The Two-way Anova

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
  influence each other.
• We need to test for the independent and combined
  effects of multiple variables.

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The Two-way Anova

• Definitions
• Main effect
• effect of one treatment variable considered in
  isolation (ignoring other variables in the study)
• Interaction
• when the effect of one variable is different
  across levels of one or more other variables

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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

4
Interaction of variables

• I vs III Effect of B at level A1 of variable A
• II vs IV Effect of B at A2
• I vs II Effect of A at B1
• III vs IV Effect of A at B2
• Two variables A B interact if the effect of A
  varies at levels of B (equivalently, if the
  effect of B varies at levels of A).

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B1 B2
B1 B2
A1 A2
A1 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 showing all 4 data points.

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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 its almost impossible to interpret an
  interaction among more than 4 variables!

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Two-way Anova - Computation

• 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

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Cell Totals (Tij) and marginal totals (Ti
Tj) A1 A2 A3 Tj B1 6 9 30
45 B2 15 12 75 102 Ti 21 21 105 147
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SXi2 12 22 302 2427 (SXi)2/N
(147)2/18 1200.5 S(Ti)2/ni (21)2 (21)2
(105)2 1984.5 6 6 6 S(Tj)2/ni
452 1022 1381 9 9 S(Tij)2/nij
62 152 302 752 2337 3
3 3 3
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SSA S(Ti)2 CM ni SSB S(Tj)2 CM
nj SSTotal SX2 CM SSE SX2 S(Tij)2
nij SSAB S(Tij)2 S(Tj)2 S(Ti)2
(SX)2 nij nj ni N
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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
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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
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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 lt F(2, 12, .05) 3.89
• Decision Reject H0 variable A has an effect.

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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 lt F(1, 12, .05) 4.75
• Decision Reject H0 variable B has an effect.

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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 MS(AB)
• MSE
• Rej. region Fobt lt F(2, 12, .05) 3.89
• Decision Reject H0 A B do interact...

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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 lt F(1, 24, .05) 4.26

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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 lt F(2, 24, .05) 3.40

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Example 2 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 MS(AB)
• MSE
• Rej. region Fobt lt F(2, 24, .05) 3.40

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Two-way Anova Example 1

• Compute
• CM (12375)2 5406007.5
• 30
• SSA 62052 65302 CM 3520.833
• 15

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Two-way Anova Example 1

• SSB 40252 43252 43852 CM
• 10
• 7440.0
• S(Tij)2 19102 21152 21362 22052
• nij 5 5 5 5
• 5380634.2

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Two-way Anova Example 1

• SSAB 5380634.2 5409528.33 5413447.5
  5406007.5
• 1646.667
• SSTotal SX2 CM 5437581111.0 5406007.5
• 311111573.5
• SSE SSTotal SSA SSB SSAB 18966.0

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Two-way Anova Example 1

• 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 18966.0 790.25
• Total 29 31573.5
• Reject H0.

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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.

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Two-way Anova Example 2

• Teaching Method
• Ability A B C
• High X 37.6 32.4 33.2
• s2 2.8 9.3 11.7
• Low X 20.0 18.4 17.6
• s2 10.0 4.3 4.3

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Two-way Anova Example 2

• (a) Do the appropriate analysis to answer the
  questions posed by the researcher (all as .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 a .01,
  determine whether this belief is supported by
  these data.

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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
• Rej. region Fobt lt F(2, 12, .05) 3.89
• Decision Reject H0 variable A has an effect.

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Two-way Anova Example 1