dfba_gamma {DFBA} | R Documentation |

## Goodman-Kruskal Gamma

### Description

Given bivariate data in the form of either a rank-ordered table or a matrix,
returns the number of concordant and discordant changes between the variates,
the Goodman-Kruskal gamma statistic, and a Bayesian analysis of the
population concordance proportion parameter *phi*.

### Usage

```
dfba_gamma(x, a0 = 1, b0 = 1, prob_interval = 0.95)
```

### Arguments

`x` |
Cross-tabulated matrix or table where cell [I, J] represents the frequency of observations where the rank of measure 1 is I and the rank of measure 2 is J. |

`a0` |
First shape parameter for the prior beta distribution (default is 1) |

`b0` |
Second shape parameter for the prior beta distribution (default is 1) |

`prob_interval` |
Desired width for interval estimates (default is 0.95) |

### Details

For bivariate data where two measures are restricted on an ordinal scale,
such as when the two variates are ranked data over a limited set of integers,
then an ordered contingency table is often a convenient data representation.
For such a case the element in the `[I, J]`

cell of the matrix is the
frequency of occasions where one variate has a rank value of `I`

and the
corresponding rank for the other variate is `J`

. This situation is a
special case of the more general case where there are two continuous
bivariate measures. For the special case of a rank-order matrix with
frequencies, there is a distribution-free concordance correlation that is in
common usage: Goodman and Kruskal's gamma `G`

(Siegel & Castellan, 1988).

Chechile (2020) showed that Goodman and Kruskal's gamma is equivalent to the
more general `\tau_A`

nonparametric correlation coefficient.
Historically, gamma was considered a different metric from `\tau`

because
typically the version of `\tau`

in standard use was `\tau_B`

, which
is a flawed metric because it does not properly correct for ties. Note:
`cor(... ,method = "kendall")`

returns the `\tau_B`

correlation, which
is incorrect when there are ties. The correct `\tau_A`

is computed by the
`dfba_bivariate_concordance()`

function.

The gamma statistic is equal to `(n_c-n_d)/(n_c+n_d)`

, where `n_c`

is
the number of occasions when the variates change in a concordant way and `n_d`

is the number of occasions when the variates change in a discordant fashion.
The value of `n_c`

for an order matrix is the sum of terms for each `[I, J]`

that are equal to `n_{ij}N^{+}_{ij}`

, where `n_{ij}`

is the frequency
for cell `[I, J]`

and `N^{+}_{ij}`

is the sum of a frequencies in the
matrix where the row value is greater than `I`

and where the column value is
greater than `J`

. The value `n_d`

is the sum of terms for each `[I, J]`

that
are `n_{ij}N^{-}_{ij}`

, where `N^{-}_{ij}`

is the sum of the frequencies
in the matrix where row value is greater than `I`

and the column value is
less than `J`

. The `n_c`

and `n_d`

values computed in this fashion
are, respectively, equal to `n_c`

and `n_d`

values found when the bivariate
measures are entered as paired vectors into the `dfba_bivariate_concordance()`

function.

As with the `dfba_bivariate_concordance()`

function, the Bayesian analysis focuses on the
population concordance proportion phi `(\phi)`

; and `G=2\phi-1`

. The
likelihood function is proportional to `\phi^{n_c}(1-\phi)^{n_d}`

. The
prior distribution is a beta function, and the posterior distribution is the
conjugate beta where `a = a0 + nc`

and
`b = b0 + nd`

.

### Value

A list containing the following components:

`gamma` |
Sample Goodman-Kruskal gamma statistic; equivalent to the sample rank correlation coefficient tau_A |

`a0` |
First shape parameter for prior beta |

`b0` |
Second shape parameter for prior beta |

`sample_p` |
Sample estimate for proportion concordance |

`nc` |
Number of concordant comparisons between the paired measures |

`nd` |
Number of discordant comparisons between the paired measures |

`a_post` |
First shape parameter for the posterior beta distribution for the phi parameter |

`b_post` |
Second shape parameter for the posterior beta distribution for the phi parameter |

`post_median` |
Median of the posterior distribution for the phi concordance parameter |

`prob_interval` |
The probability of the interval estimate for the phi parameter |

`eti_lower` |
Lower limit of the posterior equal-tail interval for the phi parameter where the width of the interval is specified by the |

`eti_upper` |
Upper limit of the posterior equal-tail interval for the phi parameter where the width of the interval is specified by the |

### References

Chechile, R.A. (2020). Bayesian Statistics for Experimental Scientists: A General Introduction Using Distribution-Free Methods. Cambridge: MIT Press.

Siegel, S., & Castellan, N. J. (1988) Nonparametric Statistics for the Behavioral Sciences. New York: McGraw Hill.

### See Also

`dfba_bivariate_concordance`

for a more extensive discussion about the `\tau_A`

statistic and the flawed `\tau_B`

correlation

### Examples

```
# Example with matrix input
N <- matrix(c(38, 4, 5, 0, 6, 40, 1, 2, 4, 8, 20, 30),
ncol = 4,
byrow = TRUE)
colnames(N) <- c('C1', 'C2', 'C3', 'C4')
rownames(N) <- c('R1', 'R2', 'R3')
dfba_gamma(N)
# Sample problem with table input
NTable <- as.table(N)
dfba_gamma(NTable)
```

*DFBA*version 0.1.0 Index]