1. The Two-way ANOVA
We have learned how to test for the effects of independent variables considered one at a time.
However, much of human behavior is determined by the influence of several variables operating at the same time.
Sometimes these variables combine to influence performance.
Two Way ANOVA

2. The Two-way ANOVA
We need to test for the independent and combined effects of multiple variables on performance. We do this with an ANOVA that asks:
(i) how different from each other are the means for levels of Variable A?
(ii) how different from each other are the means for levels of Variable B?
(iii) how different from each other are the means for the treatment combinations produced by A and B together?
Two Way ANOVA

3. The Two-way ANOVA
The first two of those questions are questions about main effects of the respective independent variables.
The third question is about the interaction effect, the effect of the two variables considered simultaneously.
Two Way ANOVA

4. The Two-way ANOVA Main effect Interaction
A main effect is the effect on performance of one treatment variable considered in isolation (ignoring other variables in the study)
Interaction
an interaction effect occurs when the effect of one variable is different across levels of one or more other variables
Two Way ANOVA

5. Interaction of variables
In order to detect interaction effects, we must use “factorial” designs.
In a factorial design each variable is tested at every level of all of the other variables.
A1 A2
B1 I II
B2 III IV
Two Way ANOVA

6. Interaction of variablesII
III IV
I vs III
Effect of B at level A1 of variable A
II vs IV
Effect of B at A2
If these are different, then we say that A and B interact
Whether you look at the varying effects of A at levels of B or the varying effects of B at levels of A is entirely a matter of choice – you get the same answer either way.
Two Way ANOVA

7. Interaction of variablesII
III IV
I vs II
Effect of A at B1
III vs IV
Effect of A at B2
If these are different, then we say that A and B interact
Whether you look at the varying effects of A at levels of B or the varying effects of B at levels of A is entirely a matter of choice – you get the same answer either way.
Two Way ANOVA

8. B1 B2 B1 B2
A A A A2
In the graphs above, the effect of A varies at levels of B, and the effect of B varies at levels of A. How you say it is a matter of preference (and your theory).
In each case, the interaction is the whole pattern. No part of the graph shows the interaction. It can only be seen in the entire pattern (here, all 4 data points).
Two Way ANOVA

9. Interaction of variables
In order to test the hypothesis about an interaction, you must use a factorial design.
The designs shown on the previous slide (2 X 2s) are the simplest possible factorial designs.
We frequently use 3 and 4 variable designs, but beware: it’s very difficult to interpret an interaction among more than 4 variables!
Two Way ANOVA

10. Two-way ANOVA - Computation
Two variables, A & B
Raw Scores (Xi) A1 A2 A3 B1 B2 B1 B2 B1 B
2 X 3 = 6 treatments
Two Way ANOVA

11. Cell Totals (Tij) and marginal totals (Ti & Tj)
A1 A2 A3 Tj B B Ti
Cell totals are in green
B Totals
are in red
Totals for the levels of A are in blue
Two Way ANOVA

12. Two-Way ANOVA – Computational formulas
CM = (ΣXi)2/N = (147)2 = ΣX2 = … = 2427 SSTotal = ΣX2 – CM SSE = ΣX2 – Σ(Tij)2 nij
Notice that these are the original data from Slide 10
Two Way ANOVA

13. Two-Way ANOVA – Computational formulas
Σ(Ti)2/ni = (21)2 + (21)2 + (105)2 =
SSA = Σ(Ti)2 – CM ni
SSA is the sum of squared deviations for Variable A
Two Way ANOVA

14. Two-Way ANOVA – Computational formulas
Σ(Tj)2/ni = = 1381
9 9
SSB = Σ(Tj)2 – CM nj
SSB is the sum of squared deviations for Variable B. It is NOT the sum of squares for Blocks – this is not a block design!
Two Way ANOVA

15. Two-Way ANOVA – Computational formulas
Σ(Tij)2/nij = … = 2337
SSAB = Σ(Tij)2 – Σ(Tj)2 – Σ(Ti)2 + (ΣX)2
nij nj ni n
SSAB is the sum of squared deviations for the interaction of A and B
Two Way ANOVA

16. CM
We now compute:
SSA = – = 784
SSB = 1381 – = 180.5
SSTotal = 2427 – =
SSE = 2427 – 2337 = 90
SSAB = 2337 – 1381 –
= 172
Two Way ANOVA

17. Source df SS MS F A a-1 = 2 784 392 52.27 B b-1 = 1 180.5 180.5 24.07
AB (a-1)(b-1) =
Error n-ab =
Total n-1 =
Two Way ANOVA

18. Two-way ANOVA – hypothesis test for A
H0: No difference among means for levels of A HA: At least two A means differ significantly Test statistic: F = MSA MSE Rej. region: Fobt < F(2, 12, .05) = 3.89 Decision: Reject H0 – variable A has an effect.
Two Way ANOVA

19. Two-way ANOVA – hypothesis test for B
H0: No difference among means for levels of B HA: At least two B means differ significantly Test statistic: F = MSB MSE Rej. region: Fobt < F(1, 12, .05) = 4.75 Decision: Reject H0 – variable B has an effect.
Two Way ANOVA

20. Two-way ANOVA – hypothesis for AB
H0: A & B do not interact to affect mean response HA: A & B do interact to affect mean response Test statistic: F = MSAB MSE Rej. region: Fobt < F(2, 12, .05) = 3.89 Decision: Reject H0 – A & B do interact...
Two Way ANOVA

21. Two way ANOVA Example 1
1. An experiment investigates the effects of two treatments, illumination level and type size of reading speed. Two levels of illumination, 15 foot-candles and 30 foot-candles, are used. Three levels of type size are used: 6-point, 12-point, and 18-point type. Test the independent and joint effects of these treatments on reading speed (  .05).
Two Way ANOVA

22. Two-way ANOVA Example 1
Reading speed (ave. words per minute) 6 point 12 point 18 point 15 fc 30 fc 15 fc 30 fc 15 fc 30 fc
Two Way ANOVA

23. Example 1 – hypothesis test for A (illumination)
H0: No difference among means for levels of A
HA: At least two A means differ significantly
Test statistic: F = MSA
MSE
Rej. region: Fobt < F(1, 24, .05) = 4.26
Two Way ANOVA

24. Example 1 – hypothesis test for B (type size)
H0: No difference among means for levels of B
HA: At least two B means differ significantly
Test statistic: F = MSB
MSE
Rej. region: Fobt < F(2, 24, .05) = 3.40
Two Way ANOVA

25. Example 1 – hypothesis test for AB interaction
H0: A & B do not interact to affect means
HA: A & B do interact to affect means
Test statistic: F = MSAB
MSE
Rej. region: Fobt < F(2, 24, .05) = 3.40
Two Way ANOVA

26. Two-way Anova – Example 1
Compute: CM = (12735)2 = SSA = – CM =
Two Way ANOVA

27. Two-way Anova – Example 1
SSB = – CM 10 = Σ(Tij)2 = … nij =
Two Way ANOVA

28. Two-way Anova – Example 1
SSAB = – – = SSTotal = ΣX2 – CM = – = SSE = SSTotal – SSA – SSB – SSAB =
Two Way ANOVA

29. Two-way Anova – Example 1
Source df SS MS F A * B * AB Error Total * Reject H0.
Two Way ANOVA

30. Two-way Anova – Example 2
A researcher is interested in comparing the effectiveness of 3 different methods of teaching reading, and also in whether the effectiveness might vary as a function of the reading ability of the students. Fifteen students with high reading ability and fifteen students with low reading ability were divided into three equal-sized group and each group was taught by one of these methods. Listed on the next slide are the reading performance scores for the various groups at school year-end.
Two Way ANOVA

31. Two-way Anova – Example 2
Teaching Method Ability A B C High X s Low X s
Two Way ANOVA

32. Two-way Anova – Example 2
(a) Do the appropriate analysis to answer the questions posed by the researcher (all αs = .05)
(b) The London School Board is currently using Method B and, prior to this experiment, had been thinking of changing to Method A because they believed that A would be better. At α = .01, determine whether this belief is supported by these data.
Two Way ANOVA

33. Example 2 – hypothesis test for A
H0: No difference among means for levels of A
HA: At least two A means differ significantly
Test statistic: F = MSA
MSE
Rejection region: Fobt < F(2, 24, .05) = 3.40
Two Way ANOVA

34. Example 2 – hypothesis test for B
H0: No difference among means for levels of B
HA: At least two B means differ significantly
Test statistic: F = MSB
MSE
Rejection region: Fobt < F(1, 24, .05) = 4.26
Two Way ANOVA

35. Example 2 – hypothesis test for interaction
H0: A and B do not interact to affect treatment means
HA: A and B do interact to affect treatment means
Test statistic: F = MSAB
MSE
Rejection region: Fobt < F(2, 24, .05) = 3.40
Two Way ANOVA

36. Two-way ANOVA – Example 2
SSE = 4 ( ) = 4 (42.4) = CM = (Σ X)2 = (796)2 n 30 =
Two Way ANOVA

37. Two-way ANOVA – Example 2
SSMethod = CM 10 = – = – =
Two Way ANOVA

38. Two-way ANOVA – Example 2
SSAbility = CM 15 = – = – =
Two Way ANOVA

39. Two-way ANOVA – Example 2
For the interaction sum of squares, we begin with the value ΣT2ij = …882 nij 5 = =
Two Way ANOVA

40. Two-way ANOVA – Example 2
Now, we can compute SSMA:
– – =
Two Way ANOVA

41. Two-way ANOVA – Example 2
Source df SS MS F Method * Ability * M x A Error Total 29 Reject HO for Method and for Ability, not for interaction.
Two Way ANOVA

42. Two-way ANOVA – Example 2b
HO: μA – μB = 0 HA: μA – μB > 0 Test statistic: t = (XA – XB) – 0 MSE tcrit = t(24, .01) = 2.492 ( ) √ n1 n2
Two Way ANOVA

43. Two-way ANOVA – Example 2b
tobt = 28.8 – = = Reject HO. A is better than B. ( ) √
Two Way ANOVA