Editing Analysis of variance (section)

====Unit-treatment additivity====
In its simplest form, the assumption of unit-treatment additivity<ref group="nb">Unit-treatment additivity is simply termed additivity in most texts. Hinkelmann and Kempthorne add adjectives and distinguish between additivity in the strict and broad senses. This allows a detailed consideration of multiple error sources (treatment, state, selection, measurement and sampling) on page 161.</ref> states that the observed response <math>y_{i,j}</math> from experimental unit <math>i</math> when receiving treatment <math>j</math> can be written as the sum of the unit's response <math>y_i</math> and the treatment-effect <math> t_j</math>, that is <ref>Kempthorne (1979, p 30)</ref><ref name="Cox">Cox (1958, Chapter 2: Some Key Assumptions)</ref><ref>Hinkelmann and Kempthorne (2008, Volume 1, Throughout. Introduced in Section 2.3.3: Principles of experimental design; The linear model; Outline of a model)</ref>
<math display="block">y_{i,j}=y_i+t_j.</math>
The assumption of unit-treatment additivity implies that, for every treatment <math>j</math>, the <math>j</math>th treatment has exactly the same effect <math>t_j</math> on every experiment unit.

The assumption of unit treatment additivity usually cannot be directly [[Falsifiability|falsified]], according to Cox and Kempthorne. However, many ''consequences'' of treatment-unit additivity can be falsified. For a randomized experiment, the assumption of unit-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.

The use of unit treatment additivity and randomization is similar to the design-based inference that is standard in finite-population [[survey sampling]].