Editing Interaction (statistics) (section)

===Qualitative and quantitative interactions===
In many applications it is useful to distinguish between qualitative and quantitative interactions.<ref>{{cite book | last=Peto | first=D. P. | year=1982 | chapter=Statistical aspects of cancer trials |title=Treatment of Cancer |edition=First | publisher=Chapman and Hall |location=London |isbn=0-412-21850-X }}</ref>  A quantitative interaction between ''A'' and ''B'' is a situation where the magnitude of the effect of ''B'' depends on the value of ''A'', but the direction of the effect of ''B'' is constant for all ''A''.  A qualitative interaction between ''A'' and ''B'' refers to a situation where both the magnitude and direction of each variable's effect can depend on the value of the other variable.

The table of means on the left, below, shows a quantitative interaction &mdash; treatment ''A'' is beneficial both when ''B'' is given, and when ''B'' is not given, but the benefit is greater when ''B'' is not given (i.e. when ''A'' is given alone).  The table of means on the right shows a qualitative interaction.  ''A'' is harmful when ''B'' is given, but it is beneficial when ''B'' is not given.  Note that the same interpretation would hold if we consider the benefit of ''B'' based on whether ''A'' is given.

{| 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;" | ''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;" | 2
! style="border-right:1px solid black;" | 1
!
!
!
!
!
! style="background:#ffdead;border-left:1px solid black;border-top:1px solid black;" | ''A''&nbsp;=&nbsp;0
! style="border-left:1px solid black;" | 2
! style="border-right:1px solid black;" | 6
|-
! 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;" | 5
! style="border-bottom:1px solid black;border-right:1px solid black;" | 3
!
!
!
!
!
! style="background:#ffdead;border-left:1px solid black;border-bottom:1px solid black;" | ''A''&nbsp;=&nbsp;1
! style="border-left:1px solid black;border-bottom:1px solid black;" | 5
! style="border-right:1px solid black;border-bottom:1px solid black;" | 3
|}

The distinction between qualitative and quantitative interactions depends on the order in which the variables are considered (in contrast, the property of additivity is invariant to the order of the variables). In the following table, if we focus on the effect of treatment ''A'', there is a quantitative interaction &mdash; giving treatment ''A'' will improve the outcome on average regardless of whether treatment ''B'' is or is not already being given (although the benefit is greater if treatment ''A'' is given alone). However, if we focus on the effect of treatment ''B'', there is a qualitative interaction &mdash; giving treatment ''B'' to a subject who is already receiving treatment ''A'' will (on average) make things worse, whereas giving treatment ''B'' to a subject who is not receiving treatment ''A'' will improve the outcome on average.

{| 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
|}