Firearms and Violent Crime Lethality in the United States

Firearms and Violent Crime Lethality in the United States

- Sean Boldt, May 2015

Firearms occupy a unique place in the American identity. They were not only present, but

instrumental in the country’s defining moments. One has difficulty imagining the Revolution,

Civil War, westward expansion, or the two World Wars without firearms - and the Americans

who carried them – front and center in the mind’s eye. Despite this, the suitability of widespread

private firearm ownership is heavily debated in the U.S. body politic. One side argues that

firearms, especially the modern variants, pose a clear danger to civil society. The other

maintains that not only is firearm possession guaranteed by the Bill of Rights, but it is an armed

citizenry that guards against tyranny. This side contends, moreover, that the presence of

privately-owned firearms in America acts as a deterrent to crime.

Gun control in the U.S. has an interesting history. Old Western tales speak of local ordinances

forbidding carrying firearms in town and shootouts between rowdy cowboys, gunslingers, bank

robbers and lawmen. In the modern era the first gun case heard by the U.S. Supreme Court,

United States v. Miller, resulted in judgment against Mr. Miller, whose sawed-off shotgun was

deemed to possess no military value. More contemporaneously, in the 1960s and 70s California

governor Ronald Reagan proposed gun control measures in response to the presence of armed

blacks protesting in and around that state’s government buildings. Coincidentally, Reagan, then

president, was shot by a deranged man in 1981. His press secretary, James Brady, was struck as

well and later became an advocate for gun control. In recent years, shootings at stateside

military bases, movie theaters and elementary schools have kept the debate going, at least for

those who wish to see greater regulation of firearms.

Purpose

This analysis, then, seeks to determine which side of the gun debate is right, insofar as whether

or not firearms pose a significant danger to society. More specifically, we seek to determine if

guns make violent crime – defined by the FBI as murder, rape, robbery and aggravated assault -

deadlier. To conduct this, we will look at murder rate by state per 100,000 residents and the

percentage of four weapon types (firearms, cutting instruments, other weapons, hands-feet-fists)

in all violent crime categories combined, as well as the demographic factors of region,

population density, wealth disparity, rates of gun ownership and levels of state gun controls.

Data and variables

Data for murder rate (dependent variable, or y) was gathered from U.S. Census Bureau (USCB)

population reports for the years 2005 to 2012 and from Federal Bureau of Investigation (FBI)

Uniform Crime Reports (UCR) for the same years. The USCB data was collated to produce the

average population by state for the years 2005 to 2012. The FBI data was assembled to produce

the average of murders by state for the years 2005 to 2012. A murder rate was then calculated

for each state as: y = murders/(population average/100,000). This variable will be expressed as

“Murders 100K” in the testing portion of this report.

Data for the percentage of violent crimes (murder, robbery and aggravated assault) committed

with various types of weapons (predictor variables) was assembled from USCB reports and FBI

UCRs for the years 2005 to 2012. Instead of murder rate, however, the data was used to

Firearms, Violent Crime, Lethality Sean Boldt

assemble a by-state average for the years 2005 to 2012 of violent crime rate by percent weapon

type. This was calculated for each state as: x = total violent crime by weapon type(population

average/100,000). These variables will be expressed as “Percent Violent Crime by Firearm

100K”, “Percent Violent Crime by Cutting Instrument 100K”, “Percent Violent Crime by Other

Weapon 100K”, and “Percent Violent Crime by Hands-Feet-Fists 100K” in the testing portion of

this report.

Data for the demographic variables was in the cases of population density and Gini Coefficient

gathered from the USCB. States were assigned a region based on the FBI’s UCRs. Percentage

of gun ownership by state was taken from the report “The Relationship Between Gun Ownership

and Firearms Homicide Rates in the United States, 1981 – 2010” by Michael Siegel, Craig S.

Ross and Charles King III in American Journal of Public Health, November 2013, volume 103,

number 11. State gun control levels were taking from the “Laws for Firearms: Best States for

Gun Owners in 2013” by James Tarr in Guns & Ammo, March 14th, 2013. These variables will

be expressed as “Pop Density per sq mi”, “Gini Coefficient of Wealth Disparity 2010”,

“Region”, “Percent Gun Ownership AVG 2001-2010”, and “G&A Best States for Gun Owners

2013” in the testing portion of this report.

Additional elements regarding data:

While the FBI categorizes forcible rape as a violent crime, it is not present in the data

because by state numbers were not available in the FBI’s Uniform Crime Reports (UCR);

Reliable data on violent crime was not available in the UCRs for Florida and Illinois and

so these states are not present in the analysis;

Data was not available in the UCRs for all the years between 2005 and 2012 for Alabama,

Hawaii and the District of Columbia, but in each case data for at least three years was

present and so an adequate sample, average and rate was determined;

Preliminary analysis showed that the District of Columbia was a significant outlier with a

violent crime rate approximately five times the national average;

This could be addressed by eliminating Washington DC from the analysis entirely or

splitting its violent crime rates between the states of Virginia and Maryland as it sits

squarely between the two;

Firearms are defined as any device that uses chemical energy to power a projectile;

Cutting instruments are defined as knives or any edged device;

Other weapons are defined as blunt instruments, poison, drowning, etc.;

Hands-feet-fists is defined by the use of one’s own body to overpower or cause harm to a

victim

The FBI uses a hierarchy (respectively murder, forcible rape, robbery and aggravated

assault) when categorizing violent crime in the UCR. So, for example, if a robbery results


in the death of the victim, then that incident is categorized as a murder/non-negligent

homicide and not a robbery. The same holds true for incidents that may have begun as

aggravated assaults but resulted in the death of the victim.

Hypotheses

H0: The type of weapon used in the commission of a violent crime will not predict the U.S.

murder rate more accurately than the mean.

H1: The use of a firearm in the commission of a violent crime will predict the U.S. murder

rate more accurately than the mean.

H2: The use of a cutting instrument in the commission of a violent crime will predict the

U.S. murder rate more accurately than the mean.

H3: The use of other weapons in the commission of a violent crime will predict the U.S.


H4: The use of hands-feet-fists in the commission of a violent crime will predict the U.S.


H5: Region will predict the U.S. murder rate more accurately than the mean.

H6: Population density will predict the U.S. murder rate more accurately than the mean.

H7: The Gini Coefficient of Wealth Disparity will predict the U.S. murder rate more

accurately than the mean.

H8: The percentage of gun ownership will predict the U.S. murder rate more accurately

than the mean.

H9: State gun friendliness will predict the U.S. murder rate more accurately than the mean.

H10: Any combination of the above predictor variables will predict the U.S. murder rate

more accurately than the mean.

Method of analysis

A linear multiple regression analysis was conducted and then retested with outliers removed.

Conclusion

The analysis showed that a combination of the predictor variables (Percent Violent Crime by

Firearm 100K, Region and Gini Coefficient of Wealth Disparity 2010) predicted the value of

Murders 100K better than the mean. This answers in the affirmative the question posed at the

beginning of the paper as to whether or not firearms increase the lethality of violent crime in the

U.S.

Violent crimes perpetrated with firearms have a very strong and direct correlation with murder

rate. Guns used in violent crimes account for approximately 55% of the variance in murder rate.

Violent crimes perpetrated with knives, other weapons or hands and feet did not show near the

relationship to murder as do firearms. In fact, the multiple regression showed an inverse


relationship between murder rate and violent crimes committed with knives, other weapons or

hands and feet when firearms were taken into account.

Wealth disparity and region also had a measurable relationship with murder rate, though not

nearly as great as the presence of firearms.

Further studies might find it useful to split the population and predictor variable values for

Washington DC between Maryland and Virginia. As previously noted, Washington DC has an

extremely high murder rate (per 100K) despite being a high gun control jurisdiction. Given that

many of the people in Washington DC are actually residents of Virginia and Maryland, both of

which have relatively lax gun law, it would seem that this would be an appropriate solution.


TEST 1: Normality assumption - Dependent variable

Of 51 cases (U.S. states plus the District of Columbia)

only two – Florida and Illinois – were removed due to

lack of available crime data in the FBI’s UCR. The mean

and median for the remaining 49 cases were, respectively,

4.51 and 4.18 with a standard deviation of 3.22.

As the histogram below shows, the data values were not

normally distributed across the 49 cases and the skewness

and kurtosis values of 3.03 and 14.5, respectively,

reflected a positive skew and a leptokurtic (pointy or

peak-like) distribution. A skewness and kurtosis values

of 0, for example, would result in a distribution that

matched exactly the normality curve.

Z-skew and z-kurtosis scores were 9.92 and 21.7, both of

which, were greater that the normality standard (|1.96|)

for a large sample.

The data for Murders 100K was non-parametric.

z = s/Se skew; z = s/Se kurtosis; 9.92 = 3.033/0.340; 21.7 =

14.503/0.668; 9.92 > |1.96|; 21.7 > |1.96|


Test 1: Normality assumption – independent (predictor) variables

The above table displays distribution data for the predictor scale variables: 1) Percent Violent

Crime by Firearm 100K, 2) Cutting Instruments 100K, 3) Other Weapons 100K, and 4) Hands-

Feet-Fists 100K; 5) the categorical predictor variable, Region; 6) the scale predictor variables

Pop Density per sq mi, 7) Gini Coefficient Wealth Disparity 2010, 8) Percent Gun Ownership

AVG 2001-2010; and 9) the ordinal predictor variable Guns & Ammo Best States For Gun

Owners 2013.

The means, or average, for percent of violent crimes by mechanism per 100,000 residents were

25.25; 16.51; 24.98; and 33.24 for the variables Percent Violent Crime by Firearm 100K, Cutting

Instrument 100K, Other Weapons 100K, and Hands-Feet-Fists 100K respectively. The median

values were close to those of the means at 25.22; 16.00; 23.91; and 31.48. The relatively small

differences between the mean and median indicated that the value distribution for these predictor

variables should be somewhat normal. The standard deviations, however, indicated otherwise.

For the remaining five predictor variables the table showed the means at 2.69; 384.40; 0.4537;

52.77; and 1.31 for Region, Pop Density per sq mi, Gini Coefficient of Wealth Disparity 2010,

Percent Gun Ownership AVG 2001-2010, and Guns & Ammo Best States For Gun Owners 2013

respectively. The median values for those variables were 3.00; 101.2; 0.454; 53.1; and 1.00. As

with the first four predictor variables, the differences between the means and medians for the last

five variables was relatively small with the exception of population density and gun control by

state.

The skewness and kurtosis scores for the first predictor variable, -0.25 and -0.54, indicated a

positive skew of the data and a somewhat platykurtic (flat) distribution. This implied that a high

portion of the perpetrators of violent crimes used firearms for the period analyzed. The relatively

flat distribution indicated that the ratio of violent crimes committed with firearms is fairly well

distributed across all 49 cases.


The z-skew and z-kurtosis scores for the predictor variables of violent crimes were -0.078 and -

0.81 for Percent Violent Crimes by Firearm 100K; 3.33 and 3.43 for Cutting Instruments 100K;

2.50 and 1.56 for Other Weapons 100K; and 2.73 and 2.96 for Hands-Feet-Fists 100K.

z = s/Se skew;

z = s/Se kurtosis

Percent Violent Crime by Firearm 100K

-0.78 = -0.260/0.340; -0.78 < |1.96|

-0.81 = -0.542/0.668; -0.81 < |1.96|

Percent Violent Crime by Cutting Instrument 100K

3.33 = 1.134/0.340; 3.33 > |1.96|

3.43 = 2.296/0.668; 3.43> |1.96|

Percent Violent Crime by Other Weapons 100K

2.50 = 0.852/0.340; 2.50 > |1.96|

1.56 = 1.045/0.668; 1.56 < |1.96|

Percent Violent Crime Hands-Feet-Fists 100K

2.73 = 0.931/0.340; 2.73 > |1.96|

2.96 = 1.981/0.668; 2.96 > |1.96|

Data for violent crimes by firearm met the normality standard, but the other three

variables did not and are non-parametric.

The z-skew and z-kurtosis scores were -0.90 and -1.61 for Region; 20.32 and 72.03 for Pop

Density per sq mi; 3.03 and 4.13 for Gini Coefficient of Wealth Disparity 2010; -2.71 and 1.07

for Percent Gun Ownership AVG 2001-2010; and 5.77 and 3.24 for Guns & Ammo Best States

for Gun Owners 2013.

z = s/Se skew;

z = s/Se kurtosis

Region

-0.90 = -0.301/0.333; -0.90 < |1.96|

-1.61 = -1.061/0.656; -1.61 < |1.96|

Pop Density per sq mi

20.32 = 6.768/0.333; 20.32 > |1.96|

72.03 = 47.257/0.656; 72.03 > |1.96|

Gini Coefficient of Wealth Disparity 2010

3.03 = 1.011/0.333; 303 > |1.96|

4.13 = 2.713/0.656; 4.13 > |1.96|

Percent Gun Ownership AVG 2001-2010

-2.71 = -0.904/0.333; -2.71 > |1.96|

1.07 = 0.705/0.656; 1.07 < |1.96|

Guns & Ammo Best States for Gun Owners 2013

5.77 = 1.924/0.333; 5.77 > |1.96|

3.24 = 2.127/0.656; 3.24 > |1.96|


Based on the z-skew and z-kurtosis

scores only two predictor variables –

Percent Violent Crime by Firearm

100K and Region – had parametric

data. All others were non-parametric.


The Kologorav-Smirnov (KS) Test suggested that the data was not normally distributed, despite the z-skew

and z-kurtosis scores that suggested otherwise. Given this fact the analysis may need to be run as non-

parametric.


Test 2: Outliers

The District of Columbia stood out as

outlier among states with strict gun

control measures. The District’s high

murder might be explained by the fact

that the effective population on any

given day is much higher than a count

of residents. It might be more useful in

future studies to split the District’s data

evenly between Virginia and Maryland.

Again the District of Columbia stood

out among states in Region South as

did, to a lesser extent, Louisiana.


Test 4: Analysis:

Descriptive statistics:

And outliers (District of Columbia and Louisiana) removed:

After removing the District of Columbia and Louisiana the range for Murders 100K dropped

from 20.27 to 7.27. The mean also dropped from 4.51 to 4.03. Both skewness and kurtosis

dropped significantly, as well, from 3.03 to 0.131 and 14.50 to -1.22.

The mean and skew change indicated a normalization of the data. The mean Percent Violent

Crimes by Firearm 100K dropped by a value of 0.3. The means for of Percent Violent Crimes by

Cutting Instruments 100K, Other Weapons 100K and Hands-Feet-Fists 100K increased by 0.05;

0.16; and 0.08 points respectively. This result suggested that the District of Columbia and

Louisiana carried disproportionate weight in the sample. The concomitant decrease in the means

of Murders 100K and Percent Violent Crime by Firearm 100K along with the increase of means


for Percent Violent Crimes by Cutting Instruments, Other Weapons, and Hands-Feet-Fists 100K

suggested a strong relationship between firearms and murder.

Further, the z-skew and z-kurtosis for Murders 100K and Percent Violent Crime by Firearm

100K met the normality standard (|1.96|).

Murders 100K

0.37 = 0.131/0.347; 0.37 < |1.96|

0.17 = -1.224/0.681; 0.17 < |1.96|

The change in skewness and kurtosis for the predictor variables Percent Violent Crimes by

Firearm 100K, Cutting Instruments 100K, Other Weapons 100K and Hands-Feet-Fists 100K was

mixed with a decline in skewness and kurtosis for Firearms. The z-skew and z-kurtosis scores

were:

Percent Violent Crime by Firearm 100K

-0.51 = -0.179/0.347; -0.51 < |1.96|

-0.80 = -0.550/0.681; -0.80 < |1.96|

Percent Violent Crime by Cutting Instrument 100K

3.14 = 1.09/0.347; 3.14 > |1.96|

3.10 = 2.117/0.681; 3.10 > |1.96|

Percent Violent Crime by Other Weapons 100K

2.31 = 0.805/0.347; 2.31 > |1.96|

1.46 = 0.996/0.681; 1.46 < |1.96|

Percent Violent Crime by Hands-Feet-Fists 100K

2.55 = 0.888/0.347; 2.55 < |1.96|

2.59 = 1.769/0.681; 2.59 < |1.96|

With outliers removed the Murders 100K and Percent Violent Crime by Firearm 100k

were parametric. Percent Violent Crime by Cutting Instruments 100K, Other Weapons

100K and Hands-Feet-Fists 100K remained non-parametric.

The z-skew and z- kurtosis scores for the predictor variables Region, Pop Density per sq mi, Gini

Coefficient of Wealth Disparity 2010, Percent Gun Ownership AVG 2001-2010 and Guns &

Ammo Best States for Gun Owners 2013 were:

Region

-0.76 = -0.261/0.340; -0.76 < |1.96|

-1.71 = -1.143/0.668; -1.71 < |1.96|


6.87 = 2.339/0.340; 6.87 > |1.96|

7.85 = 5.404/0.688; 7.85 > |1.96|



0.53 = 0.181/0.340; 0.53 < |1.96|

-0.40 = -0.270/0.668; -0.40 < |1.96|


-2.65 = -0.902/0.340; -2.65 > |1.96|

0.99 = 0.663/0.668; 0.99 < |1.96|

Guns & Ammo Best States for Gun Owners 2013

6.10 = 2.077/0.340; 6.10 > |1.96|

4.27 = 2.853/0.668; 4.27 > |1.96|

Region and Gini Coefficient of Wealth Disparity 2010 were normal, or parametric. All

others were non-parametric.

Check for additional outliers


A check for additional outliers revealed for the predictor variable Percent Violent Crimes by

Cutting Instruments 100K both South Dakota and New Hampshire stood out. With Pop Density

per sq mi, the states of New Jersey, Rhode Island, Massachusetts, Connecticut and Maryland

were outliers; and Rhode Island, Massachusetts and Hawaii stood out from the rest of the data in

Percent Gun Ownership AVG 2001-2010.

Additional outliers removed:

After the additional outliers were removed, the z-skew and z-kurtosis scores were:

Murders 100K

-0.54 = -0.207/0.383; -0.54 < |1.96|

-1.93 = -1.451/0.75; -1.93 < |1.96|

Percent Violent Crimes by Firearm 100K

-0.52 = -0.202/0.383; -0.52 < |1.96|

-0.94 = -0.706/0.75; 0.94 < |1.96|


Percent Violent Crimes by Cutting Instruments 100K

0.66 = 0.253/0.383; 0.66 < |1.96|

0.36 = 0.27/0.75; 0.36 < |1.96|

Percent Violent Crimes by Other Weapons 100K

0.78 = 0.3/0.383; 0.78 < |1.96|

-0.84 = -0.634/0.75; -0.84 < |1.96|

Percent Violent Crime by Hands-Feet-Fists 100K

0.38 = 0.146/0.383; 0.38 < |1.96|

-0.61 = -0.462/0.75; -0.61 < |1.96|


3.60 = 1.381/0.383; 3.6 > |1.96|

2.00 = 1.504/0.75; 2 > |1.96|


0.33 = 0.13/0.383; 0.33 < |1.96|

0.21 = 0.163/0.75; 0.21 < |1.96|


-0.89 = -0.344/0.383; -0.89 < |1.96|

-0.08 = -0.067/0.75; -0.08 < |1.96|

According to z-skew and z- kurtosis scores all scale variables were parametric, or normal,

except population density after removal of all outliers. See histograms on the next page(s).




Test 5: One-way analysis of variance (ANOVA) with all cases:

H0: The means of Murders 100K are not significantly different among U.S. regions.

H1: The means of Murders 100K are significantly different among U.S. regions.

The standard deviation of Region

South was more than two times the

value of the standard deviation for

the other regions. This suggests

heterogeneity of variance.

The Levene Test statistic (p = 0.554; p >

0.01) was significant, however and

indicates homogeneity of variance.

The Robust Test of Equality of Means

suggests, however, that there was a

significant difference in the means.

Welch statistic (4.6; df1 = 3; df2 =

24.37; p < 0.05) and Brown-Forsyth

statistic (9.12; df1 = 3; df2 = 34.88; p <

0.05)


The Games-Howell

statistic showed a

significant difference in

the means of region

South with all other

regions.

The graphic above shows the difference of the

Region South mean versus the other region

means.

The harmonic means table to the right

excluded Region South from Regions

Northeast, Midwest and West.


One-way ANOVA with outliers removed:


The same conclusion was reached without outliers as was with the outliers. The null

hypothesis is rejected, the alternative hypothesis as assumed - there is significant

difference in the means of rate of murder for the four U.S. regions.


The mean error bar chart confirms what the ANOVA tests have shown. The mean of

murder rate for the region South exceeds those of the other three regions by a significant

margin.


Summary of tests 1 through 5:

After removing the District of Columbia, Louisiana, South Dakota, New Hampshire, New

Jersey, Rhode Island, Massachusetts, Connecticut, Maryland and Hawaii as outliers the data for

the scale variables was parametric, or normally distributed, with the exception of population

density. (It is worth noting here that while both Washington DC and Louisiana had high murder

rates - very high in the case of DC - South Dakota, the Northeastern states and Hawaii had

relatively low rates of murder and violent crimes perpetrated with firearms) The ANOVA tests,

on the other hand, showed a heterogeneity of variance among the means of murder rate by

region. Since we are not testing the impact of region on murder rate, but rather the impact of

firearms on violent crime, this is somewhat a moot point at this stage of analysis.

Ideally we should keep the model with outliers removed because this provides more normally

distributed data while still leaving a large sample size. Further study as to the efficacy of gun

control laws on reducing violent crime, or the rate of violent crime by region would be

worthwhile.

Confirming the results of tests:

The Regions had Mean Ranks from

12.25 to 17.00 and the Kruskal-Wallis

Test indicated that we can maintain the

alternate hypothesis: there is variance

of means for Region. (Chi-Square

11.26; df 3; p < 0.05)


Multiple Regression:

Hypotheses:

H0: The type of weapon used in the commission of a violent crime will not predict the U.S.


H1: The use of a firearm in the commission of a violent crime will predict the U.S. murder rate

more accurately than the mean.

H2: The use of a cutting instrument in the commission of a violent crime will predict the U.S.


H3: The use of other weapons in the commission of a violent crime will predict the U.S. murder


H4: The use of hands-feet-fists in the commission of a violent crime will predict the U.S. murder


H5: Region will predict the U.S. murder rate more accurately than the mean.

H6: Population density will predict the U.S. murder rate more accurately than the mean.

H7: The Gini Coefficient of Wealth Disparity will predict the U.S. murder rate more accurately

than the mean.

H8: The percentage of gun ownership will predict the U.S. murder rate more accurately than the

mean.

H9: State gun friendliness will predict the U.S. murder rate more accurately than the mean.

H10: Any combination of the above predictor variables will predict the U.S. murder rate more

accurately than the mean.


Correlations:

This Pearson correlation table showed strong positive correlation (rp = 0.744) between the

dependent variable, Murders 100K, and the predictor variable Percent Violent Crime by Firearm

100K. It also showed a strong correlation (rp = 0.640) between the dependent variable and the

predictor variable Gini Coefficient of Wealth Disparity 2010. Pop Density per sq mi had a weak

correlation to the dependent variable as did Percent Gun Ownership AVG 2001-2010.

Surprisingly, the predictor variables Percent Violent Crimes by Cutting Instruments, Other

Weapons and Hands-Feet-Fists 100K all had moderate to weak negative correlation to the

dependent variable.

Likewise there was a strong correlation between Percent Violent Crimes by Firearm 100K and

Gini Coefficient of Wealth Disparity 2010. This should not be surprising as they both have a

strong correlation with the dependent variable. Percent Violent Crimes by Firearm 100K shared

a strong negative correlation with Hands-Feet-Fists 100K.

●


Pop Density per sq mi was the only non-parametric scale variable and a Spearman’s rho

correlation test shows a weak correlation (rs = 0.352; p < 0.05).

●

●

The scatter plots on the following page(s) provide a visual representation of the correlations.


As displayed in the scatter plots above there was a strong positive correlation between the

dependent variable, Murders 100K, and Percent Violent Crime by Firearm 100K. The other

three weapon variables had weak, moderate and strong negative correlation with Murders 100K.

In the upper right corner of each plot is the r2, or the square of the correlation coefficient. This

value tells us the amount of shared variance between the two variables. The r2 value for Murders

100K and Percent of Violent Crimes by Firearms 100K was 0.554 meaning that approximately

55% of the variance in Murders 100K was attributable to the presence of firearms. (Note: this

number by itself does not suggest if the attribution is positive or negative)

●


Here we see a positive moderate

correlation between Murders 100K and

Pop Density per sq mi.

There was a strong positive correlation

between the Gini Coefficient of Wealth

Disparity and Murders 100K; and the

r2 value of 0.41 suggest that

approximately 41% of the variance in

Murders 100K can be explained by this

predictor variable.


●

Delaware and New York appeared to be outliers for Pop Density per sq mi. The boxes for

the other variables are comparatively concentrated.


Stepwise regression:

Here we have a general overview of the

dependent variable and predictor

variables. Again, after removing the

outliers the scale variables were all

parametric with the exception of

population density.


Again, the correlation table showing the covariance between the predictor variables and the dependent

variable. The second row shows the significance, or confidence, of each correlation. Notably, the predictor

variables of Percent Violent Crime by Other Weapons 100K, Region and Percent Gun Ownership AVG 2001-

2010 all have significance values greater than 0.05, implying that we cannot be certain that the correlation

values for those variables are accurate or indicative of a true relationship.


The Model Summary displays the most salient point of the multiple regression. Here we see the r and r2

values shown in the correlation table. The table also shows the adjusted r2 value which corresponds, roughly,

to the r2. Most importantly, this tables shows us how each of the predictor variables interact with the

dependent variable independently of one another.

So we see that in Model 1 that Percent Violent Crime by Firearm 100K has the strongest correlation with

Murders 100K of all the predictor variables and was tested first. As already seen in the correlation table this

predictor variable had an r value, or covariance, with Murders 100K of 0.744. With an r2 value of 0.554 it

could account for roughly 55% of the variance (up or down movement) in Murders 100K.

Model 2 tests both Percent Violent Crimes by Firearm 100K and the Gini Coefficient of Wealth Disparity

2010. The results show a change in the r and r2 to 0.797 and 0.635. Looking in the r2 changed column we see

that including the second variable adds about 0.081, or 8%, to the explanatory power of the first predictor

variable.

Model 3 includes Region in the test, which adds about 4% to explaining the variance in murder rate.

The Sig. F change column tells us the significance of adding the next variable and the Durbin-Watson value

tells us if the predictor variables are auto-correlating. The value of 2.02 in that column, however, tells us that

this is unlikely.

This table displays the order in which

the three predictor variables were

entered into the analysis.


This coefficients table provides us with data necessary to test our model.

Our multiple linear regression formula is y = B0 + b1(x1) + b2(x2) + b3(x3)

In the far right two columns is the measure to determine whether or not our data violates the assumption of

multicollinearity (whether or not there is too strong a relationship between the predictor variables). The

values of which suggest that we can assume multicollinearity.

Tolerance b1 > 0.2 VIF b1 < 10



The ANOVA table confirms the

findings on the previous tables. The

Significance column’s (Sig.) values are

all 0.000, or 0.001, indicating that the

models predict the murder rate than

does the mean.


The excluded variables

table tells us which

variables were not used

in the Stepwise analysis

due to their low

correlation and

confidence values.

These figures are seen

in the Sig and Partial

Correlation columns.

Our residuals, or the difference

between the predicted and observed

outcomes, appear fairly small.


In the histogram to the left we see that

the Regression Standardized Residual

somewhat follows the normalcy curve.

Additionally, the scatter plot below

shows that most of the cases track the

linear regression line pretty closely.


These three scatter plots show the

correlations between the various

predictor variables used in the Stepwise

analysis and the dependent variable.

Here we can see the strongest

correlation with Percent Violent Crime

by Firearm 100K.


Testing the model

Three cases were selected from random by SPSS (Georgia, Indiana, and Missouri) to test the

accuracy of the regression model. The model gives predicted values and by using the formula

we can see how close these predicted values come to the actual values of murder rate for each

case.

Model: y = B0 + b1(x1) + b2(x2) + b3(x3)

Georgia predicted: 6.63 = -17.642 + 0.120(40.31) + 39.137(0.468) + 0.376(3)

Georgia actual: 5.83

Difference between predicted and actual: +0.80; or approximately 14%

Difference between mean and actual: -1.54; or approximately 26%

Indiana predicted: 3.794 = -17.642 + 0.120(28.95) + 39.137(0.44) + 0.376(2)

Indiana actual: 4.61

Difference between predicted and actual: -0.81; or approximately 18%


Missouri predicted: 5.29 = -17.642 + 0.120(33.38) + 39.137(0.455) + 0.376(3)

Missouri actual: 6.52



Testing high, middle and low

South Carolina predicted: 5.41 = -17.642 + 0.120(32.36) + 39.137(0.461) + 0.376(3)

South Carolina actual: 7.03



New York predicted: 4.73 = -17.642 + 0.120(20.63) + 39.137(0.499) + 0.376(1)

New York actual: 4.19


Difference between mean and actual: +0.1; or approximately 2%

Iowa Predicted: 1.28 = -17.642 + 0.120(12.21) + 39.137(0.427) + 0.376(2)

Iowa Actual: 1.5



In four of the six cases the model predicted the dependent variable value more accurately

than the mean. The two cases where the mean was closer to the dependent variable value

were when that value was already close to the mean. So far, this seems to be a good model.

●


Check for outliers:

The Cook’s Distances are all less than the threshold value of 1, so no outliers are indicated

with that measure. However, the Mahalonobis Distance for case 1, New York, suggests that

it may be an outlier. The next highest in that category was case 11, Maine. The next three

cases with the highest Mahalanobis Distance values were case 4, Wisconsin, case 12, Utah,

and case 3, Mississippi. We will remove these five cases (removing more might make the

sample too small) and run the Stepwise regression again in order to determine if their

removal makes for a more accurate model.

●


Regression with outliers removed:

After removing five cases our new

number of cases was 33. The mean has

increased by approximately 0.15. The

correlation, or r value, for the

dependent variable and the predictor

variables, Percent Violent Crime by

Firearm 100K and Gini Coefficient of

Wealth Disparity 2010, remain strong.

The predictor variable Region,

however, decreased markedly.


Only Percent Violent Crimes by

Firearms 100K and Region were

entered into the Stepwise analysis.

Their combined r2 values amount to

approximately 63% of the variance

seen in Murders 100K with Region

adding very little.




Retesting the model:

Y = B0 + b1(x1) + b2(x2)

Georgia predicted: 7.00 = -2.128 + 0.190(40.31) + 0.493(3)

Georgia actual: 5.83



Indiana predicted: 4.25 = -2.128 + 0.190(28.95) + 0.493(2)

Indiana actual: 4.61



Missouri predicted: 5.69 = -2.128 + 0.190(33.38) + 0.493(3)

Missouri actual: 6.52



Retesting high, middle, low:

South Carolina predicted: 5.49 = -2.128 + 0.190(32.36) + 0.493(3)

South Carolina actual: 7.03



Alaska predicted: 3.47 = -2.128 + 0.190(19.11) + 0.493(4)

Alaska actual: 4.44


Difference between mean and actual: 0

Iowa predicted: 1.17 = -2.128 + 0.190(12.21) + 0.493(2)

Iowa actual: 1.5



As with the previous model, four of the six predicted the value of Murders 100K better

than the mean.


Determining the better fit:

In the first model the aggregate value of percent difference between the predicted and actual

values of Murders 100K for all six case tests was 102. The aggregate value of percent difference

between the mean and actual values was 294. The ratio between the two values was

approximately 1:2.88.

In the second model the aggregate value of percent difference between the predicted and actual

values of Murders 100K for all six case tests was 107. The aggregate value of percent difference

between the mean and actual values was 293. The ratio between the two values was 1:2.73.

The predictor is closer to the actual in the first model relative to the mean than in the second

model. The first model is more accurate. (2.88 > 2.73)

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Documents

Firearms and Violent Crime Lethality in the United States