BUILT-IN MATHEMATICA SYMBOL

# GoodmanKruskalGamma

GoodmanKruskalGamma[v1, v2]
gives the Goodman-Kruskal coefficient for the vectors and .

gives the Goodman-Kruskal coefficients for the matrix m.

GoodmanKruskalGamma[m1, m2]
gives the Goodman-Kruskal coefficients for the matrices and .

GoodmanKruskalGamma[dist]
gives the coefficient matrix for the multivariate symbolic distribution dist.

GoodmanKruskalGamma[dist, i, j]
gives the coefficient for the multivariate symbolic distribution dist.

## DetailsDetails

• GoodmanKruskalGamma[v1, v2] gives the Goodman-Kruskal coefficient between and .
• Goodman-Kruskal is a measure of monotonic association based on the relative order of consecutive elements in the two lists.
• Goodman-Kruskal between and is given by , where is the number of concordant pairs of observations and is the number of discordant pairs.
• A concordant pair of observations and is one such that both and or both and . A discordant pair of observations is one such that and or and .
• If no ties are present, is equivalent to KendallTau.
• The arguments and can be any real-valued vectors of equal length.
• For a matrix m with columns, is a × matrix of the -coefficients between columns of m.
• For an × matrix and an × matrix , GoodmanKruskalGamma[m1, m2] is a × matrix of the -coefficients between columns of and columns of .
• GoodmanKruskalGamma[dist, i, j] gives where is equal to Probability[(x1-x2)(y1-y2)>0, {{x1, y1}disti, j, {x2, y2}disti, j}] and is equal to Probability[(x1-x2)(y1-y2)<0, {{x1, y1}disti, j, {x2, y2}disti, j}] where is the marginal of dist.
• GoodmanKruskalGamma[dist] gives a matrix where the entry is given by GoodmanKruskalGamma[dist, i, j].

## ExamplesExamplesopen allclose all

### Basic Examples (4)Basic Examples (4)

Goodman-Kruskal for two vectors:

 Out[2]=

Goodman-Kruskal for a matrix:

 Out[2]//MatrixForm=

Goodman-Kruskal for two matrices:

 Out[3]//MatrixForm=

Compute the Goodman-Kruskal for a bivariate distribution:

 Out[2]//MatrixForm=

Compare to a simulated value:

 Out[3]//MatrixForm=