1. LECTURE 14 ANALYSIS OF VARIANCE EPSY 640 Texas A&M University

2. Multigroup experimental design PURPOSES: –COMPARE 3 OR MORE GROUPS SIMULTANEOUSLY –TAKE ADVANTAGE OF POWER OF LARGER TOTAL SAMPLE SIZE –CONSTRUCT MORE COMPLEX HYPOTHESES THAT BETTER REPRESENT OUR PREDICTIONS

3. Multigroup experimental design PROCEDURES –DEFINE GROUPS TO BE STUDIES: Experimental Assignment VS Intact or Existing Groups –OPERATIONALIZE NOMINAL, ORDINAL, OR INTERVAL/RATIO MEASUREMENT OF GROUPS eg. Nominal: SPECIAL ED, LD, AND NON- LABELED Ordinal: Warned, Acceptable, Exemplary Schools Interval: 0 years’, 1 years’, 2 years’ experience

4. Multigroup experimental design PATH REPRESENTATION Treat y e R y.T

5. Multigroup experimental design PATH REPRESENTATION Treat y e R y.T = √(493.87/39986) =.111  101.262 = 10.1 = std dev. Of errors

6. Multigroup experimental design VENN DIAGRAM REPRESENTATION SSy Treat SS SStreat SSerror R 2 =SStreat/SSy

7. Multigroup experimental design VENN DIAGRAM REPRESENTATION SSy = 39986 Treat SS SStreat = 493 SSerror = 39492 R 2 =SStreat/SSy =.111

8. Multigroup experimental design dummy coding. Since the values are arbitrary we can use any two numerical values, much as we can name things arbitrarily: 0 or 1 (you are in a group or not) –compares each group to a baseline group) Another nominal assignment of values is 1 and –1, called contrast coding: -1 = control, 1=experimental group Places groups above (+1) or below (-1) the average of all groups (grand mean)

9. Multigroup experimental design dummy coding. Since the values are arbitrary we can use any two numerical values, much as we can name things arbitrarily: 0 or 1 (you are in a group or not) Example: Hispanics=2, African Americans= 3, Whites=5 Recode: H AA W 1 0 0 codes for Hispanics 0 1 0 codes for AA’s 0 0 0 codes for Whites Need only two of the columns to specify a person

10. Multigroup experimental design Another nominal assignment of values is 1,0, and –1, called contrast coding: -1 = control, 1=experimental group Places groups above (+1) or below (-1) the average of all groups (grand mean) HAAW 1 0 0 0 1 0 -1 -1 0 Only two columns needed

11. Multigroup experimental design Another nominal assignment of values is 1,0, and –1, called contrast coding: -1 = control, 1=experimental group Places groups above (+1) or below (-1) the average of all groups (grand mean) HAAW 1 0 0 0 1 0 -1 -1 0 Only two columns needed: EQUIVALENT TO TWO PREDICTORS, THE HISPANIC VS. WHITE DIFFERENCE AND THE AFRICAN-AMERICAN VS. WHITE DIFFERENCE H A Y

12. Multigroup experimental design NOMINAL: If the three are simply different treatments or conditions then there is no preferred labeling, and we can give them values 1, 2, and 3 Forms: –arbitrary (A,B,C) –interval (1,2,3) assumes interval quality to groups such as amount of treatment –Contrast (-2, 1, 1) compares groups –Dummy (1, 0, 0), different for each group

13. Dummy Coding Regression Vars Subject Treatmentx 1 x 2 y 01A1017 02A1019 03B0122 04B0127 05C0033 06C0021

14. Contrast Coding Regression Vars Subject Treatmentx 1 x 2 y 01A1017 02A1019 03B0 122 04B0 127 05C -1 -133 06C -1 -121

15. Hypotheses about Means The usual null hypothesis about three group means is that they are all equal: H 0 :  1 =  2 =  3 while the alternative hypothesis is typically represented as H 1 :  i   j for some i,j.

16. ANOVA TABLE SOURCEdf Sum Mean SquareF of Squares Treatment…k-1Ss treat SS treat / k-1(SS treat / k )/(SS e /k(n-1)) error k(n-1)SseSS e / k(n-1) no test total kn-1SsySS y / (n-1) Table 9.2: Analysis of variance table for Sums of Squares

17. ANOVA concepts 1. Compare Variance(treatment + error) to Variance(error): MS treat /MS error 2. If treatment variance=0, then both estimate sampling variation in the population of individuals; the group means (recall sampling lecture) have a variance equal to error variance/kgroups, so 3. VAR(group means) = Var(error)/n, n=#scores per group and 4. n*Var(group means) = MS treat = Var(error)

19. F-DISTRIBUTION Fig. 9.5: Central and noncentral F-distributions alpha Central F-distribution power

20. ANOVA TABLE QUIZ SOURCEDFSSMSF PROB GROUP__10050_____ ERROR_____20 TOTAL20R 2 = ____