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1The Two-way ANOVA
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.
2The Two-way ANOVA
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?
3The Two-way ANOVA
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.
4The Two-way ANOVA
Two Way ANOVA
Main effect 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
5Interaction of variables
Two Way ANOVA
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
6Interaction of variables
Two Way ANOVA
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
I II
III IV
7Interaction of variables
Two Way ANOVA
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
I II
III IV
Two Way ANOVA
A1 A2 A1 A2
B1
B2
B1
B2
• 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).
9Interaction of variables
Two Way ANOVA
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!
10Two-way ANOVA - Computation
Two Way ANOVA
Raw Scores (Xi)
A1 A2 A3
B1 B2 B1 B2 B1 B2
1 4 1 2 8 19 2 5 3 4 10 26 3 6 5 6 12 30
Two variables, A & B
2 X 3 = 6 treatments
Two Way ANOVA
Cell Totals (Tij) and marginal totals (Ti & Tj)
A1 A2 A3 Tj
B1 6 9 30 45
B2 15 12 75 102
Ti 21 21 105 147Totals for the levels of A are in blue
B Totalsare in red
Cell totals are in green
12Two-Way ANOVA – Computational formulas
Two Way ANOVA
CM = (ΣXi)2/N = (147)2 = 1200.5 18
ΣX2 = 12 + 22 + … + 302 = 2427
SSTotal = ΣX2 – CM
SSE = ΣX2 – Σ(Tij)2
nij
Notice that these are the original data from Slide 10
13Two-Way ANOVA – Computational formulas
Two Way ANOVA
Σ(Ti)2/ni = (21)2 + (21)2 + (105)2 = 1984.5
6 6 6
SSA = Σ(Ti)2 – CM
ni
SSA is the sum of squared deviations for Variable A
14Two-Way ANOVA – Computational formulas
Two Way ANOVA
Σ(Tj)2/ni = 452 + 1022 = 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!
15Two-Way ANOVA – Computational formulas
Two Way ANOVA
Σ(Tij)2/nij = 62 + 152 + … + 302 + 752 = 2337 3 3 3 3
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
We now compute:
SSA = 1984.5 – 1200.5 = 784
SSB = 1381 – 1200.5 = 180.5
SSTotal = 2427 – 1200.5 = 1226.5
SSE = 2427 – 2337 = 90
SSAB = 2337 – 1381 – 1984.5 + 1200.5
= 172
CM
Two Way ANOVA
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) = 2 172 86 11.47
Error n-ab = 12 90 7.5
Total n-1 = 17 1226.5
18Two-way ANOVA – hypothesis test for A
Two Way ANOVA
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.
19Two-way ANOVA – hypothesis test for B
Two Way ANOVA
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.
20Two-way ANOVA – hypothesis for AB
Two Way ANOVA
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...
21Two 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
22Two-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
378 415 454 439 432 426
408 396 394 467 411 428
357 451 452 477 466 464
353 455 396 410 411 412
414 398 419 417 460 475
Two Way ANOVA
23Example 1 – hypothesis test for A (illumination)
Two Way ANOVA
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
24Example 1 – hypothesis test for B (type size)
Two Way ANOVA
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
25Example 1 – hypothesis test for AB interaction
Two Way ANOVA
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
26Two-way Anova – Example 1
Two Way ANOVA
Compute:
CM = (12735)2 = 5406007.5 30
SSA = 62052 + 65302 – CM = 3520.833
15
27Two-way Anova – Example 1
Two Way ANOVA
SSB = 40252 + 43252 + 43852 – CM10= 7440.0
Σ(Tij)2 = 19102 + 21152 + … + 21802 + 22052
nij 5 5 5 5
= 5380634.2
28Two-way Anova – Example 1
Two Way ANOVA
SSAB = 5380634.2 – 5409528.33 – 5413447.5 + 5406007.5
= 1646.667
SSTotal = ΣX2 – CM = 5437581.0 – 5406007.5
= 31573.5
SSE = SSTotal – SSA – SSB – SSAB = 18966.0
29Two-way Anova – Example 1
Two Way ANOVA
Source df SS MS F
A 1 3520.83 3520.83 4.46*
B 2 7440.00 3720.00 4.71*
AB 2 1646.67 823.33 1.04
Error 24 18966.0 790.25Total 29 31573.5
* Reject H0.
30Two-way Anova – Example 2
Two Way ANOVA
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.
31Two-way Anova – Example 2
Two Way ANOVA
Teaching MethodAbility A B C
High X 37.6 32.4 33.2s2 2.8 9.3 11.7
Low X 20.0 18.4 17.6s2 10.0 4.3 4.3
32Two-way Anova – Example 2
Two Way ANOVA
(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.
33Example 2 – hypothesis test for A
Two Way ANOVA
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
34Example 2 – hypothesis test for B
Two Way ANOVA
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
35Example 2 – hypothesis test for interaction
Two Way ANOVA
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
36Two-way ANOVA – Example 2
Two Way ANOVA
SSE = 4 (2.8 + 9.3 + 11.7 + 10.0 + 4.3 + 4.3)= 4 (42.4)= 169.6
CM = (Σ X)2 = (796)2 n 30
= 21120.533
37Two-way ANOVA – Example 2
Two Way ANOVA
SSMethod = 2382 + 2542 + 2542 - CM
10
= 211976 – 21120.533 10= 21197.6 – 21120.533= 77.066
38Two-way ANOVA – Example 2
Two Way ANOVA
SSAbility = 5162 + 2802 - CM
15
= 344656 – 21120.533 15= 22977.066 – 21120.533= 1856.533
39Two-way ANOVA – Example 2
Two Way ANOVA
For the interaction sum of squares, we begin with the value
ΣT2ij = 1382 + 1622 + 1662 + …882
nij 5
= 115352 5
= 23070.4
40Two-way ANOVA – Example 2
Two Way ANOVA
Now, we can compute SSMA:
23070.4 – 21197.6 – 22977.066 + 21120.533
= 16.267
41Two-way ANOVA – Example 2
Two Way ANOVA
Source df SS MS FMethod 2 77.066 38.533 5.45*Ability 1 1856.533 1856.533 262.72*M x A 2 16.267 8.134
1.15Error 24 169.6 7.067Total 29
Reject HO for Method and for Ability, not for interaction.
42Two-way ANOVA – Example 2b
Two Way ANOVA
HO: μA – μB = 0
HA: μA – μB > 0
Test statistic: t = (XA – XB) – 0
MSE 1 + 1
tcrit = t(24, .01) = 2.492
√ ( )n1 n2
43Two-way ANOVA – Example 2b
Two Way ANOVA
tobt = 28.8 – 25.4
7.067 1 + 1 10 10
= 3.4 1.189
= 2.86 - Reject HO. A is better than B.
√ ( )
