ch11

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


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.


• 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


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: 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)




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


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

( )

( . ) ( . ) ( . ) ( . ).

( )

( . ) ( . ) ( . ) ( . ).


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)


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



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:



Interaction Plots


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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ch11

Documents

oneway anova

example anova

twoway anova

sum of squares calculations

independent variables

sum average variance

randomized design subjects

randomized block design