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**Chapter 11 Analysis of Variance**

Business Statistics: A Decision-Making Approach 6th Edition Chapter 11 Analysis of Variance Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc.

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**Chapter Goals After completing this chapter, you should be able to:**

Recognize situations in which to use analysis of variance Understand different analysis of variance designs Perform a single-factor hypothesis test and interpret results Conduct and interpret post-analysis of variance pairwise comparisons procedures Set up and perform randomized blocks analysis Analyze two-factor analysis of variance test with replications results Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc.

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**Analysis of Variance (ANOVA)**

Chapter Overview Analysis of Variance (ANOVA) One-Way ANOVA Randomized Complete Block ANOVA Two-factor ANOVA with replication F-test F-test Tukey- Kramer test Fisher’s Least Significant Difference test Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc.

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General ANOVA Setting Investigator controls one or more independent variables Called factors (or treatment variables) Each factor contains two or more levels (or categories/classifications) Observe effects on dependent variable Response to levels of independent variable Experimental design: the plan used to test hypothesis Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc.

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**One-Way Analysis of Variance**

Evaluate 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 Assumptions Populations are normally distributed Populations have equal variances Samples are randomly and independently drawn Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc.

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**Completely Randomized Design**

Experimental units (subjects) are assigned randomly to treatments Only one factor or independent variable With two or more treatment levels Analyzed by One-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.

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**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.

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**The Null Hypothesis is True**

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.

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**One-Factor ANOVA At least one mean is different:**

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

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**Partitioning the Variation**

Total variation can be split into two parts: SST = SSB + SSW SST = Total Sum of Squares SSB = Sum of Squares Between SSW = Sum of Squares Within Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc.

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**Partitioning the Variation**

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

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**Partition of Total Variation**

Total Variation (SST) Variation Due to Factor (SSB) + Variation Due to Random Sampling (SSW) = Commonly referred to as: Sum of Squares Between Sum of Squares Among Sum of Squares Explained Among Groups Variation Commonly referred to as: Sum of Squares Within Sum of Squares Error Sum of Squares Unexplained Within Groups Variation Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc.

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**Total Sum of Squares SST = SSB + SSW**

Where: SST = Total sum of squares k = number of populations (levels or treatments) ni = sample size from population i xij = jth measurement from population i x = grand mean (mean of all data values) Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc.

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**Total Variation (continued)**

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

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**Sum of Squares Between SST = SSB + SSW k = number of populations**

Where: SSB = Sum of squares between k = number of populations ni = sample size from population i xi = sample mean from population i x = grand mean (mean of all data values) Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc.

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**Between-Group Variation**

Variation Due to Differences Among Groups Mean Square Between = SSB/degrees of freedom Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc.

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**Between-Group Variation**

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

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**Sum of Squares Within SST = SSB + SSW k = number of populations**

Where: SSW = Sum of squares within k = number of populations ni = sample size from population i xi = sample mean from population i xij = jth measurement from population i Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc.

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**Within-Group Variation**

Summing the variation within each group and then adding over all groups Mean Square Within = SSW/degrees of freedom Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc.

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**Within-Group Variation**

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

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**One-Way ANOVA Table Source of Variation SS df MS F ratio**

Between Samples SSB MSB SSB k - 1 MSB = F = k - 1 MSW Within Samples SSW SSW N - k MSW = N - k SST = SSB+SSW Total N - 1 k = number of populations N = sum of the sample sizes from all populations df = degrees of freedom Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc.

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**One-Factor ANOVA F Test Statistic**

H0: μ1= μ2 = … = μ k HA: At least two population means are different Test statistic MSB is mean squares between variances MSW is mean squares within variances Degrees of freedom df1 = k – (k = number of populations) df2 = N – k (N = sum of sample sizes from all populations) Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc.

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**Interpreting One-Factor ANOVA F Statistic**

The F statistic is the ratio of the between estimate of variance and the within estimate of variance The 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: μ1= μ2 = … = μk is true The ratio will be larger than 1 if H0: μ1= μ2 = … = μk is false Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc.

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**One-Factor ANOVA F Test Example**

Club Club 2 Club You 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? Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc.

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**One-Factor ANOVA Example: Scatter Diagram**

Distance 270 260 250 240 230 220 210 200 190 Club Club 2 Club • • • • • • • • • • • • • • • Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Club

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**One-Factor ANOVA Example Computations**

Club Club 2 Club x1 = 249.2 x2 = 226.0 x3 = 205.8 x = 227.0 n1 = 5 n2 = 5 n3 = 5 N = 15 k = 3 SSB = 5 [ (249.2 – 227)2 + (226 – 227)2 + (205.8 – 227)2 ] = SSW = (254 – 249.2)2 + (263 – 249.2)2 +…+ (204 – 205.8)2 = MSB = / (3-1) = MSW = / (15-3) = 93.3 Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc.

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**One-Factor ANOVA Example Solution**

Test Statistic: Decision: Conclusion: H0: μ1 = μ2 = μ3 HA: μi not all equal = .05 df1= df2 = 12 Critical Value: F = 3.885 Reject H0 at = 0.05 = .05 There is evidence that at least one μi differs from the rest Do not reject H0 Reject H0 F = F.05 = 3.885 Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc.

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**ANOVA -- Single Factor: Excel Output**

EXCEL: tools | data analysis | ANOVA: single factor SUMMARY Groups Count Sum Average Variance Club 1 5 1246 249.2 108.2 Club 2 1130 226 77.5 Club 3 1029 205.8 94.2 ANOVA Source of Variation SS df MS F P-value F crit Between Groups 4716.4 2 2358.2 25.275 4.99E-05 3.885 Within 1119.6 12 93.3 Total 5836.0 14 Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc.

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**The Tukey-Kramer Procedure**

Tells which population means are significantly different e.g.: μ1 = μ2 μ3 Done after rejection of equal means in ANOVA Allows pair-wise comparisons Compare absolute mean differences with critical range μ μ μ x = 1 2 3 Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc.

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**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.

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**The Tukey-Kramer Procedure: Example**

1. Compute absolute mean differences: Club Club 2 Club 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.

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**The Tukey-Kramer Procedure: Example**

3. Compute Critical Range: 4. Compare: 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. Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc.

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**Tukey-Kramer in PHStat**

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

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**Randomized Complete Block ANOVA**

Like 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.

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**Partitioning the Variation**

Total variation can now be split into three parts: SST = SSB + SSBL + SSW SST = Total sum of squares SSB = Sum of squares between factor levels SSBL = Sum of squares between blocks SSW = Sum of squares within levels Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc.

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**Sum of Squares for Blocking**

SST = SSB + SSBL + SSW Where: k = number of levels for this factor b = number of blocks xj = sample mean from the jth block x = grand mean (mean of all data values) Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc.

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**Partitioning the Variation**

Total variation can now be split into three parts: SST = SSB + SSBL + SSW SST and SSB are computed as they were in One-Way ANOVA SSW = SST – (SSB + SSBL) Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc.

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

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**Randomized Block ANOVA Table**

Source of Variation SS df MS F ratio MSBL Between Blocks SSBL b - 1 MSBL MSW Between Samples MSB SSB k - 1 MSB MSW Within Samples SSW (k–1)(b-1) MSW Total SST N - 1 k = number of populations N = sum of the sample sizes from all populations b = number of blocks df = degrees of freedom Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc.

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**Blocking Test Reject H0 if F > F MSBL F = MSW**

Blocking test: df1 = b - 1 df2 = (k – 1)(b – 1) MSW Reject H0 if F > F Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc.

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**Main Factor Test Reject H0 if F > F MSB F = MSW**

Main Factor test: df1 = k - 1 df2 = (k – 1)(b – 1) MSW Reject H0 if F > F Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc.

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**Fisher’s Least Significant Difference Test**

To test which population means are significantly different e.g.: μ1 = μ2 ≠ μ3 Done after rejection of equal means in randomized block ANOVA design Allows pair-wise comparisons Compare absolute mean differences with critical range x = 1 2 3 Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc.

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**Fisher’s Least Significant Difference (LSD) Test**

where: t/2 = Upper-tailed value from Student’s 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.

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**Fisher’s Least Significant Difference (LSD) Test**

(continued) Compare: 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. Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc.

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**Two-Way ANOVA Examines the effect of**

Two or more factors of interest on the dependent variable e.g.: Percent carbonation and line speed on soft drink bottling process Interaction between the different levels of these two factors e.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.

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**Two-Way ANOVA Assumptions Populations are normally distributed**

(continued) Assumptions Populations are normally distributed Populations have equal variances Independent random samples are drawn Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc.

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**Two-Way ANOVA Sources of Variation**

Two Factors of interest: A and B a = number of levels of factor A b = number of levels of factor B N = total number of observations in all cells Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc.

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**Two-Way ANOVA Sources of Variation**

(continued) SST = SSA + SSB + SSAB + SSE Degrees of Freedom: SSA Variation due to factor A a – 1 SST Total Variation SSB Variation due to factor B b – 1 SSAB Variation due to interaction between A and B (a – 1)(b – 1) N - 1 SSE Inherent variation (Error) N – ab Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc.

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**Two Factor ANOVA Equations**

Total Sum of Squares: Sum of Squares Factor A: Sum of Squares Factor B: Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc.

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**Two Factor ANOVA Equations**

(continued) Sum of Squares Interaction Between A and B: Sum of Squares Error: Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc.

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**Two Factor ANOVA Equations**

(continued) where: a = number of levels of factor A b = number of levels of factor B n’ = number of replications in each cell Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc.

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**Mean Square Calculations**

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

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**Two-Way ANOVA: The F Test Statistic**

F Test for Factor A Main Effect H0: μA1 = μA2 = μA3 = • • • HA: Not all μAi are equal Reject H0 if F > F F Test for Factor B Main Effect H0: μB1 = μB2 = μB3 = • • • HA: Not all μBi are equal Reject H0 if F > F F Test for Interaction Effect H0: factors A and B do not interact to affect the mean response HA: factors A and B do interact Reject H0 if F > F Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc.

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**Two-Way ANOVA Summary Table**

Source of Variation Sum of Squares Degrees of Freedom Mean Squares F Statistic Factor A SSA a – 1 MSA = SSA /(a – 1) MSA MSE Factor B SSB b – 1 MSB = SSB /(b – 1) MSB MSE AB (Interaction) SSAB (a – 1)(b – 1) MSAB = SSAB / [(a – 1)(b – 1)] MSAB MSE Error SSE N – ab MSE = SSE/(N – ab) Total SST N – 1 Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc.

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**Features of Two-Way ANOVA F Test**

Degrees of freedom always add up N-1 = (N-ab) + (a-1) + (b-1) + (a-1)(b-1) Total = error + factor A + factor B + interaction The denominator of the F Test is always the same but the numerator is different The sums of squares always add up SST = SSE + SSA + SSB + SSAB Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc.

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**Examples: Interaction vs. No Interaction**

Interaction is present: No interaction: Factor B Level 1 Factor B Level 1 Factor B Level 3 Mean Response Mean Response Factor B Level 2 Factor B Level 2 Factor B Level 3 Factor A Levels 1 Factor A Levels 1 2 2 Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc.

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**Chapter Summary Described one-way analysis of variance**

The logic of ANOVA ANOVA assumptions F test for difference in k means The Tukey-Kramer procedure for multiple comparisons Described randomized complete block designs F test Fisher’s least significant difference test for multiple comparisons Described two-way analysis of variance Examined effects of multiple factors and interaction Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc.

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