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Testing if the interaction term is significant can be difficult because of the potentially-large number of [[degrees of freedom (statistics)|degrees of freedom]].<ref>{{cite journal |author=Yi-An Ko|date=September 2013 |title=Novel Likelihood Ratio Tests for Screening Gene-Gene and Gene-Environment Interactions with Unbalanced Repeated-Measures Data |journal=Genetic Epidemiology |volume=37 |issue=6 |pages=581–591 |doi=10.1002/gepi.21744 |pmid=23798480 |display-authors=etal|pmc=4009698}}</ref>
==Example==
The following hypothetical example gives the yields of 15 plants subject to two different environmental variations, and three different fertilisers.
{| class="wikitable"
|-
!
! Extra CO<sub>2</sub>
! Extra humidity
|-
| No fertiliser
| 7, 2, 1
| 7, 6
|-
| Nitrate
| 11, 6
| 10, 7, 3
|-
| Phosphate
| 5, 3, 4
| 11, 4
|}
Five sums of squares are calculated:
{| class="wikitable"
|-
! Factor
! Calculation
! Sum
! <math>\sigma^2</math>
|-
| Individual
| <math>7^2+2^2+1^2 + 7^2+6^2 + 11^2+6^2 + 10^2+7^2+3^2 + 5^2+3^2+4^2 + 11^2+4^2</math>
| 641
| 15
|-
| Fertilizer × Environment
| <math>\frac{(7+2+1)^2}{3} + \frac{(7+6)^2}{2} + \frac{(11+6)^2}{2} + \frac{(10+7+3)^2}{3} + \frac{(5+3+4)^2}{3} + \frac{(11+4)^2}{2}</math>
| 556.1667
| 6
|-
| Fertilizer
| <math>\frac{(7+2+1+7+6)^2}{5} + \frac{(11+6+10+7+3)^2}{5} + \frac{(5+3+4+11+4)^2}{5}</math>
| 525.4
| 3
|-
| Environment
| <math>\frac{(7+2+1+11+6+5+3+4)^2}{8} + \frac{(7+6+10+7+3+11+4)^2}{7} </math>
| 519.2679
| 2
|-
| Composite
| <math>\frac{(7+2+1+11+6+5+3+4+7+6+10+7+3+11+4)^2}{15} </math>
| 504.6
| 1
|}
Finally, the sums of squared deviations required for the [[analysis of variance]] can be calculated.
{| class="wikitable"
|-
! Factor
! Sum
! <math>\sigma^2</math>
! Total
! Environment
! Fertiliser
! Fertiliser × Environment
! Residual
|-
| Individual
| 641
| 15
| 1
|
|
|
| 1
|-
| Fertiliser × Environment
| 556.1667
| 6
|
|
|
| 1
| −1
|-
| Fertiliser
| 525.4
| 3
|
|
| 1
| −1
|
|-
| Environment
| 519.2679
| 2
|
| 1
|
| −1
|
|-
| Composite
| 504.6
| 1
| −1
| −1
| −1
| 1
|
|-
|
|
|
|
|
|
|
|
|-
| Squared deviations
|
|
| 136.4
| 14.668
| 20.8
| 16.099
| 84.833
|-
| Degrees of freedom
|
|
| 14
| 1
| 2
| 2
| 9
|}
==See also==
| |||