Editing Interaction (statistics) (section)

===In ANOVA===
A simple setting in which interactions can arise is a [[factorial experiment|two-factor experiment]] analyzed using [[Analysis of Variance]] (ANOVA).  Suppose we have two binary factors ''A'' and ''B''.  For example, these factors might indicate whether either of two treatments were administered to a patient, with the treatments applied either singly, or in combination.  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:

{| cellpadding="5" cellspacing="0" align="center"
|-
! 
! style="background:#ffdead;border-left:1px solid black;border-top:1px solid black;" | ''B''&nbsp;=&nbsp;0
! style="background:#ffdead;border-top:1px solid black;border-right:1px solid black;" | ''B''&nbsp;=&nbsp;1
|-
! style="background:#ffdead;border-left:1px solid black;border-top:1px solid black;" | ''A''&nbsp;=&nbsp;0
! style="border-left:1px solid black;" | 6
! style="border-right:1px solid black;" | 7
|-
! style="background:#ffdead;border-bottom:1px solid black;border-left:1px solid black;" | ''A''&nbsp;=&nbsp;1
! style="border-bottom:1px solid black;border-left:1px solid black;" | 4
! style="border-bottom:1px solid black;border-right:1px solid black;" | 5
|}

In this example, there is no interaction between the two treatments &mdash; their effects are additive.  The reason for this is that the difference in mean response between those subjects receiving treatment ''A'' and those not receiving treatment ''A'' is &minus;2 regardless of whether treatment ''B'' is administered (&minus;2&nbsp;=&nbsp;4&nbsp;&minus;&nbsp;6) or not (&minus;2&nbsp;=&nbsp;5&nbsp;&minus;&nbsp;7). Note that it automatically follows that the difference in mean response between those subjects receiving treatment ''B'' and those not receiving treatment ''B'' is the same regardless of whether treatment ''A'' is administered (7&nbsp;&minus;&nbsp;6&nbsp;=&nbsp;5&nbsp;&minus;&nbsp;4).

In contrast, if the following average responses are observed

{| cellpadding="5" cellspacing="0" align="center"
|-
! 
! style="background:#ffdead;border-left:1px solid black;border-top:1px solid black;" | ''B''&nbsp;=&nbsp;0
! style="background:#ffdead;border-top:1px solid black;border-right:1px solid black;" | ''B''&nbsp;=&nbsp;1
|-
! style="background:#ffdead;border-left:1px solid black;border-top:1px solid black;" | ''A''&nbsp;=&nbsp;0
! style="border-left:1px solid black;" | 1
! style="border-right:1px solid black;" | 4
|-
! style="background:#ffdead;border-bottom:1px solid black;border-left:1px solid black;" | ''A''&nbsp;=&nbsp;1
! style="border-bottom:1px solid black;border-left:1px solid black;" | 7
! style="border-bottom:1px solid black;border-right:1px solid black;" | 6
|}

then there is an interaction between the treatments &mdash; their effects are not additive.  Supposing that greater numbers correspond to a better response, in this situation treatment ''B'' is helpful on average if the subject is not also receiving treatment ''A'', but is detrimental on average if given in combination with treatment ''A''. Treatment ''A'' is helpful on average regardless of whether treatment ''B'' is also administered, but it is more helpful in both absolute and relative terms if given alone, rather than in combination with treatment ''B''. Similar observations are made for this particular example in the next section.