Reproduce the covariance matrix in the upper right part of Table 4.1 from the correlations and standard deviations in the upper left part of the table.
# Install Packages
library(mvtnorm)
# Enter the covariance/correlation matrix
x <- as.matrix(data.frame(c(1.000, .2721, .6858),
c(.2721, 1.0000, .4991),
c(.6858, .4991, 1.0000)))
V <- diag(c(3.0070, 2.8172, 10.8699))
S <- V %*% x %*% V
# Covariance Matrix
S## [,1] [,2] [,3]
## [1,] 9.042049 2.305046 22.41591
## [2,] 2.305046 7.936616 15.28378
## [3,] 22.415914 15.283781 118.15473cor(rmvnorm(100, sigma=S))## [,1] [,2] [,3]
## [1,] 1.0000000 0.3058924 0.6866995
## [2,] 0.3058924 1.0000000 0.5313669
## [3,] 0.6866995 0.5313669 1.0000000# Get eigenvalues and eigenvectors
eigen(x)## eigen() decomposition
## $values
## [1] 1.9906969 0.7441315 0.2651717
##
## $vectors
## [,1] [,2] [,3]
## [1,] -0.5835121 0.5545839 0.5932540
## [2,] -0.4874832 -0.8234582 0.2903044
## [3,] -0.6495180 0.1198052 -0.7508482Given Cov(𝑋,𝑌)=13.00, 𝑠𝑋2=12.00, and 𝑠𝑌2=10.00, show that the corresponding correlation is out of bounds.
sqrt(12*10)## [1] 10.95445The covariance is found by Rxy * 10.95445 = 13.00
Solving for correlation: 13 / 10.95445 = 1.186732
Since this correlation is greater than one (1.186732), data shows the corresponding correlation is out of bounds. In addition, since the covariance (cov = 13) is greater that 10.95, the value is out of bounds.
Find a normalizing transformation for the data in Figure 4.2. Explain why.
The data is positively skewed based on the box and whisker plot. More variables are found in the left side of the data set. A positive skew will add a constant of 1.0 as the lowest value. The transformation will be accomplished compressing the differences between scores in the upper end more than the differences between lower scores.
##a) List the cases that should be excluded when calculating corresponding covariances using pairwise deletion and listwise deletion.
Listwise Deletion: Exclude C, D, F, G
Pairwise Deletion: Exclude CW, DX, FY, GY, (X n=6, W n=6, Y n = 5)
##b) Calculate the covariance matrix using pairwise deletion. Show that the corresponding correlation matrix has an element that is out of bounds.
pairwise <- cbind(c(42, 34, 22, NA, 24, 16, 30),
c(13, 12, NA, 8, 7, 10, 10),
c(8, 10, 12, 14, 16, NA, NA))
pairwise## [,1] [,2] [,3]
## [1,] 42 13 8
## [2,] 34 12 10
## [3,] 22 NA 12
## [4,] NA 8 14
## [5,] 24 7 16
## [6,] 16 10 NA
## [7,] 30 10 NApairwise.cor <- cor(pairwise, use = "pairwise.complete.obs")
pairwise.cov <- cov(pairwise, use = "pairwise.complete.obs")
eigen(pairwise.cor)## eigen() decomposition
## $values
## [1] 2.68743416 0.32436846 -0.01180262
##
## $vectors
## [,1] [,2] [,3]
## [1,] -0.5404144 0.8153712 0.2076584
## [2,] -0.5817871 -0.5404064 0.6078525
## [3,] 0.6078454 0.2076793 0.7664159eigen(pairwise.cov)## eigen() decomposition
## $values
## [1] 98.229327 7.041785 -3.671112
##
## $vectors
## [,1] [,2] [,3]
## [1,] 0.9333177 0.3493531 0.08288858
## [2,] 0.1941479 -0.6852301 0.70197316
## [3,] -0.3020342 0.6390713 0.70736354pairwise.cov## [,1] [,2] [,3]
## [1,] 86.40000 15.90000 -26.33333
## [2,] 15.90000 5.20000 -10.66667
## [3,] -26.33333 -10.66667 10.00000pairwise.cor## [,1] [,2] [,3]
## [1,] 1.0000000 0.7005289 -0.8297422
## [2,] 0.7005289 1.0000000 -0.9922779
## [3,] -0.8297422 -0.9922779 1.0000000According to the eigen values, there is one covariance out of bounds in the third variable.
sqrt(10*86.4)## [1] 29.39388The covariance (cov = 26.33333) is not greater that 29.39388, the value is not out of bounds.
sqrt(10*5.2)## [1] 7.211103Since the covariance (cov = 10.66667) is greater that 7.211103, the value is out of bounds.
listwise.cor <- cor(pairwise, use = "complete.obs")
listwise.cov <- cov(pairwise, use = "complete.obs")
eigen(listwise.cov)## eigen() decomposition
## $values
## [1] 1.076844e+02 1.315572e+00 -2.147130e-16
##
## $vectors
## [,1] [,2] [,3]
## [1,] 0.8674003 0.4880625 0.09701425
## [2,] 0.3008023 -0.6695831 0.67909975
## [3,] -0.3964022 0.5598692 0.72760688listwise.cov## [,1] [,2] [,3]
## [1,] 81.33333 27.66667 -36.66667
## [2,] 27.66667 10.33333 -13.33333
## [3,] -36.66667 -13.33333 17.33333listwise.cor## [,1] [,2] [,3]
## [1,] 1.0000000 0.9543383 -0.9765536
## [2,] 0.9543383 1.0000000 -0.9962710
## [3,] -0.9765536 -0.9962710 1.0000000sqrt(81.33333*17.33333)## [1] 37.547The covariance (cov = 36.66667) is not greater that 37.547, the value is not out of bounds.
sqrt(17.33333*10.33333)## [1] 13.38324Since the covariance (cov = 13.33333) is not greater that 13.38324, the value is not out of bounds.
sqrt(10.33333*81.33333)## [1] 28.99041Since the covariance (cov = 27.66667) is not greater that 28.99041, the value is not out of bounds.