典型相关 Canonical correlation

关键词：典型相关Canonical correlation统计方法SAS代码

典型相关是一种用于检验两组变量间关系的多元分析技术。对于每组变量，典型相关创建潜在变量并检验其关系。它假定模型中所有的变量都是正态分布的连续变量。在SAS中，一组变量放在var语句内，另一组放在with语句内。两组变量数目无需相同。proc cancorr语句的all选项为许多研究者提供更多有益的输出结果。

(SAS代码)

proc cancorr data = "c:\mydata\hsb2" all;

  var read write;

  with math science;

run;

输出结果如下：



The CANCORR Procedure



VAR Variables              2

WITH Variables             2

Observations             200



              Means and Standard Deviations



                                Standard

Variable            Mean       Deviation    Label

read           52.230000       10.252937    reading score

write          52.775000        9.478586    writing score

math           52.645000        9.368448    math score

science        51.850000        9.900891    science score

Correlations Among the Original Variables



  Correlations Among the VAR Variables



                   read             write

read             1.0000            0.5968

write            0.5968            1.0000



   Correlations Among the WITH Variables



                     math           science

math               1.0000            0.6307

science            0.6307            1.0000



Correlations Between the VAR Variables and the WITH Variables



                   math           science

read             0.6623            0.6302

write            0.6174            0.5704

Canonical Correlation Analysis



                           Adjusted    Approximate        Squared

           Canonical      Canonical       Standard      Canonical

           Correlation    Correlation          Error    Correlation

       1    0.772841       0.771003       0.028548       0.597283

       2    0.023478        .                 0.070849       0.000551



                                                      Test of H0: The canonical correlations in

                                                               the current row and all

                   Eigenvalues of Inv(E)*H                       that follow are zero

                     = CanRsq/(1-CanRsq)

                                                     Likelihood Approximate

         Eigenvalue Difference Proportion Cumulative      Ratio     F Value Num DF Den DF Pr > F

            1     1.4831     1.4826   0.9996     0.9996 0.40249498       56.47      4         392     <.0001

            2     0.0006                 0.0004     1.0000  0.99944876        0.11      1          197    0.7420



                 Multivariate Statistics and F Approximations



                             S=2    M=-0.5    N=97



Statistic                        Value    F Value    Num DF    Den DF    Pr > F

Wilks' Lambda               0.40249498      56.47         4       392    <.0001

Pillai's Trace                   0.59783426      42.00         4       394    <.0001

Hotelling-Lawley Trace      1.48368501      72.58         4    234.16    <.0001

Roy's Greatest Root         1.48313347     146.09         2       197    <.0001



NOTE: F Statistic for Roy's Greatest Root is an upper bound.

NOTE: F Statistic for Wilks' Lambda is exact.



      Raw Canonical Coefficients for the VAR Variables



                                        V1                V2

read       reading score       0.063261313      0.1037907932

write      writing score      0.0492491834       -0.12190836



      Raw Canonical Coefficients for the WITH Variables



                                               W1                W2

math         math score         0.0669826768      -0.120142451

science      science score      0.0482406314      0.1208859811

Standardized Canonical Coefficients for the VAR Variables



                                            V1            V2

read       reading score        0.6486        1.0642

write      writing score        0.4668       -1.1555



Standardized Canonical Coefficients for the WITH Variables



                                              W1            W2

math         math score           0.6275       -1.1255

science      science score        0.4776        1.1969

Canonical Structure



Correlations Between the VAR Variables and Their Canonical Variables



                                            V1            V2

read       reading score        0.9272        0.3746

write      writing score        0.8539       -0.5205



Correlations Between the WITH Variables and Their Canonical Variables



                                      W1            W2

math         math score           0.9288       -0.3706

science      science score        0.8734        0.4870



Correlations Between the VAR Variables and the Canonical Variables of the WITH Variables



                                    W1            W2

read       reading score        0.7166        0.0088

write      writing score        0.6599       -0.0122



Correlations Between the WITH Variables and the Canonical Variables of the VAR Variables



                                             V1            V2

math         math score           0.7178       -0.0087

science      science score        0.6750        0.0114

Canonical Redundancy Analysis



                 Raw Variance of the VAR Variables Explained by

                     Their Own                               The Opposite

                Canonical Variables                       Canonical Variables

Canonical

 Variable                  Cumulative     Canonical                  Cumulative

   Number    Proportion    Proportion      R-Square    Proportion    Proportion

        1          0.7995          0.7995            0.5973          0.4775         0.4775

        2          0.2005          1.0000            0.0006          0.0001         0.4777





                Raw Variance of the WITH Variables Explained by

                     Their Own                               The Opposite

                Canonical Variables                       Canonical Variables

Canonical

 Variable                  Cumulative     Canonical                  Cumulative

   Number    Proportion    Proportion      R-Square    Proportion    Proportion

        1          0.8100          0.8100           0.5973          0.4838          0.4838

        2          0.1900          1.0000           0.0006          0.0001          0.4839



            Standardized Variance of the VAR Variables Explained by

                     Their Own                               The Opposite

                Canonical Variables                       Canonical Variables

Canonical

 Variable                  Cumulative     Canonical                  Cumulative

   Number    Proportion    Proportion      R-Square    Proportion    Proportion

        1          0.7944           0.7944           0.5973         0.4745           0.4745

        2          0.2056           1.0000           0.0006         0.0001           0.4746





            Standardized Variance of the WITH Variables Explained by

                     Their Own                               The Opposite

                Canonical Variables                       Canonical Variables

Canonical

 Variable                  Cumulative     Canonical                  Cumulative

   Number    Proportion    Proportion      R-Square    Proportion    Proportion

        1          0.8127          0.8127            0.5973        0.4854           0.4854

        2          0.1873          1.0000            0.0006        0.0001           0.4855



                  Squared Multiple Correlations Between the VAR Variables and

                     the First M Canonical Variables of the WITH Variables



M                                          1             2

read       reading score        0.5135        0.5136

write      writing score        0.4355        0.4356



                    Squared Multiple Correlations Between the WITH Variables

                    and the First M Canonical Variables of the VAR Variables



M                                            1             2

math         math score           0.5152        0.5153

science      science score        0.4557        0.4558

上述输出结果显示，线性组合与第一个典型相关一致。在输出结果的底部有两个典型相关。这些结果表明第一个典型相关为0.772841。输出结果中的F检验进行第一个典型相关等于0的假设检验。显然，F = 56.47是有统计学意义。然而，第二个典型相关0.0235则与0间的差异没有统计学意义(F = 0.11, p = 0.7420)。