Multiple Linear Regression

Information

• Author: Jose Carlos Molano de Oro

• University: Pontificia Universidad Javeriana

• Course: Linear Regression Analysis

• Semester: 2022-3

• Professor: Mario Gregorio Saavedra Rodriguez

• Author Email: jose_molano@javeriana.edu.co

• Professor Email: saavedrarmg@javeriana.edu.co

R Libraries

library(DT)
library(Hmisc)
library(reactablefmtr)
library(dplyr)
library(aplpack)
library(corrplot)
library(ggplot2)
library(MASS)
library(broom)
library(tidyverse)
library(ggfortify)
library(mosaic)
library(jtools)
library(latex2exp)
library(pubh)
library(sjlabelled)
library(sjPlot)
library(sjmisc)
library(Ecdat)
library(PerformanceAnalytics)
library(regclass)
library(pdp)
library(gridExtra)
library(olsrr)

FBI Arrest Data - Reported Number of Arrests by Crime

About the Dataset

This dataset contains the monthly number of reported arrests in the US for various offenses reported by participating law enforcement agencies. The arrests are by offense and broken down by age and sex or age and race. Not all agencies report race and/or ethnicity for arrests but they must report age and sex. Note that only agencies that have reported arrests for 12 months of the year are represented in the annual counts that are included in the database. Download this dataset to see totals of reported arrests for the nation from 1995–2016.

The dataset was taken from de Federal Bureo of Investigation (FBI) Crime Data Explorer

Data Curation

Initial Data Table

FBI<-read.csv(url("https://s3-us-gov-west-1.amazonaws.com/cg-d3f0433b-a53e-4934-8b94-c678aa2cbaf3/arrests_national.csv"),row.names = 2,h = T)

reactable(FBI,rownames = TRUE)

Since the id attribute is not used, it is removed from the dataset

FBI$id<-NULL
reactable(FBI,rownames = TRUE)

Dataset Summary

summary(FBI)

##    population        total_arrests         homicide          rape      
##  Min.   :262803276   Min.   :10662252   Min.   :10231   Min.   :16863  
##  1st Qu.:282395819   1st Qu.:12586911   1st Qu.:11348   1st Qu.:21701  
##  Median :297952772   Median :13839754   Median :13331   Median :25032  
##  Mean   :295487602   Mean   :13418226   Mean   :13710   Mean   :25205  
##  3rd Qu.:311023417   3rd Qu.:14180570   3rd Qu.:14134   3rd Qu.:28083  
##  Max.   :323127513   Max.   :15284300   Max.   :21230   Max.   :34650  
##     robbery       aggravated_assault    burglary         larceny       
##  Min.   : 94403   Min.   :358860     Min.   :207325   Min.   :1050058  
##  1st Qu.:105863   1st Qu.:400402     1st Qu.:288660   1st Qu.:1160498  
##  Median :108921   Median :442990     Median :295372   Median :1210490  
##  Mean   :116045   Mean   :445464     Mean   :294936   Mean   :1241126  
##  3rd Qu.:126438   3rd Qu.:478265     3rd Qu.:304564   3rd Qu.:1279616  
##  Max.   :171870   Max.   :568480     Max.   :386500   Max.   :1530200  
##  motor_vehicle_theft     arson       violent_crime    property_crime   
##  Min.   : 64566      Min.   : 8834   Min.   :480360   Min.   :1353283  
##  1st Qu.: 78934      1st Qu.:11519   1st Qu.:539047   1st Qu.:1606177  
##  Median :139978      Median :15834   Median :597236   Median :1630406  
##  Mean   :120921      Mean   :14733   Mean   :600415   Mean   :1671715  
##  3rd Qu.:148814      3rd Qu.:16759   3rd Qu.:626632   3rd Qu.:1677062  
##  Max.   :191900      Max.   :20000   Max.   :796250   Max.   :2128600  
##  other_assault        forgery           fraud         embezzlement  
##  Min.   :1078808   Min.   : 55333   Min.   :128531   Min.   :15200  
##  1st Qu.:1242966   1st Qu.: 72184   1st Qu.:173134   1st Qu.:16065  
##  Median :1293424   Median :107777   Median :281816   Median :17100  
##  Mean   :1259624   Mean   : 95762   Mean   :273572   Mean   :17620  
##  3rd Qu.:1310566   3rd Qu.:115451   3rd Qu.:343650   3rd Qu.:18852  
##  Max.   :1395800   Max.   :122300   Max.   :465000   Max.   :22381  
##  stolen_property    vandalism         weapons        prostitution   
##  Min.   : 88576   Min.   :191015   Min.   :137779   Min.   : 38306  
##  1st Qu.: 95519   1st Qu.:241417   1st Qu.:157338   1st Qu.: 58676  
##  Median :121936   Median :275064   Median :167153   Median : 78640  
##  Mean   :118191   Mean   :265275   Mean   :174857   Mean   : 74418  
##  3rd Qu.:128090   3rd Qu.:289934   3rd Qu.:190173   3rd Qu.: 87809  
##  Max.   :166500   Max.   :320900   Max.   :243900   Max.   :101600  
##  other_sex_offenses   drug_abuse         gambling     against_family  
##  Min.   : 51063     Min.   :1476100   Min.   : 3705   Min.   : 88748  
##  1st Qu.: 70076     1st Qu.:1533853   1st Qu.: 8900   1st Qu.:111938  
##  Median : 89082     Median :1576072   Median :10630   Median :127032  
##  Mean   : 81231     Mean   :1617127   Mean   :10736   Mean   :126231  
##  3rd Qu.: 93149     3rd Qu.:1674540   3rd Qu.:11916   3rd Qu.:143487  
##  Max.   :101900     Max.   :1889810   Max.   :21000   Max.   :155800  
##       dui           liquor_laws      drunkenness     disorderly_conduct
##  Min.   :1017808   Min.   :234899   Min.   :376433   Min.   :369733    
##  1st Qu.:1305198   1st Qu.:503684   1st Qu.:537818   1st Qu.:590412    
##  Median :1434117   Median :611335   Median :566726   Median :647346    
##  Mean   :1364988   Mean   :548852   Mean   :573149   Mean   :626903    
##  3rd Qu.:1461434   3rd Qu.:635714   3rd Qu.:632832   3rd Qu.:693571    
##  Max.   :1511300   Max.   :683124   Max.   :734800   Max.   :842600    
##     vagrancy         other           suspicion     curfew_loitering
##  Min.   :24851   Min.   :3218880   Min.   :  576   Min.   : 34176  
##  1st Qu.:27316   1st Qu.:3553687   1st Qu.: 1451   1st Qu.: 81406  
##  Median :29076   Median :3724251   Median : 3018   Median :139116  
##  Mean   :29909   Mean   :3668659   Mean   : 3909   Mean   :122666  
##  3rd Qu.:33056   3rd Qu.:3832337   3rd Qu.: 5562   3rd Qu.:152130  
##  Max.   :36471   Max.   :4022068   Max.   :12100   Max.   :187800

1. Aspects of Multivariate Analysis

FBI Arrest Data - Multivariate Data Description and Graphical Techniques

Mean

datatable(as.matrix(sapply(FBI,function(x) mean(x, na.rm=TRUE))))

Variance

reactable(var(FBI,use = "complete.obs"))

Covariance

reactable(cov(FBI,use = "complete.obs"))

Correlation

c=cor(FBI)
y=as.data.frame(c)
y[y==1]<-" "
y <- mutate_all(y, function(x) as.numeric(as.character(x)))


reactable(as.data.frame.array((y)),
defaultColDef = colDef(
style = highlight_min_max(as.data.frame.array((y)))))

Correlation Plot

corrplot(cor(FBI,use = "complete.obs"),method="circle")

Correlation Test: P-Values

FBI.rcorr = rcorr(as.matrix(FBI))
FBI.p=FBI.rcorr$P

reactable(as.data.frame.array(FBI.p),
          defaultColDef = colDef(
              style = highlight_min_max(as.data.frame.array(FBI.p))))

• As can be seen in the correlation plot, it can be seen that almost all the variables are correlated with each other.

• The values highlighted on the correlation matrix in green, represent the variables that are most correlated with each other.

• The values highlighted on the correlation test table in green, represent the highest p-values of variables that are most correlated with each other. This p-values non-reject the nule hypothesis.

Graphic Representation

Considering the attributes in the correlation matrix that are most correlated positively with each other (values greater than 0.95):

• aggravated_assault with violent_crime
• violent_crime with homicide
• violent_crime with stolen_property
• aggravated_assault with fraud
• fraud with arson
• homicide with weapons
• forgery with other_sex_offenses
• prostitution with drunkeness
• prostitution with curfew_loite_ring
• drug_abuse with embezzlement
• drug_abuse with vagrancy
• dui with liquor_laws

Some of the mentioned attributes are taken to perform several graphs.

Scatter Plots

ggplot(FBI, aes(x=violent_crime, y=aggravated_assault)) + geom_point()+labs(title = "Violent Crime vs Aggravated Assault")

ggplot(FBI, aes(x=violent_crime, y=aggravated_assault,label=rownames(FBI))) + geom_text()+labs(title = "Violent Crime vs Aggravated Assault using Row Names")

ggplot(FBI, aes(x=violent_crime, y=homicide)) + geom_point()+labs(title = "Violent Crime vs Homicide")

ggplot(FBI, aes(x=violent_crime, y=homicide,label=rownames(FBI))) + geom_text()+labs(title = "Violent Crime vs Homicide using Row Names")

Bubble Charts

ggplot(FBI, aes(x=violent_crime, y=aggravated_assault,size=homicide)) + geom_point(alpha=0.5)+scale_size(range=c(.1,15))+labs(title = "Violent Crime vs Aggravated Assault and Homicide")

ggplot(FBI, aes(x=violent_crime, y=aggravated_assault,size=fraud)) + geom_point(alpha=0.5)+scale_size(range=c(.1,15))+labs(title = "Violent Crime vs Aggravated Assault and Fraud")

Chernoff Faces

df<-data.frame(FBI$aggravated_assault,FBI$violent_crime,FBI$homicide,FBI$stolen_property,FBI$fraud,FBI$arson,FBI$prostitution,FBI$other_sex_offenses,FBI$drunkenness,FBI$dui,FBI$liquor_laws,FBI$drug_abuse,FBI$curfew_loitering,FBI$embezzlement,FBI$vagrancy)

faces(df, main="United States FBI Arrest Data",face.type=0, print.info=TRUE,labels = rownames(FBI))

## effect of variables:
##  modified item       Var                     
##  "height of face   " "FBI.aggravated_assault"
##  "width of face    " "FBI.violent_crime"     
##  "structure of face" "FBI.homicide"          
##  "height of mouth  " "FBI.stolen_property"   
##  "width of mouth   " "FBI.fraud"             
##  "smiling          " "FBI.arson"             
##  "height of eyes   " "FBI.prostitution"      
##  "width of eyes    " "FBI.other_sex_offenses"
##  "height of hair   " "FBI.drunkenness"       
##  "width of hair   "  "FBI.dui"               
##  "style of hair   "  "FBI.liquor_laws"       
##  "height of nose  "  "FBI.drug_abuse"        
##  "width of nose   "  "FBI.curfew_loitering"  
##  "width of ear    "  "FBI.embezzlement"      
##  "height of ear   "  "FBI.vagrancy"

2. Data Training,Testing and Correlation)

Data Partition

FBI.train <- sample_frac(tbl = df, replace = FALSE, size = 0.80)
FBI.test <- anti_join(df, FBI.train)

## Joining, by = c("FBI.aggravated_assault", "FBI.violent_crime", "FBI.homicide",
## "FBI.stolen_property", "FBI.fraud", "FBI.arson", "FBI.prostitution",
## "FBI.other_sex_offenses", "FBI.drunkenness", "FBI.dui", "FBI.liquor_laws",
## "FBI.drug_abuse", "FBI.curfew_loitering", "FBI.embezzlement", "FBI.vagrancy")

Correlation Analysis

chart.Correlation(df)

3. Linear Model (aggravated_assault with FBI.violent_crime + FBI.homicide)

Data Adjust for Linear Model (aggravated_assault with FBI.violent_crime + FBI.homicide)

Lineal Model Diagnostic

model_norm <- lm(FBI.aggravated_assault ~ FBI.violent_crime + FBI.homicide, data = FBI.train)
autoplot(model_norm)

Summary

model_norm %>% augment() %>% as_tibble()

FBI.aggravated_assault	FBI.violent_crime	FBI.homicide	.fitted	.resid	.hat	.sigma	.cooksd	.std.resid
372685	498666	10571	3.71e+05	2.04e+03	0.203	9.17e+03	0.00565	0.258
376154	505681	11092	3.75e+05	1.6e+03	0.231	9.18e+03	0.00424	0.206
472290	620510	14158	4.6e+05	1.21e+04	0.07	8.56e+03	0.0505	1.42
383977	515151	11788	3.8e+05	4.12e+03	0.302	9.1e+03	0.0444	0.555
429969	594911	12955	4.43e+05	-1.31e+04	0.08	8.43e+03	0.069	-1.54
421215	581765	12418	4.34e+05	-1.28e+04	0.0916	8.46e+03	0.0772	-1.52
438033	586558	13467	4.34e+05	4.05e+03	0.0735	9.13e+03	0.00593	0.474
408488	552077	11201	4.14e+05	-5.11e+03	0.144	9.08e+03	0.0217	-0.622
433945	597447	13480	4.43e+05	-9.25e+03	0.0563	8.84e+03	0.0229	-1.07
397707	534704	10832	4e+05	-2.57e+03	0.131	9.17e+03	0.00484	-0.31
388362	521196	11075	3.88e+05	541	0.128	9.19e+03	0.000209	0.0652
447948	611523	13435	4.55e+05	-7.4e+03	0.0971	8.96e+03	0.0276	-0.877
449297	603503	14062	4.46e+05	3.25e+03	0.0775	9.15e+03	0.00405	0.38
478417	625132	13227	4.68e+05	1.07e+04	0.259	8.58e+03	0.226	1.39
534920	717750	18290	5.27e+05	8.37e+03	0.353	8.77e+03	0.249	1.17
521570	729900	19020	5.34e+05	-1.24e+04	0.47	7.98e+03	1.1	-1.92
477809	627132	13653	4.68e+05	1e+04	0.166	8.71e+03	0.101	1.24
449933	597026	13190	4.44e+05	5.95e+03	0.0656	9.05e+03	0.0112	0.693

Model Coeficients

model_norm %>% tidy()

term	estimate	std.error	statistic	p.value
(Intercept)	-1.19e+04	2.99e+04	-0.398	0.696
FBI.violent_crime	0.851	0.115	7.4	2.2e-06
FBI.homicide	-3.95	3.22	-1.23	0.238

Confidence Intervals

model_norm %>% confint() %>% as_tibble()

2.5 %	97.5 %
-7.56e+04	5.18e+04
0.606	1.1
-10.8	2.91

Model Plot

plot_model(model_norm,colors = "Accent", 
           show.values = TRUE,
           value.offset = .4,
           value.size = 4,
           dot.size = 3,
           line.size = 1.5,
           vline.color = "blue",
           width = 1.5
)

Model Multicollinearity

model_norm %>% VIF() %>% as_tibble()

value
11.8
11.8

Adjusted Model Residuals

p1 <- ggplot(FBI.train, aes(FBI.train[,2], residuals(model_norm))) +
    geom_point() + geom_smooth(color = "blue")
p2 <- ggplot(FBI.train, aes(FBI.train[,3], residuals(model_norm))) +
    geom_point() + geom_smooth(color = "blue")
p3 <- ggplot(FBI.train, aes(FBI.train[,10], residuals(model_norm))) +
    geom_point() + geom_smooth(color = "blue")
grid.arrange(p1, p2, p3)

## `geom_smooth()` using method = 'loess' and formula 'y ~ x'
## `geom_smooth()` using method = 'loess' and formula 'y ~ x'
## `geom_smooth()` using method = 'loess' and formula 'y ~ x'

Outliers

model_norm %>% ols_plot_cooksd_bar()

Adjustment Model

model_norm %>%
  glm_coef(se_rob = TRUE, labels = model_labels(model_norm))

Parameter	Coefficient	Pr(>|t|)
Constant	-11887.45 (-71104.23, 47329.34)	0.675
FBI.violent_crime	0.85 (0.56, 1.14)	< 0.001
FBI.homicide	-3.95 (-13.81, 5.91)	0.406

Adjustment Criteria Model

model_norm %>% glance()

r.squared	adj.r.squared	sigma	statistic	p.value	df	logLik	AIC	BIC	deviance	df.residual	nobs
0.968	0.964	8.88e+03	230	5.58e-12	2	-188	383	387	1.18e+09	15	18

Variance Analysis

model_norm %>% aov() %>% tidy()

term	df	sumsq	meansq	statistic	p.value
FBI.violent_crime	1	3.62e+10	3.62e+10	458	1.17e-12
FBI.homicide	1	1.19e+08	1.19e+08	1.51	0.238
Residuals	15	1.18e+09	7.89e+07

Criteria Comparison

model_norm_AIC <- stepAIC(model_norm, trace = 0)
AIC(model_norm, model_norm_AIC)

df	AIC
4	383
3	383

4. Linear Model (FBI.homicide with FBI.drug_abuse + FBI.violent_crime)

Data Adjust for Linear Model (FBI.homicide with FBI.drug_abuse + FBI.violent_crime)

Lineal Model Diagnostic

model_norm <- lm(FBI.homicide ~ FBI.drug_abuse + FBI.violent_crime, data = FBI.train)
autoplot(model_norm)

Summary

model_norm %>% augment() %>% as_tibble()

FBI.homicide	FBI.drug_abuse	FBI.violent_crime	.fitted	.resid	.hat	.sigma	.cooksd	.std.resid
10571	1561231	498666	1.02e+04	403	0.188	710	0.0319	0.642
11092	1488707	505681	1.05e+04	596	0.228	697	0.0936	0.975
14158	1538813	620510	1.44e+04	-228	0.114	717	0.00518	-0.348
11788	1572579	515151	1.07e+04	1.07e+03	0.146	651	0.157	1.66
12955	1702537	594911	1.33e+04	-355	0.0713	713	0.00716	-0.529
12418	1663582	581765	1.29e+04	-486	0.0592	708	0.0109	-0.72
13467	1746570	586558	1.3e+04	497	0.102	706	0.0216	0.754
11201	1638846	552077	1.19e+04	-712	0.0762	692	0.0311	-1.06
13480	1841182	597447	1.32e+04	249	0.216	716	0.015	0.404
10832	1531251	534704	1.14e+04	-612	0.137	698	0.0473	-0.946
11075	1552432	521196	1.1e+04	121	0.144	719	0.00199	0.188
13435	1889810	611523	1.37e+04	-222	0.304	717	0.0213	-0.382
14062	1846351	603503	1.34e+04	629	0.225	694	0.102	1.03
13227	1579566	625132	1.45e+04	-1.27e+03	0.0907	626	0.122	-1.91
18290	1583600	717750	1.77e+04	613	0.312	693	0.171	1.06
19020	1506200	729900	1.82e+04	833	0.435	657	0.651	1.59
13653	1586902	627132	1.46e+04	-903	0.0892	674	0.0604	-1.36
13190	1678192	597026	1.34e+04	-221	0.0618	718	0.00237	-0.329

Model Coeficients

model_norm %>% tidy()

term	estimate	std.error	statistic	p.value
(Intercept)	-5.12e+03	2.63e+03	-1.95	0.0705
FBI.drug_abuse	-0.0012	0.00139	-0.861	0.403
FBI.violent_crime	0.0344	0.00263	13.1	1.34e-09

Confidence Intervals

model_norm %>% confint() %>% as_tibble()

2.5 %	97.5 %
-1.07e+04	484
-0.00416	0.00176
0.0288	0.04

Model Plot

plot_model(model_norm,colors = "Accent", 
           show.values = TRUE,
           value.offset = .4,
           value.size = 4,
           dot.size = 3,
           line.size = 1.5,
           vline.color = "blue",
           width = 1.5
)

Model Multicollinearity

model_norm %>% VIF() %>% as_tibble()

value
1.01
1.01

Adjusted Model Residuals

p1 <- ggplot(FBI.train, aes(FBI.train[,2], residuals(model_norm))) +
    geom_point() + geom_smooth(color = "blue")
p2 <- ggplot(FBI.train, aes(FBI.train[,3], residuals(model_norm))) +
    geom_point() + geom_smooth(color = "blue")
p3 <- ggplot(FBI.train, aes(FBI.train[,10], residuals(model_norm))) +
    geom_point() + geom_smooth(color = "blue")
grid.arrange(p1, p2, p3)

## `geom_smooth()` using method = 'loess' and formula 'y ~ x'
## `geom_smooth()` using method = 'loess' and formula 'y ~ x'
## `geom_smooth()` using method = 'loess' and formula 'y ~ x'

Outliers

model_norm %>% ols_plot_cooksd_bar()

Adjustment Model

model_norm %>%
  glm_coef(se_rob = TRUE, labels = model_labels(model_norm))

Parameter	Coefficient	Pr(>|t|)
Constant	-5115.68 (-10372.07, 140.7)	0.056
FBI.drug_abuse	0 (0, 0)	0.43
FBI.violent_crime	0.03 (0.03, 0.04)	< 0.001

Adjustment Criteria Model

model_norm %>% glance()

r.squared	adj.r.squared	sigma	statistic	p.value	df	logLik	AIC	BIC	deviance	df.residual	nobs
0.919	0.909	696	85.4	6.33e-09	2	-142	291	295	7.26e+06	15	18

Variance Analysis

model_norm %>% aov() %>% tidy()

term	df	sumsq	meansq	statistic	p.value
FBI.drug_abuse	1	1.09e+05	1.09e+05	0.225	0.642
FBI.violent_crime	1	8.26e+07	8.26e+07	171	1.34e-09
Residuals	15	7.26e+06	4.84e+05

Criteria Comparison

model_norm_AIC <- stepAIC(model_norm, trace = 0)
AIC(model_norm, model_norm_AIC)

df	AIC
4	291
3	290

5. Linear Model (FBI.homicide with FBI.prostitution + FBI.drug_abuse)

Data Adjust for Linear Model (FBI.homicide with FBI.prostitution + FBI.drug_abuse)

Lineal Model Diagnostic

model_norm <- lm(FBI.homicide ~ FBI.prostitution + FBI.drug_abuse, data = FBI.train)
autoplot(model_norm)

Summary

model_norm %>% augment() %>% as_tibble()

FBI.homicide	FBI.prostitution	FBI.drug_abuse	.fitted	.resid	.hat	.sigma	.cooksd	.std.resid
10571	47598	1561231	1.07e+04	-148	0.167	1.24e+03	0.00122	-0.135
11092	41877	1488707	1.04e+04	687	0.253	1.23e+03	0.0494	0.661
14158	79733	1538813	1.45e+04	-374	0.123	1.24e+03	0.0052	-0.333
11788	38306	1572579	9.59e+03	2.19e+03	0.261	1.04e+03	0.532	2.12
12955	75004	1702537	1.32e+04	-252	0.0713	1.24e+03	0.00121	-0.217
12418	71355	1663582	1.3e+04	-553	0.0589	1.23e+03	0.0047	-0.475
13467	87872	1746570	1.45e+04	-1.01e+03	0.122	1.21e+03	0.0375	-0.9
11201	62668	1638846	1.21e+04	-886	0.0743	1.22e+03	0.0157	-0.767
13480	77607	1841182	1.28e+04	634	0.22	1.23e+03	0.0335	0.598
10832	57345	1531251	1.2e+04	-1.15e+03	0.121	1.2e+03	0.0484	-1.02
11075	56575	1552432	1.18e+04	-722	0.113	1.23e+03	0.0172	-0.637
13435	79673	1889810	1.29e+04	583	0.307	1.23e+03	0.0501	0.583
14062	84891	1846351	1.37e+04	401	0.226	1.24e+03	0.014	0.379
13227	87620	1579566	1.52e+04	-2.02e+03	0.132	1.1e+03	0.166	-1.81
18290	101600	1583600	1.68e+04	1.45e+03	0.264	1.16e+03	0.237	1.41
19020	99000	1506200	1.69e+04	2.11e+03	0.335	1.03e+03	0.778	2.15
13653	80854	1586902	1.44e+04	-779	0.0893	1.22e+03	0.0151	-0.68
13190	75190	1678192	1.33e+04	-154	0.0617	1.24e+03	0.000384	-0.132

Model Coeficients

model_norm %>% tidy()

term	estimate	std.error	statistic	p.value
(Intercept)	1.27e+04	3.92e+03	3.23	0.0056
FBI.prostitution	0.115	0.0168	6.87	5.28e-06
FBI.drug_abuse	-0.00477	0.00251	-1.9	0.0772

Confidence Intervals

model_norm %>% confint() %>% as_tibble()

2.5 %	97.5 %
4.31e+03	2.1e+04
0.0796	0.151
-0.0101	0.000589

Model Plot

plot_model(model_norm,colors = "Accent", 
           show.values = TRUE,
           value.offset = .4,
           value.size = 4,
           dot.size = 3,
           line.size = 1.5,
           vline.color = "blue",
           width = 1.5
)

Model Multicollinearity

model_norm %>% VIF() %>% as_tibble()

value
1.11
1.11

Adjusted Model Residuals

p1 <- ggplot(FBI.train, aes(FBI.train[,2], residuals(model_norm))) +
    geom_point() + geom_smooth(color = "blue")
p2 <- ggplot(FBI.train, aes(FBI.train[,3], residuals(model_norm))) +
    geom_point() + geom_smooth(color = "blue")
p3 <- ggplot(FBI.train, aes(FBI.train[,10], residuals(model_norm))) +
    geom_point() + geom_smooth(color = "blue")
grid.arrange(p1, p2, p3)

## `geom_smooth()` using method = 'loess' and formula 'y ~ x'
## `geom_smooth()` using method = 'loess' and formula 'y ~ x'
## `geom_smooth()` using method = 'loess' and formula 'y ~ x'

Outliers

model_norm %>% ols_plot_cooksd_bar()

Adjustment Model

model_norm %>%
  glm_coef(se_rob = TRUE, labels = model_labels(model_norm))

Parameter	Coefficient	Pr(>|t|)
Constant	12671.62 (3431.43, 21911.81)	0.01
FBI.prostitution	0.12 (0.05, 0.18)	0.002
FBI.drug_abuse	0 (-0.01, 0)	0.166

Adjustment Criteria Model

model_norm %>% glance()

r.squared	adj.r.squared	sigma	statistic	p.value	df	logLik	AIC	BIC	deviance	df.residual	nobs
0.759	0.727	1.2e+03	23.7	2.29e-05	2	-152	311	315	2.17e+07	15	18

Variance Analysis

model_norm %>% aov() %>% tidy()

term	df	sumsq	meansq	statistic	p.value
FBI.prostitution	1	6.31e+07	6.31e+07	43.7	8.25e-06
FBI.drug_abuse	1	5.2e+06	5.2e+06	3.6	0.0772
Residuals	15	2.17e+07	1.44e+06

Criteria Comparison

model_norm_AIC <- stepAIC(model_norm, trace = 0)
AIC(model_norm, model_norm_AIC)

df	AIC
4	311
4	311

6. References

• Federal Bureau Investigation, ed. “Crime Data Explorer.” FBI. crime-data-explorer.fr.cloud.gov. Accessed August 21, 2022. https://crime-data-explorer.fr.cloud.gov/pages/downloads#datasets

• Federal Bureau Investigation, ed. “FBI National Arrests.” FBI. Accessed August 21, 2022. https://s3-us-gov-west-1.amazonaws.com/cg-d3f0433b-a53e-4934-8b94-c678aa2cbaf3/arrests_national.csv

• GeeksforGeeks. “Conditioning Plot - GeeksforGeeks.” www.geeksforgeeks.org, March 7, 2021. https://www.geeksforgeeks.org/conditioning-plot/

• Yi, Mike. “Scatter Plots | A Complete Guide to Scatter Plots.” Chartio. chartio.com. Accessed August 21, 2022. https://chartio.com/learn/charts/what-is-a-scatter-plot/

• Viliam Simko, Taiyun Wei. “An Introduction to Corrplot Package.” An Introduction to corrplot Package. cran.r-project.org, November 18, 2021. https://cran.r-project.org/web/packages/corrplot/vignettes/corrplot-intro.html

• Kabacoff, Rob. “Data Visualization with R.” Data Visualization with R. rkabacoff.github.io. Accessed August 21, 2022. https://rkabacoff.github.io/datavis/Multivariate.html

• Vitor A. A. Marchi, Francisco A. R. Rojas, Francisco Louzada, The Chi-plot and Its Asymptotic Confidence Interval for Analyzing Bivariate Dependence: An Application to the Average Intelligence and Atheism Rates across Nations Data, J. data sci. 10(2022), no. 4, 711-722, DOI 10.6339/JDS.2012.10(4).1094

• Saavedra, Mario. “Análisis de Regresión”. RPubs by RStudio . Accessed October 17, 2022. https://rpubs.com/mgsaavedraro/925582