This is a repost of the original article that was posted as an embedded PDF file.

##### Douglas M. Wiig

April 8, 2018

Abstract

#####
This article is part of my series of articles exploring the use of R

packages that allow for visualization of complex relationships among variables.

Other articles have examined visual representations produced by

the *qgraph* package in both large and small samples with more than three

variables. In this article I look specifically at the R *qgraph* package with a small

dataset of N=10, but a large number (14) of variables. Specifically, the R

*qgraph.pca* function is examined.

**1 The Problem**

In two previous blog posts I discussed some techniques for visualizing relationships

involving two or three variables and a large number of cases. In this

tutorial I will extend that discussion to show some techniques that can be used

on datasets with complex multivariate relationships involving three or more

variables.

In this post I will use a dataset called ‘Detroit.’ This data set was originally

used in the book *‘Subset selection in regression’* by Alan J. Miller published in

the Chapman and Hall series of monographs on *Statistics and Applied Probability,*

*no. 40.* It was also used in other research and appeared in appendix A

of *‘Regression analysis and its application: A data-oriented approach’* by Gunst

and Mason, *Statistics textbooks and monographs no. 24,* Marcel Dekker. Editor.

The Detroit dataset contains 14 variables and 10 cases. Each case represents

a year during the time period 1961-1973. The variables on which data was

collected are seen as possible predictors of homicide rate in Detroit during each

of the years studied.

These data are shown below

FTP UEMP MAN LIC GR CLEAR WM NMAN GOV HE WE HOM ACC ASR

260.35 11.0 455.5 178.15 215.98 93.4 558724. 538.1 133.9 2.98 117.18 8.60 9.17 306.18

269.80 7.0 480.2 156.41 180.48 88.5 538584. 547.6 137.6 3.09 134.02 8.90 40.27 315.16

272.04 5.2 506.1 198.02 209.57 94.4 519171. 562.8 143.6 3.23 141.68 8.52 45.31 277.53

272.96 4.3 535.8 222.10 231.67 92.0 500457. 591.0 150.3 3.33 147.98 8.89 49.51 234.07

272.51 3.5 576.0 301.92 297.65 91.0 482418. 626.1 164.3 3.46 159.85 13.0 55.05 30.84

261.34 3.2 601.7 391.22 367.62 87.4 465029. 659.8 179.5 3.60 157.19 14.57 53.90 17.99

268.89 4.1 577.3 665.56 616.54 88.3 448267. 686.2 187.5 3.73 155.29 21.36 50.62 86.11

295.99 3.9 596.9 1131.21 1029.75 86.1 432109. 699.6 195.4 2.91 131.75 28.03 51.47 91.59

319.87 3.6 613.5 837.60 786.23 79.0 416533. 729.9 210.3 4.25 178.74 31.49 49.16 20.39

341.43 7.1 569.3 794.90 713.77 73.9 401518. 757.8 223.8 4.47 178.30 37.39 45.80 23.03

The variables are as follows:

FTP – Full-time police per 100,000 population

UEMP – % unemployed in the population

MAN – number of manufacturing workers in thousands

LIC – Number of handgun licenses per 100,000 population

GR – Number of handgun registrations per 100,000 population

CLEAR – % homicides cleared by arrests

WM – Number of white males in the population

NMAN – Number of non-manufacturing workers in thousands

GOV – Number of government workers in thousands

HE – Average hourly earnings

WE – Average weekly earnings

HOM – Number of homicides per 100,000 of population

ACC – Death rate in accidents per 100,000 population

ASR – Number of assaults per 100,000 population

*[J.C. Fisher ”Homicide in Detroit: The Role of Firearms”, Criminology, vol.14,*

*387-400 (1976)]*

**2 Analysis**

As I have noted in previous tutorials, social science research projects often start

out with many potential independent predictor variables for a given dependent

variable. If these are all measured at the interval or ratio level, a correlation

matrix often serves as a starting point to begin analyzing relationships among

variables. In this particular case a researcher might be interested in looking at

factors that are related to total homicides. There are many R techniques to

enter data for analysis. In this case I entered the data into an Excel spreadsheet

and then loaded the file into the R environment. Install and load the following

packages:

*Hmisc*

*stats*

*qgraph*

*readxl* (only needed if importing data from Excel)

A correlation matrix can be generated using the cor function which is contained

in the stats package. To produce a matrix using all 14 variables use the

following code:

**#the data file has been loaded as ’detroit’**

**#the file has 14 columns**

**#run a pearson correlation and #run a pearson correlation and put into the object ’detcor’**

**detcor=cor(as.matrix(detroit[c(1:14)]), method=”pearson”)**

**#**

**#round the correlation matrix to 2 decimal places for better viewing**

**round(detcor, 2)**

**#**

**#The resulting matrix will be displayed on the screen**

Examination of the matrix shows a number of the predictors correlate with the

dependent variable ’HOM.’ There are also a large number of inter-correlations

among the predictor variables. This fact makes it difficult to make any generalizations

based on the correlation matrix only. As demonstrated in previous

tutorials, the qgraph function can be used to produce a visual representation of

the correlation matrix. Use the following code:

**#basic graph with 14 vars zero order correlations**

**qgraph(detcor, shape=”circle”, posCol=”darkgreen”, negCol=”darkred”, layout=”spring”)**

This will produce graph as seen below:

The graph displays positive correlations among variable as a green line, and

negative as a red line. The color intensity indicates the relative strength of the

correlation. While this approach provides an improvement over the raw matrix

it still rather difficult to interpret. There are many options other than those

used in the above example that allow qgraph to have a great deal of flexibility in

creating visual representation of complex relationships among variables. In the

next section I will examine one of these options that uses principal component

analysis of the data.

**2.1 Using qgraph Principal Component Analysis**

A discussion of the theory behind principal component exploratory analysis is

beyond the scope of this discussion. Suffice it to say that it allows for simplification

of a large number of inter-correlations by identifying factors or dimensions

that individual correlations relate to. This grouping of variables on specific factors

allows qgraph to create a visual representation of these relationships. An

excellent discussion of the theory of PCA along with R scripts can be found in

*Principal Components Analysis (PCA)*, Steven M. Holland Department of Geology,

University of Georgia, Athens, GA, 2008.

To produce a graph using the ’detcor’ correlation matrix used above use the

following code:

**#correlation matrix used is ’detcor’**

**#qgraph with loadings from principal components**

**#basic options used; many other options available**

**qgraph.pca(detcor, factor=3, rotation=”varimax”)**

**#this will yield 3 factors**

This code produces the output shown below:

As noted above the red and green arrows indicate negative and positive loadings

on the factors, and the color intensity indicates the strength. The qgraph.pca

function produces a useful visual interpretation of the clustering of variables relative

to the three factors extracted. This would be very difficult if not impossible

with only the correlation matrix or the basic qgraph visual representation.

In a future tutorial I will explore more qgraph options that can be used to

explore the Detroit dataset as well as options for a larger datasets. In future

articles I will also explore other R packages that are also useful for analyzing

large numbers of complex variable interrelationships in very large, medium, and

small samples.

** When developing R code I strongly recommend using an IDE such as

RStudio. This is a powerful coding environment and is free for personal use as

well as being open source software. RStudio will run on a variety of platforms.

If you are developing code for future publication or sharing I would also recommend

TeXstudio, a LaTex based document development environment which is also free for personal use. This document was produced using TeXstudio 2.12.6

and RStudio 1.0.136.