4.1 Adj R Square

summary(gdpA)$adj.r.squared

## [1] 0.8628926

summary(gdpB)$adj.r.squared

## [1] 0.884257

summary(gdpF)$adj.r.squared

## [1] 0.8820182

4.2 Multicolinearity

(vif < 10)

vif(gdpA)

##            percent_m             is_south       mean_education 
##             3.170944             5.064479             6.568520 
##         police_exp60         police_exp59 labour_participation 
##           115.774639           115.953281             3.658815 
##           m_per1000f            state_pop    nonwhites_per1000 
##             3.849536             2.536364             4.515073 
##         unemploy_m24         unemploy_m39           inequality 
##             6.427948             5.723608             7.360566 
##          prob_prison          time_prison           crime_rate 
##             3.219434             2.690628             4.941213

vif(gdpB)

##            percent_m             is_south         police_exp59 
##             2.293891             4.023964             5.531691 
## labour_participation    nonwhites_per1000           inequality 
##             1.575476             3.521800             4.862590 
##          time_prison           crime_rate 
##             1.139803             2.785642

vif(gdpF)

##        inequality      police_exp60 nonwhites_per1000        crime_rate 
##          4.051937          5.303696          2.653074          2.686759 
##         percent_m 
##          2.268081

4.3 Residual Pattern

(p-value > 0.05)

bptest(gdpA)

## 
##  studentized Breusch-Pagan test
## 
## data:  gdpA
## BP = 11.421, df = 15, p-value = 0.7222

bptest(gdpB)

## 
##  studentized Breusch-Pagan test
## 
## data:  gdpB
## BP = 8.1288, df = 8, p-value = 0.421

bptest(gdpF)

## 
##  studentized Breusch-Pagan test
## 
## data:  gdpF
## BP = 1.7128, df = 5, p-value = 0.8873

4.4 Residual Normality

(p-value > 0.05)

shapiro.test(gdpA$residuals)

## 
##  Shapiro-Wilk normality test
## 
## data:  gdpA$residuals
## W = 0.96012, p-value = 0.1087

shapiro.test(gdpB$residuals)

## 
##  Shapiro-Wilk normality test
## 
## data:  gdpB$residuals
## W = 0.95329, p-value = 0.05833

shapiro.test(gdpF$residuals)

## 
##  Shapiro-Wilk normality test
## 
## data:  gdpF$residuals
## W = 0.97169, p-value = 0.3068

The result from Backward and Both direction give the same variables, so one of them is eliminated.

The AIC result from the Forward direction (334.66) is bigger than Backward direction (336.19), but the Adj R Squared result from Forward (0.8820182) is smaller than from Backward (0.884257).

Model F and B has passed assumption checking on Multicolinearity, Error Pattern, and Residual Normality. Model A has not passed the Multicolinearity, but I continue to investigate it and see if it gave a better result in stepwise after outlier removal.

Continue to investigate model A, B, and F.

5.1 Model A

crmA <- crm %>% 
   gather(key = "variable_name", 
             value = "variable_value")

ggplot(crmA,aes(y=variable_value))+
  geom_boxplot()+
  facet_wrap(~variable_name,scales="free")

Observe boxplot for each dependent and independent variables, find the outlier threshold then try remove each one to see if their make the model have a better Adj R Square.

Initial Adj R Square for Backward model : 0.8628926

1. crime_rate <= 1800

crmA1 <- crm %>% 
  filter(crime_rate <= 1800)

gdpA1 <- lm(gdp ~ .,crmA1)

summary(gdpA1)$adj.r.square

## [1] 0.8460174

Decrease in Adj R Square, keep the outlier.

2. nonwhites_per1000 <= 300

crmA2 <- crm %>% 
  filter(nonwhites_per1000 <= 300)

gdpA2 <- lm(gdp ~ .,crmA2)

summary(gdpA2)$adj.r.square

## [1] 0.8131418

Decrease in Adj R Square, keep the outlier.

3. percent_m <= 170

crmA3 <- crm %>% 
  filter(percent_m <= 170)

gdpA3 <- lm(gdp ~ .,crmA3)

summary(gdpA3)$adj.r.square

## [1] 0.8542544

Decrease in Adj R Square, keep the outlier.

4. police_exp59 <= 150

crmA4 <- crm %>% 
  filter(police_exp59 <= 150)

gdpA4 <- lm(gdp ~ .,crmA4)

summary(gdpA4)$adj.r.square

## [1] 0.8637712

Increase in Adj R Square, remove the outlier.

5. time_prison <= 42

crmA5 <- crm %>% 
  filter(time_prison<=42)

gdpA5 <- lm(gdp ~ .,crmA5)

summary(gdpA5)$adj.r.square

## [1] 0.8779584

Increase in Adj R Square, remove the outlier.

6. mper1000f <= 1060

crmA6 <- crm %>%
  filter(m_per1000f <= 1060)

gdpA6 <- lm(gdp ~ .,crmA6)

summary(gdpA6)$adj.r.square

## [1] 0.8563324

Decrease in Adj R Square, keep the outlier.

7. prob_prison <= 0.1

crmA7 <- crm %>%
  filter(prob_prison <= 0.1)

gdpA7 <- lm(gdp ~ .,crmA7)

summary(gdpA7)$adj.r.square

## [1] 0.8730323

Increase in Adj R Square, remove the outlier.

8.state_pop <= 140

crmA8 <- crm %>%
  filter(state_pop <= 100)

gdpA8 <- lm(gdp ~ .,crmA8)

summary(gdpA8)$adj.r.square

## [1] 0.8504931

Decrease in Adj R Square, keep the outlier.

9. unemploy_m24 <= 140

crmA9 <- crm %>%
  filter(unemploy_m24 <= 140)

gdpA9 <- lm(gdp ~ .,crmA9)

summary(gdpA9)$adj.r.square

## [1] 0.8622883

Decrease in Adj R Square, keep the outlier.

10. unemploy_m39 <= 55

crmA10 <- crm %>%
  filter(unemploy_m39 <= 55)

gdpA10 <- lm(gdp ~ .,crmA10)

summary(gdpA10)$adj.r.square

## [1] 0.8584968

Decrease in Adj R Square, keep the outlier.

Result from Model A after eliminate outlier

crmAF <- crm %>% 
  filter(police_exp59 <= 150,
         time_prison <= 42,
         prob_prison <= 0.1)

gdpA1 <- lm(gdp~.,crmAF)
gdpN1 <- lm(gdp~1,crmAF)

5.2 Model B

GDP ~ percent_m + is_south + police_exp59 + labour_participation + nonwhites_per1000 + inequality + time_prison + crime_rate

crmB <- crm %>% 
   gather(key = "variable_name", 
             value = "variable_value",
             gdp, percent_m, is_south, police_exp59, labour_participation, nonwhites_per1000, inequality, time_prison, crime_rate)
            
ggplot(crmB,aes(y=variable_value))+
  geom_boxplot()+
  facet_wrap(~variable_name,scales="free")

Observe boxplot for each dependent and independent variables, find the outlier threshold then try remove each one to see if their make the model have a better Adj R Square.

Initial Adj R Square for Backward model : 0.884257

1. crime_rate <= 1800

crmB1 <- crm %>%
  filter(crime_rate<=1800)

gdpB1 <- lm(gdp ~ percent_m + is_south + police_exp59 + labour_participation + nonwhites_per1000 + inequality + time_prison + crime_rate,crmB1)

summary(gdpB1)$adj.r.square

## [1] 0.8709542

Decrease in Adj R Square, keep the outlier.

2. nonwhites_per1000 <= 300

crmB2 <- crm %>%
  filter(nonwhites_per1000 <= 300)

gdpB2 <- lm(gdp ~ percent_m + is_south + police_exp59 + labour_participation + nonwhites_per1000 + inequality + time_prison + crime_rate,crmB2)

summary(gdpB2)$adj.r.square

## [1] 0.8450358

Decrease in Adj R Square, keep the outlier.

3. percent_m <= 170

crmB3 <- crm %>%
  filter(percent_m <= 170)

gdpB3 <- lm(gdp ~ percent_m + is_south + police_exp59 + labour_participation + nonwhites_per1000 + inequality + time_prison + crime_rate,crmB3)

summary(gdpB3)$adj.r.square

## [1] 0.8777601

Decrease in Adj R Square, keep the outlier.

4. police_exp59 <= 150

crmB4 <- crm %>%
  filter(police_exp59 <= 150)

gdpB4 <- lm(gdp ~ percent_m + is_south + police_exp59 + labour_participation + nonwhites_per1000 + inequality + time_prison + crime_rate,crmB4)

summary(gdpB4)$adj.r.square

## [1] 0.8843402

Increase in Adj R Square, remove the outlier.

5. time_prison <= 42

crmB5 <- crm %>%
  filter(time_prison <= 42)

gdpB5 <- lm(gdp ~ percent_m + is_south + police_exp59 + labour_participation + nonwhites_per1000 + inequality + time_prison + crime_rate,crmB5)

summary(gdpB5)$adj.r.square

## [1] 0.8963129

Increase in Adj R Square, remove the outlier.

Result from Model B after eliminate outlier

crmB6 <- crm %>% 
  filter(police_exp59 <= 150,
         time_prison <= 42)

gdpB6 <- lm(gdp ~ percent_m + is_south + police_exp59 + labour_participation + nonwhites_per1000 + inequality + time_prison + crime_rate,crmB6)
gdpB6n <- lm(gdp ~1,crmB6)

5.3 Model F

GDP ~ inequality + police_exp60 + nonwhites_per1000 + crime_rate + percent_m

crmF <- crm %>% 
   gather(key = "variable_name", 
             value = "variable_value",
             gdp, percent_m, nonwhites_per1000, inequality, crime_rate, police_exp60)

ggplot(crmF,aes(y=variable_value))+
  geom_boxplot()+
  facet_wrap(~variable_name,scales="free")

Observe boxplot for each dependent and independent variables, find the outlier threshold then try remove each one to see if their make the model have a better Adj R Square.

Initial Adj R Square for Forward model : 0.8820182

1. crime_rate <= 1800

crmF1 <- crm %>%
  filter(crime_rate<=1800)

gdpF1 <- lm(gdp ~ inequality + police_exp60 + nonwhites_per1000 + crime_rate + percent_m,crmF1)

summary(gdpF1)$adj.r.square

## [1] 0.8690229

Decrease in Adj R Square, keep the outlier.

2. nonwhites_per1000 <= 300

crmF2 <- crm %>%
  filter(nonwhites_per1000 <= 300)

gdpF2 <- lm(gdp ~ percent_m + is_south + police_exp59 + labour_participation + nonwhites_per1000 + inequality + time_prison + crime_rate,crmF2)

summary(gdpF2)$adj.r.square

## [1] 0.8450358

Decrease in Adj R Square, keep the outlier.

3. percent_m <= 170

crmF3 <- crm %>%
  filter(percent_m <= 170)

gdpF3 <- lm(gdp ~ percent_m + is_south + police_exp59 + labour_participation + nonwhites_per1000 + inequality + time_prison + crime_rate,crmF3)

summary(gdpF3)$adj.r.square

## [1] 0.8777601

Decrease in Adj R Square, keep the outlier.

Every outlier give low influence to the model, so there is no outlier removal in the Model F

6.1 Model A

back3 <- step(gdpA1,direction="backward") #AIC 303.83

## Start:  AIC=318.87
## gdp ~ percent_m + is_south + mean_education + police_exp60 + 
##     police_exp59 + labour_participation + m_per1000f + state_pop + 
##     nonwhites_per1000 + unemploy_m24 + unemploy_m39 + inequality + 
##     prob_prison + time_prison + crime_rate
## 
##                        Df Sum of Sq   RSS    AIC
## - state_pop             1       0.1 29849 316.87
## - police_exp59          1       0.7 29850 316.87
## - labour_participation  1       9.9 29859 316.88
## - percent_m             1     152.5 30002 317.09
## - is_south              1     153.2 30002 317.09
## - police_exp60          1     196.3 30046 317.16
## - m_per1000f            1     212.7 30062 317.18
## - mean_education        1     214.8 30064 317.18
## - crime_rate            1     452.1 30301 317.53
## - unemploy_m24          1     687.0 30536 317.87
## - time_prison           1     804.1 30653 318.04
## - unemploy_m39          1    1205.7 31055 318.61
## <none>                              29849 318.87
## - nonwhites_per1000     1    1609.6 31459 319.18
## - prob_prison           1    1901.8 31751 319.59
## - inequality            1   14356.8 44206 334.15
## 
## Step:  AIC=316.87
## gdp ~ percent_m + is_south + mean_education + police_exp60 + 
##     police_exp59 + labour_participation + m_per1000f + nonwhites_per1000 + 
##     unemploy_m24 + unemploy_m39 + inequality + prob_prison + 
##     time_prison + crime_rate
## 
##                        Df Sum of Sq   RSS    AIC
## - police_exp59          1       0.7 29850 314.87
## - labour_participation  1       9.7 29859 314.88
## - is_south              1     153.2 30003 315.09
## - percent_m             1     154.2 30004 315.09
## - police_exp60          1     199.2 30049 315.16
## - mean_education        1     216.1 30066 315.19
## - m_per1000f            1     250.2 30100 315.24
## - crime_rate            1     453.1 30303 315.53
## - unemploy_m24          1     688.6 30538 315.87
## - time_prison           1     872.3 30722 316.14
## - unemploy_m39          1    1221.0 31070 316.63
## <none>                              29849 316.87
## - nonwhites_per1000     1    1627.2 31477 317.20
## - prob_prison           1    1902.4 31752 317.59
## - inequality            1   14394.8 44244 332.18
## 
## Step:  AIC=314.87
## gdp ~ percent_m + is_south + mean_education + police_exp60 + 
##     labour_participation + m_per1000f + nonwhites_per1000 + unemploy_m24 + 
##     unemploy_m39 + inequality + prob_prison + time_prison + crime_rate
## 
##                        Df Sum of Sq   RSS    AIC
## - labour_participation  1      12.6 29863 312.89
## - percent_m             1     155.9 30006 313.10
## - is_south              1     157.4 30008 313.10
## - mean_education        1     223.0 30073 313.20
## - m_per1000f            1     250.1 30100 313.24
## - crime_rate            1     461.5 30312 313.54
## - unemploy_m24          1     711.1 30561 313.91
## - time_prison           1    1008.9 30859 314.33
## - unemploy_m39          1    1232.7 31083 314.65
## <none>                              29850 314.87
## - nonwhites_per1000     1    1793.7 31644 315.44
## - police_exp60          1    2075.0 31925 315.83
## - prob_prison           1    2095.3 31945 315.85
## - inequality            1   14481.8 44332 330.27
## 
## Step:  AIC=312.89
## gdp ~ percent_m + is_south + mean_education + police_exp60 + 
##     m_per1000f + nonwhites_per1000 + unemploy_m24 + unemploy_m39 + 
##     inequality + prob_prison + time_prison + crime_rate
## 
##                     Df Sum of Sq   RSS    AIC
## - is_south           1     155.2 30018 311.12
## - percent_m          1     165.8 30029 311.13
## - mean_education     1     282.1 30145 311.30
## - m_per1000f         1     437.0 30300 311.53
## - crime_rate         1     453.7 30316 311.55
## - unemploy_m24       1     979.9 30843 312.31
## - time_prison        1    1022.4 30885 312.37
## - unemploy_m39       1    1283.1 31146 312.74
## <none>                           29863 312.89
## - nonwhites_per1000  1    1839.9 31703 313.52
## - police_exp60       1    2064.8 31927 313.83
## - prob_prison        1    2212.1 32075 314.03
## - inequality         1   15246.2 45109 329.04
## 
## Step:  AIC=311.12
## gdp ~ percent_m + mean_education + police_exp60 + m_per1000f + 
##     nonwhites_per1000 + unemploy_m24 + unemploy_m39 + inequality + 
##     prob_prison + time_prison + crime_rate
## 
##                     Df Sum of Sq   RSS    AIC
## - percent_m          1     117.8 30136 309.29
## - mean_education     1     271.3 30289 309.51
## - m_per1000f         1     388.4 30406 309.68
## - crime_rate         1     478.6 30497 309.81
## - time_prison        1     902.9 30921 310.42
## - unemploy_m24       1    1318.9 31337 311.01
## <none>                           30018 311.12
## - nonwhites_per1000  1    1689.1 31707 311.52
## - unemploy_m39       1    1692.7 31711 311.53
## - police_exp60       1    1951.1 31969 311.89
## - prob_prison        1    2289.2 32307 312.35
## - inequality         1   15306.1 45324 327.25
## 
## Step:  AIC=309.29
## gdp ~ mean_education + police_exp60 + m_per1000f + nonwhites_per1000 + 
##     unemploy_m24 + unemploy_m39 + inequality + prob_prison + 
##     time_prison + crime_rate
## 
##                     Df Sum of Sq   RSS    AIC
## - m_per1000f         1     321.4 30457 307.76
## - mean_education     1     382.6 30518 307.84
## - crime_rate         1     383.0 30519 307.84
## - time_prison        1     873.5 31009 308.55
## - unemploy_m24       1    1394.5 31530 309.28
## <none>                           30136 309.29
## - unemploy_m39       1    2181.6 32317 310.36
## - nonwhites_per1000  1    2380.7 32516 310.63
## - police_exp60       1    2413.1 32549 310.68
## - prob_prison        1    2437.3 32573 310.71
## - inequality         1   15189.0 45325 325.25
## 
## Step:  AIC=307.76
## gdp ~ mean_education + police_exp60 + nonwhites_per1000 + unemploy_m24 + 
##     unemploy_m39 + inequality + prob_prison + time_prison + crime_rate
## 
##                     Df Sum of Sq   RSS    AIC
## - mean_education     1     511.9 30969 306.49
## - crime_rate         1     600.8 31058 306.61
## - time_prison        1     687.4 31145 306.74
## - unemploy_m24       1    1073.1 31530 307.28
## <none>                           30457 307.76
## - unemploy_m39       1    1864.0 32321 308.37
## - police_exp60       1    2391.4 32849 309.08
## - prob_prison        1    2454.9 32912 309.17
## - nonwhites_per1000  1    2567.0 33024 309.32
## - inequality         1   14873.8 45331 323.25
## 
## Step:  AIC=306.49
## gdp ~ police_exp60 + nonwhites_per1000 + unemploy_m24 + unemploy_m39 + 
##     inequality + prob_prison + time_prison + crime_rate
## 
##                     Df Sum of Sq   RSS    AIC
## - unemploy_m24       1     732.3 31701 305.52
## - time_prison        1     785.9 31755 305.59
## - unemploy_m39       1    1359.3 32328 306.38
## <none>                           30969 306.49
## - crime_rate         1    1690.7 32660 306.83
## - prob_prison        1    2064.8 33034 307.33
## - police_exp60       1    2327.9 33297 307.68
## - nonwhites_per1000  1    4013.0 34982 309.85
## - inequality         1   27740.9 58710 332.63
## 
## Step:  AIC=305.52
## gdp ~ police_exp60 + nonwhites_per1000 + unemploy_m39 + inequality + 
##     prob_prison + time_prison + crime_rate
## 
##                     Df Sum of Sq   RSS    AIC
## - unemploy_m39       1     680.8 32382 304.45
## - time_prison        1    1389.3 33091 305.40
## - crime_rate         1    1444.8 33146 305.48
## <none>                           31701 305.52
## - prob_prison        1    1767.1 33469 305.90
## - nonwhites_per1000  1    3778.3 35480 308.47
## - police_exp60       1    3945.1 35647 308.68
## - inequality         1   27104.2 58806 330.70
## 
## Step:  AIC=304.45
## gdp ~ police_exp60 + nonwhites_per1000 + inequality + prob_prison + 
##     time_prison + crime_rate
## 
##                     Df Sum of Sq   RSS    AIC
## - crime_rate         1    1335.0 33717 304.23
## <none>                           32382 304.45
## - time_prison        1    1520.3 33902 304.47
## - prob_prison        1    1637.4 34020 304.62
## - nonwhites_per1000  1    3885.6 36268 307.44
## - police_exp60       1    4857.6 37240 308.60
## - inequality         1   26438.0 58820 328.71
## 
## Step:  AIC=304.23
## gdp ~ police_exp60 + nonwhites_per1000 + inequality + prob_prison + 
##     time_prison
## 
##                     Df Sum of Sq   RSS    AIC
## - time_prison        1    1248.2 34965 303.83
## <none>                           33717 304.23
## - prob_prison        1    2393.2 36110 305.25
## - nonwhites_per1000  1    3902.5 37620 307.05
## - police_exp60       1   20774.5 54492 323.35
## - inequality         1   27802.7 61520 328.69
## 
## Step:  AIC=303.83
## gdp ~ police_exp60 + nonwhites_per1000 + inequality + prob_prison
## 
##                     Df Sum of Sq   RSS    AIC
## <none>                           34965 303.83
## - nonwhites_per1000  1    2739.0 37704 305.15
## - prob_prison        1    7753.4 42719 310.64
## - police_exp60       1   19554.4 54520 321.37
## - inequality         1   27348.3 62314 327.25

forward3 <- step(gdpN1, scope = list(lower = gdpN1,upper = gdpA1),direction="forward") #AIC 303.83

## Start:  AIC=403.29
## gdp ~ 1
## 
##                        Df Sum of Sq    RSS    AIC
## + inequality            1    314018  88091 338.49
## + police_exp59          1    256505 145604 360.60
## + police_exp60          1    251745 150365 362.01
## + mean_education        1    236636 165474 366.22
## + prob_prison           1    219031 183078 370.67
## + is_south              1    191021 211089 376.94
## + percent_m             1    183238 218871 378.53
## + nonwhites_per1000     1    156201 245908 383.66
## + crime_rate            1     78071 324039 395.79
## + labour_participation  1     52576 349534 399.13
## + state_pop             1     30449 371661 401.83
## + m_per1000f            1     24180 377930 402.56
## <none>                              402110 403.29
## + unemploy_m39          1      2861 399249 404.98
## + unemploy_m24          1      1721 400389 405.10
## + time_prison           1       458 401652 405.24
## 
## Step:  AIC=338.49
## gdp ~ inequality
## 
##                        Df Sum of Sq   RSS    AIC
## + police_exp60          1     39557 48534 314.26
## + crime_rate            1     38272 49819 315.41
## + police_exp59          1     37234 50857 316.31
## + prob_prison           1     33556 54535 319.39
## + state_pop             1     16918 71173 331.10
## + time_prison           1      8603 79488 335.96
## + percent_m             1      5608 82483 337.59
## + unemploy_m39          1      5242 82849 337.79
## <none>                              88091 338.49
## + mean_education        1      3318 84774 338.80
## + labour_participation  1      1769 86322 339.59
## + m_per1000f            1       595 87496 340.19
## + unemploy_m24          1        68 88023 340.45
## + nonwhites_per1000     1        41 88051 340.47
## + is_south              1        17 88074 340.48
## 
## Step:  AIC=314.26
## gdp ~ inequality + police_exp60
## 
##                        Df Sum of Sq   RSS    AIC
## + prob_prison           1   10830.1 37704 305.15
## + nonwhites_per1000     1    5815.6 42719 310.64
## + time_prison           1    3725.6 44809 312.74
## + crime_rate            1    3024.0 45510 313.43
## + percent_m             1    2176.3 46358 314.24
## <none>                              48534 314.26
## + state_pop             1    1238.3 47296 315.12
## + mean_education        1    1028.0 47506 315.31
## + labour_participation  1     954.3 47580 315.38
## + is_south              1     799.8 47735 315.53
## + police_exp59          1     517.8 48017 315.79
## + unemploy_m39          1     470.9 48064 315.83
## + m_per1000f            1     309.8 48225 315.98
## + unemploy_m24          1      80.9 48454 316.18
## 
## Step:  AIC=305.15
## gdp ~ inequality + police_exp60 + prob_prison
## 
##                        Df Sum of Sq   RSS    AIC
## + nonwhites_per1000     1   2738.95 34965 303.83
## <none>                              37704 305.15
## + crime_rate            1   1254.38 36450 305.66
## + percent_m             1   1234.23 36470 305.68
## + mean_education        1   1114.82 36590 305.83
## + unemploy_m39          1    696.06 37008 306.33
## + police_exp59          1    501.20 37203 306.56
## + m_per1000f            1    441.35 37263 306.63
## + labour_participation  1    215.23 37489 306.89
## + unemploy_m24          1    114.91 37589 307.01
## + is_south              1    113.55 37591 307.01
## + time_prison           1     84.58 37620 307.05
## + state_pop             1     46.38 37658 307.09
## 
## Step:  AIC=303.83
## gdp ~ inequality + police_exp60 + prob_prison + nonwhites_per1000
## 
##                        Df Sum of Sq   RSS    AIC
## <none>                              34965 303.83
## + time_prison           1   1248.17 33717 304.23
## + crime_rate            1   1062.94 33902 304.47
## + unemploy_m39          1    690.97 34274 304.95
## + percent_m             1    257.61 34708 305.50
## + is_south              1    248.45 34717 305.51
## + mean_education        1    248.04 34717 305.51
## + state_pop             1    243.89 34722 305.52
## + police_exp59          1    154.10 34811 305.63
## + labour_participation  1     26.85 34939 305.80
## + m_per1000f            1     20.18 34945 305.80
## + unemploy_m24          1     16.94 34948 305.81

both3 <- step(gdpA1,scope = list(lower = gdpN1,upper = gdpA1),direction="both") #AIC 303.83

## Start:  AIC=318.87
## gdp ~ percent_m + is_south + mean_education + police_exp60 + 
##     police_exp59 + labour_participation + m_per1000f + state_pop + 
##     nonwhites_per1000 + unemploy_m24 + unemploy_m39 + inequality + 
##     prob_prison + time_prison + crime_rate
## 
##                        Df Sum of Sq   RSS    AIC
## - state_pop             1       0.1 29849 316.87
## - police_exp59          1       0.7 29850 316.87
## - labour_participation  1       9.9 29859 316.88
## - percent_m             1     152.5 30002 317.09
## - is_south              1     153.2 30002 317.09
## - police_exp60          1     196.3 30046 317.16
## - m_per1000f            1     212.7 30062 317.18
## - mean_education        1     214.8 30064 317.18
## - crime_rate            1     452.1 30301 317.53
## - unemploy_m24          1     687.0 30536 317.87
## - time_prison           1     804.1 30653 318.04
## - unemploy_m39          1    1205.7 31055 318.61
## <none>                              29849 318.87
## - nonwhites_per1000     1    1609.6 31459 319.18
## - prob_prison           1    1901.8 31751 319.59
## - inequality            1   14356.8 44206 334.15
## 
## Step:  AIC=316.87
## gdp ~ percent_m + is_south + mean_education + police_exp60 + 
##     police_exp59 + labour_participation + m_per1000f + nonwhites_per1000 + 
##     unemploy_m24 + unemploy_m39 + inequality + prob_prison + 
##     time_prison + crime_rate
## 
##                        Df Sum of Sq   RSS    AIC
## - police_exp59          1       0.7 29850 314.87
## - labour_participation  1       9.7 29859 314.88
## - is_south              1     153.2 30003 315.09
## - percent_m             1     154.2 30004 315.09
## - police_exp60          1     199.2 30049 315.16
## - mean_education        1     216.1 30066 315.19
## - m_per1000f            1     250.2 30100 315.24
## - crime_rate            1     453.1 30303 315.53
## - unemploy_m24          1     688.6 30538 315.87
## - time_prison           1     872.3 30722 316.14
## - unemploy_m39          1    1221.0 31070 316.63
## <none>                              29849 316.87
## - nonwhites_per1000     1    1627.2 31477 317.20
## - prob_prison           1    1902.4 31752 317.59
## + state_pop             1       0.1 29849 318.87
## - inequality            1   14394.8 44244 332.18
## 
## Step:  AIC=314.87
## gdp ~ percent_m + is_south + mean_education + police_exp60 + 
##     labour_participation + m_per1000f + nonwhites_per1000 + unemploy_m24 + 
##     unemploy_m39 + inequality + prob_prison + time_prison + crime_rate
## 
##                        Df Sum of Sq   RSS    AIC
## - labour_participation  1      12.6 29863 312.89
## - percent_m             1     155.9 30006 313.10
## - is_south              1     157.4 30008 313.10
## - mean_education        1     223.0 30073 313.20
## - m_per1000f            1     250.1 30100 313.24
## - crime_rate            1     461.5 30312 313.54
## - unemploy_m24          1     711.1 30561 313.91
## - time_prison           1    1008.9 30859 314.33
## - unemploy_m39          1    1232.7 31083 314.65
## <none>                              29850 314.87
## - nonwhites_per1000     1    1793.7 31644 315.44
## - police_exp60          1    2075.0 31925 315.83
## - prob_prison           1    2095.3 31945 315.85
## + police_exp59          1       0.7 29849 316.87
## + state_pop             1       0.1 29850 316.87
## - inequality            1   14481.8 44332 330.27
## 
## Step:  AIC=312.89
## gdp ~ percent_m + is_south + mean_education + police_exp60 + 
##     m_per1000f + nonwhites_per1000 + unemploy_m24 + unemploy_m39 + 
##     inequality + prob_prison + time_prison + crime_rate
## 
##                        Df Sum of Sq   RSS    AIC
## - is_south              1     155.2 30018 311.12
## - percent_m             1     165.8 30029 311.13
## - mean_education        1     282.1 30145 311.30
## - m_per1000f            1     437.0 30300 311.53
## - crime_rate            1     453.7 30316 311.55
## - unemploy_m24          1     979.9 30843 312.31
## - time_prison           1    1022.4 30885 312.37
## - unemploy_m39          1    1283.1 31146 312.74
## <none>                              29863 312.89
## - nonwhites_per1000     1    1839.9 31703 313.52
## - police_exp60          1    2064.8 31927 313.83
## - prob_prison           1    2212.1 32075 314.03
## + labour_participation  1      12.6 29850 314.87
## + police_exp59          1       3.5 29859 314.88
## + state_pop             1       0.1 29863 314.89
## - inequality            1   15246.2 45109 329.04
## 
## Step:  AIC=311.12
## gdp ~ percent_m + mean_education + police_exp60 + m_per1000f + 
##     nonwhites_per1000 + unemploy_m24 + unemploy_m39 + inequality + 
##     prob_prison + time_prison + crime_rate
## 
##                        Df Sum of Sq   RSS    AIC
## - percent_m             1     117.8 30136 309.29
## - mean_education        1     271.3 30289 309.51
## - m_per1000f            1     388.4 30406 309.68
## - crime_rate            1     478.6 30497 309.81
## - time_prison           1     902.9 30921 310.42
## - unemploy_m24          1    1318.9 31337 311.01
## <none>                              30018 311.12
## - nonwhites_per1000     1    1689.1 31707 311.52
## - unemploy_m39          1    1692.7 31711 311.53
## - police_exp60          1    1951.1 31969 311.89
## - prob_prison           1    2289.2 32307 312.35
## + is_south              1     155.2 29863 312.89
## + labour_participation  1      10.4 30008 313.10
## + police_exp59          1       1.3 30017 313.11
## + state_pop             1       0.7 30017 313.11
## - inequality            1   15306.1 45324 327.25
## 
## Step:  AIC=309.29
## gdp ~ mean_education + police_exp60 + m_per1000f + nonwhites_per1000 + 
##     unemploy_m24 + unemploy_m39 + inequality + prob_prison + 
##     time_prison + crime_rate
## 
##                        Df Sum of Sq   RSS    AIC
## - m_per1000f            1     321.4 30457 307.76
## - mean_education        1     382.6 30518 307.84
## - crime_rate            1     383.0 30519 307.84
## - time_prison           1     873.5 31009 308.55
## - unemploy_m24          1    1394.5 31530 309.28
## <none>                              30136 309.29
## - unemploy_m39          1    2181.6 32317 310.36
## - nonwhites_per1000     1    2380.7 32516 310.63
## - police_exp60          1    2413.1 32549 310.68
## - prob_prison           1    2437.3 32573 310.71
## + percent_m             1     117.8 30018 311.12
## + is_south              1     107.2 30029 311.13
## + police_exp59          1       4.8 30131 311.28
## + labour_participation  1       1.8 30134 311.29
## + state_pop             1       1.7 30134 311.29
## - inequality            1   15189.0 45325 325.25
## 
## Step:  AIC=307.76
## gdp ~ mean_education + police_exp60 + nonwhites_per1000 + unemploy_m24 + 
##     unemploy_m39 + inequality + prob_prison + time_prison + crime_rate
## 
##                        Df Sum of Sq   RSS    AIC
## - mean_education        1     511.9 30969 306.49
## - crime_rate            1     600.8 31058 306.61
## - time_prison           1     687.4 31145 306.74
## - unemploy_m24          1    1073.1 31530 307.28
## <none>                              30457 307.76
## - unemploy_m39          1    1864.0 32321 308.37
## - police_exp60          1    2391.4 32849 309.08
## - prob_prison           1    2454.9 32912 309.17
## + m_per1000f            1     321.4 30136 309.29
## - nonwhites_per1000     1    2567.0 33024 309.32
## + is_south              1      82.2 30375 309.64
## + labour_participation  1      59.2 30398 309.67
## + state_pop             1      51.0 30406 309.68
## + percent_m             1      50.8 30406 309.68
## + police_exp59          1      28.8 30428 309.71
## - inequality            1   14873.8 45331 323.25
## 
## Step:  AIC=306.49
## gdp ~ police_exp60 + nonwhites_per1000 + unemploy_m24 + unemploy_m39 + 
##     inequality + prob_prison + time_prison + crime_rate
## 
##                        Df Sum of Sq   RSS    AIC
## - unemploy_m24          1     732.3 31701 305.52
## - time_prison           1     785.9 31755 305.59
## - unemploy_m39          1    1359.3 32328 306.38
## <none>                              30969 306.49
## - crime_rate            1    1690.7 32660 306.83
## - prob_prison           1    2064.8 33034 307.33
## - police_exp60          1    2327.9 33297 307.68
## + mean_education        1     511.9 30457 307.76
## + m_per1000f            1     450.8 30518 307.84
## + labour_participation  1     227.3 30742 308.16
## + percent_m             1     134.5 30835 308.30
## + state_pop             1      83.5 30886 308.37
## + is_south              1      52.0 30917 308.41
## + police_exp59          1      10.3 30959 308.47
## - nonwhites_per1000     1    4013.0 34982 309.85
## - inequality            1   27740.9 58710 332.63
## 
## Step:  AIC=305.52
## gdp ~ police_exp60 + nonwhites_per1000 + unemploy_m39 + inequality + 
##     prob_prison + time_prison + crime_rate
## 
##                        Df Sum of Sq   RSS    AIC
## - unemploy_m39          1     680.8 32382 304.45
## - time_prison           1    1389.3 33091 305.40
## - crime_rate            1    1444.8 33146 305.48
## <none>                              31701 305.52
## - prob_prison           1    1767.1 33469 305.90
## + unemploy_m24          1     732.3 30969 306.49
## + is_south              1     278.3 31423 307.13
## + percent_m             1     231.2 31470 307.19
## + mean_education        1     171.2 31530 307.28
## + labour_participation  1     158.2 31543 307.30
## + m_per1000f            1      14.8 31687 307.50
## + state_pop             1      12.3 31689 307.50
## + police_exp59          1       2.4 31699 307.51
## - nonwhites_per1000     1    3778.3 35480 308.47
## - police_exp60          1    3945.1 35647 308.68
## - inequality            1   27104.2 58806 330.70
## 
## Step:  AIC=304.45
## gdp ~ police_exp60 + nonwhites_per1000 + inequality + prob_prison + 
##     time_prison + crime_rate
## 
##                        Df Sum of Sq   RSS    AIC
## - crime_rate            1    1335.0 33717 304.23
## <none>                              32382 304.45
## - time_prison           1    1520.3 33902 304.47
## - prob_prison           1    1637.4 34020 304.62
## + unemploy_m39          1     680.8 31701 305.52
## + percent_m             1     578.9 31803 305.66
## + is_south              1     344.3 32038 305.98
## + unemploy_m24          1      53.9 32328 306.38
## + m_per1000f            1      24.3 32358 306.42
## + state_pop             1       6.8 32375 306.44
## + mean_education        1       1.5 32381 306.45
## + labour_participation  1       1.2 32381 306.45
## + police_exp59          1       0.0 32382 306.45
## - nonwhites_per1000     1    3885.6 36268 307.44
## - police_exp60          1    4857.6 37240 308.60
## - inequality            1   26438.0 58820 328.71
## 
## Step:  AIC=304.23
## gdp ~ police_exp60 + nonwhites_per1000 + inequality + prob_prison + 
##     time_prison
## 
##                        Df Sum of Sq   RSS    AIC
## - time_prison           1    1248.2 34965 303.83
## <none>                              33717 304.23
## + crime_rate            1    1335.0 32382 304.45
## - prob_prison           1    2393.2 36110 305.25
## + unemploy_m39          1     571.0 33146 305.48
## + mean_education        1     357.2 33360 305.76
## + is_south              1     340.0 33377 305.78
## + m_per1000f            1     307.7 33410 305.83
## + percent_m             1     157.9 33559 306.02
## + unemploy_m24          1      64.6 33653 306.14
## + labour_participation  1      43.8 33673 306.17
## + police_exp59          1       7.6 33710 306.22
## + state_pop             1       0.6 33717 306.23
## - nonwhites_per1000     1    3902.5 37620 307.05
## - police_exp60          1   20774.5 54492 323.35
## - inequality            1   27802.7 61520 328.69
## 
## Step:  AIC=303.83
## gdp ~ police_exp60 + nonwhites_per1000 + inequality + prob_prison
## 
##                        Df Sum of Sq   RSS    AIC
## <none>                              34965 303.83
## + time_prison           1    1248.2 33717 304.23
## + crime_rate            1    1062.9 33902 304.47
## + unemploy_m39          1     691.0 34274 304.95
## - nonwhites_per1000     1    2739.0 37704 305.15
## + percent_m             1     257.6 34708 305.50
## + is_south              1     248.4 34717 305.51
## + mean_education        1     248.0 34717 305.52
## + state_pop             1     243.9 34722 305.52
## + police_exp59          1     154.1 34811 305.63
## + labour_participation  1      26.8 34939 305.79
## + m_per1000f            1      20.2 34945 305.80
## + unemploy_m24          1      16.9 34948 305.81
## - prob_prison           1    7753.4 42719 310.64
## - police_exp60          1   19554.4 54520 321.37
## - inequality            1   27348.3 62314 327.25

gdpAF <- lm(gdp ~ police_exp60 + nonwhites_per1000 + inequality + prob_prison,crmAF)

All 3 direction give the same AIC and same variables.

6.2 Assumption with New A Model

vif(gdpAF)

##      police_exp60 nonwhites_per1000        inequality       prob_prison 
##          2.417687          2.322902          3.311513          1.973156

bptest(gdpAF)

## 
##  studentized Breusch-Pagan test
## 
## data:  gdpAF
## BP = 1.8475, df = 4, p-value = 0.7638

shapiro.test(gdpAF$residuals)

## 
##  Shapiro-Wilk normality test
## 
## data:  gdpAF$residuals
## W = 0.99044, p-value = 0.9715

The New A Model passed all 3 assumption checking.

6.3 Model B

back2 <- step(gdpB6,direction = "backward") # AIC 313.74

## Start:  AIC=316.78
## gdp ~ percent_m + is_south + police_exp59 + labour_participation + 
##     nonwhites_per1000 + inequality + time_prison + crime_rate
## 
##                        Df Sum of Sq   RSS    AIC
## - labour_participation  1     574.2 34989 315.53
## - percent_m             1     956.9 35372 316.02
## <none>                              34415 316.78
## - is_south              1    1675.3 36090 316.92
## - crime_rate            1    1842.5 36257 317.13
## - nonwhites_per1000     1    4983.6 39398 320.87
## - police_exp59          1    5787.7 40202 321.78
## - time_prison           1    5841.4 40256 321.84
## - inequality            1   25988.1 60403 340.10
## 
## Step:  AIC=315.53
## gdp ~ percent_m + is_south + police_exp59 + nonwhites_per1000 + 
##     inequality + time_prison + crime_rate
## 
##                     Df Sum of Sq   RSS    AIC
## - percent_m          1     789.9 35779 314.53
## - is_south           1    1139.2 36128 314.97
## <none>                           34989 315.53
## - crime_rate         1    2475.5 37464 316.60
## - nonwhites_per1000  1    4919.0 39908 319.45
## - police_exp59       1    5470.0 40459 320.06
## - time_prison        1    5735.6 40725 320.36
## - inequality         1   26032.2 61021 338.55
## 
## Step:  AIC=314.53
## gdp ~ is_south + police_exp59 + nonwhites_per1000 + inequality + 
##     time_prison + crime_rate
## 
##                     Df Sum of Sq   RSS    AIC
## - is_south           1     987.9 36767 313.76
## <none>                           35779 314.53
## - crime_rate         1    1913.3 37692 314.87
## - time_prison        1    6313.9 42093 319.84
## - nonwhites_per1000  1    7531.2 43310 321.13
## - police_exp59       1    8326.0 44105 321.94
## - inequality         1   25732.1 61511 336.91
## 
## Step:  AIC=313.76
## gdp ~ police_exp59 + nonwhites_per1000 + inequality + time_prison + 
##     crime_rate
## 
##                     Df Sum of Sq   RSS    AIC
## - crime_rate         1    1659.1 38426 313.74
## <none>                           36767 313.76
## - time_prison        1    5685.1 42452 318.23
## - nonwhites_per1000  1    6773.6 43540 319.37
## - police_exp59       1    8921.5 45688 321.53
## - inequality         1   25480.5 62247 335.45
## 
## Step:  AIC=313.74
## gdp ~ police_exp59 + nonwhites_per1000 + inequality + time_prison
## 
##                     Df Sum of Sq   RSS    AIC
## <none>                           38426 313.74
## - time_prison        1      6535 44961 318.81
## - nonwhites_per1000  1      8385 46811 320.62
## - inequality         1     26011 64437 335.01
## - police_exp59       1     44154 82580 346.17

forward2 <- step(gdpB6n,scope=list(lower=gdpB6n,upper=gdpB6),direction="forward") # AIC 313.74

## Start:  AIC=412.74
## gdp ~ 1
## 
##                        Df Sum of Sq    RSS    AIC
## + inequality            1    324265  89961 346.02
## + police_exp59          1    268616 145610 367.69
## + percent_m             1    187224 227001 387.67
## + is_south              1    164529 249697 391.96
## + nonwhites_per1000     1    146374 267851 395.12
## + crime_rate            1     78901 335325 405.23
## + labour_participation  1     47336 366890 409.28
## <none>                              414226 412.74
## + time_prison           1      1680 412546 414.55
## 
## Step:  AIC=346.02
## gdp ~ inequality
## 
##                        Df Sum of Sq   RSS    AIC
## + police_exp59          1     39058 50902 322.39
## + crime_rate            1     38194 51766 323.15
## + time_prison           1      7159 82802 344.29
## + percent_m             1      5322 84639 345.28
## <none>                              89961 346.02
## + labour_participation  1      1255 88706 347.39
## + is_south              1       215 89745 347.91
## + nonwhites_per1000     1        19 89942 348.01
## 
## Step:  AIC=322.39
## gdp ~ inequality + police_exp59
## 
##                        Df Sum of Sq   RSS    AIC
## + nonwhites_per1000     1    5940.8 44961 318.81
## + time_prison           1    4091.3 46811 320.62
## + crime_rate            1    3876.2 47026 320.83
## + percent_m             1    2553.7 48348 322.08
## <none>                              50902 322.39
## + labour_participation  1    1405.9 49496 323.13
## + is_south              1     800.6 50101 323.68
## 
## Step:  AIC=318.81
## gdp ~ inequality + police_exp59 + nonwhites_per1000
## 
##                        Df Sum of Sq   RSS    AIC
## + time_prison           1    6535.5 38426 313.74
## + crime_rate            1    2509.5 42452 318.23
## <none>                              44961 318.81
## + percent_m             1     444.4 44517 320.36
## + labour_participation  1     432.5 44529 320.38
## + is_south              1     162.6 44799 320.65
## 
## Step:  AIC=313.74
## gdp ~ inequality + police_exp59 + nonwhites_per1000 + time_prison
## 
##                        Df Sum of Sq   RSS    AIC
## <none>                              38426 313.74
## + crime_rate            1   1659.09 36767 313.76
## + is_south              1    733.71 37692 314.87
## + labour_participation  1    245.79 38180 315.45
## + percent_m             1    183.60 38242 315.53

both2 <- step(gdpB6,scope=list(lower=gdpB6n,upper=gdpB6),direction="both") # AIC 313.74

## Start:  AIC=316.78
## gdp ~ percent_m + is_south + police_exp59 + labour_participation + 
##     nonwhites_per1000 + inequality + time_prison + crime_rate
## 
##                        Df Sum of Sq   RSS    AIC
## - labour_participation  1     574.2 34989 315.53
## - percent_m             1     956.9 35372 316.02
## <none>                              34415 316.78
## - is_south              1    1675.3 36090 316.92
## - crime_rate            1    1842.5 36257 317.13
## - nonwhites_per1000     1    4983.6 39398 320.87
## - police_exp59          1    5787.7 40202 321.78
## - time_prison           1    5841.4 40256 321.84
## - inequality            1   25988.1 60403 340.10
## 
## Step:  AIC=315.53
## gdp ~ percent_m + is_south + police_exp59 + nonwhites_per1000 + 
##     inequality + time_prison + crime_rate
## 
##                        Df Sum of Sq   RSS    AIC
## - percent_m             1     789.9 35779 314.53
## - is_south              1    1139.2 36128 314.97
## <none>                              34989 315.53
## - crime_rate            1    2475.5 37464 316.60
## + labour_participation  1     574.2 34415 316.78
## - nonwhites_per1000     1    4919.0 39908 319.45
## - police_exp59          1    5470.0 40459 320.06
## - time_prison           1    5735.6 40725 320.36
## - inequality            1   26032.2 61021 338.55
## 
## Step:  AIC=314.53
## gdp ~ is_south + police_exp59 + nonwhites_per1000 + inequality + 
##     time_prison + crime_rate
## 
##                        Df Sum of Sq   RSS    AIC
## - is_south              1     987.9 36767 313.76
## <none>                              35779 314.53
## - crime_rate            1    1913.3 37692 314.87
## + percent_m             1     789.9 34989 315.53
## + labour_participation  1     407.2 35372 316.02
## - time_prison           1    6313.9 42093 319.84
## - nonwhites_per1000     1    7531.2 43310 321.13
## - police_exp59          1    8326.0 44105 321.94
## - inequality            1   25732.1 61511 336.91
## 
## Step:  AIC=313.76
## gdp ~ police_exp59 + nonwhites_per1000 + inequality + time_prison + 
##     crime_rate
## 
##                        Df Sum of Sq   RSS    AIC
## - crime_rate            1    1659.1 38426 313.74
## <none>                              36767 313.76
## + is_south              1     987.9 35779 314.53
## + percent_m             1     638.6 36128 314.97
## + labour_participation  1      18.0 36749 315.73
## - time_prison           1    5685.1 42452 318.23
## - nonwhites_per1000     1    6773.6 43540 319.37
## - police_exp59          1    8921.5 45688 321.53
## - inequality            1   25480.5 62247 335.45
## 
## Step:  AIC=313.74
## gdp ~ police_exp59 + nonwhites_per1000 + inequality + time_prison
## 
##                        Df Sum of Sq   RSS    AIC
## <none>                              38426 313.74
## + crime_rate            1      1659 36767 313.76
## + is_south              1       734 37692 314.87
## + labour_participation  1       246 38180 315.45
## + percent_m             1       184 38242 315.53
## - time_prison           1      6535 44961 318.81
## - nonwhites_per1000     1      8385 46811 320.62
## - inequality            1     26011 64437 335.01
## - police_exp59          1     44154 82580 346.17

gdpBB <- lm(gdp ~ police_exp59 + nonwhites_per1000 + inequality + time_prison,crmB6)

All 3 direction give the same AIC and same variables.

The outlier removal on New B Model make different result in Stepwise Regression. The result in Stepwise Regression better give better AIC (313.74) and better Adj R Square (0.8979581).

6.4 Assumption with New B Model

vif(gdpBB)

##      police_exp59 nonwhites_per1000        inequality       time_prison 
##          2.006182          2.253062          3.525548          1.104807

bptest(gdpBB)

## 
##  studentized Breusch-Pagan test
## 
## data:  gdpBB
## BP = 3.9826, df = 4, p-value = 0.4084

shapiro.test(gdpBB$residuals)

## 
##  Shapiro-Wilk normality test
## 
## data:  gdpBB$residuals
## W = 0.95694, p-value = 0.09334

The New B Model passed all 3 assumption checking.

7.1 Summary

summary(gdpAF)

## 
## Call:
## lm(formula = gdp ~ police_exp60 + nonwhites_per1000 + inequality + 
##     prob_prison, data = crmAF)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -75.088 -16.432   0.997  20.392  67.007 
## 
## Coefficients:
##                     Estimate Std. Error t value Pr(>|t|)    
## (Intercept)        702.23274   55.90397  12.561 2.77e-15 ***
## police_exp60         1.22349    0.26198   4.670 3.52e-05 ***
## nonwhites_per1000   -0.11567    0.06618  -1.748  0.08836 .  
## inequality          -1.14714    0.20770  -5.523 2.38e-06 ***
## prob_prison       -968.35171  329.28581  -2.941  0.00548 ** 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 29.94 on 39 degrees of freedom
## Multiple R-squared:  0.913,  Adjusted R-squared:  0.9041 
## F-statistic: 102.4 on 4 and 39 DF,  p-value: < 2.2e-16

summary(gdpBB)

## 
## Call:
## lm(formula = gdp ~ police_exp59 + nonwhites_per1000 + inequality + 
##     time_prison, data = crmB6)
## 
## Residuals:
##    Min     1Q Median     3Q    Max 
## -74.13 -11.74   4.94  17.52  81.94 
## 
## Coefficients:
##                    Estimate Std. Error t value Pr(>|t|)    
## (Intercept)       580.97746   57.10407  10.174 1.17e-12 ***
## police_exp59        1.73117    0.25535   6.780 3.81e-08 ***
## nonwhites_per1000  -0.19836    0.06714  -2.954  0.00523 ** 
## inequality         -1.14629    0.22029  -5.204 6.18e-06 ***
## time_prison         1.95530    0.74965   2.608  0.01273 *  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 30.99 on 40 degrees of freedom
## Multiple R-squared:  0.9072, Adjusted R-squared:  0.898 
## F-statistic:  97.8 on 4 and 40 DF,  p-value: < 2.2e-16

summary(gdpF)

## 
## Call:
## lm(formula = gdp ~ inequality + police_exp60 + nonwhites_per1000 + 
##     crime_rate + percent_m, data = crm)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -88.011 -18.410   6.833  19.612  54.021 
## 
## Coefficients:
##                    Estimate Std. Error t value Pr(>|t|)    
## (Intercept)       834.38597  104.92045   7.953 7.71e-10 ***
## inequality         -1.39397    0.24656  -5.654 1.34e-06 ***
## police_exp60        0.71623    0.37868   1.891   0.0657 .  
## nonwhites_per1000  -0.07977    0.07741  -1.031   0.3088    
## crime_rate          0.04445    0.02071   2.146   0.0378 *  
## percent_m          -0.94976    0.58559  -1.622   0.1125    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 33.14 on 41 degrees of freedom
## Multiple R-squared:  0.8948, Adjusted R-squared:  0.882 
## F-statistic: 69.78 on 5 and 41 DF,  p-value: < 2.2e-16

7.2 Result Summary

• New A Model = GDP ~ inequality + nonwhites_per1000 + prob_prison + police_exp60

  AIC = 303.83*

  Adj R Square = 0.9041266*

  RMSE = 34.9665153

• New B Model = GDP ~ inequality + nonwhites_per1000 + time_prison + police_exp59

  AIC = 313.74

  Adj R Square = 0.8979581

  RMSE = 34.7180657

• Model F = GDP ~ inequality + nonwhites_per1000 + crime_rate + police_exp60 + percent_m

  AIC = 334.66

  Adj R Square = 0.8820182

  RMSE = 30.9554492*

1. From the result summary above, we can conclude that New A Model give the least information loss (smallest AIC) and biggest Adj R Square. Thus, model F give the smallest error RMSE.

2. All of 3 model have Inequality variable and Number of Non Whites Resident every 1000 people variable.

3. The differences between 3 models are in:

• Prison related variable: each model use different prison related variable. All 3 variables tell us the same insight, how severe the population in crime is.

  Model A: Probability of Imprisonment(prob_prison)

  Model B: Average time served in prison (time_prison)

  Model F: Crime Rate (crime_rate)

• Police Expenditures: The difference between 2 variables is just the time of the expenditures

  Model A: Police Expenditures in 1960

  Model B: Police Expenditures in 1960

  Model F: Police Expenditures in 1959

• Only have model F use Percentage of Male Aged 14-24 (percent m)

4. Considering the statistic result and the variable differences so New A Model is selected for final model.

New A Model = GDP ~ inequality + nonwhites_per1000 + prob_prison + police_exp60

7.3 Error Visualization

plot(gdpAF)