Chapter 10: Analysis of Variance: Comparing More Than Two Means.

Chapter 10: Analysis of Variance: Comparing More Than Two Means

Statistics for Business and Economics, 11th ed. Chapter 10: Analysis of Variance

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Where We’ve Been

Presented methods for estimating and testing hypotheses about a single population mean

Presented methods for comparing two population means

Where We’re Going

Discuss the critical elements in the design of a sampling experiment

Investigate completely randomized, randomized block, and factorial designs

Show how to analyze data using a technique called analysis of variance


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10.1: Elements of a Designed Experiment


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Factor Levels are the values of the factors


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Treatments are the factor-level combinations In the example above, a variety of

different GPA – Hours Studied combinations could occur within each subset (Yes or No) of the Study Group factor


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An experimental unit is the object on which the response and factors are observed or measured In the example above, an individual

student would be the experimental unit


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The method by which the

experimental units are selected

determines the type of experiment


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10.2: The Completely Randomized Design

The completely randomized design is a design in which treatments are randomly assigned to the experimental units or in which independent random samples of experimental units are selected for each treatment.


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Very often the object is to determine whether the varying treatments result in different means:

H0: µ1 = µ2 = µ3 = µ4 = ··· = µk

Ha: At least two of the k treatment means differ

Testing the equity of the means involves comparing the variability among the different treatments as well as within the treatments, adjusted for degrees of freedom.


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1 2

2

1

2 21 1 2 2

1 1

Variation between the treatment means:

Sum of Squares for Treatments (SST)

( )

Variation within the treatments:

Sum of Squares for Error (SSE)

( ) ( ) + (

k

i ii

n n

j jj j

SST n x x

SSE x x x x

2

1

2 2 21 1 2 2

)

( 1) ( 1) ( 1)

kn

kj kj

k k

x x

n s n s n s


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Adjusting for degrees of freedom produces comparable measures of variability

Mean Square for Treatments (MST)

1Mean Square for Error (MSE)

SSTMST

k

SSEMSE

n k


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The ratio of the variability among the treatment means to that within the treatment means is an F -statistic:

with k-1 numerator and n-k denominator degrees of freedom.

1,k n k

MSTF

MSE

10.2: The CompletelyRandomized Design

If F* 1, the difference between the treatment means may be attributable to sampling error.

If F* > 1 (significantly), there is support for the alternative hypothesis that the treatments themselves produce different results.


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0 1 2

ANOVA -Test to Compare Treatment Means:

Completely Randomized Design

:

: At least two treatment means differ.

Test Statistic: =

Rejection region: * , with 1 numerator

and

k

a

F k

H

H

MSTF

MSEF F k

- denominator degrees of freedom.n k


Conditions required for a Valid ANOVA F-Test: Completely Randomized Design1. The samples are randomly selected in an

independent manner from the k treatment populations.

2. All k sampled populations have distributions that are approximately normal.

3. The k population variances are equal.


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The USGA compares the driving distance of four brands of golf balls. H0: µ1 = µ2 = µ3 = µ4

Ha: The mean distances differ for at least two of the brands

= .10 Test Statistic: F = MST/MSE Rejection region: F > 2.25 = F.10 with v1 = 3 and v2 = 36


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Ha: The mean distances differ for at least two of the brands = .10 Test Statistic: F = MST/MSE

Rejection region: F > 2.25 = F.10 with v1 = 3 and v2 = 36

Source Degrees of Freedom

Sum of Squares

Mean Square

F p-value

Brands 3 2,794.39 931.46 43.99 .000

Error 36 762.30 21.18


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Ha: The mean distances differ for at least two of the brands = .10 Test Statistic: F = MST/MSE

Rejection region: F > 2.25 = F.10 with v1 = 3 and v2 = 36

Source Degrees of Freedom

Sum of Squares

Mean Square

F p-value

Brands 3 2,794.39 931.46 43.99 .000

Error 36 762.30 21.18


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Since the calculated F > 2.25, we reject the null hypothesis.



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If the conditions for ANOVA are not met, a nonparametric procedure is recommended (see Chapter 14).

If the null hypothesis is not rejected, that is not conclusive proof that the treatment means are all equal.


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10.3: Multiple Comparisons of Means

Suppose the ANOVA F-test indicates differences in the means. To determine the differences, we would compare the differences of the means.

With k treatment means, there are

c = k(k – 1)/2

pairs of means to be compared, and each would have a significance level smaller than the overall .


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10.3: Multiple Comparisons of Means To retain the overall confidence level,

various techniques are available for pair wise comparisons: Tukey – treatment sample sizes are

equal Bonferroni - treatment sample sizes may

be unequal Scheffé – general procedure for all linear

combinations of treatment means


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Let’s go back to the four brands of golf balls in the previous example: Rank the treatment means with an overall

95% level of confidence using Tukey’s procedure.

Estimate the highest ranked golf ball's mean driving distance.


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Brand Comparison 95% Confidence Interval

µA - µB -15.82 < µA - µB < - 4.74

µA - µC -24.71 < µA - µC < -13.63

µA - µD -4.08 < µA - µD < 7.00

µB - µC -14.43 < µB - µC < -3.35

µB - µD 6.2 < µB - µD < 17.28

µC - µD 15.09 < µC - µD < 26.17

Pair wise Comparisons for Four Golf ball BrandsBased on a SAS ANOVA report (see pages 506-7)

Brand C outperforms each of the other brands.


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10.3: Multiple Comparisons of Means To construct a confidence interval on Brand

C, we can use the descriptive statistics from the ANOVA and a straightforward one-sample t-based confidence interval (see section 7.3):

/2, 36

/2

95% sure 1/ ,

where , 10 and 2,

270 (2)(4.60) 1/10 270 2.9

C dfx t s n

s MSE n t


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10.4: The Randomized Block Design

The randomized block design: Blocks (matched sets of experimental

units) are formed. Each of the b blocks has k experimental

units, one for each treatment. One experimental unit from each block is

randomly assigned to each treatment, for a total of n = bk responses.


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10.4: The Randomized Block Design To test the equity of the means, we use the

ratio MST/MSE ~ F

2

1

2 2 2

1 1 1

( )

1 1( )

1 1

( ) ( ) ( )

1

i

i i

k

Ti

n k b

i T Bi i i

b x xSST

MSTk kSSE SS total SST SSB

MSEn b k n b k

x x b x x k x x

n b k


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0 1 2

ANOVA -Test to Compare Treatment Means:

Randomized Block Design

:

: At least two treatment means differ

Test Statistic: =

Rejection Region: , with ( -1) numerator and

( -

p

a

F k

H

H

MSTF

MSEF F k

n b

- 1) [= ( -1)( -1)] denominator degrees of freedom.k b k


Conditions required for a valid ANOVA F – Test The b blocks are randomly selected and all k

treatments are applied (in random order) to each block.

The distribution of observations corresponding to all bk block-treatment combinations are approximately normal.

The bk block-treatment distributions have equal variances.


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Completely Randomized Design

Randomized Block Design



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Suppose the golf balls analyzed above are analyzed again using ten real golfers instead of a machine. Each golfer is a block Each brand is a treatment assigned in random

order to each golfer The ten drives for each brand produce the

following means: Brand A Brand B Brand C Brand D

227 yards 233.2 yards 245.3 yards 220.7 yards



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Source df SS MS F p

Treatment (Brand) 3 2,298.7 1,099.6 54.31 .000

Block (Golfer) 9 12,073.9 1,341.5

Error 27 546.6 20.2

Total 39 15,919.2

ANOVA Table for the Golf Ball Tests



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Equity of Means 95% Confidence Intervals for the Golf Balls’ Distance

µA - µB µA - µC µA - µD µB - µC µB - µD µC - µD

(-11.9,--.4) (-24.0, -2.6) (.6, 12.0) (-17.9, -6.4) (6.7, 18.2) (18.9, 30.3)

None of the confidence intervals contain zero, so we can be 95% certain all of the brand means differ.



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Source df SS MS F p

Treatment (Brand) 3 2,298.7 1,099.6 54.31 .000

Block (Golfer) 9 12,073.9 1,341.5

Error 27 546.6 20.2

Total 39 15,919.2

ANOVA Table for the Golf Ball Tests

To test for block mean differences, use the ratio of MSB to MEE

MSB MS(Golfers) 1,341.566.26

MSE MS(Error) 20.2F


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10.5: Factorial Experiments

A complete factorial experiment is a factorial experiment in which every factor-level combination is utilized. That is, the number of treatments in the experiment equals the total number of factor-level combinations.


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Stage 1 Stage 2



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10.5: Factorial Experiments Tests Conducted in Analyses of Factorial

Experiments: Completely Randomized Design, r Replicates per Treatment

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Test for Treatment Means

: No difference among the treatment means

: At least two treatment means differ

MST : =

MSERejection Region : , with ( - 1) numerator

and ( - ) denom

a

H ab

H

Test Statistic F

F F ab

n ab

inator degrees of freedom


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0

Test for Factor Interaction

: Factors A and B do not interact to affect the response mean

: Factors A and B do interact to affect the response mean

MS( ) : =

MSERejection Region : ,

a

H

H

ABTest Statistic F

F F with ( - 1)( - 1) numerator and

( - ) denominator degrees of freedom

a b

n ab


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0

Test for Main Effect of Factor A

: No difference among the mean levels of factor

: At least two factor mean levels differ

MS( ) : =

MSERejection Region : , with ( - 1) numerat

a

H a A

H A

ATest Statistic F

F F a or and

( - ) denominator degrees of freedomn ab


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0

Test for Main Effect of Factor

: No difference among the mean levels of factor

: At least two factor mean levels differ

MS( ) : =

MSERejection Region : , with ( - 1) numerat

a

B

H b B

H B

BTest Statistic F

F F b or and

( - ) denominator degrees of freedomn ab


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Experiments: Completely Randomized Design, r Replicates per Treatment Conditions Required:

Response distribution for each factor-level combination is normal.

Response variance is constant for all treatments. Random and independent samples of

experimental units are associated with each treatment.


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The four brands of golf balls are tested again, this time with a driver and a 5 iron. Each brand-club combination (eight in all) is assigned randomly to four experimental units in a sequence of swings by Iron Byron. Are the treatment means equal? Do the factors “brand“ and “club” interact?


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Source df SS MS F

Model 7 33659.81 4808.54 140.35

Brand 1 32,092.11 32,093.11 936.75

Club 3 800.74 266.91 7.79

Interaction 3 765.96 255.32 7.45

Error 24 822.24 34.26

Total 31 34,482.05

TABLE 10.13: ANOVA Summary Table for Example 10.10

0 : The treatment means are equal.

: At least two of the eight means differ.

MST 33,659.81/ 7Test Statistic: 140.35; .0001

MSE 822.24 / 24

a

H

H

F p


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Source df SS MS F

Model 7 33659.81 4808.54 140.35

Brand 1 32,092.11 32,093.11 936.75

Club 3 800.74 266.91 7.79

Interaction 3 765.96 255.32 7.45

Error 24 822.24 34.26

Total 31 34,482.05


Reject the null hypothesis

0 : The treatment means are equal.

: At least two of the eight means differ.

MST 33,659.81/ 7Test Statistic: 140.35; .0001

MSE 822.24 / 24

a

H

H

F p


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Source df SS MS F

Model 7 33659.81 4808.54 140.35

Brand 1 32,092.11 32,093.11 936.75

Club 3 800.74 266.91 7.79

Interaction 3 765.96 255.32 7.45

Error 24 822.24 34.26

Total 31 34,482.05


0 : The factors Club and Brand do not interact to affect the mean response.

: The factors Club and Brand interact to affect the mean response

MS(AB) 255.32Test Statistic: 7.45; .0011

MSE 34.26

a

H

H

F p


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Source df SS MS F

Model 7 33659.81 4808.54 140.35

Brand 1 32,092.11 32,093.11 936.75

Club 3 800.74 266.91 7.79

Interaction 3 765.96 255.32 7.45

Error 24 822.24 34.26

Total 31 34,482.05


0 : The factors Club and Brand do not interact to affect the mean response.

: The factors Club and Brand interact to affect the mean response

MS(AB) 255.32Test Statistic: 7.45; .0011

MSE 34.26

a

H

H

F p

Reject the null hypothesis


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Further analysis (see text) suggests that, although the factor “Club” clearly has an impact on distance, the results for “Brand “ are more ambiguous: Brand B hit with a 5 iron outdistances the others, but not when hit with the driver.

Chapter 10: Analysis of Variance: Comparing More Than Two Means.

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Transcript of Chapter 10: Analysis of Variance: Comparing More Than Two Means.