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Introduction to ANOVA in Social Sciences Fall 2015

Understand ANOVA, t-test, and Chi-Square for statistical analysis using examples. Explore cognitive dissonance, memory biases, and experimental design. Hands-on Excel assignments and lab sessions. Learn to compare means and interpret results.

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Introduction to ANOVA in Social Sciences Fall 2015

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  1. Introduction to Statistics for the Social SciencesSBS200, COMM200, GEOG200, PA200, POL200, or SOC200Lecture Section 001, Fall 2015Room 150 Harvill Building10:00 - 10:50 Mondays, Wednesdays & Fridays. Welcome http://courses.eller.arizona.edu/mgmt/delaney/d15s_database_weekone_screenshot.xlsx

  2. By the end of lecture today11/6/15 Introduction to ANOVA

  3. Before next exam (November 20th) 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

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

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

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

  7. Study Type 3: One-way ANOVA Single Independent Variable comparing more than two groups Single Dependent Variable (numerical/continuous) Used to test the effect of the IV on the DV 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 Dependent variable is always quantitative Sales per Girl scout Sales per Girl scout New Bike None Trip Hawaii New Bike None Trip Hawaii In an ANOVA, independent variable is qualitative (& more than two groups)

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

  9. One-way ANOVA Number of cookies sold Bike None Hawaii trip Incentives • 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

  10. 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 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 MSBetween F = Mean square within groups: sum of squares within groups divided by its degrees of freedom MSWithin

  11. 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 Tells us generally about number of participants / observations Tells us generally about number of groups / levels of IV • 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 Review

  12. 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. How many levels of the Independent Variable? What is Independent Variable? Troop 3 (Hawaii) 14 9 19 13 15 Troop 1 (nada) 10 8 12 7 13 Troop 2 (bicycle) 12 14 10 11 13 What is Dependent Variable? How many groups? n = 5 x = 10 n = 5 x = 12 n = 5 x = 14

  13. 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 • To get an effect we want: • Large “F” - big effect and small variability • Small “p” - less than 0.05 (whatever our alpha is) We don’t know which means are different from which …. just that they are not all the same

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

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

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

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

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

  19. ANOVA table MSbetween MSwithin 40 88 SSbetween 12 2 ANOVA table dfbetween F Source df MS SS ? Between 40 2 ? 2.73 20 Within 88 12 ? 7.33 Total 128 14 SSwithin dfwithin 88 20 =2.73 =7.33 40 7.33 12 =20 2

  20. Make decision whether or not to reject null hypothesis Observed F = 2.73 Critical F(2,12) = 3.89 2.73 is not farther out on the curve than 3.89 so, we do not reject the null hypothesis F(2,12) = 2.73; n.s. Conclusion: There appears to be no effect of type of incentive on number of girl scout cookies sold 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.

  21. Let’s do same problemUsing 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

  22. Let’s do same problemUsing MS Excel

  23. Let’s do same problemUsing MS Excel

  24. Let’s do oneReplication of study(new data)

  25. Let’s do same problemUsing MS Excel

  26. Let’s do same problemUsing MS Excel

  27. MSbetween MSwithin 40 88 SSbetween 12 2 dfbetween 3-1=2 # groups - 1 SSwithin dfwithin # scores - number of groups 15-3=12 88 20 =2.73 =7.33 40 # scores - 1 7.33 12 =20 2 15- 1=14

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

  29. Make decision whether or not to reject null hypothesis Observed F = 2.73 Critical F(2,12) = 3.89 2.7 is not farther out on the curve than 3.89 so, we do not reject the null hypothesis 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

  30. Make decision whether or not to reject null hypothesis Observed F = 2.72727272 F(2,12) = 2.73; n.s. Critical F(2,12) = 3.88529 2.7 is not farther out on the curve than 3.89 so, we do not reject the null hypothesis Conclusion: There appears to be no effect of type of incentive on number of girl scout cookies sold 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.

  31. One way analysis of varianceVariance is divided Remember, one-way = one IV Total variability Between group variability (only one factor) Within group variability (error variance) 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

  32. 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)? Still, difference between means • Critical statistic (e.g. z or t or F or r) value? Step 3: Calculations MSBetween F = MSWithin Still, variabilityof curve(s) 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

  33. Thank you! See you next time!!

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