1. Differences Among Groups
chapter 9
Differences Among Groups

2. Chapter Outline How statistics test differences Types of t tests
Interpreting t
Relationship of t and r
Analysis of variance
Analysis of covariance
Experimentwise error rate
Understanding multivariate techniques

3. How Statistics Test Differences
Independent and dependent variables
Evaluating the null hypothesis
Establishing two points:
Are the groups different?
How meaningful are the differences?
Assumptions for F and t distributions
Distribution is normal.
Samples from population are random.
Numerator and denominator are estimates of same thing.
Numerator and denominator are independent.

4. Differences Among Groups: t= M 1 – 2 s / n + d f ( )
Omega squared w 2 = ( t – 1 ) / + n

5. Dependent t Test t = M p o s – r e 2 + ( ) / N 1 é ë ê ù û ú

6. Components of Variance
Total variance = True variance + Error variance
Test of significance = True variance ÷ Error variance
– t and F
Test of meaningfulness = True variance ÷ Total variance
r2 and omega squared

7. ANOVA Formulas
(continued)

8. ANOVA Formulas (continued)
Summary table
Source SS df MS F
Between C–B K–1 (C–B)/(K–1) MSB/MSW (true)
Within A–C N–K (A–C)/(N–K) (error)
Total C–B N–1

9. Data for ANOVA Group 1 X X2 10 100 11 121 9 81 8 64 48 466 Group 2
10 100
11 121
9 81
8 64
48 466
Group 2
X X2
7 49
8 64
6 36
34 234
Group 3
X X2
3 9
6 36
4 16
5 25
21 95

10. Analysis of Variance (ANOVA)
Summary table for ANOVA
Source SS df MS F
Between *
Within (error)
Total
*p < .01

11. Follow-Ups to ANOVA
Scheffé: Contrast all 3 pairs (1 and 2, 1 and 3, 2 and 3); a difference must exceed calculated Scheffé critical value (CV) to be significant, p < .05. k – 1 ( ) F a; ; N é ë ù û
2(MSW /n 3 . 8 = 2 7 9
2(1
.23
/ 5 ) 5
CV =

12. Model for Factorial ANOVA

13. Factorial ANOVA IV1: are the 3 levels (rows) different?
IV2: are the 2 levels (columns) different?
Interaction: does one IV change as a function of the other?
Three F ratios to test significance:
IV1
IV2
Interaction

14. Interaction From Factorial ANOVA
(continued)

15. Interaction From Factorial ANOVA
(continued)

16. Interaction From Factorial ANOVA (continued)

17. Repeated-Measures ANOVA
Typical use: do two or more groups change differently over trials (over time)?
Example: Two groups (exercise and control) are measured every 2 weeks for 12 weeks.
Analysis is a between (group)–by–within (trials) ANOVA with repeated measures on trials (trials are time, every 2 weeks).

18. Interaction of Groups With Repeated Measures
Three age levels and four levels of movement difficulty as repeated measures. Used with permission from Albers, Thomas, & Thomas, 2005.

19. Experimenterwise Error
Bonferroni technique to adjust alpha
EW =  ÷ number of comparisons
If  = .05 and three t tests (or ANOVAs) are to be done, then
Adjusted  = .05/3 = .017

20. Discriminant Analysis
One independent variable with two or more levels and several (two or more) measures (dependent variables).
Can a linear combination be made of dependent variables that will identify group membership?

21. Multivariate Analysis of Variance (MANOVA)
Two or more independent variables and two or more dependent variables
Independent variables = two age groups (10 & 12 years) and two levels of expertise (experts and novices)
Dependent variables = knowledge and performance
(continued)

22. Multivariate Analysis of Variance (MANOVA) (continued)
MANOVA F ratios
Age
Expertise
Age  expertise
Follow-ups

23. MANOVA With Repeated Measures
2 or more groups with 2 or more repeated measures on 2 or more variables
Groups Trials 1 2 3
Groups = Exp 1, Exp 2, Control (3)
Trials = time periods
dvs = heart rate and body fat
Sphericity assumption – equal r across trials

24. Analysis of Covariance (ANCOVA)
Relationship between one (or more) covariates and dependent variable is removed.
ANOVA is calculated on remaining dependent variables after variance due to covariate is removed.

25. Multivariate Analysis of Covariance (MANCOVA)
Relationship between groups of covariates and multiple dependent variables is removed.
MANCOVA is done on remaining dependent variables.