analysis of variance
DESCRIPTION
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 - PowerPoint PPT PresentationTRANSCRIPT
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 = ____