analysis of variance

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

Ry.T

Multigroup experimental design

• VENN DIAGRAM REPRESENTATION

SSy

Treat SS

SStreatSSerror

R2=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 Treatment x1 x2 y

• 01 A 1 0 17

• 02 A 1 0 19

• 03 B 0 1 22

• 04 B 0 1 27

• 05 C 0 0 33

• 06 C 0 0 21

Contrast CodingRegression

Vars

• Subject Treatment x1 x2 y

• 01 A 1 0 17

• 02 A 1 0 19

• 03 B 0 1 22

• 04 B 0 1 27

• 05 C -1 -1 33

• 06 C -1 -1 21

Hypotheses about Means

• The usual null hypothesis about three group means is that they are all equal:

• H0 : 1 = 2 = 3

• while the alternative hypothesis is typically represented as

• H1 : i j for some i,j .

ANOVA TABLE• SOURCE df Sum of Mean F

Squares Square

• Treatment… k-1 SStreat SStreat SStreat/ k(k-1) SSe /k(n-1)

• error k(n-1) SSe SSe / k(n-1)

• total kn-1 SSy SSy / (n-1)

• Table 9.2: Analysis of variance table for Sums of Squares

ANOVA

DEPRESSION

167.079 2 83.539 .901 .407

36238.477 391 92.682

36405.556 393

Between Groups

Within Groups

Total

Sum ofSquares df Mean Square F Sig.

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

SOURCE DF SS MS F PROB

GROUP 2 ___ 50 __ .05

ERROR __ ___ ___

TOTAL 20 R2 = ____