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Kendall tau是用來度量關聯關系的。

(引自wikipedia:http://en.wikipedia.org/wiki/Kendall_tau_rank_correlation_coefficient)

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Let (x₁, y₁), (x₂, y₂), …, (x_n, y_n) be a set of joint observations from two random variables X and Y respectively, such that all the values of (x_i) and (y_i) are unique. Any pair of observations (x_i, y_i) and (x_j, y_j) are said to be concordant if the ranks for both elements agree: that is, if both x_i > x_j and y_i > y_j or if both x_i < x_j and y_i < y_j. They are said to be discordant, if x_i > x_j and y_i < y_j or if x_i < x_j and y_i > y_j. If x_i = x_j or y_i = y_j, the pair is neither concordant nor discordant.

The Kendall τ coefficient is defined as:

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同一篇文章繼續引用關于ties:

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A pair {(x_i, y_i), (x_j, y_j)} is said to be tied if x_i = x_j or y_i = y_j; a tied pair is neither concordant nor discordant. When tied pairs arise in the data, the coefficient may be modified in a number of ways to keep it in the range [-1, 1]:

Tau-b statistic, unlike tau-a, makes adjustments for ties and is suitable for square tables. Values of tau-b range from ?1 (100% negative association, or perfect inversion) to +1 (100% positive association, or perfect agreement). A value of zero indicates the absence of association.

The Kendall tau-b coefficient is defined as:

where

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靠，搞了半天才理解，上面公式中所謂nc, nd里面的c和d，指的是concordant和discordant.

在sas中計算Kendall tau-2比較簡單，直接用proc freq就行，原來proc freq如此強大啊。

sas程序舉例：

data color;
   input Region Eyes $ Hair $ Count @@;
   label Eyes  ='Eye Color'
         Hair  ='Hair Color'
         Region='Geographic Region';
   datalines;
1 blue  fair   23  1 blue  red     7  1 blue  medium 24
1 blue  dark   11  1 green fair   19  1 green red     7
1 green medium 18  1 green dark   14  1 brown fair   34
1 brown red     5  1 brown medium 41  1 brown dark   40
1 brown black   3  2 blue  fair   46  2 blue  red    21
2 blue  medium 44  2 blue  dark   40  2 blue  black   6
2 green fair   50  2 green red    31  2 green medium 37
2 green dark   23  2 brown fair   56  2 brown red    42
2 brown medium 53  2 brown dark   54  2 brown black  13
;

proc freq data = color noprint ;                                                                                             
tables  eyes*hair / measures  noprint ;                                                                                   
weight count;                                                                                                     
output out=output KENTB;                                                                                          
test KENTB;                                                                                                            
run;

另外跟Kendall tau有點兒關聯的是Somer’s D，但是搜索了一下沒看到公式，反正Somer’s D也可以用sas proc freq直接算，方法類似。

Somers' D(C|R) and Somers' D(R|C) are asymmetric modifications of tau-b.Somers' D differs from tau-b in that it uses a correction only for pairs that are tied on the independent variable.