1

Power Fifteen

Analysis of Variance (ANOVA)

2

Analysis of Variance

One-Way ANOVA• Tabular• Regression

Two-Way ANOVA• Tabular • Regression

3

One-Way ANOVA

Apple Juice Concentrate Example, Data File xm 15-01

New product Try 3 different advertising strategies, one in

each of three cities• City 1: convenience of use• City 2: quality of product• City 3: price

Record Weekly Sales

4

Advertising Strategies & Weekly Sales for 20 Weeks

Convenience Quality Price

529 804 672

658 630 531

793 774 443

- - -

614 624 532

Mean: 577.5 Mean: 653.0 Mean: 608.65

5

Figure 1: Mean Apple Juice Sales By Advertising Strategy

520

540

560

580

600

620

640

660

convenience quality price

Advertising Strategy

Is There a Significant Difference in Average Sales?

Null Hypothesis, H0 :

Alternative Hypothesis: ≠ or≠ or ≠

6

Table 3: 1-Way ANOVA of Apple Juice Sales By Advertising Strategy

Source of Variation Sum of Squares Degrees of

Freedom

Mean

Square

Explained(between

treatments)

ESS =

k

j 1nj ( x

j - x )2 k-1 ESS/(k-1)

Unexplained(withi

n

treatments)

USS =

k

j 1

)(

1

jn

i(xij - x j)2 n-k USS/(n-k)

Total TSS =

k

j 1

)(

1

jn

i(xij - x )2 n-1

Fk-1, n-k = [ESS/(k-1)]/[USS/(n-k)]

7

Apple Juice Concentrate ANOVASource ofVariation

Sum ofSquares

Degrees ofFreedom

MeanSquare

Explained(BetweenTreatments)

ESS=57,512.23

k-1 = 2 ESS/(k-1)=28,756.12

Unexplained(WithinTreatments)

USS=506,984

n-k = 57 USS/(n-k)=8894.45

Total TSS=564,496

n-1 = 59

F2, 57 = 28,756.12/8894.45 = 3.23

8

0.0

0.2

0.4

0.6

0.8

1.0

0 2 4 6 8 10

F Variable

DE

NS

ITY

F igure 2: F-Distribution Density For 2 DOF, 57 DOF

F-Distribution Test of the Null Hypothesis of No Difference in Mean Sales with Advertising Strategy

F2, 60 (critical) @ 5% =3.15

9

y(1)

y(2)

y(3)

1 0 0

0 1 0

0 0 1

Regression Set-Up: y(1) is column of 20 sales observationsFor city 1, 1 is a column of 20 ones, 0 is a column of 20 Zeros. Regression of a quantitative variable on three dummies

Y = C(1)*Dummy(city 1) + C(2)*Dummy(city 2) + C(3)*Dummy(city 3) + e

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Table 5: One-Way ANOVA Estimated Using RegressionDependent Variable: SALESAJMethod: Least Squares

Sample: 1 60Included observations: 60

Variable Coefficient Std. Error t-Statistic Prob.

CONVENIENCE 577.5500 21.08844 27.38704 0.0000QUALITY 653.0000 21.08844 30.96483 0.0000

PRICE 608.6500 21.08844 28.86178 0.0000

R-squared 0.101882 Mean dependent var 613.0667Adjusted R-squared

0.070370 S.D. dependent var 97.81474

S.E. of regression 94.31038 Akaike info criterion 11.97977Sum squaredresid

506983.5 Schwarz criterion 12.08448

Log likelihood -356.3930 F-statistic 3.233041Durbin-Watsonstat

1.525930 Prob(F-statistic) 0.046773

One-Way ANOVA and Regression

Regression Coefficients are the City Means; F statistic

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Table 6: Test of the Null Hypothesis: All Treatment Means Are EqualWald Test:Equation: Untitled

NullHypothesis:

C(1)=C(3)

C(2)=C(3)

F-statistic 3.233041 Probability 0.046773Chi-square 6.466083 Probability 0.039437

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Two-Way ANOVA Apple Juice Concentrate Two Factors

• 3 advertising strategies• 2 advertising media: TV & Newspapers

6 cities• City 1: convenience on TV• City 2: convenience in Newspapers• City 3: quality on TV• Etc.

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Table 7: Apple Juice Concentrate Sales in Six Cities

City 1 City 2 City 3 City 4 City 5 City 6

491 464 677 689 575 803

712 559 627 650 614 584

558 759 590 704 706 525

447 557 632 652 484 498

479 528 683 576 478 812

624 670 760 836 650 565

546 534 690 628 583 708

444 657 548 798 536 546

582 557 579 497 579 616

672 474 644 841 795 587

Advertising Strategies In Two Media: Weekly Sales

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Mean Weekly Sales By Strategy and Medium

Table 9: Mean Weekly Sales, Apple Juice Concentrate, Six Cities

Convenience Quality Price

Television city1: 555.5 city3: 643 city5: 600

Newspapers city2: 575.9 city4: 687.1 city 6: 624.4

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Figure 3; Mean Apple Juice Sales by Advertising

Strategy and Medium

Strategy

Avera

ge

0100200300400500600700

convenienc

e

quality price

television

newspapers

Average Weekly Sales By Strategy & Medium

400

450

500

550

600

650

700

750

conveniencequality

Ave

rag

e S

ales

NewspapersTelevision

price

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Is There Any Difference In Mean Sales Among the Six Cities?Table 8: 1-Way ANOVA of Apple Juice Sales, Six Cities

Source of Variation Sum of Squares Degrees of Freedom Mean Square

Explained(between

treatments)

ESS = 113,620 k-1 = 5 ESS/(k-1) =

22,724

Unexplained(within

treatments)

USS = 501,137 n-k = 54 USS/(n-k) =

9280

Total TSS = 614,757 n-1 = 59

F5, 54 = (22,724/9,280) = 2.45, critical value at 5% = 2.38

------------------------------------------------------------------------

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Table 10: Schematic For 2-Way ANOVA of Apple Juice Sales

Source of Variation Sum of Squares Degrees of

Freedom

Mean Square

Explained(between

treatments)

ESS = ESS/(k-1)

Strategy ESS(Strategy) a-1 ESS(Strat.)/(a-1)

Medium ESS(Medium) b-1 ESS(Med)/(b-1)

Interaction ESS(Interaction) (a-1)(b-1) ESS(I)/(a-1)(b-1)

Unexplained(within

treatments)

USS n-ab USS/(n-k)

Total TSS n-1

Table of ANOVA for Two-Way

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TSS =

a

i1

b

j1

r

k1(xijk - x)2

Formulas For Sums of Squares

a is the # of treatments for strategies =3

b is the # of treatments for media =2

r is the # of replicates or observations =10

x= {

a

i1

b

j1

r

k1xijk }/n

The Grand Mean:

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ESS(Strategy) = r b

a

i1(xiS - x)2

Formulas For Sums of Squares (Cont.)

Where the mean for treatment i, strategy, is:

xiS = {

b

j1

r

k1xijk }/r b

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Mean Weekly Sales By Strategy and Medium

Table 9: Mean Weekly Sales, Apple Juice Concentrate, Six Cities

Convenience Quality Price

Television city1: 555.5 city3: 643 city5: 600

Newspapers city2: 575.9 city4: 687.1 city 6: 624.4

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Formulas For Sums of Squares (Cont.)

ESS(Medium) = r a

b

j1(xjM - x)2

Where the mean for treatment j, medium, is:

xjM = {

a

i1

r

k1xijk }/r a

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Formulas For Sums of Squares (Cont.)

ESS(Interaction) = r

a

i1

b

j1(xijSM - xiS - xjM + x)2

USS =

a

i1

b

j1

r

k1(xijk - ijx)2

Where is the mean for each cityijx

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Table 11: 2-Way ANOVA of Apple Juice Sales

Source of Variation Sum of Squares Degrees of

Freedom

Mean Square

Explained(between

treatments)

ESS =

Strategy ESS(Strat) = 98838.6 (a-1) = 2 49419.3

Medium ESS(Med) = 13172.0 (b-1) = 1 13172.0

Interaction ESS(I) = 1609.6 (a-1)(b-1) = 2 804.8

Unexplained(within

treatments)

USS = 501136.7 (n-ab) = 60 – 6

= 54

9280.3

Total TSS = 614756.98 (n-1) = 59

Table of Two-Way ANOVA for Apple Juice Sales

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F-Distribution Tests

F2, 54 = 804.8/9280.3 = 0.09

Test for Interaction:

Test for Advertising Medium:

F1, 54 = 13172/9280.3 = 1.42, and the critical value at the 5% level is 4.02,

Test for Advertising Strategy:

F2, 54 = 49419.3/9280.3 = 5.32, with a critical value of 3.17 at the 5% level,

26

)3(

)2(

)1(

y

y

y

= 110

001

101

Regression Set-Up

)6(

)5(

)4(

y

y

y

000

100

010

Convenience dummyQuality dummy

TV dummy

1

1

1

1

1

1

constant

SALESAPJ CONVENIENCE QUALITY PRICETELEVISION NEWSPAPERS491 1 0 0 1 0712 1 0 0 1 0558 1 0 0 1 0447 1 0 0 1 0479 1 0 0 1 0624 1 0 0 1 0546 1 0 0 1 0444 1 0 0 1 0582 1 0 0 1 0672 1 0 0 1 0464 1 0 0 0 1559 1 0 0 0 1759 1 0 0 0 1557 1 0 0 0 1528 1 0 0 0 1670 1 0 0 0 1534 1 0 0 0 1657 1 0 0 0 1557 1 0 0 0 1474 1 0 0 0 1677 0 1 0 1 0627 0 1 0 1 0

Dependent Variable: SALESAPJMethod: Least Squares

Sample: 1 60Included observations: 60

Variable Coefficient Std. Error t-Statistic Prob.

CONVENIENCE -48.50000 43.08204 -1.125759 0.2652QUALITY 62.70000 43.08204 1.455363 0.1514TELEVISION -24.40000 43.08204 -0.566361 0.5735C 624.4000 30.46360 20.49659 0.0000CONVENIENCE*TELEVISION 4.000000 60.92720 0.0656520.9479QUALITY*TELEVISION -19.70000 60.92720 -0.323337 0.7477

R-squared 0.184821 Mean dependent var 614.3167Adjusted R-squared 0.109342 S.D. dependent var 102.0765S.E. of regression 96.33436 Akaike info criterion 12.06817Sum squared resid 501136.7 Schwarz criterion 12.27760Log likelihood -356.0450 F-statistic 2.448631Durbin-Watson stat 2.452725 Prob(F-statistic) 0.045165

Dependent Variable: SALESAPJMethod: Least SquaresSample: 1 60Included observations: 60Variable Coefficient Std. Error t-Statistic Prob. CONVENIENCE -46.50000 29.96267 -1.551931 0.1263QUALITY 52.85000 29.96267 1.763862 0.0832TELEVISION -29.63333 24.46441 -1.211283 0.2309C 627.0167 24.46441 25.62974 0.0000R-squared 0.182203 Mean dependent var 614.31Adjusted R-squared 0.138393 S.D. dependent var102.0765S.E. of regression 94.75027 Akaike info criterion12.00471Sum squared resid 502746.3 Schwarz criterion12.14433Log likelihood-356.1412 F-statistic 4.158888Durbin-Watson stat 2.456222 Prob(F-statistic)

0.009921

Dependent Variable: SALESAPJMethod: Least Squares

Sample: 1 60Included observations: 60Variable Coefficient Std. Error t-Statistic Prob. CONVENIENCE -46.50000 30.08521 -1.545610 0.1277QUALITY 52.85000 30.08521 1.756677 0.0843C 612.2000 21.27346 28.77765 0.0000R-squared 0.160777 Mean dependent var 614.31Adjusted R-squared 0.131330 S.D. dependent var 102.07S.E. of regression 95.13779 Akaike info criterion11.99724Sum squared resid 515918.3 Schwarz criterion 12.101Log likelihood-356.9171 F-statistic 5.459975Durbin-Watson stat 2.379774 Prob(F-statistic)0.006769

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Wald Test:Equation: UntitledNull Hypothesis: C(2)=C(3)F-statistic 138.2678 Probability 0.000000

Chi-square 138.2678 Probability 0.000000

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ANOVA By Difference Regression with interaction terms, USS =

501,136.7 Regression dropping interaction terms<

USS = 502746.3 Difference is 1,609.6 and is the sum of

squares explained by interaction terms F-test of the interaction terms:

F2, 54 = [1609.6/2]/[501,136.7/54]