1. ANALYSIS OF VARIANCE

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 VENN DIAGRAM REPRESENTATION SSy Treat SS SStreat SSerror R 2 =SStreat/SSy

6. 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 A or B Another nominal assignment of values is 1 and –1, called contrast coding: -1 = control, 1=experimental group Compares exp. with control: 1(E) -1(C)

7. 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

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

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

10. 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.

11. ANOVA TABLE SOURCEdfSum of Mean F SquaresSquare Treatment…k-1SS treat SS treat SS treat / k (k-1) SS e /k(n-1) error k(n-1)SSeSS e / k(n-1) total kn-1SSySS y / (n-1) Table 9.2: Analysis of variance table for Sums of Squares

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

14. POWER for ANOVA Power nomographs- available from some texts on statistics Simulations- tryouts using SPSS –requires creating a known set of differences among groups –best understanding using means and SDs comparable to those to be used in the study –post hoc results from previous studies are useful; summary data can be used

15. ANOVA TABLE QUIZ SOURCEDFSSMSF PROB GROUP2___50__.05 ERROR________ TOTAL20R 2 = ____