Editing Interaction (statistics) (section)

===Unit treatment additivity===
In its simplest form, the assumption of treatment unit additivity states that the observed response ''y''<sub>''ij''</sub> from experimental unit ''i'' when receiving treatment ''j'' can be written as the sum ''y''<sub>''ij''</sub>&nbsp;=&nbsp;''y''<sub>''i''</sub>&nbsp;+&nbsp;''t''<sub>''j''</sub>.<ref name="Kempthorne (1979)">{{cite book |author-link=Oscar Kempthorne |last=Kempthorne |first=Oscar |year=1979 |title=The Design and Analysis of Experiments |edition=Corrected reprint of (1952) Wiley |publisher=Robert E. Krieger |isbn=978-0-88275-105-4 }}</ref><ref name=Cox1958_2>{{cite book |author-link=David R. Cox |last=Cox |first=David R. |year=1958 |title=Planning of experiments |publisher=Wiley |isbn=0-471-57429-5 |at=Chapter 2 }}</ref><ref>{{cite book
|author=Hinkelmann, Klaus and [[Oscar Kempthorne|Kempthorne, Oscar]]
|year=2008
|title=Design and Analysis of Experiments, Volume I: Introduction to Experimental Design
|edition=Second
|publisher=Wiley
|isbn=978-0-471-72756-9
|at=Chapters 5-6 }}</ref> The assumption of unit treatment additivity implies that every treatment has exactly the same additive effect on each experimental unit. Since any given experimental unit can only undergo one of the treatments, the assumption of unit treatment additivity is a hypothesis that is not directly falsifiable, according to Cox{{Citation needed|date=April 2010}} and Kempthorne.{{Citation needed|date=April 2010}}

However, many consequences of treatment-unit additivity can be falsified.{{Citation needed|date=April 2010}} For a randomized experiment, the assumption of treatment additivity implies that the variance is constant for all treatments. Therefore, by contraposition, a necessary condition for unit treatment additivity is that the variance is constant.{{Citation needed|date=April 2010}}

The property of unit treatment additivity is not invariant under a change of scale,{{Citation needed|date=April 2010}} so statisticians often use transformations to achieve unit treatment additivity. If the response variable is expected to follow a parametric family of probability distributions, then the statistician may specify (in the protocol for the experiment or observational study) that the responses be transformed to stabilize the variance.<ref>{{cite book
|author=Hinkelmann, Klaus and [[Oscar Kempthorne|Kempthorne, Oscar]]
|year=2008
|title=Design and Analysis of Experiments, Volume I: Introduction to Experimental Design
|edition=Second
|publisher=Wiley
|isbn=978-0-471-72756-9
|at=Chapters 7-8 }}</ref> In many cases, a statistician may specify that logarithmic transforms be applied to the responses, which are believed to follow a multiplicative model.<ref name=Cox1958_2/><ref name="Bailey on eelworms">{{cite book |last=Bailey |first=R. A.|title=Design of Comparative Experiments|url=http://www.maths.qmul.ac.uk/~rab/DOEbook/|publisher=Cambridge University Press |year=2008 |isbn=978-0-521-68357-9}} Pre-publication chapters are available on-line.</ref>

The assumption of unit treatment additivity was enunciated in experimental design by Kempthorne{{Citation needed|date=April 2010}} and Cox{{Citation needed|date=April 2010}}. Kempthorne's use of unit treatment additivity and randomization is similar to the design-based analysis of finite population survey sampling.

In recent years, it has become common{{Citation needed|date=April 2010}} to use the terminology of Donald Rubin, which uses counterfactuals. Suppose we are comparing two groups of people with respect to some attribute ''y''.  For example, the first group might consist of people who are given a standard treatment for a medical condition, with the second group consisting of people who receive a new treatment with unknown effect.  Taking a "counterfactual" perspective, we can consider an individual whose attribute has value ''y'' if that individual belongs to the first group, and whose attribute has value ''τ''(''y'') if the individual belongs to the second group.  The assumption of "unit treatment additivity" is that ''τ''(''y'')&nbsp;=&nbsp;''τ'', that is, the "treatment effect" does not depend on ''y''.  Since we cannot observe both ''y'' and τ(''y'') for a given individual, this is not testable at the individual level.  However, unit treatment additivity implies that the [[cumulative distribution function]]s ''F''<sub>1</sub> and ''F''<sub>2</sub> for the two groups satisfy 
''F''<sub>2</sub>(''y'') &nbsp;=&nbsp;''F''<sub>1</sub>(''y&nbsp;&minus;&nbsp;τ''), as long as the assignment of individuals to groups 1 and 2 is independent of all other factors influencing ''y'' (i.e. there are no [[confounding variable|confounders]]).  Lack of unit treatment additivity can be viewed as a form of interaction between the treatment assignment (e.g. to groups 1 or 2), and the baseline, or untreated value of ''y''.