Appendix
Consider a classification of \(n\) individuals based on five categorical variables \({A}_{1},{A}_{2},{A}_{3},{A}_{4}\) and \({A}_{5}\). Let \({j}_{k}\) be the number of categories of the \({k}^{th}\) variable, for \(k=1, 2, \dots , 5\) and the joint frequencies be summarised into a five-way contingency table \({\varvec{N}}=({n}_{{i}_{1}{i}_{2}{i}_{3}{i}_{4}{i}_{5}})\) of dimension \({j}_{1}\times {j}_{2}\times {j}_{3}\times {j}_{4}\times {j}_{5}\). Let \({\varvec{P}}=\left({p}_{{i}_{1}{i}_{2}{i}_{3}{i}_{4}{i}_{5}}\right)\) be the table of relative proportions such that \({\varvec{P}}=\frac{1}{n}{\varvec{N}}\).
The total inertia \({\Phi }^{2}\) based on the deviations from the five-way complete independence model is
$${\Phi }^{2} = \mathop \sum \limits_{{i_{1} ,i_{2} ,i_{3} ,i_{4} ,i_{5} }}^{{}} \frac{{\left( {p_{{i_{1} i_{2} i_{3} i_{4} i_{5} }} - p_{{i_{1} ....}} p_{{.i_{2} ...}} p_{{..i_{3} ..}} p_{{ \ldots i_{4} .}} p_{{ \ldots .i_{5} }} } \right)^{2} }}{{p_{{i_{1} ...}} p_{{.i_{2} ..}} p_{{..i_{3} .}} p_{{ \ldots i_{4} }} p_{{ \ldots .i_{5} }} }}.$$
The total inertia \({\Phi }^{2}\) can be expressed as a squared norm of \(\boldsymbol{\Pi }=({\pi }_{{i}_{1}{i}_{2}{i}_{3}{i}_{4}{i}_{5}})\) in \({\mathbb{R}}^{{j}_{1}\times {j}_{2}\times {j}_{3}\times {j}_{4}\times {j}_{5}}\) where
$$\pi_{{i_{1} i_{2} i_{3} i_{4} i_{5} }} = \frac{{p_{{i_{1} i_{2} i_{3} i_{4} i_{5} }} }}{{p_{{i_{1} ...}} p_{{.i_{2} ..}} p_{{..i_{3} .}} p_{{ \ldots i_{4} }} p_{{ \ldots .i_{5} }} }} - 1,$$
And \({\Vert \boldsymbol{\Pi }\Vert }^{2}=\sum_{{i}_{1},{i}_{2},{i}_{3},{i}_{4},{i}_{5}}{p}_{{i}_{1}\dots .}{p}_{.{i}_{2}...}{p}_{..{i}_{3}..}{p}_{\dots {i}_{4}.}{p}_{\dots .{i}_{5}}{\left({\pi }_{{i}_{1}{i}_{2}{i}_{3}{i}_{4}{i}_{5}}\right)}^{2}\).
Now, \(\boldsymbol{\Pi }\) can be expressed as the addition of twenty-six orthogonal arrays that consist of 5C2 = 10 two-way, 5C3 = 10 three-way, 5C4 = 5 four-way and 5C5 = 1 five-way association terms. That is
$$\begin{aligned}{\varvec{\varPi}}=\, &{\varvec{\varPi}}_{{{\varvec{i}}_{1} {\varvec{i}}_{2} ...}} +{\varvec{\varPi}}_{{{\varvec{i}}_{1} .{\varvec{i}}_{3} ..}} +{\varvec{\varPi}}_{{{\varvec{i}}_{1} ..{\varvec{i}}_{4} .}} +{\varvec{\varPi}}_{{{\varvec{i}}_{1} ...{\varvec{i}}_{5} }} +{\varvec{\varPi}}_{{.{\varvec{i}}_{2} {\varvec{i}}_{3} ..}} +{\varvec{\varPi}}_{{.{\varvec{i}}_{2} .{\varvec{i}}_{4} .}} +{\varvec{\varPi}}_{{.{\varvec{i}}_{2} ..{\varvec{i}}_{5} }} +{\varvec{\varPi}}_{{..{\varvec{i}}_{3} {\varvec{i}}_{4} .}} +{\varvec{\varPi}}_{{..{\varvec{i}}_{3} .{\varvec{i}}_{5} }} \\ \quad & + {{\varvec{\Pi}}}_{{...{\varvec{i}}_{4} {\varvec{i}}_{5} }} + {}_{{\varvec{\alpha}}}{\varvec{\varPi}}_{{{\varvec{i}}_{1} {\varvec{i}}_{2} {\varvec{i}}_{3} ..}} + {}_{{\varvec{\alpha}}}{\varvec{\varPi}}_{{{\varvec{i}}_{1} {\varvec{i}}_{2} .{\varvec{i}}_{4} .}} + {}_{{\varvec{\alpha}}}{\varvec{\varPi}}_{{{\varvec{i}}_{1} {\varvec{i}}_{2} ..{\varvec{i}}_{5} }} + {}_{{\varvec{\alpha}}}{\varvec{\varPi}}_{{{\varvec{i}}_{1} .{\varvec{i}}_{3} {\varvec{i}}_{4} .}} + {}_{{\varvec{\alpha}}}{\varvec{\varPi}}_{{{\varvec{i}}_{1} .{\varvec{i}}_{3} .{\varvec{i}}_{5} }} + {}_{{\varvec{\alpha}}}{\varvec{\varPi}}_{{{\varvec{i}}_{1} ..{\varvec{i}}_{4} {\varvec{i}}_{5} }} \\ \quad & + {}_{{\varvec{\alpha}}}{\varvec{\varPi}}_{{.{\varvec{i}}_{2} {\varvec{i}}_{3} {\varvec{i}}_{4} .}} + {}_{{\varvec{\alpha}}}{\varvec{\varPi}}_{{.{\varvec{i}}_{2} {\varvec{i}}_{3} .{\varvec{i}}_{5} }} + {}_{{\varvec{\alpha}}}{\varvec{\varPi}}_{{.{\varvec{i}}_{2} .{\varvec{i}}_{4} {\varvec{i}}_{5} }} + {}_{{\varvec{\alpha}}}{\varvec{\varPi}}_{{..{\varvec{i}}_{3} {\varvec{i}}_{4} {\varvec{i}}_{5} }} + {}_{{\varvec{\alpha}}}{\varvec{\varPi}}_{{{\varvec{i}}_{1} {\varvec{i}}_{2} {\varvec{i}}_{3} {\varvec{i}}_{4} .}} \\ \quad & + {}_{{\varvec{\alpha}}}{\varvec{\varPi}}_{{{\varvec{i}}_{1} {\varvec{i}}_{2} {\varvec{i}}_{3} .{\varvec{i}}_{5} }} + {}_{{\varvec{\alpha}}}{\varvec{\varPi}}_{{{\varvec{i}}_{1} {\varvec{i}}_{2} .{\varvec{i}}_{4} {\varvec{i}}_{5} }} + {}_{{\varvec{\alpha}}}{\varvec{\varPi}}_{{{\varvec{i}}_{1} .{\varvec{i}}_{3} {\varvec{i}}_{4} {\varvec{i}}_{5} }} + {}_{{\varvec{\alpha}}}{\varvec{\varPi}}_{{.{\varvec{i}}_{2} {\varvec{i}}_{3} {\varvec{i}}_{4} {\varvec{i}}_{5} }} + {}_{{\varvec{\alpha}}}{\varvec{\varPi}}_{{{\varvec{i}}_{1} {\varvec{i}}_{2} {\varvec{i}}_{3} {\varvec{i}}_{4} {\varvec{i}}_{5} }} \\ \end{aligned}$$
where
$$\begin{aligned} {\uppi }_{{i_{1} i_{2} ...}} = \,& \frac{{p_{{i_{1} i_{2} ...}} - p_{{i_{1} . \ldots }} p_{{.i_{2} ...}} }}{{p_{{i_{1} \ldots .}} p_{{.i_{2} \ldots }} }},\;{\uppi }_{{i_{1} .i_{3} ..}} = \frac{{p_{{i_{1} ..i_{3} ..}} - p_{{i_{1} . \ldots }} p_{{..i_{3} ..}} }}{{p_{{i_{1} . \ldots }} p_{{..i_{3} ..}} }},\;{\uppi }_{{i_{1} ..i_{4} .}} = \frac{{p_{{i_{1} ..i_{4} .}} - p_{{i_{1} . \ldots }} p_{{...i_{4} .}} }}{{p_{{i_{1} . \ldots }} p_{{...i_{4} .}} }}, \\ {\uppi }_{{i_{1} ...i_{5} }} = & \frac{{p_{{i_{1} ...i_{5} }} - p_{{i_{1} . \ldots }} p_{{ \ldots .i_{5} }} }}{{p_{{i_{1} . \ldots }} p_{{ \ldots .i_{5} }} }},\;{\uppi }_{{.i_{2} i_{3} ..}} = \frac{{p_{{.i_{2} i_{3} ...}} - p_{{.i_{2} ...}} p_{{..i_{3} ..}} }}{{p_{{.i_{2} ...}} p_{{..i_{3} ..}} }},\;{\uppi }_{{.i_{2} .i_{4} .}} = \frac{{p_{{.i_{2} .i_{4} .}} - p_{{.i_{2} ...}} p_{{...i_{4} .}} }}{{p_{{.i_{2} ...}} p_{{...i_{4} .}} }}, \\ {\uppi }_{{.i_{2} ..i_{5} }} = & \frac{{p_{{.i_{2} ..i_{5} }} - p_{{.i_{2} \ldots .}} p_{{....i_{5} }} }}{{p_{{.i_{2} ....}} p_{{....i_{5} }} }},\;{\uppi }_{{..i_{3} i_{4} .}} = \frac{{p_{{..i_{3} i_{4} .}} - p_{{..i_{3} ..}} p_{{...i_{4} .}} }}{{p_{{..i_{3} ..}} p_{{...i_{4} .}} }},\;{\uppi }_{{..i_{3} .i_{5} }} = \frac{{p_{{..i_{3} .i_{5} }} - p_{{..i_{3} ..}} p_{{ \ldots .i_{5} }} }}{{p_{{..i_{3} ..}} p_{{....i_{5} }} }}, \\ {\uppi }_{{...i_{4} i_{5} }} = & \frac{{p_{{ \ldots i_{4} i_{5} }} - p_{{...i_{4} .}} p_{{ \ldots .i_{5} }} }}{{p_{{...i_{4} .}} p_{{....i_{5} }} }},\; \\ \end{aligned}$$
$$\begin{aligned} {}_{\alpha }\pi_{{i_{1} i_{2} i_{3} ..}} = \,& \frac{{p_{{i_{1} i_{2} i_{3} ..}} - {}_{\alpha }^{{}} p_{{i_{1} i_{2} i_{3} ..}} }}{{p_{{i_{1} \ldots .}} p_{{.i_{2} ...}} p_{{..i_{3} ..}} }},\;{}_{\alpha }\pi_{{i_{1} i_{2} .i_{4} .}} = \frac{{p_{{i_{1} i_{2} .i_{4} .}} - {}_{\alpha }^{{}} p_{{i_{1} i_{2} .i_{4} .}} }}{{p_{{i_{1} . \ldots }} p_{{.i_{2} ...}} p_{{...i_{4} .}} }}, \\ {}_{\alpha }\pi_{{i_{1} i_{2} ..i_{5} }} = & \frac{{p_{{i_{1} i_{2} ..i_{5} }} - {}_{\alpha }^{{}} p_{{i_{1} i_{2} ..i_{5} }} }}{{p_{{i_{1} . \ldots }} p_{{.i_{2} ...}} p_{{....i_{5} }} }},\;{}_{\alpha }\pi_{{i_{1} .i_{3} i_{4} .}} = \frac{{p_{{i_{1} .i_{3} i_{4} .}} - {}_{\alpha }^{{}} p_{{i_{1} .i_{3} i_{4} .}} }}{{p_{{i_{1} . \ldots }} p_{{..i_{3} ..}} p_{{...i_{4} .}} }}, \\ {}_{\alpha }\pi_{{i_{1} .i_{3} .i_{5} }} = & \frac{{p_{{i_{1} .i_{3} .i_{5} }} - {}_{\alpha }^{{}} p_{{i_{1} .i_{3} .i_{5} }} }}{{p_{{i_{1} . \ldots }} p_{{..i_{3} ..}} p_{{....i_{5} }} }},\;{}_{\alpha }\pi_{{i_{1} ..i_{4} i_{5} }} = \frac{{p_{{i_{1} ..i_{4} i_{5} }} - {}_{\alpha }^{{}} p_{{i_{1} ..i_{4} i_{5} }} }}{{p_{{i_{1} . \ldots }} p_{{...i_{4} .}} p_{{....i_{5} }} }}, \\ {}_{\alpha }\pi_{{.i_{2} i_{3} i_{4} .}} = & \frac{{p_{{.i_{2} i_{3} i_{4} .}} - {}_{\alpha }^{{}} p_{{.i_{2} i_{3} i_{4} .}} }}{{p_{{.i_{2} ...}} p_{{..i_{3} ..}} p_{{...i_{4} .}} }},\;{}_{\alpha }\pi_{{.i_{2} i_{3} .i_{5} }} = \frac{{p_{{.i_{2} i_{3} .i_{5} }} - {}_{\alpha }^{{}} p_{{.i_{2} i_{3} .i_{5} }} }}{{p_{{.i_{2} ...}} p_{{..i_{3} ..}} p_{{. \ldots i_{5} }} }}, \\ {}_{\alpha }\pi_{{.i_{2} .i_{4} i_{5} }} = & \frac{{p_{{.i_{2} .i_{4} i_{5} }} - {}_{\alpha }^{{}} p_{{.i_{2} .i_{4} i_{5} }} }}{{p_{{.i_{2} ...}} p_{{...i_{4} .}} p_{{. \ldots i_{5} }} }},\;{}_{\alpha }\pi_{{..i_{3} i_{4} i_{5} }} = \frac{{p_{{..i_{3} i_{4} i_{5} }} - {}_{\alpha }^{{}} p_{{..i_{3} i_{4} i_{5} }} }}{{p_{{..i_{3} ..}} p_{{...i_{4} .}} p_{{. \ldots i_{5} }} }}, \\ {}_{\alpha }\Pi_{{i_{1} i_{2} i_{3} i_{4} .}} = & \frac{{p_{{i_{1} i_{2} i_{3} i_{4} .}} - {}_{\alpha }^{{}} p_{{i_{1} i_{2} i_{3} i_{4} .}} }}{{p_{{i_{1} \ldots .}} p_{{.i_{2} ...}} p_{{..i_{3} ..}} p_{{ \ldots i_{4} .}} }},\;{}_{\alpha }\Pi_{{i_{1} i_{2} i_{3} .i_{5} }} = \frac{{p_{{i_{1} i_{2} i_{3} .i_{5} }} - {}_{\alpha }^{{}} p_{{i_{1} i_{2} i_{3} .i_{5} }} }}{{p_{{i_{1} \ldots .}} p_{{.i_{2} ...}} p_{{..i_{3} ..}} p_{{ \ldots .i_{5} }} }}, \\ {}_{\alpha }\Pi_{{i_{1} i_{2} .i_{4} i_{5} }} = & \frac{{p_{{i_{1} i_{2} .i_{4} i_{5} }} - {}_{\alpha }^{{}} p_{{i_{1} i_{2} i_{4} .i_{5} }} }}{{p_{{i_{1} \ldots .}} p_{{.i_{2} ...}} p_{{...i_{4} .}} p_{{ \ldots .i_{5} }} }},\;\Pi_{{i_{1} i_{3} .i_{4} i_{5} }} = \frac{{p_{{i_{1} .i_{3} i_{4} i_{5} }} - {}_{\alpha }^{{}} p_{{i_{1} .i_{3} i_{4} i_{5} }} }}{{p_{{i_{1} \ldots .}} p_{{..i_{3} ..}} p_{{...i_{4} .}} p_{{ \ldots .i_{5} }} }}, \\ {}_{\alpha }\Pi_{{.i_{2} i_{3} i_{4} i_{5} }} = & \frac{{p_{{.i_{2} i_{3} i_{4} i_{5} }} - {}_{\alpha }^{{}} p_{{.i_{2} i_{3} i_{4} i_{5} }} }}{{p_{{.i_{2} ...}} p_{{..i_{3} ..}} p_{{...i_{4} .}} p_{{ \ldots .i_{5} }} }},\;{}_{\alpha }\pi_{{i_{1} i_{2} i_{3} i_{4} i_{5} }} = \frac{{p_{{i_{1} i_{2} i_{3} i_{4} i_{5} }} - {}_{\alpha }^{{}} p_{{i_{1} i_{2} i_{3} i_{4} i_{5} }} }}{{p_{{i_{1} . \ldots }} p_{{.i_{2} ...}} p_{{..i_{3} ..}} p_{{...i_{4} .}} p_{{. \ldots i_{5} }} }}, \\ \end{aligned}$$
$$\begin{aligned} {}_{\alpha }p_{{i_{1} i_{2} i_{3} i_{4} .}} = \,& p_{{i_{1} i_{2} \ldots }} p_{{..i_{3} ..}} p_{{ \ldots i_{4} .}} + p_{{i_{1} .i_{3} ..}} p_{{.i_{2} \ldots }} p_{{ \ldots i_{4} .}} + p_{{i_{1} ..i_{4} .}} p_{{.i_{2} \ldots }} p_{{..i_{3} ..}} + p_{{.i_{2} i_{3} ..}} p_{{i_{1} \ldots .}} p_{{ \ldots i_{4} .}} \\ \quad & + ~p_{{.i_{2} .i_{4} .}} p_{{i_{1} \ldots .}} p_{{..i_{3} ..}} + p_{{..i_{3} i_{4} .}} p_{{i_{1} \ldots .}} p_{{.i_{2} \ldots }} + p_{{i_{1} i_{2} i_{3} ..}} p_{{ \ldots i_{4} .}} + p_{{i_{1} i_{2} .i_{4} .}} p_{{..i_{3} ..}} \\ \quad & + p_{{.i_{2} i_{3} i_{4} .}} p_{{i_{1} \ldots .}} + ~p_{{i_{1} .i_{3} i_{4} .}} p_{{.i_{2} ...}} + 9p_{{i_{1} \ldots .}} p_{{.i_{2} ...}} p_{{..i_{3} ..}} p_{{ \ldots i_{4} .}} , \\ {}_{\alpha }p_{{i_{1} i_{2} i_{3} .i_{5} }} = & p_{{i_{1} i_{2} ...}} p_{{..i_{3} ..}} p_{{ \ldots .i_{5} }} + p_{{i_{1} .i_{3} ..}} p_{{.i_{2} ...}} p_{{ \ldots .i_{5} }} + p_{{i_{1} ...i_{5} }} p_{{.i_{2} ...}} p_{{..i_{3} ..}} + p_{{.i_{2} i_{3} ..}} p_{{i_{1} \ldots .}} p_{{ \ldots .i_{5} }} \\ \quad & + ~p_{{.i_{2} ..i_{5} }} p_{{i_{1} \ldots .}} p_{{..i_{3} ..}} + p_{{..i_{3} .i_{5} }} p_{{i_{1} \ldots .}} p_{{.i_{2} \ldots }} + p_{{i_{1} i_{2} i_{3} ..}} p_{{ \ldots .i_{5} }} + p_{{i_{1} i_{2} ..i_{5} }} p_{{..i_{3} ..}} \\ \quad & + p_{{.i_{2} i_{3} .i_{5} }} p_{{i_{1} \ldots .}} + ~p_{{i_{1} .i_{3} .i_{5} }} p_{{.i_{2} ...}} + 9p_{{i_{1} \ldots .}} p_{{.i_{2} ...}} p_{{..i_{3} ..}} p_{{ \ldots .i_{5} }} , \\ {}_{\alpha }p_{{i_{1} i_{2} .i_{4} i_{5} }} = & p_{{i_{1} i_{2} ...}} p_{{ \ldots i_{4} .}} p_{{ \ldots .i_{5} }} + p_{{i_{1} ..i_{4} .}} p_{{.i_{2} ...}} p_{{ \ldots .i_{5} }} + p_{{i_{1} ...i_{5} }} p_{{.i_{2} ...}} p_{{...i_{4} .}} + p_{{.i_{2} .i_{4} .}} p_{{i_{1} \ldots .}} p_{{ \ldots .i_{5} }} \\ \quad & + ~p_{{.i_{2} ..i_{5} }} p_{{i_{1} \ldots .}} p_{{ \ldots i_{4} .}} + p_{{ \ldots i_{4} i_{5} }} p_{{i_{1} \ldots .}} p_{{.i_{2} \ldots }} + p_{{i_{1} i_{2} .i_{4} .}} p_{{ \ldots .i_{5} }} + p_{{i_{1} i_{2} ..i_{5} }} p_{{...i_{4} .}} \\ \quad & + ~p_{{.i_{2} .i_{4} i_{5} }} p_{{i_{1} \ldots .}} + ~p_{{i_{1} ..i_{4} i_{5} }} p_{{.i_{2} ...}} + 9p_{{i_{1} \ldots .}} p_{{.i_{2} ...}} p_{{ \ldots i_{4} .}} p_{{ \ldots .i_{5} }} , \\ {}_{\alpha }p_{{i_{1} .i_{3} i_{4} i_{5} }} = & p_{{i_{1} .i_{3} ..}} p_{{ \ldots i_{4} .}} p_{{ \ldots .i_{5} }} + p_{{i_{1} ..i_{4} .}} p_{{..i_{3} ..}} p_{{ \ldots .i_{5} }} + p_{{i_{1} ...i_{5} }} p_{{..i_{3} ..}} p_{{...i_{4} .}} + p_{{..i_{3} i_{4} .}} p_{{i_{1} \ldots .}} p_{{ \ldots .i_{5} }} \\ \quad & + ~p_{{..i_{3} .i_{5} }} p_{{i_{1} \ldots .}} p_{{ \ldots i_{4} .}} + p_{{ \ldots i_{4} i_{5} }} p_{{i_{1} \ldots .}} p_{{..i_{3} }} + p_{{i_{1} .i_{3} i_{4} .}} p_{{ \ldots .i_{5} }} + p_{{i_{1} .i_{3} .i_{5} }} p_{{...i_{4} .}} \\ \quad & + ~p_{{..i_{3} i_{4} i_{5} }} p_{{i_{1} \ldots .}} + ~p_{{i_{1} ..i_{4} i_{5} }} p_{{..i_{3} ..}} + 9p_{{i_{1} \ldots .}} p_{{..i_{3} ..}} p_{{ \ldots i_{4} .}} p_{{ \ldots .i_{5} }} , \\ {}_{\alpha }p_{{.i_{2} i_{3} i_{4} i_{5} }} = & p_{{.i_{2} i_{3} ..}} p_{{ \ldots i_{4} .}} p_{{ \ldots .i_{5} }} + p_{{.i_{2} .i_{4} .}} p_{{..i_{3} ..}} p_{{ \ldots .i_{5} }} + p_{{.i_{2} ..i_{5} }} p_{{..i_{3} ..}} p_{{...i_{4} .}} + p_{{..i_{3} i_{4} .}} p_{{.i_{2} \ldots }} p_{{ \ldots .i_{5} }} \\ \quad & + ~p_{{..i_{3} .i_{5} }} p_{{.i_{2} \ldots }} p_{{ \ldots i_{4} .}} + p_{{ \ldots i_{4} i_{5} }} p_{{.i_{2} \ldots }} p_{{..i_{3} }} + p_{{.i_{2} i_{3} i_{4} .}} p_{{ \ldots .i_{5} }} + p_{{.i_{2} i_{3} .i_{5} }} p_{{...i_{4} .}} \\ \quad & + ~p_{{..i_{3} i_{4} i_{5} }} p_{{.i_{2} \ldots }} + ~p_{{.i_{2} .i_{4} i_{5} }} p_{{..i_{3} ..}} + 9p_{{.i_{2} \ldots }} p_{{..i_{3} ..}} p_{{ \ldots i_{4} .}} p_{{ \ldots .i_{5} }} , \\ {\text{and}}\;{}_{\alpha }p_{{i_{1} i_{2} i_{3} i_{4} i_{5} }} = & p_{{i_{1} i_{2} ...}} p_{{..i_{3} ..}} p_{{ \ldots i_{4} .}} p_{{ \ldots .i_{5} }} + p_{{i_{1} .i_{3} ..}} p_{{.i_{2} \ldots }} p_{{ \ldots i_{4} .}} p_{{ \ldots .i_{5} }} + p_{{i_{1} ..i_{4} .}} p_{{.i_{2} \ldots }} p_{{..i_{3} ..}} p_{{ \ldots .i_{5} }} \\ \quad & + ~p_{{i_{1} ...i_{5} }} p_{{.i_{2} \ldots }} p_{{..i_{3} ..}} p_{{ \ldots i_{4} .}} + p_{{.i_{2} i_{3} ..}} p_{{i_{1} . \ldots }} p_{{ \ldots i_{4} .}} p_{{ \ldots .i_{5} }} + p_{{.i_{2} i_{4} .}} p_{{i_{1} . \ldots }} p_{{..i_{3} ..}} p_{{ \ldots .i_{5} }} + \\ \quad & + ~p_{{.i_{2} ..i_{5} }} p_{{i_{1} . \ldots }} p_{{..i_{3} ..}} p_{{ \ldots i_{4} .}} + p_{{i_{1} i_{2} i_{3} ..}} p_{{ \ldots i_{4} .}} p_{{ \ldots .i_{5} }} + p_{{i_{1} i_{2} i_{4} .}} p_{{..i_{3} ..}} p_{{ \ldots .i_{5} }} \\ \quad & + ~p_{{i_{1} i_{2} ..i_{5} }} p_{{..i_{3} ..}} p_{{ \ldots i_{4} .}} + p_{{i_{1} .i_{3} i_{4} .}} p_{{.i_{2} \ldots }} p_{{ \ldots .i_{5} }} + p_{{i_{1} .i_{3} .i_{5} }} p_{{.i_{2} \ldots }} p_{{ \ldots i_{4} .}} \\ \quad & + ~p_{{i_{1} ..i_{4} i_{5} }} p_{{.i_{2} \ldots }} p_{{..i_{3} ..}} + p_{{.i_{2} i_{3} i_{4} .}} p_{{i_{1} . \ldots }} p_{{ \ldots .i_{5} }} + p_{{.i_{2} i_{3} .i_{5} }} p_{{i_{1} . \ldots }} p_{{ \ldots i_{4} .}} \\ \quad & + ~p_{{.i_{2} .i_{4} i_{5} }} p_{{i_{1} . \ldots }} p_{{..i_{3} ..}} + p_{{..i_{3} i_{4} i_{5} }} p_{{i_{1} . \ldots }} p_{{.i_{2} ...}} + p_{{i_{1} i_{2} i_{3} i_{4} .}} p_{{....i_{5} }} + p_{{i_{1} i_{2} i_{3} .i_{5} }} p_{{...i_{4} .}} \\ \quad & + ~p_{{i_{1} i_{2} .i_{4} i_{5} }} p_{{..i_{3} ..}} + p_{{i_{1} .i_{3} i_{4} i_{5} }} p_{{.i_{2} ...}} + p_{{i_{2} i_{3} i_{4} i_{5} }} p_{{i_{1} ....}} \\ \quad & + ~24p_{{i_{1} \ldots .}} p_{{.i_{2} \ldots }} p_{{..i_{3} ..}} p_{{ \ldots i_{4} .}} p_{{....i_{5} }} . \\ \end{aligned}$$
Thus, an additive partition of the squared norm of a five-way array \(\Pi\) can be expressed as
$$\begin{aligned}{\varvec{\varPi}}^{2} = \,&{\varvec{\varPi}}_{{{\varvec{i}}_{1} {\varvec{i}}_{2} ...}}^{2} +{\varvec{\varPi}}_{{{\varvec{i}}_{1} .{\varvec{i}}_{3} ..}}^{2} +{\varvec{\varPi}}_{{{\varvec{i}}_{1} ..{\varvec{i}}_{4} .}}^{2} +{\varvec{\varPi}}_{{{\varvec{i}}_{1} ...{\varvec{i}}_{5} }}^{2} +{\varvec{\varPi}}_{{.{\varvec{i}}_{2} {\varvec{i}}_{3} ..}}^{2} +{\varvec{\varPi}}_{{.{\varvec{i}}_{2} .{\varvec{i}}_{4} .}}^{2} \\ \quad & + {{\varvec{\Pi}}}_{{.{\varvec{i}}_{2} ..{\varvec{i}}_{5} }}^{2} + {{\varvec{\Pi}}}_{{..{\varvec{i}}_{3} {\varvec{i}}_{4} .}}^{2} + {{\varvec{\Pi}}}_{{..{\varvec{i}}_{3} .{\varvec{i}}_{5} }}^{2} + {{\varvec{\Pi}}}_{{...{\varvec{i}}_{4} {\varvec{i}}_{5} }}^{2} + {}_{{\varvec{\alpha}}}^{{}}{\varvec{\varPi}}_{{{\varvec{i}}_{1} {\varvec{i}}_{2} {\varvec{i}}_{3} ..}}^{2} \\ \quad & + {}_{{\varvec{\alpha}}}^{{}}{\varvec{\varPi}}_{{{\varvec{i}}_{1} {\varvec{i}}_{2} .{\varvec{i}}_{4} .}}^{2} + {}_{{\varvec{\alpha}}}^{{}}{\varvec{\varPi}}_{{{\varvec{i}}_{1} {\varvec{i}}_{2} ..{\varvec{i}}_{5} }}^{2} + {}_{{\varvec{\alpha}}}^{{}}{\varvec{\varPi}}_{{{\varvec{i}}_{1} .{\varvec{i}}_{3} {\varvec{i}}_{4} .}}^{2} + {}_{{\varvec{\alpha}}}^{{}}{\varvec{\varPi}}_{{{\varvec{i}}_{1} .{\varvec{i}}_{3} .{\varvec{i}}_{5} }}^{2} \\ \quad & + {}_{{\varvec{\alpha}}}^{{}}{\varvec{\varPi}}_{{{\varvec{i}}_{1} ..{\varvec{i}}_{4} {\varvec{i}}_{5} }}^{2} + {}_{{\varvec{\alpha}}}^{{}}{\varvec{\varPi}}_{{.{\varvec{i}}_{2} {\varvec{i}}_{3} {\varvec{i}}_{4} .}}^{2} + {}_{{\varvec{\alpha}}}^{{}}{\varvec{\varPi}}_{{.{\varvec{i}}_{2} {\varvec{i}}_{3} .{\varvec{i}}_{5} }}^{2} + {}_{{\varvec{\alpha}}}^{{}}{\varvec{\varPi}}_{{.{\varvec{i}}_{2} .{\varvec{i}}_{4} {\varvec{i}}_{5} }}^{2} \\ \quad & + {}_{{\varvec{\alpha}}}^{{}}{\varvec{\varPi}}_{{..{\varvec{i}}_{3} {\varvec{i}}_{4} {\varvec{i}}_{5} }}^{2} + {}_{{\varvec{\alpha}}}^{{}}{\varvec{\varPi}}_{{{\varvec{i}}_{1} {\varvec{i}}_{2} {\varvec{i}}_{3} {\varvec{i}}_{4} .}}^{2} + {}_{{\varvec{\alpha}}}^{{}}{\varvec{\varPi}}_{{{\varvec{i}}_{1} {\varvec{i}}_{2} {\varvec{i}}_{3} .{\varvec{i}}_{5} }}^{2} + {}_{{\varvec{\alpha}}}^{{}}{\varvec{\varPi}}_{{{\varvec{i}}_{1} {\varvec{i}}_{2} .{\varvec{i}}_{4} {\varvec{i}}_{5} }}^{2} \\ \quad & + {}_{{\varvec{\alpha}}}^{{}}{\varvec{\varPi}}_{{{\varvec{i}}_{1} .{\varvec{i}}_{3} {\varvec{i}}_{4} {\varvec{i}}_{5} }}^{2} + {}_{{\varvec{\alpha}}}^{{}}{\varvec{\varPi}}_{{.{\varvec{i}}_{2} {\varvec{i}}_{3} {\varvec{i}}_{4} {\varvec{i}}_{5} }}^{2} + {}_{{\varvec{\alpha}}}^{{}}{\varvec{\varPi}}_{{{\varvec{i}}_{1} {\varvec{i}}_{2} {\varvec{i}}_{3} {\varvec{i}}_{4} {\varvec{i}}_{5} }}^{2} . \\ \end{aligned}$$
This partition can be equivalently expressed in terms of table’s inertia such that
$$\begin{aligned} \Phi ^{2} = \,& \mathop \sum \limits_{{i_{1} ,i_{2} }} p_{{i_{1} \ldots .}} p_{{.i_{2} ...}} \left( {\frac{{p_{{i_{1} i_{2} ...}} - p_{{i_{1} ....}} p_{{.i_{2} ...}} }}{{p_{{i_{1} ....}} p_{{.i_{2} ...}} }}} \right)^{2} + \mathop \sum \limits_{{i_{1} ,i_{3} }} p_{{i_{1} ....}} p_{{..i_{3} ..}} \left( {\frac{{p_{{i_{1} .i_{3} ..}} - p_{{i_{1} ....}} p_{{..i_{3} ..}} }}{{p_{{i_{1} ....}} p_{{..i_{3} ..}} }}} \right)^{2} \\ \quad & + \mathop \sum \limits_{{i_{1} ,i_{4} }} p_{{i_{1} ....}} p_{{ \ldots i_{4} .}} \left( {\frac{{p_{{i_{1} ..i_{4} .}} - p_{{i_{1} ....}} p_{{ \ldots i_{4} .}} }}{{p_{{i_{1} ....}} p_{{ \ldots i_{4} .}} }}} \right)^{2} + \mathop \sum \limits_{{i_{1} ,i_{5} }} p_{{i_{1} ....}} p_{{ \ldots .i_{5} }} \left( {\frac{{p_{{i_{1} ...i_{5} }} - p_{{i_{1} ....}} p_{{ \ldots .i_{5} }} }}{{p_{{i_{1} ....}} p_{{ \ldots .i_{5} }} }}} \right)^{2} \\ \quad & + \mathop \sum \limits_{{i_{2} ,i_{3} }} p_{{.i_{2} ...}} p_{{..i_{3} ..}} \left( {\frac{{p_{{.i_{2} i_{3} ..}} - p_{{.i_{2} ...}} p_{{..i_{3} ..}} }}{{p_{{.i_{2} \ldots }} p_{{..i_{3} ..}} }}} \right)^{2} + \mathop \sum \limits_{{i_{2} ,i_{4} }} p_{{.i_{2} ...}} p_{{ \ldots i_{4} .}} \left( {\frac{{p_{{.i_{2} .i_{4} .}} - p_{{.i_{2} ...}} p_{{ \ldots i_{4} .}} }}{{p_{{.i_{2} ...}} p_{{ \ldots i_{4} .}} }}} \right)^{2} \\ \quad & + \mathop \sum \limits_{{i_{2} ,i_{5} }} p_{{.i_{2} ...}} p_{{....i_{5} }} \left( {\frac{{p_{{.i_{2} ..i_{5} }} - p_{{.i_{2} ...}} p_{{ \ldots .i_{5} }} }}{{p_{{.i_{2} ...}} p_{{ \ldots .i_{5} }} }}} \right)^{2} + \mathop \sum \limits_{{i_{3} ,i_{4} }} p_{{..i_{3} ..}} p_{{ \ldots i_{4} .}} \left( {\frac{{p_{{..i_{3} i_{4} .}} - p_{{..i_{3} ..}} p_{{ \ldots i_{4} .}} }}{{p_{{..i_{3} ..}} p_{{ \ldots i_{4} .}} }}} \right)^{2} \\ \quad & + \mathop \sum \limits_{{i_{3} ,i_{5} }} p_{{..i_{3} ..}} p_{{....i_{5} }} \left( {\frac{{p_{{..i_{3} .i_{5} }} - p_{{..i_{3} ..}} p_{{ \ldots .i_{5} }} }}{{p_{{..i_{3} ..}} p_{{ \ldots .i_{5} }} }}} \right)^{2} + \mathop \sum \limits_{{i_{4} ,i_{5} }} p_{{ \ldots i_{4} .}} p_{{ \ldots .i_{5} }} \left( {\frac{{p_{{ \ldots i_{4} i_{5} }} - p_{{ \ldots i_{4} .}} p_{{ \ldots .i_{5} }} }}{{p_{{ \ldots i_{4} .}} p_{{ \ldots .i_{5} }} }}} \right)^{2} \\ \quad & + \mathop \sum \limits_{{i_{1} ,i_{2} ,i_{3} }} p_{{i_{1} \ldots .}} p_{{.i_{2} \ldots }} p_{{..i_{3} ..}} \left( {\frac{{p_{{i_{1} i_{2} i_{3} ..}} - {}_{\alpha }p_{{i_{1} i_{2} i_{3} ..}} }}{{p_{1} \ldots .p_{{.i_{2} \ldots }} p_{{..i_{3} ..}} }}} \right)^{2} + \mathop \sum \limits_{{i_{1} ,i_{2} ,i_{4} }} p_{{i_{1} ....}} p_{{.i_{2} ...}} p_{{ \ldots i_{4} .}} \left( {\frac{{p_{{i_{1} i_{2} .i_{4} .}} - {}_{\alpha }p_{{i_{1} i_{2} .i_{4} .}} }}{{p_{{i_{1} ....}} p_{{.i_{2} ...}} p_{{ \ldots i_{4} .}} }}} \right)^{2} \\ \quad & + \mathop \sum \limits_{{i_{1} ,i_{2} ,i_{5} }} p_{{i_{1} ....}} p_{{.i_{2} ...}} p_{{. \ldots i_{5} }} \left( {\frac{{p_{{i_{1} i_{2} ..i_{5} }} - {}_{\alpha }p_{{i_{1} i_{2} ..i_{5} }} }}{{p_{{i_{1} \ldots .}} p_{{.i_{2} ...}} p_{{. \ldots i_{5} }} }}} \right)^{2} + \mathop \sum \limits_{{i_{1} ,i_{3} ,i_{4} }} p_{{i_{1} \ldots .}} p_{{..i_{3} ..}} p_{{ \ldots i_{4} .}} \left( {\frac{{p_{{i_{1} .i_{3} i_{4} .}} - {}_{\alpha }p_{{i_{1} .i_{3} i_{4} .}} }}{{p_{{i_{1} \ldots .}} p_{{..i_{3} ..}} p_{{ \ldots i_{4} .}} }}} \right)^{2} \\ \quad & + \mathop \sum \limits_{{i_{1} ,i_{3} ,i_{5} }} p_{{i_{1} \ldots .}} p_{{..i_{3} ..}} p_{{ \ldots .i_{5} }} \left( {\frac{{p_{{i_{1} .i_{3} .i_{5} }} - {}_{\alpha }p_{{i_{1} .i_{3} .i_{5} }} }}{{p_{{i_{1} \ldots .}} p_{{..i_{3} ..}} p_{{ \ldots .i_{5} }} }}} \right)^{2} + \mathop \sum \limits_{{i_{1} ,i_{4} ,i_{5} }} p_{{i_{1} \ldots .}} p_{{ \ldots i_{4} .}} p_{{ \ldots .i_{5} }} \left( {\frac{{p_{{i_{1} ..i_{4} i_{5} }} - {}_{\alpha }p_{{i_{1} ..i_{4} i_{5} }} }}{{p_{{i_{1} \ldots .}} p_{{ \ldots i_{4} .}} p_{{ \ldots .i_{5} }} }}} \right)^{2} \\ \quad & + \mathop \sum \limits_{{i_{2} ,i_{3} ,i_{4} }} p_{{.i_{2} ...}} p_{{..i_{3} ..}} p_{{ \ldots i_{4} .}} \left( {\frac{{p_{{.i_{2} i_{3} i_{4} .}} - {}_{\alpha }p_{{.i_{2} i_{3} i_{4} .}} }}{{p_{{.i_{2} ...}} p_{{..i_{3} ..}} p_{{ \ldots i_{4} .}} }}} \right)^{2} + \mathop \sum \limits_{{i_{2} ,i_{3} ,i_{5} }} p_{{.i_{2} ...}} p_{{..i_{3} ..}} p_{{ \ldots .i_{5} }} \left( {\frac{{p_{{.i_{2} i_{3} .i_{5} }} - {}_{\alpha }p_{{.i_{2} i_{3} .i_{5} }} }}{{p_{{.i_{2} ...}} p_{{..i_{3} ..}} p_{{ \ldots .i_{5} }} }}} \right)^{2} \\ \quad & + \mathop \sum \limits_{{i_{2} ,i_{4} ,i_{5} }} p_{{.i_{2} ...}} p_{{ \ldots i_{4} .}} p_{{ \ldots .i_{5} }} \left( {\frac{{p_{{.i_{2} .i_{4} i_{5} }} - {}_{\alpha }p_{{.i_{2} .i_{4} i_{5} }} }}{{p_{{.i_{2} ...}} p_{{...i_{4} .}} p_{{ \ldots .i_{5} }} }}} \right)^{2} + \mathop \sum \limits_{{i_{3} ,i_{4} ,i_{5} }} p_{{..i_{3} ..}} p_{{ \ldots i_{4} .}} p_{{ \ldots .i_{5} }} \left( {\frac{{p_{{..i_{3} i_{4} i_{5} }} - {}_{\alpha }p_{{..i_{3} i_{4} i_{5} }} }}{{p_{{..i_{3} ..}} p_{{...i_{4} .}} p_{{ \ldots .i_{5} }} }}} \right)^{2} \\ \quad & + \mathop \sum \limits_{{i_{1} ,i_{2} ,i_{3} ,i_{4} }} p_{{i_{1} ....}} p_{{.i_{2} ...}} p_{{..i_{3} ..}} p_{{ \ldots i_{4} .}} \left( {\frac{{p_{{i_{1} i_{2} i_{3} i_{4} .}} - {}_{\alpha }p_{{i_{1} i_{2} i_{3} i_{4} .}} }}{{p_{{i_{1} \ldots .}} p_{{.i_{2} ...}} p_{{..i_{3} ..}} p_{{ \ldots i_{4} .}} }}} \right)^{2} \\ \quad & + \mathop \sum \limits_{{i_{1} ,i_{2} ,i_{3} ,i_{5} }} p_{{i_{1} ....}} p_{{.i_{2} ...}} p_{{..i_{3} ..}} p_{{ \ldots .i_{5} }} \left( {\frac{{p_{{i_{1} i_{2} i_{3} .i_{5} }} - {}_{\alpha }p_{{i_{1} i_{2} i_{3} .i_{5} }} }}{{p_{{i_{1} \ldots .}} p_{{.i_{2} ...}} p_{{..i_{3} ..}} p_{{ \ldots ,i_{5} }} }}} \right)^{2} \\ \quad & + \mathop \sum \limits_{{i_{1} ,i_{2} ,i_{4} ,i_{5} }} p_{{i_{1} ....}} p_{{.i_{2} ...}} p_{{ \ldots i_{4} .}} p_{{ \ldots .i_{5} }} \left( {\frac{{p_{{i_{1} i_{2} .i_{4} i_{5} }} - {}_{\alpha }p_{{i_{1} i_{2} .i_{4} i_{5} }} }}{{p_{{i_{1} \ldots .}} p_{{.i_{2} ...}} p_{{...i_{4} .}} p_{{ \ldots ,i_{5} }} }}} \right)^{2} \\ \quad & + \mathop \sum \limits_{{i_{1} ,i_{3} ,i_{4} ,i_{5} }} p_{{i_{1} ....}} p_{{..i_{3} ..}} p_{{ \ldots i_{4} .}} p_{{ \ldots .i_{5} }} \left( {\frac{{p_{{i_{1} .i_{3} i_{4} i_{5} }} - {}_{\alpha }p_{{i_{1} .i_{3} i_{4} i_{5} }} }}{{p_{{i_{1} \ldots .}} p_{{..i_{3} ..}} p_{{...i_{4} .}} p_{{ \ldots ,i_{5} }} }}} \right)^{2} \\ \quad & + \mathop \sum \limits_{{i_{2} ,i_{3} ,i_{4} ,i_{5} }} p_{{.i_{2} ...}} p_{{..i_{3} ..}} p_{{ \ldots i_{4} .}} p_{{ \ldots .i_{5} }} \left( {\frac{{p_{{.i_{2} i_{3} i_{4} i_{5} }} - {}_{\alpha }p_{{.i_{2} i_{3} i_{4} i_{5} }} }}{{p_{{.i_{2} ...}} p_{{..i_{3} ..}} p_{{...i_{4} .}} p_{{ \ldots ,i_{5} }} }}} \right)^{2} \\ \quad & + \mathop \sum \limits_{{i_{1} ,i_{2} ,i_{3} ,i_{4} ,i_{5} }} p_{{i_{1} \ldots .}} p_{{.i_{2} ...}} p_{{..i_{3} ..}} p_{{ \ldots i_{4} .}} p_{{ \ldots .i_{5} }} \left( {\frac{{p_{{i_{1} i_{2} i_{3} i_{4} i_{5} }} - {}_{\alpha }p_{{i_{1} i_{2} i_{3} i_{4} i_{5} }} }}{{p_{{i_{1} \ldots .}} p_{{.i_{2} ...}} p_{{..i_{3} ..}} p_{{...i_{4} .}} p_{{ \ldots ,i_{5} }} }}} \right)^{2} . \\ \end{aligned}$$
Thus, the orthogonal partition of \(n{\Phi }^{2}={\chi }_{Total}^{2}\) for the five-way contingency table N is therefore
$$\begin{aligned} \chi_{Total}^{2} = \,& \chi_{{A_{1} A_{2} }}^{2} + \chi_{{A_{1} A_{3} }}^{2} + \chi_{{A_{1} A_{4} }}^{2} + \chi_{{A_{1} A_{5} }}^{2} + \chi_{{A_{2} A_{3} }}^{2} + \chi_{{A_{2} A_{4} }}^{2} + \chi_{{A_{2} A_{5} }}^{2} + \chi_{{A_{3} A_{4} }}^{2} + \chi_{{A_{3} A_{5} }}^{2} + \chi_{{A_{4} A_{5} }}^{2} \\ \quad & + \chi_{{A_{1} A_{2} A_{3} }}^{2} + \chi_{{A_{1} A_{2} A_{4} }}^{2} + \chi_{{A_{1} A_{2} A_{5} }}^{2} + \chi_{{A_{1} A_{3} A_{4} }}^{2} + \chi_{{A_{1} A_{3} A_{5} }}^{2} + \chi_{{A_{1} A_{4} A_{5} }}^{2} + \chi_{{A_{2} A_{3} A_{4} }}^{2} \\ \quad & + \chi_{{A_{2} A_{3} A_{5} }}^{2} + \chi_{{A_{2} A_{4} A_{5} }}^{2} + \chi_{{A_{3} A_{4} A_{5} }}^{2} + \chi_{{A_{1} A_{2} A_{3} A_{4} }}^{2} + \chi_{{A_{1} A_{2} A_{3} A_{5} }}^{2} + \chi_{{A_{1} A_{2} A_{4} A_{5} }}^{2} \\ \quad & + \chi_{{A_{1} A_{3} A_{4} A_{5} }}^{2} + \chi_{{A_{2} A_{3} A_{4} A_{5} }}^{2} + \chi_{{A_{1} A_{2} A_{3} A_{4} A_{5} }}^{2} \\ \end{aligned}$$
