Chapter 11 - ANOVA

Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.Chap 11-*Business Statistics: A Decision-Making Approach6th EditionChapter 11Analysis of Variance

Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.

Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.Chap 11-*Chapter GoalsAfter completing this chapter, you should be able to: Recognize situations in which to use analysis of varianceUnderstand different analysis of variance designsPerform a single-factor hypothesis test and interpret resultsConduct and interpret post-analysis of variance pairwise comparisons proceduresSet up and perform randomized blocks analysisAnalyze two-factor analysis of variance test with replications results


Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.Chap 11-*Chapter OverviewAnalysis of Variance (ANOVA)F-testF-testTukey-Kramer testFishers Least SignificantDifference testOne-Way ANOVARandomized Complete Block ANOVATwo-factor ANOVA with replication


Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.Chap 11-*General ANOVA SettingInvestigator controls one or more independent variablesCalled factors (or treatment variables)Each factor contains two or more levels (or categories/classifications)Observe effects on dependent variableResponse to levels of independent variableExperimental design: the plan used to test hypothesis


Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.Chap 11-*

One-Way Analysis of VarianceEvaluate the difference among the means of three or more populations

Examples: Accident rates for 1st, 2nd, and 3rd shift Expected mileage for five brands of tires

AssumptionsPopulations are normally distributedPopulations have equal variancesSamples are randomly and independently drawn


Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.Chap 11-*Completely Randomized DesignExperimental units (subjects) are assigned randomly to treatmentsOnly one factor or independent variableWith two or more treatment levelsAnalyzed byOne-factor analysis of variance (one-way ANOVA)Called a Balanced Design if all factor levels have equal sample size


Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.Chap 11-*Hypotheses of One-Way ANOVA All population means are equal i.e., no treatment effect (no variation in means among groups)

At least one population mean is different i.e., there is a treatment effect Does not mean that all population means are different (some pairs may be the same)


Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.Chap 11-*One-Factor ANOVA All Means are the same:The Null Hypothesis is True (No Treatment Effect)


Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.Chap 11-*One-Factor ANOVA At least one mean is different:The Null Hypothesis is NOT true (Treatment Effect is present)or(continued)


Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.Chap 11-*Partitioning the VariationTotal variation can be split into two parts:

SST = Total Sum of SquaresSSB = Sum of Squares BetweenSSW = Sum of Squares WithinSST = SSB + SSW


Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.Chap 11-*Partitioning the VariationTotal Variation = the aggregate dispersion of the individual data values across the various factor levels (SST)Within-Sample Variation = dispersion that exists among the data values within a particular factor level (SSW)Between-Sample Variation = dispersion among the factor sample means (SSB)SST = SSB + SSW(continued)


Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.Chap 11-*Partition of Total Variation

Variation Due to Factor (SSB/SSG)Variation Due to Random Sampling (SSW)Total Variation (SST)Commonly referred to as:Sum of Squares WithinSum of Squares ErrorSum of Squares UnexplainedWithin Groups VariationCommonly referred to as:Sum of Squares Between Sum of Squares AmongSum of Squares ExplainedAmong Groups Variation

=+



Total Sum of SquaresWhere:SST = Total sum of squaresk = number of populations (levels or treatments)ni = sample size from population ixij = jth measurement from population ix = grand mean (mean of all data values)SST = SSB + SSW


Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.Chap 11-*Total Variation(continued)


Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.Chap 11-*Sum of Squares BetweenWhere:SSB = Sum of squares betweenk = number of populationsni = sample size from population ixi = sample mean from population ix = grand mean (mean of all data values)

SST = SSB + SSW



Between-Group VariationVariation Due to Differences Among GroupsMean Square Between = SSB/degrees of freedom


Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.Chap 11-*Between-Group Variation(continued)


Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.Chap 11-*Sum of Squares WithinWhere:SSW = Sum of squares withink = number of populationsni = sample size from population ixi = sample mean from population ixij = jth measurement from population i

SST = SSB + SSW



Within-Group VariationSumming the variation within each group and then adding over all groupsMean Square Within = SSW/degrees of freedom


Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.Chap 11-*Within-Group Variation(continued)



One-Way ANOVA TableSource of VariationdfSSMSBetween SamplesSSBMSB =Within SamplesN - kSSWMSW =TotalN - 1SST =SSB+SSWk - 1MSBMSWF ratiok = number of populationsN = sum of the sample sizes from all populationsdf = degrees of freedomSSBk - 1SSWN - kF =



One-Factor ANOVAF Test StatisticTest statistic

MSB is mean squares between variancesMSW is mean squares within variancesDegrees of freedomdf1 = k 1 (k = number of populations)df2 = N k (N = sum of sample sizes from all populations)

H0: 1= 2 = = kHA: At least two population means are different



Interpreting One-Factor ANOVA F StatisticThe F statistic is the ratio of the between estimate of variance and the within estimate of varianceThe ratio must always be positive df1 = k -1 will typically be small df2 = N - k will typically be large

The ratio should be close to 1 if H0: 1= 2 = = k is true

The ratio will be larger than 1 if H0: 1= 2 = = k is false



One-Factor ANOVA F Test ExampleYou want to see if three different golf clubs yield different distances. You randomly select five measurements from trials on an automated driving machine for each club. At the .05 significance level, is there a difference in mean distance?Club 1 Club 2 Club 3254 234 200263 218 222241 235 197237 227 206251 216 204



One-Factor ANOVA Example: Scatter Diagram270260250240230220210200190Distance

Club 1 Club 2 Club 3254 234 200263 218 222241 235 197237 227 206251 216 204Club1 2 3



One-Factor ANOVA Example Computations

Club 1 Club 2 Club 3254 234 200263 218 222241 235 197237 227 206251 216 204x1 = 249.2x2 = 226.0x3 = 205.8

x = 227.0n1 = 5n2 = 5n3 = 5N = 15k = 3SSB = 5 [ (249.2 227)2 + (226 227)2 + (205.8 227)2 ] = 4716.4SSW = (254 249.2)2 + (263 249.2)2 ++ (204 205.8)2 = 1119.6MSB = 4716.4 / (3-1) = 2358.2MSW = 1119.6 / (15-3) = 93.3


Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.Chap 11-*F = 25.275

One-Factor ANOVA Example SolutionH0: 1 = 2 = 3HA: i not all equal = .05df1= 2 df2 = 12 Test Statistic:

Decision:

Conclusion:Reject H0 at = 0.05There is evidence that at least one i differs from the rest0 = .05F.05 = 3.885Reject H0Do not reject H0Critical Value: F = 3.885


Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.Chap 11-*ANOVA -- Single Factor:Excel OutputEXCEL: tools | data analysis | ANOVA: single factor

SUMMARYGroupsCountSumAverageVarianceClub 151246249.2108.2Club 25113022677.5Club 351029205.894.2ANOVASource of VariationSSdfMSFP-valueF critBetween Groups4716.422358.225.2754.99E-053.885Within Groups1119.61293.3Total5836.014


Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.Chap 11-*The Tukey-Kramer ProcedureTells which population means are significantly differente.g.: 1 = 2 3Done after rejection of equal means in ANOVAAllows pair-wise comparisonsCompare absolute mean differences with critical range

x1 = 23


Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.Chap 11-*Tukey-Kramer Critical Range

where:q = Value from standardized range table with k and N - k degrees of freedom for the desired level of MSW = Mean Square Within ni and nj = Sample sizes from populations (levels) i and j


Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.Chap 11-*The Tukey-Kramer Procedure: Example

1. Compute absolute mean differences:

Club 1 Club 2 Club 3254 234 200263 218 222241 235 197237 227 206251 216 204

2. Find the q value from the table in appendix J with k and N - k degrees of freedom for the desired level of


Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.Chap 11-*The Tukey-Kramer Procedure: Example

5. All of the absolute mean differences are greater than critical range. Therefore there is a significant difference between each pair of means at 5% level of significance. 3. Compute Critical Range:4. Compare:


Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.Chap 11-*Tukey-Kramer in PHStat


Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.Chap 11-*Randomized Complete Block ANOVALike One-Way ANOVA, we test for equal population means (for different factor levels, for example)...

...but we want to control for possible variation from a second factor (with two or more levels)

Used when more than one factor may influence the value of the dependent variable, but only one is of key interest

Levels of the secondary factor are called blocks


Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.Chap 11-*Partitioning the VariationTotal variation can now be split into three parts:

SST = Total sum of squaresSSB = Sum of squares between factor levelsSSBL = Sum of squares between blocksSSW = Sum of squares within levelsSST = SSB + SSBL + SSW


Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.Chap 11-*Sum of Squares for BlockingWhere:k = number of levels for this factorb = number of blocksxj = sample mean from the jth block x = grand mean (mean of all data values)

SST = SSB + SSBL + SSW



Partitioning the VariationTotal variation can now be split into three parts:

SST and SSB are computed as they were in One-Way ANOVASST = SSB + SSBL + SSWSSW = SST (SSB + SSBL)


Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.Chap 11-*Mean Squares



Randomized Block ANOVA TableSource of VariationdfSSMSBetween SamplesSSBMSB Within Samples(k1)(b-1)SSWMSWTotalN - 1SSTk - 1MSBLMSWF ratiok = number of populationsN = sum of the sample sizes from all populationsb = number of blocksdf = degrees of freedomBetween BlocksSSBLb - 1MSBL MSBMSW


Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.Chap 11-*Blocking TestBlocking test: df1 = b - 1

df2 = (k 1)(b 1)MSBLMSWF =Reject H0 if F > F


Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.Chap 11-*Main Factor test: df1 = k - 1

df2 = (k 1)(b 1)MSBMSWF =Reject H0 if F > F Main Factor Test


Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.Chap 11-*Fishers Least Significant Difference TestTo test which population means are significantly differente.g.: 1 = 2 3Done after rejection of equal means in randomized block ANOVA designAllows pair-wise comparisonsCompare absolute mean differences with critical range

x = 123


Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.Chap 11-*Fishers Least Significant Difference (LSD) Testwhere: t/2 = Upper-tailed value from Students t-distribution for /2 and (k -1)(n - 1) degrees of freedom MSW = Mean square within from ANOVA table b = number of blocks k = number of levels of the main factor


Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.Chap 11-*Fishers Least Significant Difference (LSD) Test(continued)If the absolute mean difference is greater than LSD then there is a significant difference between that pair of means at the chosen level of significance. Compare:


Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.Chap 11-*Two-Way ANOVAExamines the effect ofTwo or more factors of interest on the dependent variablee.g.: Percent carbonation and line speed on soft drink bottling processInteraction between the different levels of these two factorse.g.: Does the effect of one particular percentage of carbonation depend on which level the line speed is set?


Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.Chap 11-*Two-Way ANOVAAssumptions

Populations are normally distributedPopulations have equal variancesIndependent random samples are drawn

(continued)


Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.Chap 11-*Two-Way ANOVA Sources of VariationTwo Factors of interest: A and Ba = number of levels of factor Ab = number of levels of factor BN = total number of observations in all cells


Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.Chap 11-*Two-Way ANOVA Sources of VariationSSTTotal VariationSSAVariation due to factor ASSBVariation due to factor BSSABVariation due to interaction between A and BSSEInherent variation (Error)Degrees of Freedom:a 1b 1(a 1)(b 1)N abN - 1SST = SSA + SSB + SSAB + SSE(continued)


Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.Chap 11-*Two Factor ANOVA EquationsTotal Sum of Squares:Sum of Squares Factor A:Sum of Squares Factor B:


Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.Chap 11-*Two Factor ANOVA EquationsSum of Squares Interaction Between A and B:Sum of Squares Error:(continued)


Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.Chap 11-*Two Factor ANOVA Equationswhere:a = number of levels of factor Ab = number of levels of factor Bn = number of replications in each cell(continued)


Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.Chap 11-*Mean Square Calculations


Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.Chap 11-*Two-Way ANOVA:The F Test StatisticF Test for Factor B Main EffectF Test for Interaction EffectH0: A1 = A2 = A3 = HA: Not all Ai are equalH0: factors A and B do not interact to affect the mean response HA: factors A and B do interactF Test for Factor A Main EffectH0: B1 = B2 = B3 = HA: Not all Bi are equalReject H0 if F > F

Reject H0 if F > FReject H0 if F > F


Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.Chap 11-*Two-Way ANOVASummary Table

Source of VariationSum of SquaresDegrees of FreedomMean SquaresF StatisticFactor ASSAa 1MSA = SSA /(a 1)MSA MSEFactor BSSBb 1MSB= SSB /(b 1)MSB MSEAB (Interaction)SSAB(a 1)(b 1)MSAB = SSAB / [(a 1)(b 1)]MSAB MSEErrorSSEN abMSE = SSE/(N ab)TotalSSTN 1


Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.Chap 11-*Features of Two-Way ANOVA F TestDegrees of freedom always add up

N-1 = (N-ab) + (a-1) + (b-1) + (a-1)(b-1)Total = error + factor A + factor B + interactionThe denominator of the F Test is always the same but the numerator is differentThe sums of squares always add upSST = SSE + SSA + SSB + SSAB

Total = error + factor A + factor B + interaction


Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.Chap 11-*Examples:Interaction vs. No InteractionNo interaction:

12Factor B Level 1Factor B Level 3Factor B Level 2Factor A Levels12Factor B Level 1Factor B Level 3Factor B Level 2Factor A LevelsMean ResponseMean ResponseInteraction is present:


Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.Chap 11-*Chapter SummaryDescribed one-way analysis of varianceThe logic of ANOVAANOVA assumptionsF test for difference in k meansThe Tukey-Kramer procedure for multiple comparisonsDescribed randomized complete block designsF testFishers least significant difference test for multiple comparisonsDescribed two-way analysis of varianceExamined effects of multiple factors and interaction


Chapter 11 - ANOVA

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