Correlation Analysis

Motor Trend Car Road Tests

head(mtcars)

cor(mtcars)

            mpg        cyl       disp         hp        drat         wt        qsec         vs          am       gear
mpg   1.0000000 -0.8521620 -0.8475514 -0.7761684  0.68117191 -0.8676594  0.41868403  0.6640389  0.59983243  0.4802848
cyl  -0.8521620  1.0000000  0.9020329  0.8324475 -0.69993811  0.7824958 -0.59124207 -0.8108118 -0.52260705 -0.4926866
disp -0.8475514  0.9020329  1.0000000  0.7909486 -0.71021393  0.8879799 -0.43369788 -0.7104159 -0.59122704 -0.5555692
hp   -0.7761684  0.8324475  0.7909486  1.0000000 -0.44875912  0.6587479 -0.70822339 -0.7230967 -0.24320426 -0.1257043
drat  0.6811719 -0.6999381 -0.7102139 -0.4487591  1.00000000 -0.7124406  0.09120476  0.4402785  0.71271113  0.6996101
wt   -0.8676594  0.7824958  0.8879799  0.6587479 -0.71244065  1.0000000 -0.17471588 -0.5549157 -0.69249526 -0.5832870
qsec  0.4186840 -0.5912421 -0.4336979 -0.7082234  0.09120476 -0.1747159  1.00000000  0.7445354 -0.22986086 -0.2126822
vs    0.6640389 -0.8108118 -0.7104159 -0.7230967  0.44027846 -0.5549157  0.74453544  1.0000000  0.16834512  0.2060233
am    0.5998324 -0.5226070 -0.5912270 -0.2432043  0.71271113 -0.6924953 -0.22986086  0.1683451  1.00000000  0.7940588
gear  0.4802848 -0.4926866 -0.5555692 -0.1257043  0.69961013 -0.5832870 -0.21268223  0.2060233  0.79405876  1.0000000
carb -0.5509251  0.5269883  0.3949769  0.7498125 -0.09078980  0.4276059 -0.65624923 -0.5696071  0.05753435  0.2740728
            carb
mpg  -0.55092507
cyl   0.52698829
disp  0.39497686
hp    0.74981247
drat -0.09078980
wt    0.42760594
qsec -0.65624923
vs   -0.56960714
am    0.05753435
gear  0.27407284
carb  1.00000000

library(corrplot)

corrplot 0.92 loaded

corrplot(cor(mtcars), method="color")

library(corrplot)
corrplot(cor(mtcars), method="number")

library(corrplot)
corrplot(cor(mtcars), method="pie")

library(corrplot)
corrplot(cor(mtcars), method="circle")

corrplot.mixed(cor(mtcars))

corrplot(cor(mtcars), order="hclust")

mtcarsCorMtest <- cor.mtest(mtcars, conf.level = .95)
mtcarsCorMtest

$p
              mpg          cyl         disp           hp         drat           wt         qsec           vs           am
mpg  0.000000e+00 6.112687e-10 9.380327e-10 1.787835e-07 1.776240e-05 1.293959e-10 1.708199e-02 3.415937e-05 2.850207e-04
cyl  6.112687e-10 0.000000e+00 1.802838e-12 3.477861e-09 8.244636e-06 1.217567e-07 3.660533e-04 1.843018e-08 2.151207e-03
disp 9.380327e-10 1.802838e-12 0.000000e+00 7.142679e-08 5.282022e-06 1.222320e-11 1.314404e-02 5.235012e-06 3.662114e-04
hp   1.787835e-07 3.477861e-09 7.142679e-08 0.000000e+00 9.988772e-03 4.145827e-05 5.766253e-06 2.940896e-06 1.798309e-01
drat 1.776240e-05 8.244636e-06 5.282022e-06 9.988772e-03 0.000000e+00 4.784260e-06 6.195826e-01 1.167553e-02 4.726790e-06
wt   1.293959e-10 1.217567e-07 1.222320e-11 4.145827e-05 4.784260e-06 0.000000e+00 3.388683e-01 9.798492e-04 1.125440e-05
qsec 1.708199e-02 3.660533e-04 1.314404e-02 5.766253e-06 6.195826e-01 3.388683e-01 0.000000e+00 1.029669e-06 2.056621e-01
vs   3.415937e-05 1.843018e-08 5.235012e-06 2.940896e-06 1.167553e-02 9.798492e-04 1.029669e-06 0.000000e+00 3.570439e-01
am   2.850207e-04 2.151207e-03 3.662114e-04 1.798309e-01 4.726790e-06 1.125440e-05 2.056621e-01 3.570439e-01 0.000000e+00
gear 5.400948e-03 4.173297e-03 9.635921e-04 4.930119e-01 8.360110e-06 4.586601e-04 2.425344e-01 2.579439e-01 5.834043e-08
carb 1.084446e-03 1.942340e-03 2.526789e-02 7.827810e-07 6.211834e-01 1.463861e-02 4.536949e-05 6.670496e-04 7.544526e-01
             gear         carb
mpg  5.400948e-03 1.084446e-03
cyl  4.173297e-03 1.942340e-03
disp 9.635921e-04 2.526789e-02
hp   4.930119e-01 7.827810e-07
drat 8.360110e-06 6.211834e-01
wt   4.586601e-04 1.463861e-02
qsec 2.425344e-01 4.536949e-05
vs   2.579439e-01 6.670496e-04
am   5.834043e-08 7.544526e-01
gear 0.000000e+00 1.290291e-01
carb 1.290291e-01 0.000000e+00

$lowCI
             mpg        cyl        disp         hp       drat          wt        qsec         vs         am        gear
mpg   1.00000000 -0.9257694 -0.92335937 -0.8852686  0.4360484 -0.93382641  0.08195487  0.4103630  0.3175583  0.15806177
cyl  -0.92576936  1.0000000  0.80724418  0.6816016 -0.8429083  0.59657947 -0.77927809 -0.9039393 -0.7369979 -0.71802597
disp -0.92335937  0.8072442  1.00000000  0.6106794 -0.8487237  0.78115863 -0.67961513 -0.8488377 -0.7792690 -0.75751468
hp   -0.88526861  0.6816016  0.61067938  1.0000000 -0.6895522  0.40251134 -0.84759984 -0.8559675 -0.5456270 -0.45447743
drat  0.43604838 -0.8429083 -0.84872374 -0.6895522  1.0000000 -0.84997951 -0.26594700  0.1081948  0.4843991  0.46414402
wt   -0.93382641  0.5965795  0.78115863  0.4025113 -0.8499795  1.00000000 -0.49335358 -0.7571117 -0.8386752 -0.77446381
qsec  0.08195487 -0.7792781 -0.67961513 -0.8475998 -0.2659470 -0.49335358  1.00000000  0.5346428 -0.5356240 -0.52261830
vs    0.41036301 -0.9039393 -0.84883771 -0.8559675  0.1081948 -0.75711174  0.53464277  1.0000000 -0.1915957 -0.15371324
am    0.31755830 -0.7369979 -0.77926901 -0.5456270  0.4843991 -0.83867523 -0.53562398 -0.1915957  1.0000000  0.61589632
gear  0.15806177 -0.7180260 -0.75751468 -0.4544774  0.4641440 -0.77446381 -0.52261830 -0.1537132  0.6158963  1.00000000
carb -0.75464796  0.2184331  0.05367539  0.5431200 -0.4259976  0.09273981 -0.81780480 -0.7661329 -0.2971204 -0.08250603
            carb
mpg  -0.75464796
cyl   0.21843307
disp  0.05367539
hp    0.54311998
drat -0.42599760
wt    0.09273981
qsec -0.81780480
vs   -0.76613289
am   -0.29712041
gear -0.08250603
carb  1.00000000

$uppCI
            mpg        cyl       disp         hp       drat         wt       qsec         vs         am       gear
mpg   1.0000000 -0.7163171 -0.7081376 -0.5860994  0.8322010 -0.7440872  0.6696186  0.8223262  0.7844520  0.7100628
cyl  -0.7163171  1.0000000  0.9514607  0.9154223 -0.4646481  0.8887052 -0.3055388 -0.6442689 -0.2126675 -0.1738615
disp -0.7081376  0.9514607  1.0000000  0.8932775 -0.4805193  0.9442902 -0.1001493 -0.4808327 -0.3055178 -0.2565810
hp   -0.5860994  0.9154223  0.8932775  1.0000000 -0.1186280  0.8192573 -0.4774331 -0.5006318  0.1152646  0.2332119
drat  0.8322010 -0.4646481 -0.4805193 -0.1186280  1.0000000 -0.4839784  0.4263400  0.6839680  0.8501319  0.8427222
wt   -0.7440872  0.8887052  0.9442902  0.8192573 -0.4839784  1.0000000  0.1852649 -0.2556982 -0.4532461 -0.2944887
qsec  0.6696186 -0.3055388 -0.1001493 -0.4774331  0.4263400  0.1852649  1.0000000  0.8679076  0.1291876  0.1469065
vs    0.8223262 -0.6442689 -0.4808327 -0.5006318  0.6839680 -0.2556982  0.8679076  1.0000000  0.4883712  0.5175379
am    0.7844520 -0.2126675 -0.3055178  0.1152646  0.8501319 -0.4532461  0.1291876  0.4883712  1.0000000  0.8949546
gear  0.7100628 -0.1738615 -0.2565810  0.2332119  0.8427222 -0.2944887  0.1469065  0.5175379  0.8949546  1.0000000
carb -0.2503183  0.7397479  0.6536467  0.8708249  0.2663358  0.6755700 -0.3988165 -0.2756654  0.3982389  0.5684422
           carb
mpg  -0.2503183
cyl   0.7397479
disp  0.6536467
hp    0.8708249
drat  0.2663358
wt    0.6755700
qsec -0.3988165
vs   -0.2756654
am    0.3982389
gear  0.5684422
carb  1.0000000

corrplot(
  cor(mtcars),
  p.mat = mtcarsCorMtest$p, 
  sig.level = .05,
  order = "hclust", 
  addrect = 3
)

Pearson’s product-moment correlation

cor.test(
  mtcars$mpg, 
  mtcars$wt,
  alternative = "two.sided",
  method = "pearson",
  conf.level = 0.95
)

    Pearson's product-moment correlation

data:  mtcars$mpg and mtcars$wt
t = -9.559, df = 30, p-value = 1.294e-10
alternative hypothesis: true correlation is not equal to 0
95 percent confidence interval:
 -0.9338264 -0.7440872
sample estimates:
       cor 
-0.8676594 

Kendall’s rank correlation

cor.test(
  mtcars$mpg, 
  mtcars$wt,
  alternative = "greater",
  method = "kendall",
  conf.level = 0.9,
  exact = FALSE
)

    Kendall's rank correlation tau

data:  mtcars$mpg and mtcars$wt
z = -5.7981, p-value = 1
alternative hypothesis: true tau is greater than 0
sample estimates:
       tau 
-0.7278321 

Spearman’s rank correlation

cor.test(
  mtcars$mpg, 
  mtcars$wt,
  alternative = "less",
  method = "spearman",
  conf.level = 0.8,
  exact = FALSE
)

    Spearman's rank correlation rho

data:  mtcars$mpg and mtcars$wt
S = 10292, p-value = 7.438e-12
alternative hypothesis: true rho is less than 0
sample estimates:
      rho 
-0.886422