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

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

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

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

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)

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 codes for Hispanics codes for AA’s codes for Whites Need only two of the columns to specify a person

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 Only two columns needed

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 Only two columns needed: EQUIVALENT TO TWO PREDICTORS, THE HISPANIC VS. WHITE DIFFERENCE AND THE AFRICAN-AMERICAN VS. WHITE DIFFERENCE H A Y

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 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

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)

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

ANOVA TABLE QUIZ SOURCEDFSSMSF PROB GROUP__10050_____ ERROR_____20 TOTAL20R 2 = ____