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

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

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

Where We’ve Been
• Presented methods for estimating and testing hypotheses about a single population mean
• Presented methods for comparing two population means

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

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

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

10.1: Elements of a Designed Experiment

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

10.1: Elements of a Designed Experiment

Factor Levels are the values of the factors

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

10.1: Elements of a Designed Experiment
• 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

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

10.1: Elements of a Designed Experiment
• 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

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

10.1: Elements of a Designed Experiment

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

10.1: Elements of a Designed Experiment

The method by which the experimental units are selected determines the type of experiment

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

10.1: Elements of a Designed Experiment

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

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.

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

10.2: The Completely Randomized Design

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

10.2: The Completely Randomized Design
• 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

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

10.2: The Completely Randomized Design
• 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.

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

10.2: The Completely Randomized Design
• Adjusting for degrees of freedom produces comparable measures of variability

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

10.2: The Completely Randomized Design

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

10.2: The Completely Randomized Design
• 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.

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

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.

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

10.2: The CompletelyRandomized Design

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

10.2: The CompletelyRandomized Design
• Conditions required for a Valid ANOVA F-Test: Completely Randomized Design
• The samples are randomly selected in an independent manner from the k treatment populations.
• All k sampled populations have distributions that are approximately normal.
• The k population variances are equal.

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

10.2: The Completely Randomized Design
• 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

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

10.2: The Completely Randomized Design

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

10.2: The Completely Randomized Design
• 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

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

10.2: The Completely Randomized Design
• 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

Since the calculated

F > 2.25, we reject the null hypothesis.

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

10.2: The CompletelyRandomized Design
• 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.

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

10.3: Multiple Comparisons of Means
• Suppose the ANOVA F-testindicates 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 .

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

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

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

10.3: Multiple Comparisons of Means
• 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.

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

10.3: Multiple Comparisons of Means

Pair wise Comparisons for Four Golf ball Brands

Based on a SAS ANOVA report (see pages 506-7)

Brand C outperforms each of the other brands.

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

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):

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

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.

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

10.4: The Randomized Block Design
• To test the equity of the means, we use the ratio MST/MSE ~ F

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

10.4: The Randomized Block Design

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

10.4: The Randomized Block Design

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.

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

10.4: The Randomized Block Design

Completely Randomized Design

Randomized Block Design

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

10.4: The Randomized Block Design
• 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:

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

10.4: The Randomized Block Design

ANOVA Table for the Golf Ball Tests

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

10.4: The Randomized Block Design

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

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

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

10.4: The Randomized Block Design

ANOVA Table for the Golf Ball Tests

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

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

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.

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

10.5: Factorial Experiments

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

10.5: Factorial Experiments

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

10.5: Factorial Experiments

Stage 1

Stage 2

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

10.5: Factorial Experiments
• Tests Conducted in Analyses of Factorial Experiments: Completely Randomized Design, r Replicates per Treatment

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

10.5: Factorial Experiments
• Tests Conducted in Analyses of Factorial Experiments: Completely Randomized Design, r Replicates per Treatment

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

10.5: Factorial Experiments
• Tests Conducted in Analyses of Factorial Experiments: Completely Randomized Design, r Replicates per Treatment

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

10.5: Factorial Experiments
• Tests Conducted in Analyses of Factorial Experiments: Completely Randomized Design, r Replicates per Treatment

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

10.5: Factorial Experiments
• Tests Conducted in Analyses of Factorial 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.

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

10.5: Factorial Experiments
• 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?

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

10.5: Factorial Experiments

TABLE 10.13: ANOVA Summary Table for Example 10.10

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

10.5: Factorial Experiments

TABLE 10.13: ANOVA Summary Table for Example 10.10

Reject the null hypothesis

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

10.5: Factorial Experiments

TABLE 10.13: ANOVA Summary Table for Example 10.10

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

10.5: Factorial Experiments

TABLE 10.13: ANOVA Summary Table for Example 10.10

Reject the null hypothesis

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

10.5: Factorial Experiments

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.

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