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# NG BB 34 Analysis of Variance (ANOVA)

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• 1. UNCLASSIFIED / FOUO UNCLASSIFIED / FOUO National Guard Black Belt Training Module 34 Analysis of Variance (ANOVA) UNCLASSIFIED / FOUO UNCLASSIFIED / FOUO
• 2. UNCLASSIFIED / FOUOCPI Roadmap – Analyze 8-STEP PROCESS 6. See 1.Validate 2. Identify 3. Set 4. Determine 5. Develop 7. Confirm 8. Standardize Counter- the Performance Improvement Root Counter- Results Successful Measures Problem Gaps Targets Cause Measures & Process Processes Through Define Measure Analyze Improve Control ACTIVITIES TOOLS • Value Stream Analysis • Identify Potential Root Causes • Process Constraint ID • Reduce List of Potential Root • Takt Time Analysis Causes • Cause and Effect Analysis • Brainstorming • Confirm Root Cause to Output • 5 Whys Relationship • Affinity Diagram • Estimate Impact of Root Causes • Pareto on Key Outputs • Cause and Effect Matrix • FMEA • Prioritize Root Causes • Hypothesis Tests • Complete Analyze Tollgate • ANOVA • Chi Square • Simple and Multiple Regression Note: Activities and tools vary by project. Lists provided here are not necessarily all-inclusive. UNCLASSIFIED / FOUO
• 3. UNCLASSIFIED / FOUO Learning Objectives  Gain a conceptual understanding of Analysis of Variance (ANOVA) and the ANOVA table  Be able to design and perform a one or two factor experiment  Recognize and interpret interactions  Fully understand the ANOVA model assumptions and how to validate them  Understand and apply multiple pair-wise comparisons  Establish a sound basis on which to learn more complex experimental designs Analysis of Variance (ANOVA) UNCLASSIFIED / FOUO 3
• 4. UNCLASSIFIED / FOUO Applications for ANOVA  Administrative – A manager wants to understand how different attendance policies may affect productivity.  Transportation – An AAFES manager wants to know if the average shipping costs are higher between three distribution centers. Analysis of Variance (ANOVA) UNCLASSIFIED / FOUO 4
• 5. UNCLASSIFIED / FOUO When To Use ANOVA Independent Variable (X) Continuous Categorical Categorical Continuous Dependent Variable (Y) Regression ANOVA Logistic Chi-Square (2) Regression Test The tool depends on the data type. ANOVA is used with an attribute (categorical) input and a continuous response. Analysis of Variance (ANOVA) UNCLASSIFIED / FOUO 5
• 6. UNCLASSIFIED / FOUO ANOVA Output Boxplot of Processing Time by Facility 12 10 Processing Time 8 6 4 2 0 Facility A Facility B Facility C Facility Analysis of Variance (ANOVA) UNCLASSIFIED / FOUO 6
• 7. UNCLASSIFIED / FOUO One-Way ANOVA vs. Two-Sample t-Test Let’s compare sets of data taken on different methods of processing invoices which vary a Factor A A two-sample t-test: What if we compare several methods? Old Method New Method Method 1 Method 2 Method 3 Method 4 13.6 15.3 16.3 17.2 19.4 20.5 14.9 17.6 15.2 17.3 17.9 18.8 15.2 15.6 14.9 16.0 18.1 21.3 13.2 16.2 19.2 20.5 22.8 25.0 19.5 21.7 20.1 22.6 24.7 26.4 13.2 15.1 13.2 14.3 17.3 18.5 15.8 17.2 15.8 17.6 19.7 23.2 Q: Is there a difference Q: Are there any statistically significant in the average for differences in the averages for the each method? methods? Q: If so, which are different from which others? Analysis of Variance (ANOVA) UNCLASSIFIED / FOUO 7
• 8. UNCLASSIFIED / FOUO Is There a Difference? 30 25 x 20 x Response x 15 x 10 5 Method 1 Method 2 Method 3 Method 4 Factor A Plotting the averages for the different methods shows a difference, but is it statistically significant? Analysis of Variance (ANOVA) UNCLASSIFIED / FOUO 8
• 9. UNCLASSIFIED / FOUO Is There a Difference Now? 30 25 x 20 x Response x 15 x 10 5 Method 1 Method 2 Method 3 Method 4 Factor A Now that we have a bit more data, does factor A make a difference? Why or why not? Analysis of Variance (ANOVA) UNCLASSIFIED / FOUO 9
• 10. UNCLASSIFIED / FOUO What About Now? 30 25 x 20 x Response x 15 x 10 5 Method 1 Method 2 Method 3 Method 4 Factor A Now what do you think? Does factor A make a difference? Why or why not? Analysis of Variance (ANOVA) UNCLASSIFIED / FOUO 10
• 11. UNCLASSIFIED / FOUO One-Way ANOVA Fundamentals  One-Way Analysis of Variance (ANOVA) is a statistical method for comparing the means of more than two levels when a single factor is varied  The hypothesis tested is: Ho: µ1 = µ2 = µ3 = µ4 =…= µk Ha: At least one µ is different  Simply speaking, an ANOVA tests whether any of the means are different. ANOVA does not tell us which ones are different (we‟ll supplement ANOVA with multiple comparison procedures for that) Analysis of Variance (ANOVA) UNCLASSIFIED / FOUO 11
• 12. UNCLASSIFIED / FOUO Sources of Variability  ANOVA looks at three sources of variability:  Total – Total variability among all observations  Between – Variation between subgroup means (factor)  Within – Random (chance) variation within each subgroup (noise, or statistical error) “Between “Within Subgroup Subgroup Variation” Variation” Total = Between + Within Analysis of Variance (ANOVA) UNCLASSIFIED / FOUO 12
• 13. UNCLASSIFIED / FOUO Questions Asked by ANOVA Ho : 1  2  3   4 Ha : At least one k is different Are any of the 4 population means different? Analysis of Variance (ANOVA) UNCLASSIFIED / FOUO 13
• 14. UNCLASSIFIED / FOUO Sums of Squares yj = Mean of Group 70 y = Grand Mean of the Response experiment 65 60 yi,j = Individual measurement 55 1 2 3 4 i =represents a data point Factor/Level within the jth group j = represents the jth group Analysis of Variance (ANOVA) UNCLASSIFIED / FOUO 14
• 15. UNCLASSIFIED / FOUO Sums of Squares Formula g nj g g nj  (y ij  y )2   n j (y j  y )2   (y ij  y j )2 j 1 i 1 j 1 j 1 i 1 SS(Total)  SS(Factor)  SS(Error) SS(Total) = Total Sum of Squares of the Experiment (individuals - Grand Mean) SS(Factor) = Sum of Squares of the Factor (Group Mean - Grand Mean) SS(Error) = Sum of Squares within the Group (individuals - Group Mean) By comparing the Sums of Squares, we can tell if the observed difference is due to a true difference or random chance Analysis of Variance (ANOVA) UNCLASSIFIED / FOUO 15
• 16. UNCLASSIFIED / FOUO ANOVA Sum of Squares  We can separate the total sum of squares into two components (“within” and “between”).  If the factor we are interested in has little or no effect on the average response, then these two estimates (within and between) should be fairly equal and we will conclude all subgroups could have come from one larger population.  As these two estimates (within and between) become significantly different, we will attribute this difference as originating from a difference in subgroup means.  Minitab will calculate this! Analysis of Variance (ANOVA) UNCLASSIFIED / FOUO 16
• 17. UNCLASSIFIED / FOUO Null and Alternate Hypothesis Ho : 1   2  3   4 Ha : At least one  k is different To determine whether we can reject the null hypothesis, or not, we must calculate the Test Statistic (F-ratio) using the Analysis of Variance table as described on the following slide Analysis of Variance (ANOVA) UNCLASSIFIED / FOUO 17
• 18. UNCLASSIFIED / FOUO Developing the ANOVA Table SOURCE SS df MS (=SS/df) F {=MS(Factor)/MS(Error)} BETWEEN SS(Factor) a-1 SS (factor)/df factor MS(Factor) / MS(Error)  n  1 a WITHIN SS(Error) j SS(Error) / df error j 1  a  TOTAL SS(Total)   nj   1    j 1  i = represents a data point within the jth group (factor level) j = represents the jth group (factor level) a = total # of groups (factor levels) Why is Source “Within” called the Error or Noise? In practical terms, what is the F-ratio telling us? What do you think large F-ratios mean? Analysis of Variance (ANOVA) UNCLASSIFIED / FOUO 18
• 19. UNCLASSIFIED / FOUO ANOVA Example: Invoice Processing CT  A Six Sigma team wants to compare the invoice processing times at three different facilities.  If one facility is better than the others, they can look for opportunities to implement the best practice across the organization.  Open the Minitab worksheet: Invoice ANOVA.mtw.  The data shows invoice processing cycle times at Facility A, B, and C. Analysis of Variance (ANOVA) UNCLASSIFIED / FOUO 19
• 20. UNCLASSIFIED / FOUO Is One Facility Better Than The Others?  How might we determine which, if any, of the three facilities has a shorter cycle time?  What other concerns might you have about this experiment? Minitab Tip: Minitab usually likes data in columns (List the numerical response data in one single column, and the factor you want to investigate beside it). ANOVA is one tool that breaks that rule – ANOVA (unstacked) can analyze unstacked data. Analysis of Variance (ANOVA) UNCLASSIFIED / FOUO 20
• 21. UNCLASSIFIED / FOUO One-Way ANOVA in Minitab Select Stat>ANOVA>One-Way Analysis of Variance (ANOVA) UNCLASSIFIED / FOUO 21
• 22. UNCLASSIFIED / FOUO One-Way ANOVA in Minitab Enter the Response and the Factor Select Graphs to go to the Graphs dialog box Analysis of Variance (ANOVA) UNCLASSIFIED / FOUO 22
• 23. UNCLASSIFIED / FOUO ANOVA-Boxplots Select > Boxplots of data Let‟s look at some Boxplots while we are here Analysis of Variance (ANOVA) UNCLASSIFIED / FOUO 23
• 24. UNCLASSIFIED / FOUO ANOVA – Multiple Comparisons Select Comparisons>Tukey’s We will get into more detail on these later in this session Analysis of Variance (ANOVA) UNCLASSIFIED / FOUO 24
• 25. UNCLASSIFIED / FOUO Boxplots – What Do You Think? Boxplot of Processing Time 12 10 Processing Time 8 6 4 2 0 Facility A Facility B Facility C Facility What would you conclude? Which facility has the best cycle time? Analysis of Variance (ANOVA) UNCLASSIFIED / FOUO 25
• 26. UNCLASSIFIED / FOUO ANOVA Table – Session Window What would we conclude from the ANOVA table? Analysis of Variance (ANOVA) UNCLASSIFIED / FOUO 26
• 27. UNCLASSIFIED / FOUO Pairwise Comparisons – Tukey This is the output for the Tukey Test Pairwise Comparisons are simply confidence intervals for the difference between the tabulated pairs, with alpha being determined by the individual error rate Tukey pairwise comparisons answer the question “Which ones are Statistically Significantly Different?” How do we interpret these paired tests? Analysis of Variance (ANOVA) UNCLASSIFIED / FOUO 27
• 28. UNCLASSIFIED / FOUOTukey Pairwise Interpretation Since we have 3 Facilities, there are 3 Two-Way Comparisons in this analysis First: We subtract the mean for cycle time for Facility A from the means for Facilities B & C. Minitab then calculates confidence intervals around these differences. If the interval contains zero, then there is Not a Statistically Significant Difference between that pair. Here the intervals for Facility B and C do Not contain zero so there is a Statistically Significant Difference between Facility A and the other two Facilities. Facility A is Statistically Significantly Different from Facilities B & C Analysis of Variance (ANOVA) UNCLASSIFIED / FOUO 28
• 29. UNCLASSIFIED / FOUOTukey Pairwise Interpretation (Cont.) Second: We subtract the mean for cycle time for Facility B from the mean for Facility C. Minitab then calculates the confidence interval around that difference. If the interval contains zero, then there is Not a Statistically Significant Difference between the pair. Here the interval Does Not contain zero so there is a Statistically Significant Difference between B and C. Facility B is Statistically Significantly Different from Facility C Analysis of Variance (ANOVA) UNCLASSIFIED / FOUO 29
• 30. UNCLASSIFIED / FOUO Example: Pay for Performance  In this study, the number of 411 calls processed in a given day was measured under one of five different pay-for-performance incentive plans  The null hypothesis would be that the different pay plans would have no significant effect on productivity levels.  Open the data set: One Way ANOVA Example.mtw. Analysis of Variance (ANOVA) UNCLASSIFIED / FOUO 30
• 31. UNCLASSIFIED / FOUO Example Data – Pay for Performance  We want to determine if there is a significant difference in the level of production between the different plans.  What concerns might you have about this experiment? Analysis of Variance (ANOVA) UNCLASSIFIED / FOUO 31
• 32. UNCLASSIFIED / FOUO One Way ANOVA in Minitab Select Stat>ANOVA>One-way Analysis of Variance (ANOVA) UNCLASSIFIED / FOUO 32
• 33. UNCLASSIFIED / FOUO Boxplots in Minitab Let‟s start with Graphs > Boxplots Analysis of Variance (ANOVA) UNCLASSIFIED / FOUO 33
• 34. UNCLASSIFIED / FOUO Production by Plan Boxplots Boxplot of production 1250 1200 1150 If you production were the 1100 manager, what would 1050 you do? 1000 A B C D E plan Does the incentive plan seem to matter? Analysis of Variance (ANOVA) UNCLASSIFIED / FOUO 34
• 35. UNCLASSIFIED / FOUO ANOVA Table – Pay for Performance Do we have any evidence that the incentive plan matters? Who can explain the ANOVA table to the class? Analysis of Variance (ANOVA) UNCLASSIFIED / FOUO 35
• 36. UNCLASSIFIED / FOUO Tukey Pairwise Comparisons – Pay for Perf Which plans are different? The ANOVA Table answers the question “Are all the subgroup averages the same?” Tukey Pairwise Comparisons answer the question “Which ones are different?” Analysis of Variance (ANOVA) UNCLASSIFIED / FOUO 36
• 37. UNCLASSIFIED / FOUO Tukey Pairwise Comparisons – Pay for Perf Which pairs are different? Which intervals do not contain zero? Is it possible for the ANOVA Table and the Tukey pairs to conflict? Analysis of Variance (ANOVA) UNCLASSIFIED / FOUO 37
• 38. UNCLASSIFIED / FOUO ANOVA – Main Effects Plot Select Stat>ANOVA>Main Effects Plot Analysis of Variance (ANOVA) UNCLASSIFIED / FOUO 38
• 39. UNCLASSIFIED / FOUO ANOVA – Main Effects Plot (cont.) Enter the Responses and Factors, then click on OK to go to Main Effects Plot Analysis of Variance (ANOVA) UNCLASSIFIED / FOUO 39
• 40. UNCLASSIFIED / FOUO Graphical Analysis – Main Effects Plots Main Effects Plot for production Data Means 1200 1175 1150 Mean 1125 1100 1075 1050 A B C D E plan What does the plot tell us? Analysis of Variance (ANOVA) UNCLASSIFIED / FOUO 40
• 41. UNCLASSIFIED / FOUO Graphical Analysis – Interval Plots Select Stat>ANOVA>Interval Plot What do the Interval plots tell us about our experiment? Analysis of Variance (ANOVA) UNCLASSIFIED / FOUO 41
• 42. UNCLASSIFIED / FOUO ANOVA – Interval Plots First select With Groups since we have five groups, and then click on OK to go to the next dialog box Then enter Graph variable And Categorical variable and click on OK to go to the Interval Plot Analysis of Variance (ANOVA) UNCLASSIFIED / FOUO 42
• 43. UNCLASSIFIED / FOUO ANOVA – Interval Plots Another graphical way to present your findings ! Interval Plot of production 95% CI for the Mean 1200 1150 production 1100 1050 A B C D E plan How might the interval plot have looked differently if the confidence interval level (percent) were changed? Analysis of Variance (ANOVA) UNCLASSIFIED / FOUO 43
• 44. UNCLASSIFIED / FOUO ANOVA Table – A Quick Quiz Source DF SS MS F p Factor 3 ? 1542.0 ? 0.000 Error ? 2,242 ? Total 23 6,868 Could you complete the above ANOVA table? Analysis of Variance (ANOVA) UNCLASSIFIED / FOUO 44
• 45. UNCLASSIFIED / FOUO Exercise: Degrees of Freedom Step One:  Let‟s go around the room and have everyone give a number which we will flipchart  The numbers need to add up to 100 Step Two:  How many degrees of freedom did I have?  How many would I have if, in addition to adding to 100, I added one more mathematical requirement? Analysis of Variance (ANOVA) UNCLASSIFIED / FOUO 45
• 46. UNCLASSIFIED / FOUO What Are “Degrees of Freedom?” degrees of freedom  currency in statistics We earn a degree of freedom for every data point we collect We spend a degree of freedom for each parameter we estimate Analysis of Variance (ANOVA) UNCLASSIFIED / FOUO 46
• 47. UNCLASSIFIED / FOUO What Are “Degrees of Freedom”?  In ANOVA, the degrees of freedom are as follows:  dftotal = N-1 = # of observations - 1  dffactor = L-1 = # of levels - 1  dfinteraction = dffactorA X dffactorB  dferror = dftotal - dfeverything else Let‟s say we are testing a factor that has five levels and we collect seven data points at each factor level… How many observations would we have? 5 levels x 7 observations per level =35 total observations How many total degrees of freedom would we have? 35 - 1 = 34 How many degrees of freedom to estimate the factor effect? 5 levels - 1 = 4 How many degrees of freedom do we have to estimate error? 34 total - 4 factor = 30 degrees of freedom Analysis of Variance (ANOVA) UNCLASSIFIED / FOUO 47
• 48. UNCLASSIFIED / FOUO Key ANOVA Assumptions  Model errors are assumed to be normally distributed with a mean of zero, and are to be randomly distributed (no patterns).  The samples are assumed to come from normally distributed populations. We can investigate these assumptions with residual plots.  The variance is assumed constant for all factor levels. We can investigate this assumption with a statistical test for equal variances. Analysis of Variance (ANOVA) UNCLASSIFIED / FOUO 48
• 49. UNCLASSIFIED / FOUO ANOVA – Residual Analysis  Residual plots should show no pattern relative to any factor, including the fitted response.  Residuals vs. the fitted response should have an average of about zero.  Residuals should be fairly normally distributed. Practical Note: Moderate departures from normality of the residuals are of little concern. We always want to check the residuals, though, because they are an opportunity to learn more about the data. Analysis of Variance (ANOVA) UNCLASSIFIED / FOUO 49
• 50. UNCLASSIFIED / FOUO Constant Variance Assumption  There are two tests we can use to test the assumption of constant (equal) variance:  Bartletts Test is frequently used to test this hypothesis for data that is normally distributed.  Levenes Test can be used when the data is not normally distributed. Note: Minitab will perform this analysis for us with the procedure called „Test for Equal Variances’ Practical Note: Balanced designs (consistent sample size for all factor levels) are very robust to the constant variance assumption. Still, make a habit of checking for constant variances. It is an opportunity to learn if factor levels have different amounts of variability, which is useful information. Analysis of Variance (ANOVA) UNCLASSIFIED / FOUO 50
• 51. UNCLASSIFIED / FOUO Test for Equal Variances Select: Stat>ANOVA>Test for Equal Variances Then place production in response & Plan in factor Press OK Analysis of Variance (ANOVA) UNCLASSIFIED / FOUO 51
• 52. UNCLASSIFIED / FOUO Constant Variance Assumption (Cont.). Test for Equal Variances for production Bartletts Test A Both Bartlett’s Test Test Statistic 4.95 P-Value 0.292 and Levene’s Test Levenes Test are run on the data B Test Statistic 0.46 and are reported P-Value 0.764 at the same time. plan C D E 0 20 40 60 80 100 120 95% Bonferroni Confidence Intervals for StDevs Analysis of Variance (ANOVA) UNCLASSIFIED / FOUO 52
• 53. UNCLASSIFIED / FOUO Model Adequacy – More Good News  By selecting an adequate sample size and randomly conducting the trials, your experiment should be robust to the normality assumption (remember the Central Limit Theorem)  Although there are certain assumptions that need to be verified, there are precautions you can take when designing and conducting your experiment to safeguard against some common mistakes  Protect the integrity of your experiment right from the start  Often, problems can be easily corrected by collecting a larger sample size of data Analysis of Variance (ANOVA) UNCLASSIFIED / FOUO 53
• 54. UNCLASSIFIED / FOUO One-Way ANOVA Wrap-Up  We will formally address the checking of model assumptions during the Two-Way ANOVA analysis.  Re-capping One-Way ANOVA methodology: 1. Select a sound sample size and factor levels 2. Randomly conduct your trials and collect the data 3. Conduct your ANOVA analysis 4. Follow up with pairwise comparisons, if indicated 5. Examine the residuals, variance and normality assumptions 6. Generate main effects plots, interval plots, etc. 7. Draw conclusions  This short procedure is not meant to be an exhaustive methodology.  What other items would you add? Analysis of Variance (ANOVA) UNCLASSIFIED / FOUO 54
• 55. UNCLASSIFIED / FOUO Individual Exercise  A market research firm for the Defense Commissary Agency (DECA) believed that the sales of a given product in units was dependent upon its placement  Items placed at eye level tended to have higher sales than items placed near the floor  Using the data in the Minitab file Sales vs Product Placement.mtw, draw some conclusions about the relationship between sales and product placement  You will have 10 minutes to complete this exercise Analysis of Variance (ANOVA) UNCLASSIFIED / FOUO 55
• 56. UNCLASSIFIED / FOUO UNCLASSIFIED / FOUO National Guard Black Belt Training Two-Way ANOVA UNCLASSIFIED / FOUO UNCLASSIFIED / FOUO
• 57. UNCLASSIFIED / FOUO One-Way vs. Two-Way ANOVA  In One-Way ANOVA, we looked at how different levels of a single factor impacted a response variable.  In Two-Way ANOVA, we will examine how different levels of two factors and their interaction impact a response variable. Analysis of Variance (ANOVA) UNCLASSIFIED / FOUO 57
• 58. UNCLASSIFIED / FOUO Now We Can Consider Two Factors A Low High  At a high level, a Two- 69 80 Way ANOVA (two Low 65 82 factor) can be viewed as a two-factor B experiment 59  The factors can take 42 High on many levels; you 44 63 are not limited to two levels for each Analysis of Variance (ANOVA) UNCLASSIFIED / FOUO 58
• 59. UNCLASSIFIED / FOUO Two-Way ANOVA  Experiments often involve the study of more than one factor.  Factorial designs are very efficient methods to investigate various combinations of levels of the factors.  These designs evaluate the effect on the response caused by different levels of factors and their interaction.  As in the case of One-Way ANOVA, we will be building a model and verifying some assumptions. Analysis of Variance (ANOVA) UNCLASSIFIED / FOUO 59
• 60. UNCLASSIFIED / FOUO Two-Factor Factorial Design  The general two-factor factorial experiment takes the following form. As in the case of a one-factor ANOVA, randomizing the experiment is important: Factor B 1 2 ... b 1 Factor A 2 . a  In this experiment, Factor A has levels ranging from 1 to a, Factor B has levels ranging from 1 to b, while the replications have replicates 1 to n A balanced design is always preferred (same number of observations for each treatment) because it buffers against any inequality of variances Analysis of Variance (ANOVA) UNCLASSIFIED / FOUO 60
• 61. UNCLASSIFIED / FOUO Two-Factor Factorial Design  Just as in the One-Factor ANOVA, the total variability can be segmented into its component sum of squares: SST= SSA+ SSB + SSAB + SSe Given:  SST is the total sum of squares,  SSA is the sum of squares from factor A,  SSB is the sum of squares from factor B,  SSAB is the sum of squares due to the interaction between A&B  SSe is the sum of squares from error Analysis of Variance (ANOVA) UNCLASSIFIED / FOUO 61
• 62. UNCLASSIFIED / FOUO Degrees of Freedom – Two Factor ANOVA  Each Sum of Squares has associated degrees of freedom: Source Sum of Squares Degrees of Freedom Mean Square F0 Factor A SSA a-1 SS A MS A MS A  F0  a 1 MS E Factor B SSB b-1 SS MS B MS B  B F0  b 1 MS E Interaction SSAB (a - 1)(b - 1) SS AB MS AB MS AB  F0  (a  1)(b  1) MS E SS E Error SSE ab(n - 1) MS E  ab(n  1) Total SST abn - 1 Analysis of Variance (ANOVA) UNCLASSIFIED / FOUO 62
• 63. UNCLASSIFIED / FOUO Marketing Example  AAFES is trying to introduce their own brand of candy and wants to find out which product packaging or regions will yield the highest sales.  They sold their candy in either a plain brown bag, a colorful bag or a clear plastic bag at the cash register (point of sale).  AAFES had stores in regions which varied economically and the information was captured to see if different regions affect sales.  The data set is: Two Way ANOVA Marketing.mtw.  As a class we will analyze the data. Analysis of Variance (ANOVA) UNCLASSIFIED / FOUO 63
• 64. UNCLASSIFIED / FOUO Marketing Example Data The team collected sales data for three different packaging styles in three geographic regions. They are interested in knowing if the packaging affects sales in any of the regions. Analysis of Variance (ANOVA) UNCLASSIFIED / FOUO 64
• 65. UNCLASSIFIED / FOUO Marketing Example Data (Cont.) Selection Stat>ANOVA>Two-Way Analysis of Variance (ANOVA) UNCLASSIFIED / FOUO 65
• 66. UNCLASSIFIED / FOUO Marketing Example Data (Cont.) Check both boxes for Display means Enter the Response and Factors – the choice of row vs. column for factors is unimportant Click on OK to get the analysis in your Session Window Analysis of Variance (ANOVA) UNCLASSIFIED / FOUO 66
• 67. UNCLASSIFIED / FOUO Marketing Example – ANOVA Table What is significant? Who wants to give it a try? Analysis of Variance (ANOVA) UNCLASSIFIED / FOUO 67
• 68. UNCLASSIFIED / FOUO Generating a Main Effects Plot Select Stat>ANOVA>Main Effects Plot Let‟s look at a Main Effects Plot Analysis of Variance (ANOVA) UNCLASSIFIED / FOUO 68
• 69. UNCLASSIFIED / FOUOSelecting Main Effects Fill in Responses and Factors, then click on OK to get Plots Analysis of Variance (ANOVA) UNCLASSIFIED / FOUO 69
• 70. UNCLASSIFIED / FOUO Main Effects Plot Main Effects Plot for sales Data Means region packaging 800 750 700 Mean 650 600 550 1 2 3 color plain point of sale Which factor has the stronger effect? Analysis of Variance (ANOVA) UNCLASSIFIED / FOUO 70
• 71. UNCLASSIFIED / FOUO Generating an Interaction Plot Select Stat>ANOVA>Interactions plot Analysis of Variance (ANOVA) UNCLASSIFIED / FOUO 71
• 72. UNCLASSIFIED / FOUO Selecting Interactions Enter the Responses and Factors, then click on OK to get Plot Analysis of Variance (ANOVA) UNCLASSIFIED / FOUO 72
• 73. UNCLASSIFIED / FOUO Interactions Plot Interaction Plot for sales Data Means region 1100 1 2 1000 3 900 800 Mean 700 600 500 400 color plain point of sale packaging How do we interpret Interaction Plots? Analysis of Variance (ANOVA) UNCLASSIFIED / FOUO 73
• 74. UNCLASSIFIED / FOUO Residual Analysis Select Store residuals and Store fits, then select Graphs and select Four in one (under Residual Plots) and click on OK and OK again so we can do some model confirmation Analysis of Variance (ANOVA) UNCLASSIFIED / FOUO 74
• 75. UNCLASSIFIED / FOUO Residual Four Pack Residual Plots for sales Normal Probability Plot Versus Fits 99.9 N 270 200 99 AD 0.409 90 P-Value 0.343 100 Residual Percent What are we 50 0 10 1 -100 looking for? 0.1 -200 -200 -100 0 100 200 400 600 800 1000 1200 Residual Fitted Value What are Histogram Versus Order 200 the 40 100 assumptions Frequency we want to Residual 30 0 20 10 -100 verify? 0 -200 -120 -60 0 60 120 1 20 40 60 80 00 20 40 60 80 00 20 40 60 1 1 1 1 1 2 2 2 2 Residual Observation Order Analysis of Variance (ANOVA) UNCLASSIFIED / FOUO 75
• 76. UNCLASSIFIED / FOUOTest for Equal Variances Select Stat>Basic Statistics>2 Variances Here is another option for checking for Equal Variances Analysis of Variance (ANOVA) UNCLASSIFIED / FOUO 76
• 77. UNCLASSIFIED / FOUOTest for Equal Variances We can only check one factor at a Now go back and repeat the analysis. time in this dialog box. First do Sales This time do Sales by Packaging. by Region. Then click on OK to get Then click on OK to get this comparison this comparison of variances. of variances. Analysis of Variance (ANOVA) UNCLASSIFIED / FOUO 77
• 78. UNCLASSIFIED / FOUO Test for Equal Variances Test for Equal Variances for sales Bartletts Test Test Statistic 42.13 1 P-Value 0.000 Lev enes Test Test Statistic 17.02 P-Value 0.000 region 2 3 100 150 200 250 300 95% Bonferroni Confidence Intervals for StDevs Do the factor levels have equal variances? Analysis of Variance (ANOVA) UNCLASSIFIED / FOUO 78
• 79. UNCLASSIFIED / FOUOTest for Equal Variances Test for Equal Variances for sales Bartletts Test Test Statistic 70.97 color P-Value 0.000 Lev enes Test Test Statistic 34.49 P-Value 0.000 packaging plain point of sale 100 150 200 250 300 350 95% Bonferroni Confidence Intervals for StDevs What about the Variances for these factor levels? Analysis of Variance (ANOVA) UNCLASSIFIED / FOUO 79
• 80. UNCLASSIFIED / FOUO ANOVA Conclusions  Did our model assumptions hold up?  How comfortable are we with the conclusions drawn?  Questions? Analysis of Variance (ANOVA) UNCLASSIFIED / FOUO 80
• 81. UNCLASSIFIED / FOUO Individual Exercise - Employee Productivity A manager wanted to increase productivity due to the organization‟s slim margins.  The hope was to increase productivity by 8%-10% and reduce payroll through attrition.  The manager piloted a program across three departments that involved 99 employees.  The manager was evaluating the effect on productivity of a four day work week, flextime, and the status quo.  Using the data collected in Two Way ANOVA.mtw, help the manager interpret the results. Analysis of Variance (ANOVA) UNCLASSIFIED / FOUO 81
• 82. UNCLASSIFIED / FOUO Takeaways  Conceptual ANOVA  Main Effects Plots  Sums of Squares  Interactions Plot  ANOVA Table  Two-Factor ANOVA  ANOVA Boxplots,  Two-Factor Model Multiple Comparisons  Residual Analysis  Tukey Pairwise Comparisons  Test for Equal Variances Analysis of Variance (ANOVA) UNCLASSIFIED / FOUO 82
• 83. UNCLASSIFIED / FOUO What other comments or questions do you have? UNCLASSIFIED / FOUO
• 84. UNCLASSIFIED / FOUO UNCLASSIFIED / FOUO National Guard Black Belt Training APPENDIX UNCLASSIFIED / FOUO UNCLASSIFIED / FOUO
• 85. UNCLASSIFIED / FOUOAnalysis of Variance Using Minitab Another look at the ANOVA table 1093 = 33.1 Analysis of Variance (ANOVA) UNCLASSIFIED / FOUO 85
• 86. UNCLASSIFIED / FOUO Reading the ANOVA Table One-Way Analysis of Variance If P is small (say, less Analysis of Variance on Response than 5%), then we Source DF SS MS F p conclude that at least one subgroup mean is Factor 3 4,626 1542.0 13.76 0.000 different. In this case, Error 20 2,242 112.1 we reject the hypothesis Total 23 6,868 that all the subgroup means are equal  12   22   32   42 The F-test is close to 1.00  2 Pooled  when subgroup means are 4 (*Only if subgroup sizes are equal) similar. In this case, This F-test ratio is much greater than 1.00, hence subgroup means are NOT similar. Analysis of Variance (ANOVA) UNCLASSIFIED / FOUO 86
• 87. UNCLASSIFIED / FOUO Multiple Comparisons  Tukey’s – Family error rate controlled  Fisher’s – Individual error rate controlled  Dunnett’s – Compares all results to a control group  Hsu’s MCB – Compares all results to a known best group  Which one do you use? In general, Tukey’s is recommended because it‟s „tighter‟. In other words, you will be less likely to find a difference between means (less statistical power), but you will be protected against a “false positive”, especially when there are a lot of groups.  Tukey‟s makes each test at a higher level of significance (a‟ > .05) and holds the family error rate to a = .05  Fisher‟s makes all tests at the specified significance level (usually a = .05) and reports the “family” error rate, a‟ Analysis of Variance (ANOVA) UNCLASSIFIED / FOUO 87
• 88. UNCLASSIFIED / FOUO What the F-Distribution Explains Here we see the F-Distribution and the F-test dynamics illustrated. This is the distribution of F-ratios that would occur if all methods produced the same results. Notice that the F-ratio we observed from the experiment is way out in the tail of the distribution. For this distribution, 3 is the d.f. for the numerator and 20 is the d.f. for the denominator. F - D is t r i b u t io n f o r 3 a n d 2 0 d e g r e e s o f F r e e d o m 0 .7 10% Point 0 .6 0 .5 5% Point Observed Point 0 .4 Prob 0 .3 0 .2 1% Point 0 .1 0 .0 0 2 4 6 8 10 12 14 F - V a lu e Analysis of Variance (ANOVA) UNCLASSIFIED / FOUO 88
• 89. UNCLASSIFIED / FOUO F Distribution: Probability Distribution Function (PDF) Plots 1.0 N1* N2 0.9 1, 1 d.f. 0.8 3, 3 d.f. 0.7 5, 8 d.f. 0.6 8, 8 d.f 0.5 * N1 refers to the d.f. in 0.4 the numerator 0.3 0.2 0.1 0.0 0 1 2 3 F Analysis of Variance (ANOVA) UNCLASSIFIED / FOUO 89