CUNY 621 HW 4
Joel Park
4/24/2018
Overview
In this homework assignment, you will explore, analyze and model a data set containing approximately 8000 records representing a customer at an auto insurance company. Each record has two response variables. The first response variable, TARGET_FLAG is a 1 or a 0. A “1” means that the person was in a car crash. A zero means that the person was not in a car crash. The second response variable is TARGET_AMT. This value is zero if the person did not crash their car. But if they did crash their car, this number will be a value greater than zero.
Your objective is to build a multiple linear regression and binary logistic regression models on the training data to predict the probability that a person will crash their car and also the amount of money it will cost if the person does crash their car. You can only use the variables given to you (or variables that you derive from the variables provided). Below is a short description of the variables of interest in the data set:
Let’s perform some data exploration and see what insight we can develop from this dataset.
## TARGET_FLAG TARGET_AMT KIDSDRIV AGE HOMEKIDS YOJ PARENT1 TRAVTIME
## 1 0 0 0 60 0 11 No 14
## 2 0 0 0 43 0 11 No 22
## 3 0 0 0 35 1 10 No 5
## 4 0 0 0 51 0 14 No 32
## 5 0 0 0 50 0 NA No 36
## 6 1 2946 0 34 1 12 Yes 46
## CAR_USE TIF RED_CAR CLM_FREQ REVOKED MVR_PTS CAR_AGE INCOME HOME_VAL
## 1 Private 11 yes 2 No 3 18 67349 0
## 2 Commercial 1 yes 0 No 0 1 91449 257252
## 3 Private 4 no 2 No 3 10 16039 124191
## 4 Private 7 yes 0 No 0 6 NA 306251
## 5 Private 1 no 2 Yes 3 17 114986 243925
## 6 Commercial 1 no 0 No 0 7 125301 0
## BLUEBOOK OLDCLAIM MSTATUS SEX EDUCATION JOB CAR_TYPE
## 1 14230 4461 No M PhD Professional Minivan
## 2 14940 0 No M High School Blue Collar Minivan
## 3 4010 38690 Yes F High School Clerical SUV
## 4 15440 0 Yes M <High School Blue Collar Minivan
## 5 18000 19217 Yes F PhD Doctor SUV
## 6 17430 0 No F Bachelors Blue Collar Sports Car
## URBANICITY
## 1 Highly Urban/ Urban
## 2 Highly Urban/ Urban
## 3 Highly Urban/ Urban
## 4 Highly Urban/ Urban
## 5 Highly Urban/ Urban
## 6 Highly Urban/ Urban## TARGET_FLAG TARGET_AMT KIDSDRIV AGE
## Min. :0.0000 Min. : 0 Min. :0.0000 Min. :16.00
## 1st Qu.:0.0000 1st Qu.: 0 1st Qu.:0.0000 1st Qu.:39.00
## Median :0.0000 Median : 0 Median :0.0000 Median :45.00
## Mean :0.2638 Mean : 1504 Mean :0.1711 Mean :44.79
## 3rd Qu.:1.0000 3rd Qu.: 1036 3rd Qu.:0.0000 3rd Qu.:51.00
## Max. :1.0000 Max. :107586 Max. :4.0000 Max. :81.00
## NA's :6
## HOMEKIDS YOJ PARENT1 TRAVTIME
## Min. :0.0000 Min. : 0.0 No :7084 Min. : 5.00
## 1st Qu.:0.0000 1st Qu.: 9.0 Yes:1077 1st Qu.: 22.00
## Median :0.0000 Median :11.0 Median : 33.00
## Mean :0.7212 Mean :10.5 Mean : 33.49
## 3rd Qu.:1.0000 3rd Qu.:13.0 3rd Qu.: 44.00
## Max. :5.0000 Max. :23.0 Max. :142.00
## NA's :454
## CAR_USE TIF RED_CAR CLM_FREQ REVOKED
## Commercial:3029 Min. : 1.000 no :5783 Min. :0.0000 No :7161
## Private :5132 1st Qu.: 1.000 yes:2378 1st Qu.:0.0000 Yes:1000
## Median : 4.000 Median :0.0000
## Mean : 5.351 Mean :0.7986
## 3rd Qu.: 7.000 3rd Qu.:2.0000
## Max. :25.000 Max. :5.0000
##
## MVR_PTS CAR_AGE INCOME HOME_VAL
## Min. : 0.000 Min. :-3.000 Min. : 0 Min. : 0
## 1st Qu.: 0.000 1st Qu.: 1.000 1st Qu.: 28097 1st Qu.: 0
## Median : 1.000 Median : 8.000 Median : 54028 Median :161160
## Mean : 1.696 Mean : 8.328 Mean : 61898 Mean :154867
## 3rd Qu.: 3.000 3rd Qu.:12.000 3rd Qu.: 85986 3rd Qu.:238724
## Max. :13.000 Max. :28.000 Max. :367030 Max. :885282
## NA's :510 NA's :445 NA's :464
## BLUEBOOK OLDCLAIM MSTATUS SEX EDUCATION
## Min. : 1500 Min. : 0 No :3267 F:4375 <High School:1203
## 1st Qu.: 9280 1st Qu.: 0 Yes:4894 M:3786 Bachelors :2242
## Median :14440 Median : 0 High School :2330
## Mean :15710 Mean : 4037 Masters :1658
## 3rd Qu.:20850 3rd Qu.: 4636 PhD : 728
## Max. :69740 Max. :57037
##
## JOB CAR_TYPE URBANICITY
## Blue Collar :1825 Minivan :2145 Highly Rural/ Rural:1669
## Clerical :1271 Panel Truck: 676 Highly Urban/ Urban:6492
## Professional:1117 Pickup :1389
## Manager : 988 Sports Car : 907
## Lawyer : 835 SUV :2294
## Student : 712 Van : 750
## (Other) :1413
Above is the numerical summaries and visualizations associated with the dataset. As with any data, it is important to look and explore some of the data. Already, we start to notice some details to this dataset including the numerous amounts of missing data, as well as skew in the histograms. However, when we develop our models, we can keep this into consideration. Let’s continue on and see if there are any correlations with the dataset. Any correlation between the numerical variables?
## TARGET_FLAG TARGET_AMT KIDSDRIV AGE HOMEKIDS
## TARGET_FLAG 1.00000000 0.53992929 0.097295404 -0.103649398 0.115481537
## TARGET_AMT 0.53992929 1.00000000 0.056857228 -0.046745810 0.068857861
## KIDSDRIV 0.09729540 0.05685723 1.000000000 -0.072361631 0.463046635
## AGE -0.10364940 -0.04674581 -0.072361631 1.000000000 -0.442383841
## HOMEKIDS 0.11548154 0.06885786 0.463046635 -0.442383841 1.000000000
## YOJ -0.06658661 -0.02144631 0.048112090 0.139566052 0.090416449
## TRAVTIME 0.05149130 0.03270817 0.008979590 0.004555303 -0.007787772
## TIF -0.07718644 -0.04525925 -0.003423442 0.002871951 0.004673246
## CLM_FREQ 0.22338169 0.11515694 0.035087170 -0.026312189 0.030695809
## MVR_PTS 0.22526236 0.13770829 0.055019621 -0.073523273 0.062776101
## CAR_AGE -0.10435770 -0.06283345 -0.055877063 0.182184524 -0.156534495
## YOJ TRAVTIME TIF CLM_FREQ
## TARGET_FLAG -0.06658661 0.051491295 -0.077186438 0.223381685
## TARGET_AMT -0.02144631 0.032708168 -0.045259254 0.115156936
## KIDSDRIV 0.04811209 0.008979590 -0.003423442 0.035087170
## AGE 0.13956605 0.004555303 0.002871951 -0.026312189
## HOMEKIDS 0.09041645 -0.007787772 0.004673246 0.030695809
## YOJ 1.00000000 -0.015762889 0.029302946 -0.030658029
## TRAVTIME -0.01576289 1.000000000 -0.009343232 0.009306981
## TIF 0.02930295 -0.009343232 1.000000000 -0.024972898
## CLM_FREQ -0.03065803 0.009306981 -0.024972898 1.000000000
## MVR_PTS -0.03917262 0.009937566 -0.037174513 0.400121265
## CAR_AGE 0.06122969 -0.037055196 0.009125709 -0.011538390
## MVR_PTS CAR_AGE
## TARGET_FLAG 0.225262361 -0.104357704
## TARGET_AMT 0.137708292 -0.062833451
## KIDSDRIV 0.055019621 -0.055877063
## AGE -0.073523273 0.182184524
## HOMEKIDS 0.062776101 -0.156534495
## YOJ -0.039172617 0.061229694
## TRAVTIME 0.009937566 -0.037055196
## TIF -0.037174513 0.009125709
## CLM_FREQ 0.400121265 -0.011538390
## MVR_PTS 1.000000000 -0.019363647
## CAR_AGE -0.019363647 1.000000000
Let’s start breaking down the data. Let’s review some information what may potentially lead to higher insurance premiums. According to Money Crashers, there are 11 factors that seem to affect the insurance premiums.
- Gender and Age
“Young men usually incur higher rates than young women as statistically, more male teenagers have accidents than female teenagers. However, older men generally have better rates than older women. Some evidence suggests that older women are in more minor accidents than older men – though the difference in premium costs usually isn’t drastic.”
## Min. 1st Qu. Median Mean 3rd Qu. Max. NA's
## 16.00 39.00 45.00 44.79 51.00 81.00 6
Q-Q Plot:
Age appears to be normally distributed (which is important for regression).
Let’s confirm the normal distribution with a Shapiro-Wilk normality test:
##
## Shapiro-Wilk normality test
##
## data: sample(train$AGE, 500, replace = TRUE)
## W = 0.99639, p-value = 0.3223From the output, the p-value is greater than the significance level 0.05 implying that the distribution of the data are not significantly different from normal distribution. In other words, we can assume the normality.
As of note, “An issue with the Shapiro-Wilk’s test is that when you feed it more data, the chances of the null hypothesis being rejected becomes larger. So what happens is that for large amounts of data even very small deviations from normality can be detected, leading to rejection of the null hypothesis event though for practical purposes the data is more than normal enough.”The R function shapiro.test protects the user from this test by limiting the data size to 5000.
Reference: https://stackoverflow.com/questions/15427692/perform-a-shapiro-wilk-normality-test
Any correlation between age and target amount payout?
## TARGET_FLAG
## AGE 0 1
## Young 222 187
## Middle 5603 1882
## Old 182 79## [1] "Percentage of Young Drivers (30 and younger) in a Crash? 0.457"## [1] "Percentage of Middle Age Drivers (30 to 60) in a Crash? 0.251"## [1] "Percentage of Older Drivers (60 and older) in a Crash? 0.303"R-squared correlation between Age and Payout?
## [1] "Correlation between Age and if it involved in car crash? -0.103"## [1] "Correlation between Age and Amount Paid if involved in car crash? -0.042"There doesn’t seem to be obvious correlation with the Pearson correlation, though from an absolute percentage count, it does appear that the middle aged drivers the less likely to be involved in an accident. And if there was an accident, there does not appear to be a correlation between age and the amount paid.
Let’s take a look at the linear regression model using only age as the predictor variable and TARGET_AMT as the response variable.
##
## Call:
## lm(formula = TARGET_AMT ~ AGE, data = train)
##
## Residuals:
## Min 1Q Median 3Q Max
## -2158 -1566 -1407 -516 106225
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 2522.162 275.257 9.163 < 2e-16 ***
## AGE -22.757 6.035 -3.771 0.000164 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 4701 on 8153 degrees of freedom
## (6 observations deleted due to missingness)
## Multiple R-squared: 0.001741, Adjusted R-squared: 0.001619
## F-statistic: 14.22 on 1 and 8153 DF, p-value: 0.0001637There does seem to be statistical significance, but a very poor adjusted R-squared value, which suggests that there are more factors to play.
Let’s take a look at the logistic regression for Age vs. TARGET_FLAG:
##
## Call:
## glm(formula = TARGET_FLAG ~ AGE, family = "binomial", data = train)
##
## Deviance Residuals:
## Min 1Q Median 3Q Max
## -1.0723 -0.8035 -0.7395 1.4260 2.0183
##
## Coefficients:
## Estimate Std. Error z value Pr(>|z|)
## (Intercept) 0.186140 0.132052 1.410 0.159
## AGE -0.027408 0.002956 -9.272 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for binomial family taken to be 1)
##
## Null deviance: 9404.0 on 8154 degrees of freedom
## Residual deviance: 9316.8 on 8153 degrees of freedom
## (6 observations deleted due to missingness)
## AIC: 9320.8
##
## Number of Fisher Scoring iterations: 4Again, statistical significance is noted. We’ll consider this when we build our final models.
- Marital Status
“Married people tend to have fewer accidents than single people; therefore, getting married (especially for men) can significantly lower your rate. How much your rate decreases depends on your previous driving history – if you are a man who has never been in an accident and has a clean driving record, you could see your rates nearly halved.”
## No Yes
## 3267 4894
It seems in theory that married people tend to drive more carefully. In general, married people tend to be older and may worry more about their own health and their family. It does appear at an initial glance that the percentage of people who tend to be in accidents are single.
## TARGET_FLAG
## MSTATUS 0 1
## No 2167 1100
## Yes 3841 1053## [1] "Percentage of Single People Involved in Car Crashes: 0.337"## [1] "Percentage of Married People Involved in Car Crashes: 0.215"Let’s take a look at the logistic regression for MSTATUS vs. TARGET_FLAG:
##
## Call:
## glm(formula = TARGET_FLAG ~ MSTATUS, family = "binomial", data = train)
##
## Deviance Residuals:
## Min 1Q Median 3Q Max
## -0.9061 -0.9061 -0.6961 1.4755 1.7529
##
## Coefficients:
## Estimate Std. Error z value Pr(>|z|)
## (Intercept) -0.67803 0.03702 -18.32 <2e-16 ***
## MSTATUSYes -0.61606 0.05080 -12.13 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for binomial family taken to be 1)
##
## Null deviance: 9418.0 on 8160 degrees of freedom
## Residual deviance: 9270.8 on 8159 degrees of freedom
## AIC: 9274.8
##
## Number of Fisher Scoring iterations: 4Again, let’s take a look at the linear regression for MSTATUS vs. TARGET_AMT:
##
## Call:
## lm(formula = TARGET_AMT ~ MSTATUS, data = train)
##
## Residuals:
## Min 1Q Median 3Q Max
## -2009 -2009 -1167 -591 106419
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 2009.00 81.99 24.504 < 2e-16 ***
## MSTATUSYes -841.56 105.87 -7.949 2.14e-15 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 4686 on 8159 degrees of freedom
## Multiple R-squared: 0.007684, Adjusted R-squared: 0.007563
## F-statistic: 63.18 on 1 and 8159 DF, p-value: 2.135e-15Both appear to have statistical significance.
- Where You Live
“Because most traffic accidents occur close to home, the area you live in greatly affects your rates. More densely populated neighborhoods with more cars mean you are at a higher risk of accidents, theft, and collisions with injuries.
Repairing your car also costs more in some areas, and some areas have higher rates of theft. Plus, in this economy, many urban areas with high unemployment rates have a lot of uninsured drivers, as many people can’t afford to insure their cars. Detroit and Philadelphia are two of the most expensive cities in which to insure a car, as they both have high traffic density and high rates of uninsured drivers."
The data collects information regarding urban areas vs. non-urban areas.
## TARGET_FLAG
## URBANICITY 0 1
## Highly Rural/ Rural 1554 115
## Highly Urban/ Urban 4454 2038## [1] "Percentage of People Living in Rural Areas Involved in Car Crashes: 0.069"## [1] "Percentage of People Living in Urban Areas Involved in Car Crashes: 0.314"It seems to make sense that in areas where there is high amounts of congestion and traffic that there will more likely be accidents.
## Highly Rural/ Rural Highly Urban/ Urban
## 1669 6492
Let’s take a look at the logistic regression for URBANICITY vs. TARGET_FLAG:
##
## Call:
## glm(formula = TARGET_FLAG ~ URBANICITY, family = "binomial",
## data = train)
##
## Deviance Residuals:
## Min 1Q Median 3Q Max
## -0.8681 -0.8681 -0.8681 1.5222 2.3130
##
## Coefficients:
## Estimate Std. Error z value Pr(>|z|)
## (Intercept) -2.60366 0.09664 -26.94 <2e-16 ***
## URBANICITYHighly Urban/ Urban 1.82182 0.10027 18.17 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for binomial family taken to be 1)
##
## Null deviance: 9418.0 on 8160 degrees of freedom
## Residual deviance: 8915.9 on 8159 degrees of freedom
## AIC: 8919.9
##
## Number of Fisher Scoring iterations: 5. Credit Score
“Many insurance companies take your credit score into account when determining your rate. There is no specific point at which your credit score begins to affect your rate, but in general, lower scores mean higher insurance premiums.”
Interestingly, before we start investigating the data, we should look at some of the social issues regarding insurance premiums and credit score. There is some arguments out there that credit scores in itself is a systematic bias against minorities. An interesting article by FiscalTiger discuss the nuances of credit scores. While it is against the law for credit bureaus to take someone’s race into consideration when creating a credit report, “this does not mean that credit scores don’t reflect deep economic divisions among people of different races and backgrounds. A 2013 poll found that 62 percent of indebted white households has ‘excellent’ or ‘good’ credit scores. Compare this with 44 percent of African-American households that also carried debt.”
That being said, we will not be examining credit score, as there is no credit score in this dataset. It is important to note that as data scientists, it is important that we use data reliably and do not create more harm with what we do.
Reference: https://www.fiscaltiger.com/are-credit-scores-racist/
- Profession
“Auto insurance companies may also make correlations between a person’s risk of accident and their profession, and they can adjust your premium accordingly if they think you’re more likely to get in an accident. For example, delivery drivers and journalists are on the road constantly, and thus are more likely to be in an accident, whereas airline pilots often just drive between the airport and home, and don’t spend much time on the road. Others, such as police officers, paramedics, nuns, and insurance underwriters, often receive a good rate, as they are seen to be more careful than the average driver.”
What type of jobs are there listed in this dataset?
## [1] "" "Blue Collar" "Clerical" "Doctor"
## [5] "Home Maker" "Lawyer" "Manager" "Professional"
## [9] "Student"Let’s look at each individual job:
## 0 1 Percent_Crashed
## 390 136 0.2585551
## Blue Collar 1191 634 0.3473973
## Clerical 900 371 0.2918961
## Doctor 217 29 0.1178862
## Home Maker 461 180 0.2808112
## Lawyer 682 153 0.1832335
## Manager 851 137 0.1386640
## Professional 870 247 0.2211280
## Student 446 266 0.3735955It appears that jobs such as doctor, manager, lawyer, and professional seem to have less percent crashed. Let’s categorize the data into two buckets, Professional and NonProfessional (I realize that the nomenclature may be an oversimplification, but I mean no harm by the way I bucket the jobs.)
In this particular case, I’ll subset Doctor, Lawyer, Manager, and Professional as Professional and designate the rest as NonProfessional.
## TARGET_FLAG
## JOB_CAT 0 1
## NonProfessional 5354 2030
## Professional 654 123What does the statistics and visualizations look like for professionals?
## TARGET_FLAG
## Min. :0.0000
## 1st Qu.:0.0000
## Median :0.0000
## Mean :0.1583
## 3rd Qu.:0.0000
## Max. :1.0000## TARGET_AMT
## Min. : 343.9
## 1st Qu.: 2506.5
## Median : 3993.0
## Mean : 4221.7
## 3rd Qu.: 5060.5
## Max. :32363.2How many and what percentage of professionals were in a car accident?
## [1] "There are a total of 777 professionals in this dataset."## [1] "How many were involved in a car accident? 123"## [1] "Percent Crashed: 0.158"Visualization:
Let’s look at Nonprofessionals.
## TARGET_FLAG
## Min. :0.0000
## 1st Qu.:0.0000
## Median :0.0000
## Mean :0.2749
## 3rd Qu.:1.0000
## Max. :1.0000## TARGET_AMT
## Min. : 30.28
## 1st Qu.: 2635.50
## Median : 4114.57
## Mean : 5791.88
## 3rd Qu.: 5811.75
## Max. :107586.14How many and what percentage of Nonprofessionals were in a car accident?
## [1] "There are a total of 7384 nonprofessionals in this dataset."## [1] "How many were involved in a car accident? 2030"## [1] "Percent Crashed: 0.275"Visualization: How does the histogram plot compare with TARGET_AMT in regards to Professionals vs. Nonprofessionals.
It appears that the median/mean payout for both the Professionals and Nonprofessionals appear similar.
- Safety Rating
“Owning a vehicle with a high safety rating means there is a lower chance of needing to pay for your or your passengers’ medical bills – therefore, your rate will be lower. Owning a car with a lower safety rating, however, will usually result in a higher cost.
The safety rating is based on several factors, including the likelihood of the car’s owner getting into an accident, and also how likely a passenger is to be injured in an accident. Safety features such as airbags, automatic seat belts, and traction control help make your car safer, which makes you less likely to get in an accident, as well as making it less dangerous."
This likely makes more sense, as cars that tend to have a higher safety rating tends to minimize the amount of damage that a passenger (or even the car) takes. As of note, this dataset does not contain any information regarding car safety ratings.
- Vehicle Size
“Larger cars are generally safer than smaller cars in an accident. Therefore, many larger cars with good safety ratings have lower premiums than smaller cars with otherwise similar ratings. However, cars with larger engines relative to body size tend to have higher rates – for instance, insurance for a sports car with a V8 engine costs much more than a small car with a V4 engine.”
What type of cars are there in our dataset?
## [1] "Minivan" "Panel Truck" "Pickup" "Sports Car" "SUV"
## [6] "Van"All of the cars in this list tend to be on the larger size, with the exception of the sports car. Let’s divide this dataset into Large vs. Small cars.
## TARGET_FLAG
## CAR_TYPE 0 1
## Minivan 1796 349
## Panel Truck 498 178
## Pickup 946 443
## Sports Car 603 304
## SUV 1616 678
## Van 549 201## 0 1 Percent_Crashed
## Large 5405 1849 0.2548939
## Small 603 304 0.3351709- Age of the Car
“Though the repair costs of an older vehicle are similar to the costs of a newer vehicle, an older car is more likely to be “totaled” in an accident. This is because the cost of significant repairs needed for an older car can often be higher than the vehicle’s entire worth. Therefore, it is likely that the owner would simply discard the vehicle and replace it, rather than paying for repairs.
Since the cost to replace a new car is much higher than to replace an old car, newer cars are not considered to be totaled as often, and generally have much higher collision coverage rates than older autos. The higher coverage translates to a higher premium for a newer car.
However, if your car is quite old, you could probably drop the collision coverage altogether and simply save the money to buy a replacement jalopy if you get in an accident."
- Likelihood of Theft
“Some cars are more attractive to thieves than others, and a car model that hits the top 10 most stolen list is likely to have higher rates than one that’s not a likely target. But if you have a car alarm or other anti-theft features, this can lower the premium.”
This is information that makes sense, but there is no information regarding likelihood of theft in this dataset.
- Driving History
“If you have been in accidents, received any tickets, or made previous auto insurance claims, the insurance company has learned that you’re more likely to make another claim than a similar driver who doesn’t have any blemishes on their record. If your driving record is bad enough, some insurance companies will refuse to give you insurance at all.
Fortunately, these blemishes tend to become less important over time. So if you had some wild years with a few tickets or accidents in your past, making an effort to drive more slowly and carefully in order to avoid future problems can pay off in time. Most tickets and non-injury accidents stop affecting your rate after three years, and injury accidents generally do not affect your rate after five years. A DUI ticket can affect your premium for up to 10 years, however, and many companies won’t insure someone with one."
In this particular dataset, we will take OLDCLAIM as a proxy for driving history. This predictor variable is the total payout over the past five year. The higher the number, the more likely that the person will have a payout.
It is hard to see if there is any correlation at all. Let’s see what correlation exists.
## OLDCLAIM TARGET_AMT
## OLDCLAIM 1.00000000 0.07095329
## TARGET_AMT 0.07095329 1.00000000There does not appear to be much correlation.
- Driving Activity
“Some companies can alter your rate based on what you use your car for, the distance you drive, and where and when you drive. Business commuters usually put more miles on their car, and the more you’re on the road, the more likely you are to get in an accident. You may be able to get a discount on your insurance if you don’t drive the car much, or don’t use it to commute to work. Plus, if you can keep your car in a secure location, such as a garage, it’s less likely to incur damage, which lowers your rate further.”
There is no data in this dataset that would be an adequate proxy for driving activity.
Reference: https://www.moneycrashers.com/factors-affect-car-insurance-rates/
Data Preparation and Model Building
When we had reviewed the data from before, we had noted that there were a significant amount of missing data points.
How many data points are missing altogether?
## [1] 1879Let’s see which data points are missing.
## CAR_AGE HOME_VAL YOJ INCOME AGE URBANICITY
## 510 464 454 445 6 0
## CAR_TYPE JOB EDUCATION SEX MSTATUS OLDCLAIM
## 0 0 0 0 0 0
## BLUEBOOK MVR_PTS REVOKED CLM_FREQ RED_CAR TIF
## 0 0 0 0 0 0
## CAR_USE TRAVTIME PARENT1 HOMEKIDS KIDSDRIV TARGET_AMT
## 0 0 0 0 0 0
## TARGET_FLAG
## 0Let’s see if we can impute these values. Let’s start with CAR_AGE. We will take only complete cases and create a histogram to see how it is distributed.
It appears right skew, with one bin appearing to be the most frequent. It may be best to impute the missing data with the mode value, as opposed to the mean or median value.
What is the mode (or most frequent value) car age? And then take this value and impute it into the Age of the Car.
## [1] 1Now, let’s perform the same tasks for the following predictor variables, HOME_VAL, YOJ, and INCOME.
Looking at the histograms, the best imputation for HOME_VAL, YOJ, and INCOME would be the mode value, median value, and median value respectively. Let’s go ahead and impute these values into the dataset.
Lastly, there are 6 missing data points for age. We can substitute that with the median as well, given that there are very few missing data points.
Now that we have our data imputed. Let’s start transforming some of our predictor variables. As we have noticed above, some of the data points are normally distributed, and some others are not. By performing a BoxCox transformation, I hope to normalize some of these predictors, which could potentially yield a better model.
We can first start by performing a BoxCox Transformation. However, we must remember the limitations of a BoxCox Transformation. It can only be done on non-zero positive values, otherwise the transformation cannot occur.
Let’s first look at the minimum values of each variables.
## TARGET_FLAG TARGET_AMT KIDSDRIV
## "0" " 0.00000" "0"
## AGE HOMEKIDS YOJ
## "16" "0" " 0"
## PARENT1 TRAVTIME CAR_USE
## "No" " 5" "Commercial"
## TIF RED_CAR CLM_FREQ
## " 1" "no" "0"
## REVOKED MVR_PTS CAR_AGE
## "No" " 0" " 0"
## INCOME HOME_VAL BLUEBOOK
## " 0" " 0" " 1500"
## OLDCLAIM MSTATUS SEX
## " 0" "No" "F"
## EDUCATION JOB CAR_TYPE
## "<High School" "" "Minivan"
## URBANICITY
## "Highly Rural/ Rural"Predictor variables TIF, BLUEBOOK, TRAVTIME, AGE all have values that are positive and non-zero. However, for other values such as CAR_AGE, YOJ, INCOME, we may be able to create a transformation if we simply add one to all of the values. From a practical stance, if we take age of the car for example, we can simply state that the first day a person owns a car, we can say that value is 1, as in, this is the first year that the patient owns the car. Likewise, we can make the same adjustment to years on the job. Adding another $1 to the income is unlikely to create a drastic difference to the variable.
## $TIF
## Box-Cox Transformation
##
## 8160 data points used to estimate Lambda
##
## Input data summary:
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 1.000 1.000 4.000 5.351 7.000 25.000
##
## Largest/Smallest: 25
## Sample Skewness: 0.891
##
## Estimated Lambda: 0.2
##
##
## $BLUEBOOK
## Box-Cox Transformation
##
## 8160 data points used to estimate Lambda
##
## Input data summary:
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 1500 9280 14440 15710 20850 69740
##
## Largest/Smallest: 46.5
## Sample Skewness: 0.794
##
## Estimated Lambda: 0.5
##
##
## $TRAVTIME
## Box-Cox Transformation
##
## 8160 data points used to estimate Lambda
##
## Input data summary:
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 5.00 22.00 33.00 33.48 44.00 142.00
##
## Largest/Smallest: 28.4
## Sample Skewness: 0.447
##
## Estimated Lambda: 0.7
##
##
## $AGE
## Box-Cox Transformation
##
## 8160 data points used to estimate Lambda
##
## Input data summary:
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 16.00 39.00 45.00 44.79 51.00 81.00
##
## Largest/Smallest: 5.06
## Sample Skewness: -0.029
##
## Estimated Lambda: 1
## With fudge factor, no transformation is applied
##
##
## $CAR_AGE
## Box-Cox Transformation
##
## 8160 data points used to estimate Lambda
##
## Input data summary:
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 1.000 2.000 9.000 8.872 13.000 29.000
##
## Largest/Smallest: 29
## Sample Skewness: 0.363
##
## Estimated Lambda: 0.4
##
##
## $YOJ
## Box-Cox Transformation
##
## 8160 data points used to estimate Lambda
##
## Input data summary:
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 1.00 10.00 12.00 11.53 14.00 24.00
##
## Largest/Smallest: 24
## Sample Skewness: -1.26
##
## Estimated Lambda: 1.6
##
##
## $INCOME
## Box-Cox Transformation
##
## 8160 data points used to estimate Lambda
##
## Input data summary:
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 1 29707 54029 61472 83308 367031
##
## Largest/Smallest: 367000
## Sample Skewness: 1.24
##
## Estimated Lambda: 0.4The BoxCox Tranformation had created some suggestions regarding the data. We will considere these transformations when we create our final model.
There are other potential transformations or modifications we can make to the data, but at this time, it is unclear what predictor variables would do well for the interaction terms. Likewise, anytime we take continuous variables and put them into buckets, we risk losing valuable information about the data, and at this time, will hold from doing so.
Let’s now proceed and build a few models for both logistic regression (to determine whether or not a crash will occur) and a linear regression (to determine how much money is paid out if a crash did indeed occur). We will build the models and evaluate them in terms of their efficacy.
Let’s start with the logistic regression model first.
Binary Logistic Regression Model:
Let’s start with building the full model without any modifications or shrinkage to the predictor variables. We will call this model, Logistic Regression Model Number 1.
##
## Call:
## glm(formula = TARGET_FLAG ~ ., family = "binomial", data = full_bin_df)
##
## Deviance Residuals:
## Min 1Q Median 3Q Max
## -2.5827 -0.7114 -0.3982 0.6266 3.1578
##
## Coefficients:
## Estimate Std. Error z value Pr(>|z|)
## (Intercept) -2.901e+00 3.405e-01 -8.519 < 2e-16 ***
## KIDSDRIV 3.861e-01 6.119e-02 6.311 2.77e-10 ***
## AGE -1.231e-03 4.016e-03 -0.307 0.759182
## HOMEKIDS 5.084e-02 3.711e-02 1.370 0.170667
## YOJ -1.116e-02 8.583e-03 -1.301 0.193401
## PARENT1Yes 3.794e-01 1.096e-01 3.463 0.000534 ***
## TRAVTIME 1.462e-02 1.883e-03 7.762 8.37e-15 ***
## CAR_USEPrivate -7.566e-01 9.170e-02 -8.251 < 2e-16 ***
## TIF -5.538e-02 7.343e-03 -7.543 4.61e-14 ***
## RED_CARyes -7.382e-03 8.633e-02 -0.086 0.931852
## CLM_FREQ 1.967e-01 2.854e-02 6.894 5.41e-12 ***
## REVOKEDYes 8.884e-01 9.128e-02 9.733 < 2e-16 ***
## MVR_PTS 1.130e-01 1.361e-02 8.305 < 2e-16 ***
## CAR_AGE -3.945e-03 6.867e-03 -0.575 0.565619
## INCOME -3.624e-06 1.075e-06 -3.372 0.000747 ***
## HOME_VAL -1.084e-06 3.171e-07 -3.419 0.000627 ***
## BLUEBOOK -2.072e-05 5.260e-06 -3.939 8.20e-05 ***
## OLDCLAIM -1.389e-05 3.910e-06 -3.553 0.000380 ***
## MSTATUSYes -5.222e-01 8.176e-02 -6.388 1.68e-10 ***
## SEXM 8.015e-02 1.120e-01 0.715 0.474379
## EDUCATIONBachelors -3.694e-01 1.141e-01 -3.238 0.001205 **
## EDUCATIONHigh School 1.925e-02 9.493e-02 0.203 0.839295
## EDUCATIONMasters -2.662e-01 1.756e-01 -1.516 0.129557
## EDUCATIONPhD -1.419e-01 2.113e-01 -0.672 0.501883
## JOBBlue Collar 3.082e-01 1.855e-01 1.662 0.096601 .
## JOBClerical 4.115e-01 1.966e-01 2.093 0.036345 *
## JOBDoctor -4.445e-01 2.668e-01 -1.666 0.095662 .
## JOBHome Maker 2.368e-01 2.100e-01 1.127 0.259589
## JOBLawyer 1.093e-01 1.694e-01 0.645 0.518722
## JOBManager -5.566e-01 1.714e-01 -3.247 0.001167 **
## JOBProfessional 1.581e-01 1.784e-01 0.887 0.375226
## JOBStudent 2.307e-01 2.143e-01 1.076 0.281781
## CAR_TYPEPanel Truck 5.538e-01 1.617e-01 3.425 0.000615 ***
## CAR_TYPEPickup 5.511e-01 1.007e-01 5.473 4.42e-08 ***
## CAR_TYPESports Car 1.023e+00 1.299e-01 7.875 3.40e-15 ***
## CAR_TYPESUV 7.651e-01 1.113e-01 6.877 6.12e-12 ***
## CAR_TYPEVan 6.129e-01 1.264e-01 4.849 1.24e-06 ***
## URBANICITYHighly Urban/ Urban 2.392e+00 1.129e-01 21.193 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for binomial family taken to be 1)
##
## Null deviance: 9418.0 on 8160 degrees of freedom
## Residual deviance: 7300.4 on 8123 degrees of freedom
## AIC: 7376.4
##
## Number of Fisher Scoring iterations: 5The second model will take this full model and attempt to shrink the amount of predictor variables needed by using a backwards stepwise approach. We will call this Logistic Regression Model 2.
## Start: AIC=7376.36
## TARGET_FLAG ~ KIDSDRIV + AGE + HOMEKIDS + YOJ + PARENT1 + TRAVTIME +
## CAR_USE + TIF + RED_CAR + CLM_FREQ + REVOKED + MVR_PTS +
## CAR_AGE + INCOME + HOME_VAL + BLUEBOOK + OLDCLAIM + MSTATUS +
## SEX + EDUCATION + JOB + CAR_TYPE + URBANICITY
##
## Df Deviance AIC
## - RED_CAR 1 7300.4 7374.4
## - AGE 1 7300.5 7374.5
## - CAR_AGE 1 7300.7 7374.7
## - SEX 1 7300.9 7374.9
## - YOJ 1 7302.1 7376.1
## - HOMEKIDS 1 7302.2 7376.2
## <none> 7300.4 7376.4
## - INCOME 1 7311.9 7385.9
## - HOME_VAL 1 7312.1 7386.1
## - PARENT1 1 7312.4 7386.4
## - OLDCLAIM 1 7313.2 7387.2
## - EDUCATION 4 7321.6 7389.6
## - BLUEBOOK 1 7316.1 7390.1
## - KIDSDRIV 1 7340.2 7414.2
## - MSTATUS 1 7340.9 7414.9
## - CLM_FREQ 1 7347.3 7421.3
## - JOB 8 7362.5 7422.5
## - TIF 1 7359.4 7433.4
## - TRAVTIME 1 7360.8 7434.8
## - CAR_USE 1 7369.5 7443.5
## - MVR_PTS 1 7369.9 7443.9
## - CAR_TYPE 5 7391.5 7457.5
## - REVOKED 1 7393.5 7467.5
## - URBANICITY 1 7950.0 8024.0
##
## Step: AIC=7374.37
## TARGET_FLAG ~ KIDSDRIV + AGE + HOMEKIDS + YOJ + PARENT1 + TRAVTIME +
## CAR_USE + TIF + CLM_FREQ + REVOKED + MVR_PTS + CAR_AGE +
## INCOME + HOME_VAL + BLUEBOOK + OLDCLAIM + MSTATUS + SEX +
## EDUCATION + JOB + CAR_TYPE + URBANICITY
##
## Df Deviance AIC
## - AGE 1 7300.5 7372.5
## - CAR_AGE 1 7300.7 7372.7
## - SEX 1 7300.9 7372.9
## - YOJ 1 7302.1 7374.1
## - HOMEKIDS 1 7302.2 7374.2
## <none> 7300.4 7374.4
## - INCOME 1 7311.9 7383.9
## - HOME_VAL 1 7312.1 7384.1
## - PARENT1 1 7312.4 7384.4
## - OLDCLAIM 1 7313.3 7385.3
## - EDUCATION 4 7321.6 7387.6
## - BLUEBOOK 1 7316.1 7388.1
## - KIDSDRIV 1 7340.3 7412.3
## - MSTATUS 1 7340.9 7412.9
## - CLM_FREQ 1 7347.3 7419.3
## - JOB 8 7362.5 7420.5
## - TIF 1 7359.4 7431.4
## - TRAVTIME 1 7360.8 7432.8
## - CAR_USE 1 7369.5 7441.5
## - MVR_PTS 1 7369.9 7441.9
## - CAR_TYPE 5 7391.6 7455.6
## - REVOKED 1 7393.5 7465.5
## - URBANICITY 1 7950.0 8022.0
##
## Step: AIC=7372.46
## TARGET_FLAG ~ KIDSDRIV + HOMEKIDS + YOJ + PARENT1 + TRAVTIME +
## CAR_USE + TIF + CLM_FREQ + REVOKED + MVR_PTS + CAR_AGE +
## INCOME + HOME_VAL + BLUEBOOK + OLDCLAIM + MSTATUS + SEX +
## EDUCATION + JOB + CAR_TYPE + URBANICITY
##
## Df Deviance AIC
## - CAR_AGE 1 7300.8 7370.8
## - SEX 1 7301.0 7371.0
## - YOJ 1 7302.3 7372.3
## <none> 7300.5 7372.5
## - HOMEKIDS 1 7303.0 7373.0
## - INCOME 1 7311.9 7381.9
## - HOME_VAL 1 7312.3 7382.3
## - PARENT1 1 7312.8 7382.8
## - OLDCLAIM 1 7313.3 7383.3
## - EDUCATION 4 7321.7 7385.7
## - BLUEBOOK 1 7316.8 7386.8
## - KIDSDRIV 1 7341.0 7411.0
## - MSTATUS 1 7341.0 7411.0
## - CLM_FREQ 1 7347.3 7417.3
## - JOB 8 7363.1 7419.1
## - TIF 1 7359.4 7429.4
## - TRAVTIME 1 7360.8 7430.8
## - CAR_USE 1 7369.5 7439.5
## - MVR_PTS 1 7370.2 7440.2
## - CAR_TYPE 5 7391.7 7453.7
## - REVOKED 1 7393.7 7463.7
## - URBANICITY 1 7950.9 8020.9
##
## Step: AIC=7370.8
## TARGET_FLAG ~ KIDSDRIV + HOMEKIDS + YOJ + PARENT1 + TRAVTIME +
## CAR_USE + TIF + CLM_FREQ + REVOKED + MVR_PTS + INCOME + HOME_VAL +
## BLUEBOOK + OLDCLAIM + MSTATUS + SEX + EDUCATION + JOB + CAR_TYPE +
## URBANICITY
##
## Df Deviance AIC
## - SEX 1 7301.3 7369.3
## - YOJ 1 7302.7 7370.7
## <none> 7300.8 7370.8
## - HOMEKIDS 1 7303.3 7371.3
## - INCOME 1 7312.4 7380.4
## - HOME_VAL 1 7312.5 7380.5
## - PARENT1 1 7313.1 7381.1
## - OLDCLAIM 1 7313.7 7381.7
## - BLUEBOOK 1 7317.1 7385.1
## - EDUCATION 4 7326.6 7388.6
## - MSTATUS 1 7341.4 7409.4
## - KIDSDRIV 1 7341.4 7409.4
## - CLM_FREQ 1 7347.5 7415.5
## - JOB 8 7363.4 7417.4
## - TIF 1 7359.9 7427.9
## - TRAVTIME 1 7361.1 7429.1
## - CAR_USE 1 7369.7 7437.7
## - MVR_PTS 1 7370.5 7438.5
## - CAR_TYPE 5 7392.3 7452.3
## - REVOKED 1 7394.0 7462.0
## - URBANICITY 1 7951.3 8019.3
##
## Step: AIC=7369.34
## TARGET_FLAG ~ KIDSDRIV + HOMEKIDS + YOJ + PARENT1 + TRAVTIME +
## CAR_USE + TIF + CLM_FREQ + REVOKED + MVR_PTS + INCOME + HOME_VAL +
## BLUEBOOK + OLDCLAIM + MSTATUS + EDUCATION + JOB + CAR_TYPE +
## URBANICITY
##
## Df Deviance AIC
## - YOJ 1 7303.2 7369.2
## <none> 7301.3 7369.3
## - HOMEKIDS 1 7303.7 7369.7
## - HOME_VAL 1 7313.0 7379.0
## - INCOME 1 7313.1 7379.1
## - PARENT1 1 7313.6 7379.6
## - OLDCLAIM 1 7314.2 7380.2
## - EDUCATION 4 7327.0 7387.0
## - BLUEBOOK 1 7324.6 7390.6
## - KIDSDRIV 1 7341.9 7407.9
## - MSTATUS 1 7341.9 7407.9
## - CLM_FREQ 1 7348.2 7414.2
## - JOB 8 7363.6 7415.6
## - TIF 1 7360.4 7426.4
## - TRAVTIME 1 7361.7 7427.7
## - CAR_USE 1 7370.2 7436.2
## - MVR_PTS 1 7371.0 7437.0
## - REVOKED 1 7394.8 7460.8
## - CAR_TYPE 5 7409.3 7467.3
## - URBANICITY 1 7952.0 8018.0
##
## Step: AIC=7369.16
## TARGET_FLAG ~ KIDSDRIV + HOMEKIDS + PARENT1 + TRAVTIME + CAR_USE +
## TIF + CLM_FREQ + REVOKED + MVR_PTS + INCOME + HOME_VAL +
## BLUEBOOK + OLDCLAIM + MSTATUS + EDUCATION + JOB + CAR_TYPE +
## URBANICITY
##
## Df Deviance AIC
## - HOMEKIDS 1 7305.0 7369.0
## <none> 7303.2 7369.2
## - HOME_VAL 1 7315.1 7379.1
## - INCOME 1 7315.7 7379.7
## - PARENT1 1 7315.8 7379.8
## - OLDCLAIM 1 7316.3 7380.3
## - EDUCATION 4 7328.6 7386.6
## - BLUEBOOK 1 7327.0 7391.0
## - KIDSDRIV 1 7344.3 7408.3
## - MSTATUS 1 7345.7 7409.7
## - CLM_FREQ 1 7350.1 7414.1
## - JOB 8 7365.5 7415.5
## - TIF 1 7362.5 7426.5
## - TRAVTIME 1 7363.3 7427.3
## - CAR_USE 1 7372.7 7436.7
## - MVR_PTS 1 7373.8 7437.8
## - REVOKED 1 7396.9 7460.9
## - CAR_TYPE 5 7411.5 7467.5
## - URBANICITY 1 7953.0 8017.0
##
## Step: AIC=7369.01
## TARGET_FLAG ~ KIDSDRIV + PARENT1 + TRAVTIME + CAR_USE + TIF +
## CLM_FREQ + REVOKED + MVR_PTS + INCOME + HOME_VAL + BLUEBOOK +
## OLDCLAIM + MSTATUS + EDUCATION + JOB + CAR_TYPE + URBANICITY
##
## Df Deviance AIC
## <none> 7305.0 7369.0
## - INCOME 1 7317.2 7379.2
## - HOME_VAL 1 7317.3 7379.3
## - OLDCLAIM 1 7318.2 7380.2
## - EDUCATION 4 7330.8 7386.8
## - PARENT1 1 7329.0 7391.0
## - BLUEBOOK 1 7329.2 7391.2
## - MSTATUS 1 7346.0 7408.0
## - CLM_FREQ 1 7352.1 7414.1
## - JOB 8 7368.6 7416.6
## - KIDSDRIV 1 7362.3 7424.3
## - TIF 1 7364.0 7426.0
## - TRAVTIME 1 7364.7 7426.7
## - CAR_USE 1 7374.3 7436.3
## - MVR_PTS 1 7376.1 7438.1
## - REVOKED 1 7399.6 7461.6
## - CAR_TYPE 5 7413.8 7467.8
## - URBANICITY 1 7955.0 8017.0##
## Call: glm(formula = TARGET_FLAG ~ KIDSDRIV + PARENT1 + TRAVTIME + CAR_USE +
## TIF + CLM_FREQ + REVOKED + MVR_PTS + INCOME + HOME_VAL +
## BLUEBOOK + OLDCLAIM + MSTATUS + EDUCATION + JOB + CAR_TYPE +
## URBANICITY, family = "binomial", data = full_bin_df)
##
## Coefficients:
## (Intercept) KIDSDRIV
## -2.985e+00 4.187e-01
## PARENT1Yes TRAVTIME
## 4.614e-01 1.451e-02
## CAR_USEPrivate TIF
## -7.566e-01 -5.532e-02
## CLM_FREQ REVOKEDYes
## 1.969e-01 8.941e-01
## MVR_PTS INCOME
## 1.140e-01 -3.710e-06
## HOME_VAL BLUEBOOK
## -1.110e-06 -2.298e-05
## OLDCLAIM MSTATUSYes
## -1.405e-05 -5.003e-01
## EDUCATIONBachelors EDUCATIONHigh School
## -3.911e-01 1.383e-02
## EDUCATIONMasters EDUCATIONPhD
## -3.121e-01 -1.882e-01
## JOBBlue Collar JOBClerical
## 3.060e-01 4.135e-01
## JOBDoctor JOBHome Maker
## -4.482e-01 2.784e-01
## JOBLawyer JOBManager
## 9.984e-02 -5.659e-01
## JOBProfessional JOBStudent
## 1.495e-01 2.898e-01
## CAR_TYPEPanel Truck CAR_TYPEPickup
## 6.031e-01 5.477e-01
## CAR_TYPESports Car CAR_TYPESUV
## 9.711e-01 7.128e-01
## CAR_TYPEVan URBANICITYHighly Urban/ Urban
## 6.420e-01 2.392e+00
##
## Degrees of Freedom: 8160 Total (i.e. Null); 8129 Residual
## Null Deviance: 9418
## Residual Deviance: 7305 AIC: 7369The stepwise model has produced a model that is not better in regards to fit as evidenced by the residual deviance and AIC, but it does contain less coefficients, which may be ultimately a better choice for us (Occam’s razor).
Before we continue, we have noticed that there are a significant amount of variables. In fact, it can seem quite overwhelming. One possible way to judge which variable may be important may be called the “Variable Importance” test. Let’s investigate this further.
Variable Importance:
“To assess the relative importance of individual predictors in the model, we can also look at the absolute value of the t-statistic for each model parameter. This technique is utilized by the varImp function in the caret package for general and generalized linear models.”
Reference: https://www.r-bloggers.com/evaluating-logistic-regression-models/
Looking at the first 6 rows (most important).
## Overall
## URBANICITYHighly Urban/ Urban 21.195625
## REVOKEDYes 9.806389
## CAR_TYPESports Car 9.041959
## MVR_PTS 8.394753
## CAR_TYPESUV 8.293251
## CAR_USEPrivate 8.261928If we use Logistic Regression Model 2, it is interesting to note that these are the top six variables that is considered likely the most important. This may not ultimately impact our final model for logistic regression, but it is also interesting to look out for this type of information.
Let’s continue onto building Logistic Regression Model 3. Remember, how we performed the BoxCox Transformation and noted the lambda values for the variables. Well, we will be using the transformed dataset (derived from the BoxCox Transformation) and see if this could ultimately create a better model. Specificially, we will be looking into: TARGET_FLAG ~ KIDSDRIV + AGE + HOMEKIDS + I(YOJ^1.6) + PARENT1 + I(TRAVTIME^0.7) + CAR_USE + I(TIF^0.2) + RED_CAR + CLM_FREQ + REVOKED + MVR_PTS + I(CAR_AGE^0.4) + I(INCOME^0.4) + HOME_VAL + I(BLUEBOOK^0.5) + OLDCLAIM + MSTATUS + SEX + EDUCATION + JOB + CAR_TYPE + URBANICITY.
##
## Call:
## glm(formula = TARGET_FLAG ~ KIDSDRIV + AGE + HOMEKIDS + I(YOJ^1.6) +
## PARENT1 + I(TRAVTIME^0.7) + CAR_USE + I(TIF^0.2) + RED_CAR +
## CLM_FREQ + REVOKED + MVR_PTS + I(CAR_AGE^0.4) + I(INCOME^0.4) +
## HOME_VAL + I(BLUEBOOK^0.5) + OLDCLAIM + MSTATUS + SEX + EDUCATION +
## JOB + CAR_TYPE + URBANICITY, family = "binomial", data = full_bin_df)
##
## Deviance Residuals:
## Min 1Q Median 3Q Max
## -2.5595 -0.7117 -0.3984 0.6201 3.1439
##
## Coefficients:
## Estimate Std. Error z value Pr(>|z|)
## (Intercept) -1.455e+00 4.078e-01 -3.567 0.000360 ***
## KIDSDRIV 3.930e-01 6.144e-02 6.397 1.59e-10 ***
## AGE -2.059e-03 4.067e-03 -0.506 0.612705
## HOMEKIDS 3.538e-02 3.772e-02 0.938 0.348204
## I(YOJ^1.6) 8.961e-04 1.609e-03 0.557 0.577457
## PARENT1Yes 3.850e-01 1.098e-01 3.506 0.000455 ***
## I(TRAVTIME^0.7) 6.032e-02 7.615e-03 7.921 2.35e-15 ***
## CAR_USEPrivate -7.575e-01 9.192e-02 -8.241 < 2e-16 ***
## I(TIF^0.2) -9.424e-01 1.205e-01 -7.818 5.37e-15 ***
## RED_CARyes -1.201e-02 8.643e-02 -0.139 0.889493
## CLM_FREQ 1.969e-01 2.858e-02 6.888 5.66e-12 ***
## REVOKEDYes 8.885e-01 9.142e-02 9.719 < 2e-16 ***
## MVR_PTS 1.131e-01 1.365e-02 8.290 < 2e-16 ***
## I(CAR_AGE^0.4) -4.034e-02 4.667e-02 -0.864 0.387420
## I(INCOME^0.4) -8.746e-03 1.764e-03 -4.957 7.14e-07 ***
## HOME_VAL -1.029e-06 3.122e-07 -3.295 0.000983 ***
## I(BLUEBOOK^0.5) -5.465e-03 1.214e-03 -4.502 6.74e-06 ***
## OLDCLAIM -1.411e-05 3.918e-06 -3.602 0.000316 ***
## MSTATUSYes -5.462e-01 8.204e-02 -6.658 2.77e-11 ***
## SEXM 7.075e-02 1.109e-01 0.638 0.523523
## EDUCATIONBachelors -3.071e-01 1.152e-01 -2.665 0.007699 **
## EDUCATIONHigh School 5.745e-02 9.555e-02 0.601 0.547717
## EDUCATIONMasters -2.106e-01 1.720e-01 -1.224 0.220925
## EDUCATIONPhD -1.183e-01 2.058e-01 -0.575 0.565250
## JOBBlue Collar 3.203e-01 1.854e-01 1.728 0.084031 .
## JOBClerical 3.817e-01 1.963e-01 1.944 0.051870 .
## JOBDoctor -4.317e-01 2.671e-01 -1.617 0.105960
## JOBHome Maker 1.847e-02 2.205e-01 0.084 0.933219
## JOBLawyer 1.204e-01 1.694e-01 0.711 0.477124
## JOBManager -5.499e-01 1.713e-01 -3.209 0.001331 **
## JOBProfessional 1.600e-01 1.784e-01 0.897 0.369802
## JOBStudent 1.529e-02 2.235e-01 0.068 0.945455
## CAR_TYPEPanel Truck 5.539e-01 1.567e-01 3.534 0.000410 ***
## CAR_TYPEPickup 5.473e-01 1.009e-01 5.425 5.79e-08 ***
## CAR_TYPESports Car 1.009e+00 1.299e-01 7.766 8.07e-15 ***
## CAR_TYPESUV 7.630e-01 1.103e-01 6.919 4.55e-12 ***
## CAR_TYPEVan 6.446e-01 1.264e-01 5.098 3.43e-07 ***
## URBANICITYHighly Urban/ Urban 2.399e+00 1.130e-01 21.229 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for binomial family taken to be 1)
##
## Null deviance: 9415.3 on 8159 degrees of freedom
## Residual deviance: 7274.2 on 8122 degrees of freedom
## (1 observation deleted due to missingness)
## AIC: 7350.2
##
## Number of Fisher Scoring iterations: 5Unfortunately, the AIC has not really improved. And now with all of these transformed variables, this model seems less intuitive than model 2. So while it was worthwhile to take a look, we will not be using model 3 as our final model.
For academic exercise, let’s take a look at Logistic Regression Model 3’s “Variable Importance”.
## Overall
## URBANICITYHighly Urban/ Urban 21.229229
## REVOKEDYes 9.719202
## MVR_PTS 8.289974
## CAR_USEPrivate 8.240706
## I(TRAVTIME^0.7) 7.921223
## I(TIF^0.2) 7.817932In both model 3 and model 2, URBANCITY seemed to have the highest variable importance, then if a license was revoked and if the driver had points. This does make logical sense as any person living in a congested area with a poor driving record is likely at a higher risk for a crash.
Before we continue onto the linear regression portion, let’s make one more attempt. Can we reduce the complexity of our logistic model? We can try to use Lasso regression as a means to reduce (or decrease the weight) of some of these variables.
Reference: https://stats.stackexchange.com/questions/159316/logistic-regression-in-r-with-many-predictors, https://stackoverflow.com/questions/23459701/logistic-regression-model-multicolinearity-of-categorical-variables-in-r
In order to perform lasso, let’s rid of categorical variables, and only focus on the numerical predictor variables.
By using cross-validation, I had split this dataset into a training dataset and a testing dataset, which then a model was developed and a prediction was made. Below are the intercepts of the coefficients and the first six predictions (made by the predict() function). To evaluate how well this model does, we had used the mean squared error.
## (Intercept) KIDSDRIV AGE HOMEKIDS YOJ
## 0.2769658621 0.0471199536 -0.0002694515 0.0241038450 0.0000000000
## TRAVTIME
## 0.0004831707## 1
## 2 0.1539869
## 3 0.4166842
## 5 0.3082434
## 6 0.2512818
## 7 0.2554711
## 12 0.1313635## [1] "MSE for lasso estimate: 0.176410792322809"It’s interesting what Lasso had reduced the variables to. However, given the complexity of lasso (and a part of my lack of knowledge of utilizing lasso), we will be using Logistic Regression Model Number 2 as our final model.
Let’s switch gears and now develop a linear regression model for the amount paid if the person did indeed get into an accident. Given that we are only looking at people who have ultimately crashed, we need to filter out the results for the zero dollar payout (as these are the people who have not crashed). We are interested in creating a model for the amount paid out in the event of a crash.
Like the Logistic Regression Model Number 1, we will be using all of the variables (untransformed) and build the Linear Regression Model Number 1. Let’s see what this model creates.
##
## Call:
## lm(formula = TARGET_AMT ~ ., data = amt_df)
##
## Residuals:
## Min 1Q Median 3Q Max
## -9069 -3182 -1483 478 99672
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 2.571e+03 1.987e+03 1.294 0.1959
## KIDSDRIV -1.810e+02 3.170e+02 -0.571 0.5680
## AGE 1.930e+01 2.123e+01 0.909 0.3635
## HOMEKIDS 2.176e+02 2.074e+02 1.049 0.2942
## YOJ 1.947e+01 4.921e+01 0.396 0.6924
## PARENT1Yes 2.746e+02 5.876e+02 0.467 0.6403
## TRAVTIME 7.601e-01 1.108e+01 0.069 0.9453
## CAR_USEPrivate -4.674e+02 5.219e+02 -0.896 0.3706
## TIF -1.575e+01 4.253e+01 -0.370 0.7112
## RED_CARyes -1.868e+02 4.967e+02 -0.376 0.7068
## CLM_FREQ -1.163e+02 1.581e+02 -0.736 0.4619
## REVOKEDYes -1.137e+03 5.167e+02 -2.201 0.0279 *
## MVR_PTS 1.137e+02 6.858e+01 1.658 0.0975 .
## CAR_AGE -8.022e+01 4.085e+01 -1.964 0.0497 *
## INCOME -8.433e-03 6.728e-03 -1.253 0.2102
## HOME_VAL 1.601e-03 1.918e-03 0.835 0.4037
## BLUEBOOK 1.248e-01 3.054e-02 4.086 4.55e-05 ***
## OLDCLAIM 2.519e-02 2.265e-02 1.112 0.2661
## MSTATUSYes -7.316e+02 4.882e+02 -1.498 0.1342
## SEXM 1.412e+03 6.566e+02 2.150 0.0317 *
## EDUCATIONBachelors 1.559e+02 6.337e+02 0.246 0.8057
## EDUCATIONHigh School -4.282e+02 5.140e+02 -0.833 0.4049
## EDUCATIONMasters 1.042e+03 1.073e+03 0.971 0.3318
## EDUCATIONPhD 2.233e+03 1.301e+03 1.716 0.0863 .
## JOBBlue Collar 5.384e+02 1.146e+03 0.470 0.6387
## JOBClerical 3.475e+02 1.204e+03 0.289 0.7728
## JOBDoctor -2.070e+03 1.763e+03 -1.174 0.2405
## JOBHome Maker -2.549e+01 1.267e+03 -0.020 0.9839
## JOBLawyer 3.221e+02 1.030e+03 0.313 0.7546
## JOBManager -7.829e+02 1.066e+03 -0.734 0.4628
## JOBProfessional 1.086e+03 1.129e+03 0.962 0.3363
## JOBStudent 8.088e+01 1.286e+03 0.063 0.9499
## CAR_TYPEPanel Truck -6.420e+02 9.609e+02 -0.668 0.5041
## CAR_TYPEPickup -7.582e+01 5.970e+02 -0.127 0.8990
## CAR_TYPESports Car 1.070e+03 7.505e+02 1.425 0.1542
## CAR_TYPESUV 9.086e+02 6.670e+02 1.362 0.1733
## CAR_TYPEVan 4.957e+01 7.714e+02 0.064 0.9488
## URBANICITYHighly Urban/ Urban 9.313e+01 7.565e+02 0.123 0.9020
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 7693 on 2115 degrees of freedom
## Multiple R-squared: 0.02986, Adjusted R-squared: 0.01289
## F-statistic: 1.759 on 37 and 2115 DF, p-value: 0.003273The adjusted R-squared value is quite low here: 0.01289. Or in other words, the model does not appear to explain the variation in the response variable quite well. However, before we continue, we must check to see if the assumptions of linear regresion were violated or not.
Let’s see if there are any underlying patterns in the model’s plot.
There appears to be significant outliers in this dataset, and given the skew in this data (as demonstrated in the Q-Q plot), this may be affecting the data quite adversely. Would the response variable benefit from a transformation? Let’s try performing a Box Cox transformation to the response variable and see what the results are.
## Box-Cox Transformation
##
## 2153 data points used to estimate Lambda
##
## Input data summary:
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 30.28 2609.78 4104.00 5702.18 5787.00 107586.14
##
## Largest/Smallest: 3550
## Sample Skewness: 5.63
##
## Estimated Lambda: 0
## With fudge factor, Lambda = 0 will be used for transformationsWith lambda = 0, let’s create a logarithmic transformation of the response variable. Let’s see if we can improve on this model with the response variable being a logarithmic transformation, and call this Linear Regression Model 2.
##
## Call:
## lm(formula = l_TARGET_AMT ~ ., data = amt_df_trans)
##
## Residuals:
## Min 1Q Median 3Q Max
## -4.6581 -0.4056 0.0305 0.4078 3.2910
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 7.936e+00 2.089e-01 37.989 < 2e-16 ***
## KIDSDRIV -3.251e-02 3.333e-02 -0.975 0.329508
## AGE 1.935e-03 2.233e-03 0.867 0.386172
## HOMEKIDS 2.272e-02 2.180e-02 1.042 0.297539
## YOJ -8.763e-04 5.174e-03 -0.169 0.865528
## PARENT1Yes 2.454e-02 6.178e-02 0.397 0.691261
## TRAVTIME -2.795e-04 1.165e-03 -0.240 0.810509
## CAR_USEPrivate -1.420e-02 5.487e-02 -0.259 0.795832
## TIF -1.961e-03 4.472e-03 -0.438 0.661158
## RED_CARyes 2.146e-02 5.222e-02 0.411 0.681119
## CLM_FREQ -3.659e-02 1.662e-02 -2.201 0.027827 *
## REVOKEDYes -9.675e-02 5.433e-02 -1.781 0.075071 .
## MVR_PTS 1.477e-02 7.211e-03 2.049 0.040586 *
## CAR_AGE -1.240e-03 4.295e-03 -0.289 0.772920
## INCOME -1.205e-06 7.074e-07 -1.704 0.088567 .
## HOME_VAL 2.079e-08 2.016e-07 0.103 0.917876
## BLUEBOOK 1.200e-05 3.212e-06 3.735 0.000193 ***
## OLDCLAIM 4.489e-06 2.381e-06 1.885 0.059540 .
## MSTATUSYes -8.309e-02 5.133e-02 -1.619 0.105683
## SEXM 9.462e-02 6.904e-02 1.370 0.170695
## EDUCATIONBachelors -3.515e-02 6.663e-02 -0.528 0.597834
## EDUCATIONHigh School 7.069e-03 5.405e-02 0.131 0.895957
## EDUCATIONMasters 1.406e-01 1.128e-01 1.246 0.212834
## EDUCATIONPhD 2.378e-01 1.368e-01 1.738 0.082299 .
## JOBBlue Collar 6.276e-02 1.205e-01 0.521 0.602631
## JOBClerical 5.525e-02 1.266e-01 0.437 0.662440
## JOBDoctor -3.585e-02 1.854e-01 -0.193 0.846668
## JOBHome Maker -5.047e-02 1.332e-01 -0.379 0.704811
## JOBLawyer -1.214e-02 1.083e-01 -0.112 0.910737
## JOBManager 1.987e-02 1.121e-01 0.177 0.859314
## JOBProfessional 1.083e-01 1.187e-01 0.912 0.361655
## JOBStudent 1.316e-02 1.352e-01 0.097 0.922510
## CAR_TYPEPanel Truck 3.761e-03 1.010e-01 0.037 0.970313
## CAR_TYPEPickup 2.818e-02 6.277e-02 0.449 0.653494
## CAR_TYPESports Car 5.762e-02 7.891e-02 0.730 0.465372
## CAR_TYPESUV 9.353e-02 7.013e-02 1.334 0.182478
## CAR_TYPEVan -1.152e-02 8.111e-02 -0.142 0.887095
## URBANICITYHighly Urban/ Urban 5.586e-02 7.955e-02 0.702 0.482577
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.8089 on 2115 degrees of freedom
## Multiple R-squared: 0.02605, Adjusted R-squared: 0.009017
## F-statistic: 1.529 on 37 and 2115 DF, p-value: 0.02192The adjusted R-squared is even worse. However, let’s look at the model plots again.
The fit is certainly better, and the distribution does indeed appear to be improved. However, with a poor adjusted R-squared value, it may benefit us to also look at the transformed predictive values as well.
Let’s utilize the transformed values (from our previous BoxCox Transformation above) and the logarithmic transformation of the response variable and see if we can create a better fit for model 3 (Linear Regression Model 3).
##
## Call:
## lm(formula = l_TARGET_AMT ~ KIDSDRIV + AGE + HOMEKIDS + I(YOJ^1.6) +
## PARENT1 + I(TRAVTIME^0.7) + CAR_USE + I(TIF^0.2) + RED_CAR +
## CLM_FREQ + REVOKED + MVR_PTS + I(CAR_AGE^0.4) + I(INCOME^0.4) +
## HOME_VAL + I(BLUEBOOK^0.5) + OLDCLAIM + MSTATUS + SEX + EDUCATION +
## JOB + CAR_TYPE + URBANICITY, data = amt_df_trans)
##
## Residuals:
## Min 1Q Median 3Q Max
## -4.6689 -0.4022 0.0332 0.4102 3.2922
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 7.799e+00 2.479e-01 31.466 < 2e-16 ***
## KIDSDRIV -3.549e-02 3.339e-02 -1.063 0.2879
## AGE 1.805e-03 2.254e-03 0.801 0.4233
## HOMEKIDS 2.422e-02 2.211e-02 1.096 0.2733
## I(YOJ^1.6) -4.923e-04 9.816e-04 -0.502 0.6160
## PARENT1Yes 2.248e-02 6.174e-02 0.364 0.7158
## I(TRAVTIME^0.7) -1.439e-03 4.673e-03 -0.308 0.7582
## CAR_USEPrivate -2.888e-03 5.479e-02 -0.053 0.9580
## I(TIF^0.2) -5.609e-02 7.291e-02 -0.769 0.4418
## RED_CARyes 2.089e-02 5.215e-02 0.401 0.6888
## CLM_FREQ -3.674e-02 1.660e-02 -2.212 0.0270 *
## REVOKEDYes -9.497e-02 5.426e-02 -1.750 0.0802 .
## MVR_PTS 1.493e-02 7.206e-03 2.072 0.0384 *
## I(CAR_AGE^0.4) 2.479e-03 2.827e-02 0.088 0.9301
## I(INCOME^0.4) -8.181e-04 1.102e-03 -0.743 0.4578
## HOME_VAL -2.136e-08 1.972e-07 -0.108 0.9138
## I(BLUEBOOK^0.5) 3.241e-03 7.234e-04 4.480 7.87e-06 ***
## OLDCLAIM 4.601e-06 2.379e-06 1.934 0.0533 .
## MSTATUSYes -7.663e-02 5.136e-02 -1.492 0.1359
## SEXM 1.033e-01 6.793e-02 1.521 0.1284
## EDUCATIONBachelors -5.461e-02 6.712e-02 -0.814 0.4159
## EDUCATIONHigh School 2.505e-03 5.432e-02 0.046 0.9632
## EDUCATIONMasters 1.057e-01 1.106e-01 0.956 0.3393
## EDUCATIONPhD 1.753e-01 1.320e-01 1.329 0.1841
## JOBBlue Collar 7.379e-02 1.198e-01 0.616 0.5381
## JOBClerical 7.536e-02 1.255e-01 0.600 0.5482
## JOBDoctor -5.359e-02 1.852e-01 -0.289 0.7723
## JOBHome Maker -3.460e-02 1.383e-01 -0.250 0.8025
## JOBLawyer -1.484e-02 1.080e-01 -0.137 0.8908
## JOBManager 1.672e-02 1.119e-01 0.149 0.8813
## JOBProfessional 1.161e-01 1.183e-01 0.982 0.3265
## JOBStudent 3.023e-02 1.401e-01 0.216 0.8292
## CAR_TYPEPanel Truck -2.101e-03 9.683e-02 -0.022 0.9827
## CAR_TYPEPickup 3.779e-02 6.277e-02 0.602 0.5472
## CAR_TYPESports Car 7.241e-02 7.842e-02 0.923 0.3559
## CAR_TYPESUV 1.018e-01 6.911e-02 1.473 0.1408
## CAR_TYPEVan -2.738e-02 8.084e-02 -0.339 0.7349
## URBANICITYHighly Urban/ Urban 5.258e-02 7.940e-02 0.662 0.5079
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.8078 on 2114 degrees of freedom
## (1 observation deleted due to missingness)
## Multiple R-squared: 0.02836, Adjusted R-squared: 0.01135
## F-statistic: 1.668 on 37 and 2114 DF, p-value: 0.00721
Again, the adjusted R-squared is doing a poor job here. And interestingly, when very little of the individual predictors are significant, as it is here, we should consider the possibility of collinearity.
Let’s take a step to the side and take a moment to look at the pairwise correlations for all of the numerical predictor variables.
## KIDSDRIV AGE HOMEKIDS YOJ TRAVTIME TIF CLM_FREQ MVR_PTS
## KIDSDRIV 1.00 0.00 0.48 0.07 -0.02 -0.01 0.01 0.02
## AGE 0.00 1.00 -0.41 0.12 0.06 -0.02 0.02 -0.03
## HOMEKIDS 0.48 -0.41 1.00 0.07 -0.04 0.00 -0.01 0.04
## YOJ 0.07 0.12 0.07 1.00 0.03 -0.02 -0.02 -0.04
## TRAVTIME -0.02 0.06 -0.04 0.03 1.00 -0.02 0.03 0.02
## TIF -0.01 -0.02 0.00 -0.02 -0.02 1.00 0.02 -0.03
## CLM_FREQ 0.01 0.02 -0.01 -0.02 0.03 0.02 1.00 0.30
## MVR_PTS 0.02 -0.03 0.04 -0.04 0.02 -0.03 0.30 1.00
## CAR_AGE 0.01 0.15 -0.07 0.05 0.06 -0.01 0.05 -0.04
## INCOME 0.05 0.17 -0.10 0.35 0.05 -0.03 0.00 -0.05
## HOME_VAL 0.04 0.19 -0.06 0.30 0.04 -0.02 -0.01 -0.07
## BLUEBOOK 0.04 0.15 -0.08 0.16 0.03 -0.01 -0.02 -0.03
## OLDCLAIM 0.01 0.02 0.01 0.02 -0.03 -0.01 0.41 0.15
## CAR_AGE INCOME HOME_VAL BLUEBOOK OLDCLAIM
## KIDSDRIV 0.01 0.05 0.04 0.04 0.01
## AGE 0.15 0.17 0.19 0.15 0.02
## HOMEKIDS -0.07 -0.10 -0.06 -0.08 0.01
## YOJ 0.05 0.35 0.30 0.16 0.02
## TRAVTIME 0.06 0.05 0.04 0.03 -0.03
## TIF -0.01 -0.03 -0.02 -0.01 -0.01
## CLM_FREQ 0.05 0.00 -0.01 -0.02 0.41
## MVR_PTS -0.04 -0.05 -0.07 -0.03 0.15
## CAR_AGE 1.00 0.40 0.16 0.18 0.02
## INCOME 0.40 1.00 0.45 0.42 -0.02
## HOME_VAL 0.16 0.45 1.00 0.21 -0.01
## BLUEBOOK 0.18 0.42 0.21 1.00 -0.03
## OLDCLAIM 0.02 -0.02 -0.01 -0.03 1.00As we can see that for the most part, there are not too many strong correlations between the independent variables. However, there are some with a higher degree of correlation (approximately 0.4), such as HOMEKIDS and KIDSDRIV. This likely makes sense as the more kids there are at home, the more will drive when they get older. Let’s see if we can utilize another tool to help us see which variables have high multicolinearity.
“In statistics, the variance inflation factor (VIF) is the ratio of variance in a model with multiple terms, divided by the variance of a model with one term alone. It quantifies the severity of multicollinearity in an ordinary least squares regression analysis. It provides an index that measures how much the variance of an estimated regression coefficient is increased because of collinearity.”
When analyzing the magnitude of multicollinearity, “a rule of thumb is that if VIF > 5, then multicollinearity is high.”
Reference: https://en.wikipedia.org/wiki/Variance_inflation_factor
Let’s look at the first three models we analyzed with VIF, and see if we can derive some insight into this dataset and models.
Linear Regression Model 1:
## KIDSDRIV AGE
## 1.436027 1.496272
## HOMEKIDS YOJ
## 2.234314 1.694815
## PARENT1Yes TRAVTIME
## 2.162926 1.031280
## CAR_USEPrivate TIF
## 2.474947 1.017530
## RED_CARyes CLM_FREQ
## 1.833052 1.416592
## REVOKEDYes MVR_PTS
## 1.587162 1.137529
## CAR_AGE INCOME
## 1.888068 2.859254
## HOME_VAL BLUEBOOK
## 1.855777 2.336869
## OLDCLAIM MSTATUSYes
## 1.891359 2.166634
## SEXM EDUCATIONBachelors
## 3.876110 2.686288
## EDUCATIONHigh School EDUCATIONMasters
## 2.236481 5.394925
## EDUCATIONPhD JOBBlue Collar
## 3.367247 9.932732
## JOBClerical JOBDoctor
## 7.516017 1.502302
## JOBHome Maker JOBLawyer
## 4.472591 2.548098
## JOBManager JOBProfessional
## 2.463689 4.712299
## JOBStudent CAR_TYPEPanel Truck
## 6.516638 2.547406
## CAR_TYPEPickup CAR_TYPESports Car
## 2.118701 2.484660
## CAR_TYPESUV CAR_TYPEVan
## 3.491773 1.832417
## URBANICITYHighly Urban/ Urban
## 1.052710Linear Regression Model 2:
## KIDSDRIV AGE
## 1.436027 1.496272
## HOMEKIDS YOJ
## 2.234314 1.694815
## PARENT1Yes TRAVTIME
## 2.162926 1.031280
## CAR_USEPrivate TIF
## 2.474947 1.017530
## RED_CARyes CLM_FREQ
## 1.833052 1.416592
## REVOKEDYes MVR_PTS
## 1.587162 1.137529
## CAR_AGE INCOME
## 1.888068 2.859254
## HOME_VAL BLUEBOOK
## 1.855777 2.336869
## OLDCLAIM MSTATUSYes
## 1.891359 2.166634
## SEXM EDUCATIONBachelors
## 3.876110 2.686288
## EDUCATIONHigh School EDUCATIONMasters
## 2.236481 5.394925
## EDUCATIONPhD JOBBlue Collar
## 3.367247 9.932732
## JOBClerical JOBDoctor
## 7.516017 1.502302
## JOBHome Maker JOBLawyer
## 4.472591 2.548098
## JOBManager JOBProfessional
## 2.463689 4.712299
## JOBStudent CAR_TYPEPanel Truck
## 6.516638 2.547406
## CAR_TYPEPickup CAR_TYPESports Car
## 2.118701 2.484660
## CAR_TYPESUV CAR_TYPEVan
## 3.491773 1.832417
## URBANICITYHighly Urban/ Urban
## 1.052710Linear Regression Model 3:
## KIDSDRIV AGE
## 1.444711 1.529467
## HOMEKIDS I(YOJ^1.6)
## 2.302231 1.699048
## PARENT1Yes I(TRAVTIME^0.7)
## 2.165043 1.031685
## CAR_USEPrivate I(TIF^0.2)
## 2.472873 1.015939
## RED_CARyes CLM_FREQ
## 1.832338 1.415384
## REVOKEDYes MVR_PTS
## 1.584465 1.138920
## I(CAR_AGE^0.4) I(INCOME^0.4)
## 1.625873 4.046382
## HOME_VAL I(BLUEBOOK^0.5)
## 1.780144 2.063749
## OLDCLAIM MSTATUSYes
## 1.886852 2.173745
## SEXM EDUCATIONBachelors
## 3.760053 2.729023
## EDUCATIONHigh School EDUCATIONMasters
## 2.264175 5.194129
## EDUCATIONPhD JOBBlue Collar
## 3.142537 9.841941
## JOBClerical JOBDoctor
## 7.410804 1.503319
## JOBHome Maker JOBLawyer
## 4.833634 2.540431
## JOBManager JOBProfessional
## 2.463331 4.671913
## JOBStudent CAR_TYPEPanel Truck
## 7.012269 2.345808
## CAR_TYPEPickup CAR_TYPESports Car
## 2.120154 2.459893
## CAR_TYPESUV CAR_TYPEVan
## 3.398621 1.824953
## URBANICITYHighly Urban/ Urban
## 1.051448As we can see, there are some values with high VIF suggestive of multicollinearity. We have too many variables that are trying to do the same job of explaining the response. “We can reduce the collinearity by carefully removing some of the variables. We can then make a more stable estimation of the coefficients and come to a more secure conclusion regarding the effect of the reamining predictors on the response.” (Reference: LMR Page 109)
Judging from all three models’ VIF and the correlation matrix, we can start to eliminate some variables. In this case, let’s remove HOMEKIDS (as the effect is unknown, and KIDSDRIV likely is a better representation of risk), EDUCATION, and JOB.
Let’s create model 4 with the elimination of these variables and with the addition of the other transformed variables.
Linear Regression Model 4:
##
## Call:
## lm(formula = l_TARGET_AMT ~ KIDSDRIV + AGE + I(YOJ^1.6) + PARENT1 +
## I(TRAVTIME^0.7) + CAR_USE + I(TIF^0.2) + RED_CAR + CLM_FREQ +
## REVOKED + MVR_PTS + I(CAR_AGE^0.4) + I(INCOME^0.4) + HOME_VAL +
## I(BLUEBOOK^0.5) + OLDCLAIM + MSTATUS + SEX + CAR_TYPE + URBANICITY,
## data = amt_df_trans)
##
## Residuals:
## Min 1Q Median 3Q Max
## -4.6889 -0.3988 0.0301 0.4088 3.2635
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 7.823e+00 1.921e-01 40.726 < 2e-16 ***
## KIDSDRIV -1.966e-02 2.948e-02 -0.667 0.5049
## AGE 1.043e-03 2.037e-03 0.512 0.6086
## I(YOJ^1.6) -1.551e-04 9.224e-04 -0.168 0.8665
## PARENT1Yes 5.030e-02 5.530e-02 0.910 0.3631
## I(TRAVTIME^0.7) -1.526e-03 4.657e-03 -0.328 0.7432
## CAR_USEPrivate 4.707e-03 4.200e-02 0.112 0.9108
## I(TIF^0.2) -4.900e-02 7.261e-02 -0.675 0.4999
## RED_CARyes 1.756e-02 5.194e-02 0.338 0.7354
## CLM_FREQ -3.598e-02 1.648e-02 -2.183 0.0291 *
## REVOKEDYes -8.875e-02 5.396e-02 -1.645 0.1002
## MVR_PTS 1.568e-02 7.153e-03 2.192 0.0285 *
## I(CAR_AGE^0.4) 6.700e-03 2.358e-02 0.284 0.7763
## I(INCOME^0.4) -1.959e-04 7.838e-04 -0.250 0.8026
## HOME_VAL -1.399e-08 1.935e-07 -0.072 0.9423
## I(BLUEBOOK^0.5) 3.225e-03 7.199e-04 4.480 7.85e-06 ***
## OLDCLAIM 4.397e-06 2.367e-06 1.858 0.0634 .
## MSTATUSYes -5.864e-02 4.863e-02 -1.206 0.2280
## SEXM 1.068e-01 6.714e-02 1.591 0.1119
## CAR_TYPEPanel Truck 1.423e-02 9.152e-02 0.155 0.8765
## CAR_TYPEPickup 4.366e-02 6.112e-02 0.714 0.4750
## CAR_TYPESports Car 7.125e-02 7.806e-02 0.913 0.3615
## CAR_TYPESUV 9.717e-02 6.883e-02 1.412 0.1582
## CAR_TYPEVan -2.347e-02 7.843e-02 -0.299 0.7648
## URBANICITYHighly Urban/ Urban 4.652e-02 7.866e-02 0.591 0.5543
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.8069 on 2127 degrees of freedom
## (1 observation deleted due to missingness)
## Multiple R-squared: 0.02472, Adjusted R-squared: 0.01371
## F-statistic: 2.246 on 24 and 2127 DF, p-value: 0.0004957
So far, this model has outperformed the first three models with a modest increase in the adjusted R-square to 0.01371.
Perhaps there are too many predictor variables, and this over-complex model is in fact creating issues with the model. Can we still improve on this by finding the most optimal number or combination of predictors? We will be using the leaps package. For each size of model p, it finds the variables that produces the minimum RSS.
## [1] "How many variables that maximizes the adjusted R-squared value? 8"
Looks like according to the model that would satisfy the Cp Statistic and increases the adjusted R-squared would be 8 variables. Which variables should be included?
## (Intercept) KIDSDRIV
## TRUE FALSE
## AGE HOMEKIDS
## FALSE FALSE
## YOJ PARENT1Yes
## FALSE FALSE
## TRAVTIME CAR_USEPrivate
## FALSE FALSE
## TIF RED_CARyes
## FALSE FALSE
## CLM_FREQ REVOKEDYes
## TRUE TRUE
## MVR_PTS CAR_AGE
## TRUE FALSE
## INCOME HOME_VAL
## FALSE FALSE
## BLUEBOOK OLDCLAIM
## TRUE TRUE
## MSTATUSYes SEXM
## TRUE TRUE
## EDUCATIONBachelors EDUCATIONHigh School
## TRUE FALSE
## EDUCATIONMasters EDUCATIONPhD
## FALSE FALSE
## JOBBlue Collar JOBClerical
## FALSE FALSE
## JOBDoctor JOBHome Maker
## FALSE FALSE
## JOBLawyer JOBManager
## FALSE FALSE
## JOBProfessional JOBStudent
## FALSE FALSE
## CAR_TYPEPanel Truck CAR_TYPEPickup
## FALSE FALSE
## CAR_TYPESports Car CAR_TYPESUV
## FALSE FALSE
## CAR_TYPEVan URBANICITYHighly Urban/ Urban
## FALSE FALSESo this includes the Intercept, CLM_FREQ, MVR_PTS, BLUEBOOK, MSTATUS, EDUCATION, SEX, CAR_AGE. Let’s create a Linear Regression Model 5 from just using these variables and the logairthmic response variable of TARGET_AMT.
##
## Call:
## lm(formula = l_TARGET_AMT ~ CLM_FREQ + MVR_PTS + BLUEBOOK + MSTATUS +
## EDUCATION + SEX + CAR_AGE, data = amt_df_trans)
##
## Residuals:
## Min 1Q Median 3Q Max
## -4.7107 -0.4026 0.0361 0.4002 3.2884
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 8.139e+00 5.970e-02 136.330 < 2e-16 ***
## CLM_FREQ -2.233e-02 1.464e-02 -1.525 0.1274
## MVR_PTS 1.728e-02 7.077e-03 2.442 0.0147 *
## BLUEBOOK 9.790e-06 2.192e-06 4.465 8.41e-06 ***
## MSTATUSYes -7.726e-02 3.499e-02 -2.208 0.0273 *
## EDUCATIONBachelors -5.064e-02 5.854e-02 -0.865 0.3871
## EDUCATIONHigh School 5.173e-03 5.035e-02 0.103 0.9182
## EDUCATIONMasters 4.416e-02 7.520e-02 0.587 0.5571
## EDUCATIONPhD 8.359e-02 9.567e-02 0.874 0.3823
## SEXM 5.881e-02 3.516e-02 1.673 0.0945 .
## CAR_AGE -1.116e-03 4.252e-03 -0.262 0.7930
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.8064 on 2142 degrees of freedom
## Multiple R-squared: 0.01973, Adjusted R-squared: 0.01515
## F-statistic: 4.311 on 10 and 2142 DF, p-value: 5.49e-06
Model 5 appears to be even more improved from model 4, with an adjusted R-squared value of 0.0.1515. Now, let’s try to simplify this and try to perform a backwards stepwise approach and attempt to eliminiate more variables to create Linear Regression Model 6.
## Start: AIC=-915.67
## l_TARGET_AMT ~ CLM_FREQ + MVR_PTS + BLUEBOOK + MSTATUS + EDUCATION +
## SEX + CAR_AGE
##
## Df Sum of Sq RSS AIC
## - EDUCATION 4 2.6416 1395.5 -919.59
## - CAR_AGE 1 0.0448 1392.9 -917.60
## <none> 1392.8 -915.67
## - CLM_FREQ 1 1.5126 1394.3 -915.33
## - SEX 1 1.8198 1394.7 -914.86
## - MSTATUS 1 3.1713 1396.0 -912.77
## - MVR_PTS 1 3.8785 1396.7 -911.68
## - BLUEBOOK 1 12.9660 1405.8 -897.72
##
## Step: AIC=-919.59
## l_TARGET_AMT ~ CLM_FREQ + MVR_PTS + BLUEBOOK + MSTATUS + SEX +
## CAR_AGE
##
## Df Sum of Sq RSS AIC
## - CAR_AGE 1 0.0202 1395.5 -921.56
## <none> 1395.5 -919.59
## - CLM_FREQ 1 1.5386 1397.0 -919.22
## - SEX 1 1.7610 1397.2 -918.87
## - MSTATUS 1 2.9488 1398.4 -917.04
## - MVR_PTS 1 3.7967 1399.3 -915.74
## - BLUEBOOK 1 14.5721 1410.0 -899.22
##
## Step: AIC=-921.56
## l_TARGET_AMT ~ CLM_FREQ + MVR_PTS + BLUEBOOK + MSTATUS + SEX
##
## Df Sum of Sq RSS AIC
## <none> 1395.5 -921.56
## - CLM_FREQ 1 1.5215 1397.0 -921.21
## - SEX 1 1.7492 1397.2 -920.86
## - MSTATUS 1 2.9807 1398.5 -918.96
## - MVR_PTS 1 3.7775 1399.3 -917.74
## - BLUEBOOK 1 15.2806 1410.8 -900.11##
## Call:
## lm(formula = l_TARGET_AMT ~ CLM_FREQ + MVR_PTS + BLUEBOOK + MSTATUS +
## SEX, data = amt_df_trans)
##
## Residuals:
## Min 1Q Median 3Q Max
## -4.7064 -0.4032 0.0398 0.3993 3.2398
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 8.126e+00 4.645e-02 174.951 < 2e-16 ***
## CLM_FREQ -2.232e-02 1.459e-02 -1.530 0.1262
## MVR_PTS 1.703e-02 7.063e-03 2.411 0.0160 *
## BLUEBOOK 1.020e-05 2.103e-06 4.849 1.33e-06 ***
## MSTATUSYes -7.447e-02 3.478e-02 -2.141 0.0323 *
## SEXM 5.755e-02 3.508e-02 1.640 0.1011
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.8062 on 2147 degrees of freedom
## Multiple R-squared: 0.01785, Adjusted R-squared: 0.01557
## F-statistic: 7.806 on 5 and 2147 DF, p-value: 2.72e-07The adjusted R-squared certainly has improved modestly at 0.01557. We will be using model 6 as our linear regression model as there are 5 predictor variables needed for prediction (as opposed to the full set) and has the highest adjusted R-squared value.
Before we move onto evaluating the models, for academic practice, let’s take a look at the marginal plots for the 6th model to get a better sense of what the predictor variables are doing. The marginal plots will help us visualize how each independent variable influences the response variable if all other variables were controlled.
Now that we have chosen our logistic regression model (#2) and our linear regression model (#6) as our final models. Let’s see if we can evaluate the models and make some predictions on our test set. We had already evaluated the performance strength on the linear regression models (by using adjusted R-square values), so let’s focus on evaluating the logistic regression model.
By using cross-validation, let’s see how the logistic regression model performs.
## Confusion Matrix and Statistics
##
## Reference
## Prediction 0 1
## 0 1665 374
## 1 125 284
##
## Accuracy : 0.7962
## 95% CI : (0.7796, 0.812)
## No Information Rate : 0.7312
## P-Value [Acc > NIR] : 5.284e-14
##
## Kappa : 0.411
## Mcnemar's Test P-Value : < 2.2e-16
##
## Sensitivity : 0.4316
## Specificity : 0.9302
## Pos Pred Value : 0.6944
## Neg Pred Value : 0.8166
## Prevalence : 0.2688
## Detection Rate : 0.1160
## Detection Prevalence : 0.1671
## Balanced Accuracy : 0.6809
##
## 'Positive' Class : 1
## ## Sensitivity Specificity Pos Pred Value
## 0.4316109 0.9301676 0.6943765
## Neg Pred Value Precision Recall
## 0.8165768 0.6943765 0.4316109
## F1 Prevalence Detection Rate
## 0.5323336 0.2687908 0.1160131
## Detection Prevalence Balanced Accuracy
## 0.1670752 0.6808893
## Area under the curve: 0.821So the model itself is not great, but not terrible. The accuracy is at 0.7978, sensitivity is at 0.4441, specificity at 0.9241, positive predictive value at 0.9241, and negative predictive value at 0.8232. The F1 rate is 0.5360825.
Before we move on, we had assumed that any predicted value greater than 0.5 was positive, and that a value less than 0.5 was negative. However, this number does not necessarily maximize the sensitivity and specificity. For example, making a cutoff at 0.2 made actually improve the model. Let’s take a closer look into this.
hist(pred)
print(paste0("Median: ",median(pred)))## [1] "Median: 0.202"Just judging by the histogram alone, it is not entirely clear that 0.5 may have been the best choice for this model. Let’s look at some more visualizations and see what value may be best in creating the best predictions.
## cutoff sens spec
## 1 0.10 0.93617021 0.4089385
## 2 0.15 0.89361702 0.5301676
## 3 0.20 0.83738602 0.6189944
## 4 0.25 0.79179331 0.7011173
## 5 0.30 0.72492401 0.7564246
## 6 0.35 0.65501520 0.8290503
## 7 0.40 0.57598784 0.8726257
## 8 0.45 0.50000000 0.9050279
## 9 0.50 0.43161094 0.9301676
## 10 0.55 0.35258359 0.9513966
## 11 0.60 0.28571429 0.9648045
## 12 0.65 0.21884498 0.9754190
## 13 0.70 0.14589666 0.9843575
## 14 0.75 0.11550152 0.9882682
## 15 0.80 0.06231003 0.9932961
## 16 0.85 0.03343465 0.9988827
## 17 0.90 0.01215805 0.9994413
We will be using a value of 0.28 as our cutoff, keeping in mind that we are trying to balance out both sensitivity and specificity.
## Confusion Matrix and Statistics
##
## Reference
## Prediction 0 1
## 0 1324 171
## 1 489 464
##
## Accuracy : 0.7304
## 95% CI : (0.7123, 0.7479)
## No Information Rate : 0.7406
## P-Value [Acc > NIR] : 0.8799
##
## Kappa : 0.3965
## Mcnemar's Test P-Value : <2e-16
##
## Sensitivity : 0.7307
## Specificity : 0.7303
## Pos Pred Value : 0.4869
## Neg Pred Value : 0.8856
## Prevalence : 0.2594
## Detection Rate : 0.1895
## Detection Prevalence : 0.3893
## Balanced Accuracy : 0.7305
##
## 'Positive' Class : 1
## ## Sensitivity Specificity Pos Pred Value
## 0.7307087 0.7302813 0.4868835
## Neg Pred Value Precision Recall
## 0.8856187 0.4868835 0.7307087
## F1 Prevalence Detection Rate
## 0.5843829 0.2593954 0.1895425
## Detection Prevalence Balanced Accuracy
## 0.3892974 0.7304950
## Area under the curve: 0.8127This value of 0.28 may be a better cutoff altogether. Let’s try this for our predictions.
Prediction
Before we start making predictions with the data, let’s clean up the testset. We want to modify the testset such that it will be able to be utilized in our models for prediction.
## TARGET_FLAG KIDSDRIV AGE HOMEKIDS YOJ PARENT1 TRAVTIME CAR_USE TIF
## 1 NA 0 48 0 11 No 26 Private 1
## 2 NA 1 40 1 11 Yes 21 Private 6
## 3 NA 0 44 2 12 Yes 30 Commercial 10
## 4 NA 0 35 2 NA Yes 74 Private 6
## 5 NA 0 59 0 12 No 45 Private 1
## 6 NA 0 46 0 14 No 7 Commercial 1
## RED_CAR CLM_FREQ REVOKED MVR_PTS CAR_AGE INCOME HOME_VAL BLUEBOOK
## 1 yes 0 No 2 10 52881 0 21970
## 2 no 1 No 2 1 50815 0 18930
## 3 no 0 No 0 10 43486 0 5900
## 4 no 0 Yes 0 4 21204 0 9230
## 5 yes 2 No 4 1 87460 0 15420
## 6 no 1 No 2 12 NA 207519 25660
## OLDCLAIM MSTATUS SEX EDUCATION JOB CAR_TYPE
## 1 0 No M Bachelors Manager Van
## 2 3295 No M High School Manager Minivan
## 3 0 No F High School Blue Collar SUV
## 4 0 No M High School Clerical Pickup
## 5 44857 No M High School Manager Minivan
## 6 2119 Yes M Bachelors Professional Panel Truck
## URBANICITY
## 1 Highly Urban/ Urban
## 2 Highly Urban/ Urban
## 3 Highly Rural/ Rural
## 4 Highly Rural/ Rural
## 5 Highly Urban/ Urban
## 6 Highly Urban/ UrbanNow, let’s make the binary predictions using our model. Did the person crash his or her car?
## 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
## 0 0 0 0 0 NA 1 1 0 0 0 1 NA 0 0
## 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
## NA 1 0 1 1 0 1 0 1 1 1 1 1 0 0
## 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45
## 0 1 0 0 0 0 0 0 0 1 1 1 0 1 0
## 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
## 0 0 1 0 1 0 1 1 0 NA 0 1 1 0 1
## 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75
## 0 0 1 0 0 1 1 1 0 0 NA 0 1 1 1
## 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90
## 0 1 1 0 0 1 1 1 0 1 1 1 1 0 NA
## 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105
## 0 0 NA 0 0 0 0 1 1 0 1 NA 1 1 NA
## 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120
## 0 0 0 1 0 1 0 0 0 NA 0 0 1 1 0
## 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135
## 0 1 1 NA 0 1 1 0 0 0 0 0 0 0 0
## 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150
## NA 1 1 0 0 0 1 0 0 0 1 0 NA 1 NA
## 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165
## 1 NA NA 1 1 1 NA 0 1 1 0 0 0 0 1
## 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180
## 0 0 NA 1 0 0 1 NA 1 1 1 NA 1 1 1
## 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195
## 1 0 0 1 0 1 0 0 0 0 1 1 NA 1 0
## 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210
## 1 1 0 0 0 0 1 0 0 0 0 NA 0 0 0
## 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225
## 0 0 1 1 0 1 1 0 0 0 0 0 1 1 0
## 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240
## 1 1 1 1 0 0 0 1 0 0 0 0 0 0 1
## 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255
## 0 NA 1 0 0 1 1 1 0 1 NA NA 0 1 1
## 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270
## 1 NA 0 1 0 0 0 0 NA 0 0 0 0 1 1
## 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285
## 1 0 1 NA 0 0 NA NA 0 0 0 0 0 1 1
## 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300
## 1 0 1 1 1 1 0 0 1 0 1 0 1 0 0
## 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315
## 0 0 1 1 1 1 0 1 0 0 NA NA 0 1 0
## 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330
## 0 0 0 1 0 0 1 0 0 1 1 NA NA NA 0
## 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345
## NA NA 1 0 0 1 NA 1 NA 0 1 1 1 1 0
## 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360
## 0 0 0 0 0 0 1 NA 1 0 1 1 0 0 0
## 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375
## 1 0 0 1 0 1 0 1 NA 0 0 0 1 0 0
## 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390
## 1 0 NA 0 0 0 1 1 1 0 NA NA 0 0 1
## 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405
## 0 0 0 0 0 1 0 1 1 NA 1 NA 0 0 0
## 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420
## 0 0 1 0 0 NA 1 1 0 1 1 0 1 0 0
## 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435
## 1 1 1 0 NA 1 0 0 1 1 1 0 0 NA 1
## 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450
## 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1
## 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465
## 0 0 1 0 NA 1 1 1 0 1 0 0 0 1 0
## 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480
## 0 1 1 0 NA NA 1 0 0 0 0 1 1 0 NA
## 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495
## NA 0 0 1 1 1 0 1 0 1 1 0 0 1 0
## 496 497 498 499 500 501 502 503 504 505 506 507 508 509 510
## 1 NA 0 NA 1 0 NA 1 NA 1 0 1 0 0 0
## 511 512 513 514 515 516 517 518 519 520 521 522 523 524 525
## 0 0 NA 0 0 0 NA 1 1 1 0 0 0 0 0
## 526 527 528 529 530 531 532 533 534 535 536 537 538 539 540
## 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0
## 541 542 543 544 545 546 547 548 549 550 551 552 553 554 555
## 0 0 1 1 0 0 NA 1 1 0 0 0 0 1 0
## 556 557 558 559 560 561 562 563 564 565 566 567 568 569 570
## 0 1 0 1 1 0 0 1 0 1 0 1 1 1 1
## 571 572 573 574 575 576 577 578 579 580 581 582 583 584 585
## 0 NA 0 0 0 0 0 0 NA 0 0 1 0 1 0
## 586 587 588 589 590 591 592 593 594 595 596 597 598 599 600
## 0 0 0 1 1 1 0 0 1 1 NA NA 1 0 1
## 601 602 603 604 605 606 607 608 609 610 611 612 613 614 615
## 1 0 0 0 1 1 1 NA 0 0 0 1 0 0 1
## 616 617 618 619 620 621 622 623 624 625 626 627 628 629 630
## 0 0 NA NA 1 0 0 0 NA 0 1 1 0 0 1
## 631 632 633 634 635 636 637 638 639 640 641 642 643 644 645
## 0 1 NA 0 0 0 NA 1 0 0 0 1 0 0 0
## 646 647 648 649 650 651 652 653 654 655 656 657 658 659 660
## 1 1 1 0 1 NA 0 1 0 0 0 1 0 0 0
## 661 662 663 664 665 666 667 668 669 670 671 672 673 674 675
## 1 1 1 0 0 1 1 NA 1 NA 0 1 1 0 1
## 676 677 678 679 680 681 682 683 684 685 686 687 688 689 690
## 0 0 1 0 0 1 0 1 0 1 0 0 1 0 0
## 691 692 693 694 695 696 697 698 699 700 701 702 703 704 705
## NA 0 0 0 0 1 NA 1 1 0 0 0 1 0 0
## 706 707 708 709 710 711 712 713 714 715 716 717 718 719 720
## 0 1 1 0 0 0 0 NA 0 0 0 0 0 0 NA
## 721 722 723 724 725 726 727 728 729 730 731 732 733 734 735
## 1 0 0 0 0 0 0 0 0 0 1 1 1 0 1
## 736 737 738 739 740 741 742 743 744 745 746 747 748 749 750
## NA 0 0 1 0 1 0 1 0 1 1 NA 0 0 1
## 751 752 753 754 755 756 757 758 759 760 761 762 763 764 765
## 0 0 1 1 NA 1 1 1 0 1 0 1 1 1 1
## 766 767 768 769 770 771 772 773 774 775 776 777 778 779 780
## 1 1 0 1 0 1 1 0 NA 0 1 0 1 0 0
## 781 782 783 784 785 786 787 788 789 790 791 792 793 794 795
## 1 1 0 0 0 1 0 NA 1 0 0 0 1 0 0
## 796 797 798 799 800 801 802 803 804 805 806 807 808 809 810
## 1 0 1 1 1 NA 0 0 1 0 0 0 0 0 1
## 811 812 813 814 815 816 817 818 819 820 821 822 823 824 825
## 1 0 NA 0 1 0 0 1 1 1 1 0 NA 1 1
## 826 827 828 829 830 831 832 833 834 835 836 837 838 839 840
## NA NA 0 0 NA 0 0 1 0 0 0 NA NA 0 0
## 841 842 843 844 845 846 847 848 849 850 851 852 853 854 855
## 0 0 0 0 0 NA 1 1 1 1 1 0 0 0 1
## 856 857 858 859 860 861 862 863 864 865 866 867 868 869 870
## NA 0 0 1 0 0 1 0 0 1 0 1 0 0 NA
## 871 872 873 874 875 876 877 878 879 880 881 882 883 884 885
## 0 1 NA 1 1 0 0 0 0 1 1 0 0 0 1
## 886 887 888 889 890 891 892 893 894 895 896 897 898 899 900
## 1 1 0 0 0 1 0 0 0 0 1 0 0 NA 1
## 901 902 903 904 905 906 907 908 909 910 911 912 913 914 915
## 0 0 1 0 0 0 1 0 1 1 1 0 0 0 0
## 916 917 918 919 920 921 922 923 924 925 926 927 928 929 930
## NA 1 1 0 1 0 0 0 0 0 0 0 0 1 0
## 931 932 933 934 935 936 937 938 939 940 941 942 943 944 945
## 0 1 NA 1 NA 1 0 NA 1 1 1 0 1 0 1
## 946 947 948 949 950 951 952 953 954 955 956 957 958 959 960
## NA 0 0 0 1 0 0 1 1 0 1 0 0 1 0
## 961 962 963 964 965 966 967 968 969 970 971 972 973 974 975
## 1 1 1 0 0 1 1 1 0 NA 1 NA 0 1 0
## 976 977 978 979 980 981 982 983 984 985 986 987 988 989 990
## 1 1 0 0 1 0 0 1 1 1 1 NA 0 1 1
## 991 992 993 994 995 996 997 998 999 1000 1001 1002 1003 1004 1005
## 0 NA 0 0 1 0 0 0 0 NA 1 1 1 0 1
## 1006 1007 1008 1009 1010 1011 1012 1013 1014 1015 1016 1017 1018 1019 1020
## 0 0 NA 1 0 1 0 0 NA 0 1 0 1 0 0
## 1021 1022 1023 1024 1025 1026 1027 1028 1029 1030 1031 1032 1033 1034 1035
## 0 0 1 1 1 1 1 0 NA 0 0 0 0 0 0
## 1036 1037 1038 1039 1040 1041 1042 1043 1044 1045 1046 1047 1048 1049 1050
## 0 0 0 0 0 0 1 1 0 1 0 1 1 1 NA
## 1051 1052 1053 1054 1055 1056 1057 1058 1059 1060 1061 1062 1063 1064 1065
## 1 1 1 0 1 0 1 0 1 1 0 NA 1 NA 0
## 1066 1067 1068 1069 1070 1071 1072 1073 1074 1075 1076 1077 1078 1079 1080
## 0 0 0 0 1 0 NA 1 1 0 0 0 0 1 0
## 1081 1082 1083 1084 1085 1086 1087 1088 1089 1090 1091 1092 1093 1094 1095
## 1 1 0 1 1 1 0 0 0 0 0 0 0 1 NA
## 1096 1097 1098 1099 1100 1101 1102 1103 1104 1105 1106 1107 1108 1109 1110
## 0 0 0 1 1 0 0 1 1 1 1 0 0 NA 1
## 1111 1112 1113 1114 1115 1116 1117 1118 1119 1120 1121 1122 1123 1124 1125
## 1 NA 0 0 0 0 1 1 1 0 1 1 0 0 0
## 1126 1127 1128 1129 1130 1131 1132 1133 1134 1135 1136 1137 1138 1139 1140
## 0 1 0 0 0 NA 0 1 0 1 0 0 0 0 0
## 1141 1142 1143 1144 1145 1146 1147 1148 1149 1150 1151 1152 1153 1154 1155
## 0 0 1 1 1 0 0 1 0 1 1 1 0 NA 1
## 1156 1157 1158 1159 1160 1161 1162 1163 1164 1165 1166 1167 1168 1169 1170
## 1 0 0 0 0 0 0 0 0 0 1 0 1 1 0
## 1171 1172 1173 1174 1175 1176 1177 1178 1179 1180 1181 1182 1183 1184 1185
## 1 NA 1 1 0 0 NA 0 1 1 NA 1 0 1 1
## 1186 1187 1188 1189 1190 1191 1192 1193 1194 1195 1196 1197 1198 1199 1200
## NA 0 0 NA 0 1 0 NA 1 0 0 0 0 1 1
## 1201 1202 1203 1204 1205 1206 1207 1208 1209 1210 1211 1212 1213 1214 1215
## 0 0 0 1 0 0 NA 0 0 1 0 0 1 0 0
## 1216 1217 1218 1219 1220 1221 1222 1223 1224 1225 1226 1227 1228 1229 1230
## 0 1 1 0 0 0 0 1 0 NA 0 0 0 1 1
## 1231 1232 1233 1234 1235 1236 1237 1238 1239 1240 1241 1242 1243 1244 1245
## 0 1 1 1 0 0 0 1 0 0 1 0 0 0 0
## 1246 1247 1248 1249 1250 1251 1252 1253 1254 1255 1256 1257 1258 1259 1260
## 1 1 0 0 0 0 1 0 1 0 0 1 0 1 0
## 1261 1262 1263 1264 1265 1266 1267 1268 1269 1270 1271 1272 1273 1274 1275
## 0 1 0 1 NA 1 1 0 1 0 1 0 1 1 0
## 1276 1277 1278 1279 1280 1281 1282 1283 1284 1285 1286 1287 1288 1289 1290
## 0 1 0 0 0 1 0 0 NA 1 0 1 0 0 0
## 1291 1292 1293 1294 1295 1296 1297 1298 1299 1300 1301 1302 1303 1304 1305
## 1 NA 0 1 NA 0 1 NA 1 NA 0 0 NA 0 0
## 1306 1307 1308 1309 1310 1311 1312 1313 1314 1315 1316 1317 1318 1319 1320
## 0 1 1 0 1 1 0 NA 0 0 0 0 NA 0 1
## 1321 1322 1323 1324 1325 1326 1327 1328 1329 1330 1331 1332 1333 1334 1335
## 0 NA 1 0 0 1 0 1 0 NA 0 0 0 0 1
## 1336 1337 1338 1339 1340 1341 1342 1343 1344 1345 1346 1347 1348 1349 1350
## NA 0 0 1 0 0 1 0 0 1 0 0 1 0 NA
## 1351 1352 1353 1354 1355 1356 1357 1358 1359 1360 1361 1362 1363 1364 1365
## 0 1 0 0 0 0 0 NA 0 0 0 0 1 0 NA
## 1366 1367 1368 1369 1370 1371 1372 1373 1374 1375 1376 1377 1378 1379 1380
## 0 1 1 1 1 0 0 0 0 0 1 1 NA 0 NA
## 1381 1382 1383 1384 1385 1386 1387 1388 1389 1390 1391 1392 1393 1394 1395
## 1 1 1 0 1 0 0 NA 0 1 1 1 0 1 0
## 1396 1397 1398 1399 1400 1401 1402 1403 1404 1405 1406 1407 1408 1409 1410
## 0 0 1 1 1 0 0 1 1 NA 1 0 0 0 1
## 1411 1412 1413 1414 1415 1416 1417 1418 1419 1420 1421 1422 1423 1424 1425
## 0 0 NA 0 0 0 0 0 1 0 0 1 1 0 1
## 1426 1427 1428 1429 1430 1431 1432 1433 1434 1435 1436 1437 1438 1439 1440
## 1 0 0 1 0 1 0 0 0 0 1 1 1 0 0
## 1441 1442 1443 1444 1445 1446 1447 1448 1449 1450 1451 1452 1453 1454 1455
## 0 1 1 0 NA 1 1 1 0 0 0 0 0 0 NA
## 1456 1457 1458 1459 1460 1461 1462 1463 1464 1465 1466 1467 1468 1469 1470
## NA 1 NA 0 0 0 0 NA 1 0 1 0 0 0 1
## 1471 1472 1473 1474 1475 1476 1477 1478 1479 1480 1481 1482 1483 1484 1485
## 0 0 0 0 1 0 1 0 0 0 1 NA 1 0 1
## 1486 1487 1488 1489 1490 1491 1492 1493 1494 1495 1496 1497 1498 1499 1500
## 1 0 1 1 0 0 1 0 0 1 1 0 0 0 0
## 1501 1502 1503 1504 1505 1506 1507 1508 1509 1510 1511 1512 1513 1514 1515
## 0 1 1 1 0 NA 1 NA 0 0 0 NA 0 0 1
## 1516 1517 1518 1519 1520 1521 1522 1523 1524 1525 1526 1527 1528 1529 1530
## 1 0 1 0 0 1 0 0 NA NA 1 0 NA 1 1
## 1531 1532 1533 1534 1535 1536 1537 1538 1539 1540 1541 1542 1543 1544 1545
## 1 1 0 0 1 1 0 1 1 1 0 0 NA 1 NA
## 1546 1547 1548 1549 1550 1551 1552 1553 1554 1555 1556 1557 1558 1559 1560
## 0 0 1 0 NA NA 0 0 1 NA 0 0 0 1 0
## 1561 1562 1563 1564 1565 1566 1567 1568 1569 1570 1571 1572 1573 1574 1575
## 0 1 1 1 1 NA NA 0 0 1 1 0 0 NA NA
## 1576 1577 1578 1579 1580 1581 1582 1583 1584 1585 1586 1587 1588 1589 1590
## 1 1 0 0 0 1 0 0 0 0 0 0 1 0 1
## 1591 1592 1593 1594 1595 1596 1597 1598 1599 1600 1601 1602 1603 1604 1605
## 1 1 NA NA 0 0 1 NA 0 1 1 0 1 1 0
## 1606 1607 1608 1609 1610 1611 1612 1613 1614 1615 1616 1617 1618 1619 1620
## 1 0 0 0 1 1 0 0 NA 1 1 0 1 1 NA
## 1621 1622 1623 1624 1625 1626 1627 1628 1629 1630 1631 1632 1633 1634 1635
## 1 0 1 0 0 0 1 0 0 1 0 1 0 1 0
## 1636 1637 1638 1639 1640 1641 1642 1643 1644 1645 1646 1647 1648 1649 1650
## 0 1 0 0 1 0 0 NA 0 1 0 0 0 0 1
## 1651 1652 1653 1654 1655 1656 1657 1658 1659 1660 1661 1662 1663 1664 1665
## 0 0 0 0 1 1 0 0 0 0 NA 1 1 NA 1
## 1666 1667 1668 1669 1670 1671 1672 1673 1674 1675 1676 1677 1678 1679 1680
## 0 NA 1 1 NA 1 0 1 NA 0 0 0 0 0 0
## 1681 1682 1683 1684 1685 1686 1687 1688 1689 1690 1691 1692 1693 1694 1695
## 0 1 NA 0 1 0 0 1 0 0 0 0 0 1 0
## 1696 1697 1698 1699 1700 1701 1702 1703 1704 1705 1706 1707 1708 1709 1710
## 1 0 1 1 0 1 0 0 1 0 0 NA NA 1 0
## 1711 1712 1713 1714 1715 1716 1717 1718 1719 1720 1721 1722 1723 1724 1725
## 1 0 1 0 1 0 0 0 1 NA 1 1 0 1 1
## 1726 1727 1728 1729 1730 1731 1732 1733 1734 1735 1736 1737 1738 1739 1740
## NA NA 1 1 1 1 0 0 1 0 0 1 NA 0 0
## 1741 1742 1743 1744 1745 1746 1747 1748 1749 1750 1751 1752 1753 1754 1755
## 1 0 0 0 1 0 0 0 1 0 0 1 0 1 0
## 1756 1757 1758 1759 1760 1761 1762 1763 1764 1765 1766 1767 1768 1769 1770
## 0 0 1 0 0 1 0 0 0 1 NA 0 1 0 0
## 1771 1772 1773 1774 1775 1776 1777 1778 1779 1780 1781 1782 1783 1784 1785
## 0 NA 0 1 1 0 NA 1 1 1 NA 0 1 0 0
## 1786 1787 1788 1789 1790 1791 1792 1793 1794 1795 1796 1797 1798 1799 1800
## 0 0 0 1 1 0 0 0 0 NA 1 0 0 0 1
## 1801 1802 1803 1804 1805 1806 1807 1808 1809 1810 1811 1812 1813 1814 1815
## 1 NA 0 0 0 NA NA 1 0 0 0 0 1 1 1
## 1816 1817 1818 1819 1820 1821 1822 1823 1824 1825 1826 1827 1828 1829 1830
## 0 0 0 0 0 0 0 0 1 1 1 0 1 0 0
## 1831 1832 1833 1834 1835 1836 1837 1838 1839 1840 1841 1842 1843 1844 1845
## 0 0 0 0 1 0 0 1 0 1 0 0 1 NA 0
## 1846 1847 1848 1849 1850 1851 1852 1853 1854 1855 1856 1857 1858 1859 1860
## 0 0 0 NA 0 1 0 1 NA 0 1 0 0 0 0
## 1861 1862 1863 1864 1865 1866 1867 1868 1869 1870 1871 1872 1873 1874 1875
## 1 0 0 1 1 0 0 0 1 1 NA 1 NA 0 1
## 1876 1877 1878 1879 1880 1881 1882 1883 1884 1885 1886 1887 1888 1889 1890
## 0 0 1 1 1 NA NA 0 0 1 1 0 0 1 1
## 1891 1892 1893 1894 1895 1896 1897 1898 1899 1900 1901 1902 1903 1904 1905
## 0 1 0 0 1 NA 0 0 0 1 0 1 0 1 0
## 1906 1907 1908 1909 1910 1911 1912 1913 1914 1915 1916 1917 1918 1919 1920
## 0 0 0 1 1 1 1 0 0 1 NA 0 0 NA 0
## 1921 1922 1923 1924 1925 1926 1927 1928 1929 1930 1931 1932 1933 1934 1935
## 0 1 0 0 1 0 NA 0 0 0 NA 0 1 1 0
## 1936 1937 1938 1939 1940 1941 1942 1943 1944 1945 1946 1947 1948 1949 1950
## 1 1 0 0 0 0 0 NA 0 0 0 NA NA 1 0
## 1951 1952 1953 1954 1955 1956 1957 1958 1959 1960 1961 1962 1963 1964 1965
## 1 0 1 1 0 NA 0 0 1 NA 1 NA 1 1 0
## 1966 1967 1968 1969 1970 1971 1972 1973 1974 1975 1976 1977 1978 1979 1980
## NA 0 1 0 NA 0 0 1 1 0 1 0 1 1 NA
## 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995
## 0 1 NA 0 0 0 0 0 NA 1 0 NA 1 1 0
## 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010
## 0 1 1 1 NA 1 1 NA NA 1 0 1 0 1 1
## 2011 2012 2013 2014 2015 2016 2017 2018 2019 2020 2021 2022 2023 2024 2025
## 1 0 1 0 0 1 0 1 1 0 0 0 NA 0 1
## 2026 2027 2028 2029 2030 2031 2032 2033 2034 2035 2036 2037 2038 2039 2040
## 0 0 0 0 1 0 0 0 0 1 1 0 0 1 0
## 2041 2042 2043 2044 2045 2046 2047 2048 2049 2050 2051 2052 2053 2054 2055
## 0 0 0 1 0 0 0 1 1 0 1 1 1 0 NA
## 2056 2057 2058 2059 2060 2061 2062 2063 2064 2065 2066 2067 2068 2069 2070
## 0 0 0 0 0 0 1 1 0 1 0 1 1 1 0
## 2071 2072 2073 2074 2075 2076 2077 2078 2079 2080 2081 2082 2083 2084 2085
## 0 0 NA 1 0 0 1 0 0 1 0 1 NA 0 0
## 2086 2087 2088 2089 2090 2091 2092 2093 2094 2095 2096 2097 2098 2099 2100
## 0 1 1 NA 1 0 0 1 0 1 1 0 1 1 1
## 2101 2102 2103 2104 2105 2106 2107 2108 2109 2110 2111 2112 2113 2114 2115
## 1 1 1 0 NA 0 1 NA 1 0 1 0 1 0 0
## 2116 2117 2118 2119 2120 2121 2122 2123 2124 2125 2126 2127 2128 2129 2130
## NA 1 1 1 0 0 0 1 1 1 0 NA 0 0 0
## 2131 2132 2133 2134 2135 2136 2137 2138 2139 2140 2141
## 0 0 0 0 1 0 0 0 0 0 0Interestingly, there are NA values in the prediction? Why is that occurring? Let’s take a further look into this.
## [1] 2601There are a significant number of missing points. This may potentially explain why there are so many missing response variables, but interestingly, after looking closer at the data, it appears that the NA responses are coming from rows that have income values that are listed as NA. If we take a look at the histogram of the income distribution.
We notice that this is significantly right skew. Therefore, we will replace all of the NA values with the median and try the prediction again.
Let’s see if we can make a prediction without having NA responses.
## [1] "How many NA values are left? 111"There are still a 111 values left. However, if we look at the data one more time, it appears that home value appears to have NA values that may be impacting the response variables.
Given that there the most frequent count for home value is 0. We will assume that all NA values are of 0 value as well.
Now let’s try again and make the predictions for the test dataset.
## 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
## 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0
## 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
## 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0
## 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45
## 0 0 0 0 0 0 0 0 0 1 0 1 0 1 0
## 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
## 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1
## 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75
## 0 0 0 0 0 0 1 1 0 0 0 0 1 0 1
## 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90
## 0 0 0 0 0 1 0 0 0 0 1 0 0 0 1
## 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105
## 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0
## 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120
## 0 0 0 0 0 1 0 0 0 1 0 0 1 1 0
## 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135
## 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0
## 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150
## 0 1 1 0 0 0 1 0 0 0 1 0 0 0 0
## 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165
## 1 0 1 0 0 0 0 0 1 1 0 0 0 0 1
## 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180
## 0 0 0 0 0 0 1 0 1 0 0 1 1 1 1
## 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195
## 1 0 0 0 0 0 0 0 0 0 1 1 0 0 0
## 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210
## 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0
## 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225
## 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0
## 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240
## 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1
## 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255
## 0 0 1 0 0 0 0 0 0 1 0 1 0 0 0
## 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270
## 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0
## 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285
## 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0
## 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300
## 0 0 0 1 1 0 0 0 0 0 0 0 1 0 0
## 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315
## 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0
## 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330
## 0 0 0 1 0 0 1 0 0 1 0 1 0 0 0
## 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345
## 0 0 1 0 0 0 0 1 0 0 1 1 0 1 0
## 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360
## 0 0 0 0 0 0 0 1 1 0 1 0 0 0 0
## 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375
## 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0
## 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390
## 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1
## 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405
## 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
## 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420
## 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0
## 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435
## 1 1 1 0 0 0 0 0 1 0 0 0 0 0 0
## 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450
## 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1
## 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465
## 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0
## 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480
## 0 1 1 0 0 0 1 0 0 0 0 1 1 0 0
## 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495
## 0 0 0 0 1 1 0 0 0 1 1 0 0 0 0
## 496 497 498 499 500 501 502 503 504 505 506 507 508 509 510
## 1 0 0 0 0 0 0 1 0 1 0 0 0 0 0
## 511 512 513 514 515 516 517 518 519 520 521 522 523 524 525
## 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0
## 526 527 528 529 530 531 532 533 534 535 536 537 538 539 540
## 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
## 541 542 543 544 545 546 547 548 549 550 551 552 553 554 555
## 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0
## 556 557 558 559 560 561 562 563 564 565 566 567 568 569 570
## 0 0 0 0 0 0 0 0 0 1 0 1 0 0 1
## 571 572 573 574 575 576 577 578 579 580 581 582 583 584 585
## 0 0 0 0 0 0 0 0 1 0 0 1 0 1 0
## 586 587 588 589 590 591 592 593 594 595 596 597 598 599 600
## 0 0 0 1 0 0 0 0 0 1 1 1 0 0 1
## 601 602 603 604 605 606 607 608 609 610 611 612 613 614 615
## 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0
## 616 617 618 619 620 621 622 623 624 625 626 627 628 629 630
## 0 0 0 0 1 0 0 0 0 0 1 1 0 0 1
## 631 632 633 634 635 636 637 638 639 640 641 642 643 644 645
## 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0
## 646 647 648 649 650 651 652 653 654 655 656 657 658 659 660
## 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0
## 661 662 663 664 665 666 667 668 669 670 671 672 673 674 675
## 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0
## 676 677 678 679 680 681 682 683 684 685 686 687 688 689 690
## 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
## 691 692 693 694 695 696 697 698 699 700 701 702 703 704 705
## 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
## 706 707 708 709 710 711 712 713 714 715 716 717 718 719 720
## 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0
## 721 722 723 724 725 726 727 728 729 730 731 732 733 734 735
## 1 0 0 0 0 0 0 0 0 0 1 1 0 0 0
## 736 737 738 739 740 741 742 743 744 745 746 747 748 749 750
## 0 0 0 0 0 1 0 1 0 0 0 1 0 0 0
## 751 752 753 754 755 756 757 758 759 760 761 762 763 764 765
## 0 0 1 1 0 0 0 0 0 0 0 1 0 0 1
## 766 767 768 769 770 771 772 773 774 775 776 777 778 779 780
## 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0
## 781 782 783 784 785 786 787 788 789 790 791 792 793 794 795
## 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0
## 796 797 798 799 800 801 802 803 804 805 806 807 808 809 810
## 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0
## 811 812 813 814 815 816 817 818 819 820 821 822 823 824 825
## 0 0 0 0 0 0 0 1 1 0 1 0 0 0 1
## 826 827 828 829 830 831 832 833 834 835 836 837 838 839 840
## 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
## 841 842 843 844 845 846 847 848 849 850 851 852 853 854 855
## 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0
## 856 857 858 859 860 861 862 863 864 865 866 867 868 869 870
## 0 0 0 1 0 0 1 0 0 0 0 1 0 0 1
## 871 872 873 874 875 876 877 878 879 880 881 882 883 884 885
## 0 1 0 1 0 0 0 0 0 0 0 0 0 0 1
## 886 887 888 889 890 891 892 893 894 895 896 897 898 899 900
## 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0
## 901 902 903 904 905 906 907 908 909 910 911 912 913 914 915
## 0 0 1 0 0 0 1 0 0 0 1 0 0 0 0
## 916 917 918 919 920 921 922 923 924 925 926 927 928 929 930
## 0 1 1 0 0 0 0 0 0 0 0 0 0 1 0
## 931 932 933 934 935 936 937 938 939 940 941 942 943 944 945
## 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0
## 946 947 948 949 950 951 952 953 954 955 956 957 958 959 960
## 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0
## 961 962 963 964 965 966 967 968 969 970 971 972 973 974 975
## 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0
## 976 977 978 979 980 981 982 983 984 985 986 987 988 989 990
## 0 0 0 0 0 0 0 1 0 1 1 0 0 0 1
## 991 992 993 994 995 996 997 998 999 1000 1001 1002 1003 1004 1005
## 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0
## 1006 1007 1008 1009 1010 1011 1012 1013 1014 1015 1016 1017 1018 1019 1020
## 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0
## 1021 1022 1023 1024 1025 1026 1027 1028 1029 1030 1031 1032 1033 1034 1035
## 0 0 1 0 1 1 0 0 0 0 0 0 0 0 0
## 1036 1037 1038 1039 1040 1041 1042 1043 1044 1045 1046 1047 1048 1049 1050
## 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0
## 1051 1052 1053 1054 1055 1056 1057 1058 1059 1060 1061 1062 1063 1064 1065
## 1 1 1 0 0 0 0 0 1 1 0 1 0 0 0
## 1066 1067 1068 1069 1070 1071 1072 1073 1074 1075 1076 1077 1078 1079 1080
## 0 0 0 0 1 0 0 0 1 0 0 0 0 1 0
## 1081 1082 1083 1084 1085 1086 1087 1088 1089 1090 1091 1092 1093 1094 1095
## 1 1 0 0 1 1 0 0 0 0 0 0 0 0 0
## 1096 1097 1098 1099 1100 1101 1102 1103 1104 1105 1106 1107 1108 1109 1110
## 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0
## 1111 1112 1113 1114 1115 1116 1117 1118 1119 1120 1121 1122 1123 1124 1125
## 1 0 0 0 0 0 0 1 1 0 0 1 0 0 0
## 1126 1127 1128 1129 1130 1131 1132 1133 1134 1135 1136 1137 1138 1139 1140
## 0 1 0 0 0 0 0 1 0 1 0 0 0 0 0
## 1141 1142 1143 1144 1145 1146 1147 1148 1149 1150 1151 1152 1153 1154 1155
## 0 0 0 0 1 0 0 1 0 1 0 1 0 0 1
## 1156 1157 1158 1159 1160 1161 1162 1163 1164 1165 1166 1167 1168 1169 1170
## 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
## 1171 1172 1173 1174 1175 1176 1177 1178 1179 1180 1181 1182 1183 1184 1185
## 0 1 1 1 0 0 0 0 0 1 0 1 0 0 1
## 1186 1187 1188 1189 1190 1191 1192 1193 1194 1195 1196 1197 1198 1199 1200
## 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0
## 1201 1202 1203 1204 1205 1206 1207 1208 1209 1210 1211 1212 1213 1214 1215
## 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0
## 1216 1217 1218 1219 1220 1221 1222 1223 1224 1225 1226 1227 1228 1229 1230
## 0 1 0 0 0 0 0 1 0 1 0 0 0 0 1
## 1231 1232 1233 1234 1235 1236 1237 1238 1239 1240 1241 1242 1243 1244 1245
## 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0
## 1246 1247 1248 1249 1250 1251 1252 1253 1254 1255 1256 1257 1258 1259 1260
## 1 0 0 0 0 0 1 0 0 0 0 1 0 0 0
## 1261 1262 1263 1264 1265 1266 1267 1268 1269 1270 1271 1272 1273 1274 1275
## 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0
## 1276 1277 1278 1279 1280 1281 1282 1283 1284 1285 1286 1287 1288 1289 1290
## 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0
## 1291 1292 1293 1294 1295 1296 1297 1298 1299 1300 1301 1302 1303 1304 1305
## 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0
## 1306 1307 1308 1309 1310 1311 1312 1313 1314 1315 1316 1317 1318 1319 1320
## 0 1 0 0 1 1 0 1 0 0 0 0 0 0 1
## 1321 1322 1323 1324 1325 1326 1327 1328 1329 1330 1331 1332 1333 1334 1335
## 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0
## 1336 1337 1338 1339 1340 1341 1342 1343 1344 1345 1346 1347 1348 1349 1350
## 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0
## 1351 1352 1353 1354 1355 1356 1357 1358 1359 1360 1361 1362 1363 1364 1365
## 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0
## 1366 1367 1368 1369 1370 1371 1372 1373 1374 1375 1376 1377 1378 1379 1380
## 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0
## 1381 1382 1383 1384 1385 1386 1387 1388 1389 1390 1391 1392 1393 1394 1395
## 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0
## 1396 1397 1398 1399 1400 1401 1402 1403 1404 1405 1406 1407 1408 1409 1410
## 0 0 0 0 0 0 0 1 1 0 0 0 0 0 1
## 1411 1412 1413 1414 1415 1416 1417 1418 1419 1420 1421 1422 1423 1424 1425
## 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0
## 1426 1427 1428 1429 1430 1431 1432 1433 1434 1435 1436 1437 1438 1439 1440
## 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0
## 1441 1442 1443 1444 1445 1446 1447 1448 1449 1450 1451 1452 1453 1454 1455
## 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0
## 1456 1457 1458 1459 1460 1461 1462 1463 1464 1465 1466 1467 1468 1469 1470
## 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0
## 1471 1472 1473 1474 1475 1476 1477 1478 1479 1480 1481 1482 1483 1484 1485
## 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0
## 1486 1487 1488 1489 1490 1491 1492 1493 1494 1495 1496 1497 1498 1499 1500
## 0 0 0 1 0 0 1 0 0 1 0 0 0 0 0
## 1501 1502 1503 1504 1505 1506 1507 1508 1509 1510 1511 1512 1513 1514 1515
## 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0
## 1516 1517 1518 1519 1520 1521 1522 1523 1524 1525 1526 1527 1528 1529 1530
## 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1
## 1531 1532 1533 1534 1535 1536 1537 1538 1539 1540 1541 1542 1543 1544 1545
## 0 0 0 0 0 0 0 0 1 1 0 0 0 0 1
## 1546 1547 1548 1549 1550 1551 1552 1553 1554 1555 1556 1557 1558 1559 1560
## 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0
## 1561 1562 1563 1564 1565 1566 1567 1568 1569 1570 1571 1572 1573 1574 1575
## 0 1 0 0 1 0 0 0 0 0 1 0 0 0 0
## 1576 1577 1578 1579 1580 1581 1582 1583 1584 1585 1586 1587 1588 1589 1590
## 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
## 1591 1592 1593 1594 1595 1596 1597 1598 1599 1600 1601 1602 1603 1604 1605
## 0 1 0 0 0 0 0 0 0 0 1 0 0 1 0
## 1606 1607 1608 1609 1610 1611 1612 1613 1614 1615 1616 1617 1618 1619 1620
## 1 0 0 0 1 0 0 0 0 0 1 0 1 0 0
## 1621 1622 1623 1624 1625 1626 1627 1628 1629 1630 1631 1632 1633 1634 1635
## 1 0 1 0 0 0 0 0 0 1 0 1 0 0 0
## 1636 1637 1638 1639 1640 1641 1642 1643 1644 1645 1646 1647 1648 1649 1650
## 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1
## 1651 1652 1653 1654 1655 1656 1657 1658 1659 1660 1661 1662 1663 1664 1665
## 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1
## 1666 1667 1668 1669 1670 1671 1672 1673 1674 1675 1676 1677 1678 1679 1680
## 0 0 0 1 0 1 0 1 0 0 0 0 0 0 0
## 1681 1682 1683 1684 1685 1686 1687 1688 1689 1690 1691 1692 1693 1694 1695
## 0 1 1 0 1 0 0 0 0 0 0 0 0 0 0
## 1696 1697 1698 1699 1700 1701 1702 1703 1704 1705 1706 1707 1708 1709 1710
## 1 0 1 1 0 0 0 0 0 0 0 1 0 1 0
## 1711 1712 1713 1714 1715 1716 1717 1718 1719 1720 1721 1722 1723 1724 1725
## 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
## 1726 1727 1728 1729 1730 1731 1732 1733 1734 1735 1736 1737 1738 1739 1740
## 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0
## 1741 1742 1743 1744 1745 1746 1747 1748 1749 1750 1751 1752 1753 1754 1755
## 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0
## 1756 1757 1758 1759 1760 1761 1762 1763 1764 1765 1766 1767 1768 1769 1770
## 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0
## 1771 1772 1773 1774 1775 1776 1777 1778 1779 1780 1781 1782 1783 1784 1785
## 0 0 0 1 0 0 1 0 1 0 0 0 0 0 0
## 1786 1787 1788 1789 1790 1791 1792 1793 1794 1795 1796 1797 1798 1799 1800
## 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0
## 1801 1802 1803 1804 1805 1806 1807 1808 1809 1810 1811 1812 1813 1814 1815
## 0 0 0 0 0 0 1 1 0 0 0 0 1 0 0
## 1816 1817 1818 1819 1820 1821 1822 1823 1824 1825 1826 1827 1828 1829 1830
## 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0
## 1831 1832 1833 1834 1835 1836 1837 1838 1839 1840 1841 1842 1843 1844 1845
## 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0
## 1846 1847 1848 1849 1850 1851 1852 1853 1854 1855 1856 1857 1858 1859 1860
## 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
## 1861 1862 1863 1864 1865 1866 1867 1868 1869 1870 1871 1872 1873 1874 1875
## 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0
## 1876 1877 1878 1879 1880 1881 1882 1883 1884 1885 1886 1887 1888 1889 1890
## 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0
## 1891 1892 1893 1894 1895 1896 1897 1898 1899 1900 1901 1902 1903 1904 1905
## 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0
## 1906 1907 1908 1909 1910 1911 1912 1913 1914 1915 1916 1917 1918 1919 1920
## 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0
## 1921 1922 1923 1924 1925 1926 1927 1928 1929 1930 1931 1932 1933 1934 1935
## 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0
## 1936 1937 1938 1939 1940 1941 1942 1943 1944 1945 1946 1947 1948 1949 1950
## 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0
## 1951 1952 1953 1954 1955 1956 1957 1958 1959 1960 1961 1962 1963 1964 1965
## 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0
## 1966 1967 1968 1969 1970 1971 1972 1973 1974 1975 1976 1977 1978 1979 1980
## 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0
## 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995
## 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0
## 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010
## 0 0 1 0 0 1 0 1 0 1 0 0 0 0 0
## 2011 2012 2013 2014 2015 2016 2017 2018 2019 2020 2021 2022 2023 2024 2025
## 0 0 1 0 0 0 0 1 1 0 0 0 0 0 0
## 2026 2027 2028 2029 2030 2031 2032 2033 2034 2035 2036 2037 2038 2039 2040
## 0 0 0 0 1 0 0 0 0 1 0 0 0 1 0
## 2041 2042 2043 2044 2045 2046 2047 2048 2049 2050 2051 2052 2053 2054 2055
## 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
## 2056 2057 2058 2059 2060 2061 2062 2063 2064 2065 2066 2067 2068 2069 2070
## 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0
## 2071 2072 2073 2074 2075 2076 2077 2078 2079 2080 2081 2082 2083 2084 2085
## 0 0 1 0 0 0 1 0 0 1 0 0 0 0 0
## 2086 2087 2088 2089 2090 2091 2092 2093 2094 2095 2096 2097 2098 2099 2100
## 0 0 0 0 1 0 0 0 0 0 1 0 0 1 1
## 2101 2102 2103 2104 2105 2106 2107 2108 2109 2110 2111 2112 2113 2114 2115
## 1 0 1 0 0 0 1 0 0 0 1 0 0 0 0
## 2116 2117 2118 2119 2120 2121 2122 2123 2124 2125 2126 2127 2128 2129 2130
## 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0
## 2131 2132 2133 2134 2135 2136 2137 2138 2139 2140 2141
## 0 0 0 0 0 0 0 0 0 0 0Are there any more missing values?
## [1] "How many NA values are left? 0"Now that we have no NA values left, let’s continue and now predict the payoff. Remember, if the person has not been in an accident as noted above in the logistic regression, then the payout is $0.
## 1 2 3 4 5 6 7 8
## 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
## 9 10 11 12 13 14 15 16
## 0.000 0.000 0.000 4056.041 3720.827 0.000 0.000 3998.122
## 17 18 19 20 21 22 23 24
## 3572.338 0.000 3410.820 0.000 0.000 0.000 0.000 0.000
## 25 26 27 28 29 30 31 32
## 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
## 33 34 35 36 37 38 39 40
## 0.000 0.000 0.000 0.000 0.000 0.000 0.000 3885.467
## 41 42 43 44 45 46 47 48
## 0.000 3919.282 0.000 4132.979 0.000 0.000 0.000 0.000
## 49 50 51 52 53 54 55 56
## 0.000 3662.108 0.000 0.000 3900.010 0.000 0.000 0.000
## 57 58 59 60 61 62 63 64
## 0.000 0.000 0.000 3439.035 0.000 0.000 0.000 0.000
## 65 66 67 68 69 70 71 72
## 0.000 0.000 3686.349 4340.280 0.000 0.000 0.000 0.000
## 73 74 75 76 77 78 79 80
## 3690.686 0.000 3775.585 0.000 0.000 0.000 0.000 0.000
## 81 82 83 84 85 86 87 88
## 3237.992 0.000 0.000 0.000 0.000 4277.822 0.000 0.000
## 89 90 91 92 93 94 95 96
## 0.000 3879.180 0.000 0.000 0.000 0.000 0.000 0.000
## 97 98 99 100 101 102 103 104
## 0.000 0.000 0.000 0.000 0.000 4254.481 3799.540 4213.587
## 105 106 107 108 109 110 111 112
## 0.000 0.000 0.000 0.000 0.000 0.000 3831.029 0.000
## 113 114 115 116 117 118 119 120
## 0.000 0.000 3905.464 0.000 0.000 3742.758 3145.917 0.000
## 121 122 123 124 125 126 127 128
## 0.000 3804.922 3803.689 0.000 0.000 0.000 0.000 0.000
## 129 130 131 132 133 134 135 136
## 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
## 137 138 139 140 141 142 143 144
## 4466.042 4538.710 0.000 0.000 0.000 3698.948 0.000 0.000
## 145 146 147 148 149 150 151 152
## 0.000 3572.338 0.000 0.000 0.000 0.000 4013.398 0.000
## 153 154 155 156 157 158 159 160
## 4034.256 0.000 0.000 0.000 0.000 0.000 3793.276 3943.556
## 161 162 163 164 165 166 167 168
## 0.000 0.000 0.000 0.000 4556.384 0.000 0.000 0.000
## 169 170 171 172 173 174 175 176
## 0.000 0.000 0.000 4601.809 0.000 4759.054 0.000 0.000
## 177 178 179 180 181 182 183 184
## 4740.024 3568.228 4706.852 3512.804 3846.853 0.000 0.000 0.000
## 185 186 187 188 189 190 191 192
## 0.000 0.000 0.000 0.000 0.000 0.000 3216.296 3576.747
## 193 194 195 196 197 198 199 200
## 0.000 0.000 0.000 3613.883 0.000 0.000 0.000 0.000
## 201 202 203 204 205 206 207 208
## 0.000 0.000 0.000 0.000 0.000 0.000 4052.044 0.000
## 209 210 211 212 213 214 215 216
## 0.000 0.000 0.000 0.000 3754.081 3665.377 0.000 0.000
## 217 218 219 220 221 222 223 224
## 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
## 225 226 227 228 229 230 231 232
## 0.000 0.000 4154.580 0.000 0.000 0.000 0.000 0.000
## 233 234 235 236 237 238 239 240
## 0.000 0.000 0.000 0.000 0.000 0.000 0.000 4049.365
## 241 242 243 244 245 246 247 248
## 0.000 0.000 4079.049 0.000 0.000 0.000 0.000 0.000
## 249 250 251 252 253 254 255 256
## 0.000 4068.693 0.000 3826.482 0.000 0.000 0.000 0.000
## 257 258 259 260 261 262 263 264
## 0.000 0.000 3327.392 0.000 0.000 0.000 0.000 0.000
## 265 266 267 268 269 270 271 272
## 0.000 0.000 0.000 0.000 4731.360 0.000 3959.608 0.000
## 273 274 275 276 277 278 279 280
## 0.000 4172.947 0.000 0.000 4495.088 0.000 0.000 0.000
## 281 282 283 284 285 286 287 288
## 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
## 289 290 291 292 293 294 295 296
## 3804.612 3481.812 0.000 0.000 0.000 0.000 0.000 0.000
## 297 298 299 300 301 302 303 304
## 0.000 3971.053 0.000 0.000 0.000 0.000 0.000 0.000
## 305 306 307 308 309 310 311 312
## 3432.580 0.000 0.000 0.000 0.000 0.000 4457.895 0.000
## 313 314 315 316 317 318 319 320
## 0.000 3776.129 0.000 0.000 0.000 0.000 4139.416 0.000
## 321 322 323 324 325 326 327 328
## 0.000 3387.786 0.000 0.000 3726.473 0.000 3932.303 0.000
## 329 330 331 332 333 334 335 336
## 0.000 0.000 0.000 0.000 4065.394 0.000 0.000 0.000
## 337 338 339 340 341 342 343 344
## 0.000 3963.253 0.000 0.000 3352.724 4381.015 0.000 3763.031
## 345 346 347 348 349 350 351 352
## 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
## 353 354 355 356 357 358 359 360
## 4132.835 3862.426 0.000 3320.540 0.000 0.000 0.000 0.000
## 361 362 363 364 365 366 367 368
## 3716.656 0.000 0.000 3260.394 0.000 0.000 0.000 0.000
## 369 370 371 372 373 374 375 376
## 0.000 0.000 0.000 0.000 0.000 0.000 0.000 3896.873
## 377 378 379 380 381 382 383 384
## 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
## 385 386 387 388 389 390 391 392
## 0.000 0.000 0.000 0.000 0.000 3537.694 0.000 0.000
## 393 394 395 396 397 398 399 400
## 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
## 401 402 403 404 405 406 407 408
## 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
## 409 410 411 412 413 414 415 416
## 0.000 0.000 0.000 3817.678 0.000 0.000 3716.671 0.000
## 417 418 419 420 421 422 423 424
## 0.000 0.000 0.000 0.000 3846.547 4160.016 4113.188 0.000
## 425 426 427 428 429 430 431 432
## 0.000 0.000 0.000 0.000 3805.388 0.000 0.000 0.000
## 433 434 435 436 437 438 439 440
## 0.000 0.000 0.000 4519.269 0.000 0.000 0.000 0.000
## 441 442 443 444 445 446 447 448
## 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
## 449 450 451 452 453 454 455 456
## 4105.150 4144.647 0.000 0.000 0.000 0.000 0.000 4449.710
## 457 458 459 460 461 462 463 464
## 0.000 4500.229 0.000 0.000 0.000 0.000 0.000 0.000
## 465 466 467 468 469 470 471 472
## 0.000 0.000 4109.604 3507.037 0.000 0.000 0.000 3691.599
## 473 474 475 476 477 478 479 480
## 0.000 0.000 0.000 0.000 4235.781 3639.486 0.000 0.000
## 481 482 483 484 485 486 487 488
## 0.000 0.000 0.000 0.000 3704.952 3453.173 0.000 0.000
## 489 490 491 492 493 494 495 496
## 0.000 4037.999 3906.650 0.000 0.000 0.000 0.000 4093.843
## 497 498 499 500 501 502 503 504
## 0.000 0.000 0.000 0.000 0.000 0.000 3669.236 0.000
## 505 506 507 508 509 510 511 512
## 4262.719 0.000 0.000 0.000 0.000 0.000 0.000 0.000
## 513 514 515 516 517 518 519 520
## 0.000 0.000 0.000 0.000 4173.712 0.000 0.000 4249.813
## 521 522 523 524 525 526 527 528
## 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
## 529 530 531 532 533 534 535 536
## 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
## 537 538 539 540 541 542 543 544
## 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
## 545 546 547 548 549 550 551 552
## 0.000 0.000 0.000 4531.288 0.000 0.000 0.000 0.000
## 553 554 555 556 557 558 559 560
## 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
## 561 562 563 564 565 566 567 568
## 0.000 0.000 0.000 0.000 3205.546 0.000 4292.642 0.000
## 569 570 571 572 573 574 575 576
## 0.000 3513.711 0.000 0.000 0.000 0.000 0.000 0.000
## 577 578 579 580 581 582 583 584
## 0.000 0.000 3565.442 0.000 0.000 3843.360 0.000 3432.577
## 585 586 587 588 589 590 591 592
## 0.000 0.000 0.000 0.000 3931.177 0.000 0.000 0.000
## 593 594 595 596 597 598 599 600
## 0.000 0.000 3447.862 3712.126 4045.409 0.000 0.000 3492.024
## 601 602 603 604 605 606 607 608
## 3912.259 0.000 0.000 0.000 0.000 0.000 4344.736 0.000
## 609 610 611 612 613 614 615 616
## 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
## 617 618 619 620 621 622 623 624
## 0.000 0.000 0.000 4438.034 0.000 0.000 0.000 0.000
## 625 626 627 628 629 630 631 632
## 0.000 4899.863 4132.464 0.000 0.000 3883.749 0.000 0.000
## 633 634 635 636 637 638 639 640
## 0.000 0.000 0.000 0.000 0.000 4148.388 0.000 0.000
## 641 642 643 644 645 646 647 648
## 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
## 649 650 651 652 653 654 655 656
## 0.000 0.000 0.000 0.000 3670.927 0.000 0.000 0.000
## 657 658 659 660 661 662 663 664
## 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
## 665 666 667 668 669 670 671 672
## 0.000 0.000 0.000 0.000 0.000 0.000 0.000 3358.704
## 673 674 675 676 677 678 679 680
## 3722.578 0.000 0.000 0.000 0.000 0.000 0.000 0.000
## 681 682 683 684 685 686 687 688
## 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
## 689 690 691 692 693 694 695 696
## 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
## 697 698 699 700 701 702 703 704
## 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
## 705 706 707 708 709 710 711 712
## 0.000 0.000 0.000 3777.881 0.000 0.000 0.000 0.000
## 713 714 715 716 717 718 719 720
## 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
## 721 722 723 724 725 726 727 728
## 3647.980 0.000 0.000 0.000 0.000 0.000 0.000 0.000
## 729 730 731 732 733 734 735 736
## 0.000 0.000 3280.802 4254.739 0.000 0.000 0.000 0.000
## 737 738 739 740 741 742 743 744
## 0.000 0.000 0.000 0.000 3627.948 0.000 3461.248 0.000
## 745 746 747 748 749 750 751 752
## 0.000 0.000 4024.122 0.000 0.000 0.000 0.000 0.000
## 753 754 755 756 757 758 759 760
## 3630.387 3648.093 0.000 0.000 0.000 0.000 0.000 0.000
## 761 762 763 764 765 766 767 768
## 0.000 3836.009 0.000 0.000 3501.215 3953.984 0.000 0.000
## 769 770 771 772 773 774 775 776
## 0.000 0.000 0.000 0.000 0.000 4328.821 0.000 0.000
## 777 778 779 780 781 782 783 784
## 0.000 0.000 0.000 0.000 0.000 4560.383 0.000 0.000
## 785 786 787 788 789 790 791 792
## 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
## 793 794 795 796 797 798 799 800
## 0.000 0.000 0.000 0.000 0.000 3493.489 4047.157 0.000
## 801 802 803 804 805 806 807 808
## 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
## 809 810 811 812 813 814 815 816
## 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
## 817 818 819 820 821 822 823 824
## 0.000 4300.549 4384.528 0.000 3479.474 0.000 0.000 0.000
## 825 826 827 828 829 830 831 832
## 3758.093 0.000 0.000 0.000 0.000 0.000 0.000 0.000
## 833 834 835 836 837 838 839 840
## 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
## 841 842 843 844 845 846 847 848
## 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
## 849 850 851 852 853 854 855 856
## 4431.210 4140.230 4462.282 0.000 0.000 0.000 0.000 0.000
## 857 858 859 860 861 862 863 864
## 0.000 0.000 3842.392 0.000 0.000 3796.705 0.000 0.000
## 865 866 867 868 869 870 871 872
## 0.000 0.000 3776.046 0.000 0.000 3870.199 0.000 4612.326
## 873 874 875 876 877 878 879 880
## 0.000 3921.472 0.000 0.000 0.000 0.000 0.000 0.000
## 881 882 883 884 885 886 887 888
## 0.000 0.000 0.000 0.000 3598.874 0.000 4290.067 0.000
## 889 890 891 892 893 894 895 896
## 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
## 897 898 899 900 901 902 903 904
## 0.000 0.000 0.000 0.000 0.000 0.000 4229.218 0.000
## 905 906 907 908 909 910 911 912
## 0.000 0.000 4219.742 0.000 0.000 0.000 4517.187 0.000
## 913 914 915 916 917 918 919 920
## 0.000 0.000 0.000 0.000 4790.970 3714.218 0.000 0.000
## 921 922 923 924 925 926 927 928
## 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
## 929 930 931 932 933 934 935 936
## 3817.641 0.000 0.000 3711.629 0.000 0.000 0.000 0.000
## 937 938 939 940 941 942 943 944
## 0.000 0.000 0.000 0.000 3843.304 0.000 0.000 0.000
## 945 946 947 948 949 950 951 952
## 0.000 0.000 0.000 0.000 0.000 3573.094 0.000 0.000
## 953 954 955 956 957 958 959 960
## 0.000 3629.990 0.000 0.000 0.000 0.000 0.000 0.000
## 961 962 963 964 965 966 967 968
## 0.000 0.000 0.000 0.000 0.000 4251.133 0.000 0.000
## 969 970 971 972 973 974 975 976
## 0.000 0.000 4060.846 0.000 0.000 0.000 0.000 0.000
## 977 978 979 980 981 982 983 984
## 0.000 0.000 0.000 0.000 0.000 0.000 3731.981 0.000
## 985 986 987 988 989 990 991 992
## 4003.229 3897.668 0.000 0.000 0.000 4218.746 0.000 0.000
## 993 994 995 996 997 998 999 1000
## 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
## 1001 1002 1003 1004 1005 1006 1007 1008
## 0.000 4562.241 4162.445 0.000 0.000 0.000 0.000 0.000
## 1009 1010 1011 1012 1013 1014 1015 1016
## 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
## 1017 1018 1019 1020 1021 1022 1023 1024
## 0.000 3230.437 0.000 0.000 0.000 0.000 4559.846 0.000
## 1025 1026 1027 1028 1029 1030 1031 1032
## 4148.592 3507.718 0.000 0.000 0.000 0.000 0.000 0.000
## 1033 1034 1035 1036 1037 1038 1039 1040
## 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
## 1041 1042 1043 1044 1045 1046 1047 1048
## 0.000 0.000 3318.132 0.000 3294.117 0.000 0.000 0.000
## 1049 1050 1051 1052 1053 1054 1055 1056
## 0.000 0.000 3730.758 4105.836 3316.897 0.000 0.000 0.000
## 1057 1058 1059 1060 1061 1062 1063 1064
## 0.000 0.000 3263.261 3855.849 0.000 3576.546 0.000 0.000
## 1065 1066 1067 1068 1069 1070 1071 1072
## 0.000 0.000 0.000 0.000 0.000 3494.678 0.000 0.000
## 1073 1074 1075 1076 1077 1078 1079 1080
## 0.000 4383.849 0.000 0.000 0.000 0.000 4089.432 0.000
## 1081 1082 1083 1084 1085 1086 1087 1088
## 4202.676 4003.167 0.000 0.000 3203.508 4572.345 0.000 0.000
## 1089 1090 1091 1092 1093 1094 1095 1096
## 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
## 1097 1098 1099 1100 1101 1102 1103 1104
## 0.000 0.000 0.000 3497.604 0.000 0.000 0.000 0.000
## 1105 1106 1107 1108 1109 1110 1111 1112
## 0.000 0.000 0.000 0.000 0.000 0.000 3726.166 0.000
## 1113 1114 1115 1116 1117 1118 1119 1120
## 0.000 0.000 0.000 0.000 0.000 4049.711 3780.466 0.000
## 1121 1122 1123 1124 1125 1126 1127 1128
## 0.000 3820.162 0.000 0.000 0.000 0.000 5019.780 0.000
## 1129 1130 1131 1132 1133 1134 1135 1136
## 0.000 0.000 0.000 0.000 3986.227 0.000 3843.352 0.000
## 1137 1138 1139 1140 1141 1142 1143 1144
## 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
## 1145 1146 1147 1148 1149 1150 1151 1152
## 3518.736 0.000 0.000 4230.188 0.000 4249.976 0.000 4192.333
## 1153 1154 1155 1156 1157 1158 1159 1160
## 0.000 0.000 3872.830 0.000 0.000 0.000 0.000 0.000
## 1161 1162 1163 1164 1165 1166 1167 1168
## 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
## 1169 1170 1171 1172 1173 1174 1175 1176
## 0.000 0.000 0.000 4053.309 3901.157 4109.003 0.000 0.000
## 1177 1178 1179 1180 1181 1182 1183 1184
## 0.000 0.000 0.000 4002.590 0.000 3528.558 0.000 0.000
## 1185 1186 1187 1188 1189 1190 1191 1192
## 3609.087 0.000 0.000 0.000 0.000 0.000 0.000 0.000
## 1193 1194 1195 1196 1197 1198 1199 1200
## 0.000 3787.911 0.000 0.000 0.000 0.000 0.000 0.000
## 1201 1202 1203 1204 1205 1206 1207 1208
## 0.000 0.000 0.000 0.000 0.000 0.000 3265.838 0.000
## 1209 1210 1211 1212 1213 1214 1215 1216
## 0.000 0.000 0.000 0.000 4965.767 0.000 0.000 0.000
## 1217 1218 1219 1220 1221 1222 1223 1224
## 4035.283 0.000 0.000 0.000 0.000 0.000 4664.114 0.000
## 1225 1226 1227 1228 1229 1230 1231 1232
## 3860.731 0.000 0.000 0.000 0.000 4171.923 0.000 0.000
## 1233 1234 1235 1236 1237 1238 1239 1240
## 0.000 4375.013 0.000 0.000 0.000 4103.514 0.000 0.000
## 1241 1242 1243 1244 1245 1246 1247 1248
## 0.000 0.000 0.000 0.000 0.000 4716.232 0.000 0.000
## 1249 1250 1251 1252 1253 1254 1255 1256
## 0.000 0.000 0.000 3896.297 0.000 0.000 0.000 0.000
## 1257 1258 1259 1260 1261 1262 1263 1264
## 4038.793 0.000 0.000 0.000 0.000 0.000 0.000 3583.676
## 1265 1266 1267 1268 1269 1270 1271 1272
## 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
## 1273 1274 1275 1276 1277 1278 1279 1280
## 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
## 1281 1282 1283 1284 1285 1286 1287 1288
## 3690.970 0.000 0.000 0.000 0.000 0.000 3576.782 0.000
## 1289 1290 1291 1292 1293 1294 1295 1296
## 0.000 0.000 4006.678 0.000 0.000 0.000 0.000 0.000
## 1297 1298 1299 1300 1301 1302 1303 1304
## 0.000 0.000 4144.022 0.000 0.000 0.000 0.000 0.000
## 1305 1306 1307 1308 1309 1310 1311 1312
## 0.000 0.000 4479.542 0.000 0.000 3689.827 3497.095 0.000
## 1313 1314 1315 1316 1317 1318 1319 1320
## 3930.299 0.000 0.000 0.000 0.000 0.000 0.000 4062.230
## 1321 1322 1323 1324 1325 1326 1327 1328
## 0.000 0.000 3493.222 0.000 0.000 4299.095 0.000 0.000
## 1329 1330 1331 1332 1333 1334 1335 1336
## 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
## 1337 1338 1339 1340 1341 1342 1343 1344
## 0.000 0.000 0.000 0.000 0.000 3766.654 0.000 0.000
## 1345 1346 1347 1348 1349 1350 1351 1352
## 4785.084 0.000 0.000 0.000 0.000 0.000 0.000 3301.337
## 1353 1354 1355 1356 1357 1358 1359 1360
## 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
## 1361 1362 1363 1364 1365 1366 1367 1368
## 0.000 0.000 4512.102 0.000 0.000 0.000 4267.627 3237.293
## 1369 1370 1371 1372 1373 1374 1375 1376
## 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
## 1377 1378 1379 1380 1381 1382 1383 1384
## 0.000 0.000 0.000 0.000 4823.310 3886.065 0.000 0.000
## 1385 1386 1387 1388 1389 1390 1391 1392
## 0.000 0.000 0.000 0.000 0.000 0.000 3704.771 3863.617
## 1393 1394 1395 1396 1397 1398 1399 1400
## 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
## 1401 1402 1403 1404 1405 1406 1407 1408
## 0.000 0.000 4036.744 3732.036 0.000 0.000 0.000 0.000
## 1409 1410 1411 1412 1413 1414 1415 1416
## 0.000 4049.153 0.000 0.000 0.000 0.000 0.000 0.000
## 1417 1418 1419 1420 1421 1422 1423 1424
## 0.000 0.000 0.000 0.000 0.000 4088.061 0.000 0.000
## 1425 1426 1427 1428 1429 1430 1431 1432
## 0.000 4323.999 0.000 0.000 0.000 0.000 0.000 0.000
## 1433 1434 1435 1436 1437 1438 1439 1440
## 0.000 0.000 0.000 0.000 0.000 3446.415 0.000 0.000
## 1441 1442 1443 1444 1445 1446 1447 1448
## 0.000 3670.232 0.000 0.000 0.000 3584.522 0.000 0.000
## 1449 1450 1451 1452 1453 1454 1455 1456
## 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
## 1457 1458 1459 1460 1461 1462 1463 1464
## 3776.487 0.000 0.000 0.000 0.000 0.000 0.000 0.000
## 1465 1466 1467 1468 1469 1470 1471 1472
## 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
## 1473 1474 1475 1476 1477 1478 1479 1480
## 0.000 0.000 0.000 0.000 3322.266 0.000 0.000 0.000
## 1481 1482 1483 1484 1485 1486 1487 1488
## 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
## 1489 1490 1491 1492 1493 1494 1495 1496
## 3955.388 0.000 0.000 3742.933 0.000 0.000 4003.742 0.000
## 1497 1498 1499 1500 1501 1502 1503 1504
## 0.000 0.000 0.000 0.000 0.000 0.000 0.000 3548.604
## 1505 1506 1507 1508 1509 1510 1511 1512
## 0.000 0.000 3434.211 0.000 0.000 0.000 0.000 0.000
## 1513 1514 1515 1516 1517 1518 1519 1520
## 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
## 1521 1522 1523 1524 1525 1526 1527 1528
## 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
## 1529 1530 1531 1532 1533 1534 1535 1536
## 3808.462 3543.040 0.000 0.000 0.000 0.000 0.000 0.000
## 1537 1538 1539 1540 1541 1542 1543 1544
## 0.000 0.000 3883.679 4130.431 0.000 0.000 0.000 0.000
## 1545 1546 1547 1548 1549 1550 1551 1552
## 3985.261 0.000 0.000 0.000 0.000 0.000 0.000 0.000
## 1553 1554 1555 1556 1557 1558 1559 1560
## 0.000 3852.088 0.000 0.000 0.000 0.000 0.000 0.000
## 1561 1562 1563 1564 1565 1566 1567 1568
## 0.000 4774.632 0.000 0.000 4692.453 0.000 0.000 0.000
## 1569 1570 1571 1572 1573 1574 1575 1576
## 0.000 0.000 3742.135 0.000 0.000 0.000 0.000 0.000
## 1577 1578 1579 1580 1581 1582 1583 1584
## 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
## 1585 1586 1587 1588 1589 1590 1591 1592
## 0.000 0.000 0.000 0.000 0.000 0.000 0.000 4149.065
## 1593 1594 1595 1596 1597 1598 1599 1600
## 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
## 1601 1602 1603 1604 1605 1606 1607 1608
## 3885.911 0.000 0.000 4128.276 0.000 4873.397 0.000 0.000
## 1609 1610 1611 1612 1613 1614 1615 1616
## 0.000 3923.644 0.000 0.000 0.000 0.000 0.000 3830.032
## 1617 1618 1619 1620 1621 1622 1623 1624
## 0.000 4102.827 0.000 0.000 3639.474 0.000 4222.293 0.000
## 1625 1626 1627 1628 1629 1630 1631 1632
## 0.000 0.000 0.000 0.000 0.000 4181.373 0.000 3815.750
## 1633 1634 1635 1636 1637 1638 1639 1640
## 0.000 0.000 0.000 0.000 3800.687 0.000 0.000 0.000
## 1641 1642 1643 1644 1645 1646 1647 1648
## 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
## 1649 1650 1651 1652 1653 1654 1655 1656
## 0.000 3469.662 0.000 0.000 0.000 0.000 0.000 0.000
## 1657 1658 1659 1660 1661 1662 1663 1664
## 0.000 0.000 0.000 0.000 3475.903 5080.903 3500.980 3953.879
## 1665 1666 1667 1668 1669 1670 1671 1672
## 4647.058 0.000 0.000 0.000 3998.160 0.000 3646.943 0.000
## 1673 1674 1675 1676 1677 1678 1679 1680
## 3861.982 0.000 0.000 0.000 0.000 0.000 0.000 0.000
## 1681 1682 1683 1684 1685 1686 1687 1688
## 0.000 3576.167 3861.133 0.000 4146.234 0.000 0.000 0.000
## 1689 1690 1691 1692 1693 1694 1695 1696
## 0.000 0.000 0.000 0.000 0.000 0.000 0.000 3719.679
## 1697 1698 1699 1700 1701 1702 1703 1704
## 0.000 4184.179 3895.539 0.000 0.000 0.000 0.000 0.000
## 1705 1706 1707 1708 1709 1710 1711 1712
## 0.000 0.000 3875.813 0.000 3995.177 0.000 0.000 0.000
## 1713 1714 1715 1716 1717 1718 1719 1720
## 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
## 1721 1722 1723 1724 1725 1726 1727 1728
## 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
## 1729 1730 1731 1732 1733 1734 1735 1736
## 3992.678 0.000 0.000 0.000 0.000 0.000 0.000 0.000
## 1737 1738 1739 1740 1741 1742 1743 1744
## 0.000 0.000 0.000 0.000 3779.413 0.000 0.000 0.000
## 1745 1746 1747 1748 1749 1750 1751 1752
## 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
## 1753 1754 1755 1756 1757 1758 1759 1760
## 0.000 3314.138 0.000 0.000 0.000 0.000 0.000 0.000
## 1761 1762 1763 1764 1765 1766 1767 1768
## 4515.487 0.000 0.000 0.000 0.000 3871.194 0.000 0.000
## 1769 1770 1771 1772 1773 1774 1775 1776
## 0.000 0.000 0.000 0.000 0.000 4703.573 0.000 0.000
## 1777 1778 1779 1780 1781 1782 1783 1784
## 3377.242 0.000 3856.955 0.000 0.000 0.000 0.000 0.000
## 1785 1786 1787 1788 1789 1790 1791 1792
## 0.000 0.000 0.000 0.000 3895.095 0.000 0.000 0.000
## 1793 1794 1795 1796 1797 1798 1799 1800
## 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
## 1801 1802 1803 1804 1805 1806 1807 1808
## 0.000 0.000 0.000 0.000 0.000 0.000 3594.508 4045.056
## 1809 1810 1811 1812 1813 1814 1815 1816
## 0.000 0.000 0.000 0.000 3844.573 0.000 0.000 0.000
## 1817 1818 1819 1820 1821 1822 1823 1824
## 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
## 1825 1826 1827 1828 1829 1830 1831 1832
## 0.000 3534.011 0.000 0.000 0.000 0.000 0.000 0.000
## 1833 1834 1835 1836 1837 1838 1839 1840
## 0.000 0.000 0.000 0.000 0.000 4510.229 0.000 3704.256
## 1841 1842 1843 1844 1845 1846 1847 1848
## 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
## 1849 1850 1851 1852 1853 1854 1855 1856
## 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
## 1857 1858 1859 1860 1861 1862 1863 1864
## 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
## 1865 1866 1867 1868 1869 1870 1871 1872
## 0.000 0.000 0.000 0.000 0.000 0.000 3836.064 0.000
## 1873 1874 1875 1876 1877 1878 1879 1880
## 0.000 0.000 0.000 0.000 0.000 4015.542 0.000 0.000
## 1881 1882 1883 1884 1885 1886 1887 1888
## 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
## 1889 1890 1891 1892 1893 1894 1895 1896
## 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
## 1897 1898 1899 1900 1901 1902 1903 1904
## 0.000 0.000 0.000 3401.781 0.000 0.000 0.000 0.000
## 1905 1906 1907 1908 1909 1910 1911 1912
## 0.000 0.000 0.000 0.000 0.000 4156.546 0.000 0.000
## 1913 1914 1915 1916 1917 1918 1919 1920
## 0.000 0.000 3963.253 0.000 0.000 0.000 0.000 0.000
## 1921 1922 1923 1924 1925 1926 1927 1928
## 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
## 1929 1930 1931 1932 1933 1934 1935 1936
## 0.000 0.000 0.000 0.000 0.000 4062.875 0.000 0.000
## 1937 1938 1939 1940 1941 1942 1943 1944
## 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
## 1945 1946 1947 1948 1949 1950 1951 1952
## 0.000 0.000 3305.541 0.000 3170.962 0.000 0.000 0.000
## 1953 1954 1955 1956 1957 1958 1959 1960
## 3417.402 0.000 0.000 0.000 0.000 0.000 0.000 0.000
## 1961 1962 1963 1964 1965 1966 1967 1968
## 4502.417 0.000 0.000 0.000 0.000 0.000 0.000 0.000
## 1969 1970 1971 1972 1973 1974 1975 1976
## 0.000 0.000 0.000 0.000 0.000 4061.154 0.000 0.000
## 1977 1978 1979 1980 1981 1982 1983 1984
## 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
## 1985 1986 1987 1988 1989 1990 1991 1992
## 0.000 0.000 0.000 0.000 0.000 0.000 0.000 3431.435
## 1993 1994 1995 1996 1997 1998 1999 2000
## 4020.546 4058.588 0.000 0.000 0.000 4214.913 0.000 0.000
## 2001 2002 2003 2004 2005 2006 2007 2008
## 3796.436 0.000 3686.385 0.000 3844.482 0.000 0.000 0.000
## 2009 2010 2011 2012 2013 2014 2015 2016
## 0.000 0.000 0.000 0.000 3680.511 0.000 0.000 0.000
## 2017 2018 2019 2020 2021 2022 2023 2024
## 0.000 3470.107 4073.661 0.000 0.000 0.000 0.000 0.000
## 2025 2026 2027 2028 2029 2030 2031 2032
## 0.000 0.000 0.000 0.000 0.000 3480.293 0.000 0.000
## 2033 2034 2035 2036 2037 2038 2039 2040
## 0.000 0.000 4240.918 0.000 0.000 0.000 3615.644 0.000
## 2041 2042 2043 2044 2045 2046 2047 2048
## 0.000 0.000 0.000 0.000 0.000 0.000 0.000 3808.612
## 2049 2050 2051 2052 2053 2054 2055 2056
## 0.000 0.000 0.000 0.000 0.000 0.000 3697.068 0.000
## 2057 2058 2059 2060 2061 2062 2063 2064
## 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
## 2065 2066 2067 2068 2069 2070 2071 2072
## 0.000 0.000 0.000 3568.180 0.000 0.000 0.000 0.000
## 2073 2074 2075 2076 2077 2078 2079 2080
## 4857.049 0.000 0.000 0.000 4154.792 0.000 0.000 3960.377
## 2081 2082 2083 2084 2085 2086 2087 2088
## 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
## 2089 2090 2091 2092 2093 2094 2095 2096
## 0.000 3596.915 0.000 0.000 0.000 0.000 0.000 3802.817
## 2097 2098 2099 2100 2101 2102 2103 2104
## 0.000 0.000 4898.136 3381.754 3618.065 0.000 3958.452 0.000
## 2105 2106 2107 2108 2109 2110 2111 2112
## 0.000 0.000 3669.958 0.000 0.000 0.000 3882.047 0.000
## 2113 2114 2115 2116 2117 2118 2119 2120
## 0.000 0.000 0.000 0.000 0.000 0.000 3685.214 0.000
## 2121 2122 2123 2124 2125 2126 2127 2128
## 0.000 0.000 3989.684 0.000 0.000 0.000 0.000 0.000
## 2129 2130 2131 2132 2133 2134 2135 2136
## 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
## 2137 2138 2139 2140 2141
## 0.000 0.000 0.000 0.000 0.000And this concludes the assignment. Thank you!
Appendix
# Loading packages
requiredpackages <- c("ggplot2", "reshape2", "dplyr","tidyr","corrplot", "caret", "glmnet","leaps", "pROC","car", "faraway")
for (i in requiredpackages){
if(!require(i,character.only=T)) install.packages(i)
library(i,character.only=T)
}
# Loading dataset and cleaning
url_train <- 'https://raw.githubusercontent.com/jcp9010/MSDA/master/CUNY%20621/Homework%204/insurance_training_data.csv'
url_eval <- 'https://raw.githubusercontent.com/jcp9010/MSDA/master/CUNY%20621/Homework%204/insurance-evaluation-data.csv'
train <- read.csv(url_train, header=TRUE)
eval <- read.csv(url_eval, header=TRUE)
# Get rid of the index column
train <- train[,-1]
# Data will requiring cleaning (i.e. need to rid of Z_, $ signs, commas, etc.)
new_df <- train[,names(train) %in% c("INCOME","OLDCLAIM", "HOME_VAL", "BLUEBOOK")]
new_df <- apply(new_df, 2, function(y) gsub("\\$","",y))
new_df <- apply(new_df, 2, function(y) gsub(",","",y))
new_df <- apply(new_df, 2, as.integer)
train <- train[,!names(train) %in% c("INCOME","OLDCLAIM", "HOME_VAL", "BLUEBOOK")]
train <- cbind(train, new_df)
new_df <- train[,names(train) %in% c("MSTATUS","SEX", "EDUCATION", "JOB", "CAR_TYPE", "URBANICITY")]
new_df <- apply(new_df, 2, function(y) gsub("z_","",y))
train <- train[,!names(train) %in% c("MSTATUS","SEX", "EDUCATION", "JOB", "CAR_TYPE", "URBANICITY")]
train <- cbind(train, new_df)
head(train)
summary(train)
# Boxplot
ggplot(melt(train), aes(x=factor(variable), y=value)) + facet_wrap(~variable, scale="free") + geom_boxplot()
# Histogram
ggplot(melt(train), aes(x=value)) + facet_wrap(~variable, scale="free") + geom_histogram(bins=50)
# Correlation
num_train <- train[,!names(train) %in% c("PARENT1","MSTATUS","EDUCATION","JOB","CAR_USE","CAR_TYPE","RED_CAR", "REVOKED","URBANICITY","INCOME","OLDCLAIM","HOME_VAL","BLUEBOOK","SEX")]
a <- cor(num_train, method="pearson", use="complete.obs")
a
corrplot(a, method="circle")
# Age
summary(train$AGE)
ggplot(train, aes(x=AGE)) + geom_histogram(binwidth = 1, col="black", fill="blue") + xlab("Age") + ylab("Count")
ggplot(train, aes(x=AGE, y=TARGET_AMT)) + geom_point() + xlab("Age") + ylab("Amount Paid Out by Insurance in Dollars")
qqnorm(train$AGE)
qqline(train$AGE)
# Shapiro-Wilk Test
shapiro.test(sample(train$AGE,500, replace=TRUE))
# Correlation between age and payout?
age_accident <- train %>%
select(AGE,TARGET_FLAG) %>%
mutate(AGE = cut(AGE, breaks = c(0,30,60,110), include.lowest=TRUE, labels=c("Young", "Middle", "Old")))
age_accident_table <- table(age_accident)
age_accident_table
print(paste0("Percentage of Young Drivers (30 and younger) in a Crash? ", round(age_accident_table[1,2]/(age_accident_table[1,1] + age_accident_table[1,2]),3)))
print(paste0("Percentage of Middle Age Drivers (30 to 60) in a Crash? ", round(age_accident_table[2,2]/(age_accident_table[2,1] + age_accident_table[2,2]),3)))
print(paste0("Percentage of Older Drivers (60 and older) in a Crash? ", round(age_accident_table[3,2]/(age_accident_table[3,1] + age_accident_table[3,2]),3)))
# R-squared correlation?
a <- cor(train$AGE, train$TARGET_FLAG, method = c("pearson"), use = "complete.obs")
b <- cor(train$AGE, train$TARGET_AMT, method = c("pearson"), use = "complete.obs")
print(paste0("Correlation between Age and if it involved in car crash? ", round(a,3)))
print(paste0("Correlation between Age and Amount Paid if involved in car crash? ", round(b,3)))
# One variable linear regression with age
summary(lm(TARGET_AMT ~ AGE, data = train))
# Logistic regresion with age and TARGET_FLAG
summary(glm(TARGET_FLAG ~ AGE, family="binomial", data = train))
# Marital Status
summary(train$MSTATUS)
ggplot(train, aes(x=MSTATUS, group = TARGET_FLAG)) +
geom_bar(aes(fill = factor(TARGET_FLAG)), stat="count") +
ylab("Count") +
xlab("Marriage Status")
ggplot(train, aes(x=MSTATUS, y=TARGET_AMT)) + geom_jitter() + xlab("Marriage Status") + ylab("Amount Paid Out by Insurance in Dollars")
mstatus_table <- train %>%
select(MSTATUS, TARGET_FLAG) %>%
table()
mstatus_table
print(paste0("Percentage of Single People Involved in Car Crashes: ", round(mstatus_table[1,2]/(mstatus_table[1,1] + mstatus_table[1,2]),3)))
print(paste0("Percentage of Married People Involved in Car Crashes: ", round(mstatus_table[2,2]/(mstatus_table[2,1] + mstatus_table[2,2]),3)))
# Logistic regresion with marital status
summary(glm(TARGET_FLAG ~ MSTATUS, family="binomial", data = train))
# Linear regresion with marital status
summary(lm(TARGET_AMT ~ MSTATUS, data = train))
# Living Location
living_area <- train %>%
select(URBANICITY, TARGET_FLAG) %>%
table()
living_area
print(paste0("Percentage of People Living in Rural Areas Involved in Car Crashes: ", round(living_area[1,2]/(living_area[1,1] + living_area[1,2]),3)))
print(paste0("Percentage of People Living in Urban Areas Involved in Car Crashes: ", round(living_area[2,2]/(living_area[2,1] + living_area[2,2]),3)))
summary(train$URBANICITY)
ggplot(train, aes(x=URBANICITY, group = TARGET_FLAG)) +
geom_bar(aes(fill = factor(TARGET_FLAG)), stat="count") +
ylab("Count") +
xlab("Where people are living")
ggplot(train, aes(x=URBANICITY, y=TARGET_AMT)) + geom_jitter() + xlab("Where people are living") + ylab("Amount Paid Out by Insurance in Dollars")
# Logistic regression for Urbanicity
summary(glm(TARGET_FLAG ~ URBANICITY, family="binomial", data = train))
# Profession
levels(train$JOB)
jobs <- train %>%
select(JOB, TARGET_FLAG) %>%
table() %>%
as.data.frame.matrix()
jobs$Percent_Crashed <- apply(jobs, 1, function(y) y[2]/sum(y))
jobs
prof_df <- train %>%
select(JOB, TARGET_AMT, TARGET_FLAG)
prof_df$JOB_CAT <- ifelse(prof_df$JOB == c('Doctor', 'Manager', 'Lawyer', 'Professional'),'Professional','NonProfessional')
prof_df %>%
select(JOB_CAT, TARGET_FLAG) %>%
table()
# Stats for Profession
professional <- train %>%
filter(JOB == c('Doctor', 'Manager', 'Lawyer', 'Professional')) %>%
select(TARGET_FLAG, TARGET_AMT)
summary(professional[1])
# Get rid of the zeros from TARGET_AMT as that will very heavily skew the TARGET_AMT data.
professional %>% select(TARGET_AMT) %>% filter(TARGET_AMT > 0) %>% summary()
# Percentage of professionals in car accidents
print(paste0("There are a total of ", nrow(professional), " professionals in this dataset."))
print(paste0("How many were involved in a car accident? ", sum(professional$TARGET_FLAG)))
print(paste0("Percent Crashed: ", round(sum(professional$TARGET_FLAG)/nrow(professional),3)))
# Visualization
ggplot(prof_df, aes(x=JOB_CAT, group = TARGET_FLAG)) + geom_bar(aes(fill = factor(TARGET_FLAG)), stat="count") + xlab("Job Category") + ylab("Frequency")
prof_df %>% filter(JOB_CAT == 'Professional' & TARGET_AMT > 0) %>% select(TARGET_AMT) %>% melt() %>%
ggplot(aes(x=value)) + geom_histogram(bins=50) + xlab("Dollar Amount Paid") + ylab("Count")
# Nonprofessional
Nonprofessional <- train %>%
filter(JOB != c('Doctor', 'Manager', 'Lawyer', 'Professional')) %>%
select(TARGET_FLAG, TARGET_AMT)
summary(Nonprofessional[1])
# Get rid of the zeros from TARGET_AMT as that will very heavily skew the TARGET_AMT data.
Nonprofessional %>% select(TARGET_AMT) %>% filter(TARGET_AMT > 0) %>% summary()
print(paste0("There are a total of ", nrow(Nonprofessional), " nonprofessionals in this dataset."))
print(paste0("How many were involved in a car accident? ", sum(Nonprofessional$TARGET_FLAG)))
print(paste0("Percent Crashed: ", round(sum(Nonprofessional$TARGET_FLAG)/nrow(Nonprofessional),3)))
# Visualization
prof_df %>% select(JOB_CAT, TARGET_AMT) %>% filter(TARGET_AMT > 0) %>%
ggplot(aes(x=TARGET_AMT, fill=JOB_CAT)) + geom_histogram(bins=50)
# Vehicle Size
levels(train$CAR_TYPE)
car_size <- train %>% select(CAR_TYPE, TARGET_FLAG, TARGET_AMT) %>%
mutate(CAR_SIZE = ifelse(CAR_TYPE != 'Sports Car', 'Large', 'Small'))
car_size %>% select(CAR_TYPE, TARGET_FLAG) %>% table()
a <- car_size %>% select(CAR_SIZE, TARGET_FLAG) %>% table() %>% as.data.frame.matrix()
a$Percent_Crashed <- apply(a, 1, function(y) y[2]/sum(y))
a
# Age of the Car
train %>% select(TARGET_FLAG, CAR_AGE) %>%
ggplot(aes(x=factor(TARGET_FLAG), y=CAR_AGE, fill=TARGET_FLAG)) + geom_boxplot() + ylab("Age of the Car") + xlab("No accident vs. Yes accident")
train %>% select(CAR_AGE, TARGET_AMT, TARGET_FLAG) %>%
ggplot(aes(x=CAR_AGE, y=TARGET_AMT, color=factor(TARGET_FLAG))) + geom_jitter()
# Driving History
ggplot(train, aes(x=OLDCLAIM, y=TARGET_AMT, color=factor(TARGET_FLAG))) + geom_point()
# Correlation for the Driving History
train %>% select(OLDCLAIM, TARGET_AMT) %>% cor()
# Any number of missing data points?
sum(is.na(train))
rev(sort(colSums(sapply(train, is.na))))
# CAR_AGE
complete_car_age <- train[complete.cases(train$CAR_AGE),]$CAR_AGE
hist(complete_car_age,col="green",main="Age of Car Histogram",xlab="Age of the Car")
# Creating the mode
# Reference: https://stackoverflow.com/questions/2547402/is-there-a-built-in-function-for-finding-the-mode
Mode <- function(x) {
ux <- unique(x)
ux[which.max(tabulate(match(x, ux)))]
}
Mode(complete_car_age)
train[is.na(train$CAR_AGE),]$CAR_AGE <- Mode(complete_car_age)
# Imputing other NA variables
train %>% select(HOME_VAL, YOJ, INCOME) %>% melt() %>%
ggplot(aes(x=value)) + geom_histogram(bins=50) + facet_wrap(~variable, scale="free")
train[is.na(train$HOME_VAL),]$HOME_VAL <- Mode(train[complete.cases(train$HOME_VAL),]$HOME_VAL)
train[is.na(train$YOJ),]$YOJ <- median(train[complete.cases(train$YOJ),]$YOJ)
train[is.na(train$INCOME),]$INCOME <- median(train[complete.cases(train$INCOME),]$INCOME)
train[is.na(train$AGE),]$AGE <- median(train[complete.cases(train$AGE),]$AGE)
# Looking at the minimum values in order to adjust for the BoxCox Transformation
apply(train,2,min)
# BoxCox Transformation
# BoxCoxTrans from caret package only works on positive numeric values. Must take out the categorical variables
boxcox_df <- train %>%
filter(CAR_AGE >= 0) %>%
mutate(CAR_AGE = CAR_AGE + 1, YOJ = YOJ + 1, INCOME = INCOME + 1) %>%
select(TIF, BLUEBOOK, TRAVTIME, AGE, CAR_AGE, YOJ, INCOME)
apply(boxcox_df, 2, BoxCoxTrans)
# Logistic Regression Model Number 1
full_bin_df <- train %>% select(-TARGET_AMT)
bin_glm_1 <- glm(TARGET_FLAG ~ ., family="binomial", full_bin_df)
summary(bin_glm_1)
# Logistic Regression Model Number 2
bin_glm_2 <- step(bin_glm_1, direction="backward")
bin_glm_2
# Variable Importance
# Reference: https://www.r-bloggers.com/evaluating-logistic-regression-models/
a <- varImp(bin_glm_2)
head(a[order(-a$Overall),,drop=FALSE])
# Logistic Regression Model Number 3
bin_glm_3 <- glm(TARGET_FLAG ~ KIDSDRIV + AGE + HOMEKIDS + I(YOJ^1.6) + PARENT1 + I(TRAVTIME^0.7) + CAR_USE + I(TIF^0.2) + RED_CAR + CLM_FREQ + REVOKED + MVR_PTS + I(CAR_AGE^0.4) + I(INCOME^0.4) + HOME_VAL + I(BLUEBOOK^0.5) + OLDCLAIM + MSTATUS + SEX + EDUCATION + JOB + CAR_TYPE + URBANICITY,family="binomial",full_bin_df)
summary(bin_glm_3)
# Variable Importance for Model Number 3
# Reference: https://www.r-bloggers.com/evaluating-logistic-regression-models/
a <- varImp(bin_glm_3)
head(a[order(-a$Overall),,drop=FALSE])
# Lasso Regression
# Reference: https://www.r-bloggers.com/ridge-regression-and-the-lasso/
lasso_full_bin_df <- full_bin_df %>%
select(TARGET_FLAG, KIDSDRIV, AGE, HOMEKIDS, YOJ, TRAVTIME, TIF, CLM_FREQ, MVR_PTS, CAR_AGE, INCOME, HOME_VAL, BLUEBOOK, OLDCLAIM)
x <- model.matrix(TARGET_FLAG ~ ., lasso_full_bin_df)[,-1]
y <- lasso_full_bin_df$TARGET_FLAG
# Creating a vector of lambda values
lambda <- 10^seq(10,-2, length=1000)
# Create a training and testing dataset
# Reference: https://www.r-bloggers.com/ridge-regression-and-the-lasso/
lasso_full_bin_df <- full_bin_df %>%
select(TARGET_FLAG, KIDSDRIV, AGE, HOMEKIDS, YOJ, TRAVTIME, TIF, CLM_FREQ, MVR_PTS, CAR_AGE, INCOME, HOME_VAL, BLUEBOOK, OLDCLAIM)
x <- model.matrix(TARGET_FLAG ~ ., lasso_full_bin_df)[,-1]
y <- lasso_full_bin_df$TARGET_FLAG
# Creating a vector of lambda values
lambda <- 10^seq(10,-2, length=1000)
# Create a training and testing dataset
train1 <- sample(1:nrow(x), nrow(x)/2)
test <- (-train1)
ytest = y[test]
# Fitting a lasso model
lasso.mod <- glmnet(x[train1,], y[train1], alpha = 1, lambda = lambda)
# Find the best lambda from our list via cross-validation
cv.out <- cv.glmnet(x[train1,], y[train1], alpha = 1)
bestlam <- cv.out$lambda.min
# Making the prediction
lasso.coef
head(lasso.pred)
print(paste0("MSE for lasso estimate: ", mean((lasso.pred-ytest)^2)))
# Creating a dataset for linear regression
amt_df <- train %>% select(-TARGET_FLAG) %>% filter(TARGET_AMT > 0)
# Creating Linear Regression Model Number 1
amt_lm_1 <- lm(TARGET_AMT ~ ., data = amt_df)
summary(amt_lm_1)
# Checking residual plot for model number 1
par(mfrow=c(2,2))
plot(amt_lm_1)
# BoxCox For response variable
BoxCoxTrans(amt_df$TARGET_AMT)
# Linear Regression Model number 2
l_TARGET_AMT <- log(amt_df$TARGET_AMT)
amt_df_trans <- cbind(amt_df, l_TARGET_AMT)
amt_df_trans <- amt_df_trans %>% select(-TARGET_AMT)
amt_lm_2 <- lm(l_TARGET_AMT ~ ., data=amt_df_trans)
summary(amt_lm_2)
# Linear Regression Model Number 3
amt_lm_3 <- lm(l_TARGET_AMT ~ KIDSDRIV + AGE + HOMEKIDS + I(YOJ^1.6) + PARENT1 + I(TRAVTIME^0.7) + CAR_USE + I(TIF^0.2) + RED_CAR + CLM_FREQ + REVOKED + MVR_PTS + I(CAR_AGE^0.4) + I(INCOME^0.4) + HOME_VAL + I(BLUEBOOK^0.5) + OLDCLAIM + MSTATUS + SEX + EDUCATION + JOB + CAR_TYPE + URBANICITY, data=amt_df_trans)
summary(amt_lm_3)
par(mfrow=c(2,2))
plot(amt_lm_3)
# Correlations
round(cor(amt_df_trans[,names(amt_df_trans) %in% c("KIDSDRIV", "AGE","HOMEKIDS","YOJ","TRAVTIME","TIF","CLM_FREQ","MVR_PTS","CAR_AGE","INCOME","HOME_VAL","BLUEBOOK","OLDCLAIM")]),2)
# VIF
vif(amt_lm_1)
vif(amt_lm_2)
vif(amt_lm_3)
# Linear Regression Model 4
amt_lm_4 <- lm(l_TARGET_AMT ~ KIDSDRIV + AGE + I(YOJ^1.6) + PARENT1 + I(TRAVTIME^0.7) + CAR_USE + I(TIF^0.2) + RED_CAR + CLM_FREQ + REVOKED + MVR_PTS + I(CAR_AGE^0.4) + I(INCOME^0.4) + HOME_VAL + I(BLUEBOOK^0.5) + OLDCLAIM + MSTATUS + SEX + CAR_TYPE + URBANICITY, data=amt_df_trans)
summary(amt_lm_4)
par(mfrow=c(2,2))
plot(amt_lm_4)
# leaps package, finding RSS
b <- regsubsets(l_TARGET_AMT ~ ., data=amt_df_trans)
rs <- summary(b)
plot(2:9, rs$adjr2,xlab="No. of Parameters", ylab="Adjusted R-square")
print(paste0("How many variables that maximizes the adjusted R-squared value? ", which.max(rs$adjr2)))
plot(2:9, rs$cp,xlab="No. of Parameters", ylab="Cp Statistic")
abline(0,1)
# Which variables should be included?
rs$which[which.max(rs$adjr),]
# Linear Regression Model 5
amt_lm_5 <- lm(l_TARGET_AMT ~ CLM_FREQ + MVR_PTS + BLUEBOOK + MSTATUS + EDUCATION + SEX + CAR_AGE, data=amt_df_trans)
summary(amt_lm_5)
par(mfrow=c(2,2))
plot(amt_lm_5)
# Linear Regression Model 6
amt_lm_6 <- step(amt_lm_5, direction="backward")
summary(amt_lm_6)
# Added-Variable Plot
avPlots(amt_lm_6)
# Testing the logistic regression model
Train <- createDataPartition(full_bin_df$TARGET_FLAG, p=0.7, list=FALSE)
training <- full_bin_df[Train, ]
testing <- full_bin_df[-Train, ]
pred <- round(predict(bin_glm_2, newdata=testing, type="response"), 3)
pred1 <- ifelse(pred > 0.5, 1, 0)
#Confusion Matrix
ans <- confusionMatrix(data=pred1, testing$TARGET_FLAG, positive='1')
ans
ans$byClass
plot(roc(testing$TARGET_FLAG, pred), main="ROC Curve from pROC Package")
# Please note that the X axis is in Specificity (as opposed to 1 - Specificity in the above function)
# Area Under the Curve
auc(roc(testing$TARGET_FLAG, pred))
# Histogram of the prediction values
hist(pred)
print(paste0("Median: ",median(pred)))
# Balancing out sensitivity and specificity
Cutoff <- seq(.1, .9, by=0.05)
cutoff <- c()
sens <- c()
spec <- c()
for (i in 1:length(Cutoff)) {
tmp_pred <- ifelse(pred > Cutoff[i], 1, 0)
ans <- confusionMatrix(data=tmp_pred, testing$TARGET_FLAG, positive='1')
cutoff <- c(cutoff, Cutoff[i])
sens <- c(sens, ans$byClass['Sensitivity'][[1]])
spec <- c(spec, ans$byClass['Specificity'][[1]])
}
cutoff_df <- as.data.frame(cbind(cutoff, sens, spec))
cutoff_df
ggplot(cutoff_df, aes(x=cutoff)) +
geom_line(aes(y = sens, colour = "Sensitivity")) +
geom_line(aes(y = spec, colour = "Specificity")) +
xlab("Cutoff Values") + ylab("")
# Redoing the logistic regression analysis
Train <- createDataPartition(full_bin_df$TARGET_FLAG, p=0.7, list=FALSE)
training <- full_bin_df[Train, ]
testing <- full_bin_df[-Train, ]
pred <- round(predict(bin_glm_2, newdata=testing, type="response"), 3)
pred1 <- ifelse(pred > 0.28, 1, 0)
#Confusion Matrix
ans <- confusionMatrix(data=pred1, testing$TARGET_FLAG, positive='1')
ans
ans$byClass
plot(roc(testing$TARGET_FLAG, pred), main="ROC Curve from pROC Package")
# Please note that the X axis is in Specificity (as opposed to 1 - Specificity in the above function)
# Area Under the Curve
auc(roc(testing$TARGET_FLAG, pred))
# Cleaning up the testing dataset.
# Get rid of the index column
eval <- eval[,-1]
# Data will requiring cleaning (i.e. need to rid of Z_, $ signs, commas, etc.)
new_df <- eval[,names(eval) %in% c("INCOME","OLDCLAIM", "HOME_VAL", "BLUEBOOK")]
new_df <- apply(new_df, 2, function(y) gsub("\\$","",y))
new_df <- apply(new_df, 2, function(y) gsub(",","",y))
new_df <- apply(new_df, 2, as.integer)
eval <- eval[,!names(eval) %in% c("INCOME","OLDCLAIM", "HOME_VAL", "BLUEBOOK")]
eval <- cbind(eval, new_df)
new_df <- eval[,names(eval) %in% c("MSTATUS","SEX", "EDUCATION", "JOB", "CAR_TYPE", "URBANICITY")]
new_df <- apply(new_df, 2, function(y) gsub("z_","",y))
eval <- eval[,!names(eval) %in% c("MSTATUS","SEX", "EDUCATION", "JOB", "CAR_TYPE", "URBANICITY")]
eval <- cbind(eval, new_df)
eval <- eval[,-c(2)]
head(eval)
# Making the logistic regression predictions
test_ans <- round(predict(bin_glm_2, newdata=eval[,-1], type="response"), 3)
test_ans <- ifelse(test_ans > 0.28, 1, 0)
test_ans
# Missing data points
sum(is.na(eval))
# Histogram of income
hist(eval$INCOME)
# Replacing income with the median
eval[is.na(eval$INCOME),]$INCOME <- median(eval[complete.cases(eval$INCOME),]$INCOME)
# How many NAs left?
test_ans <- round(predict(bin_glm_2, newdata=eval[,-1], type="response"), 3)
test_ans <- ifelse(test_ans > 0.5, 1, 0)
print(paste0("How many NA values are left? ", sum(is.na(test_ans))))
# Histogram of Home Values
hist(eval$HOME_VAL)
# Replacing Home Value NA with 0
eval[is.na(eval$HOME_VAL),]$HOME_VAL <- 0
# Predicting again
test_ans <- round(predict(bin_glm_2, newdata=eval[,-1], type="response"), 3)
test_ans <- ifelse(test_ans > 0.5, 1, 0)
test_ans
# Any more NAs left?
print(paste0("How many NA values are left? ", sum(is.na(test_ans))))
# Linear Regression Predictions
test_ans_2 <- ifelse(test_ans == 1, round(exp(predict(amt_lm_6, newdata=eval)), 3), 0)
test_ans_2
# You may un-comment this section if you would like to download a csv file of the predictions
# write.csv(test_ans, file="LogisticPrediction.csv")
# write.csv(test_ans_2, file="LinearPrediction.csv")