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Applied Statistics in Business & Applied Statistics in Business & Economics, Economics, 44thth edition edition

David P. Doane and Lori E. Seward David P. Doane and Lori E. Seward

Prepared by Lloyd R. Jaisingh Prepared by Lloyd R. Jaisingh

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Analysis of VarianceAnalysis of Variance

Chapter ContentsChapter Contents

11.1 Overview of ANOVA11.1 Overview of ANOVA

11.2 One-Factor ANOVA (Completely Randomized Model)11.2 One-Factor ANOVA (Completely Randomized Model)

11.3 Multiple Comparisons11.3 Multiple Comparisons

11.4 Tests for Homogeneity of Variances11.4 Tests for Homogeneity of Variances

11.5 Two-Factor ANOVA without Replication (Randomized Block Model)11.5 Two-Factor ANOVA without Replication (Randomized Block Model)

11.6 Two-Factor ANOVA with Replication (Full Factorial Model)11.6 Two-Factor ANOVA with Replication (Full Factorial Model)

11.7 Higher Order ANOVA Models (Optional)11.7 Higher Order ANOVA Models (Optional)

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Chapter Learning Objectives Chapter Learning Objectives

LO11-1:LO11-1: Use basic ANOVA terminology correctly.Use basic ANOVA terminology correctly.

LO11-2:LO11-2: Recognize from data format when one-factor ANOVA is appropriate.Recognize from data format when one-factor ANOVA is appropriate.

LO11-3:LO11-3: Interpret sums of squares and calculations in an ANOVA table.Interpret sums of squares and calculations in an ANOVA table.

LO11-4:LO11-4: Use Excel or other software for ANOVA calculations.Use Excel or other software for ANOVA calculations.

LO11-5: LO11-5: Use a table or Excel to find critical values for the Use a table or Excel to find critical values for the F distribution.F distribution.

LO11-6:LO11-6: Explain the assumptions of ANOVA and why they are important.Explain the assumptions of ANOVA and why they are important.

LO11-7: LO11-7: Understand and perform Tukey's test for paired means.Understand and perform Tukey's test for paired means.

LO11-8:LO11-8: Use Hartley's test for equal variances in Use Hartley's test for equal variances in c treatment groups.c treatment groups.

LO11-9:LO11-9: Recognize from data format when two-factor ANOVA is needed.Recognize from data format when two-factor ANOVA is needed.

LO11-10:LO11-10: Interpret main effects and interaction effects in two-factor ANOVA.Interpret main effects and interaction effects in two-factor ANOVA.

LO11-11:LO11-11: Recognize the need for experimental design and GLM (optional).Recognize the need for experimental design and GLM (optional).

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Analysis of VarianceAnalysis of Variance

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• Analysis of variance (ANOVA) is a comparison of means.Analysis of variance (ANOVA) is a comparison of means.• ANOVA allows you to compare more than two means simultaneously.ANOVA allows you to compare more than two means simultaneously.• Proper experimental design efficiently uses limited data to draw the strongest Proper experimental design efficiently uses limited data to draw the strongest

possible inferences.possible inferences.

11.1 Overview of ANOVA11.1 Overview of ANOVA

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The Goal: Explaining VariationThe Goal: Explaining Variation

• ANOVA seeks to identify sources of variation in a numerical dependent variable Y (the response variable).

• Variation in Y about its mean is explained by one or more categorical independent variables (the factors) or is unexplained (random error).

LO11-1LO11-1

LO11-1: LO11-1: Use basic ANOVA terminology correctly.Use basic ANOVA terminology correctly.

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The Goal: Explaining VariationThe Goal: Explaining Variation

• Each possible value of a factor or combination of factors is a treatment.• We test to see if each factor has a significant effect on Y using (for example) the

hypotheses: H0: 1 = 2 = 3 = 4 (e.g. mean defect rates are the same for all

four plants)H1: Not all the means are equal

• The test uses the F distribution.

• If we cannot reject H0, we conclude that observations within each treatment have a common mean .

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11.1 Overview of ANOVA11.1 Overview of ANOVALO11-1LO11-1

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The Goal: Explaining VariationThe Goal: Explaining Variation

Figure 11.3

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11.1 Overview of ANOVA11.1 Overview of ANOVALO11-1LO11-1

11-7

ANOVA AssumptionsANOVA Assumptions

• Analysis of Variance assumes that theAnalysis of Variance assumes that the

- observations on - observations on YY are independent, are independent,

- populations being sampled are normal,- populations being sampled are normal,

- populations being sampled have equal - populations being sampled have equal variances. variances.

• ANOVA is somewhat robust to departures from normality and equal variance ANOVA is somewhat robust to departures from normality and equal variance assumptions.assumptions.

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ANOVA CalculationsANOVA Calculations

• Software (e.g., Excel, MegaStat, MINITAB, SPSS) is used to analyze data. • Large samples increase the power of the test,

but power also depends on the degree of variation in Y.

• Lowest power would be in a small sample with high variation in Y.

11.1 Overview of ANOVA11.1 Overview of ANOVALO11-6LO11-6

LO11-6: LO11-6: Explain the assumptions of ANOVA and why they are important.Explain the assumptions of ANOVA and why they are important.

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• An equivalent way to express the one-factor model is to say that treatment An equivalent way to express the one-factor model is to say that treatment jj came came from a population with a common mean (from a population with a common mean () plus a treatment effect () plus a treatment effect (AAjj) plus ) plus

random error (random error (ijij):):

yyijij = = + + AAjj + + ijij

jj = 1, 2, …, = 1, 2, …, cc and and ii = 1, 2, …, = 1, 2, …, nn

• Random error is assumed to be normally distributed with zero mean and the same Random error is assumed to be normally distributed with zero mean and the same variance for all treatments.variance for all treatments.

One-Factor ANOVA as a Linear ModelOne-Factor ANOVA as a Linear Model

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11.2 One-Factor ANOVA11.2 One-Factor ANOVA(Completely Randomized Model)(Completely Randomized Model)

• A fixed effects model only looks at what happens to the response for particular levels of the factor.

H0: A1 = A2 = … = Ac = 0

H1: Not all Aj are zero• If the H0 is true, then the ANOVA model collapses to yij = + ij

• One can use Excel’s one-factor ANOVA menu using Data One can use Excel’s one-factor ANOVA menu using Data Analysis to analyze data.Analysis to analyze data.

LO11-2LO11-2

LO11-2: LO11-2: Recognize from data format when one-factor ANOVA is appropriate.Recognize from data format when one-factor ANOVA is appropriate.

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Partitioned Sum of SquaresPartitioned Sum of Squares

Table 11.2

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• Use Appendix F or Excel to obtain the critical value of F for a given

• For ANOVA, the F test is a right-tailed test.

LO11-3LO11-311.2 One-Factor ANOVA11.2 One-Factor ANOVA(Completely Randomized Model)(Completely Randomized Model)

LO11-3: LO11-3: Interpret sums of squares and calculations in an ANOVA table.Interpret sums of squares and calculations in an ANOVA table.

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Decision Rule for an F-testDecision Rule for an F-test

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11.2 One-Factor ANOVA11.2 One-Factor ANOVA(Completely Randomized Model)(Completely Randomized Model)LO11-5LO11-5

LO11-5: LO11-5: Use a table or Excel to find critical values for the Use a table or Excel to find critical values for the F distributionF distribution.

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11.3 Multiple Comparisons11.3 Multiple Comparisons

• After rejecting the hypothesis of equal mean, we naturally want to know: Which After rejecting the hypothesis of equal mean, we naturally want to know: Which means differ significantly?means differ significantly?

• In order to maintain the desired overall probability of type I error, a In order to maintain the desired overall probability of type I error, a simultaneous simultaneous confidence intervalconfidence interval for the difference of means must be obtained. for the difference of means must be obtained.

• For For cc groups, there are groups, there are cc((cc – 1) distinct pairs of means to be compared. – 1) distinct pairs of means to be compared.• These types of comparisons are called These types of comparisons are called Multiple Comparison TestsMultiple Comparison Tests..• Tukey’s studentized range testTukey’s studentized range test (or (or HSDHSD for “honestly significant difference” test) is for “honestly significant difference” test) is

a multiple comparison test that has good power and is widely used.a multiple comparison test that has good power and is widely used.• Named for statistician John Wilder Tukey (1915 – 2000)Named for statistician John Wilder Tukey (1915 – 2000)• This test is not available in Excel’s Tools > Data Analysis but is available in This test is not available in Excel’s Tools > Data Analysis but is available in

MegaStat MegaStat andand Minitab Minitab

Tukey’s TestTukey’s Test

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LO11-7LO11-7

LO11-7: LO11-7: Understand and perform Tukey's test for paired means.Understand and perform Tukey's test for paired means.

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11.4 Tests for Homogeneity of Variances11.4 Tests for Homogeneity of Variances

• ANOVA assumes that observations on the response variable are from normally distributed populations that have the same variance.

• The one-factor ANOVA test is only slightly affected by inequality of variance when group sizes are equal.

• Test this assumption of homogeneous variances, using Hartley’s Fmax Test.

ANOVA AssumptionsANOVA Assumptions

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• The hypotheses are

The test statistic is the ratio of the largest sample variance to the smallest sample variance.

LO11-8LO11-8

LO11-8: LO11-8: Use Hartley's test for equal variances in Use Hartley's test for equal variances in c c treatment groups.treatment groups.

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• The decision rule is:

Hartley’s TestHartley’s Test

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11.4 Tests for Homogeneity of Variances11.4 Tests for Homogeneity of VariancesLO11-8LO11-8

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• Levene’s test is a more robust alternative to Hartley’s Levene’s test is a more robust alternative to Hartley’s FF test. test.• Levene’s test does not assume a normal distribution.Levene’s test does not assume a normal distribution.• It is based on the distances of the observations from their sample It is based on the distances of the observations from their sample mediansmedians rather rather

than their sample than their sample means.means.• A computer program (e.g., MINITAB) is needed to perform this test.A computer program (e.g., MINITAB) is needed to perform this test.

Levene’s TestLevene’s Test

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11.4 Tests for Homogeneity of Variances11.4 Tests for Homogeneity of VariancesLO11-6LO11-6

LO11-6: LO11-6: Explain the assumptions of ANOVA and why they are important.Explain the assumptions of ANOVA and why they are important.