Statistics for business decision making and analysis robert stine and foster chapter 26

Chapter 26 Analysis of Variance Copyright © 2011 Pearson Education, Inc 26.1 Comparing Several Groups Did agricultural yield go up this year because of more fertilizer or more rain? Or is it the result of temperature or type of seed used?  Use regression analysis with dummy variables to compare the averages of several groups  This approach is also known as analysis of variance of 42 Copyright © 2011 Pearson Education, Inc 26.1 Comparing Several Groups Which Wheat Variety Should a Farmer Plant?  Five varieties of wheat are being considered: Endurance, Hatcher, NuHills, RonL, and Ripper  Each variety was grown in randomly chosen plots and yield was measured as bushels per acre  Balanced experiment: experiment with an equal number of observations for each treatment of 42 Copyright © 2011 Pearson Education, Inc 26.1 Comparing Several Groups Steps to Follow in the Analysis  Plot the data to find patterns  Propose a regression model for the data  Check conditions associated with the model  Test hypotheses and draw a conclusion of 42 Copyright © 2011 Pearson Education, Inc 26.1 Comparing Several Groups Comparing Groups in Plots – Boxplots of Yield of 42 Copyright © 2011 Pearson Education, Inc 26.1 Comparing Several Groups Comparing Groups in Plots – Summary Statistics of 42 Copyright © 2011 Pearson Education, Inc 26.1 Comparing Several Groups Relating the t-Test to Regression  Is there a significant difference between the average yield of Endurance and the others?  Since the variances among groups appear similar, use the two sample t-test and pool the variances of 42 Copyright © 2011 Pearson Education, Inc 26.1 Comparing Several Groups Relating the t-Test to Regression  The t-statistic and p-value show that Endurance has a significantly higher mean yield per acre than the combination of other varieties of 42 Copyright © 2011 Pearson Education, Inc 26.1 Comparing Several Groups Relating the t-Test to Regression  The t-test can be formulated as a regression with a dummy variable D(Endurance) that is coded if plot is seeded with Endurance and otherwise 10 of 42 Copyright © 2011 Pearson Education, Inc 26.3 Multiple Comparisons Tukey Confidence Intervals - Wheat Example  Note that the width of the 95% Tukey confidence interval is the same for any pairwise comparison  The difference in yield between any two varieties compared must be more than 6.07 bushels/acre in order to be statistically significant 28 of 42 Copyright © 2011 Pearson Education, Inc 26.3 Multiple Comparisons Bonferroni Confidence Intervals  These intervals adjust for multiple comparisons by changing the α level used in the standard interval to α/M for M intervals  For the comparison among wheat varieties, Bonferroni confidence intervals reduce α = 0.05 to α/10 = 0.005 and replaced t = 2.08 with t = 3.00 29 of 42 Copyright © 2011 Pearson Education, Inc 26.4 Groups of Different Size  With groups of different sizes, unbalanced data produce confidence intervals of different widths  Compute the estimated standard error for a pairwise comparison using the following formula with relevant sample sizes: se( y1  y2 ) se 1  n1 n2 30 of 42 Copyright © 2011 Pearson Education, Inc 4M Example 26.1: JUDGING THE CREDIBILITY OF ADVERTISEMENTS Motivation Advertising executives want to compare four commercials for a retail item that make claims of varying strengths Specifically, they want to know how over-the-top an ad can be before customers turn away in disbelief 31 of 42 Copyright © 2011 Pearson Education, Inc 4M Example 26.1: JUDGING THE CREDIBILITY OF ADVERTISEMENTS Method The data consist of reactions for a sample of 80 customers who viewed commercials with claims in one of four categories: Tame, Plausible, Stretch and Outrageous Each customer was randomly assigned to a commercial The response variable is Credibility obtained by customers’ responses to items on a questionnaire they completed after viewing the ad 32 of 42 Copyright © 2011 Pearson Education, Inc 4M Example 26.1: JUDGING THE CREDIBILITY OF ADVERTISEMENTS Method Use regression with three dummy variables to capture the four types of claims made in the commercials Check the conditions for ANOVA Linearity is not an issue and there are no obvious lurking variables because randomization was used in designing the study 33 of 42 Copyright © 2011 Pearson Education, Inc 4M Example 26.1: JUDGING THE CREDIBILITY OF ADVERTISEMENTS Mechanics - Results 34 of 42 Copyright © 2011 Pearson Education, Inc 4M Example 26.1: JUDGING THE CREDIBILITY OF ADVERTISEMENTS Mechanics – Results 35 of 42 Copyright © 2011 Pearson Education, Inc 4M Example 26.1: JUDGING THE CREDIBILITY OF ADVERTISEMENTS Mechanics – Check remaining conditions before proceeding with inference Similar variances condition is satisfied 36 of 42 Copyright © 2011 Pearson Education, Inc 4M Example 26.1: JUDGING THE CREDIBILITY OF ADVERTISEMENTS Mechanics – Check remaining conditions before proceeding with inference Nearly normal condition is satisfied 37 of 42 Copyright © 2011 Pearson Education, Inc 4M Example 26.1: JUDGING THE CREDIBILITY OF ADVERTISEMENTS Mechanics – The F-test has a p-value 0.0251; reject H0 The mean credibility of the four commercials is not equal Performing pairwise comparisons using Tukey intervals, the difference between average credibility must be more than 3.25 to be statistically significant 38 of 42 Copyright © 2011 Pearson Education, Inc 4M Example 26.1: JUDGING THE CREDIBILITY OF ADVERTISEMENTS Message Based on the Tukey intervals, there is only one statistically significant pairwise difference (between commercials making tame claims and those that make outrageous) Customers place less credibility in ads that make outrageous claims than ads that make tame claims 39 of 42 Copyright © 2011 Pearson Education, Inc Best Practices  Use a randomized experiment to obtain data  Check the assumptions of multiple regression when using ANOVA regression  Use Tukey or Bonferroni confidence intervals to identify groups that are significantly different  Recognize the cost of snooping in the data to choose hypotheses 40 of 42 Copyright © 2011 Pearson Education, Inc Pitfalls  Don’t compare the means of several groups using lots of t-tests  Don’t forget confounding factors  Never pretend you have only two groups 41 of 42 Copyright © 2011 Pearson Education, Inc Pitfalls (Continued)  Do not add or subtract standard errors  Do not use a one-way ANOVA to analyze data with repeated measurements 42 of 42 Copyright © 2011 Pearson Education, Inc ... Inc 26. 3 Multiple Comparisons Bonferroni Confidence Intervals  These intervals adjust for multiple comparisons by changing the α level used in the standard interval to α/M for M intervals  For. .. NuHills, RonL, and Ripper  Each variety was grown in randomly chosen plots and yield was measured as bushels per acre  Balanced experiment: experiment with an equal number of observations for each.. .Chapter 26 Analysis of Variance Copyright © 2011 Pearson Education, Inc 26. 1 Comparing Several Groups Did agricultural yield go up