Frequency analysis of an Australian Motor Third Party Liability dataset

Introduction

# Graphs----
face_text='plain'
face_title='plain'
size_title = 14
size_text = 11
legend_size = 11

global_theme <- function() {
  theme_minimal() %+replace%
    theme(
      text = element_text(size = size_text, face = face_text),
      legend.position = "bottom",
      legend.direction = "horizontal", 
      legend.box = "vertical",
      legend.key = element_blank(),
      legend.text = element_text(size = legend_size),
      axis.text = element_text(size = size_text, face = face_text), 
      plot.title = element_text(
        size = size_title, 
        hjust = 0.5
      ),
      plot.subtitle = element_text(hjust = 0.5)
    )
}

# Outputs
options("digits" = 2)

In Brief

The objective of this vignette is to demonstrate the application of Quasi-Poisson regression in analyzing insurance data, specifically focusing on the ausprivauto0405 dataset from Charpentier (2014). This dataset provides information on insurance contracts and claims related to Australian motor third-party liability insurance.

By leveraging Quasi-Poisson regression, our goal is to model the frequency of claims and investigate the factors influencing claim occurrence within the insurance data.

Required Packages

required_libraries <- c(
  "tidyverse", 
  "CASdatasets",
  "MASS",
  "AER",
  "broom",
  "knitr",
  "kableExtra"
)
invisible(lapply(required_libraries, library, character.only = TRUE))

Data

The data used in this vignette come from the Australian motor third-party liability insurance portfolio.

The first dataset, ausprivauto0405, encompasses details regarding contracts and clients obtained from an Australian insurance company, related to some motor insurance portfolio. Third-party insurance is a compulsory insurance for vehicle owners in Australia. It insures vehicle owners against injury caused to other drivers, passengers, or pedestrians as a result of an accident.

For clarity purposes, the ausprivauto0405 table will be named CLAIMS.

Dictionaries

The list of the 9 variables from the freMTPLfreq dataset is reported in Table 1.

Table 1: Content of the CLAIMS dataset: ausprivauto0405

Attribute	Type	Description
Exposure	Numeric	The number of policy years
VehValue	Numeric	The vehicle value in thousands of AUD
VehAge	Factor	The vehicle age group
VehBody	Factor	The vehicle body group
Gender	Factor	The gender of the policyholder
DrivAge	Numeric	The age of the policyholder
ClaimOcc	Factor	Indicates occurrence of a claim
ClaimNb	Numeric	The number of claims
ClaimAmount	Numeric	The sum of claim payments

Importation

data(ausprivauto0405)


CLAIMS <- ausprivauto0405 |>
  filter(Exposure > 0.70)

CLAIMS$VehAge <- CLAIMS$VehAge |> 
  factor(levels = c("youngest cars", "young cars", "old cars", "oldest cars"))


age_mapping <- c("youngest people" = 1, "young people" = 2, "working people" = 3,
                 "older work. people" = 4, "old people" = 5, "oldest people" = 6)

drivage_levels <- levels(CLAIMS$DrivAge)


CLAIMS$DrivAge <- CLAIMS$DrivAge |>
  levels() |>
  (\(drivage_levels) order(sapply(drivage_levels, function(x) age_mapping[x])))() |>
  (\(drivage_order) factor(CLAIMS$DrivAge, levels = drivage_levels[drivage_order]))()

Models

Purpose

In the domain of automobile insurance, Quasi-Poisson Regression emerges as a potent tool for understanding and forecasting accident frequencies, repair costs, and claims trends.

Through Quasi-Poisson Regression, insurers gain the ability to not only anticipate forthcoming challenges but also refine pricing strategies and ensure resilience in a dynamic landscape of risk.

Pay Attention

The results from QuasiPoisson regression models are valid if:

• the responses are independent.
  
• the responses are distributed according to a Poisson distribution with parameter Lambda.
  
• There may be overdispersion present in the data. Quasi-Poisson regression models are appropriate for handling situations where the variance exceeds the mean in the data.

In this analysis, we explore the relationship between the response variable ClaimNb (the number of insurance claims) and the explanatory variables DrivAge (driver age) and VehAge (vehicle age). This modeling framework aligns with the principles outlined by Agresti (2013), a prominent figure in statistical methodology, who emphasizes the significance of considering multiple explanatory factors in regression analysis.

Introduction to Quasi-Poisson Regression

To model the frequency of insurance claims, we employ a Quasi-Poisson regression approach. The response variable, ClaimNb, represents the count of insurance claims and is assumed to follow a Quasi-Poisson distribution:

\[ \text{ClaimNb} \sim \text{QuasiPoisson}(\lambda), \]

where \(\lambda\) is the mean rate of claims. Unlike the standard Poisson regression, the Quasi-Poisson model is particularly useful when the data exhibits overdispersion—meaning the variance of ClaimNb is greater than the mean. This model adjusts for this overdispersion, ensuring that the estimates of variance are accurate, leading to more reliable inferences.

Model Specification

The Quasi-Poisson regression model relates \(\lambda\) to a set of predictor variables and an additional term accounting for exposure through a logarithmic link function. The logarithmic link function ensures that the predicted rate of claims is always positive, as required by the Quasi-Poisson distribution. More precisely, the natural logarithm of \(\lambda\) is expressed as a linear combination of the predictors:

\[ \log(\lambda) = \beta_0 + \beta_1 \times \text{DrivAge} + \beta_2 \times \text{VehAge} + \log(\text{Exposure}), \]

Explanation of the Model Components

• \(\beta_0\): This is the intercept term, representing the log of the expected number of claims when all predictors are at their reference levels.
• \(\beta_1\): The coefficient for DrivAge, representing the change in the log expected number of claims for each one-unit increase in the driver’s age.
• \(\beta_2\): The coefficient for VehAge, indicating the change in the log expected number of claims for each one-unit increase in the vehicle’s age.
• \(\log(\text{Exposure})\): An offset term to adjust for varying levels of exposure across observations. This could represent differences in policy duration, the amount of coverage, or other factors that influence the level of risk exposure.

Addressing Overdispersion

In many real-world datasets, especially in insurance claims data, the variance often exceeds the mean, leading to overdispersion. The Quasi-Poisson model accounts for this by introducing a dispersion parameter \(\phi\), which scales the variance:

\[ \text{Var}(Y) = \phi \cdot \lambda, \]

where \(\phi > 1\) indicates the presence of overdispersion. This adjustment makes the model more robust and ensures that standard errors and confidence intervals are correctly estimated.

Practical Applications

• Risk Assessment: By understanding the relationship between claim frequency and variables such as DrivAge and VehAge, insurers can more accurately assess risk levels across different policyholders.
• Pricing Strategies: The insights gained from the Quasi-Poisson regression model can inform pricing strategies, helping insurers set premiums that reflect the underlying risk more accurately.
• Claims Management: Identifying key factors that drive claim frequencies allows insurers to implement targeted interventions, such as promoting safer driving habits among younger drivers or encouraging the use of newer, safer vehicles.

Conclusion

The Quasi-Poisson regression model is a powerful tool for analyzing insurance claims data, particularly when dealing with overdispersion. By adjusting for the extra variability in the data, this model provides more reliable and accurate estimates, which are crucial for effective risk management and decision-making in the insurance industry.

The coefficients \(\beta_0\), \(\beta_1\), and \(\beta_2\) are estimated through regression to quantify their impact on the expected rate of claims. This model not only improves the understanding of the factors influencing claim frequencies but also enhances the insurer’s ability to make informed decisions.

The estimated lambda parameter, which represents the mean of claims, is: 0.12.

set.seed(1234) 

theoretic_count <- rpois(nrow(CLAIMS), mean(CLAIMS$ClaimNb))

tc_df <- tibble(theoretic_count)

freq_theoretic <- prop.table(table(tc_df$theoretic_count))

freq_claim <- prop.table(table(CLAIMS$ClaimNb))

freq_theoretic_df <- tibble(
  Count = as.numeric(names(freq_theoretic)),
  Frequency = as.numeric(freq_theoretic),
  Source = "Theoretical Count"
)

freq_claim_df <- tibble(
  Count = as.numeric(names(freq_claim)),
  Frequency = as.numeric(freq_claim),
  Source = "Empirical Count"
)

freq_combined <- freq_theoretic_df |> 
  rbind(freq_claim_df)

The theoretical and empirical histograms associated with a Poisson distribution are shown in Figure 1.

ggplot(freq_combined, aes(x = Count, y = Frequency, fill = Source)) +
  geom_bar(stat = "identity", position = "dodge2", width = 0.3) +
  labs(x = "Claim Number", y = "Frequency", fill = "Legend") +
  theme(legend.position = "right") +
  scale_fill_manual(
    NULL,
    values = c("Empirical Count" = "black", "Theoretical Count" = "#1E88E5")
  ) +
  labs(fill = "Legend") +
  labs(x = "Claim Number", y = NULL) +
  theme(legend.position = "right")+
  global_theme()

Figure 1: Theoretical and empirical histogram of claims in frequence

freg <- formula(ClaimNb ~ DrivAge + VehAge + offset(log(Exposure)))
  
reg <- glm(freg, family = quasipoisson, data = CLAIMS)

summary(reg)

Call:
glm(formula = freg, family = quasipoisson, data = CLAIMS)

Coefficients:
                          Estimate Std. Error t value Pr(>|t|)    
(Intercept)                -1.6570     0.0868  -19.09  < 2e-16 ***
DrivAgeyoung people        -0.0318     0.0887   -0.36  0.71960    
DrivAgeworking people      -0.1535     0.0872   -1.76  0.07823 .  
DrivAgeolder work. people  -0.1175     0.0865   -1.36  0.17424    
DrivAgeold people          -0.3317     0.0947   -3.50  0.00046 ***
DrivAgeoldest people       -0.2878     0.1054   -2.73  0.00631 ** 
VehAgeyoung cars           -0.0698     0.0684   -1.02  0.30743    
VehAgeold cars             -0.1885     0.0678   -2.78  0.00547 ** 
VehAgeoldest cars          -0.2817     0.0700   -4.03  5.7e-05 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

(Dispersion parameter for quasipoisson family taken to be 1.1)

    Null deviance: 9444.4  on 17519  degrees of freedom
Residual deviance: 9396.8  on 17511  degrees of freedom
AIC: NA

Number of Fisher Scoring iterations: 6

This is a Quasi-Poisson regression model predicting ClaimNb (number of claims) using DrivAge (driver age) and VehAge (vehicle age) as predictors. The model coefficients indicate the change in the log count of claims associated with each predictor level compared to a reference level.

For instance, as DrivAge transitions from the youngest category to the old people category, the log count of claims decreases by 0.33. This suggests that older drivers are associated with fewer claims compared to younger drivers.

Similarly, as VehAge increases within each category, the log count of claims also decreases, indicating that older vehicles tend to be involved in fewer claims.

Most of the coefficients in the model are statistically significant.

Coefficients

reg_coef <- tidy(reg)


reg_coef <- reg_coef |>
  mutate(significance = case_when(
    p.value < 0.001 ~ "***",
    p.value < 0.01 ~ "**",
    p.value < 0.05 ~ "*",
    p.value < 0.01 ~ ".",
    TRUE ~ ""
  ))

kable(reg_coef, format = "html", escape = FALSE) |>
  kable_styling(full_width = FALSE) |>
  add_footnote(c("Significance levels: *** p < 0.001, ** p < 0.01, * p < 0.05, . p.value < 0.1"), notation = "none")

Table 2: Coefficients

term	estimate	std.error	statistic	p.value	significance
(Intercept)	-1.66	0.09	-19.09	0.00	***
DrivAgeyoung people	-0.03	0.09	-0.36	0.72
DrivAgeworking people	-0.15	0.09	-1.76	0.08
DrivAgeolder work. people	-0.12	0.09	-1.36	0.17
DrivAgeold people	-0.33	0.09	-3.50	0.00	***
DrivAgeoldest people	-0.29	0.11	-2.73	0.01	**
VehAgeyoung cars	-0.07	0.07	-1.02	0.31
VehAgeold cars	-0.19	0.07	-2.78	0.01	**
VehAgeoldest cars	-0.28	0.07	-4.03	0.00	***
Significance levels: *** p < 0.001, ** p < 0.01, * p < 0.05, . p.value < 0.1

Count-Ratio

reg_count_ratio <- tidy(exp(coef(reg)[-1]))

reg_count_ratio <- reg_count_ratio |>
  mutate(p.value = reg_coef$p.value[-1]) |>
  mutate(significance = case_when(
    p.value < 0.001 ~ "***",
    p.value < 0.01 ~ "**",
    p.value < 0.05 ~ "*",
    TRUE ~ ""
  )) |>
  dplyr::select(-p.value)

kable(reg_count_ratio, format = "html", escape = FALSE) |>
  kable_styling(full_width = FALSE) |>
  add_footnote(c("Significance levels: *** p < 0.001, ** p < 0.01, * p < 0.05"), notation = "none")

Table 3: Count Ratio

names	x	significance
DrivAgeyoung people	0.97
DrivAgeworking people	0.86
DrivAgeolder work. people	0.89
DrivAgeold people	0.72	***
DrivAgeoldest people	0.75	**
VehAgeyoung cars	0.93
VehAgeold cars	0.83	**
VehAgeoldest cars	0.75	***
Significance levels: *** p < 0.001, ** p < 0.01, * p < 0.05

Each count ratio represents the change in the count of making a claim associated with a one-unit increase in the predictor variable, compared to the reference category DrivAge youngest people. For example, a count ratio of -0.33 for DrivAge old people implies that the count of making a claim for individuals considered as old is approximately 28% lower compared to the reference category.

Similarly, count ratios below 1 for VehAge categories suggest a decrease in the count of making a claim as the vehicle age increases.

Confidence intervals

reg_conf_int <- as.data.frame(exp(confint(reg))[-1, ])

Waiting for profiling to be done...

colnames(reg_conf_int) <- c("2.5 %", "97.5 %")

reg_conf_int <- reg_conf_int |>
  mutate(p.value = reg_coef$p.value[-1]) |>
  mutate(significance = case_when(
    p.value < 0.001 ~ "***",
    p.value < 0.01 ~ "**",
    p.value < 0.05 ~ "*",
    TRUE ~ ""
  )) |>
  dplyr::select(-p.value)

kable(reg_conf_int, format = "html", escape = FALSE) |>
  kable_styling(full_width = FALSE) |>
  add_footnote(c("Significance levels : *** p < 0.001, ** p < 0.01, * p < 0.05"), notation = "none")

Table 4: Confidence intervals

	2.5 %	97.5 %	significance
DrivAgeyoung people	0.82	1.15
DrivAgeworking people	0.72	1.02
DrivAgeolder work. people	0.75	1.06
DrivAgeold people	0.60	0.87	***
DrivAgeoldest people	0.61	0.92	**
VehAgeyoung cars	0.82	1.07
VehAgeold cars	0.73	0.95	**
VehAgeoldest cars	0.66	0.87	***
Significance levels : *** p < 0.001, ** p < 0.01, * p < 0.05

Graphs

DriverAge

count_ratio <- exp(coef(reg)[-1])
conf_int <- exp(confint(reg))[-1, ]

driver_age_vars <- grep("^DrivAge", names(count_ratio), value = TRUE)

data_age <- tibble(
  variable = driver_age_vars,
  coefficient = count_ratio[driver_age_vars], 
  lower_bound = conf_int[driver_age_vars, 1], 
  upper_bound = conf_int[driver_age_vars, 2]
)

driver_vehage_vars <- grep("^VehAge", names(count_ratio), value = TRUE)

data_vehage <- tibble(
  variable = driver_vehage_vars,
  coefficient = count_ratio[driver_vehage_vars], 
  lower_bound = conf_int[driver_vehage_vars, 1], 
  upper_bound = conf_int[driver_vehage_vars, 2]
)

data_age$variable <- factor(data_age$variable, levels = rev(driver_age_vars))

ggplot(
  data_age, 
  aes(
    x = coefficient,
    y = variable,
    xmin = lower_bound,
    xmax = upper_bound
  )
) +
  geom_point(stat = "identity", size = 3, color = "#1E88E5") +
  geom_errorbar(
    width = 0.2,
    position = position_dodge(width = 0.6),
    color = "#1E88E5"
  ) +
  labs(
    x = NULL,
    y = NULL
  ) +
  global_theme()

Figure 2: Count ratio and confidence interval of Driver Age

VehicleAge

data_vehage <- data_vehage |> 
  mutate(variable = reorder(variable, coefficient, decreasing = FALSE))

ggplot(
  data_vehage, 
  aes(
    x = coefficient,
    y = variable,
    xmin = lower_bound,
    xmax = upper_bound
  )
) +
  geom_point(
    stat = "identity",
    size = 3,
    color = "#1E88E5"
  ) +
  geom_errorbar(
    width = 0.2,
    position = position_dodge(width = 0.6),
    color = "#1E88E5"
  ) +
  labs(
    x = NULL,
    y = NULL
  ) +
  global_theme()

Figure 3: Count ratio and confidence interval of vehicle Age

References

Agresti, Alan. 2013. Categorical Data Analysis, 3rd Edition.

Charpentier, Arthur. 2014. Computational Actuarial Science with R. The R Series. Chapman; Hall/CRC. https://www.routledge.com/Computational-Actuarial-Science-with-R/Charpentier/p/book/9781138033788.

See also

For more similar claim frequency datasets with a Poisson-like distribution, see freMTPL (import with data("freMTPLfreq")): French automobile dataset, norauto: Norwegian automobile dataset (import with data("norauto")), beMTPL16: Belgian automobile dataset (import with data("beMTPL16")), or pg17trainpol (import with data("pg17trainpol")).