lack of balance leads to further complications. "The orthogonality property of main effects and interactions present in balanced data does not carry over to the unbalanced case. This means that the usual analysis of variance techniques do not apply. Consequently, the analysis of unbalanced factorials is much more difficult than that for balanced designs." In the general case, "The analysis of variance can also be applied to unbalanced data, but then the sums of squares, mean squares, and F-ratios will depend on the order in which the sources of variation are considered." ANOVA is (in part) a test of statistical significance. The American Psychological Association (and many other organisations) holds the view that simply reporting statistical significance is insufficient and that reporting confidence bounds is preferred. Generalizations ANOVA is considered to be a special case of linear regression which in turn is a special case of the general linear model. All consider the observations to be the sum of a model (fit) and a residual (error) to be minimized. The Kruskal–Wallis test and the Friedman test are nonparametric tests, which do not rely on an assumption of normality. Connection to linear regression Below we make clear the connection between multi-way ANOVA and linear regression. Linearly re-order the data so that -th observation is associated with a response and factors where denotes the different factors and is the total number of factors. In one-way ANOVA and in two-way ANOVA . Furthermore, we assume the -th factor has levels, namely . Now, we can one-hot encode the factors into the dimensional vector . The one-hot encoding function is defined such that the -th entry of is The vector is the concatenation of all of the above vectors for all . Thus, . In order to obtain a fully general -way interaction ANOVA we must also concatenate every additional interaction term in the vector and then add an intercept term. Let that vector be . With this notation in place, we now have the exact connection with linear regression. We simply regress response against the vector . However, there is a concern about identifiability. In order to overcome such issues we assume that the sum of the parameters within each set of interactions is equal to zero. From here, one can use F-statistics or other methods to determine the relevance of the individual factors. Example We can consider the 2-way interaction example where