Copyright © 2013, 2009, and 2007, Pearson Education, Inc. Chapter 14 Comparing Groups: Analysis of Variance Methods Section 14.1 One-Way ANOVA: Comparing Several Means

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 3 Analysis of Variance The analysis of variance method compares means of several groups.  Let g denote the number of groups.  Each group has a corresponding population of subjects.  The means of the response variable for the g populations are denoted by.

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 4 Hypotheses and Assumptions for the ANOVA Test Comparing Means The analysis of variance is a significance test of the null hypothesis of equal population means:  The alternative hypothesis is:  : at least two of the population means are unequal.

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 5 The assumptions for the ANOVA test comparing population means are as follows:  The population distributions of the response variable for the g groups are normal with the same standard deviation for each group.  Randomization (depends on data collection method):  In a survey sample, independent random samples are selected from each of the g populations.  For an experiment, subjects are randomly assigned separately to the g groups. Hypotheses and Assumptions for the ANOVA Test Comparing Means

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 6 Example: Tolerance of Being on Hold? An airline has a toll-free telephone number for reservations. Often the call volume is heavy, and callers are placed on hold until a reservation agent is free to answer. The airline hopes a caller remains on hold until the call is answered, so as not to lose a potential customer.

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 7 The airline recently conducted a randomized experiment to analyze whether callers would remain on hold longer, on the average, if they heard:  An advertisement about the airline and its current promotion  Muzak (“elevator music”)  Classical music Example: Tolerance of Being on Hold?

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 8 The company randomly selected one out of every 1000 calls in a week. For each call, they randomly selected one of the three recorded messages. They measured the number of minutes that the caller stayed on hold before hanging up (these calls were purposely not answered). Example: Tolerance of Being on Hold?

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 9 Table 14.1 Telephone Holding Times by Type of Recorded Message. Each observation is the number of minutes a caller remained on hold before hanging up, rounded to the nearest minute. Example: Tolerance of Being on Hold?

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 10 Denote the holding time means for the populations that these three random samples represent by:  = mean for the advertisement  = mean for the Muzak  = mean for the classical music Example: Tolerance of Being on Hold?

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 11 The hypotheses for the ANOVA test are:   : at least two of the population means are different Example: Tolerance of Being on Hold?

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 12 Here is a display of the sample means: Figure 14.1 Sample Means of Telephone Holding Times for Callers Who Hear One of Three Recorded Messages. Question: Since the sample means are quite different, can we conclude that the population means differ? Example: Tolerance of Being on Hold?

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 13 As you can see from the output on the previous page, the sample means are quite different. But even if the population means are equal, we expect the sample means to differ some because of sampling variability. This alone is not sufficient evidence to enable us to reject. Example: Tolerance of Being on Hold?

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 14 Variability Between Groups and Within Groups Is the Key to Significance The ANOVA method is used to compare population means. It is called analysis of variance because it uses evidence about two types of variability.

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 15 Two examples of data sets with equal means but unequal variability: Figure 14.2 Data from Table 14.1 in Figure 14.2a and Hypothetical Data in Figure 14.2b That Have the Same Means but Less Variability Within Groups Variability Between Groups and Within Groups Is the Key to Significance

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 16 Which case do you think gives stronger evidence against ? What is the difference between the data in these two cases? Variability Between Groups and Within Groups Is the Key to Significance

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 17 In both cases the variability between pairs of means is the same. In ‘Case b’ the variability within each sample is much smaller than in ‘Case a.’ The fact that ‘Case b’ has less variability within each sample gives stronger evidence against. Variability Between Groups and Within Groups Is the Key to Significance

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 18 ANOVA F-Test Statistic The analysis of variance (ANOVA) F-test statistic is: The larger the variability between groups relative to the variability within groups, the larger the F test statistic tends to be.

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 19 The test statistic for comparing means has the F-distribution. The larger the F-test statistic value, the stronger the evidence against. ANOVA F-Test Statistic

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 20 SUMMARY: ANOVA F-test for Comparing Population Means of Several Groups 1. Assumptions:  Independent random samples  Normal population distributions with equal standard deviations 2. Hypotheses:   : at least two of the population means are different

Copyright © 2013, 2009, and 2007, Pearson Education, Inc Test statistic:  F- sampling distribution has, (total sample size – no. of groups) SUMMARY: ANOVA F-test for Comparing Population Means of Several Groups

Copyright © 2013, 2009, and 2007, Pearson Education, Inc P-value: Right-tail probability above the observed F- value 5. Conclusion: If decision is needed, reject if P-value significance level (such as 0.05) SUMMARY: ANOVA F-test for Comparing Population Means of Several Groups

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 23 The Variance Estimates and the ANOVA Table Let denote the standard deviation for each of the g population distributions  One assumption for the ANOVA F-test is that each population has the same standard deviation,.  The F-test statistic is the ratio of two estimates of, the population variance for each group.

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 24  The estimate of in the denominator of the F-test statistic uses the variability within each group.  The estimate of in the numerator of the F-test statistic uses the variability between each sample mean and the overall mean for all the data. The Variance Estimates and the ANOVA Table

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 25  Computer software displays the two estimates of in the ANOVA table similar to tables displayed in regression.  The MS column contains the two estimates, which are called mean squares.  The ratio of the two mean squares is the F- test statistic.  This F- statistic has a P-value. The Variance Estimates and the ANOVA Table

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 26 Example: Telephone Holding Times This example is a continuation of a previous example in which an airline conducted a randomized experiment to analyze whether callers would remain on hold longer, on the average, if they heard:  An advertisement about the airline and its current promotion  Muzak (“elevator music”)  Classical music

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 27 Denote the holding time means for the populations that these three random samples represent by:  = mean for the advertisement  = mean for the Muzak  = mean for the classical music Example: Telephone Holding Times

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 28 The hypotheses for the ANOVA test are:   : at least two of the population means are different Example: Telephone Holding Times

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 29 Table 14.2 ANOVA Table for F Test Using Data From Table 14.1 Example: Telephone Holding Times

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 30 Since P-value < 0.05, there is sufficient evidence to reject. We conclude that a difference exists among the three types of messages in the population mean amount of time that customers are willing to remain on hold. Example: Telephone Holding Times

Copyright © 2013, 2009, and 2007, Pearson Education, Inc Population distributions are normal  Moderate violations of the normality assumption are not serious. 2.These distributions have the same standard deviation  Moderate violations are also not serious. 3.The data resulted from randomization. Assumptions and the Effects of Violating Them

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 32 You can construct box plots or dot plots for the sample data distributions to check for extreme violations of normality. Misleading results may occur with the F-test if the distributions are highly skewed and the sample size N is small. Assumptions and the Effects of Violating Them

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 33  Misleading results may also occur with the F-test if there are relatively large differences among the standard deviations (the largest sample standard deviation being more than double the smallest one).  The ANOVA methods presented here are for independent samples. For dependent methods, other techniques must be used. Assumptions and the Effects of Violating Them

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 34 Using One F Test or Several t Tests to Compare the Means Why Not Use Multiple t-tests?  When there are several groups, using the F test instead of multiple t tests allows us to control the probability of a type I error.  If separate t tests are used, the significance level applies to each individual comparison, not the overall type I error rate for all the comparisons.  However, the F test does not tell us which groups differ or how different they are.