Introduction

On this chapter ,we will use R-Programming Language in order to conduct inferential analysis to analyse the data as much as we can to help decision making

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Data Set and Data types view

Data Set (1st five rows) :

veh_value	exposure	veh_body	veh_age	gender	area	agecat
1.06	0.3039014	HBACK	3	F	C	2
1.03	0.6488706	HBACK	2	F	A	4
3.26	0.5694730	UTE	2	F	E	2
4.14	0.3175907	STNWG	2	F	D	2
0.72	0.6488706	HBACK	4	F	C	2
2.01	0.8542094	HDTOP	3	M	C	4

data set dimensions

lets have a look at the data set dimension :

## [1] 67856    10

The Data set has 10 columns and 67856 rows

data set variables types

lets have a look at the variables types:

## Rows: 67,856
## Columns: 10
## $ veh_value <dbl> 1.06, 1.03, 3.26, 4.14, 0.72, 2.01, 1.60, 1.47, 0.52, 0.38, …
## $ exposure  <dbl> 0.30390144, 0.64887064, 0.56947296, 0.31759069, 0.64887064, …
## $ clm       <dbl> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, …
## $ numclaims <dbl> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, …
## $ claimcst0 <dbl> 0.0000, 0.0000, 0.0000, 0.0000, 0.0000, 0.0000, 0.0000, 0.00…
## $ veh_body  <chr> "HBACK", "HBACK", "UTE", "STNWG", "HBACK", "HDTOP", "PANVN",…
## $ veh_age   <dbl> 3, 2, 2, 2, 4, 3, 3, 2, 4, 4, 2, 3, 2, 1, 3, 2, 3, 3, 4, 3, …
## $ gender    <chr> "F", "F", "F", "F", "F", "M", "M", "M", "F", "F", "M", "M", …
## $ area      <chr> "C", "A", "E", "D", "C", "C", "A", "B", "A", "B", "A", "C", …
## $ agecat    <dbl> 2, 4, 2, 2, 2, 4, 4, 6, 3, 4, 2, 4, 4, 5, 6, 4, 4, 4, 2, 3, …

now we will make some data transformation,will do the below :

convert agecat ,veh_age,veh_body ,gender and area types into factor

lets look again to the variables types after datatype conversion :

## Rows: 67,856
## Columns: 10
## $ veh_value <dbl> 1.06, 1.03, 3.26, 4.14, 0.72, 2.01, 1.60, 1.47, 0.52, 0.38, …
## $ exposure  <dbl> 0.30390144, 0.64887064, 0.56947296, 0.31759069, 0.64887064, …
## $ clm       <fct> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, …
## $ numclaims <fct> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, …
## $ claimcst0 <dbl> 0.0000, 0.0000, 0.0000, 0.0000, 0.0000, 0.0000, 0.0000, 0.00…
## $ veh_body  <fct> HBACK, HBACK, UTE, STNWG, HBACK, HDTOP, PANVN, HBACK, HBACK,…
## $ veh_age   <fct> 3, 2, 2, 2, 4, 3, 3, 2, 4, 4, 2, 3, 2, 1, 3, 2, 3, 3, 4, 3, …
## $ gender    <fct> F, F, F, F, F, M, M, M, F, F, M, M, F, M, M, M, F, M, F, F, …
## $ area      <fct> C, A, E, D, C, C, A, B, A, B, A, C, C, A, B, C, F, C, D, C, …
## $ agecat    <fct> 2, 4, 2, 2, 2, 4, 4, 6, 3, 4, 2, 4, 4, 5, 6, 4, 4, 4, 2, 3, …

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Data summary

##    veh_value         exposure        clm       numclaims   claimcst0      
##  Min.   : 0.000   Min.   :0.002738   0:63232   0:63232   Min.   :    0.0  
##  1st Qu.: 1.010   1st Qu.:0.219028   1: 4624   1: 4333   1st Qu.:    0.0  
##  Median : 1.500   Median :0.446270             2:  271   Median :    0.0  
##  Mean   : 1.777   Mean   :0.468651             3:   18   Mean   :  137.3  
##  3rd Qu.: 2.150   3rd Qu.:0.709103             4:    2   3rd Qu.:    0.0  
##  Max.   :34.560   Max.   :0.999316                       Max.   :55922.1  
##                                                                           
##     veh_body     veh_age   gender    area      agecat   
##  SEDAN  :22233   1:12257   F:38603   A:16312   1: 5742  
##  HBACK  :18915   2:16587   M:29253   B:13341   2:12875  
##  STNWG  :16261   3:20064             C:20540   3:15767  
##  UTE    : 4586   4:18948             D: 8173   4:16189  
##  TRUCK  : 1750                       E: 5912   5:10736  
##  HDTOP  : 1579                       F: 3578   6: 6547  
##  (Other): 2532

Numerical variables EDA

we will create new data frame that contain only The needed Numerical variables : veh_value , exposure and claimcst0

Numerical_variables summary

	claimcst0	exposure	veh_value
Mean	1.372702e+02	0.4686515	1.777021e+00
Std.Dev	1.056298e+03	0.2900254	1.205232e+00
Min	0.000000e+00	0.0027379	0.000000e+00
Q1	0.000000e+00	0.2190281	1.010000e+00
Median	0.000000e+00	0.4462697	1.500000e+00
Q3	0.000000e+00	0.7091034	2.150000e+00
Max	5.592213e+04	0.9993155	3.456000e+01
MAD	0.000000e+00	0.3612632	8.006040e-01
IQR	0.000000e+00	0.4900753	1.140000e+00
CV	7.695028e+00	0.6188509	6.782316e-01
Skewness	1.750173e+01	0.1755497	2.967891e+00
SE.Skewness	9.403100e-03	0.0094031	9.403100e-03
Kurtosis	4.798760e+02	-1.1425554	2.675256e+01
N.Valid	6.785600e+04	67856.0000000	6.785600e+04
Pct.Valid	1.000000e+02	100.0000000	1.000000e+02

Observations:

A. for claims: the mean is 137 ,median is 0 , and the internal quantile range (50% of th data) is 0 as well ==>
- A.1 mean > median and the data is right skewed
- A.2 as the data range from (0,inf) ==> the data has gamma distribution
B. for exposure: the mean is 0.469 ,median is 0.446 , and the internal quantile range (50% of th data) is 0.999 ==>
- B.1 as the data probabilities are not the same. ==> the data has non-uniform distribution
C. for veh_value: the mean is 1777 ,median is 1500 , and the internal quantile range (50% of th data) is 1140 ==>
- C.1 mean > median and the data is right skewed
- C.2 as the data range from (0,inf) ==> the data has gamma distribution

Numerical variables Visualization

Correlation between Numerical variables

pairs plot

as the data is not normally distributed , we will see the corr using ‘spearman’ method

	veh_value	exposure	claimcst0
veh_value	1.0000000	0.0091092	0.0265644
exposure	0.0091092	1.0000000	0.1328116
claimcst0	0.0265644	0.1328116	1.0000000

Categorical variables EDA

we will create new data frame that contain only The needed Categorical variables : numclaims, veh_body veh_age gender ,area and agecat

Categorical_variables summary:

## Frequencies  
## Categorical_variables$numclaims  
## Type: Factor  
## 
##                Freq    % Valid   % Valid Cum.    % Total   % Total Cum.
## ----------- ------- ---------- -------------- ---------- --------------
##           0   63232    93.1856        93.1856    93.1856        93.1856
##           1    4333     6.3856        99.5712     6.3856        99.5712
##           2     271     0.3994        99.9705     0.3994        99.9705
##           3      18     0.0265        99.9971     0.0265        99.9971
##           4       2     0.0029       100.0000     0.0029       100.0000
##        <NA>       0                               0.0000       100.0000
##       Total   67856   100.0000       100.0000   100.0000       100.0000
## 
## Categorical_variables$veh_body  
## Type: Factor  
## 
##                Freq   % Valid   % Valid Cum.   % Total   % Total Cum.
## ----------- ------- --------- -------------- --------- --------------
##         BUS      48     0.071          0.071     0.071          0.071
##       CONVT      81     0.119          0.190     0.119          0.190
##       COUPE     780     1.149          1.340     1.149          1.340
##       HBACK   18915    27.875         29.215    27.875         29.215
##       HDTOP    1579     2.327         31.542     2.327         31.542
##       MCARA     127     0.187         31.729     0.187         31.729
##       MIBUS     717     1.057         32.786     1.057         32.786
##       PANVN     752     1.108         33.894     1.108         33.894
##       RDSTR      27     0.040         33.934     0.040         33.934
##       SEDAN   22233    32.765         66.699    32.765         66.699
##       STNWG   16261    23.964         90.663    23.964         90.663
##       TRUCK    1750     2.579         93.242     2.579         93.242
##         UTE    4586     6.758        100.000     6.758        100.000
##        <NA>       0                              0.000        100.000
##       Total   67856   100.000        100.000   100.000        100.000
## 
## Categorical_variables$veh_age  
## Type: Factor  
## 
##                Freq   % Valid   % Valid Cum.   % Total   % Total Cum.
## ----------- ------- --------- -------------- --------- --------------
##           1   12257     18.06          18.06     18.06          18.06
##           2   16587     24.44          42.51     24.44          42.51
##           3   20064     29.57          72.08     29.57          72.08
##           4   18948     27.92         100.00     27.92         100.00
##        <NA>       0                               0.00         100.00
##       Total   67856    100.00         100.00    100.00         100.00
## 
## Categorical_variables$gender  
## Type: Factor  
## 
##                Freq   % Valid   % Valid Cum.   % Total   % Total Cum.
## ----------- ------- --------- -------------- --------- --------------
##           F   38603     56.89          56.89     56.89          56.89
##           M   29253     43.11         100.00     43.11         100.00
##        <NA>       0                               0.00         100.00
##       Total   67856    100.00         100.00    100.00         100.00
## 
## Categorical_variables$area  
## Type: Factor  
## 
##                Freq   % Valid   % Valid Cum.   % Total   % Total Cum.
## ----------- ------- --------- -------------- --------- --------------
##           A   16312     24.04          24.04     24.04          24.04
##           B   13341     19.66          43.70     19.66          43.70
##           C   20540     30.27          73.97     30.27          73.97
##           D    8173     12.04          86.01     12.04          86.01
##           E    5912      8.71          94.73      8.71          94.73
##           F    3578      5.27         100.00      5.27         100.00
##        <NA>       0                               0.00         100.00
##       Total   67856    100.00         100.00    100.00         100.00
## 
## Categorical_variables$agecat  
## Type: Factor  
## 
##                Freq   % Valid   % Valid Cum.   % Total   % Total Cum.
## ----------- ------- --------- -------------- --------- --------------
##           1    5742      8.46           8.46      8.46           8.46
##           2   12875     18.97          27.44     18.97          27.44
##           3   15767     23.24          50.67     23.24          50.67
##           4   16189     23.86          74.53     23.86          74.53
##           5   10736     15.82          90.35     15.82          90.35
##           6    6547      9.65         100.00      9.65         100.00
##        <NA>       0                               0.00         100.00
##       Total   67856    100.00         100.00    100.00         100.00

Observations:

A. for numclaims: ==>
- A.1 The number of Risks that have no claims are 63232 (93.19%) out of Total number of Risks 67856
- A.2 The number of Risks that have one claim are 4333 (6.39%) out of Total number of Risks 67856
- A.3 The number of Risks that have two claim are 271 (3.99%%) out of Total number of Risks 67856
- A.4 The number of Risks that have three claim are 18 (0.27%%) out of Total number of Risks 67856
- A.5 The number of Risks that have four claim are 2 (0.0029%%) out of Total number of Risks 67856
B. for veh_body: ==>
- B.1 SEDAN,HBACK,and STNWG have the most frequencies with 22233(32.76%),18915(27.86%),16261(23.97%) in sequence out of Total number of Risks 67856
- B.2 RDSTR,BUS,and CONVT have the lowest frequencies with 27(0.04%),48(0.07%),81(0.12%) in sequence out of Total number of Risks 67856
C. for veh_age: ==>
- C.1 The number of Risks with the highest frequency is veh_age==3 with 20064 (29.57%) out of Total number of Risks 67856
- C.2 The number of Risks with the lowest frequency is veh_age==1 with 12257 (18.06%) out of Total number of Risks 67856
D. for gender: ==>
- D.1 Females has the higher frequency than males with 38603 (56.89%) for females and 29253(43.11%) for males
F. for veh_age: ==>
- F.1 The number of Risks with the highest frequency is area : C with 20540 (30.27%) out of Total number of Risks 67856
- F.2 The number of Risks with the lowest frequency is area : F with 3578 (5.27%) out of Total number of Risks 67856
G. for agecat: ==>
- G.1 The number of Risks with the highest frequency is agecat == 3 and agecat == 4 C with 16189 (23.86%) ,15767 (23.24%) in sequence out of Total number of Risks 67856
- G.2 The number of Risks with the lowest frequency is agecat == 1 and agecat == 6 with 5742 (8.46%) ,6547 (9.65%) in sequence out of Total number of Risks 67856

Categorical variables Visualization

Categorical variables in terms of claims occurrence:

1. gender :

Characteristic	N	Overall, N = 67,856¹	0, N = 63,232¹	1, N = 4,624¹	p-value²
gender	67,856				0.6
F		38,603 (57%)	35,955 (57%)	2,648 (57%)
M		29,253 (43%)	27,277 (43%)	1,976 (43%)
¹ n (%)
² Pearson’s Chi-squared test

Insights

A. Out of 67856 risk , we have 4624 with positive claims claims

B. female had more claim percentage than males

C. the p value of the the pearson chisq test is 0.6 ==> more than 0.05 which statistically means there is no significant difference between genders in respect of claims

Visualizing gender against clm

now we want to assess and report the effect size and the magnitude of differences between these groups

table <- table(data$clm ,data$gender)
chi_square_test<-chisq.test(table)
phi_coefficient <- sqrt(chi_square_test$statistic / sum(table))
cat(paste("The effect size (Phi Coefficient) is",phi_coefficient ,"indicating",interpret_phi(phi_coefficient),"effect size"))

## The effect size (Phi Coefficient) is 0.0019987264321987 indicating tiny effect size

claims probabilities by gender

##  gender   prob      SE  df asymp.LCL asymp.UCL null  z.ratio p.value
##  F      0.0686 0.00129 Inf    0.0661    0.0712  0.5 -129.543  <.0001
##  M      0.0675 0.00147 Inf    0.0647    0.0705  0.5 -112.676  <.0001
## 
## Confidence level used: 0.95 
## Intervals are back-transformed from the logit scale 
## Tests are performed on the logit scale

Insights

the probability of females having claims is little higher than the probability of males with .0686 females and 0.0675 for males

To visualize our results :

2. area :

Characteristic	N	Overall, N = 67,856¹	0, N = 63,232¹	1, N = 4,624¹	p-value²
area	67,856				0.003
A		16,312 (24%)	15,227 (24%)	1,085 (23%)
B		13,341 (20%)	12,376 (20%)	965 (21%)
C		20,540 (30%)	19,128 (30%)	1,412 (31%)
D		8,173 (12%)	7,677 (12%)	496 (11%)
E		5,912 (8.7%)	5,526 (8.7%)	386 (8.3%)
F		3,578 (5.3%)	3,298 (5.2%)	280 (6.1%)
¹ n (%)
² Pearson’s Chi-squared test

Insights

A. area A has the highest number of claims with ‘31%’ percentage

B. area F has the lowest number of claims with ‘6.1%’ percentage

C. the p value of the the pearson chisq test is 0.003 ==> less than 0.05 which statistically means there is significant difference between areas in respect of claims

Visualizing area against clm

now we want to assess and report the effect size and the magnitude of differences between these groups

library(vcd)
table <- table(data$clm ,data$area)
cat(paste("The effect size (Cramér's V) is ", assocstats(table)$cramer,"indicating",interpret_cramers_v(assocstats(table)$cramer),"effect size"))

## The effect size (Cramér's V) is  0.0163593665773446 indicating tiny effect size

claims probabilities by area

##  area   prob      SE  df asymp.LCL asymp.UCL null z.ratio p.value
##  A    0.0665 0.00195 Inf    0.0628    0.0704  0.5 -84.066  <.0001
##  B    0.0723 0.00224 Inf    0.0681    0.0769  0.5 -76.337  <.0001
##  C    0.0687 0.00177 Inf    0.0654    0.0723  0.5 -94.504  <.0001
##  D    0.0607 0.00264 Inf    0.0557    0.0661  0.5 -59.130  <.0001
##  E    0.0653 0.00321 Inf    0.0593    0.0719  0.5 -50.552  <.0001
##  F    0.0783 0.00449 Inf    0.0699    0.0875  0.5 -39.621  <.0001
## 
## Confidence level used: 0.95 
## Intervals are back-transformed from the logit scale 
## Tests are performed on the logit scale

Insights

We can see that area== F has the highest probability of claimed with 0.0783% and area== D has the lowest probability with 0.0607%

To visualize our results :

3. agecat :

Characteristic	N	Overall, N = 67,856¹	0, N = 63,232¹	1, N = 4,624¹	p-value²
agecat	67,856				<0.001
1		5,742 (8.5%)	5,246 (8.3%)	496 (11%)
2		12,875 (19%)	11,943 (19%)	932 (20%)
3		15,767 (23%)	14,654 (23%)	1,113 (24%)
4		16,189 (24%)	15,085 (24%)	1,104 (24%)
5		10,736 (16%)	10,122 (16%)	614 (13%)
6		6,547 (9.6%)	6,182 (9.8%)	365 (7.9%)
¹ n (%)
² Pearson’s Chi-squared test

Insights

A. Age Category 2 and 4 have the highest number of claims with ‘24%’ percentage for each -‘noting that both categories has the highest count of total agecat observations’-

B. Age Category 6 has the lowest number of claims with ‘7.9%’ percentage

C. the p value of the the pearson chisq test < 0.001 ==> less than 0.05 which statistically means there is significant difference between Age Categories in respect of claims

Visualizing Age Category against clm

now we want to assess and report the effect size and the magnitude of differences between these groups

## The effect size (Cramér's V) is  0.0324228335586784 indicating tiny effect size

claims probabilities by agecat

##  agecat   prob      SE  df asymp.LCL asymp.UCL null z.ratio p.value
##  1      0.0864 0.00371 Inf    0.0794    0.0939  0.5 -50.210  <.0001
##  2      0.0724 0.00228 Inf    0.0680    0.0770  0.5 -74.994  <.0001
##  3      0.0706 0.00204 Inf    0.0667    0.0747  0.5 -82.904  <.0001
##  4      0.0682 0.00198 Inf    0.0644    0.0722  0.5 -83.865  <.0001
##  5      0.0572 0.00224 Inf    0.0530    0.0617  0.5 -67.429  <.0001
##  6      0.0558 0.00284 Inf    0.0504    0.0616  0.5 -52.530  <.0001
## 
## Confidence level used: 0.95 
## Intervals are back-transformed from the logit scale 
## Tests are performed on the logit scale

Insights

We can see that agecat== 1 has the highest probability of claimed with 0.0864% and agecat== 6 has the lowest probability with 0.0558%

To visualize our results :

4. veh_age :

Characteristic	N	Overall, N = 67,856¹	0, N = 63,232¹	1, N = 4,624¹	p-value²
veh_age	67,856				<0.001
1		12,257 (18%)	11,432 (18%)	825 (18%)
2		16,587 (24%)	15,328 (24%)	1,259 (27%)
3		20,064 (30%)	18,702 (30%)	1,362 (29%)
4		18,948 (28%)	17,770 (28%)	1,178 (25%)
¹ n (%)
² Pearson’s Chi-squared test

Insights

A. Vehicle Age 3 has highest number of claims with ‘29%’ percentage-‘noting that it has the highest count of total Vehicle Age categories observations’-

B. Vehicle Age 1 has the lowest number of claims with ‘18%’ percentage

C. the p value of the the pearson chisq test < 0.001 ==> less than 0.05 which statistically means there is significant difference between Vehicle Ages in respect of claims

Visualizing Vehicle Age Category against clm

now we want to assess and report the effect size and the magnitude of differences between these groups

## The effect size (Cramér's V) is  0.0197729280570061 indicating tiny effect size

claims probabilities by agecat

##  veh_age   prob      SE  df asymp.LCL asymp.UCL null z.ratio p.value
##  1       0.0673 0.00226 Inf    0.0630    0.0719  0.5 -72.921  <.0001
##  2       0.0759 0.00206 Inf    0.0720    0.0800  0.5 -85.251  <.0001
##  3       0.0679 0.00178 Inf    0.0645    0.0714  0.5 -93.341  <.0001
##  4       0.0622 0.00175 Inf    0.0588    0.0657  0.5 -90.198  <.0001
## 
## Confidence level used: 0.95 
## Intervals are back-transformed from the logit scale 
## Tests are performed on the logit scale

Insights

We can see that veh_age== 2 has the highest probability of claimed with 0.0759% and veh_age== 4 has the lowest probability with 0.0622%

To visualize our results :

5. veh_body :

Characteristic	N	Overall, N = 67,856¹	0, N = 63,232¹	1, N = 4,624¹
veh_body	67,856
BUS		48 (<0.1%)	39 (<0.1%)	9 (0.2%)
CONVT		81 (0.1%)	78 (0.1%)	3 (<0.1%)
COUPE		780 (1.1%)	712 (1.1%)	68 (1.5%)
HBACK		18,915 (28%)	17,651 (28%)	1,264 (27%)
HDTOP		1,579 (2.3%)	1,449 (2.3%)	130 (2.8%)
MCARA		127 (0.2%)	113 (0.2%)	14 (0.3%)
MIBUS		717 (1.1%)	674 (1.1%)	43 (0.9%)
PANVN		752 (1.1%)	690 (1.1%)	62 (1.3%)
RDSTR		27 (<0.1%)	25 (<0.1%)	2 (<0.1%)
SEDAN		22,233 (33%)	20,757 (33%)	1,476 (32%)
STNWG		16,261 (24%)	15,088 (24%)	1,173 (25%)
TRUCK		1,750 (2.6%)	1,630 (2.6%)	120 (2.6%)
UTE		4,586 (6.8%)	4,326 (6.8%)	260 (5.6%)
¹ n (%)

Insights

A. Sedan veh_body has highest number of claims with ‘32%’ percentage-‘noting that it has the highest count of total veh_body observations’-

B. convt and RDSTR veh_body has the lowest number of claims with less than ‘0.1%’ percentage

Visualizing veh_body Category against clm

now we want to assess and report the effect size and the magnitude of differences between these groups

## The effect size (Cramér's V) is  0.0252738312900611 indicating tiny effect size

claims probabilities by veh_body

##  veh_body   prob      SE  df asymp.LCL asymp.UCL null z.ratio p.value
##  BUS      0.1875 0.05634 Inf    0.1005    0.3227  0.5  -3.965  0.0001
##  CONVT    0.0370 0.02097 Inf    0.0120    0.1086  0.5  -5.540  <.0001
##  COUPE    0.0872 0.01010 Inf    0.0693    0.1091  0.5 -18.503  <.0001
##  HBACK    0.0668 0.00182 Inf    0.0634    0.0705  0.5 -90.549  <.0001
##  HDTOP    0.0823 0.00692 Inf    0.0697    0.0969  0.5 -26.335  <.0001
##  MCARA    0.1102 0.02779 Inf    0.0664    0.1776  0.5  -7.371  <.0001
##  MIBUS    0.0600 0.00887 Inf    0.0448    0.0799  0.5 -17.497  <.0001
##  PANVN    0.0824 0.01003 Inf    0.0648    0.1044  0.5 -18.174  <.0001
##  RDSTR    0.0741 0.05040 Inf    0.0186    0.2525  0.5  -3.437  0.0006
##  SEDAN    0.0664 0.00167 Inf    0.0632    0.0697  0.5 -98.133  <.0001
##  STNWG    0.0721 0.00203 Inf    0.0683    0.0762  0.5 -84.269  <.0001
##  TRUCK    0.0686 0.00604 Inf    0.0576    0.0814  0.5 -27.581  <.0001
##  UTE      0.0567 0.00341 Inf    0.0504    0.0638  0.5 -44.034  <.0001
## 
## Confidence level used: 0.95 
## Intervals are back-transformed from the logit scale 
## Tests are performed on the logit scale

Insights

We can see that veh_body== BUS has the highest probability of claimed with 18.75% and veh_body== CONVT has the lowest probability with 0.0370%

To visualize our results :

————————————————————————

Data summary - Positive claims

positive claims Data is the data with claims > 0

##    veh_value         exposure        numclaims   claimcst0          veh_body   
##  Min.   : 0.000   Min.   :0.002738   0:   0    Min.   :  200.0   SEDAN  :1476  
##  1st Qu.: 1.100   1st Qu.:0.410678   1:4333    1st Qu.:  353.8   HBACK  :1264  
##  Median : 1.570   Median :0.637919   2: 271    Median :  761.6   STNWG  :1173  
##  Mean   : 1.859   Mean   :0.611271   3:  18    Mean   : 2014.4   UTE    : 260  
##  3rd Qu.: 2.310   3rd Qu.:0.832307   4:   2    3rd Qu.: 2091.4   HDTOP  : 130  
##  Max.   :13.900   Max.   :0.999316             Max.   :55922.1   TRUCK  : 120  
##                                                                  (Other): 201  
##  veh_age  gender   area     agecat  
##  1: 825   F:2648   A:1085   1: 496  
##  2:1259   M:1976   B: 965   2: 932  
##  3:1362            C:1412   3:1113  
##  4:1178            D: 496   4:1104  
##                    E: 386   5: 614  
##                    F: 280   6: 365  
##

Numerical variables EDA

we will create new data frame that contain only The needed Numerical variables : veh_value , exposure and claimcst0

Numerical_variables summary

	claimcst0	exposure	veh_value
Mean	2.014404e+03	0.6112714	1.8591955
Std.Dev	3.548907e+03	0.2616474	1.1595951
Min	2.000000e+02	0.0027379	0.0000000
Q1	3.537700e+02	0.4106776	1.1000000
Median	7.615650e+02	0.6379192	1.5700000
Q3	2.091900e+03	0.8323066	2.3100000
Max	5.592213e+04	0.9993155	13.9000000
MAD	8.325763e+02	0.3125536	0.8302560
IQR	1.737655e+03	0.4216290	1.2100000
CV	1.761765e+00	0.4280381	0.6237080
Skewness	5.038509e+00	-0.3468537	1.8535754
SE.Skewness	3.601020e-02	0.0360102	0.0360102
Kurtosis	4.019591e+01	-0.8719807	6.9007387
N.Valid	4.624000e+03	4624.0000000	4624.0000000
Pct.Valid	1.000000e+02	100.0000000	100.0000000

Observations:

A. for claims: the mean is 2014 ,median is 761 , and the internal quantile range (50% of th data) is 1737 as well ==>

-noting that the the mean was 137 ,median was 0 , and the internal quantile range (50% of th data) was 0 as well for all data set-
B. for exposure: the mean is 0.611 ,median is 0.638 , and the internal quantile range (50% of th data) is 0.422 ==>

-noting that the mean was 0.469 ,median was 0.446 , and the internal quantile range (50% of th data) was 0.999 for all data set-
C. for veh_value: the mean is 1859 ,median is 1570 , and the internal quantile range (50% of th data) is 1210 ==>

-noting that the mean was 1777 ,median was 1500 , and the internal quantile range (50% of th data) was 1140 for all data set-

Numerical variables Visualization

Correlation between Numerical variables

pairs plot

as the data is not normally distributed , we will see the corr using ‘spearman’ method

	veh_value	exposure	claimcst0
veh_value	1.0000000	0.0624592	-0.0219103
exposure	0.0624592	1.0000000	-0.0418187
claimcst0	-0.0219103	-0.0418187	1.0000000

Categorical variables EDA

we will create new data frame that contain only The needed Categorical variables : numclaims, veh_body veh_age gender ,area and agecat

numclaims	veh_body	veh_age	gender	area	agecat
1	SEDAN	3	M	B	6
1	SEDAN	3	F	F	4
1	HBACK	3	M	C	4
2	STNWG	3	M	F	2
1	STNWG	2	M	F	3
1	HBACK	3	F	A	4

Categorical_variables summary:

## Frequencies  
## Categorical_variables$numclaims  
## Type: Factor  
## 
##               Freq   % Valid   % Valid Cum.   % Total   % Total Cum.
## ----------- ------ --------- -------------- --------- --------------
##           0      0     0.000          0.000     0.000          0.000
##           1   4333    93.707         93.707    93.707         93.707
##           2    271     5.861         99.567     5.861         99.567
##           3     18     0.389         99.957     0.389         99.957
##           4      2     0.043        100.000     0.043        100.000
##        <NA>      0                              0.000        100.000
##       Total   4624   100.000        100.000   100.000        100.000
## 
## Categorical_variables$veh_body  
## Type: Factor  
## 
##               Freq   % Valid   % Valid Cum.   % Total   % Total Cum.
## ----------- ------ --------- -------------- --------- --------------
##         BUS      9     0.195          0.195     0.195          0.195
##       CONVT      3     0.065          0.260     0.065          0.260
##       COUPE     68     1.471          1.730     1.471          1.730
##       HBACK   1264    27.336         29.066    27.336         29.066
##       HDTOP    130     2.811         31.877     2.811         31.877
##       MCARA     14     0.303         32.180     0.303         32.180
##       MIBUS     43     0.930         33.110     0.930         33.110
##       PANVN     62     1.341         34.451     1.341         34.451
##       RDSTR      2     0.043         34.494     0.043         34.494
##       SEDAN   1476    31.920         66.414    31.920         66.414
##       STNWG   1173    25.368         91.782    25.368         91.782
##       TRUCK    120     2.595         94.377     2.595         94.377
##         UTE    260     5.623        100.000     5.623        100.000
##        <NA>      0                              0.000        100.000
##       Total   4624   100.000        100.000   100.000        100.000
## 
## Categorical_variables$veh_age  
## Type: Factor  
## 
##               Freq   % Valid   % Valid Cum.   % Total   % Total Cum.
## ----------- ------ --------- -------------- --------- --------------
##           1    825     17.84          17.84     17.84          17.84
##           2   1259     27.23          45.07     27.23          45.07
##           3   1362     29.46          74.52     29.46          74.52
##           4   1178     25.48         100.00     25.48         100.00
##        <NA>      0                               0.00         100.00
##       Total   4624    100.00         100.00    100.00         100.00
## 
## Categorical_variables$gender  
## Type: Factor  
## 
##               Freq   % Valid   % Valid Cum.   % Total   % Total Cum.
## ----------- ------ --------- -------------- --------- --------------
##           F   2648     57.27          57.27     57.27          57.27
##           M   1976     42.73         100.00     42.73         100.00
##        <NA>      0                               0.00         100.00
##       Total   4624    100.00         100.00    100.00         100.00
## 
## Categorical_variables$area  
## Type: Factor  
## 
##               Freq   % Valid   % Valid Cum.   % Total   % Total Cum.
## ----------- ------ --------- -------------- --------- --------------
##           A   1085     23.46          23.46     23.46          23.46
##           B    965     20.87          44.33     20.87          44.33
##           C   1412     30.54          74.87     30.54          74.87
##           D    496     10.73          85.60     10.73          85.60
##           E    386      8.35          93.94      8.35          93.94
##           F    280      6.06         100.00      6.06         100.00
##        <NA>      0                               0.00         100.00
##       Total   4624    100.00         100.00    100.00         100.00
## 
## Categorical_variables$agecat  
## Type: Factor  
## 
##               Freq   % Valid   % Valid Cum.   % Total   % Total Cum.
## ----------- ------ --------- -------------- --------- --------------
##           1    496     10.73          10.73     10.73          10.73
##           2    932     20.16          30.88     20.16          30.88
##           3   1113     24.07          54.95     24.07          54.95
##           4   1104     23.88          78.83     23.88          78.83
##           5    614     13.28          92.11     13.28          92.11
##           6    365      7.89         100.00      7.89         100.00
##        <NA>      0                               0.00         100.00
##       Total   4624    100.00         100.00    100.00         100.00

Observations:

A. for numclaims: ==>
- A.1 The number of Risks that have one claim are 4333 (93.707) out of Total number of Risks with positive claims 4624
- A.2 The number of Risks that have one claim are 271 (5.861) out of Total number of Risks with positive claims 4624
- A.3 The number of Risks that have two claim are 18 (0.389%) out of Total number of Risks with positive claims 4624
- A.4 The number of Risks that have three claim are 2 (0.043%) out of Total number of Risks with positive claims 4624
B. for veh_body: ==>
- B.1 SEDAN,HBACK,and STNWG have the most claims incurred frequency with 1476(31.920%),1264(27.34%),1173(25.37%) in sequence out of Total number of Risks with positive claims 4624
- B.2 RDSTR,CONVT,and CONVT have the lowest claims incurred frequency with 2(0.043%),3(0.056%),9(0.195%) in sequence out of Total number of Risks with positive claims 4624
C. for veh_age: ==>
- C.1 The number of Risks with the highest claims incurred frequency is veh_age==3 with 1362 (29.46%) out of Total number of Risks with positive claims 4624
- C.2 The number of Risks with the lowest claims incurred frequency is veh_age==1 with 825 (17.84%) out of Total number of Risks with positive claims 4624
D. for gender: ==>
- D.1 Females has the higher claims incurred frequency than males with 2648 (57.2%) for females and 1976(42.73%) for males
F. for area: ==>
- F.1 The number of Risks with the highest claims incurred frequency is area : C with 1412 (30.54%) out of Total number of Risks with positive claims 4624
- F.2 The number of Risks with the lowest claims incurred frequency is area : F with 280 (6.06%) out of Total number of Risks with positive claims 4624
G. for agecat: ==>
- G.1 The number of Risks with the highest claims incurred frequency is agecat == 3 and agecat == 4 C with 1113 (27.7%) ,1104 (23.88%) in sequence out of Total number of Risks with positive claims 4624
- G.2 The number of Risks with the lowest claims incurred frequency is agecat == 6 with 365 (7.89%) out of Total number of Risks with positive claims 4624

Categorical variables Visualization

————————————————————————

Inferential Analysis

choosing the Right statistics tests

In Order To determine what types of statistical tests we will apply , we need to to determine Normality of the target variable (claims):

claimcst0 variable Normality check by conducting Visualization

claims variable Normality check by Applying Anderson-Darling normality test

## 
##  Anderson-Darling normality test
## 
## data:  df$claimcst0
## A = 654.01, p-value < 2.2e-16

Based on our above analysis results:

Normality: The Anderson-Darling normality test (p-value < 0.05) indicates non-normal distribution.

==> Given these results, We will be using non-parametric statistical tests

Aplying non-parametric statistical tests

A. Mann-Whitney U Test

Mann-Whitney U Test -for features that have 2 categories : equivelent to wilcox.test() wilcox rank sum test with continuity correction,better than wilcox sighned rank exct test(that used when variables are paired)

A.1 Mann-Whitney U Test on claimcst0 against gender

## 
##  Wilcoxon rank sum test with continuity correction
## 
## data:  df$claimcst0 by df$gender
## W = 2520284, p-value = 0.03234
## alternative hypothesis: true location shift is not equal to 0

After Applying the test we found that:

The pvalue is less than 0.05 –> statistically there is significant difference in the data distribution between genders

A.2 Visualization

A.3 wilcox_effsize :
now we want to assess and report the effect size and the magnitude of differences between these groups

## # A tibble: 1 × 7
##   .y.       group1 group2 effsize    n1    n2 magnitude
## * <chr>     <chr>  <chr>    <dbl> <int> <int> <ord>    
## 1 claimcst0 F      M       0.0315  2648  1976 small

The wilcox_effsize test showed that the effect size= .0315 and the indicating small effect size

A.4 Assumption Checks:
Check the homogeneity of variances using Levene’s test.

levene_test <- leveneTest(claimcst0~gender,data = df)
print(levene_test)

## Levene's Test for Homogeneity of Variance (center = median)
##         Df F value    Pr(>F)    
## group    1  12.543 0.0004016 ***
##       4622                      
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

The test indicates that the variances are significantly different across the groups (p-value < 0.05).

A.5 Post-hoc Power Analysis:
Conduct a post-hoc power analysis based on the effect size.

# Post-hoc Power Analysis
#t-test (two samples with unequal n)
library(pwr)

## Warning: package 'pwr' was built under R version 4.3.3

effect_size_value <- effect_size$effsize
power_analysis <- pwr.t2n.test(n1 = length(df$gender=="M"), n2 = length(df$gender=="F"), 
                               d = effect_size_value, 
                               sig.level = 0.05, power = NULL)
print(power_analysis)

## 
##      t test power calculation 
## 
##              n1 = 4624
##              n2 = 4624
##               d = 0.03147296
##       sig.level = 0.05
##           power = 0.3277674
##     alternative = two.sided

The power of the test is 0.3277674,This means there is a 32.77% chance of detecting a true effect.

B. kruskal.test

B.1 kruskal.test on claims against area

B.1.1 kruskal.test on claims against area as it has more than 2 factors

## 
##  Kruskal-Wallis rank sum test
## 
## data:  df$claimcst0 and df$area
## Kruskal-Wallis chi-squared = 26.616, df = 5, p-value = 6.776e-05

After Applying the test we found that:

The pvalue is less than 0.05 –> statistically there is significant difference in the data distribution between areas

B.1.2 Visualization

B.1.3 kruskal_effsize :
now we want to assess and report the effect size and the magnitude of differences between these groups

.y.	n	effsize	method	magnitude
claimcst0	4624	0.0046807	eta2[H]	small

The kruskal_effsize test showed that the effect size (epsilon_squared)= 0.004680724 and the indicating small effect size

B.1.4 to find out in details these differences we will Perform Dunn’s Test with holm correction for p-values :

##    Comparison          Z      P.unadj        P.adj
## 1       A - B  0.3929272 6.943733e-01 1.0000000000
## 2       A - C -0.1226836 9.023577e-01 0.9023576683
## 3       B - C -0.5348567 5.927489e-01 1.0000000000
## 4       A - D -1.0455938 2.957486e-01 1.0000000000
## 5       B - D -1.3404712 1.800922e-01 1.0000000000
## 6       C - D -0.9908884 3.217401e-01 1.0000000000
## 7       A - E -2.8037166 5.051729e-03 0.0505172881
## 8       B - E -3.0477582 2.305554e-03 0.0276666430
## 9       C - E -2.8067635 5.004197e-03 0.0550461713
## 10      D - E -1.6131418 1.067137e-01 0.8537095799
## 11      A - F -3.9167774 8.974053e-05 0.0011666269
## 12      B - F -4.1238931 3.725219e-05 0.0005587828
## 13      C - F -3.9375509 8.231746e-05 0.0011524445
## 14      D - F -2.7541273 5.884887e-03 0.0529639842
## 15      E - F -1.2278040 2.195206e-01 1.0000000000

From the above table we can get that :

The p.adjusted value is less than 0.05 between area B-E ,area A-F,area B-F,area C-F –> statistically there is significant difference in the data distribution between these groups,and there is no statistically significant difference in the data distribution between all other area groups

B.1.5 Assumption Checks:
Check the homogeneity of variances using Levene’s test.

levene_test <- leveneTest(claimcst0 ~area,data=df)
print(levene_test)

## Levene's Test for Homogeneity of Variance (center = median)
##         Df F value   Pr(>F)   
## group    5  3.9598 0.001384 **
##       4618                    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

The test indicates that the variances are significantly different across the groups (p-value < 0.05).

B.1.6 Post-hoc Power Analysis:
Conduct a post-hoc power analysis based on the effect size.

effect_size <- sqrt(eff_size$effsize)
power_analysis <- pwr.anova.test(k = length(unique(df$area)),
                                 n = nrow(df) / length(unique(df$area)),
                                 f = effect_size,
                                 sig.level = 0.05,
                                 power = NULL)
print(power_analysis)

## 
##      Balanced one-way analysis of variance power calculation 
## 
##               k = 6
##               n = 770.6667
##               f = 0.06841581
##       sig.level = 0.05
##           power = 0.966717
## 
## NOTE: n is number in each group

The power of the test is 0.966717,This means there is a 96% chance of detecting a true effect.

B.2 kruskal.test on claims against veh_body

B.2.1 kruskal.test on claims against veh_body as it has more than 2 factors

## 
##  Kruskal-Wallis rank sum test
## 
## data:  df$claimcst0 and df$veh_body
## Kruskal-Wallis chi-squared = 18.358, df = 12, p-value = 0.1052

After Applying the test we found that:

The pvalue is more than 0.05 –> statistically there is no significant difference in the data distribution between veh_body

B.2.2 Visualization

B.3 kruskal.test on claims against age category

B.3.1 kruskal.test on claims against age category as it has more than 2 factors

## 
##  Kruskal-Wallis rank sum test
## 
## data:  df$claimcst0 and df$agecat
## Kruskal-Wallis chi-squared = 11.099, df = 5, p-value = 0.04946

After Applying the test we found that:

The pvalue is less than 0.05 –> statistically there is significant difference in the data distribution between agecat

B.3.2 Visualization

B.3.3 kruskal_effsize :
now we want to assess and report the effect size and the magnitude of differences between these groups

.y.	n	effsize	method	magnitude
claimcst0	4624	0.0013206	eta2[H]	small

The kruskal_effsize test showed that the effect size (epsilon_squared)= 0.001320641 and the indicating small effect size

B.3.4 to find out in details these differences we will Perform Dunn’s Test with holm correction for p-values :

##    Comparison           Z     P.unadj      P.adj
## 1       1 - 2  2.25523606 0.024118516 0.28942219
## 2       1 - 3  2.50162552 0.012362461 0.16071199
## 3       2 - 3  0.21869627 0.826886659 1.00000000
## 4       1 - 4  2.74351582 0.006078512 0.08509917
## 5       2 - 4  0.51604229 0.605824873 1.00000000
## 6       3 - 4  0.31181652 0.755179965 1.00000000
## 7       1 - 5  3.16592394 0.001545912 0.02318868
## 8       2 - 5  1.26571867 0.205613823 1.00000000
## 9       3 - 5  1.11552540 0.264625342 1.00000000
## 10      4 - 5  0.85082071 0.394868957 1.00000000
## 11      1 - 6  1.94645859 0.051599677 0.56759645
## 12      2 - 6  0.14394179 0.885546439 1.00000000
## 13      3 - 6 -0.01363333 0.989122512 0.98912251
## 14      4 - 6 -0.23298594 0.815772326 1.00000000
## 15      5 - 6 -0.86090528 0.389290214 1.00000000

From the above table we can get that :

The p.adjusted value is less than 0.05 only between agecat 1-5 –> statistically there is significant difference in the data distribution between these groups,and there is no statistically significant difference in the data distribution between all other agecat groups

B.3.5 Assumption Checks:
Check the homogeneity of variances using Levene’s test.

levene_test <- leveneTest(claimcst0 ~agecat,data=df)
print(levene_test)

## Levene's Test for Homogeneity of Variance (center = median)
##         Df F value    Pr(>F)    
## group    5  4.5834 0.0003578 ***
##       4618                      
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

The test indicates that the variances are significantly different across the groups (p-value < 0.05).

B.3.6 Post-hoc Power Analysis:
Conduct a post-hoc power analysis based on the effect size.

effect_size <- sqrt(eff_size$effsize)
power_analysis <- pwr.anova.test(k = length(unique(df$agecat)),
                                 n = nrow(df) / length(unique(df$agecat)),
                                 f = effect_size,
                                 sig.level = 0.05,
                                 power = NULL)
print(power_analysis)

## 
##      Balanced one-way analysis of variance power calculation 
## 
##               k = 6
##               n = 770.6667
##               f = 0.03634062
##       sig.level = 0.05
##           power = 0.4397375
## 
## NOTE: n is number in each group

The power of the test is 0.4397375,This means there is a 43.9% chance of detecting a true effect.

B.4 kruskal.test on claims against number of claims

B.4.1 kruskal.test on claims against number of claims as it has more than 2 factors

## 
##  Kruskal-Wallis rank sum test
## 
## data:  df$claimcst0 and df$numclaims
## Kruskal-Wallis chi-squared = 159.02, df = 3, p-value < 2.2e-16

After Applying the test we found that:

The pvalue is less than 0.05 –> statistically there is significant difference in the data distribution between numclaims

B.4.2 Visualization

B.4.3 kruskal_effsize :
now we want to assess and report the effect size and the magnitude of differences between these groups

.y.	n	effsize	method	magnitude
claimcst0	4624	0.0337713	eta2[H]	small

The kruskal_effsize test showed that the effect size (epsilon_squared)= 0.0337713 and the indicating small effect size

B.4.4 to find out in details these differences we will Perform Dunn’s Test with holm correction for p-values :

##   Comparison           Z      P.unadj        P.adj
## 1      1 - 2 -11.6335467 2.782897e-31 1.669738e-30
## 2      1 - 3  -4.7312845 2.231036e-06 1.115518e-05
## 3      2 - 3  -1.5983152 1.099729e-01 3.299186e-01
## 4      1 - 4  -1.7924333 7.306358e-02 2.922543e-01
## 5      2 - 4  -0.7598618 4.473372e-01 8.946744e-01
## 6      3 - 4  -0.2015760 8.402482e-01 8.402482e-01

From the above table we can get that :

The p.adjusted value is less than 0.05 between numclaims 1-2 ,numclaims 1-3 –> statistically there is significant difference in the data distribution between these groups,and there is no statistically significant differences in the data distribution between all other numclaims groups

B.4.5 Assumption Checks:
Check the homogeneity of variances using Levene’s test.

levene_test <- leveneTest(claimcst0 ~numclaims,data=df)
print(levene_test)

## Levene's Test for Homogeneity of Variance (center = median)
##         Df F value Pr(>F)
## group    3  0.9987 0.3923
##       4620

The test indicates that the variances are not significantly different across the groups (p-value > 0.05).

B.4.6 Post-hoc Power Analysis:
Conduct a post-hoc power analysis based on the effect size.

effect_size <- sqrt(eff_size$effsize)
power_analysis <- pwr.anova.test(k = length(unique(df$numclaims)),
                                 n = nrow(df) / length(unique(df$numclaims)),
                                 f = effect_size,
                                 sig.level = 0.05,
                                 power = NULL)
print(power_analysis)

## 
##      Balanced one-way analysis of variance power calculation 
## 
##               k = 4
##               n = 1156
##               f = 0.1837697
##       sig.level = 0.05
##           power = 1
## 
## NOTE: n is number in each group

The power of the test is 1,This means there is a 100% chance of detecting a true effect.

B.5 kruskal.test on claims against veh_age

B.5.1 kruskal.test on claims against veh_age as it has more than 2 factors

## 
##  Kruskal-Wallis rank sum test
## 
## data:  df$claimcst0 and df$veh_age
## Kruskal-Wallis chi-squared = 18.668, df = 3, p-value = 0.0003201

After Applying the test we found that:

The pvalue is less than 0.05 –> statistically there is significant difference in the data distribution between veh_age

B.5.2 Visualization

B.5.3 kruskal_effsize :
now we want to assess and report the effect size and the magnitude of differences between these groups

.y.	n	effsize	method	magnitude
claimcst0	4624	0.0033914	eta2[H]	small

The kruskal_effsize test showed that the effect size (epsilon_squared)= 0.003391443 and the indicating small effect size

B.5.4 to find out in details these differences we will Perform Dunn’s Test with holm correction for p-values :

##   Comparison         Z      P.unadj        P.adj
## 1      1 - 2 -1.401384 1.610994e-01 0.3221987936
## 2      1 - 3 -2.860319 4.232150e-03 0.0169286002
## 3      2 - 3 -1.622098 1.047823e-01 0.3143468597
## 4      1 - 4 -3.988218 6.657135e-05 0.0003994281
## 5      2 - 4 -2.918060 3.522165e-03 0.0176108256
## 6      3 - 4 -1.379031 1.678852e-01 0.1678852290

From the above table we can get that :

The p.adjusted value is less than 0.05 only between veh_age 1-3 ,veh_age 1-4 ,veh_age 2-4 –> statistically there is significant difference in the data distribution between these groups,and there is no statistically significant difference in the data distribution between all other veh_age groups

B.5.5 Assumption Checks:
Check the homogeneity of variances using Levene’s test.

levene_test <- leveneTest(claimcst0 ~veh_age,data=df)
print(levene_test)

## Levene's Test for Homogeneity of Variance (center = median)
##         Df F value Pr(>F)
## group    3   0.926 0.4272
##       4620

The test indicates that the variances are not significantly different across the groups (p-value > 0.05).

B.5.6 Post-hoc Power Analysis:
Conduct a post-hoc power analysis based on the effect size.

effect_size <- sqrt(eff_size$effsize)
power_analysis <- pwr.anova.test(k = length(unique(df$veh_age)),
                                 n = nrow(df) / length(unique(df$veh_age)),
                                 f = effect_size,
                                 sig.level = 0.05,
                                 power = NULL)
print(power_analysis)

## 
##      Balanced one-way analysis of variance power calculation 
## 
##               k = 4
##               n = 1156
##               f = 0.0582361
##       sig.level = 0.05
##           power = 0.9288195
## 
## NOTE: n is number in each group

The power of the test is 0.92,This means there is a 92% chance of detecting a true effect.

————————————————————————

Insurance risk measurements by factors.

as the insurance risk level is measured by the frequency and the severity; we will calculate the the same and group it by factors.

gender factor

gender	claimcst0	numclaims	exposure	frequency	average_severity	risk_premium
F	4908749	2832	17954.60	0.1577311	1733.315	273.3978
M	4405855	2105	13846.21	0.1520271	2093.043	318.1993

we can see that :

females number of claims are more than males.
females claims cost are more than males but; the average severity is less ,and this is due to the larger number of claims(more than males)
both females and males almost have the same frequency.
female risk premium is lesser than males ,and this due to the higher exposure females have.

the below chart visualize all together for more clear understanding:

veh_age factor

veh_age	claimcst0	numclaims	exposure	frequency	average_severity	risk_premium
3	2718237	1446	9542.111	0.1515388	1879.832	284.8675
2	2486217	1354	7923.677	0.1708803	1836.202	313.7706
4	2554895	1261	8996.079	0.1401722	2026.087	284.0010
1	1555255	876	5338.951	0.1640772	1775.405	291.3034

we can see that :

the claims cost for the vehicle age 1 is the lowest while for vehicle age 3 is the highest.
vehicle age 1 has the lowest number on claims while the vehicle age 3 has the highest.
the exposure for the vehicle age 1 is the lowest while for vehicle age 3 is the highest.
the frequency for the vehicle age 4 is the lowest while for vehicle age 2 is the highest.
the average_severity for the vehicle age 1 is the lowest while for vehicle age 4 is the highest.
the risk premium for the vehicle age 3 & 4 is the lowest and almost the same, while its the highest for vehicle age 2

the below chart visualize all together for more clear understating :

veh_body factor

veh_body	claimcst0	numclaims	exposure	frequency	average_severity	risk_premium
HBACK	2589136.192	1330	8810.31348	0.1509594	1946.7189	293.8756
UTE	597208.965	276	2105.73032	0.1310709	2163.8006	283.6113
STNWG	2363091.211	1248	7638.39014	0.1633852	1893.5026	309.3703
HDTOP	294811.869	136	783.29911	0.1736246	2167.7343	376.3720
PANVN	133113.412	68	409.16085	0.1661938	1957.5502	325.3327
SEDAN	2681622.477	1598	10444.59959	0.1529977	1678.1117	256.7473
TRUCK	319496.849	130	843.96441	0.1540349	2457.6681	378.5667
COUPE	187723.251	75	319.12663	0.2350164	2502.9767	588.2406
MIBUS	116104.880	45	316.84052	0.1420273	2580.1084	366.4458
MCARA	10673.950	15	59.27995	0.2530367	711.5967	180.0601
BUS	13363.120	10	25.84805	0.3868764	1336.3120	516.9876
CONVT	6888.810	3	32.59685	0.0920334	2296.2700	211.3336
RDSTR	1369.458	3	11.66872	0.2570976	456.4861	117.3615

We can see that:

sedan,Hback,and Stnwg have the highest claims cost (), while RDSTR has the lowest claims cost.
RDSTR ,convt,Bus,MCARA,COUPE & PANVN have less than 100 claims, SEDAN,STNWG & HBACK have more than 1000 claims ,other have claims between 130 and 276.
BUS has the highest frequency with 38.3% , (RDSTR,MCARA, and COUPE have frequincies (27.7%,25.3% and 23.5%), other groups have frequencies less than 18%.
RDSTR has the lowest average severity , (MIBUS,COUPE and TRUCK ) have the highest.
the risk premium is the less than 200 for RDSTR and MCARA, the more tgan 500 for BUS and COUPE.

the below chart visualize all together for more clear understanding:

area factor

area	claimcst0	numclaims	exposure	frequency	average_severity	risk_premium
C	2865707.2	1493	9578.494	0.1558700	1919.429	299.1814
A	2071765.6	1181	7597.101	0.1554540	1754.247	272.7048
E	868822.9	413	2771.866	0.1489971	2103.687	313.4434
D	911058.2	524	3819.518	0.1371901	1738.661	238.5270
B	1795295.2	1021	6297.848	0.1621189	1758.369	285.0649
F	801955.4	305	1735.992	0.1756921	2629.362	461.9581

we can see that :

the lowest claims cost is for area F , and the highest cost is for area C.
area F has the lowest number of claims , while area C has the highest .
area D has the lowest frequency , while area F has the highest.
area D has the lowest average frequency ,while area F has the highest.
area D has the lowest risk premium , while area F has the highest.

the below chart visualize all together for more clear understanding:

age category factor

agecat	claimcst0	numclaims	exposure	frequency	average_severity	risk_premium
2	1984840.8	1000	5891.871	0.1697254	1984.841	336.8778
4	2145303.0	1185	7616.542	0.1555824	1810.382	281.6636
6	683568.5	390	3099.666	0.1258200	1752.740	220.5297
3	2132107.1	1189	7409.457	0.1604706	1793.194	287.7549
5	1061412.2	648	5171.009	0.1253140	1637.982	205.2621
1	1307372.9	525	2612.274	0.2009743	2490.234	500.4732

we can see that :

age category 6 has the lowest claims cost , while age category 4 has the highest
age category 6 has the lowest number of claims , while age category 4 & 3 has the highest (1185,1189) .
age category 6 & 5 has the lowest frequency with almost(12.5%) , while age category 1 has the highest with 20%.
age category 5 has the lowest average frequency ,while age category 1 has the highest.
age category 5 has the lowest risk premium , while age category 1 has the highest.

the below chart visualize all together for more clear understanding:

Auto Insurance Analysis

Author : Omar Soub

Date : 27-05/2024

Introduction

Data Set and Data types view

Data Set (1st five rows) :

data set dimensions

data set variables types

Data summary

Numerical variables EDA

Numerical_variables summary

Numerical variables Visualization

Correlation between Numerical variables

Categorical variables EDA

Categorical_variables summary:

Categorical variables Visualization

Categorical variables in terms of claims occurrence:

1. gender :

2. area :

3. agecat :

4. veh_age :

5. veh_body :

Data summary - Positive claims

Numerical variables EDA

Numerical_variables summary

Numerical variables Visualization

Correlation between Numerical variables

Categorical variables EDA

Categorical_variables summary:

Categorical variables Visualization

Inferential Analysis

choosing the Right statistics tests

claimcst0 variable Normality check by conducting Visualization

claims variable Normality check by Applying Anderson-Darling normality test

Aplying non-parametric statistical tests

A. Mann-Whitney U Test

B. kruskal.test

B.1 kruskal.test on claims against area

B.2 kruskal.test on claims against veh_body

B.3 kruskal.test on claims against age category

B.4 kruskal.test on claims against number of claims

B.5 kruskal.test on claims against veh_age

Insurance risk measurements by factors.

gender factor

veh_age factor

veh_body factor

area factor

age category factor