# An R Tutorial: Visual Representation of Complex Multivariate Relationships Using the R qgraph Package, Part Two Repost

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

##### 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 Rqgraph.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’
#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.

# Tutorial: Using R to Analyze NORC GSS Social Science Data, Part Six, R and ANOVA

Tutorial: Using R to Analyze NORC GSS Social Science Data, Part Six, R and ANOVA

A tutorial by Douglas M. Wiig

As discussed in previous segments of this tutorial, for anyone interested in researching social science questions there is a wealth of survey data available through the National Opinion Research Center (NORC) and its associated research universities. The Center has been conducting a national survey each year since 1972 and has compiled a massive database of data from these surveys. Most if not all of these data files can be accessed and downloaded without charge. I have been working with the 2014 edition of the data and for all part of this tutorial will use the GSS2014 data file that is available for download on the Center’s web site. (See the NORC main website at http://www.norc.org/Research/Projects/Pages/general-social-survey.aspx and at http://www3.norc.org/GSS+Website ).

```Accessing and loading the NORC GSS2014 data set was discussed in part one of this tutorial. Refer to it if you need specific information on downloading the data set in STATA or SPSS format.  In this segment we will use the subset function to select a desired set of cases from all of the cases in the data file that meet certain criteria.  As indicated in my previous tutorial the GSS2014 data set contains a
total of 2588 cases and 866 variables.

Before starting this segment of the tutorial be sure that the foreign
package is installed and loaded into your R session. As I have indicated in previous tutorials, use of an IDE such as R Studio greatly facilitates entering and debugging R code when doing research such as is discussed in my tutorials.

Import the GSS 2014 data file in SPSS format and load it into the data frame ‘gss14’ using:
#########################################################import  GSS2014 file in SPSS .sav format
#uses foreign package
########################################################require(foreign)
#################################################```

In this tutorial the analysis of this sample will focus on examining the hypothesis “An individual’s outlook on life is influenced by the amount of education the person has attained.” The GSS variables ‘educ’, education in number of years and ‘life’, whether the respondent rated life DULL, ROUTINE, or EXCITING. A simple approach to testing this hypothesis is to compare mean levels of education for each of the three categories of response. I will do this analysis in two stages. In the first stage I will use techniques discussed in a previous tutorial to select a subset of each response category and display the mean education level for each of the three categories.

The three subsets life1, life2, and life3 are generated using the following code:

###################################################
# create 3 subsets from gss14 view on life by years of education
##################################################

life1 <- subset(gss14, life == “DULL”, select=educ)

life2 <- subset(gss14, life ==”ROUTINE”, select=educ)

life3 <- subset(gss14, life == “EXCITING”, select=educ)

###################################################

#The three means of the subsets are displayed using the code:

# run summary statistics for each subgroup

##################################################

summary(life1)

summary(life2)

summary(life3)

###################################################

resulting the following output:
educ
Min. : 0.00
1st Qu.:10.00
Median :12.00
Mean :11.78
3rd Qu.:13.00
Max. :20.00

summary(life2)
educ
Min. : 0.00
1st Qu.:12.00
Median :13.00
Mean :13.22
3rd Qu.:16.00
Max. :20.00
NA’s :1

summary(life3)
educ
Min. : 2.00
1st Qu.:12.00
Median :14.00
Mean :14.31
3rd Qu.:16.00
Max. :20.00

As is seen above there does appear to be a difference among mean years of education and the corresponding outlook on life. In order to examine whether or not these observed differences are not due to chance a simple one-way Analysis of Variance can be generated.

In the next tutorial I will discuss performing the ANOVA and using pair-wise comparisons to determine which if any means are different.