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| A simple setting in which interactions can arise is a two-factor experiment analyzed using Analysis of Variance (ANOVA). Suppose we are interested in studying the effects of cocaine on sleep. We might design an experiment to simultaneously test whether both the use of cocaine and the duration of usage affect the number of hours a squirrel will sleep in a night. We might give half of the squirrels we test cocaine, and the other half a placebo (control) substance (the ''substance variable''). And we might vary the duration of usage by administering cocaine or placebo for one of two possible durations before test, 4 weeks or 12 weeks (the ''usage duration variable''). | A simple setting in which interactions can arise is a two-factor experiment analyzed using Analysis of Variance (ANOVA). Suppose we are interested in studying the effects of cocaine on sleep. We might design an experiment to simultaneously test whether both the use of cocaine and the duration of usage affect the number of hours a squirrel will sleep in a night. We might give half of the squirrels we test cocaine, and the other half a placebo substance (the ''substance'' variable). And we might vary the duration of usage by administering cocaine or placebo for one of two possible durations before test, 4 weeks or 12 weeks (the ''duration'' variable). We can then consider the average treatment response (e.g. number of hours slept) for each squirrel, as a function of the treatment combination that was administered (e.g. substance and duration). The following table shows one possible situation: |
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| || 4-Week Control || 4-Week Cocaine || 12-Week Control || 12-Week Cocaine || | || '''4-Week Placebo (Control)''' || '''4-Week Cocaine''' || '''12-Week Placebo (Control)''' || '''12-Week Cocaine''' || |
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| There are three null hypotheses to be tested: * H,,01,,: Both substance groups sleep for the same number of hours on average. * H,,02,,: Both treatment duration groups sleep for the same number of hours on average. * H,,03,,: The two factors are independent or there is no interaction effect. |
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| have two binary factors A and B. For example, these factors might indicate whether a treatment was administered to a patient, and for how long the treatment was used. We can then consider the average treatment response (e.g. the symptom levels following treatment) for each patient, as a function of the treatment combination that was administered. The following table shows one possible situation: | We can start by computing the '''group means''' (for each cell, row, and column): || || '''4-Week''' || '''12-Week''' || '''''All Durations''''' || || '''Placebo''' || 7 || 9.7|| ''8.35'' || || '''Cocaine''' || 4.9 || 4.7 || ''4.8'' || || '''''All Substances''''' || ''5.95'' || ''7.2'' || ''6.575'' || And by calculating the sum of squares (SS) for each group (cell): 4-Week Placebo Dataset = {7 8 6 7 6.5}, ''M'' = 7<<BR>> '''''SS,,4-Wk Placebo,,''''' = (7-7)^2^ + (8-7)^2^ + (6-7)^2^ + (7-7)^2^ + (6.5-7)^2^ = '''2.25'''<<BR>><<BR>> 4-Week Cocaine = {5.5 3.5 4.5 6 5}, ''M'' = 4.9<<BR>> '''''SS,,4-Wk Cocaine,,''''' = (5.5-4.9)^2^ + (3.5-4.9)^2^ + (4.5-4.9)^2^ + (6-4.9)^2^ + (5-4.9)^2^ = '''3.7'''<<BR>><<BR>> 12-Week Placebo = {8 10 13 9 8.5}, ''M'' = 9.7<<BR>> '''''SS,,12-Wk Placebo,,''''' = (8-9.7)^2^ + (10-9.7)^2^ + (13-9.7)^2^ + (9-9.7)^2^ + (8.5-9.7)^2^ = '''15.8'''<<BR>><<BR>> 12-Week Cocaine = {5 4.5 4 6 4}, ''M'' = 4.7<<BR>> '''''SS,,12-Wk Cocaine,,''''' = (5-4.7)^2^ + (4.5-4.7)^2^ + (4-4.7)^2^ + (6-4.7)^2^ + (4-4.7)^2^ = '''2.8'''<<BR>><<BR>> Now, we'll calculate the ''SS,,B,,'':<<BR>> '''''SS,,B,,''''' = ''n'' [( ''M,,4-Wk Placebo,,'' - ''M,,Group,,'' )^2^ + ( ''M,,4-Wk Cocaine,,'' - ''M,,Group,,'')^2^ + ( ''M,,12-Wk Placebo,,'' - ''M,,Group,,'' )^2^ + ( ''M,,12-Wk Cocaine,,'' - ''M,,Group,,'')^2^]<<BR>> = 5 [(7 - 6.575 )^2^ + (4.9 - 6.575)^2^ + (9.7 - 6.575)^2^ + (4.7 - 6.575)^2^]<<BR>> = 5 [0.180625 + 2.805625 + 9.765625 + 3.515625]<<BR>> = 5 [16.2675]<<BR>> = '''81.3375'''<<BR>><<BR>> Now, we'll calculate the ''SS,,W,,'':<<BR>> '''''SS,,W,,''''' = ''SS,,4-Wk Placebo,,'' + ''SS,,4-Wk Cocaine,,'' + ''SS,,12-Wk Placebo,,'' + ''SS,,12-Wk Cocaine,,''<<BR>> = 2.25 + 3.7 + 15.8 + 2.8<<BR>> = '''24.55'''<<BR>><<BR>> '''''df,,W,,''''' = ''N - rc ''<<BR>> = 20 - (2 * 2)<<BR>> = 16<<BR>><<BR>> '''''s,,W,,^2^''''' = ''SS,,W,,'' / ''df,,W,,''<<BR>> = 24.55 / 16<<BR>> = '''1.534375'''<<BR>><<BR>> Now, we'll calculate the ''SS,,R,,'':<<BR>> '''''SS,,R,,''''' = ''n'' [( ''M,,Placebo,,'' - ''M,,Group,,'' )^2^ + ( ''M,, Cocaine,,'' - ''M,,Group,,'')^2^]<<BR>> = 10 [(8.35 - 6.575)^2^ + (4.8 - 6.575)^2^]<<BR>> = 10 [3.150625 + 3.150625]<<BR>> = 10 [6.30125]<<BR>> = '''63.0125'''<<BR>><<BR>> '''''df,,R,,''''' = ''r'' - 1<<BR>> = 2-1<<BR>> = '''1'''<<BR>><<BR>> '''''s,,R,,^2^''''' = ''SS,,R,,'' / ''df,,R,,''<<BR>> = 63.0125 / 1<<BR>> = '''63.0125'''<<BR>><<BR>> Now, we'll calculate the ''SS,,C,,'':<<BR>> '''''SS,,C,,''''' = ''n'' [( ''M,,4-Week,,'' - ''M,,Group,,'' )^2^ + ( ''M,, 12-Week,,'' - ''M,,Group,,'')^2^]<<BR>> = 10 [(5.95 - 6.575)^2^ + (7.2 - 6.575)^2^]<<BR>> = 10 [0.390625 + 0.390625]<<BR>> = 10 [0.78125]<<BR>> = '''7.8125'''<<BR>><<BR>> '''''df,,C,,''''' = ''c'' - 1<<BR>> = 2-1<<BR>> = '''1'''<<BR>><<BR>> '''''s,,C,,^2^''''' = ''SS,,C,,'' / ''df,,C,,''<<BR>> = 7.8125 / 1<<BR>> = '''7.8125'''<<BR>><<BR>> Now, we'll calculate the ''SS,,RC,,'':<<BR>> '''''SS,,RC,,''''' = ''SS,,B,,'' + ''SS,,R,,'' + ''SS,,C,,''<<BR>> = 81.3375+ 63.0125 + 7.8125<<BR>> = 152.1625<<BR>><<BR>> '''''df,,RC,,''''' = (''r'' - 1)(''c'' - 1)<<BR>> = (2-1)(2-1)<<BR>> = '''1'''<<BR>><<BR>> '''''s,,RC,,^2^''''' = ''SS,,RC,,'' / ''df,,RC,,''<<BR>> = 152.1625 / 1<<BR>> = '''152.1625'''<<BR>><<BR>> Now, we'll calculate the ''SS,,T,,'':<<BR>> '''''SS,,T,,''''' = ''SS,,B,,'' + ''SS,,W,'' + ''SS,,R,,'' + ''SS,,C,,''+ ''SS,,RC,,''<<BR>> = 81.3375+ 24.55 + 63.0125 + 7.8125 + 152.1625<<BR>> = 328.875<<BR>><<BR>> '''''df,,T,,''''' = ''N'' - 1<<BR>> = 20-1<<BR>> = '''19'''<<BR>><<BR>> Now, we'll calculate the ''F'' values: '''''F,,R,,''''' = ''s,,R,,^2^'' / ''s,,W,,^2^''<<BR>> = 63.0125 / 1.534375<<BR>> = '''41.06721'''<<BR>><<BR>> '''''F,,C,,''''' = ''s,,C,,^2^'' / ''s,,W,,^2^''<<BR>> = 7.8125 / 1.534375<<BR>> = '''5.09165'''<<BR>><<BR>> '''''F,,RC,,''''' = ''s,,RC,,^2^'' / ''s,,W,,^2^''<<BR>> = 152.1625 / 1.534375<<BR>> = '''99.16904'''<<BR>><<BR>> |
When considering the relationship among three or more variables, an interaction may arise. Interactions describe a situation in which the simultaneous influence of two variables on a third is not additive. Most commonly, interactions are considered in the context of regression analyses, but they may also be evaluated using two-way ANOVA.
A simple setting in which interactions can arise is a two-factor experiment analyzed using Analysis of Variance (ANOVA). Suppose we are interested in studying the effects of cocaine on sleep. We might design an experiment to simultaneously test whether both the use of cocaine and the duration of usage affect the number of hours a squirrel will sleep in a night. We might give half of the squirrels we test cocaine, and the other half a placebo substance (the substance variable). And we might vary the duration of usage by administering cocaine or placebo for one of two possible durations before test, 4 weeks or 12 weeks (the duration variable). We can then consider the average treatment response (e.g. number of hours slept) for each squirrel, as a function of the treatment combination that was administered (e.g. substance and duration). The following table shows one possible situation:
4-Week Placebo (Control) |
4-Week Cocaine |
12-Week Placebo (Control) |
12-Week Cocaine |
7.5 |
5.5 |
8.0 |
5.0 |
8.0 |
3.5 |
10.0 |
4.5 |
6.0 |
4.5 |
13.0 |
4.0 |
7.0 |
6.0 |
9.0 |
6.0 |
6.5 |
5.0 |
8.5 |
4.0 |
There are three null hypotheses to be tested:
H01: Both substance groups sleep for the same number of hours on average.
H02: Both treatment duration groups sleep for the same number of hours on average.
H03: The two factors are independent or there is no interaction effect.
We can start by computing the group means (for each cell, row, and column):
|
4-Week |
12-Week |
All Durations |
Placebo |
7 |
9.7 |
8.35 |
Cocaine |
4.9 |
4.7 |
4.8 |
All Substances |
5.95 |
7.2 |
6.575 |
And by calculating the sum of squares (SS) for each group (cell):
4-Week Placebo Dataset = {7 8 6 7 6.5}, M = 7
SS4-Wk Placebo = (7-7)2 + (8-7)2 + (6-7)2 + (7-7)2 + (6.5-7)2 = 2.25
4-Week Cocaine = {5.5 3.5 4.5 6 5}, M = 4.9
SS4-Wk Cocaine = (5.5-4.9)2 + (3.5-4.9)2 + (4.5-4.9)2 + (6-4.9)2 + (5-4.9)2 = 3.7
12-Week Placebo = {8 10 13 9 8.5}, M = 9.7
SS12-Wk Placebo = (8-9.7)2 + (10-9.7)2 + (13-9.7)2 + (9-9.7)2 + (8.5-9.7)2 = 15.8
12-Week Cocaine = {5 4.5 4 6 4}, M = 4.7
SS12-Wk Cocaine = (5-4.7)2 + (4.5-4.7)2 + (4-4.7)2 + (6-4.7)2 + (4-4.7)2 = 2.8
Now, we'll calculate the SSB:
SSB = n [( M4-Wk Placebo - MGroup )2 + ( M4-Wk Cocaine - MGroup)2 + ( M12-Wk Placebo - MGroup )2 + ( M12-Wk Cocaine - MGroup)2]
= 5 [(7 - 6.575 )2 + (4.9 - 6.575)2 + (9.7 - 6.575)2 + (4.7 - 6.575)2]
= 5 [0.180625 + 2.805625 + 9.765625 + 3.515625]
= 5 [16.2675]
= 81.3375
Now, we'll calculate the SSW:
SSW = SS4-Wk Placebo + SS4-Wk Cocaine + SS12-Wk Placebo + SS12-Wk Cocaine
= 2.25 + 3.7 + 15.8 + 2.8
= 24.55
dfW = N - rc
= 20 - (2 * 2)
= 16
sW2 = SSW / dfW
= 24.55 / 16
= 1.534375
Now, we'll calculate the SSR:
SSR = n [( MPlacebo - MGroup )2 + ( M Cocaine - MGroup)2]
= 10 [(8.35 - 6.575)2 + (4.8 - 6.575)2]
= 10 [3.150625 + 3.150625]
= 10 [6.30125]
= 63.0125
dfR = r - 1
= 2-1
= 1
sR2 = SSR / dfR
= 63.0125 / 1
= 63.0125
Now, we'll calculate the SSC:
SSC = n [( M4-Week - MGroup )2 + ( M 12-Week - MGroup)2]
= 10 [(5.95 - 6.575)2 + (7.2 - 6.575)2]
= 10 [0.390625 + 0.390625]
= 10 [0.78125]
= 7.8125
dfC = c - 1
= 2-1
= 1
sC2 = SSC / dfC
= 7.8125 / 1
= 7.8125
Now, we'll calculate the SSRC:
SSRC = SSB + SSR + SSC
= 81.3375+ 63.0125 + 7.8125
= 152.1625
dfRC = (r - 1)(c - 1)
= (2-1)(2-1)
= 1
sRC2 = SSRC / dfRC
= 152.1625 / 1
= 152.1625
Now, we'll calculate the SST:
SST = SSB + SSW,'' + ''SSR'' + ''SSC''+ ''SSRC,,
= 81.3375+ 24.55 + 63.0125 + 7.8125 + 152.1625
= 328.875
dfT = N - 1
= 20-1
= 19
Now, we'll calculate the F values:
FR = sR2 / sW2
= 63.0125 / 1.534375
= 41.06721
FC = sC2 / sW2
= 7.8125 / 1.534375
= 5.09165
FRC = sRC2 / sW2
= 152.1625 / 1.534375
= 99.16904