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3. ANOVA for 3 Drugs Give estimate of common population standard deviation Give observed test statistic value

4. ANOVA for 3 Drugs What is the distribution of the test statistic if the three drugs are equally effective in terms of the mean response? What is the corresponding p-value? At the 5% level, what is your (decision and) conclusion? Think about this and be ready to click in …

5. What is your decision and conclusion? A) Reject H 0 and conclude that the three drugs do appear to be equally effective in terms of the average response. B) Reject H 0 and conclude that at least one of the three drugs appears to be different in terms of the average response. C) Fail to reject H 0 and conclude that the three drugs do appear to be equally effective in terms of the average response. D) Fail to reject H 0 and conclude that at least one of the three drugs appears to be different in terms of the average response.

6. We Reject H 0 … What is Next? Multiple Comparisons page 178 At least one population mean is different? Which one(s)? Pairwise Comparisons: Drug 1 vs Drug 2, Drug 1 vs Drug 3, Drug 2 vs Drug 3 Does CI contain 0? Are two population means significantly different? Use a multiple comparisons method that controls for overall Type I error rate (or for overall confidence level). TUKEY’S Method (via SPSS only)

7. Try It! Comparing 3 Drugs We rejected H 0 … which drug(s) significantly differ? a. Use output to report about the three comparisons: Does the confidence interval for  1 –  2 contain 0? Does the confidence interval for  1 –  3 contain 0? Does the confidence interval for  2 –  3 contain 0?

8. Try It! Comparing 3 Drugs b. State conclusions regarding the differences between the mean response for 3 drug groups from the Tukey multiple comparison method. The population mean responses appear to differ for … but not differ for …

9. Individual Confidence Intervals for the Population Means We assume population standard deviations are all equal, so estimate of that common population standard deviation is used in forming individual CIs. DF for t* multiplier = N – k. From Utts, Jessica M. and Robert F. Heckard. Mind on Statistics, Fourth Edition. 2012. Used with permission.

10. Try It! Comparing 3 Drugs k = 3 groups based on N = 19 obs Pooled standard deviation sp = 1.62 DF for t* multiplier is N – k = ______. For level 0.95, t* = __________________ Drug 3 was descriptively the best. Compute a 95% CI for the mean response for all subjects taking Drug 3. Drug 1:Sample mean = 8.22Sample size = 5 Drug 2:Sample mean = 9.30Sample size = 7 Drug 3:Sample mean = 6.80Sample size = 7

11. Try It! Memory Experiment Three groups of subjects – each group given list of words to remember. Lengths: 1 st group = 10 words (short), 2 nd group = 20 words (medium), 3 rd group = 40 words (long). % of words recalled for each subject recorded. Sample mean percentage of words recalled: Short: 68.3%, Medium: 48%, Long: 39.2% One-way ANOVA used to assess whether length of list had significant effect on % of words recalled.

12. Try It! Memory Experiment a. Complete the ANOVA table. b. State hypotheses that the above F statistic is testing. H 0 : _______________________________________ H a : _________________________________________ Click in for part (c)

13. Click in: Yes or No c.Using a 5% level, does it appear that the average % words recalled is the same for the three different lengths of list?

14. Try It! Memory Experiment d. Multiple Comparisons: Circle all pairs significantly different (at 1% level). short vs medium short vs long medium vs long

15. e.Calculate a 99% CI for mean % words recalled for long list group, where sample mean based on 6 subjects in long list group was 39.2 percent.

16. What if conditions do not hold? Pg 181 Section 16.3: two methods used when one or both assumptions about equal popul std devs and normal distributions are violated. When data skewed, or extreme outliers present, better to analyze median rather than mean. Tests for comparing medians:  Kruskal-Wallis Test: Compare relative rankings (sizes) of the data in the observed samples, called a rank test or nonparametric test (no assumptions made about a specific distrib for population of measurements.  Mood’s Median Test

17. Two-Way ANOVA One-way ANOVA: only one explanatory variable and one quantitative response variable. Section 16.4 = overview of two-way ANOVA  Possibility of interaction …effect of one factor on mean response depends on specific level of other factor. Want to learn more? Consider taking for Stats 401 next term Another great stat class (no prereq) = Stats 408 (winter only)