1. Confidence Limits on Mean Sample mean is a point estimateSample mean is a point estimate We want interval estimateWe want interval estimate  Probability that interval computed this way includes  = 0.95

2. For Our Data

3. Confidence Interval The interval does not include 5.65--the population mean without a violent videoThe interval does not include 5.65--the population mean without a violent video Consistent with result of t testConsistent with result of t test What can we conclude from confidence interval?What can we conclude from confidence interval?

4. 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.

5. Major Points The logicThe logic Calculations in SPSSCalculations in SPSS Magnitude of effectMagnitude of effect Xeta squared Xomega squared Cont.

6. Major Points--Assumeptions Assume:Assume: XObservations normally distributed within each population XPopulation variances are equal Homogeneity of variance or homoscedasticityHomogeneity of variance or homoscedasticity XObservations are independent

7. Assumptions--cont. Analysis of variance is generally robust to first twoAnalysis of variance is generally robust to first two XA robust test is one that is not greatly affected by violations of assumptions.

8. Logic of the Analysis of Variance Null hypothesis: Population means from different conditions are equalNull hypothesis: Population means from different conditions are equal   1 =       Alternative hypothesis: H 1Alternative hypothesis: H 1 XNot all population means equal. Cont.

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

10. Logic--cont. Create a measure of variability among group meansCreate a measure of variability among group means XMS group Create a measure of variability within groupsCreate a measure of variability within groups XMS error Cont.

11. Logic--cont. Form ratio of MS group /MS errorForm ratio of MS group /MS error XRatio approximately 1 if null true XRatio significantly larger than 1 if null false X“approximately 1” can actually be as high as 2 or 3, but not much higher

12. Grand mean = 3.78

13. Calculations Start with Sum of Squares (SS)Start with Sum of Squares (SS) XWe need: SS totalSS total SS groupsSS groups SS errorSS error Compute degrees of freedom (df )Compute degrees of freedom (df ) Compute mean squares and FCompute mean squares and F Cont.

14. Calculations--cont.

15. Degrees of Freedom (df ) Number of “observations” free to varyNumber of “observations” free to vary Xdf total = N - 1 N observationsN observations Xdf groups = g - 1 g meansg means Xdf error = g (n - 1) n observations in each group = n - 1 dfn observations in each group = n - 1 df times g groupstimes g groups

16. Summary Table

17. When there are more than two groups Significant F only shows that not all groups are equalSignificant F only shows that not all groups are equal XWe want to know what groups are different. Such procedures are designed to control familywise error rate.Such procedures are designed to control familywise error rate. XFamilywise error rate defined XContrast with per comparison error rate

18. Multiple Comparisons The more tests we run the more likely we are to make Type I error.The more tests we run the more likely we are to make Type I error. XGood reason to hold down number of tests

19. Fisher’s LSD Procedure Requires significant overall F, or no testsRequires significant overall F, or no tests Run standard t tests between pairs of groups.Run standard t tests between pairs of groups.

20. Bonferroni t Test Run t tests between pairs of groups, as usualRun t tests between pairs of groups, as usual XHold down number of t tests XReject if t exceeds critical value in Bonferroni table Works by using a more strict level of significance for each comparisonWorks by using a more strict level of significance for each comparison Cont.

21. Bonferroni t--cont. Critical value of  for each test set at.05/c, where c = number of tests runCritical value of  for each test set at.05/c, where c = number of tests run  Assuming familywise  =.05 Xe. g. with 3 tests, each t must be significant at.05/3 =.0167 level. With computer printout, just make sure calculated probability <.05/cWith computer printout, just make sure calculated probability <.05/c Necessary table is in the bookNecessary table is in the book

22. Assumptions Assume:Assume: XObservations normally distributed within each population XPopulation variances are equal Homogeneity of variance or homoscedasticityHomogeneity of variance or homoscedasticity XObservations are independent Cont.

23. Magnitude of Effect Eta squared (  2 )Eta squared (  2 ) XEasy to calculate XSomewhat biased on the high side XFormula See slide #33See slide #33 XPercent of variation in the data that can be attributed to treatment differences Cont.

24. Magnitude of Effect--cont. Omega squared (  2 )Omega squared (  2 )  Much less biased than  2 XNot as intuitive XWe adjust both numerator and denominator with MS error XFormula on next slide

25.  2 and  2 for Foa, et al.  2 =.18: 18% of variability in symptoms can be accounted for by treatment  2 =.18: 18% of variability in symptoms can be accounted for by treatment  2 =.12: This is a less biased estimate, and note that it is 33% smaller.  2 =.12: This is a less biased estimate, and note that it is 33% smaller.