相关系数 - zgw21cn
积差相关也称积矩相关，是英国统计学家皮尔逊于20世纪提出的一种计算直线相关的方法，因而又称为皮尔逊相关。其计算公式为
若|r|越接近于0，则表明x与y之间呈直线关系的密切程度较低；若｜r|越接近于１，则表明x与y之间呈直线关系的密切程度越高。
积差相关适应于
（１）两个变量之间是线性关系，都是连续数据。
（２）两个变量的总体是正态分布，或接近正态的单峰分布。
（３）两个变量的观测值是成对的，每对观测值之间相互独立。
２. Spearman等级相关
斯皮尔曼等级相关对数据条件的要求没有积差相关系数严格，只要两个变量的观测值是成对的等级评定资料，或者是由连续变量观测资料转化得到的等级资料，不论两个变量的总体分布形态、样本容量的大小如何，都可以用斯皮尔曼等级相关来进行研究。
计算等级相关系数可以将数据变换成等级以后用Pearson相关系数公式计算，也可以计算出每一对样本的等级之差d,然后计算
其中n表示样本容量，即数据的对数。
３. Kendall等级相关系数。
适应于两个分类变量均为有序分类的情况，对相关的有序变量进行非参数相关检验。相关系数定义如下
其中n为样本容量，p表示在所有的项中，排在本项之后的项数之和。
例如，有某人Ａ在8人组中，身高排第１，体重排第３。数据形式如下。
| Person | A | B | C | D | E | F | G | H |
| Rank by Height | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
| Rank by Weight | 3 | 4 | 1 | 2 | 5 | 7 | 8 | 6 |
注意到Ａ的体重排3,在它右边有4,1,2,5,7,8,6.那么排序在它右边的有4,5,7,8,6.这样的话，排序在它右边的有5个数。同理，对Ｂ，排序在它右边的数有4个。所有这些项的和加起来，即p=5 + 4 + 5 + 4 + 3 + 1 + 0 + 0 = 22.最终，Ｔ=88/56-1=0.57.附ＳＡＳ代码。
proc contents data=InData /*Dataset*/
out=InVar(keep=varnum name)
noprint;
run;
proc sql noprint;
select distinct name
into :VarName separated by \’ \’
from InVar
order by varnum;
run; %put &VarName;
ods listing close;
ods output spearmancorr=spearman
hoeffdingcorr=hoeffding;
proc Corr data
=InData spearman hoeffding rank;var &VarName;
with Flg;
run;
ods listing;
data spearman1(keep
=variable scorr spvalue ranksp);length variable $ 16;
set spearman;
array best(6) best1—best6;/*6 is number of variable*/
array r(6) r1—r6;
array p(6) p1—p6;
do i=1 to 6;
variable=best(i);
scorr=r(i);
spvalue=p(i);
ranksp=i;
output;
end;
run;
data hoeffding1(keep=variable hcorr hpvalue rankho);
length variable $ 16;
set hoeffding;
array best(6) best1—best6;;/*6 is number of variable*/
array r(
6) r1—r6;array p(6) p1—p6;
do i=1 to 6;
variable=best(i);
hcorr=r(i);
hpvalue=p(i);
rankho=i;
output;
end;
run;
proc sort data
=spearman1;by variable;
run;
proc sort data
=hoeffding1;by variable;
run;
data correlations;
merge spearman1 hoeffding1;
by variable;
run;
proc sort data
=correlations;by ranksp;
run;
proc print data
=correlations label split=\’*\’;var variable ranksp rankho scorr spvalue hcorr hpvalue;
label ranksp=\’Spearman rank*of variable\’
scorr=\’Spearman Correlation\’
spvalue=\’Spearman p-value\’
rankho=\’Hoeffding rank*of variable\’
hcorr=\’Hoeffding Correlation\’
hpvalue=\’Hoeffding p-value\’;
run;
proc contents data=InData(drop=Flg ID_CLIENT ID_Shop Sex Marital_Status Flg_Phone Area_Code_Phone Payment_Day Shop_Rank
Residence_Type Flg_M Flg_F Flg_Rw Flg_SW Jobcode Flg_RP)
out=InVar(keep=varnum name)
noprint;
run;
proc sql noprint;
select distinct name
into :VarName separated by \’ \’
from InVar
order by varnum;
run;
%put &VarName;
ods listing close;
ods output spearmancorr=spearman
hoeffdingcorr=hoeffding;
proc Corr data=InData spearman hoeffding rank;
var &VarName;
with Flg;
run;
ods listing;
data spearman1(keep=variable scorr spvalue ranksp);
length variable $ 16;
set spearman;
array best(6) best1—best6;
array r(6) r1—r6;
array p(6) p1—p6;
do i=1 to 6;
variable=best(i);
scorr=r(i);
spvalue=p(i);
ranksp=i;
output;
end;
run;
data hoeffding1(keep=variable hcorr hpvalue rankho);
length variable $ 16;
set hoeffding;
array best(6) best1—best6;
array r(6) r1—r6;
array p(6) p1—p6;
do i=1 to 6;
variable=best(i);
hcorr=r(i);
hpvalue=p(i);
rankho=i;
output;
end;
run;
proc sort data=spearman1;
by variable;
run;
proc sort data=hoeffding1;
by variable;
run;
data correlations;
merge spearman1 hoeffding1;
by variable;
run;
proc sort data=correlations;
by ranksp;
run;
proc print data=correlations label split=\’*\’;
var variable ranksp rankho scorr spvalue hcorr hpvalue;
label ranksp=\’Spearman rank*of variable\’
scorr=\’Spearman Correlation\’
spvalue=\’Spearman p-value\’
rankho=\’Hoeffding rank*of variable\’
hcorr=\’Hoeffding Correlation\’
hpvalue=\’Hoeffding p-value\’;
run;