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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
10
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
11
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
12
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.
13
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
14
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
15
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
17
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
------------------------------------------------------------------------
18
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
19
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:
20
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
21
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
22
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
23
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
24
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
25
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
32
Wald Test:Equation: UntitledNull Hypothesis: C(2)=C(3)F-statistic 138.2678 Probability 0.000000
Chi-square 138.2678 Probability 0.000000