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Given:

M =
_{g*} | the mean of any particular one of the the individual groups of measures | |

M =
_{r*} | the mean of the row to which that group belongs | |

M =
_{c*} | the mean of the column to which that group belongs | |

M =
_{T} | the mean of the total array of data |

If there is zero interaction between the row and column variables, then the difference between the mean of any particular group and the mean of the total array of data

M—_{g*}M
_{T} |

M—_{r*}M
_{T} | the difference between the mean of the row to which that group belongs and the mean of the total array of data | |

and^{T}_{T}
| ||

M—_{c*}M
_{T} | the difference between the mean of the column to which that group belongs and the mean of the total array of data |

Thus on the null hypothesis expectation of zero interaction

[null]M—_{g*}M
_{T} | = (M—_{r*}M)+(_{T}M—_{c*}M)
_{T} | |

[null]M—_{g*}M
_{T} | = M+_{r*}M—2_{c*}M
_{T} | |

[null]M
_{g*} | = M+_{r*}M—_{c*}M
_{T} |

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