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\begin{document}
\title{Is gun ownership associated with higher homicide rates? Results from the
United States, 1973-2016}
\author[1]{Jinkinson Smith}%
\affil[1]{Affiliation not available}%
\vspace{-1em}
\date{\today}
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This study examines the association between gun ownership rates and
homicide rates in the United States from 1973 to 2016. The results show
a very strong positive~correlation (Spearman's rho=0.79) between gun
ownership and homicide rates, which supports the hypothesis that higher
rates of gun ownership increase the likelihood of homicide. On average,
a 1\% increase in gun ownership is found to be associated with 2.6
additional homicides per million people.
\par\null
\textbf{Methods}
Gun ownership data~ was obtained from the 26 waves of the General Social
Survey in which respondents were asked if they had a gun in their
home.\cite{chicago}~The rate of gun ownership for a given year was
calculated by dividing the number of~respondents in that year~who
answered ``Yes'' to the~ aforementioned gun question by the number of~
respondents~ who answered ``Yes'' or ``No''~ in the same year; this
value was then converted to a percentage. Homicide data was obtained
almost entirely from the Uniform Crime Reports online data
tool.\cite{statistics} The exception was the homicide rate for 2016,
which was not available through this tool;~ I obtained this rate from a
separate FBI report.\cite{11}~Hereafter, I refer to the
yes/no-based gun ownership estimates simply as ``gun ownership''~ and
the UCR-based homicide estimates simply as ``homicide'' for simplicity.
\par\null
I used the chi-square~goodness-of-fit test in Microsoft Excel to
determine~whether~ the homicide and gun ownership data were normally
distributed, following a procedure previously described
elsewhere.\cite{goodness-of-fit} This produced p-values for both datasets
(p= 0.999869577 for gun ownership, 0.985909364 for homicide) that were
far above the level of significance (p=0.05), indicating that there is
insufficient evidence to reject the null hypothesis that both datasets
are normally distributed.
\par\null
In order to examine the~ two remaining assumptions necessary for a
Pearson correlation to be valid, namely linearity and~homoscedasticity,
I did the following. First, as noted above, I calculated the Pearson
correlation between gun ownership and homicide, which was 0.90; this
indicates the presence of a very strong linear correlation, meaning that
the assumption of linearity is justified in these data.
\par\null
Second, I examined the scatterplot (figure 1 below) to see if the data
met the assumption of homoscedasticity. If this assumption is valid, the
data should cluster mostly around the middle of the regression line, and
then taper off as one approaches either end of the
line.\cite{herschel2017} However, this was not the pattern observed in my
data: instead, as figure 1 shows, the data was clearly clustered much
more around both the high and low ends than in the~middle, with a
particular density of data points at the low end of the regression line.
Therefore, the assumption of homoscedasticity is not met here, so a
Pearson correlation is not appropriate. Therefore, I calculated a
Spearman's rank correlation on the same data and obtained a Spearman's
rho of 0.79.~\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/figure-1/figure-1}
\caption{{A scatter plot of gun ownership rates (x-axis) and national homicide
rates per 100,000 people (y-axis) in the United States.
{\label{223035}}%
}}
\end{center}
\end{figure}
In figure 1 above, the dashed trend line corresponds to the line of best
fit for the data. The equation of this line is y = 26.03x - 3.5973,
where y = homicide rate and x = gun ownership rate. Note, however, that
gun ownership rates are all less than 1 (a rate of 100\% would
correspond to an x-value of 1). Therefore, to estimate the change in
homicide rate associated with a percent change in gun ownership, you
must enter a decimal value corresponding to that percentage (e.g. 0.3
for a 30\% increase/decrease in gun ownership). Taking this into
account, these results show that a single percentage-point increase in
the rate of~gun ownership is associated with an increase in the homicide
rate of (26.03/100)=0.2603 homicides per~100,000 people, corresponding
to (0.26*10)=2.6 additional homicides per million people.
\par\null
To produce a result similar~ to~ that obtained by Siegel et al.
(2013),\cite{Siegel_2013} I converted the homicide rates into percentage
values corresponding to percent deviations from the mean. Thus each
value V (which was originally the number of homicides per 100,000
people) was converted as follows: (V-M)/M, where M is the mean of all
homicide rate values (about 7.5). An additional chi-square
goodness-of-fit test yielded another P-value (in this case, 0.985) which
again greatly exceeded the 0.05 threshold, thereby supporting the
normality assumption. Yet the scatter plot of these data, shown below in
figure 2, again revealed strong heteroscedasticity, meaning that
Spearman's rank correlation must be calculated.\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/figure-2/figure-2}
\end{center}
\end{figure}
All rankings for the homicide percentages were identical to those for
the original homicide rates, and since the data for the independent
variable (gun ownership) is unchanged, the value of Spearman's rho is
thus identical to that reported~earlier, namely 0.79. Of more interest,
however, is the slope of the trend line for this scatter plot:
3.453.~This implies that a 1\%~increase in gun ownership is associated
with a 3.453\% (rounded to 3.5\%) increase in the homicide rate.
\par\null
Of course, it should be kept in mind that these results only establish a
correlation, not a cause-and-effect relationship. Nevertheless, these
findings are consistent with those which have previously reported a
positive gun ownership-homicide
association.\cite{Miller_2002}\cite{Miller_2007}\cite{Siegel_2013}~
\par\null
\textbf{Granger Tests}
To address the issue of causality, I next performed Granger tests on
these data.\cite{test} My null hypothesis was that lagged gun
ownership values do not explain variation in homicide rates, and my
alternative hypothesis was that homicide rates (the y-variable) affect
gun ownership rates (the x-variable).~I~attempted to address the issue
of stationarity for both the gun ownership and homicide time series by
using the method of ``differencing''. In this case, this entailed the
subtracting of one value in the time series from its successor, and then
graphing the differences. Of note, this means that the difference time
series that this method produced have no data for the last year (in this
case 2016), because it had no successor to subtract from it. The
resulting graph is shown in figure 3 below.
\par\null
It has previously been shown that arbitrarily choosing lag lengths in
Granger tests, or even choosing them based on statistical criteria, can
produce misleading results.\cite{Thornton_1985} I tried to address this
issue by choosing a lag value based on the final prediction error (FPE)
method as~ described by Urbain,\cite{Urbain_1989} based on the work of
Hsiao.\cite{Hsiao_1979} This involves trying to minimize the output of
the following equation (Urbain 1989, p. 318):
m (n)=a\textsuperscript{2~}*(T+m+n)/(T-m-n)
Here,~a\textsuperscript{2}=1/T*sum(y\textsubscript{t}-y*\textsubscript{t}).
\par\null\par\null
\url{https://otexts.org/fpp2/autocorrelation.html}
\url{https://www.statisticshowto.datasciencecentral.com/stationarity/}
\par\null\par\null
~
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