Select the method or formula of your choice.

A pair is concordant if the observation ranking higher on variable X also ranks higher on variable Y. The pair is discordant if the observation ranking higher on X ranks lower on Y. The pair is tied if the subjects have the same classification on X and/or Y.

Term | Description |
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n_{ij} | observations in the cell corresponding to the i^{th} row and j^{th} column |

Goodman and Kruskal's gamma is a measure of association between the ordinal variables. Perfect association exists when |γ| = 1. If X and Y are independent, γ = 0.

Term | Description |
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C | number of concordant pairs = Σ_{i<k}Σ_{j<l} n_{ij} n_{kl} |

D | number of discordant pairs = Σ_{i<k}Σ_{j>l} n_{ij} n_{kl} |

n _{ij} | observations in the cell corresponding to i^{th} row and j^{th} column |

Somers' D measures the strength and direction of the relationship between two ordinal variables.

With Y as the response variable:

With X as the response variable:

Term | Description |
---|---|

T _{X} | number of pairs tied on X = |

T _{Y} | number of pairs tied on Y = |

C | number of concordant pairs |

D | number of discordant pairs |

n _{i+} | number of observations in the i^{th} row |

n _{+j} | number of observations in the j^{th} column |

n _{ij} | observations in the cell corresponding to the i^{th} row and j^{th} column |

n _{++} | total number of observations |

Kendall's tau-b, like gamma, measures the association between ordinal variables. One strength of Kendall's tau-b as a measure of association is that it accounts for tied pairs in its calculation. Gamma has a problem with tied pairs, so gamma almost always shows a higher association than tau-b. The tau-b values range from -1.0 to 1.0.

Term | Description |
---|---|

T _{X} | number of pairs tied on X = Σ_{i} n_{i+} (n_{i+}- 1) 0.5 |

T _{Y} | number of pairs tied on Y = Σ_{j} n_{+j} (n_{+j}- 1) 0.5 |

C | number of concordant pairs = Σ_{i<k}Σ_{j<l} n_{ij} n_{kl} |

D | number of discordant pairs = Σ_{i<k}Σ_{j>l} n_{ij} n_{kl} |

n _{i+} | number of observations in the i^{th} row |

n _{+j} | number of observations in the j^{th} column |

n _{ij} | observations in the cell corresponding to i^{th} row and j^{th} column |

n _{++} | total number of observations |

The test of concordance is a test of independence. The p-value is a probability that measures the evidence against the null hypothesis. Lower probabilities provide stronger evidence against the null hypothesis.

The test of concordance uses a standard normal test statistic:

The p-value equals the probability that Z is greater than Z*.

Term | Description |
---|---|

C | number of concordant pairs = Σ_{i<k}Σ_{j<l} n_{ij} n_{kl} |

D | number of discordant pairs = Σ_{i<k}Σ_{j>l} n_{ij} n_{kl} |

SE(C-D) | Standard error or C-D, which approximately equals |

n_{++} | the total number of observations |

n_{i+} | the number of observations in the i^{th} row. |

n_{+j} | the number of observations in the j^{th} column. |