Covariance, correlation, association measures for continuous variables
TODO
psychcor.plot()- link to diagScatter, diagMultivariate, association, associationOrder
Install required packages
wants <- c("coin", "psych")
has <- wants %in% rownames(installed.packages())
if(any(!has)) install.packages(wants[!has])Bivariate covariance and correlation
Covariance
Corrected (sample) covariance
x <- c(17, 30, 30, 25, 23, 21)
y <- c(1, 12, 8, 10, 5, 3)
cov(x, y)[1] 19.2Uncorrected (population) covariance
(cmML <- cov.wt(cbind(x, y), method="ML")$cov) x y
x 21.88889 16.00000
y 16.00000 14.91667cmML[upper.tri(cmML)][1] 16Correlation
Empirical correlation
(r <- cor(x, y))[1] 0.8854667Fisher’s \(Z\)-transformation
Used, e.g., for averaging correlations
library(psych)
(rZ <- fisherz(r))[1] 1.400533fisherz2r(rZ)[1] 0.8854667Partial and semi-partial correlation
set.seed(123)
N <- 100
z1 <- runif(N)
z2 <- runif(N)
x <- -0.3*z1 + 0.2*z2 + rnorm(N, 0, 0.3)
y <- 0.3*z1 - 0.4*z2 + rnorm(N, 0, 0.3)
cor(x, y)[1] -0.1620401Partial correlation \(r_{(xy).z}\)
x.z1 <- residuals(lm(x ~ z1))
y.z1 <- residuals(lm(y ~ z1))
cor(x.z1, y.z1)[1] -0.05298174x.z12 <- residuals(lm(x ~ z1 + z2))
y.z12 <- residuals(lm(y ~ z1 + z2))
cor(x.z12, y.z12)[1] 0.02470899Semi-partial correlation \(r_{(x.z)y}\)
cor(x.z1, y)[1] -0.04772153Covariance matrix
X1 <- c(19, 19, 31, 19, 24)
X2 <- c(95, 76, 94, 76, 76)
X3 <- c(197, 178, 189, 184, 173)
(X <- cbind(X1, X2, X3)) X1 X2 X3
[1,] 19 95 197
[2,] 19 76 178
[3,] 31 94 189
[4,] 19 76 184
[5,] 24 76 173(covX <- cov(X)) X1 X2 X3
X1 27.80 22.55 0.4
X2 22.55 102.80 82.4
X3 0.40 82.40 87.7(cML <- cov.wt(X, method="ML"))$cov
X1 X2 X3
X1 22.24 18.04 0.32
X2 18.04 82.24 65.92
X3 0.32 65.92 70.16
$center
X1 X2 X3
22.4 83.4 184.2
$n.obs
[1] 5cML$cov X1 X2 X3
X1 22.24 18.04 0.32
X2 18.04 82.24 65.92
X3 0.32 65.92 70.16Correlation matrix
cor(X) X1 X2 X3
X1 1.000000000 0.4218204 0.008100984
X2 0.421820411 1.0000000 0.867822404
X3 0.008100984 0.8678224 1.000000000cov2cor(covX)vec <- rnorm(nrow(X))
cor(vec, X) X1 X2 X3
[1,] -0.04054191 -0.1729373 -0.4405556Correlation for ordinal continuous variables
Spearman’s \(\rho\)
DV1 <- c(97, 76, 56, 99, 50, 62, 36, 69, 55, 17)
DV2 <- c(42, 74, 22, 99, 73, 44, 10, 68, 19, -34)
DV3 <- c(61, 88, 21, 29, 56, 37, 21, 70, 46, 88)
DV4 <- c(58, 65, 38, 19, 55, 23, 26, 60, 50, 91)
DVmat <- cbind(DV1, DV2, DV3, DV4)cor(DV1, DV2, method="spearman")[1] 0.7333333cor(DVmat, method="spearman") DV1 DV2 DV3 DV4
DV1 1.00000000 0.7333333 0.05487907 -0.1878788
DV2 0.73333333 1.0000000 0.11585581 -0.1636364
DV3 0.05487907 0.1158558 1.00000000 0.8963581
DV4 -0.18787879 -0.1636364 0.89635813 1.0000000Kendall’s \(\tau\)-b
cor(DV1, DV2, method="kendall")[1] 0.6444444cor(DVmat, method="kendall") DV1 DV2 DV3 DV4
DV1 1.00000000 0.64444444 0.02273314 -0.15555556
DV2 0.64444444 1.00000000 0.11366572 -0.06666667
DV3 0.02273314 0.11366572 1.00000000 0.79566006
DV4 -0.15555556 -0.06666667 0.79566006 1.00000000Correlation tests
Pearson correlation
cor.test(DV1, DV2)
Pearson's product-moment correlation
data: DV1 and DV2
t = 3.4996, df = 8, p-value = 0.008084
alternative hypothesis: true correlation is not equal to 0
95 percent confidence interval:
0.2902442 0.9447410
sample estimates:
cor
0.7777418 library(psych)
corr.test(DVmat, adjust="bonferroni")Call:corr.test(x = DVmat, adjust = "bonferroni")
Correlation matrix
DV1 DV2 DV3 DV4
DV1 1.00 0.78 -0.09 -0.35
DV2 0.78 1.00 -0.07 -0.39
DV3 -0.09 -0.07 1.00 0.89
DV4 -0.35 -0.39 0.89 1.00
Sample Size
[1] 10
Probability values (Entries above the diagonal are adjusted for multiple tests.)
DV1 DV2 DV3 DV4
DV1 0.00 0.05 1 1
DV2 0.01 0.00 1 1
DV3 0.80 0.86 0 0
DV4 0.32 0.27 0 0
To see confidence intervals of the correlations, print with the short=FALSE optionSpearman’s \(\rho\)
cor.test(DV1, DV2, method="spearman")
Spearman's rank correlation rho
data: DV1 and DV2
S = 44, p-value = 0.02117
alternative hypothesis: true rho is not equal to 0
sample estimates:
rho
0.7333333 library(coin)
spearman_test(DV1 ~ DV2, distribution=approximate(B=9999))
Approximative Spearman Correlation Test
data: DV1 by DV2
Z = 2.2, p-value = 0.0207
alternative hypothesis: true mu is not equal to 0library(psych)
corr.test(DVmat, method="spearman", adjust="bonferroni")Call:corr.test(x = DVmat, method = "spearman", adjust = "bonferroni")
Correlation matrix
DV1 DV2 DV3 DV4
DV1 1.00 0.73 0.05 -0.19
DV2 0.73 1.00 0.12 -0.16
DV3 0.05 0.12 1.00 0.90
DV4 -0.19 -0.16 0.90 1.00
Sample Size
[1] 10
Probability values (Entries above the diagonal are adjusted for multiple tests.)
DV1 DV2 DV3 DV4
DV1 0.00 0.09 1 1
DV2 0.02 0.00 1 1
DV3 0.88 0.75 0 0
DV4 0.60 0.65 0 0
To see confidence intervals of the correlations, print with the short=FALSE optionKendall’s \(\tau\)-b
cor.test(DV1, DV2, method="kendall")
Kendall's rank correlation tau
data: DV1 and DV2
T = 37, p-value = 0.009148
alternative hypothesis: true tau is not equal to 0
sample estimates:
tau
0.6444444 library(psych)
corr.test(DVmat, method="kendall", adjust="bonferroni")Call:corr.test(x = DVmat, method = "kendall", adjust = "bonferroni")
Correlation matrix
DV1 DV2 DV3 DV4
DV1 1.00 0.64 0.02 -0.16
DV2 0.64 1.00 0.11 -0.07
DV3 0.02 0.11 1.00 0.80
DV4 -0.16 -0.07 0.80 1.00
Sample Size
[1] 10
Probability values (Entries above the diagonal are adjusted for multiple tests.)
DV1 DV2 DV3 DV4
DV1 0.00 0.27 1.00 1.00
DV2 0.04 0.00 1.00 1.00
DV3 0.95 0.75 0.00 0.04
DV4 0.67 0.85 0.01 0.00
To see confidence intervals of the correlations, print with the short=FALSE optionDifference between two independent correlations
N <- length(DV1)
library(psych)
r.test(n=N, n2=N, r12=cor(DV1, DV2), r34=cor(DV3, DV4))Correlation tests
Call:r.test(n = N, r12 = cor(DV1, DV2), r34 = cor(DV3, DV4), n2 = N)
Test of difference between two independent correlations
z value 0.73 with probability 0.46Detach (automatically) loaded packages (if possible)
try(detach(package:psych))
try(detach(package:coin))
try(detach(package:survival))
try(detach(package:splines))Get the article source from GitHub
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