© 2002 Thomson / South-Western Slide 11-1 Chapter 11 Analysis of Variance and Chi-Square...

© 2002 Thomson / South-Western Slide 11-1

Chapter 11

Analysis of Variance and Chi-Square

Applications


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.


Learning Objectives, continued

• Compute and interpret the results of a two-way ANOVA.

• Understand and interpret interaction.• Understand the chi-square goodness-

of-fit test and how to use it.• Analyze data by using the chi-square

test of independence.


Introduction to Design of Experiments

Introduction to Design of Experiments

• An Experimental Design is a plan and a structure to test hypotheses in which the business analyst controls or manipulates one or more variables. It contains independent and dependent variables.

• Factors is another name for the independent variables of an experimental design.


Design of Experiments, continued

• Treatment variable is the independent variable that the experimenter either controls or modifies.

• Classification variable is the independent variable that was present prior to the experiment, and is not a result of the experimenter’s manipulations or control.


Design of Experiments, continued

• Levels or Classifications are the subcategories of the independent variable used by the business analyst in the experimental design.

• The Dependent Variable is the response to the different levels of the independent variables.


Three Types of Experimental Designs

Three Types of Experimental Designs

• Completely Randomized Design• Randomized Block Design• Factorial Experiments


Completely Randomized DesignCompletely Randomized Design

Machine Operator

1 2 3

.

.

.

.

.

.

.

.

.

Valve OpeningMeasurements


Example: Number of Foreign Freighters Docking in each Port per Day

Example: Number of Foreign Freighters Docking in each Port per Day

Long Beach

5

7

4

2

Houston

2

3

5

4

6

New York

8

4

6

7

9

8

New Orleans

3

5

3

4

2


Analysis of Variance (ANOVA): Assumptions

Analysis of Variance (ANOVA): 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 OverviewOne-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


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: Sums of Squares Definitions

One-Way ANOVA: Sums of Squares Definitions

j 2 2C 2

j=1 i=1 1 1 1

Total sum of squares = error sum of squares + between sum of squares

SST = SSC + SSE

n

ji

:

particular member of a treatment level

= a treatme

XjC C

j j i

n

ijj jj

where

i

j

X Xn X XX

j

j

ij

nt level

= number of treatment levels

number of observations in a given treatment level

X= grand mean

= mean of a treatment group or level

individual value

n

XX

C


One-Way ANOVA: Computational Formulas

One-Way ANOVA: Computational Formulas

SSC j C

SSE ij j

nN C

SST ij

nN

MSCSSC

MSESSE

FMSC

MSE

jj

C

C

ij

C

E

ij

C

T

C

E

n X X df

X X df

X X df

df

df

j

j

2

1

2

11

2

11

1

1

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


Freighter One-Way ANOVA:Preliminary Calculations

Freighter One-Way ANOVA:Preliminary Calculations

Long Beach

5742

T1 = 18

n1= 4

Houston

23546

T2 = 20

n2 = 5

New York

846798

T3 = 42

n3 = 6

New Orleans

35342

T4 = 17

n4 = 5

T = 97N = 20


Freighter One-Way ANOVA: Sum of Squares CalculationsFreighter One-Way ANOVA: Sum of Squares Calculations

1 2 3 4

1 2 3 4

1 2 3 4

: 18 20 42 42 97

: 4 5 6 5 20

: 4.5 4.0 7.0 3.4 4.85

j

j

j

T

N

X

T T T T T

n n n n n

X X X X X


SSC j

SSE ij j

n

jj

C

ij

C

n X X

X Xj

2

1

2 2 2 2

2

11

2 2 2 2

2 2 2 2

4 5 6 55 485 0 485 7 485 4 48542 35

5 45 45 45 454 0 4 0 34 34

44

[ (4. . ) (4. . ) (4. . ) (3. . ).

( . ) (7 . ) (4 . ) (2 . )(2 . ) (3 . ) (4 . ) (2 . ).

20

5 485 485 485 485 4858655

2

11

2 2 2 2 2

SST ij

n

X Xij

C j

( . ) (7 . ) (4 . ) (4 . ) (2 . ).

Freighter One-Way ANOVA: Sum of Squares Calculations, continued

Freighter One-Way ANOVA: Sum of Squares Calculations, continued


Freighter One-Way ANOVA: Mean Square and F Calculations

Freighter One-Way ANOVA: Mean Square and F Calculations

C

E

T

C

E

dfdfdf

df

df

C

N C

N

MSCSSC

MSESSE

FMSC

MSE

1 4 1 3

20 4 16

1 20 1 19

42 35

31412

44 20

162 76

1412

2 76512

..

..

.

..


Freighter Example: Analysis of VarianceFreighter Example:

Analysis of Variance

Source of Variancedf SS MS F

Between Factor 3 42.35 14.12 5.12Error 16 44.20 2.76Total 19 86.55


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 91 161.45 199.50 215.71 224.58 230.16 233.99 236.77 238.88 240.54

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .15 4.54 3.68 3.29 3.06 2.90 2.79 2.71 2.64 2.5916 4.49 3.63 3.24 3.01 2.85 2.74 2.66 2.59 2.5417 4.45 3.59 3.20 2.96 2.81 2.70 2.61 2.55 2.49

. , ,05 3 16F

DenominatorDegrees of Freedom

Numerator Degrees of Freedom


Freighter One-Way ANOVA: Procedural Summary

Freighter One-Way ANOVA: Procedural Summary

If

If

F > reject H .

F do reject H .

c o

c o

FF

3 24

3 24

. ,

. ,

Rejection Region

Critical Value. , , .05 9 11 3 24F

Non rejectionRegion

1

2

3

16

others thefromdifferent is

means theof oneleast At :H

:H

a

4321o

ocSince F = 5.12 > 3.24, reject H .F


Excel Output for the Freighter Example

Excel Output for the Freighter Example

Anova: Single Factor

SUMMARYGroups Count Sum Average Variance

Long Beach 4 18 4.5 4.3333Houston 5 20 4 2.5New York 6 42 7 3.2New Orleans 5 17 3.4 1.3

ANOVASource of Variation SS df MS F P-value F critBetween Groups 42.35 3 14.117 5.1101 0.0114 3.2389Within Groups 44.2 16 2.7625

Total 86.55 19


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.


Randomized Block Design

• An experimental design in which there is one independent variable, and a second variable known as a blocking variable, that is used to control for confounding or concomitant variables.

• Confounding or concomitant variable are not being controlled by the business analyst but can have an effect on the outcome of the treatment being studied.


Randomized Block Design, continued

• Blocking variable is a variable that the business analyst wants to control but is not the treatment variable of interest.

• Repeated measures design is a randomized block design in which each block level is an individual item or person, and that person or item is measured across all treatments.


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


Tread-Wear Example: Randomized Block Design

Tread-Wear Example: Randomized Block Design

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

X

C = 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. . ).

[

[ ]

Tread-wear Randomized Block Design: Sum of Squares Calculations (Part 1)





SSE ij j i

SST ij

X X X X

X X

i

n

j

C

i

n

j

C

2

11

2 2

2 2

2

11

2 2 2

7 354 377 356 4 354 337 3566 2 98 310 356 4 2 98 4 03 356

0143

7 356 4 356 6 3

( )

(3. . . . ) (3. . . . )(2. . . . ) (3. . . . ).

( )

(3. . ) (3. . ) (2.

. ) (3. . ).

56 4 3565176

2


Tread-wear Randomized Block Design: Mean Square Calculations

Tread-wear 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 0.387

Error 0.143 8 0.018

Total 5.176 14


Tread-wear Randomized Block Design Treatment Effects: Procedural SummaryTread-wear 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

FMSC

MSE

1742

0 01896 78

.

..

F = 96.78 > = 8.65, reject H ..01,2,8 oF


Excel Output for Tread-Wear Randomized Block Design

Excel Output for Tread-Wear Randomized Block Design

Anova: Two-Factor Without Replication

SUMMARY Count Sum Average Variance 1 3 11.3 3.7666667 0.4933333 2 3 10.1 3.3666667 0.3033333 3 3 10.6 3.5333333 0.3033333 4 3 9.3 3.1 0.21

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


Two-Way Factorial Design

• An experimental design in which two ot more independent variables are studied simultaneously and every level of treatment is studied under the conditions of every level of all other treatments.

• Also called a factorial experiment.


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


CEO Dividend 2 3 Factorial Design: Data and Measurements

CEO Dividend 2 3 Factorial Design: Data and Measurements

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


CEO Dividend 2 3 Factorial Design: Calculations (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

..

.

..

..

.

..

..

.

..

..


CEO Dividend: Analysis of Variance CEO Dividend: Analysis of Variance

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.


CEO Dividend Excel Output (Part 1)

CEO Dividend Excel Output (Part 1)

Anova: Two-Factor With Replication

SUMMARY NYSE ASE OTC TotalReports

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


CEO Dividend Excel Output (Part 2)CEO Dividend Excel Output (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


2 Goodness-of-Fit Test2 Goodness-of-Fit Test

The2 goodness-of-fit test compares expected (theoretical) frequencies of categories from a population distribution to the observed (actual) frequencies from a distribution to determine whether there is a difference between what was expected and what was observed.


2 Goodness-of-Fit Test2 Goodness-of-Fit Test

data sample thefrom estimated parameters ofnumber =

categories ofnumber

valuesexpected offrequency

valuesobserved offrequency :

- 1 - = df

2

2

c

k

where

ck

eo

ff

fff

e

o

e


Month GallonsJanuary 1,553

February 1,585March 1,649April 1,590May 1,497June 1,443July 1,410

August 1,450September 1,495

October 1,564November 1,602December 1,609

18,447

Milk Sales Data for Demonstration Problem 11.4

Milk Sales Data for Demonstration Problem 11.4


Demonstration Problem 11.4: Hypotheses and Decision Rules

Demonstration Problem 11.4: Hypotheses and Decision Rules

ddistributeuniformly not are

salesmilk for figuresmilk monthly The :H

ddistributeuniformly are

salesmilk for figuresmilk monthly The :H

a

o

.

.. ,

01

1

12 1 0

11

24 72501 11

2

df k cIf reject H .

If do not reject H .

Cal

2o

Cal

2o

24 725

24 725

. ,

. ,


Demonstration Problem 11.4: Calculations

Demonstration Problem 11.4: Calculations

Month fo fe (fo - fe)2/fe

January 1,553 1,537.25 0.16February 1,585 1,537.25 1.48March 1,649 1,537.25 8.12April 1,590 1,537.25 1.81May 1,497 1,537.25 1.05June 1,443 1,537.25 5.78July 1,410 1,537.25 10.53August 1,450 1,537.25 4.95September 1,495 1,537.25 1.16October 1,564 1,537.25 0.47November 1,602 1,537.25 2.73December 1,609 1,537.25 3.35

18,447 18,447.00 41.59

Observed Chi-square= 41.59


Demonstration Problem 11.4: Conclusion

Demonstration Problem 11.4: Conclusion

0.01

df = 11

24.725

Non Rejectionregion

2o41.59 24.725, reject H .

Cal


Defects Example: Using a 2 Goodness-of-Fit Test to Test a Population Proportion

Defects Example: Using a 2 Goodness-of-Fit Test to Test a Population Proportion

a

H : = .08

H : .08

o P

P

.

.. ,

05

1

2 1 0

1

384105 1

2

df k c

If reject H .

If do not reject H .

Cal

2o

Cal

2o

3841

3841

. ,

. ,


Defects Example: CalculationsDefects Example: Calculations

2

2

33 16 167 18416 184

18.06 + 1.57

1963

o ef ff e

=

2 2

.

.

fo fe

Defects 33 16Nondefects 167 184

200 200n =

200 .08

16

1

200 .92

184

e

e

e

e

Defects n P

Nondefects n P

ff

ff


Defects Example: Conclusion

Defects Example: Conclusion

0.05

df = 1

3.841

Non Rejectionregion

2o19.63 3.841, reject H .

Cal


Contingency Analysis:2 Test of IndependenceContingency Analysis:

2 Test of Independence

A statistical test used to analyze the frequencies of two variables with multiple categories to determine whether the two variables are independent.

Qualitative VariablesNominal Data


Investment Example2 Test of Independence


• In which region of the country do you reside?A. NortheastB. MidwestC. SouthD. West

• Which type of financial investment are you most likely to make today?E. StocksF. BondsG. Treasury bills



Type of financialInvestment

E F GA O13 nA

Geographic B nB

Region C nC

D nD

nE nF nG N

Contingency Table




Type of Financial Investment

E F GA e12 nA

Geographic B nB

Region C nC

D nD

nE nF nG N

Contingency Table

If A and F are independent,

P A F P A P F

P AN

P FN

P A FN N

A F

A F

n n

n n

AF

A F

A F

en n

n n

N P A F

NN N

N


2 Test of Independence: Formulas2 Test of Independence: Formulas

ij

i j

en nN

where

: i = the row

j = the colum n

the total of row i

the total of column j

N = the total of all fr equencies

i

j

nn

2

2

o e

where

f ff e

: df = (r - 1)(c - 1) r = the numbe r of rowsc = the numbe r of columns

ExpectedFrequencies

Calculated

(Observed )

© 2002 Thomson / South-Western Slide 11-1 Chapter 11 Analysis of Variance and Chi-Square...

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Transcript of © 2002 Thomson / South-Western Slide 11-1 Chapter 11 Analysis of Variance and Chi-Square...