1. Ch. 14: Comparisons on More Than Two Conditions

2. F and t The t test can only be used compare 2 groups.
The F test can be used to compare 2 or more groups.
F and t are statistically related in that F =t 2 when there are only 2 comparison groups.
i.e., dfbetween = 1

3. Calculating the Effect Size for F
But only if F is a focused, not an omnibus test.
i.e., dfbetween = 1

4. Logic of ANOVA ANOVA compares:
The variation of the average results per condition and
The average variation within the different conditions.
In this comparison, the F ratio or F test is formed.

5. Dividing Up the Variance
Total SS = Between SS + Within SS

6. Calculating the Degrees of Freedom
df between = k – 1
df within = N – k
df total = N – 1,

7. Summary ANOVA Table Source SS df MS F p Between conditions 138 3 46
11.50
.003
Within conditions 32 8 4

8. After the Omnibus F
Tests of simple effects are used to test differences between group means.
For example, if wanted to compare means from Group 3 and Group 1:

9. Factorial Designs Simple Effects: Differences between group means.
Main Effect: Overall effect of a factor, independent of other factors.
Interaction Effects: Leftover effects after removing all main effects.
Also called residuals

10. Effects and the Factorial ANOVA
Grand Mean: Mean of all group means.
Row Effect = Mr – MG
Column Effect = Mc – MG,
Interaction Effect = Group mean – grand mean – row effect – column effect

11. Row and Column Effects Milk treatment Vitamin Treatment Present Absent
Row means
Row effects 19 15 17
+3.0 12 10 11
-3.0
Column means
15.5
12.5 14
(grand mean)
Column effects
+1.5
-1.5

12. Interaction Effects Group mean - Grand mean Row effect Column effect =VM 19 14
3.0
1.5
0.5 V 15
(-1.5)
(-05) M 12
(-3.0)
(-0.5) O 10
Sum 56
0.0

13. Concept of Error Error = score – group mean
Group mean = grand mean + row effect + column effect + interaction
Score = grand mean + row effect + column effect + interaction + error

14. Computing the Two-Way ANOVA
Interaction SS = total SS – (row SS + column SS + within SS).

15. Summary Table for Two-Way ANOVA
Source SS df MS F p
reffect size
Vitamins (rows)
108 1
27.0
.0008
.88
Milk (columns) 27
6.75
.03
.68
Interaction 3
0.75
.41
.29
Within error 32 8 4

16. Contrasts
Useful for making focused comparisons when there are more than 2 groups.
Predictions are made using lambda coefficients, or λ weights.
These weights must sum to zero.
Contrast using a t test:

17. Linear Contrast Carved Out of Summary ANOVA Table Example
Source SS df MS F p
Between conditions
138 3 46
1.28
.35
Contrast
135 1
3.75
.089
Noncontrast 2
1.5
.04
Within conditions
288 8 36

18. Calculating Effect Size From a Contrast F

19. Other Correlational Measures
Alerting Correlation: Correlation between the means and their respective contrast weights.
Contrast Correlation: Effect size calculated based on contrast F.

20. Intrinsically vs. Nonintrinsically Repeated Measures Research
Intrinsically Repeated Measures: The study requires subjects to be measured more than once.
Nonintrinsically Repeated Measures: Repeated measures design in not required.
But used to increase the efficiency, precision, and power of the study.

21. Three Subjects’ Performance Measured on 4 Occasions

22. Three Subjects’ Performance Showing Zero Interaction