Estimation of Crimes in the USA in 1990-92 Regression Model Analysis

This report employs linear and distribution-based regression models to estimate crime rates (crm1000) in the USA during 1990-92. It addresses the issue of crime calculation based on limited offenses when the population is less than the infringement percentage. The study examines crime extent across 440 county districts in four US states, comparing Multiple Linear Regression (MLR) with Poisson and Negative Binomial regression models. The analysis reveals that the Negative Binomial Model provides a better fit for estimating crm1000. The methodology includes correlation analysis, outlier detection, and stepwise MLR, alongside Poisson and Negative Binomial regressions, using an 80:20 training/testing data split of US county demographic information from 1990-92 to ensure prediction accuracy.

RUNNING HEAD: ESTIMATION OF CRIMES IN THE USA IN 1990-92 – A COMPARATIVE ANALYSIS OF
MULTIPLE LINEAR REGRESSION WITH POISSON/NEGATIVE BINOMIAL REGRESSION MODELLING
Estimation of Crimes in the USA in 1990-92 – A Comparative
Analysis of Multiple Linear Regression with Poisson/Negative
Binomial Regression Modelling

ESTIMATION OF CRIMES IN THE USA IN 1990-92 – A COMPARATIVE ANALYSIS OF MULTIPLE LINEAR
REGRESSION WITH POISSON/NEGATIVE BINOMIAL REGRESSION MODELLING
Abstract
This document introduces the use of linear as well as distribution-based regression
models as a tool to estimate problems in an exhaustive analysis of crime rates (crm1000) in
USA during 1990-92. If the population of the unit is less than the percentage of the
infringement, the crime must be calculated on the basis of a limited number of offences.
Regression models based on criminal activity per thousand people are desirable because they
are based on misallocation assumptions that are consistent with the nature of the event count.
To illustrate the application and benefits of this method, the extent of the crimes in 440
counties districts of four states in the USA is described in this document. The Negative
Binomial Model was found to be a better fit than Poisson and Multiple Linear Regression
(MLR) models in estimation analysis for crm1000.
2

ESTIMATION OF CRIMES IN THE USA IN 1990-92 – A COMPARATIVE ANALYSIS OF MULTIPLE LINEAR
REGRESSION WITH POISSON/NEGATIVE BINOMIAL REGRESSION MODELLING
Table of Contents
Abstract......................................................................................................................................2
Introduction................................................................................................................................3
Problem Statement.....................................................................................................................5
Methodology..............................................................................................................................5
Descriptive Summary.................................................................................................................6
Multiple Regression Analysis....................................................................................................9
Outlier Detection....................................................................................................................9
Correlation Analysis.............................................................................................................11
Stepwise MLR with Addition/Removal of Covariates.........................................................13
Poisson Regression...................................................................................................................17
Negative Binomial Regression.................................................................................................19
Conclusion................................................................................................................................21
References................................................................................................................................22
Appendices...............................................................................................................................23
Introduction
3

ESTIMATION OF CRIMES IN THE USA IN 1990-92 – A COMPARATIVE ANALYSIS OF MULTIPLE LINEAR
REGRESSION WITH POISSON/NEGATIVE BINOMIAL REGRESSION MODELLING
The purpose of this document is to establish a statistical methodology for analysing
the total number of crimes, and to address the problems caused by small population size with
regression modelling. Overall, sample units are a collection of people from 4 regions
(Northeast, Midwest, South, and West) consisting of 440 counties of the United States in
1990-92, and researcher was interested in clarifying the differences in crime among these
four regions. These crime rates were defined as the number of criminal cases, which were
divided by population and were reported as crimes of 1000 by the residents. The standard
method for analysing the per capita crime rate was the use of aggregate as the variable of
population characteristics, beds in hospitals, professional physicians, poverty level, income
level, and education level of people in ordinary least square regression. For this article, the
method of least squares has been applied on the training data with a validation with the test
data, which was partitioned in an 80:20 ratio from the original dataset. This document
describes how to resolve the issue of accuracy in prediction by using a Poison, and negative
Binomial regression models that were ideal for this type of data. In criminology, regression
models have become commonplace in criminal profession analysis, but rarely apply to
general analysis of crime or other social phenomena. To make these methods available to a
wider range of researchers, this article was dedicated to developing the specific benefits of
these models to solve problems in a comprehensive data analysis (Bretz, Westfall, &
Hothorn, 2016; Wandel, Schmidli, & Neuenschwander, 2016).
4

ESTIMATION OF CRIMES IN THE USA IN 1990-92 – A COMPARATIVE ANALYSIS OF MULTIPLE LINEAR
REGRESSION WITH POISSON/NEGATIVE BINOMIAL REGRESSION MODELLING
Problem Statement
The potential crime rate for a particular population as a whole was crime
corresponding to the rate of crime in general. The crime arte in the USA during 1990-92 has
been assessed, and an effort to estimate the crime rate for future trends has been done by
regression modelling.
The aim of the research was to estimate crime per thousand people on percentage of
population aged 18–34 (“pop1834”), population aged 65 years old or older (“pop65plus”),
professionally active non-federal physicians (“phys”), total number of beds, cribs and
bassinets (“beds”), percentage of adults who completed at least 12 years of school
(“higrads”), percentage of adults with bachelor’s degree (“bachelors”), population with
income below poverty level (“poors”), unemployed labour force, per capita income (dollars)
(“percapitaincome”), and total personal income of 1990 CDI population (in millions of
dollars) (“totalincome”). The estimation was analysed based on four geographical regions
(“regions”) used by the U.S. Bureau of the Census.
Methodology
The crime rate per 1000 people was calculated, and the normality of the variable was
scrutinized for appropriateness of regression models. This was the response variable of the
study. The correlation analysis was conducted to find the associated significant predictors.
First, the Multiple Linear Regression Model was constructed with all the significant
correlated factors. Then in a backward regression approach, the non-significant predictors
have been eliminated one-by-one to find the suitable and significant model. The final linear
model was again revaluated with simple regression models, separately with gradual inclusion
of each of the predictors, and the changes in coefficient of determinations were noted.
Secondly, Poisson Regression was also used to simulate count data for crime rate. At last, a
5

ESTIMATION OF CRIMES IN THE USA IN 1990-92 – A COMPARATIVE ANALYSIS OF MULTIPLE LINEAR
REGRESSION WITH POISSON/NEGATIVE BINOMIAL REGRESSION MODELLING
negative Binomial Regression has been used as it has the same average structure as the
Poisson regression and region as an additional parameter was applied to model the over-
dispersion in Poisson regression model (Osgood, 2000).
The data set of the research was the U.S. county demographic information on 1990-
92. The data set was partitioned in 80:20 ratios for training and testing datasets. The final
linear regression model was checked against the test dataset for appropriate predicting power
of the model.
Descriptive Summary
Crime per thousand people (crm1000) was calculated as crm1000 = (crimes/popul)
∗1000. The average crime rate (crm1000) was 57.29 (SD = 27.33) per thousand people,
considering all regions together. Region wise scrutiny revealed that average crime rate was
41.23 (SD = 31.12) in Northeast, 51.23 (SD = 23.79) in Midwest, 70.74 (SD = 24.41) in
South, and 60.88 (SD = 15.94) in West USA. The Shapiro-Wilk test was used to check for
the normality of the response variable. The test result (W = 0.9, p < 0.05) indicated that
crm1000 was did not follow a normal distribution. Multiple linear regression for crm1000 on
“pop1834”, “phys”, “beds”, “poors”, “totalincome”, “higrads”, and “region”. The residuals
were found be highly right skewed (S = 2.15). Shapiro-Wilk test was conducted to confirm
the pattern of the residuals, and the test statistic (W = 0.88, p < 0.05) revealed that residuals
were not distributed normally. Boxplot in Figure 1 affirmed that the data was affected by
presence of outliers, which also adversely affected the multiple regression models. The
sample size was 440, and applying Central Limit theorem the normality was assumed. Table
1 presents the detailed descriptive summary of the continuous variables. The region wise
details revealed that West region of the USA was greater in geographical area and total
6

ESTIMATION OF CRIMES IN THE USA IN 1990-92 – A COMPARATIVE ANALYSIS OF MULTIPLE LINEAR
REGRESSION WITH POISSON/NEGATIVE BINOMIAL REGRESSION MODELLING
income compared to other three regions. Interestingly, crime rate in West was lower than
South region. From skewness of crm1000 in Northern region it was possible to infer the
presence of outliers. The Northern region was the second largest region after West, but it had
the lowest crime rates (M = 41.23, SD = 31.12). Western region also had the highest average
number of physicians with highest number of beds during 1990.
Figure 1: Boxplot for Residuals for crm1000 for the main dataset
7

ESTIMATION OF CRIMES IN THE USA IN 1990-92 – A COMPARATIVE ANALYSIS OF MULTIPLE LINEAR
REGRESSION WITH POISSON/NEGATIVE BINOMIAL REGRESSION MODELLING
Table 1: Region wise Descriptive Summary of the Variables in US Demographic Data
Note: Crime rate = crm1000 = (crimes/popul)
∗1000
8
Variables M SD Skewness M SD Skewness M SD Skewness M SD Skewness
Area 655.51 452.52 3.07 633.07 602.87 7.70 726.62 445.31 1.90 2751.75 3015.78 3.02
Population 395834.52 369150.40 2.39 346171.56 555676.61 6.42 329003.89 359229.52 3.73 581282.18 1082601.03 6.22
Population aged 18–34 (%) 27.97 3.41 1.86 28.52 4.19 1.87 29.21 4.97 0.78 28.19 3.27 -0.12
Population aged 65 years old or older (%) 13.64 2.79 0.76 11.66 2.41 -0.12 11.97 5.59 1.99 11.31 2.70 1.40
Professionally active nonfederal physicians 1094.47 1444.99 2.18 860.42 1729.86 5.82 821.13 1163.53 3.17 1353.92 2927.07 6.15
Number of beds 1508.10 1793.49 2.91 1477.65 2532.04 5.44 1352.27 1708.67 3.24 1575.73 3329.31 6.63
Higrad - 12 years of school (%) 77.57 5.82 -0.46 79.36 5.54 0.23 75.20 7.55 -0.83 79.69 7.87 -1.06
Bachelor’s degree (%) 21.78 7.22 0.30 19.78 7.46 0.88 21.04 8.20 1.53 22.05 7.25 0.54
Income below poverty level 6.53 3.37 1.25 7.90 3.34 0.63 10.57 5.65 1.66 9.15 4.05 1.03
Force which is unemployed 7.17 1.76 0.54 6.33 2.16 0.49 6.19 1.92 2.16 7.00 3.53 1.92
per capita income dollars) 20598.77 4635.07 0.94 18301.09 2745.45 1.36 17486.99 3843.83 1.13 18322.58 4276.34 1.51
Total personal income (million dollars) 8734.16 8708.12 1.83 6835.58 12047.38 6.49 6129.98 7408.97 3.45 11595.61 22827.03 6.06
Crime Rate (per 1000 people) 41.23 31.12 5.71 51.11 23.79 1.01 70.74 24.41 0.39 60.88 15.94 0.44
Northeast Midwest South West

ESTIMATION OF CRIMES IN THE USA IN 1990-92 – A COMPARATIVE ANALYSIS OF MULTIPLE LINEAR
REGRESSION WITH POISSON/NEGATIVE BINOMIAL REGRESSION MODELLING
Table 2: Correlation Matrix for the USA demographic data
Note: The variables corresponding to highlighted correlation coefficients are considered for
linear regression modelling
9
Correlations area popul pop1834 pop65plus phys "beds" higrads bachelors poors unemployed percapitaincome "totalincome" region crm1000
area 1
popul .173 ** 1
pop1834 -.055 .078 1
pop65plus .006 -.029 -.616
** 1
phys .078 .940
** .120
* -.003 1
"beds" .073 .924
** .075 .053 .950 ** 1
crimes .129 ** .886
** .090 -.035 .820 ** .857 **
higrads -.099 * -.017 .251 ** -.268
** -.004 -.112
* 1
bachelors -.137 ** .147
** .456 ** -.339
** .237 ** .100 * .708 ** 1
poors .171 ** .038 .034 .007 .064 .173 ** -.692 ** -.408 ** 1
unemployed .199 ** .005 -.279
** .236 ** -.051 .008 -.594 ** -.541 ** .437 ** 1
percapitaincome -.188 ** .236
** -.032 .019 .316 ** .195 ** .523 ** .695 ** -.602 ** -.322 ** 1
"totalincome" .127 ** .987
** .071 -.023 .948 ** .902 ** .043 .222 ** -.039 -.034 .348 ** 1
region .363 ** .069 .052 -.173
** .025 -.003 -.010 .020 .271 ** -.054 -.222 ** .038 1
crm1000 .043 .280
** .191 ** -.067 .308 ** .392 ** -.226 ** .038 .472 ** .042 -.080 .228
** .343
** 1
**. Correlation is significant at the 0.01 level (2-tailed). *. Correlation is significant at the 0.05 level (2-tailed).

ESTIMATION OF CRIMES IN THE USA IN 1990-92 – A COMPARATIVE ANALYSIS OF MULTIPLE LINEAR
REGRESSION WITH POISSON/NEGATIVE BINOMIAL REGRESSION MODELLING
Multiple Regression Analysis
Outlier Detection
Crime rate crm1000 was right skewed, primarily due to outlier values in crimes in
Northern region counties in the USA. 4 outlier values (Appendices Figure 9) of the
(crm1000) response variable were observed in the distribution. The impact of the outliers on
the fit of the regression model was apprehended. High right skewness for number of beds,
number of non-federal physicians, population in 1990, and total income were probable
problems in the estimation of crime rate of USA in 1990-92. Comparatively, population of
18-34 years and population below poverty level had well spread distributions. Distribution of
25 years or older people having education for at least 12 years in school was noted to have
negative skewness, from Figure 4.
Figure 2: Histogram of crm1000 and population of 18-34 years
10

ESTIMATION OF CRIMES IN THE USA IN 1990-92 – A COMPARATIVE ANALYSIS OF MULTIPLE LINEAR
REGRESSION WITH POISSON/NEGATIVE BINOMIAL REGRESSION MODELLING
Figure 3: Histograms for beds and non-federal physicians
Figure 4: Histograms of poor and population n in 1990
Figure 5: Histograms of HI grads and total income
11

ESTIMATION OF CRIMES IN THE USA IN 1990-92 – A COMPARATIVE ANALYSIS OF MULTIPLE LINEAR
REGRESSION WITH POISSON/NEGATIVE BINOMIAL REGRESSION MODELLING
Correlation Analysis
A correlation matrix with Pearson’s correlation coefficient was constructed. The
significant correlations can be identified from Table 2. The significantly correlated variables
with crm1000 were noted to be estimated population in 1990, percentage of population aged
18–34, professionally active non-federal physicians, total number of beds, cribs and
bassinets, percentage of adults who completed at least 12 years of school, population with
income below poverty level, regions, and total personal income of 1990 CDI population (in
millions of dollars). These predictors were fitted in the first regression model to estimated
crm1000. From correlation between the predictors, a high correlation (positive) was observed
between population in 1990 and professionally active non-federal physicians (r = 0.94, p <
0.05). Total number of beds also had a significantly high correlation (positive) with
population (r = 0.92, p < 0.05), at 5% level of significance. Also, active non-federal
physicians had a high positive and significant correlation with total number of beds (r = 0.95,
p < 0.05), whereas “poors” and “higrads” had a significantly high negative correlation (r = -
0.69, p < 0.05). These high correlation coefficients between the predictor variables were an
indication of possible multicollinearity in the regression models.
Figure 6: Scatter Diagrams for Crime on population aged 18-34 and population in 1990
12

ESTIMATION OF CRIMES IN THE USA IN 1990-92 – A COMPARATIVE ANALYSIS OF MULTIPLE LINEAR
REGRESSION WITH POISSON/NEGATIVE BINOMIAL REGRESSION MODELLING
Figure 7: Scatter Diagrams for Crime on non-federal physicians and Boxplot for Crime rate
in four regions
Figure 8: Scatter Diagrams for Crime on poor below poverty level and number of beds
Figure 9: Scatter Diagrams for Crime on HI school graduates and total personal Income
13

ESTIMATION OF CRIMES IN THE USA IN 1990-92 – A COMPARATIVE ANALYSIS OF MULTIPLE LINEAR
REGRESSION WITH POISSON/NEGATIVE BINOMIAL REGRESSION MODELLING
Stepwise MLR with Addition/Removal of Covariates
Multiple regression modelling started with eight predictors, “pop1834”, “phys”,
“beds”, “higrads”, “poors”, “totalincome”, and “region”. “Higrads” (t = 1.38, p = 0.169) was
found to be statistically non-significant in predicting crime rate in 1990 (crm1000). The
ANOVA model was statistically significant (F = 32.07, p < 0.05) at 5% level of significance.
From figure 16, the residuals were observed to be equally spread along both sides of the
estimated line. Impact of outliers was noted from the cook’s distance and residual plot with
fitted values in Figure 10. Three major outliers could be identified from Figure 10 (1st and 6t).
Cook’s distance for observation 1 was 4.98, and 0.90 for observation 6. Variation in
independent variables was able to explain 45.77% (coefficient of determination, R2
) variation
of crime rate per thousand population.
Figure 10: Residuals plot versus fitted values
Studentized Breusch-Pagan test was used (BP = 58.29, p < 0.05), and the null
hypothesis assuming homoscedasticity was rejected based on the p-value, which indicated
presence of heteroscedasticity. Leverage in this model due to presence of outlier value was
evident from Figure 17. The Q-Q plot indicated that residuals were almost normal with three
14

ESTIMATION OF CRIMES IN THE USA IN 1990-92 – A COMPARATIVE ANALYSIS OF MULTIPLE LINEAR
REGRESSION WITH POISSON/NEGATIVE BINOMIAL REGRESSION MODELLING
outlier observations (Montgomery, Peck, & Vining, 2012). The estimated linear equation was
found as,
crm 1000=0 . 77∗pop 1834−0 . 006∗phys + 0 . 013∗beds+0 . 35∗higrads+ 1. 49∗poors
−0 . 0007∗totalincome−17. 42∗North east −13. 47∗Midwest +3 . 33∗South−6 . 49
“Higrads” due to insignificance and multicollinearity (due to high correlation with
total income) was removed; and the new MLR model was constructed. All the predictors
were found to be statistically significant, where number of professionally active non-federal
physicians was a significant predictor (t = -2.08, p < 0.05) at 5% level. Total income was a
significant predictor at (t= -2.61, p < 0.01) 1% level of significance. Rest of the predictors
were significant were found to be statistically significant at 0.1% level of significance. The
model was statistically significant (F = 35.75, p < 0.05), and the variation in predictors were
found to assess 45.47% (coefficient of determination, R2
) variation in crm1000. From figure
18, the residuals were observed to be almost equally spread along both sides of the estimated
line. Impact of outliers was noted from cook’s distance in the model. Cook’s distance for
observation 1 was 5.64, and 0.99 for observation 6. Studentized Breusch-Pagan test was used
(BP = 53.16, p < 0.05), and the null hypothesis assuming homoscedasticity was rejected
based on the p-value. Leverage in this model due to presence of first observation in the
training dataset. The Q-Q plot indicated that residuals were almost normal with three outlier
observations. The estimated linear equation was found as,
crm 1000=0 . 913∗pop 1834−0 . 006∗phys + 0 . 013∗beds+1 . 097∗poors−0 . 0008∗totalincome
−19 .27∗North east −14 . 04∗Midwest+ 2. 06∗South−20 . 77
The MLR was framed using “West” as the baseline for region (categorical) variable.
Northeast (t = -5.297, p < 0.05) and Midwest (t= -3.796, p < 0.05) were the two regions
which were found to be statistically significant in predicting crime rate when compared with
15

ESTIMATION OF CRIMES IN THE USA IN 1990-92 – A COMPARATIVE ANALYSIS OF MULTIPLE LINEAR
REGRESSION WITH POISSON/NEGATIVE BINOMIAL REGRESSION MODELLING
the West region in the USA. No significant difference in crime rate for South region
compared to West was found (Hardin, Hardin, Hilbe, & Hilbe, 2007).
The fit of the final MLR model was tested by predicting the crm1000 of the “test”
dataset with “West” as the baseline region. Predicted crm1000 values for test dataset and
actual crm1000 values were compared. The root mean square error for difference between
predicted and actual values was calculated to 19.44 for the test data set with 20%
observations (n = 88) from the original dataset. From figure 11 the predicted versus actual
values of “crm100” for the test data set can be noted. Detail scrutiny of the comparative
analysis revealed that crm1000 values were sometimes over predicted, and some values got
under predicted. Overall fit was found to be quite under fit with average crm100 as 45.87 per
thousand people, whereas, actual mean was approximately 59.25. Presence of outliers in test
data could have a probable reason behind this.
Figure 11: Predicted versus actual crm1000 for test data
16

ESTIMATION OF CRIMES IN THE USA IN 1990-92 – A COMPARATIVE ANALYSIS OF MULTIPLE LINEAR
REGRESSION WITH POISSON/NEGATIVE BINOMIAL REGRESSION MODELLING
Figure 12: Residual versus predicted data for MLR with test dataset
Table 3: Multiple Regression Models with Step wise Approach
17

ESTIMATION OF CRIMES IN THE USA IN 1990-92 – A COMPARATIVE ANALYSIS OF MULTIPLE LINEAR
REGRESSION WITH POISSON/NEGATIVE BINOMIAL REGRESSION MODELLING
Note: The * represents significance at 5% level of significance,
** represents significance at 1% level, and *** represents significance at 0.1% level
Poisson Regression
The count of crimes was modelled where distribution of number of crimes committed
within a population of 1000 for an individual citizen (Cameron, & Trivedi, 2013). The
Poisson distribution with λ parameter generates the Poisson regression, and then relates this
parameter to the explanatory variables, rather than the count. Mean of crm1000 was
evaluated to be 56.79, whereas variance was evaluated as 762.21. Region wise scrutiny
revealed that variance by mean ratios for all the four region were considerably higher than 1.
Hence, Poisson regression was not an appropriate fit for the data, due to the fact that
conditional variance was higher than the mean. Still, the article tried to fit the Poisson model
with an offset “popul/1000” to make crm1000 (continuous in nature) a rate or count variable.
Figure 13: Variance by mean ration of crm1000 for four regions
18

ESTIMATION OF CRIMES IN THE USA IN 1990-92 – A COMPARATIVE ANALYSIS OF MULTIPLE LINEAR
REGRESSION WITH POISSON/NEGATIVE BINOMIAL REGRESSION MODELLING
The scholar initiated the regression model with same eight variables as of MLR model
1, and four regions as the categories of the “region” variable. Here, “higrads” was found to be
statistically significant predictor (t = -5.06, p < 0.05). Also, total income was found to be a
significant predictor (t = -31.10, p < 0.05). “Phys” (t = -0.95, p = 0.342) and “beds” (t =
0.061, p =0.95) were the two insignificant predictors. Removal of these two predictors
improved the model and all the predictors were found to be significant. The regression
equation at this stage was found to be,
crm 1000=1 . 013∗pop 1834+1 . 000∗phys+1. 000∗beds+0 . 992∗higrads+1. 011∗poors
+0 . 999∗totalincome +0 . 722∗North eact +0 . 939∗Midwest +1. 155∗South
The base region was changed to “West”, and the regression model was executed. The
final estimated equation was found as,
crm 1000=1 . 014∗pop 1834 +0 . 992∗higrads+ 0. 999∗total income +1. 011∗poors +0 . 726∗Northeast
+0 . 9433∗Midwest + 1. 159∗South
The final Poisson model was tested for residual deviance with chi square test. The
residual deviation is the difference between deviance of the ideal model (observed and
predicted values are identical) and the current model. The test statistic (res.dev = 4235.83, p <
0.05) indicated that Poisson model was not a good fit for the data, and better model was
required.
19

ESTIMATION OF CRIMES IN THE USA IN 1990-92 – A COMPARATIVE ANALYSIS OF MULTIPLE LINEAR
REGRESSION WITH POISSON/NEGATIVE BINOMIAL REGRESSION MODELLING
Table 4: Poisson Regression models
20

ESTIMATION OF CRIMES IN THE USA IN 1990-92 – A COMPARATIVE ANALYSIS OF MULTIPLE LINEAR
REGRESSION WITH POISSON/NEGATIVE BINOMIAL REGRESSION MODELLING
Negative Binomial Regression
Negative Binomial Regression was used as a modification of the Poisson model, as
the conditional variance was greater than the mean (Figure 13, crm100 given region). This
can be seen as generalization of Poisson regression because it has the same average structure
as the Poisson model and has an additional parameter to simulate excessive dispersion (Hilbe,
2011).
The first model was constructed with the predictors of the first Poisson regression
model. South region in comparison to West was found to have almost similar crime that that
of the West region, and hence was an insignificant predictor (z = 1.85, p = 0.065). The
regression equation was evaluated, where pop1834 was found to be insignificant.
crm 1000=1 . 012∗pop 1834+1 .00∗phys+0. 99∗beds+ 0 .987∗higrads+1 . 019∗poors
+0 . 99∗total income+0 . 636∗North east+0 .942∗Midwest +1 .159∗South+0 .530
In the second model “higrads” (z = -1.65, p = 0.099) was noted to be insignificant at
5% level. Crime rate in Midwest was noted to almost similar to that of South region (t = -
0.428, p = 0.668).
crm 1000=1 . 00∗phys+0 .99∗beds +0 . 99∗higrads+1 . 019∗poors
+0 . 99∗total income+0 . 657∗North east+0 .963∗Midwest +1 .159∗South+0 . 502
The third and the final model consisted of 5 predictors, where region had Northeast
and South as the two significant predictors. The expected log count for Northeast was 0.38
lower than log count of West. The expected difference in log count between West and
Midwest, the reference region was 0.26. To assess if region was itself statistically significant,
the scholar compared the model with and without region. The theta or R parameter (inverse
dispersion parameter ( α )) was equal to 4.29. The third model was,
21

ESTIMATION OF CRIMES IN THE USA IN 1990-92 – A COMPARATIVE ANALYSIS OF MULTIPLE LINEAR
REGRESSION WITH POISSON/NEGATIVE BINOMIAL REGRESSION MODELLING
crm 1000=1 . 00∗phys+ 0 .99∗beds +1. 034∗poors+ 0. 99∗total income+ 0 .694∗North east
+0 . 975∗Midwest + 1. 235∗South+ 0. 502
Comparison of Poisson and Negative Binomial regression coefficients
The likelihood ratio test was found to be significant with p-value less than 0.05,
indicating significance of regions in the previous regression models. Also, conditional
variance was noticeably greater than the mean for crm1000, considering region as a
categorical variable. Hence, Negative Binomial was an appropriate choice compared to
Poisson in estimating crm1000 (Loeys, Moerkerke, De Smet, & Buysse, 2012).
22

ESTIMATION OF CRIMES IN THE USA IN 1990-92 – A COMPARATIVE ANALYSIS OF MULTIPLE LINEAR
REGRESSION WITH POISSON/NEGATIVE BINOMIAL REGRESSION MODELLING
Table 5: Negative Binomial Regression Models
23

ESTIMATION OF CRIMES IN THE USA IN 1990-92 – A COMPARATIVE ANALYSIS OF MULTIPLE LINEAR
REGRESSION WITH POISSON/NEGATIVE BINOMIAL REGRESSION MODELLING
Conclusion
The use of regression modelling is a major step forward in the study of
comprehensive crime data. Standard analysis methods require a large accumulation of data
between attack types or total units. Otherwise, the offensive marker is too small to generate a
ratio per capita with appropriate distribution and sufficient accuracy to demonstrate at least a
least square analysis. Linear regression models provided the researcher with a suitable
method for analysing detailed data. The partitioning of the data set in any other ratio could
inflict a new set of significant predictors in the model. The models of Poisson regression and
then Negative Binomial regression model were able to estimate the crm1000 in a better way
with less residual. From Table 7, it can be noted that the Negative Binomial regression model
was able to predict the actual crm1000 values of the test data more accurately compared to
the linear models. In the present sample, analysis of the rate of crime included murder,
robbery, rape, motor vehicle theft, aggravated assault, larceny-theft, and burglary. Negative
Binomial model offered an estimation model that was very suitable for data, while the
analysis with the MLR produces outliers which had a striking impact on the results (Allison,
2014; Cox, 2018).
24

ESTIMATION OF CRIMES IN THE USA IN 1990-92 – A COMPARATIVE ANALYSIS OF MULTIPLE LINEAR
REGRESSION WITH POISSON/NEGATIVE BINOMIAL REGRESSION MODELLING
References
Allison, P. D. (2014). Event history and survival analysis: Regression for longitudinal event
data (Vol. 46). SAGE publications.
Bretz, F., Westfall, P., & Hothorn, T. (2016). Multiple comparisons using R. Chapman and
Hall/CRC.
Cameron, A. C., & Trivedi, P. K. (2013). Regression analysis of count data (Vol. 53).
Cambridge university press.
Cox, D. R. (2018). Analysis of survival data. Routledge.
Hardin, J. W., Hardin, J. W., Hilbe, J. M., & Hilbe, J. (2007). Generalized linear models and
extensions. Stata press.
Hilbe, J. M. (2011). Negative binomial regression. Cambridge University Press.
Loeys, T., Moerkerke, B., De Smet, O., & Buysse, A. (2012). The analysis of zero‐inflated
count data: Beyond zero‐inflated Poisson regression. British Journal of Mathematical
and Statistical Psychology, 65(1), 163-180.
Montgomery, D. C., Peck, E. A., & Vining, G. G. (2012). Introduction to linear regression
analysis (Vol. 821). John Wiley & Sons.
Osgood, D. W. (2000). Poisson-based regression analysis of aggregate crime rates. Journal of
quantitative criminology, 16(1), 21-43.
Wandel, S., Schmidli, H., & Neuenschwander, B. (2016). Use of Historical Data. Cancer
Clinical Trials: Current and Controversial Issues in Design and Analysis, 37.
25

ESTIMATION OF CRIMES IN THE USA IN 1990-92 – A COMPARATIVE ANALYSIS OF MULTIPLE LINEAR
REGRESSION WITH POISSON/NEGATIVE BINOMIAL REGRESSION MODELLING
Appendices
Figure 14: Boxplot of crm1000 with outliers
Figure 15: Boxplot of population in 1990 with outliers
26

ESTIMATION OF CRIMES IN THE USA IN 1990-92 – A COMPARATIVE ANALYSIS OF MULTIPLE LINEAR
REGRESSION WITH POISSON/NEGATIVE BINOMIAL REGRESSION MODELLINGFigure 16: Boxplot of pop1834 with outliers
Figure 17: Boxplot of phys with outliers
Figure 18: Boxplot of higrads with outliers
27

ESTIMATION OF CRIMES IN THE USA IN 1990-92 – A COMPARATIVE ANALYSIS OF MULTIPLE LINEAR
REGRESSION WITH POISSON/NEGATIVE BINOMIAL REGRESSION MODELLING
Figure 19: Boxplot of poors with outliers
Figure 20: Boxplot of total income with outliers
28

ESTIMATION OF CRIMES IN THE USA IN 1990-92 – A COMPARATIVE ANALYSIS OF MULTIPLE LINEAR
REGRESSION WITH POISSON/NEGATIVE BINOMIAL REGRESSION MODELLING
Figure 21: Regression plot for linear model 1
Figure 22: Regression plot for linear model 2
29

ESTIMATION OF CRIMES IN THE USA IN 1990-92 – A COMPARATIVE ANALYSIS OF MULTIPLE LINEAR
REGRESSION WITH POISSON/NEGATIVE BINOMIAL REGRESSION MODELLING
Figure 23: Regression plot for linear model 3
Figure 24: Regression plot for linear model 4
30

ESTIMATION OF CRIMES IN THE USA IN 1990-92 – A COMPARATIVE ANALYSIS OF MULTIPLE LINEAR
REGRESSION WITH POISSON/NEGATIVE BINOMIAL REGRESSION MODELLING
Figure 25: Regression plot for linear model 5
31