Statistics for the Social Sciences

1. Statistics for the Social Sciences Psychology 340 Fall 2013 Tuesday, October 15, 2013 Analysis of Variance (ANOVA)

2. Homework Assignment Due 10/22 Chapter 12: 14, 15, 21, 22 (Use SPSS for #21 & 22. Print out your output, identify the relevant statistics and probabilities on the output, and write out responses to all parts of each question) Chapter 13: 1, 4, 7, 8

3. More than two • Independent & One score per subject • 1 independent variable • The 1 factor between groups ANOVA: Statistical analysis follows design

4. Last Time Basics of ANOVA Why Computations (Definitional & Computational Formulas) Questions about any of the above before we move on?

5. Today • Brief review of last time • ANOVA table • Assumptions in ANOVA • Post-hoc and planned comparisons • Effect sizes in ANOVA • ANOVA in SPSS • Writing up ANOVA results in research reports • The structural model in ANOVA

6. Example • Effect of knowledge of prior behavior on jury decisions • Dependent variable: rate how innocent/guilty • Independent variable: 3 levels • Criminal record • Clean record • No information (no mention of a record)

7. Observed variance F-ratio = Variance from chance Analysis of Variance Test statistic • Need a measure that describes several difference scores • Variance • Variance is essentially an average squared difference MB MA MC • More than two groups

8. ANOVA Tables Generic ANOVA table: Results from criminal record study displayed as ANOVA table:

9. Assumptions in ANOVA Populations follow a normal curve Populations have equal variances

10. Why do the ANOVA? • What’s the big deal? Why not just run a bunch of t-tests instead of doing an ANOVA? • Experiment-wise error (see pg 391) • The type I error rate of the family (the entire set) of comparisons • αEW = 1 - (1 - α)c where c = # of comparisons • e.g., If you conduct two t-tests, each with an alpha level of 0.05, the combined chance of making a type I error is nearly 10 in 100 (rather than 5 in 100) • Planned comparisons and post hoc tests are procedures designed to reduce experiment-wise error

11. Testing Hypotheses with ANOVA • Hypothesis testing: a four step program • Step 1: State your hypotheses • Step 2: Set decision criteria • Step 3: Compute your test statistics • Compute your estimated variances • Compute your F-ratio • Step 4: Make a decision about your null hypothesis • Additional tests: Planned comparisons & Post hoc tests • Reconciling our multiple alternative hypotheses

12. The ANOVA tests this one!! Testing Hypotheses with ANOVA • Hypothesis testing: a five step program • Null hypothesis: H0: all the groups are equal • Step 1: State your hypotheses MB MC MA • Alternative hypotheses (HA) • Not all of the populations all have same mean Choosing between these requires additional test

13. XA XC XB 1 factor ANOVA • Planned contrasts and Post-hoc tests: • Further tests used to rule out the different alternative hypotheses • reject • reject • fail to reject • Alternative hypotheses (HA) • Not all of the populations all have same mean

14. Which follow-up test? • Planned comparisons • A set of specific comparisons that you “planned” to do in advance of conducting the overall ANOVA • General rule of thumb, don’t exceed the number of conditions that you have (or even stick with one fewer) • Post-hoc tests • A set of comparisons that you decided to examine only after you find a significant (reject H0) ANOVA • Often end up looking at all possible pair-wise comparisons

15. Planned Comparisons • Different types • Simple comparisons - testing two groups • Complex comparisons - testing combined groups • Bonferroni procedure • Use more stringent significance level for each comparison • Divide your desired α-level by the number of planned contrasts • Basic procedure: • Within-groups population variance estimate (denominator) • Between-groups population variance estimate of the two groups of interest (numerator) • Figure F in usual way

16. XA XC Planned Comparisons • Example: compare criminal record & no info grps XB 1) Within-groups population variance estimate (denominator) 2) Between-groups population variance estimate of the two groups of interest (numerator)

17. XA XC Planned Comparisons • Example: compare criminal record & no info grps XB 1) Within-groups population variance estimate (denominator) 2) Between-groups population variance estimate of the two groups of interest (numerator) 3) Figure F in usual way α = 0.05 Fcrit (1,12) = 4.75 Fail to reject H0: Criminal record and no info are not statistically different

18. Post-hoc tests • Generally, you are testing all of the possible comparisons (rather than just a specific few) • Different types • Tukey’s HSD test • Scheffe test • Others (Fisher’s LSD, Neuman-Keuls test, Duncan test) • Generally they differ with respect to how conservative they are.

19. Effect sizes in ANOVA Recall: • The effect size for ANOVA is r2 • Sometimes called η2 (“eta squared”) • The percent of the variance in the dependent variable that is accounted for by the independent variable

20. Effect sizes in ANOVA • The effect size for ANOVA is r2 • Sometimes called η2 (“eta squared”) • The percent of the variance in the dependent variable that is accounted for by the independent variable

21. One-Way ANOVA in SPSS • Enter the data: similar to independent samples t-test, observations in one column, a second column for group assignment • Analyze: compare means, 1-way ANOVA • Your grouping variable is the “factor” and your continuous (outcome) variable goes in the “dependent list” box • Specify any comparisons or post hocs at this time too • Planned Comparisons (contrasts): are entered with 1, 0, & -1 • Post-hoc tests: make sure that you enter your α-level • Under “options,” you can request descriptive statistics (e.g., to see group means)

22. ANOVA in Research Articles • F(3, 67) = 5.81, p < .01 • Means given in a table or in the text • Follow-up analyses • Planned comparisons • Using t tests

23. 1 factor ANOVA • Reporting your results • The observed difference • Kind of test • Computed F-ratio • Degrees of freedom for the test • The “p-value” of the test • Any post-hoc or planned comparison results • “The mean score of Group A was 12, Group B was 25, and Group C was 27. A one-way ANOVA was conducted and the results yielded a significant difference, F(2,25) = 5.67, p < 0.05. Post hoc tests revealed that the differences between groups A and B and A and C were statistically reliable (respectively t(13) = 5.67, p < 0.05 & t(13) = 6.02, p <0.05). Groups B and C did not differ significantly from one another”

24. Group’s mean’s deviation from grand mean (M-GM) Score’s deviation from group mean (X-M) Score’s deviation from grand mean (X-GM) The structural model and ANOVA • The structural model is all about deviations Score (X) Group mean (M) Grand mean (GM)