Chapter 16 Oneway Analysis of Variance Confidence Limits

Chapter 16 One-way Analysis of Variance Confidence Limits on Mean • Sample mean is a point estimate • We want interval estimate X Probability that interval computed this way includes m = 0. 95 1

Chapter 16 One-way Analysis of Variance For Our Data 2

Chapter 16 One-way Analysis of Variance Confidence Interval • The interval does not include 5. 65 --the population mean without a violent video • Consistent with result of t test • What can we conclude from confidence interval? 3

Analysis of Variance ANOVA is a technique for using differences between sample means to draw inferences about the presence or absence of differences between populations means. Chapter 16 One-way Analysis of Variance

Chapter 16 One-way Analysis of Variance 5 Major Points • The logic • Calculations in SPSS • Magnitude of effect X eta squared X omega squared Cont.

Chapter 16 One-way Analysis of Variance Major Points--Assumeptions • Assume: X Observations normally distributed within each population X Population variances are equal • Homogeneity of variance or homoscedasticity X Observations are independent 6

Chapter 16 One-way Analysis of Variance Assumptions--cont. • Analysis of variance is generally robust to first two X A robust test is one that is not greatly affected by violations of assumptions. 7

Chapter 16 One-way Analysis of Variance 8 Logic of the Analysis of Variance • Null hypothesis: Population means from different conditions are equal X m 1 = m 2 = m 3 = m 4 • Alternative hypothesis: H 1 X Not all population means equal. Cont.

9 Chapter 16 One-way Analysis of Variance Lets visualize total amount of variance in an experiment Total Variance = Mean Square Total Between Group Differences (Mean Square Group) Error Variance (Individual Differences + Random Variance) Mean Square Error F ratio is a proportion of the MS group/MS Error. The larger the group differences, the bigger the F

Chapter 16 One-way Analysis of Variance 10 Logic--cont. • Create a measure of variability among group means X MSgroup • Create a measure of variability within groups X MSerror Cont.

Chapter 16 One-way Analysis of Variance Logic--cont. • Form ratio of MSgroup /MSerror X Ratio approximately 1 if null true X Ratio significantly larger than 1 if null false X “approximately 1” can actually be as high as 2 or 3, but not much higher 11

12 Chapter 16 One-way Analysis of Variance Grand mean = 3. 78

Chapter 16 One-way Analysis of Variance 13 Calculations • Start with Sum of Squares (SS) X We need: • SStotal • SSgroups • SSerror • Compute degrees of freedom (df ) • Compute mean squares and F Cont.

Chapter 16 One-way Analysis of Variance Calculations--cont. 14

Chapter 16 One-way Analysis of Variance Degrees of Freedom (df ) • Number of “observations” free to vary X dftotal = N - 1 • N observations X dfgroups = g - 1 • g means X dferror = g (n - 1) • n observations in each group = n - 1 df • times g groups 15

Chapter 16 One-way Analysis of Variance Summary Table 16

Chapter 16 One-way Analysis of Variance When there are more than two groups • Significant F only shows that not all groups are equal X We want to know what groups are different. • Such procedures are designed to control familywise error rate. X Familywise error rate defined X Contrast with per comparison error rate 17

Chapter 16 One-way Analysis of Variance Multiple Comparisons • The more tests we run the more likely we are to make Type I error. X Good reason to hold down number of tests 18

Chapter 16 One-way Analysis of Variance Fisher’s LSD Procedure • Requires significant overall F, or no tests • Run standard t tests between pairs of groups. 19

Chapter 16 One-way Analysis of Variance 20 Bonferroni t Test • Run t tests between pairs of groups, as usual X Hold down number of t tests X Reject if t exceeds critical value in Bonferroni table • Works by using a more strict level of significance for each comparison Cont.

Chapter 16 One-way Analysis of Variance Bonferroni t--cont. • Critical value of a for each test set at. 05/c, where c = number of tests run X Assuming familywise a =. 05 X e. g. with 3 tests, each t must be significant at. 05/3 =. 0167 level. • With computer printout, just make sure calculated probability <. 05/c • Necessary table is in the book 21

Chapter 16 One-way Analysis of Variance 22 Assumptions • Assume: X Observations normally distributed within each population X Population variances are equal • Homogeneity of variance or homoscedasticity X Observations are independent Cont.

Chapter 16 One-way Analysis of Variance 23 Magnitude of Effect • Eta squared (h 2) X Easy to calculate X Somewhat biased on the high side X Formula • See slide #33 X Percent of variation in the data that can be attributed to treatment differences Cont.

Chapter 16 One-way Analysis of Variance Magnitude of Effect--cont. • Omega squared (w 2) X Much less biased than h 2 X Not as intuitive X We adjust both numerator and denominator with MSerror X Formula on next slide 24

Chapter 16 One-way Analysis of Variance h 2 and w 2 for Foa, et al. • h 2 =. 18: 18% of variability in symptoms can be accounted for by treatment • w 2 =. 12: This is a less biased estimate, and note that it is 33% smaller. 25