2. Introduction to Statistics for the Social Sciences SBS200, COMM200, GEOG200, PA200, POL200, or SOC200 Lecture Section 001, Fall 2015 Room 150 Harvill Building 10:00 - 10:50 Mondays, Wednesdays & Fridays. http://courses.eller.arizona.edu/mgmt/delaney/d15s_database_weekone_screenshot.xlsx

4. By the end of lecture today 11/6/15 Introduction to ANOVA

5. Before next exam (November 20 th ) Please read chapters 1 - 11 in OpenStax textbook Please read Chapters 2, 3, and 4 in Plous Chapter 2: Cognitive Dissonance Chapter 3: Memory and Hindsight Bias Chapter 4: Context Dependence

6. Homework Assignment Assignment 19 Please complete this homework worksheet Completing ANOVAs using Excel Due: Monday, November 9 th

7. Everyone will want to be enrolled in one of the lab sessions Labs continue this week, Project 3

8. Study Type 2: t-test Study Type 3: One-way Analysis of Variance (ANOVA) Comparing more than two means We are looking to compare two means

9. Single Independent Variable comparing more than two groups Study Type 3: One-way ANOVA Single Dependent Variable (numerical/continuous) Ian was interested in the effect of incentives for girl scouts on the number of cookies sold. He randomly assigned girl scouts into one of three groups. The three groups were given one of three incentives and looked to see who sold more cookies. The 3 incentives were 1) Trip to Hawaii, 2) New Bike or 3) Nothing. This is an example of a true experiment Used to test the effect of the IV on the DV None New Bike Sales per Girl scout Trip Hawaii None New Bike Trip Hawaii Dependent variable is always quantitative In an ANOVA, independent variable is qualitative (& more than two groups) Sales per Girl scout

10. Be careful you are not designing a Chi Square One-way ANOVA versus Chi Square None New Bike Sales per Girl scout Trip Hawaii This is an ANOVA None New Bike Total Number of Boxes Sold Trip Hawaii This is a Chi Square If this is just frequency you may have a problem These are means These are just frequencies

11. One-way ANOVA One-way ANOVAs test only one independent variable - although there may be many levels “Factor” = one independent variable “Level” = levels of the independent variable treatment condition groups “Main Effect” of independent variable = difference between levels Note: doesn’t tell you which specific levels (means) differ from each other A multi-factor experiment would be a multi-independent variables experiment Number of cookies sold Incentives None Bike Hawaii trip

12. Sum of squares (SS): The sum of squared deviations of some set of scores about their mean Mean squares (MS): The sum of squares divided by its degrees of freedom Mean square within groups: sum of squares within groups divided by its degrees of freedom Mean square between groups: sum of squares between groups divided by its degrees of freedom Mean square total: sum of squares total divided by its degrees of freedom MS Within MS Between F =

13. Comparing ANOVAs with t-tests Similarities still include: Using distributions to make decisions about common and rare events Using distributions to make inferences about whether to reject the null hypothesis or not The same 5 steps for testing an hypothesis The three primary differences between t-tests and ANOVAS are: 1. ANOVAs can test more than two means 2. We are comparing sample means indirectly by comparing sample variances 3. We now will have two types of degrees of freedom t(16) = 3.0; p < 0.05 F(2, 15) = 3.0; p < 0.05 Tells us generally about number of participants / observations Tells us generally about number of groups / levels of IV

14. A girl scout troop leader wondered whether providing an incentive to whomever sold the most girl scout cookies would have an effect on the number cookies sold. She provided a big incentive to one troop (trip to Hawaii), a lesser incentive to a second troop (bicycle), and no incentive to a third group, and then looked to see who sold more cookies. n = 5 x = 10 n = 5 x = 12 n = 5 x = 14 Troop 1 (nada) 10 8 12 7 13 Troop 2 (bicycle) 12 14 10 11 13 Troop 3 (Hawaii) 14 9 19 13 15 What is Independent Variable? How many groups? What is Dependent Variable? How many levels of the Independent Variable?

15. Main effect of incentive: Will offering an incentive result in more girl scout cookies being sold? If we have a “effect” of incentive then the means are significantly different from each other we reject the null we have a significant F p < 0.05 We don’t know which means are different from which …. just that they are not all the same To get an effect we want: Large “F” - big effect and small variability Small “p” - less than 0.05 (whatever our alpha is)

16. Hypothesis testing: Step 1: Identify the research problem Describe the null and alternative hypotheses Is there a significant difference in the number of cookie boxes sold between the girlscout troops that were given the different levels of incentive?

17. Hypothesis testing: Decision rule =.05 Critical F (2,12) =3.98 Degrees of freedom (between) = number of groups - 1 Degrees of freedom (within) = # of scores - # of groups = 3 - 1 = 2 = (15-3) = 12* *or = (5-1) + (5-1) + (5-1) = 12.

18. α =.05 Critical F (2,12) = 3.89 F (2,12) Appendix B.4 (pg.518)

19. ANOVA table ? dfMS F ? ? ? Source Between Within Total ? ? SS ? ? ? “SS” = “Sum of Squares” - will be given for exams - you can think of this as the numerator in a standard deviation formula

20. ? ? ? ANOVA table 128 dfMS F # groups - 1 # scores - number of groups # scores - 1 2 12 14 Source Between Within Total 88 40 SS ? ? ? ? ? ? “SS” = “Sum of Squares” - will be given for exams 3-1=2 15-3=12 15- 1=14

21. ANOVA table 128 df MS F 2 12 14 Source Between Within Total 88 40 SS MS between MS within SS within df within 20 7.33 SS between df between 88 12 =7.33 40 2 =20 20 7.33 =2.73 2.73 40 2 88 12 ? ? ?

22. Make decision whether or not to reject null hypothesis 2.73 is not farther out on the curve than 3.89 so, we do not reject the null hypothesis Observed F = 2.73 Critical F (2,12) = 3.89 Conclusion: There appears to be no effect of type of incentive on number of girl scout cookies sold F (2,12) = 2.73; n.s. The average number of cookies sold for three different incentives were compared. The mean number of cookie boxes sold for the “Hawaii” incentive was 14, the mean number of cookies boxes sold for the “Bicycle” incentive was 12, and the mean number of cookies sold for the “No” incentive was 10. An ANOVA was conducted and there appears to be no significant difference in the number of cookies sold as a result of the different levels of incentive F(2, 12) = 2.73; n.s.

23. Let ’ s do same problem Using MS Excel A girlscout troop leader wondered whether providing an incentive to whomever sold the most girlscout cookies would have an effect on the number cookies sold. She provided a big incentive to one troop (trip to Hawaii), a lesser incentive to a second troop (bicycle), and no incentive to a third group, and then looked to see who sold more cookies. Troop 1 (Nada) 10 8 12 7 13 Troop 2 (bicycle) 12 14 10 11 13 Troop 3 (Hawaii) 14 9 19 13 15 n = 5 x = 10 n = 5 x = 12 n = 5 x = 14

24. Let ’ s do same problem Using MS Excel

26. Let ’ s do one Replication of study (new data)

27. Let ’ s do same problem Using MS Excel

29. SS within df within SS between df between 88 12 =7.33 40 2 =20 20 7.33 =2.73 40 2 88 12 MS between MS within # groups - 1 # scores - number of groups # scores - 1 3-1=2 15-3=12 15- 1=14

30. F critical (is observed F greater than critical F?) P-value (is it less than.05?) No, so it is not significant Do not reject null No, so it is not significant Do not reject null

31. Make decision whether or not to reject null hypothesis 2.7 is not farther out on the curve than 3.89 so, we do not reject the null hypothesis Observed F = 2.73 Critical F (2,12) = 3.89 Also p-value is not smaller than 0.05 so we do not reject the null hypothesis Step 6: Conclusion: There appears to be no effect of type of incentive on number of girl scout cookies sold

32. Make decision whether or not to reject null hypothesis 2.7 is not farther out on the curve than 3.89 so, we do not reject the null hypothesis Observed F = 2.72727272 Critical F (2,12) = 3.88529 Conclusion: There appears to be no effect of type of incentive on number of girl scout cookies sold F (2,12) = 2.73; n.s. The average number of cookies sold for three different incentives were compared. The mean number of cookie boxes sold for the “Hawaii” incentive was 14, the mean number of cookies boxes sold for the “Bicycle” incentive was 12, and the mean number of cookies sold for the “No” incentive was 10. An ANOVA was conducted and there appears to be no significant difference in the number of cookies sold as a result of the different levels of incentive F(2, 12) = 2.73; n.s.

33. One way analysis of variance Variance is divided Total variability Within group variability (error variance) Between group variability (only one factor) Remember, 1 factor = 1 independent variable (this will be our numerator – like difference between means) Remember, error variance = random error (this will be our denominator – like within group variability Remember, one-way = one IV

34. Five steps to hypothesis testing Step 1: Identify the research problem (hypothesis) Describe the null and alternative hypotheses Step 2: Decision rule Alpha level? ( α =.05 or.01)? Step 3: Calculations Step 4: Make decision whether or not to reject null hypothesis If observed t (or F) is bigger then critical t (or F) then reject null Step 5: Conclusion - tie findings back in to research problem Critical statistic (e.g. z or t or F or r) value? MS Within MS Between F = Still, difference between means Still, variability of curve(s)