Editing Analysis of variance (section)

==For a single factor==
{{Main|One-way analysis of variance}}
The simplest experiment suitable for ANOVA analysis is the completely randomized experiment with a single factor. More complex experiments with a single factor involve constraints on randomization and include completely randomized blocks and [[Latin square|Latin squares]] (and variants: [[Mutually orthogonal Latin squares|Graeco-Latin squares]], etc.). The more complex experiments share many of the complexities of multiple factors.

There are some alternatives to conventional one-way analysis of variance, e.g.: Welch's heteroscedastic F test, Welch's heteroscedastic F test with trimmed means and Winsorized variances, Brown-Forsythe test, Alexander-Govern test, James second order test and Kruskal-Wallis test, available in [https://cran.r-project.org/web/packages/onewaytests/index.html onewaytests] [[R package|R]]

It is useful to represent each data point in the following form, called a statistical model:
<math display="block">Y_{ij} = \mu + \tau_j + \varepsilon_{ij}</math>
where
* ''i'' = 1, 2, 3, ..., ''R''
* ''j'' = 1, 2, 3, ..., ''C''
* ''μ'' = overall average (mean)
* ''τ''<sub>''j''</sub> = differential effect (response) associated with the ''j'' level of X; {{pb}} this assumes that overall the values of ''τ''<sub>''j''</sub> add to zero (that is, <math display="inline">\sum_{j = 1}^C \tau_j = 0</math>)
* ''ε''<sub>''ij''</sub> = noise or error associated with the particular ''ij'' data value

That is, we envision an additive model that says every data point can be represented by summing three quantities: the true mean, averaged over all factor levels being investigated, plus an incremental component associated with the particular column (factor level), plus a final component associated with everything else affecting that specific data value.