R code for Linear regression model

Medical Malpractice

1.Overview

medicalmalpractice <- read.csv("Documents/CHULA/1-2567/RegTS/Assignment1/A1_medicalmalpractice.csv")
names(medicalmalpractice)
## [1] "Amount"           "Severity"         "Age"              "Private.Attorney"
## [5] "Marital.Status"   "Specialty"        "Insurance"        "Gender"
head(medicalmalpractice)
##   Amount Severity Age Private.Attorney Marital.Status         Specialty
## 1  57041        7  62                1              2   Family Practice
## 2 324976        6  38                1              2             OBGYN
## 3 135383        4  34                1              2        Cardiology
## 4 829742        7  42                1              1        Pediatrics
## 5 197675        3  60                0              2             OBGYN
## 6  75368        9  71                1              2 Internal Medicine
##           Insurance Gender
## 1           Private   Male
## 2      No Insurance Female
## 3           Unknown   Male
## 4      No Insurance Female
## 5 Medicare/Medicaid Female
## 6 Medicare/Medicaid Female

2.Data exploration

Examine data graphically and numerically

Convert data from char to numeric

medicalmalpractice2<-medicalmalpractice
medicalmalpractice2$Amount <- as.numeric(medicalmalpractice2$Amount)
medicalmalpractice2$Severity <-as.numeric(medicalmalpractice2$Severity)
medicalmalpractice2$Age <-as.numeric(medicalmalpractice2$Age)
medicalmalpractice2$Private.Attorney <-as.numeric(medicalmalpractice2$Private.Attorney)
medicalmalpractice2$Marital.Status <-as.numeric(medicalmalpractice2$Marital.Status)

medicalmalpractice2$Specialty[medicalmalpractice2$Specialty=="Family Practice" ]=0
medicalmalpractice2$Specialty[medicalmalpractice2$Specialty=="OBGYN" ]=1
medicalmalpractice2$Specialty[medicalmalpractice2$Specialty=="Cardiology" ]=2
medicalmalpractice2$Specialty[medicalmalpractice2$Specialty=="Pediatrics" ]=3
medicalmalpractice2$Specialty[medicalmalpractice2$Specialty=="Internal Medicine" ]=4
medicalmalpractice2$Specialty[medicalmalpractice2$Specialty=="Anesthesiology" ]=5
medicalmalpractice2$Specialty[medicalmalpractice2$Specialty=="Emergency Medicine" ]=6
medicalmalpractice2$Specialty[medicalmalpractice2$Specialty=="Ophthamology" ]=7
medicalmalpractice2$Specialty[medicalmalpractice2$Specialty=="Urological Surgery" ]=8
medicalmalpractice2$Specialty[medicalmalpractice2$Specialty=="Orthopedic Surgery" ]=9
medicalmalpractice2$Specialty[medicalmalpractice2$Specialty=="Neurology/Neurosurgery" ]=10
medicalmalpractice2$Specialty[medicalmalpractice2$Specialty=="Occupational Medicine" ]=11
medicalmalpractice2$Specialty[medicalmalpractice2$Specialty=="Resident" ]=12
medicalmalpractice2$Specialty[medicalmalpractice2$Specialty=="Thoracic Surgery" ]=13
medicalmalpractice2$Specialty[medicalmalpractice2$Specialty=="General Surgery" ]=14
medicalmalpractice2$Specialty[medicalmalpractice2$Specialty=="Radiology" ]=15
medicalmalpractice2$Specialty[medicalmalpractice2$Specialty=="Pathology" ]=16
medicalmalpractice2$Specialty[medicalmalpractice2$Specialty=="Physical Medicine" ]=17
medicalmalpractice2$Specialty[medicalmalpractice2$Specialty=="Plastic Surgeon" ]=18
medicalmalpractice2$Specialty[medicalmalpractice2$Specialty=="Dermatology" ]=19
medicalmalpractice2$Specialty <- as.numeric(medicalmalpractice2$Specialty)

medicalmalpractice2$Insurance[medicalmalpractice2$Insurance=="Medicare/Medicaid"]=0
medicalmalpractice2$Insurance[medicalmalpractice2$Insurance=="No Insurance"]=1
medicalmalpractice2$Insurance[medicalmalpractice2$Insurance=="Private"]=2
medicalmalpractice2$Insurance[medicalmalpractice2$Insurance=="Unknown"]=3
medicalmalpractice2$Insurance[medicalmalpractice2$Insurance=="Workers Compensation"]=4
medicalmalpractice2$Insurance <- as.numeric(medicalmalpractice2$Insurance)

medicalmalpractice2$Gender[medicalmalpractice2$Gender=="Male"]=0
medicalmalpractice2$Gender[medicalmalpractice2$Gender=="Female"]=1
medicalmalpractice2$Gender <- as.numeric(medicalmalpractice2$Gender)

Examine a histogram of the response

If it appears non-normal then look for a transformation to normality, or at least symmetry.

Histogram

hist(medicalmalpractice$Amount,main = "Histogram of Amount")

Since,this histogram is a left skewness, we need to transform data by using log-function.

create new variable

medicalmalpractice2$log.Amount<-log(medicalmalpractice2$Amount)
hist(medicalmalpractice2$log.Amount,main="Histogram of log Amount")

Continuous covariates

Examine histograms.

Look out for obvious outliers and extreme skewness.

hist(medicalmalpractice2$Age)

summary(medicalmalpractice2$Age)
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##     0.0    28.0    43.0    42.7    58.0    87.0

After adjusting the data using the log function, it can be seen that the data is more symmetrical or can be called a normal distribution. We therefore chose to use log.amount as the dependent variable in the study.

Examine scatter plots of the response against each of the co variates, singly.

If the relationship appears nonlinear and/or heteroscedastic, apply a transformation that fixes this.

scatter plot

plot(log.Amount~Age,data = medicalmalpractice2)

From the scatter plot of log.Amount and Age, it was found that there is a relationship between the log.amount and the age of the claimant. It can be observed that, as the age of the claimant increases as a result, the log.amount has decreased.

Categorical variables

Creating categorical variables

Examine box plots of the response against each of the co variates, singly.

  • box plot
table(medicalmalpractice2$Severity)
## 
##     1     2     3     4     5     6     7     8     9 
##   665  1340 28251 15709  9615  3375  8873  3627  7755
boxplot(log.Amount~Severity,data = medicalmalpractice2,ylab="log.Amount",xlab="Severity")

table(medicalmalpractice2$Private.Attorney)
## 
##     0     1 
## 26861 52349
boxplot(log.Amount~Private.Attorney,data = medicalmalpractice2,ylab="log.Amount",xlab="Private.Attorney")

table(medicalmalpractice2$Marital.Status)
## 
##     0     1     2     3     4 
##  3832 22802 41220   994 10362
boxplot(log.Amount~Marital.Status,data = medicalmalpractice2,ylab="log.Amount",xlab="Marital.Status")

table(medicalmalpractice2$Specialty)
## 
##     0     1     2     3     4     5     6     7     8     9    10    11    12 
## 11436  8876  2659  1416  5223  8732  4676  3289  2027  7272  4737   725  1983 
##    13    14    15    16    17    18    19 
##   664  9412  1979   714   642  1364  1384
boxplot(log.Amount~Specialty,data = medicalmalpractice2,ylab="log.Amount",xlab="Specialty")

table(medicalmalpractice2$Insurance)
## 
##     0     1     2     3     4 
## 10882  8002 34289 24052  1985
boxplot(log.Amount~Insurance,data = medicalmalpractice2,ylab="log.Amount",xlab="Type.insurance")

table(medicalmalpractice2$Gender)
## 
##     0     1 
## 31440 47770
boxplot(log.Amount~Gender,data = medicalmalpractice2,ylab="log.Amount",xlab="Gender")

Continuous co-variate

Examine a scatter plot matrix of the continuous co-variate,to identify collinearities.

library(corrplot)
## Warning: package 'corrplot' was built under R version 4.3.3
corrplot.mixed(cor(medicalmalpractice2),lower = "number", upper = "circle",lower.col = "black")

Examine a scatter plot matrix of the continuous co-variate, to identify collinearities.

Basic Scatter plot Matrix

#library(graphics)
pairs(~log.Amount+Age,data = medicalmalpractice2,main="Simple Scatterplot Matrix")

Initial screening

Perform regressions of each co-variate singly.

Consider for inclusion in the multivariate regression:

  • co-variate for which p < 0.2;

  • any other co-variate which could be important, according to subject knowledge.

model.age = lm(log.Amount~Age,data = medicalmalpractice2)
summary(model.age)
## 
## Call:
## lm(formula = log.Amount ~ Age, data = medicalmalpractice2)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -4.2065 -0.6843  0.0801  0.5724  2.4862 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 11.760191   0.009227 1274.58   <2e-16 ***
## Age         -0.008615   0.000196  -43.95   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 1.093 on 79208 degrees of freedom
## Multiple R-squared:  0.02381,    Adjusted R-squared:  0.0238 
## F-statistic:  1932 on 1 and 79208 DF,  p-value: < 2.2e-16

Use factor() for categorical variable

Type III ANOVA - Anova(type = 3)

model.severity = lm(log.Amount~factor(Severity),data = medicalmalpractice2)
summary(model.severity)
## 
## Call:
## lm(formula = log.Amount ~ factor(Severity), data = medicalmalpractice2)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -4.6271 -0.6303  0.0658  0.7345  2.7113 
## 
## Coefficients:
##                     Estimate Std. Error t value Pr(>|t|)    
## (Intercept)       11.3712491  0.0391317 290.589  < 2e-16 ***
## factor(Severity)2 -0.0005892  0.0478667  -0.012     0.99    
## factor(Severity)3 -0.4280417  0.0395896 -10.812  < 2e-16 ***
## factor(Severity)4 -0.1842539  0.0399514  -4.612 4.00e-06 ***
## factor(Severity)5  0.2280030  0.0404623   5.635 1.76e-08 ***
## factor(Severity)6  0.7899040  0.0428137  18.450  < 2e-16 ***
## factor(Severity)7  0.8020122  0.0405716  19.768  < 2e-16 ***
## factor(Severity)8  0.6661950  0.0425682  15.650  < 2e-16 ***
## factor(Severity)9  0.2919843  0.0407750   7.161 8.09e-13 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 1.009 on 79201 degrees of freedom
## Multiple R-squared:  0.1678, Adjusted R-squared:  0.1677 
## F-statistic:  1996 on 8 and 79201 DF,  p-value: < 2.2e-16
model.severity.III <- car::Anova(model.severity, type = 3)
model.severity.III
## Anova Table (Type III tests)
## 
## Response: log.Amount
##                  Sum Sq    Df F value    Pr(>F)    
## (Intercept)       85988     1 84441.9 < 2.2e-16 ***
## factor(Severity)  16258     8  1995.7 < 2.2e-16 ***
## Residuals         80651 79201                      
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
model.private.attorney = lm(log.Amount~factor(Private.Attorney),data = medicalmalpractice2)
summary(model.private.attorney)
## 
## Call:
## lm(formula = log.Amount ~ factor(Private.Attorney), data = medicalmalpractice2)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -3.4439 -0.6418 -0.0537  0.7145  2.9302 
## 
## Coefficients:
##                            Estimate Std. Error t value Pr(>|t|)    
## (Intercept)               10.806557   0.006245  1730.6   <2e-16 ***
## factor(Private.Attorney)1  0.886291   0.007681   115.4   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 1.023 on 79208 degrees of freedom
## Multiple R-squared:  0.1439, Adjusted R-squared:  0.1439 
## F-statistic: 1.331e+04 on 1 and 79208 DF,  p-value: < 2.2e-16
model.private.attorney.III<- car::Anova(model.private.attorney, type = 3)
model.private.attorney.III
## Anova Table (Type III tests)
## 
## Response: log.Amount
##                           Sum Sq    Df F value    Pr(>F)    
## (Intercept)              3136873     1 2994827 < 2.2e-16 ***
## factor(Private.Attorney)   13945     1   13313 < 2.2e-16 ***
## Residuals                  82965 79208                      
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
model.maritial.status = lm(log.Amount~factor(Marital.Status),data = medicalmalpractice2)
summary(model.maritial.status)
## 
## Call:
## lm(formula = log.Amount ~ factor(Marital.Status), data = medicalmalpractice2)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -4.9308 -0.6734  0.0910  0.5634  2.8742 
## 
## Coefficients:
##                         Estimate Std. Error t value Pr(>|t|)    
## (Intercept)             12.34112    0.01727  714.63   <2e-16 ***
## factor(Marital.Status)1 -0.91874    0.01866  -49.23   <2e-16 ***
## factor(Marital.Status)2 -0.92033    0.01805  -50.98   <2e-16 ***
## factor(Marital.Status)3 -0.97289    0.03805  -25.57   <2e-16 ***
## factor(Marital.Status)4 -1.47694    0.02021  -73.07   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 1.069 on 79205 degrees of freedom
## Multiple R-squared:  0.06598,    Adjusted R-squared:  0.06594 
## F-statistic:  1399 on 4 and 79205 DF,  p-value: < 2.2e-16
model.maritial.status.III<- car::Anova(model.maritial.status, type = 3)
model.maritial.status.III
## Anova Table (Type III tests)
## 
## Response: log.Amount
##                        Sum Sq    Df  F value    Pr(>F)    
## (Intercept)            583626     1 510702.0 < 2.2e-16 ***
## factor(Marital.Status)   6395     4   1398.9 < 2.2e-16 ***
## Residuals               90515 79205                       
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
model.type.insurance = lm(log.Amount~factor(Insurance),data = medicalmalpractice2)
summary(model.type.insurance)
## 
## Call:
## lm(formula = log.Amount ~ factor(Insurance), data = medicalmalpractice2)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -4.4267 -0.6495  0.0495  0.7255  2.7174 
## 
## Coefficients:
##                    Estimate Std. Error  t value Pr(>|t|)    
## (Intercept)        11.05668    0.01003 1101.825  < 2e-16 ***
## factor(Insurance)1  0.22581    0.01542   14.648  < 2e-16 ***
## factor(Insurance)2  0.73392    0.01152   63.721  < 2e-16 ***
## factor(Insurance)3 -0.03708    0.01209   -3.066  0.00217 ** 
## factor(Insurance)4  0.25387    0.02555    9.937  < 2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 1.047 on 79205 degrees of freedom
## Multiple R-squared:  0.1044, Adjusted R-squared:  0.1043 
## F-statistic:  2308 on 4 and 79205 DF,  p-value: < 2.2e-16
model.type.insurance.III<- car::Anova(model.type.insurance, type = 3)
model.type.insurance.III
## Anova Table (Type III tests)
## 
## Response: log.Amount
##                    Sum Sq    Df   F value    Pr(>F)    
## (Intercept)       1330326     1 1214018.4 < 2.2e-16 ***
## factor(Insurance)   10116     4    2307.9 < 2.2e-16 ***
## Residuals           86793 79205                        
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
model.gender = lm(log.Amount~factor(Gender),data = medicalmalpractice2)
summary(model.gender)
## 
## Call:
## lm(formula = log.Amount ~ factor(Gender), data = medicalmalpractice2)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -4.1766 -0.6636  0.0527  0.5077  2.5719 
## 
## Coefficients:
##                  Estimate Std. Error t value Pr(>|t|)    
## (Intercept)     11.167147   0.006153 1815.05   <2e-16 ***
## factor(Gender)1  0.373334   0.007923   47.12   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 1.091 on 79208 degrees of freedom
## Multiple R-squared:  0.02727,    Adjusted R-squared:  0.02726 
## F-statistic:  2221 on 1 and 79208 DF,  p-value: < 2.2e-16
model.gender.III<- car::Anova(model.gender, type = 3)
model.gender.III
## Anova Table (Type III tests)
## 
## Response: log.Amount
##                 Sum Sq    Df   F value    Pr(>F)    
## (Intercept)    3920731     1 3294412.7 < 2.2e-16 ***
## factor(Gender)    2643     1    2220.6 < 2.2e-16 ***
## Residuals        94267 79208                        
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
model.specialty = lm(log.Amount~factor(Specialty),data = medicalmalpractice2)
summary(model.specialty)
## 
## Call:
## lm(formula = log.Amount ~ factor(Specialty), data = medicalmalpractice2)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -4.6653 -0.6125  0.0330  0.6106  2.6046 
## 
## Coefficients:
##                      Estimate Std. Error  t value Pr(>|t|)    
## (Intercept)         11.752570   0.009409 1249.026  < 2e-16 ***
## factor(Specialty)1   0.052545   0.014234    3.692 0.000223 ***
## factor(Specialty)2  -0.447982   0.021664  -20.679  < 2e-16 ***
## factor(Specialty)3   0.562448   0.028348   19.841  < 2e-16 ***
## factor(Specialty)4  -0.552005   0.016805  -32.849  < 2e-16 ***
## factor(Specialty)5  -1.221191   0.014300  -85.398  < 2e-16 ***
## factor(Specialty)6  -0.619976   0.017466  -35.496  < 2e-16 ***
## factor(Specialty)7  -0.692064   0.019909  -34.761  < 2e-16 ***
## factor(Specialty)8   0.228336   0.024250    9.416  < 2e-16 ***
## factor(Specialty)9  -0.430302   0.015092  -28.512  < 2e-16 ***
## factor(Specialty)10  0.107685   0.017386    6.194 5.91e-10 ***
## factor(Specialty)11 -0.413578   0.038537  -10.732  < 2e-16 ***
## factor(Specialty)12 -0.492518   0.024477  -20.122  < 2e-16 ***
## factor(Specialty)13 -0.417597   0.040167  -10.397  < 2e-16 ***
## factor(Specialty)14 -0.309552   0.014004  -22.105  < 2e-16 ***
## factor(Specialty)15 -1.434829   0.024498  -58.569  < 2e-16 ***
## factor(Specialty)16 -0.433296   0.038815  -11.163  < 2e-16 ***
## factor(Specialty)17 -0.380954   0.040812   -9.334  < 2e-16 ***
## factor(Specialty)18 -0.415773   0.028824  -14.424  < 2e-16 ***
## factor(Specialty)19  0.503631   0.028638   17.586  < 2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 1.006 on 79190 degrees of freedom
## Multiple R-squared:  0.1726, Adjusted R-squared:  0.1724 
## F-statistic: 869.6 on 19 and 79190 DF,  p-value: < 2.2e-16
model.specialty.III<-car::Anova(model.specialty,type = 3)
model.specialty.III
## Anova Table (Type III tests)
## 
## Response: log.Amount
##                    Sum Sq    Df    F value    Pr(>F)    
## (Intercept)       1579574     1 1560066.46 < 2.2e-16 ***
## factor(Specialty)   16729    19     869.61 < 2.2e-16 ***
## Residuals           80180 79190                         
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Variable selection

model1

model1 = lm(log.Amount~Age+factor(Severity)+factor(Private.Attorney)+
            factor(Marital.Status)+factor(Specialty)+factor(Insurance)+
              factor(Gender),data = medicalmalpractice2)
summary(model1)
## 
## Call:
## lm(formula = log.Amount ~ Age + factor(Severity) + factor(Private.Attorney) + 
##     factor(Marital.Status) + factor(Specialty) + factor(Insurance) + 
##     factor(Gender), data = medicalmalpractice2)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -4.9017 -0.5247  0.0813  0.5842  3.0767 
## 
## Coefficients:
##                             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)               11.2370540  0.0413480 271.768  < 2e-16 ***
## Age                       -0.0032889  0.0001739 -18.909  < 2e-16 ***
## factor(Severity)2         -0.0097951  0.0415787  -0.236   0.8138    
## factor(Severity)3          0.0779337  0.0347715   2.241   0.0250 *  
## factor(Severity)4          0.1460935  0.0349558   4.179 2.93e-05 ***
## factor(Severity)5          0.3074173  0.0352669   8.717  < 2e-16 ***
## factor(Severity)6          0.6573744  0.0372808  17.633  < 2e-16 ***
## factor(Severity)7          0.7738358  0.0356817  21.687  < 2e-16 ***
## factor(Severity)8          0.7024718  0.0372317  18.868  < 2e-16 ***
## factor(Severity)9          0.3680782  0.0355504  10.354  < 2e-16 ***
## factor(Private.Attorney)1  0.5782796  0.0088119  65.625  < 2e-16 ***
## factor(Marital.Status)1   -0.7112218  0.0163008 -43.631  < 2e-16 ***
## factor(Marital.Status)2   -0.6420651  0.0151951 -42.255  < 2e-16 ***
## factor(Marital.Status)3   -0.9084077  0.0318466 -28.525  < 2e-16 ***
## factor(Marital.Status)4   -0.7786859  0.0171209 -45.482  < 2e-16 ***
## factor(Specialty)1         0.1330818  0.0131103  10.151  < 2e-16 ***
## factor(Specialty)2        -0.4046167  0.0190074 -21.287  < 2e-16 ***
## factor(Specialty)3         0.3051253  0.0250067  12.202  < 2e-16 ***
## factor(Specialty)4        -0.1358548  0.0150662  -9.017  < 2e-16 ***
## factor(Specialty)5        -0.3831837  0.0140937 -27.188  < 2e-16 ***
## factor(Specialty)6        -0.1747114  0.0156465 -11.166  < 2e-16 ***
## factor(Specialty)7        -0.3223394  0.0177241 -18.187  < 2e-16 ***
## factor(Specialty)8         0.4965736  0.0218554  22.721  < 2e-16 ***
## factor(Specialty)9        -0.2663941  0.0132348 -20.128  < 2e-16 ***
## factor(Specialty)10        0.0654388  0.0152108   4.302 1.69e-05 ***
## factor(Specialty)11        0.1721087  0.0345701   4.979 6.42e-07 ***
## factor(Specialty)12       -0.2366024  0.0214669 -11.022  < 2e-16 ***
## factor(Specialty)13       -0.4387324  0.0351969 -12.465  < 2e-16 ***
## factor(Specialty)14       -0.1658826  0.0123499 -13.432  < 2e-16 ***
## factor(Specialty)15       -0.5545085  0.0223681 -24.790  < 2e-16 ***
## factor(Specialty)16        0.1487479  0.0348059   4.274 1.93e-05 ***
## factor(Specialty)17       -0.3911948  0.0357714 -10.936  < 2e-16 ***
## factor(Specialty)18       -0.1102876  0.0255993  -4.308 1.65e-05 ***
## factor(Specialty)19        0.6335575  0.0254137  24.930  < 2e-16 ***
## factor(Insurance)1         0.1913199  0.0129859  14.733  < 2e-16 ***
## factor(Insurance)2         0.4270579  0.0098674  43.280  < 2e-16 ***
## factor(Insurance)3         0.0713164  0.0105174   6.781 1.20e-11 ***
## factor(Insurance)4        -0.0438296  0.0215600  -2.033   0.0421 *  
## factor(Gender)1            0.2889563  0.0084051  34.379  < 2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.8764 on 79171 degrees of freedom
## Multiple R-squared:  0.3726, Adjusted R-squared:  0.3723 
## F-statistic:  1237 on 38 and 79171 DF,  p-value: < 2.2e-16
model1.III = car::Anova(model1, type = 3)
model1.III
## Anova Table (Type III tests)
## 
## Response: log.Amount
##                          Sum Sq    Df  F value    Pr(>F)    
## (Intercept)               56724     1 73857.83 < 2.2e-16 ***
## Age                         275     1   357.57 < 2.2e-16 ***
## factor(Severity)           4098     8   667.00 < 2.2e-16 ***
## factor(Private.Attorney)   3308     1  4306.62 < 2.2e-16 ***
## factor(Marital.Status)     1788     4   581.93 < 2.2e-16 ***
## factor(Specialty)          3981    19   272.81 < 2.2e-16 ***
## factor(Insurance)          2456     4   799.42 < 2.2e-16 ***
## factor(Gender)              908     1  1181.88 < 2.2e-16 ***
## Residuals                 60804 79171                       
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

After, we run a 1st model. We do a backward selection by eliminating variable which look the least promising, one at a time.

model2

  • remove gender

    model2 = lm(log.Amount~Age+factor(Severity)+factor(Private.Attorney)+
              factor(Marital.Status)+factor(Specialty)+factor(Insurance)
            ,data = medicalmalpractice2)
summary(model2)
    ## 
## Call:
## lm(formula = log.Amount ~ Age + factor(Severity) + factor(Private.Attorney) + 
##     factor(Marital.Status) + factor(Specialty) + factor(Insurance), 
##     data = medicalmalpractice2)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -4.7750 -0.5286  0.0763  0.5960  2.9935 
## 
## Coefficients:
##                             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)               11.4866526  0.0410080 280.108  < 2e-16 ***
## Age                       -0.0038593  0.0001744 -22.126  < 2e-16 ***
## factor(Severity)2         -0.0122239  0.0418876  -0.292  0.77042    
## factor(Severity)3         -0.0158017  0.0349220  -0.452  0.65092    
## factor(Severity)4          0.0635887  0.0351325   1.810  0.07030 .  
## factor(Severity)5          0.2520922  0.0354919   7.103 1.23e-12 ***
## factor(Severity)6          0.6025886  0.0375235  16.059  < 2e-16 ***
## factor(Severity)7          0.6217620  0.0356695  17.431  < 2e-16 ***
## factor(Severity)8          0.6229196  0.0374359  16.640  < 2e-16 ***
## factor(Severity)9          0.2954049  0.0357512   8.263  < 2e-16 ***
## factor(Private.Attorney)1  0.5616775  0.0088640  63.366  < 2e-16 ***
## factor(Marital.Status)1   -0.6110156  0.0161573 -37.817  < 2e-16 ***
## factor(Marital.Status)2   -0.5838802  0.0152128 -38.381  < 2e-16 ***
## factor(Marital.Status)3   -0.7600641  0.0317873 -23.911  < 2e-16 ***
## factor(Marital.Status)4   -0.7558942  0.0172351 -43.858  < 2e-16 ***
## factor(Specialty)1         0.2358909  0.0128595  18.344  < 2e-16 ***
## factor(Specialty)2        -0.3567711  0.0190972 -18.682  < 2e-16 ***
## factor(Specialty)3         0.4077837  0.0250123  16.303  < 2e-16 ***
## factor(Specialty)4        -0.1471467  0.0151745  -9.697  < 2e-16 ***
## factor(Specialty)5        -0.4084195  0.0141791 -28.804  < 2e-16 ***
## factor(Specialty)6        -0.2334260  0.0156686 -14.898  < 2e-16 ***
## factor(Specialty)7        -0.2697730  0.0177892 -15.165  < 2e-16 ***
## factor(Specialty)8         0.5286700  0.0219977  24.033  < 2e-16 ***
## factor(Specialty)9        -0.3051730  0.0132846 -22.972  < 2e-16 ***
## factor(Specialty)10        0.0439634  0.0153109   2.871  0.00409 ** 
## factor(Specialty)11        0.2775299  0.0346896   8.000 1.26e-15 ***
## factor(Specialty)12       -0.2045458  0.0216060  -9.467  < 2e-16 ***
## factor(Specialty)13       -0.3370353  0.0353329  -9.539  < 2e-16 ***
## factor(Specialty)14       -0.1240135  0.0123810 -10.016  < 2e-16 ***
## factor(Specialty)15       -0.6814787  0.0222250 -30.663  < 2e-16 ***
## factor(Specialty)16        0.2533479  0.0349303   7.253 4.11e-13 ***
## factor(Specialty)17       -0.2906494  0.0359165  -8.092 5.93e-16 ***
## factor(Specialty)18       -0.0094062  0.0256195  -0.367  0.71351    
## factor(Specialty)19        0.6315127  0.0256025  24.666  < 2e-16 ***
## factor(Insurance)1         0.2053427  0.0130759  15.704  < 2e-16 ***
## factor(Insurance)2         0.4082104  0.0099254  41.128  < 2e-16 ***
## factor(Insurance)3        -0.0080912  0.0103368  -0.783  0.43377    
## factor(Insurance)4        -0.0619223  0.0217137  -2.852  0.00435 ** 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.8829 on 79172 degrees of freedom
## Multiple R-squared:  0.3632, Adjusted R-squared:  0.3629 
## F-statistic:  1220 on 37 and 79172 DF,  p-value: < 2.2e-16
    model2.III = car::Anova(model2, type = 3)
model2.III
    ## Anova Table (Type III tests)
## 
## Response: log.Amount
##                          Sum Sq    Df  F value    Pr(>F)    
## (Intercept)               61157     1 78460.24 < 2.2e-16 ***
## Age                         382     1   489.56 < 2.2e-16 ***
## factor(Severity)           3772     8   604.89 < 2.2e-16 ***
## factor(Private.Attorney)   3130     1  4015.24 < 2.2e-16 ***
## factor(Marital.Status)     1558     4   499.54 < 2.2e-16 ***
## factor(Specialty)          5085    19   343.34 < 2.2e-16 ***
## factor(Insurance)          2860     4   917.18 < 2.2e-16 ***
## Residuals                 61712 79172                       
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

model3

  • remove Insurance

    model3 = lm(log.Amount~Age+factor(Severity)+factor(Private.Attorney)+
              factor(Marital.Status)+factor(Specialty),data = medicalmalpractice2)
summary(model3)
    ## 
## Call:
## lm(formula = log.Amount ~ Age + factor(Severity) + factor(Private.Attorney) + 
##     factor(Marital.Status) + factor(Specialty), data = medicalmalpractice2)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -4.6196 -0.5458  0.0628  0.5959  3.0439 
## 
## Coefficients:
##                             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)               11.7192219  0.0411559 284.752  < 2e-16 ***
## Age                       -0.0043400  0.0001782 -24.353  < 2e-16 ***
## factor(Severity)2         -0.0078127  0.0428443  -0.182  0.85531    
## factor(Severity)3         -0.0224104  0.0357029  -0.628  0.53021    
## factor(Severity)4          0.0678515  0.0359184   1.889  0.05889 .  
## factor(Severity)5          0.2776437  0.0362853   7.652 2.01e-14 ***
## factor(Severity)6          0.6626130  0.0383595  17.274  < 2e-16 ***
## factor(Severity)7          0.6732899  0.0364280  18.483  < 2e-16 ***
## factor(Severity)8          0.6731974  0.0382723  17.590  < 2e-16 ***
## factor(Severity)9          0.3154848  0.0365546   8.631  < 2e-16 ***
## factor(Private.Attorney)1  0.5851312  0.0089836  65.133  < 2e-16 ***
## factor(Marital.Status)1   -0.6470610  0.0164871 -39.247  < 2e-16 ***
## factor(Marital.Status)2   -0.6251202  0.0155384 -40.231  < 2e-16 ***
## factor(Marital.Status)3   -0.8050115  0.0324631 -24.798  < 2e-16 ***
## factor(Marital.Status)4   -0.8236961  0.0175881 -46.833  < 2e-16 ***
## factor(Specialty)1         0.2821569  0.0131084  21.525  < 2e-16 ***
## factor(Specialty)2        -0.3795534  0.0195218 -19.443  < 2e-16 ***
## factor(Specialty)3         0.4699420  0.0255446  18.397  < 2e-16 ***
## factor(Specialty)4        -0.1637169  0.0155165 -10.551  < 2e-16 ***
## factor(Specialty)5        -0.4544886  0.0144799 -31.387  < 2e-16 ***
## factor(Specialty)6        -0.2692053  0.0160134 -16.811  < 2e-16 ***
## factor(Specialty)7        -0.2944650  0.0181746 -16.202  < 2e-16 ***
## factor(Specialty)8         0.5831951  0.0224807  25.942  < 2e-16 ***
## factor(Specialty)9        -0.3386426  0.0135751 -24.946  < 2e-16 ***
## factor(Specialty)10        0.0414355  0.0156604   2.646  0.00815 ** 
## factor(Specialty)11        0.3279830  0.0354682   9.247  < 2e-16 ***
## factor(Specialty)12       -0.2224760  0.0220978 -10.068  < 2e-16 ***
## factor(Specialty)13       -0.3524514  0.0361226  -9.757  < 2e-16 ***
## factor(Specialty)14       -0.1280740  0.0126603 -10.116  < 2e-16 ***
## factor(Specialty)15       -0.7713810  0.0226737 -34.021  < 2e-16 ***
## factor(Specialty)16        0.2888289  0.0357216   8.086 6.27e-16 ***
## factor(Specialty)17       -0.3104816  0.0367211  -8.455  < 2e-16 ***
## factor(Specialty)18        0.0001662  0.0261904   0.006  0.99494    
## factor(Specialty)19        0.6910922  0.0261684  26.409  < 2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.9031 on 79176 degrees of freedom
## Multiple R-squared:  0.3337, Adjusted R-squared:  0.3334 
## F-statistic:  1202 on 33 and 79176 DF,  p-value: < 2.2e-16
    model3.III = car::Anova(model3, type = 3)
model3.III
    ## Anova Table (Type III tests)
## 
## Response: log.Amount
##                          Sum Sq    Df  F value    Pr(>F)    
## (Intercept)               66128     1 81083.84 < 2.2e-16 ***
## Age                         484     1   593.06 < 2.2e-16 ***
## factor(Severity)           4552     8   697.66 < 2.2e-16 ***
## factor(Private.Attorney)   3460     1  4242.35 < 2.2e-16 ***
## factor(Marital.Status)     1838     4   563.55 < 2.2e-16 ***
## factor(Specialty)          6584    19   424.93 < 2.2e-16 ***
## Residuals                 64572 79176                       
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

model4

  • remove Specialty

    model4 = lm(log.Amount~Age+factor(Severity)+factor(Private.Attorney)+
              factor(Marital.Status)
              ,data = medicalmalpractice2)
summary(model4)
    ## 
## Call:
## lm(formula = log.Amount ~ Age + factor(Severity) + factor(Private.Attorney) + 
##     factor(Marital.Status), data = medicalmalpractice2)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -4.7971 -0.5872  0.0325  0.6389  2.9534 
## 
## Coefficients:
##                             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)               11.7552018  0.0417869 281.313   <2e-16 ***
## Age                       -0.0059086  0.0001859 -31.791   <2e-16 ***
## factor(Severity)2         -0.0019601  0.0449648  -0.044   0.9652    
## factor(Severity)3         -0.0682039  0.0374458  -1.821   0.0685 .  
## factor(Severity)4          0.0635747  0.0376780   1.687   0.0915 .  
## factor(Severity)5          0.3417958  0.0380594   8.981   <2e-16 ***
## factor(Severity)6          0.7425748  0.0402323  18.457   <2e-16 ***
## factor(Severity)7          0.7508217  0.0381547  19.678   <2e-16 ***
## factor(Severity)8          0.7712936  0.0401200  19.225   <2e-16 ***
## factor(Severity)9          0.3501579  0.0383397   9.133   <2e-16 ***
## factor(Private.Attorney)1  0.5768932  0.0079917  72.186   <2e-16 ***
## factor(Marital.Status)1   -0.7184381  0.0170655 -42.099   <2e-16 ***
## factor(Marital.Status)2   -0.6930925  0.0161698 -42.863   <2e-16 ***
## factor(Marital.Status)3   -0.8844600  0.0339471 -26.054   <2e-16 ***
## factor(Marital.Status)4   -0.9811496  0.0182828 -53.665   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.9479 on 79195 degrees of freedom
## Multiple R-squared:  0.2657, Adjusted R-squared:  0.2656 
## F-statistic:  2047 on 14 and 79195 DF,  p-value: < 2.2e-16
    model4.III = car::Anova(model4, type = 3)
model4.III
    ## Anova Table (Type III tests)
## 
## Response: log.Amount
##                          Sum Sq    Df  F value    Pr(>F)    
## (Intercept)               71104     1 79137.14 < 2.2e-16 ***
## Age                         908     1  1010.64 < 2.2e-16 ***
## factor(Severity)           6520     8   907.12 < 2.2e-16 ***
## factor(Private.Attorney)   4682     1  5210.88 < 2.2e-16 ***
## factor(Marital.Status)     2631     4   732.07 < 2.2e-16 ***
## Residuals                 71156 79195                       
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

model5

  • remove Marital Status

    model5 = lm(log.Amount~Age+factor(Severity)+factor(Private.Attorney),data = medicalmalpractice2)
summary(model5)
    ## 
## Call:
## lm(formula = log.Amount ~ Age + factor(Severity) + factor(Private.Attorney), 
##     data = medicalmalpractice2)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -4.1546 -0.6050  0.0171  0.6598  2.9456 
## 
## Coefficients:
##                             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)               11.0019939  0.0390354 281.847   <2e-16 ***
## Age                       -0.0059210  0.0001743 -33.966   <2e-16 ***
## factor(Severity)2          0.0052617  0.0457854   0.115   0.9085    
## factor(Severity)3         -0.0703808  0.0381231  -1.846   0.0649 .  
## factor(Severity)4          0.0832507  0.0383566   2.170   0.0300 *  
## factor(Severity)5          0.3579685  0.0387471   9.239   <2e-16 ***
## factor(Severity)6          0.7950473  0.0409533  19.414   <2e-16 ***
## factor(Severity)7          0.8138196  0.0388078  20.971   <2e-16 ***
## factor(Severity)8          0.8693059  0.0407981  21.308   <2e-16 ***
## factor(Severity)9          0.3975908  0.0390227  10.189   <2e-16 ***
## factor(Private.Attorney)1  0.6119364  0.0080088  76.408   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.9652 on 79199 degrees of freedom
## Multiple R-squared:  0.2386, Adjusted R-squared:  0.2385 
## F-statistic:  2482 on 10 and 79199 DF,  p-value: < 2.2e-16
    model5.III = car::Anova(model5, type = 3)
model5.III
    ## Anova Table (Type III tests)
## 
## Response: log.Amount
##                          Sum Sq    Df F value    Pr(>F)    
## (Intercept)               74009     1 79437.5 < 2.2e-16 ***
## Age                        1075     1  1153.7 < 2.2e-16 ***
## factor(Severity)           7810     8  1047.9 < 2.2e-16 ***
## factor(Private.Attorney)   5439     1  5838.2 < 2.2e-16 ***
## Residuals                 73787 79199                      
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

    model6

  • remove Private Attorney

    model6 = lm(log.Amount~Age+factor(Severity),data = medicalmalpractice2)
summary(model6)
    ## 
## Call:
## lm(formula = log.Amount ~ Age + factor(Severity), data = medicalmalpractice2)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -4.5803 -0.6229  0.0567  0.7194  2.7514 
## 
## Coefficients:
##                     Estimate Std. Error t value Pr(>|t|)    
## (Intercept)       11.6500486  0.0394820 295.072  < 2e-16 ***
## Age               -0.0068022  0.0001802 -37.741  < 2e-16 ***
## factor(Severity)2  0.0061325  0.0474427   0.129    0.897    
## factor(Severity)3 -0.4033570  0.0392441 -10.278  < 2e-16 ***
## factor(Severity)4 -0.1669751  0.0395998  -4.217 2.48e-05 ***
## factor(Severity)5  0.2176989  0.0401045   5.428 5.71e-08 ***
## factor(Severity)6  0.7791488  0.0424351  18.361  < 2e-16 ***
## factor(Severity)7  0.7994931  0.0402120  19.882  < 2e-16 ***
## factor(Severity)8  0.6733387  0.0421912  15.959  < 2e-16 ***
## factor(Severity)9  0.3020775  0.0404144   7.475 7.83e-14 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 1 on 79200 degrees of freedom
## Multiple R-squared:  0.1825, Adjusted R-squared:  0.1824 
## F-statistic:  1964 on 9 and 79200 DF,  p-value: < 2.2e-16
    model6.III = car::Anova(model5, type = 3)
model6.III
    ## Anova Table (Type III tests)
## 
## Response: log.Amount
##                          Sum Sq    Df F value    Pr(>F)    
## (Intercept)               74009     1 79437.5 < 2.2e-16 ***
## Age                        1075     1  1153.7 < 2.2e-16 ***
## factor(Severity)           7810     8  1047.9 < 2.2e-16 ***
## factor(Private.Attorney)   5439     1  5838.2 < 2.2e-16 ***
## Residuals                 73787 79199                      
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

    model7

  • remove Severity

  • similar to model age

    model7 = lm(log.Amount~Age,data = medicalmalpractice2)
summary(model7)
    ## 
## Call:
## lm(formula = log.Amount ~ Age, data = medicalmalpractice2)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -4.2065 -0.6843  0.0801  0.5724  2.4862 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 11.760191   0.009227 1274.58   <2e-16 ***
## Age         -0.008615   0.000196  -43.95   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 1.093 on 79208 degrees of freedom
## Multiple R-squared:  0.02381,    Adjusted R-squared:  0.0238 
## F-statistic:  1932 on 1 and 79208 DF,  p-value: < 2.2e-16
    model7.III = car::Anova(model7, type = 3)
model7.III
    ## Anova Table (Type III tests)
## 
## Response: log.Amount
##              Sum Sq    Df F value    Pr(>F)    
## (Intercept) 1940279     1 1624552 < 2.2e-16 ***
## Age            2307     1    1932 < 2.2e-16 ***
## Residuals     94602 79208                      
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

From box plot 2 to 6, it can be seen that when reducing one variable at a time, the Adjusted R-squared value will decrease accordingly.

After,we do backward selection. We try to reduce only one variable on every model.

model8

  • remove Private.Attorney

    model8 = lm(log.Amount~Age+factor(Severity)+
              factor(Marital.Status)+factor(Specialty)+factor(Insurance)+
              factor(Gender),data = medicalmalpractice2)
summary(model8)
    ## 
## Call:
## lm(formula = log.Amount ~ Age + factor(Severity) + factor(Marital.Status) + 
##     factor(Specialty) + factor(Insurance) + factor(Gender), data = medicalmalpractice2)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -5.3630 -0.5307  0.0850  0.6044  3.0868 
## 
## Coefficients:
##                           Estimate Std. Error t value Pr(>|t|)    
## (Intercept)             11.9537187  0.0409498 291.912  < 2e-16 ***
## Age                     -0.0039958  0.0001783 -22.417  < 2e-16 ***
## factor(Severity)2       -0.0126418  0.0426943  -0.296  0.76715    
## factor(Severity)3       -0.1422028  0.0355380  -4.001 6.30e-05 ***
## factor(Severity)4       -0.0260597  0.0357926  -0.728  0.46657    
## factor(Severity)5        0.2042260  0.0361771   5.645 1.66e-08 ***
## factor(Severity)6        0.6421999  0.0382804  16.776  < 2e-16 ***
## factor(Severity)7        0.7408774  0.0366354  20.223  < 2e-16 ***
## factor(Severity)8        0.5660483  0.0381711  14.829  < 2e-16 ***
## factor(Severity)9        0.2925145  0.0364851   8.017 1.09e-15 ***
## factor(Marital.Status)1 -0.7386569  0.0167327 -44.145  < 2e-16 ***
## factor(Marital.Status)2 -0.6426351  0.0156028 -41.187  < 2e-16 ***
## factor(Marital.Status)3 -0.7847173  0.0326437 -24.039  < 2e-16 ***
## factor(Marital.Status)4 -0.8486884  0.0175461 -48.369  < 2e-16 ***
## factor(Specialty)1      -0.0152752  0.0132604  -1.152  0.24935    
## factor(Specialty)2      -0.3459458  0.0194958 -17.745  < 2e-16 ***
## factor(Specialty)3       0.3368405  0.0256729  13.120  < 2e-16 ***
## factor(Specialty)4      -0.3344122  0.0151552 -22.066  < 2e-16 ***
## factor(Specialty)5      -0.7340793  0.0133899 -54.823  < 2e-16 ***
## factor(Specialty)6      -0.3431824  0.0158486 -21.654  < 2e-16 ***
## factor(Specialty)7      -0.4639409  0.0180643 -25.683  < 2e-16 ***
## factor(Specialty)8       0.1694616  0.0218504   7.756 8.90e-15 ***
## factor(Specialty)9      -0.2724202  0.0135896 -20.046  < 2e-16 ***
## factor(Specialty)10      0.0782914  0.0156177   5.013 5.37e-07 ***
## factor(Specialty)11     -0.2798177  0.0347862  -8.044 8.82e-16 ***
## factor(Specialty)12     -0.3564365  0.0219630 -16.229  < 2e-16 ***
## factor(Specialty)13     -0.3766949  0.0361282 -10.427  < 2e-16 ***
## factor(Specialty)14     -0.2309076  0.0126404 -18.267  < 2e-16 ***
## factor(Specialty)15     -0.8086908  0.0226213 -35.749  < 2e-16 ***
## factor(Specialty)16     -0.3065071  0.0350227  -8.752  < 2e-16 ***
## factor(Specialty)17     -0.3343846  0.0367204  -9.106  < 2e-16 ***
## factor(Specialty)18     -0.3175029  0.0260854 -12.172  < 2e-16 ***
## factor(Specialty)19      0.3548371  0.0257286  13.792  < 2e-16 ***
## factor(Insurance)1       0.1427332  0.0133126  10.722  < 2e-16 ***
## factor(Insurance)2       0.4415222  0.0101296  43.587  < 2e-16 ***
## factor(Insurance)3       0.0431195  0.0107906   3.996 6.45e-05 ***
## factor(Insurance)4       0.0666996  0.0220708   3.022  0.00251 ** 
## factor(Gender)1          0.2587275  0.0086177  30.023  < 2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.8999 on 79172 degrees of freedom
## Multiple R-squared:  0.3384, Adjusted R-squared:  0.3381 
## F-statistic:  1095 on 37 and 79172 DF,  p-value: < 2.2e-16
    model8.III = car::Anova(model8, type = 3)
model8.III
    ## Anova Table (Type III tests)
## 
## Response: log.Amount
##                        Sum Sq    Df  F value    Pr(>F)    
## (Intercept)             69003     1 85212.38 < 2.2e-16 ***
## Age                       407     1   502.50 < 2.2e-16 ***
## factor(Severity)         6687     8  1032.22 < 2.2e-16 ***
## factor(Marital.Status)   2024     4   624.90 < 2.2e-16 ***
## factor(Specialty)        5104    19   331.76 < 2.2e-16 ***
## factor(Insurance)        2798     4   863.95 < 2.2e-16 ***
## factor(Gender)            730     1   901.37 < 2.2e-16 ***
## Residuals               64112 79172                       
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

model9

  • remove Severity

    model9 = lm(log.Amount~Age+factor(Private.Attorney)+
          factor(Marital.Status)+factor(Specialty)+factor(Insurance)+
          factor(Gender),data = medicalmalpractice2)
summary(model9)
    ## 
## Call:
## lm(formula = log.Amount ~ Age + factor(Private.Attorney) + factor(Marital.Status) + 
##     factor(Specialty) + factor(Insurance) + factor(Gender), data = medicalmalpractice2)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -4.5790 -0.5496  0.0589  0.5816  3.3033 
## 
## Coefficients:
##                             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)               11.5540153  0.0229902 502.563  < 2e-16 ***
## Age                       -0.0038640  0.0001794 -21.536  < 2e-16 ***
## factor(Private.Attorney)1  0.7308660  0.0086173  84.814  < 2e-16 ***
## factor(Marital.Status)1   -0.8102968  0.0167337 -48.423  < 2e-16 ***
## factor(Marital.Status)2   -0.7340057  0.0156050 -47.036  < 2e-16 ***
## factor(Marital.Status)3   -1.0441128  0.0328381 -31.796  < 2e-16 ***
## factor(Marital.Status)4   -0.8945727  0.0175808 -50.884  < 2e-16 ***
## factor(Specialty)1         0.1617523  0.0135309  11.954  < 2e-16 ***
## factor(Specialty)2        -0.4697343  0.0196083 -23.956  < 2e-16 ***
## factor(Specialty)3         0.3594479  0.0257914  13.937  < 2e-16 ***
## factor(Specialty)4        -0.1645730  0.0155513 -10.583  < 2e-16 ***
## factor(Specialty)5        -0.4467280  0.0145201 -30.766  < 2e-16 ***
## factor(Specialty)6        -0.2075833  0.0161500 -12.853  < 2e-16 ***
## factor(Specialty)7        -0.3825499  0.0182813 -20.926  < 2e-16 ***
## factor(Specialty)8         0.5553710  0.0225416  24.638  < 2e-16 ***
## factor(Specialty)9        -0.3058662  0.0136583 -22.394  < 2e-16 ***
## factor(Specialty)10        0.0645746  0.0156888   4.116 3.86e-05 ***
## factor(Specialty)11        0.1749780  0.0357094   4.900 9.60e-07 ***
## factor(Specialty)12       -0.2707255  0.0221694 -12.212  < 2e-16 ***
## factor(Specialty)13       -0.4995212  0.0363441 -13.744  < 2e-16 ***
## factor(Specialty)14       -0.1901972  0.0127528 -14.914  < 2e-16 ***
## factor(Specialty)15       -0.6515576  0.0230586 -28.257  < 2e-16 ***
## factor(Specialty)16        0.1722823  0.0359492   4.792 1.65e-06 ***
## factor(Specialty)17       -0.4643031  0.0369247 -12.574  < 2e-16 ***
## factor(Specialty)18       -0.1365591  0.0264394  -5.165 2.41e-07 ***
## factor(Specialty)19        0.7275559  0.0261996  27.770  < 2e-16 ***
## factor(Insurance)1         0.2261872  0.0134018  16.877  < 2e-16 ***
## factor(Insurance)2         0.4936135  0.0101435  48.663  < 2e-16 ***
## factor(Insurance)3         0.0826797  0.0108568   7.615 2.66e-14 ***
## factor(Insurance)4        -0.0438233  0.0222716  -1.968   0.0491 *  
## factor(Gender)1            0.2232372  0.0083814  26.635  < 2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.9054 on 79179 degrees of freedom
## Multiple R-squared:  0.3303, Adjusted R-squared:   0.33 
## F-statistic:  1302 on 30 and 79179 DF,  p-value: < 2.2e-16
    model9.III = car::Anova(model9, type = 3)
model9.III
    ## Anova Table (Type III tests)
## 
## Response: log.Amount
##                          Sum Sq    Df   F value    Pr(>F)    
## (Intercept)              207029     1 252569.71 < 2.2e-16 ***
## Age                         380     1    463.81 < 2.2e-16 ***
## factor(Private.Attorney)   5896     1   7193.41 < 2.2e-16 ***
## factor(Marital.Status)     2379     4    725.57 < 2.2e-16 ***
## factor(Specialty)          5458    19    350.43 < 2.2e-16 ***
## factor(Insurance)          3316     4   1011.31 < 2.2e-16 ***
## factor(Gender)              582     1    709.42 < 2.2e-16 ***
## Residuals                 64902 79179                        
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

model10

  • remove Age

    model10 = lm(log.Amount~factor(Severity)+factor(Private.Attorney)+
          factor(Marital.Status)+factor(Specialty)+factor(Insurance)+
          factor(Gender),data = medicalmalpractice2)
summary(model10)
    ## 
## Call:
## lm(formula = log.Amount ~ factor(Severity) + factor(Private.Attorney) + 
##     factor(Marital.Status) + factor(Specialty) + factor(Insurance) + 
##     factor(Gender), data = medicalmalpractice2)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -4.9321 -0.5269  0.0806  0.5853  3.1125 
## 
## Coefficients:
##                            Estimate Std. Error t value Pr(>|t|)    
## (Intercept)               11.062129   0.040391 273.879  < 2e-16 ***
## factor(Severity)2         -0.014854   0.041671  -0.356    0.721    
## factor(Severity)3          0.081090   0.034849   2.327    0.020 *  
## factor(Severity)4          0.150509   0.035034   4.296 1.74e-05 ***
## factor(Severity)5          0.314983   0.035344   8.912  < 2e-16 ***
## factor(Severity)6          0.663100   0.037363  17.747  < 2e-16 ***
## factor(Severity)7          0.787120   0.035755  22.014  < 2e-16 ***
## factor(Severity)8          0.709937   0.037313  19.026  < 2e-16 ***
## factor(Severity)9          0.372402   0.035630  10.452  < 2e-16 ***
## factor(Private.Attorney)1  0.588599   0.008815  66.774  < 2e-16 ***
## factor(Marital.Status)1   -0.662038   0.016128 -41.049  < 2e-16 ***
## factor(Marital.Status)2   -0.643811   0.015229 -42.275  < 2e-16 ***
## factor(Marital.Status)3   -0.923352   0.031908 -28.938  < 2e-16 ***
## factor(Marital.Status)4   -0.780304   0.017159 -45.474  < 2e-16 ***
## factor(Specialty)1         0.133842   0.013140  10.186  < 2e-16 ***
## factor(Specialty)2        -0.413883   0.019044 -21.733  < 2e-16 ***
## factor(Specialty)3         0.306106   0.025063  12.213  < 2e-16 ***
## factor(Specialty)4        -0.139792   0.015099  -9.259  < 2e-16 ***
## factor(Specialty)5        -0.395564   0.014110 -28.034  < 2e-16 ***
## factor(Specialty)6        -0.181164   0.015678 -11.555  < 2e-16 ***
## factor(Specialty)7        -0.331067   0.017758 -18.643  < 2e-16 ***
## factor(Specialty)8         0.498243   0.021904  22.746  < 2e-16 ***
## factor(Specialty)9        -0.271327   0.013262 -20.459  < 2e-16 ***
## factor(Specialty)10        0.062654   0.015244   4.110 3.96e-05 ***
## factor(Specialty)11        0.175370   0.034647   5.062 4.17e-07 ***
## factor(Specialty)12       -0.236328   0.021515 -10.984  < 2e-16 ***
## factor(Specialty)13       -0.449145   0.035272 -12.734  < 2e-16 ***
## factor(Specialty)14       -0.167283   0.012377 -13.515  < 2e-16 ***
## factor(Specialty)15       -0.565370   0.022411 -25.227  < 2e-16 ***
## factor(Specialty)16        0.150438   0.034884   4.313 1.62e-05 ***
## factor(Specialty)17       -0.401374   0.035848 -11.197  < 2e-16 ***
## factor(Specialty)18       -0.112169   0.025657  -4.372 1.23e-05 ***
## factor(Specialty)19        0.635291   0.025471  24.942  < 2e-16 ***
## factor(Insurance)1         0.193104   0.013015  14.837  < 2e-16 ***
## factor(Insurance)2         0.433524   0.009884  43.863  < 2e-16 ***
## factor(Insurance)3         0.074508   0.010540   7.069 1.57e-12 ***
## factor(Insurance)4        -0.039876   0.021607  -1.845    0.065 .  
## factor(Gender)1            0.304117   0.008386  36.266  < 2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.8783 on 79172 degrees of freedom
## Multiple R-squared:  0.3697, Adjusted R-squared:  0.3694 
## F-statistic:  1255 on 37 and 79172 DF,  p-value: < 2.2e-16
    model10.III = car::Anova(model10, type = 3)
model10.III
    ## Anova Table (Type III tests)
## 
## Response: log.Amount
##                          Sum Sq    Df  F value    Pr(>F)    
## (Intercept)               57868     1 75009.87 < 2.2e-16 ***
## factor(Severity)           4204     8   681.12 < 2.2e-16 ***
## factor(Private.Attorney)   3440     1  4458.80 < 2.2e-16 ***
## factor(Marital.Status)     1743     4   564.72 < 2.2e-16 ***
## factor(Specialty)          4116    19   280.78 < 2.2e-16 ***
## factor(Insurance)          2516     4   815.29 < 2.2e-16 ***
## factor(Gender)             1015     1  1315.25 < 2.2e-16 ***
## Residuals                 61079 79172                       
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

model11

  • remove Insurance

    model11 = lm(log.Amount~Age+factor(Severity)+factor(Private.Attorney)+
          factor(Marital.Status)+factor(Specialty)
        ,data = medicalmalpractice2)
summary(model11)
    ## 
## Call:
## lm(formula = log.Amount ~ Age + factor(Severity) + factor(Private.Attorney) + 
##     factor(Marital.Status) + factor(Specialty), data = medicalmalpractice2)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -4.6196 -0.5458  0.0628  0.5959  3.0439 
## 
## Coefficients:
##                             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)               11.7192219  0.0411559 284.752  < 2e-16 ***
## Age                       -0.0043400  0.0001782 -24.353  < 2e-16 ***
## factor(Severity)2         -0.0078127  0.0428443  -0.182  0.85531    
## factor(Severity)3         -0.0224104  0.0357029  -0.628  0.53021    
## factor(Severity)4          0.0678515  0.0359184   1.889  0.05889 .  
## factor(Severity)5          0.2776437  0.0362853   7.652 2.01e-14 ***
## factor(Severity)6          0.6626130  0.0383595  17.274  < 2e-16 ***
## factor(Severity)7          0.6732899  0.0364280  18.483  < 2e-16 ***
## factor(Severity)8          0.6731974  0.0382723  17.590  < 2e-16 ***
## factor(Severity)9          0.3154848  0.0365546   8.631  < 2e-16 ***
## factor(Private.Attorney)1  0.5851312  0.0089836  65.133  < 2e-16 ***
## factor(Marital.Status)1   -0.6470610  0.0164871 -39.247  < 2e-16 ***
## factor(Marital.Status)2   -0.6251202  0.0155384 -40.231  < 2e-16 ***
## factor(Marital.Status)3   -0.8050115  0.0324631 -24.798  < 2e-16 ***
## factor(Marital.Status)4   -0.8236961  0.0175881 -46.833  < 2e-16 ***
## factor(Specialty)1         0.2821569  0.0131084  21.525  < 2e-16 ***
## factor(Specialty)2        -0.3795534  0.0195218 -19.443  < 2e-16 ***
## factor(Specialty)3         0.4699420  0.0255446  18.397  < 2e-16 ***
## factor(Specialty)4        -0.1637169  0.0155165 -10.551  < 2e-16 ***
## factor(Specialty)5        -0.4544886  0.0144799 -31.387  < 2e-16 ***
## factor(Specialty)6        -0.2692053  0.0160134 -16.811  < 2e-16 ***
## factor(Specialty)7        -0.2944650  0.0181746 -16.202  < 2e-16 ***
## factor(Specialty)8         0.5831951  0.0224807  25.942  < 2e-16 ***
## factor(Specialty)9        -0.3386426  0.0135751 -24.946  < 2e-16 ***
## factor(Specialty)10        0.0414355  0.0156604   2.646  0.00815 ** 
## factor(Specialty)11        0.3279830  0.0354682   9.247  < 2e-16 ***
## factor(Specialty)12       -0.2224760  0.0220978 -10.068  < 2e-16 ***
## factor(Specialty)13       -0.3524514  0.0361226  -9.757  < 2e-16 ***
## factor(Specialty)14       -0.1280740  0.0126603 -10.116  < 2e-16 ***
## factor(Specialty)15       -0.7713810  0.0226737 -34.021  < 2e-16 ***
## factor(Specialty)16        0.2888289  0.0357216   8.086 6.27e-16 ***
## factor(Specialty)17       -0.3104816  0.0367211  -8.455  < 2e-16 ***
## factor(Specialty)18        0.0001662  0.0261904   0.006  0.99494    
## factor(Specialty)19        0.6910922  0.0261684  26.409  < 2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.9031 on 79176 degrees of freedom
## Multiple R-squared:  0.3337, Adjusted R-squared:  0.3334 
## F-statistic:  1202 on 33 and 79176 DF,  p-value: < 2.2e-16
    model11.III = car::Anova(model11, type = 3)
model11.III
    ## Anova Table (Type III tests)
## 
## Response: log.Amount
##                          Sum Sq    Df  F value    Pr(>F)    
## (Intercept)               66128     1 81083.84 < 2.2e-16 ***
## Age                         484     1   593.06 < 2.2e-16 ***
## factor(Severity)           4552     8   697.66 < 2.2e-16 ***
## factor(Private.Attorney)   3460     1  4242.35 < 2.2e-16 ***
## factor(Marital.Status)     1838     4   563.55 < 2.2e-16 ***
## factor(Specialty)          6584    19   424.93 < 2.2e-16 ***
## Residuals                 64572 79176                       
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

model12

  • remove Specialty

    model12 = lm(log.Amount~Age+factor(Severity)+factor(Private.Attorney)+
           factor(Marital.Status)
         ,data = medicalmalpractice2)
summary(model12)
    ## 
## Call:
## lm(formula = log.Amount ~ Age + factor(Severity) + factor(Private.Attorney) + 
##     factor(Marital.Status), data = medicalmalpractice2)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -4.7971 -0.5872  0.0325  0.6389  2.9534 
## 
## Coefficients:
##                             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)               11.7552018  0.0417869 281.313   <2e-16 ***
## Age                       -0.0059086  0.0001859 -31.791   <2e-16 ***
## factor(Severity)2         -0.0019601  0.0449648  -0.044   0.9652    
## factor(Severity)3         -0.0682039  0.0374458  -1.821   0.0685 .  
## factor(Severity)4          0.0635747  0.0376780   1.687   0.0915 .  
## factor(Severity)5          0.3417958  0.0380594   8.981   <2e-16 ***
## factor(Severity)6          0.7425748  0.0402323  18.457   <2e-16 ***
## factor(Severity)7          0.7508217  0.0381547  19.678   <2e-16 ***
## factor(Severity)8          0.7712936  0.0401200  19.225   <2e-16 ***
## factor(Severity)9          0.3501579  0.0383397   9.133   <2e-16 ***
## factor(Private.Attorney)1  0.5768932  0.0079917  72.186   <2e-16 ***
## factor(Marital.Status)1   -0.7184381  0.0170655 -42.099   <2e-16 ***
## factor(Marital.Status)2   -0.6930925  0.0161698 -42.863   <2e-16 ***
## factor(Marital.Status)3   -0.8844600  0.0339471 -26.054   <2e-16 ***
## factor(Marital.Status)4   -0.9811496  0.0182828 -53.665   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.9479 on 79195 degrees of freedom
## Multiple R-squared:  0.2657, Adjusted R-squared:  0.2656 
## F-statistic:  2047 on 14 and 79195 DF,  p-value: < 2.2e-16
    model12.III = car::Anova(model12, type = 3)
model12.III
    ## Anova Table (Type III tests)
## 
## Response: log.Amount
##                          Sum Sq    Df  F value    Pr(>F)    
## (Intercept)               71104     1 79137.14 < 2.2e-16 ***
## Age                         908     1  1010.64 < 2.2e-16 ***
## factor(Severity)           6520     8   907.12 < 2.2e-16 ***
## factor(Private.Attorney)   4682     1  5210.88 < 2.2e-16 ***
## factor(Marital.Status)     2631     4   732.07 < 2.2e-16 ***
## Residuals                 71156 79195                       
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Diagnostics