ANALYSIS OF VARIANCE

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

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

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

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

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)

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

Dummy Coding Regression Vars Subject Treatmentx 1 x 2 y 01A A B B C C0021

Contrast Coding Regression Vars Subject Treatmentx 1 x 2 y 01A A B B C C

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.

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

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

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

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