STAT 360: Computational Methods in STAT

Lab 4: Multicollinearity, Singularity, and Matrix Algebra

Load R Libraries, Import and Attach Relevant Data, and Specify Seed

library(rmarkdown); library(knitr)

library(dplyr)
library(psych); library(corrplot); library(GPArotation)
library(lavaan); library(semPlot); library(moments)

HUNGERDATA <- read.csv("https://www.dropbox.com/s/znpumq7o1fghlfb/?dl=1")
attach(HUNGERDATA)

set.seed(37)

Use some of the available variables to evaluate the efficiency of volunteer packers while accounting for common issues with multivariate data

Exercise 01:

Use cbind() along with cor() to compute the correlation matrix, R1, for the interaction between VOLUNTEERS, BOXES, and MEALSPERVOL

r <- cor(cbind(VOLUNTEERS, BOXES, MEALSPERVOL))
corrplot(r, method = "number")

Exercise 02:

Explain what the correlation matrix above means in terms of the presence or absence of multicollinearity

BOXES and VOLUNTEERS have a correlation greater than 0.9 which could imply multicollinearity between the two variables.

Exercise 03:

Compute the eigenvalues of the correlation matrix above

(ev <- eigen(r)$values)

## [1] 2.10754899 0.86161349 0.03083751

Exercise 04:

Use the eigenvalues above to compute the condition indices for the interaction between VOLUNTEERS, BOXES, and MEALPERVOL

multiColl <- F
seriousMultiColl <- F

for (value in ev) {
  multiColl <- sqrt(max(ev)/value) > 15 || multiColl
  seriousMultiColl <- sqrt(max(ev)/value) > 30 || seriousMultiColl
}

multiColl #Is there multicollinearty?

## [1] FALSE

seriousMultiColl #Is there severe mukticollinearity?

## [1] FALSE

Exercise 05:

Explain what the condition indices above mean in terms of the presence or absence of multicollinearity

The indices above do not suggest multicollinearity. 

Exercise 06:

Explain if the conclusion above supports your earlier conclusion regarding the presence of absence of multicollinearity

It does not support my earlier conclusion. 

Exercise 07:

Compute the (missing value) correlation matrix, R2, for the interaction between VOLUNTEERS, BOXES, MEALSPERVOL, and KIDSFED

r <- cor(cbind(VOLUNTEERS, BOXES, MEALSPERVOL, KIDSFED), use = "pairwise.complete.obs")
corrplot(r, method = "number")

Exercise 08:

Explain what the correlation matrix above means in terms of the presence or absence of singularity

There seems to be a singulatiry between BOXES and KIDSFED

Exercise 09:

Compute the eigenvalues of the correlation matrix above

(ev <- eigen(r)$values)

## [1]  2.9984164194  0.9673883611  0.0345186971 -0.0003234776

Exercise 10:

Explain what the eigenvalues above mean in terms of the presence or absence of singularity

The negative eigenvalue suggests a singularity between the two variables.

Exercise 11:

Compute the variance-covariance matrix, S1, for the interaction between VOLUNTEERS and MEALSPERVOL

(varcov <- cov(cbind(VOLUNTEERS, MEALSPERVOL)))

##             VOLUNTEERS MEALSPERVOL
## VOLUNTEERS    1560.023   -1650.909
## MEALSPERVOL  -1650.909    8030.970

Exercise 12:

Use sum() and diag() to compute the trace of the variance-covariance matrix above

sum(diag(varcov))

## [1] 9590.992

Exercise 13:

Compute the (missing value) variance-covariance matrix, S2, for the interaction between BOXES and KIDSFED

(varcovmiss <- cov(cbind(BOXES, KIDSFED), use = "complete.obs"))

##            BOXES  KIDSFED
## BOXES   2255.278 1270.042
## KIDSFED 1270.042  716.000

Exercise 14:

Explain how to find the first (top left) element in the product of the two variance-covariance matrices above

Take the first row of the first varcovar matrix above and multiply each of the two elements with the two elements in the first column of the second varcovar matrix and then add the two resulting products together to get the top left element in the matrix product. 

Exercise 15:

Use algebra to compute the first (top left) element in the product of the two variance-covariance matrices above

(varcov[1,1]*varcovmiss[1,1] + varcov[1,2]*varcovmiss[2,1])

## [1] 1421561

Exercise 16:

Use %*% to compute the full product, P, of the two variance-covariance matrices above

(varcovprod <- varcov %*% varcovmiss)

##               BOXES KIDSFED
## VOLUNTEERS  1421561  799243
## MEALSPERVOL 6476408 3653451

Exercise 17:

Explain how to find the transpose of the product matrix, P, above

Reflect all elements of the matrix across the diagonal of the matrix. (Or take each row (0 through n) and make it a column in the same order)

Exercise 18:

Use t() to compute the transpose of the product matrix, P

t(varcovprod)

##         VOLUNTEERS MEALSPERVOL
## BOXES      1421561     6476408
## KIDSFED     799243     3653451

Exercise 19:

Use det() to compute the determinant of the product matrix, P

det(varcovprod)

## [1] 17381236703

Exercise 20:

Use solve() to compute the inverse of the product matrix, P

solve(varcovprod)

##            VOLUNTEERS   MEALSPERVOL
## BOXES    0.0002101951 -4.598309e-05
## KIDSFED -0.0003726091  8.178712e-05