ANOVA Assumptions

The results of a one-way ANOVA can be considered reliable as long as the following assumptions are met:

• Response variables are normally distributed (or approximately normally distributed).
• Samples are independent.
• Variances of populations are equal.
• Responses for a given group are independent and identically distributed normal random variables—not a simple random sample (SRS).

Necessary assumptions for randomization-based analysis are as follows.

Randomization-Based Analysis

In a 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 C.S. Peirce and Ronald A. Fisher. This design-based analysis was developed by Francis J. Anscombe at Rothamsted Experimental Station and by Oscar Kempthorne at Iowa State University. Kempthorne and his students make an assumption of unit-treatment additivity.

Unit-Treatment Additivity

In its simplest form, the assumption of unit-treatment additivity states that the observed response from the experimental unit when receiving treatment can be written as the sum of the unit's response $y_i$ and the treatment-effect $t_j$, or

$y_{i, j} = y_i+t_j$

The assumption of unit-treatment additivity implies that for every treatment $j$, the $j$th treatment has exactly the same effect $t_j$ on every experiment unit. The assumption of unit-treatment additivity usually cannot be directly falsified; however, many consequences of unit-treatment 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 one-way ANOVA discussed previously. 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. However, there are differences. For example, the randomization-based analysis results in a small but (strictly) negative correlation between the observations. 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.

In summary, the normal model based ANOVA analysis assumes the independence, normality and homogeneity of the variances of the residuals. The randomization-based analysis assumes only the homogeneity of the variances of the residuals (as a consequence of unit-treatment additivity) and uses the randomization procedure of the experiment. Both these analyses require homoscedasticity, as an assumption for the normal model analysis and as a consequence of randomization and additivity for the randomization-based analysis.