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Kendall tau是用来度量关联关系的。

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

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

The Kendall τ coefficient is defined as:

\tau = \frac{(\text{number of concordant pairs}) - (\text{number of discordant pairs})}{\frac{1}{2} n (n-1) } .

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同一篇文章继续引用关于ties:

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A pair {(xi, yi), (xj, yj)} is said to be tied if xi = xj or yi = yj; 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:

\tau_B = \frac{n_c-n_d}{\sqrt{(n_0-n_1)(n_0-n_2)}}

where

\begin{array}{ccl}
n_0 & = & n(n-1)/2\\
n_1 & = & \sum_i t_i (t_i-1)/2 \\
n_2 & = & \sum_j u_j (u_j-1)/2 \\
t_i & = & \mbox{Number of tied values in the } i^{th} \mbox{ group of ties for the first quantity} \\
u_j & = & \mbox{Number of tied values in the } j^{th} \mbox{ group of ties for the second quantity}
\end{array}

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

在sas中计算Kendall tau-2比较简单,直接用proc freq就行,原来proc freq如此强大啊。

sas程序举例:

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有点儿关联的是Somer’s D,但是搜索了一下没看到公式,反正Somer’s D也可以用sas proc freq直接算,方法类似。

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.

posted on 2011-08-28 15:11 人在江湖 阅读(670) 评论(0)  编辑  收藏 所属分类: BI

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