Chapter 07-Basic statistics(Part3 correlations)

这一部分使用R基础已安装包中的state.x77数据集。该数据集的数据是关于美国50个州在1977年对人口，收入，文盲率，平均寿命，谋杀率，高中毕业率统计所得。

1.关联的种类(types of correlations)

(1)PEARSON,SPEARMAN,KENDALL CORRELATIONS

·Pearson：评估两个数值变量间的线性关系的程度的暂时性关联；

·Spearman’s Rank Order：评估两个有排序关系的变量的相关率；

·Kendall's Tau：是非参数参与的排序关联。

cor(x,use=,method=)

其中，缺省时，use="everything",method="pearson"

例01：

> states<-state.x77[,1:6]
> cov(states)
              Population      Income   Illiteracy     Life Exp      Murder      HS Grad
Population 19931683.7588 571229.7796  292.8679592 -407.8424612 5663.523714 -3551.509551
Income       571229.7796 377573.3061 -163.7020408  280.6631837 -521.894286  3076.768980
Illiteracy      292.8680   -163.7020    0.3715306   -0.4815122    1.581776    -3.235469
Life Exp       -407.8425    280.6632   -0.4815122    1.8020204   -3.869480     6.312685
Murder         5663.5237   -521.8943    1.5817755   -3.8694804   13.627465   -14.549616
HS Grad       -3551.5096   3076.7690   -3.2354694    6.3126849  -14.549616    65.237894

> 
> cor(states)
            Population     Income Illiteracy    Life Exp     Murder     HS Grad
Population  1.00000000  0.2082276  0.1076224 -0.06805195  0.3436428 -0.09848975
Income      0.20822756  1.0000000 -0.4370752  0.34025534 -0.2300776  0.61993232
Illiteracy  0.10762237 -0.4370752  1.0000000 -0.58847793  0.7029752 -0.65718861
Life Exp   -0.06805195  0.3402553 -0.5884779  1.00000000 -0.7808458  0.58221620
Murder      0.34364275 -0.2300776  0.7029752 -0.78084575  1.0000000 -0.48797102
HS Grad    -0.09848975  0.6199323 -0.6571886  0.58221620 -0.4879710  1.00000000
> cor(states,method="spearman")
           Population     Income Illiteracy   Life Exp     Murder    HS Grad
Population  1.0000000  0.1246098  0.3130496 -0.1040171  0.3457401 -0.3833649
Income      0.1246098  1.0000000 -0.3145948  0.3241050 -0.2174623  0.5104809
Illiteracy  0.3130496 -0.3145948  1.0000000 -0.5553735  0.6723592 -0.6545396
Life Exp   -0.1040171  0.3241050 -0.5553735  1.0000000 -0.7802406  0.5239410
Murder      0.3457401 -0.2174623  0.6723592 -0.7802406  1.0000000 -0.4367330
HS Grad    -0.3833649  0.5104809 -0.6545396  0.5239410 -0.4367330  1.0000000

·cov()函数：产生了variances & covariances；

·cor()函数：提供了Pearson Product Moment correlation coefficients；

·cor( ,method="spearman")：提供了Spearman Rank Order correlation coefficients。

例02：

> x<-states[,c("Population","Income","Illiteracy","HS Grad")]
> y<-states[,c("Life Exp","Murder")]
> cor(x,y)
              Life Exp     Murder
Population -0.06805195  0.3436428
Income      0.34025534 -0.2300776
Illiteracy -0.58847793  0.7029752
HS Grad     0.58221620 -0.4879710

用上面的方法可以产生非方矩阵(nonsquare matrices)。

注意，结果不能表明是否the correlations differ significantly from 0，故需要tests of significance.

(2)部分关联(PARTIAL CORRELATIONS)

partial correlation: a correlation between two quantitative variables,controlling for one or more other quantitative variables.

ggm包中的pcor()函数：提供partial correlation coefficients。

pcor(u,S)

·u: 数据的向量，前两个数标记需要关联的数，剩余的数标记其他conditioning variables，即variables being partialed out;

·S: covariance matrix among the variables。

例03：

> library(ggm)
> # partial correlation of population and murder rate, controlling
> # for income, illiteracy rate, and HS graduation rate
> pcor(c(1,5,2,3,6), cov(states))
[1] 0.346

注释：0.346是指人口与谋杀率之间的correlation，并且要控制收入，文盲率，高中毕业率的影响。

(3)OTHER TYPES OF CORRELATIONS

polycor()包中的函数hetcor()函数：计算一个heterogeneous correlation matrix，

在数值变量(numerical variables)间包含Pearson product-moment correlations，

在数值变量与常量(ordinal variables)间有polyserial correlations，

在常量间有polychoric correlations，

在two dichotomous variables间有tetrachoric correlations。

2. 测试关联的重要值(Testing correlations for significance)

cor.test()函数：测试一个个体的Pearson,Spearman,Kendall correlation coefficient.

cor.test(x,y,alternative=,method=)

·x与y：是要被关联的变量；

·alternative：特指two-tailed,one-tailed test, ("two.side"，"less"，"greater")；

alternative="less"：研究的假设是人口关联小于0；

alternative="greater"：研究的假设是人口关联大于0；

缺省状态是，alternative="two side".

·method：特指关联的方式，(“pearson”，“kendall”，“spearman”)。

例04：

> states<-state.x77[,1:6]
> cor(states)
            Population     Income Illiteracy    Life Exp     Murder     HS Grad
Population  1.00000000  0.2082276  0.1076224 -0.06805195  0.3436428 -0.09848975
Income      0.20822756  1.0000000 -0.4370752  0.34025534 -0.2300776  0.61993232
Illiteracy  0.10762237 -0.4370752  1.0000000 -0.58847793  0.7029752 -0.65718861
Life Exp   -0.06805195  0.3402553 -0.5884779  1.00000000 -0.7808458  0.58221620
Murder      0.34364275 -0.2300776  0.7029752 -0.78084575  1.0000000 -0.48797102
HS Grad    -0.09848975  0.6199323 -0.6571886  0.58221620 -0.4879710  1.00000000
> cor.test(states[,3],states[,5])

	Pearson's product-moment correlation

data:  states[, 3] and states[, 5]
t = 6.8479, df = 48, p-value = 1.258e-08
alternative hypothesis: true correlation is not equal to 0
95 percent confidence interval:
 0.5279280 0.8207295
sample estimates:
      cor
0.7029752 

注意：cor.test()函数一次只能测试一组关联。

corr.test()函数(psych包中)为pearson，spearman，kendall关联的矩阵提供关联与重要值。

例05：

> library(psych)
> corr.test(states,use="complete")
Call:corr.test(x = states, use = "complete")
Correlation matrix
           Population Income Illiteracy Life Exp Murder HS Grad
Population       1.00   0.21       0.11    -0.07   0.34   -0.10
Income           0.21   1.00      -0.44     0.34  -0.23    0.62
Illiteracy       0.11  -0.44       1.00    -0.59   0.70   -0.66
Life Exp        -0.07   0.34      -0.59     1.00  -0.78    0.58
Murder           0.34  -0.23       0.70    -0.78   1.00   -0.49
HS Grad         -0.10   0.62      -0.66     0.58  -0.49    1.00
Sample Size
[1] 50
Probability values (Entries above the diagonal are adjusted for multiple tests.)
           Population Income Illiteracy Life Exp Murder HS Grad
Population       0.00   0.59       1.00      1.0   0.10       1
Income           0.15   0.00       0.01      0.1   0.54       0
Illiteracy       0.46   0.00       0.00      0.0   0.00       0
Life Exp         0.64   0.02       0.00      0.0   0.00       0
Murder           0.01   0.11       0.00      0.0   0.00       0
HS Grad          0.50   0.00       0.00      0.0   0.00       0

注意：corr.test()函数中：

·use="pairwise"或"complete", pairwise deletion of missing values respectively;

·method="pearson"(缺省), "spearman", "kendall".

psych包中的pcor.test()函数：用来测试conditional independence of two variables controlling for one or more additional variables,assuming multivariate normality.

pcor.test(r,q,n)

r:pcor()函数产生的partial correlation；

q:被控制的变量数；

n:样本大小。

psych包中的r.test()函数：提供许多significance的测试。