Editing Interaction (statistics) (section)

===Model size===
Given ''n'' predictors, the number of terms in a linear model that includes a constant, every predictor, and every possible interaction is <math>\tbinom{n}{0} + \tbinom{n}{1} + \tbinom{n}{2} + \cdots + \tbinom{n}{n} = 2^n</math>. Since this quantity grows exponentially, it readily becomes impractically large. One method to limit the size of the model is to limit the order of interactions. For example, if only two-way interactions are allowed, the number of terms becomes <math>\tbinom{n}{0} + \tbinom{n}{1} + \tbinom{n}{2} = 1 + \tfrac{1}{2}n + \tfrac{1}{2}n^2</math>. The below table shows the number of terms for each number of predictors and maximum order of interaction.

{| class="wikitable" style="text-align: right;"
|+ Number of terms
! rowspan="2" | Predictors
! colspan="5" | Including up to ''m''-way interactions
|-
! 2 !! 3 !! 4 !! 5 !! ∞
|-
! scope="row" | 1
| 2 || 2 || 2 || 2 || 2
|-
! scope="row" | 2
| 4 || 4 || 4 || 4 || 4
|-
! scope="row" | 3
| 7 || 8 || 8 || 8 || 8
|-
! scope="row" | 4
| 11 || 15 || 16 || 16 || 16
|-
! scope="row" | 5
| 16 || 26 || 31 || 32 || 32
|-
! scope="row" | 6
| 22 || 42 || 57 || 63 || 64
|-
! scope="row" | 7
| 29 || 64 || 99 || 120 || 128
|-
! scope="row" | 8
| 37 || 93 || 163 || 219 || 256
|-
! scope="row" | 9
| 46 || 130 || 256 || 382 || 512
|-
! scope="row" | 10
| 56 || 176 || 386 || 638 || 1,024
|-
! scope="row" | 11
| 67 || 232 || 562 || 1,024 || 2,048
|-
! scope="row" | 12
| 79 || 299 || 794 || 1,586 || 4,096
|-
! scope="row" | 13
| 92 || 378 || 1,093 || 2,380 || 8,192
|-
! scope="row" | 14
| 106 || 470 || 1,471 || 3,473 || 16,384
|-
! scope="row" | 15
| 121 || 576 || 1,941 || 4,944 || 32,768
|-
! scope="row" | 20
| 211 || 1,351 || 6,196 || 21,700 || 1,048,576
|-
! scope="row" | 25
| 326 || 2,626 || 15,276 || 68,406 || 33,554,432
|-
! scope="row" | 50
| 1,276 || 20,876 || 251,176 || 2,369,936 || 10<sup>15</sup>
|-
! scope="row" | 100
| 5,051 || 166,751 || 4,087,976 || 79,375,496 || 10<sup>30</sup>
|-
! scope="row" | 1,000
| 500,501 || 166,667,501 || 10<sup>10</sup> || 10<sup>12</sup> || 10<sup>300</sup>
|}