Editing Analysis of variance (section)

===Randomization-based analysis===
{{See also|Random assignment|Randomization test}}
In a [[Randomized controlled trial|randomized controlled experiment]], the treatments are randomly assigned to experimental units, following the experimental protocol. This randomization is objective and declared before the experiment is carried out. The objective random-assignment is used to test the significance of the [[null hypothesis]], following the ideas of [[Charles Sanders Peirce|C. S. Peirce]] and [[Ronald Fisher]]. This design-based analysis was discussed and developed by [[Francis J. Anscombe]] at [[Rothamsted Experimental Station]] and by [[Oscar Kempthorne]] at [[Iowa State University]].<ref>Anscombe (1948)</ref> Kempthorne and his students make an assumption of ''unit treatment additivity'', which is discussed in the books of Kempthorne and [[David R. Cox]].<ref>{{cite book |last1=Hinkelmann |first1=Klaus |last2=Kempthorne |first2=Oscar |title=Design and Analysis of Experiments, Volume 2: Advanced Experimental Design |date=2005 |publisher=John Wiley |page=213 |isbn=978-0-471-70993-0 |url=https://books.google.com/books?id=GiYc5nRVKf8C&pg=PA213 |language=en}}</ref><ref>{{cite book |last1=Cox |first1=D. R. |title=Planning of Experiments |date=1992 |publisher=Wiley |isbn=978-0-471-57429-3 |language=en}}</ref>

====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]].

====Derived linear model====
Kempthorne uses the randomization-distribution and the assumption of ''unit treatment additivity'' to produce a ''derived linear model'', very similar to the textbook model discussed previously.<ref>Hinkelmann and Kempthorne (2008, Volume 1, Section 6.3: 
Completely Randomized Design; Derived Linear Model)</ref> The test statistics of this derived linear model are closely approximated by the test statistics of an appropriate normal linear model, according to approximation theorems and simulation studies.<ref name="HinkelmannKempthorne">Hinkelmann and Kempthorne (2008, Volume 1, Section 6.6: Completely randomized design; Approximating the randomization test)</ref> However, there are differences. For example, the randomization-based analysis results in a small but (strictly) negative correlation between the observations.<ref>Bailey (2008, Chapter 2.14 "A More General Model" in Bailey, pp.&nbsp;38–40)</ref><ref>Hinkelmann and Kempthorne (2008, Volume 1, Chapter 7: Comparison of Treatments)</ref> In the randomization-based analysis, there is ''no assumption'' of a ''normal'' distribution and certainly ''no assumption'' of ''independence''. On the contrary, ''the observations are dependent''!

The randomization-based analysis has the disadvantage that its exposition involves tedious algebra and extensive time. Since the randomization-based analysis is complicated and is closely approximated by the approach using a normal linear model, most teachers emphasize the normal linear model approach. Few statisticians object to model-based analysis of balanced randomized experiments.

====Statistical models for observational data====
However, when applied to data from non-randomized experiments or [[observational study|observational studies]], model-based analysis lacks the warrant of randomization.<ref>
Kempthorne (1979, pp 125–126, 
"The experimenter must decide which of the various causes that he feels will produce variations in his results must be controlled 
experimentally. Those causes that he does not control experimentally, because he is not cognizant of them, he must control by the device of randomization." "[O]nly when the treatments in the experiment are applied by the experimenter using the full randomization procedure is the chain of inductive inference sound. It is ''only'' under these circumstances that the experimenter can attribute whatever effects he observes to the treatment and the treatment only. Under these circumstances his conclusions are reliable in the statistical sense.") 
</ref> For observational data, the derivation of confidence intervals must use ''subjective'' models, as emphasized by [[Ronald Fisher]] and his followers. In practice, the estimates of treatment-effects from observational studies generally are often inconsistent. In practice, "statistical models" and observational data are useful for suggesting hypotheses that should be treated very cautiously by the public.<ref>Freedman {{full citation needed|date=November 2012}}</ref>