 1)
Anova: Two-Factor With Replication
SUMMARY NYSE ASE OTC TotalAQReport
Count 4 4 4 12Sum 6 10 14 30Average 1.5 2.5 3.5 2.5Variance 0.3333 0.3333 0.3333 1
PresentationCount 4 4 4 12Sum 8 12 15 35Average 2 3 3.75 2.9167Variance 0.6667 0.6667 0.25 0.9924
TotalCount 8 8 8Sum 14 22 29Average 1.75 2.75 3.625Variance 0.5 0.5 0.2679
Excel Output for the CEO Dividend Example (Part 2)
Excel Output for the CEO Dividend Example (Part 2)
ANOVASource of Variation SS df MS F P-value F critSample 1.0417 1 1.0417 2.4194 0.1373 4.4139Columns 14.083 2 7.0417 16.355 9E-05 3.5546Interaction 0.0833 2 0.0417 0.0968 0.9082 3.5546Within 7.75 18 0.4306
Total 22.958 23
MINITAB Output for the Demonstration Problem 11.4:
MINITAB Output for the Demonstration Problem 11.4:
MINITAB Output for the Demonstration Problem 11.4:
Interaction Plots
MINITAB Output for the Demonstration Problem 11.4:
Interaction Plots
321
4
3
2
1
4321
4
3
2
1
Warehouses
Length
1234
Warehouses
123
Length
Interaction Plot (data means) for DaysAbsent
321
4
3
2
1
4321
4
3
2
1
Warehouses
Length
1234
Warehouses
123
Length
Interaction Plot (data means) for DaysAbsent
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