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XLeratorDB/statistics Documentation

SQL Server Kendall's Tau function


KENDALLT_TV

Updated: 24 May 2013


Use the table-valued function KENDALLT_TV to calculate Kendall’s tau (t), a non-parametric measure of association based on the number of concordances and discordances in paired observations. Concordance occurs when paired observations vary together and discordance occurs when paired observations vary differently. The equation for Kendall’s tau is:

XLeratorDB formaula for KENDALLT function for SQL Server
Where
C is the number of concordant pairs
D is the number of discordant pairs
n is the number of pairs
T is the number of x ties
U is the number of y ties
 
The function also calculates


Syntax
SELECT * FROM [wctStatistics].[wct].[KENDALLT_TV](
   <@x_y_Query, nvarchar(max),>)
Arguments
@InputData_RangeQuery
a T-SQL statement, as a string, that specifies the subject, rater, and rating values.
Return Type
RETURNS TABLE (
      [Tau_b] [float] NULL,
      [Zb] [float] NULL,
      [Pb] [float] NULL,
      [SDb] [float] NULL,
      [Tau_a] [float] NULL,
      [Za] [float] NULL,
      [Pa] [float] NULL,
      [SDa] [float] NULL,
      [C] [float] NULL,
      [D] [float] NULL,
      [S] [float] NULL,
      [T] [float] NULL,
      [U] [float] NULL,
      [N] [float] NULL
)

Column
Column Description
Tau_b
the Kendall tau b statistic
Zb
the z-value for tau b
Pb
the p-value for tau b
SDb
the standard deviation for tau b
Tau_a
the Kendall tau a statistic
Za
the z-value for tau a
Pa
the p-value for tau a
SDa
the standard deviation for tau a
C
the number of concordant pairs
D
the number of discordant pairs
S
C – D
T
the number of x ties
U
the number of y ties
N
the number of pairs
Remarks
·         The function is insensitive to order
·         If x is NULL or y is NULL the pair is not included in the calculations
Examples
SELECT *
INTO #k
FROM (
      SELECT 2.5,1 UNION ALL
      SELECT 2.5,1 UNION ALL
      SELECT 2.5,1 UNION ALL
      SELECT 2.5,1 UNION ALL
      SELECT 5,2 UNION ALL
      SELECT 6.5,1 UNION ALL
      SELECT 6.5,1 UNION ALL
      SELECT 10,2 UNION ALL
      SELECT 10,1 UNION ALL
      SELECT 10,1 UNION ALL
      SELECT 10,1 UNION ALL
      SELECT 10,1 UNION ALL
      SELECT 14,1 UNION ALL
      SELECT 14,1 UNION ALL
      SELECT 14,2 UNION ALL
      SELECT 16,2 UNION ALL
      SELECT 17,2
      ) n(x,y)
 
SELECT p.*
FROM wct.KENDALLT_TV('SELECT x,y FROM #k') k
--This CROSS APPLY UNPIVOTS the tvf columns for formatting
CROSS APPLY(
      SELECT 'tau_a', tau_a UNION ALL
      SELECT 'tau_b', tau_b UNION ALL
      SELECT 'C', C UNION ALL
      SELECT 'D', D UNION ALL
      SELECT 'S', S UNION ALL
      SELECT 'T', T UNION ALL
      SELECT 'U', U UNION ALL
      SELECT 'za', za UNION ALL
      SELECT 'zb', zb UNION ALL
      SELECT 'SDa', SDa UNION ALL
      SELECT 'SDb', SDb UNION ALL
      SELECT 'pa', pa UNION ALL
      SELECT 'pb', pb UNION ALL
      SELECT 'N', N
      ) p(stat, value_stat)
DROP TABLE #k
This produces the following result.



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