6.5 Generalized linear models and rating factors

Generalized linear models (GLMs) are a powerful tool in actuarial mathematics. They extend linear regression to handle non-normal distributions, making them ideal for modeling insurance data. GLMs consist of three key components: exponential family distributions, link functions, and linear predictors.

Rating factors play a crucial role in insurance pricing using GLMs. These factors capture policyholder characteristics and risk attributes, helping actuaries develop fair and accurate prices. Understanding how to interpret GLM coefficients and select appropriate models is essential for effective actuarial analysis and decision-making.

Components of generalized linear models

• Generalized linear models (GLMs) extend the concept of linear regression to allow for response variables that have non-normal distributions, providing a flexible framework for modeling a wide range of data types in actuarial mathematics
• GLMs consist of three main components: the exponential family of distributions, link functions, and linear predictors, which work together to define the relationship between the response variable and the explanatory variables

Exponential family of distributions

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• The exponential family of distributions includes a broad class of probability distributions that share certain properties, such as the normal, binomial, Poisson, and gamma distributions
• These distributions are characterized by their mean and variance, which can be expressed as functions of the natural parameter and the dispersion parameter
• The choice of distribution depends on the nature of the response variable (continuous, binary, count, etc.) and the underlying assumptions about the data generating process
• For example, the normal distribution is used for continuous response variables, while the Poisson distribution is used for modeling count data (number of claims)

Link functions

• Link functions establish a connection between the linear predictor and the expected value of the response variable, allowing for non-linear relationships
• The link function transforms the expected value of the response variable to the scale of the linear predictor, ensuring that the model's predictions are consistent with the properties of the chosen distribution
• Common link functions include the identity link for normal distribution, logit link for binomial distribution, and log link for Poisson distribution
• The choice of link function depends on the distribution and the desired interpretation of the model coefficients (additive effects, multiplicative effects, or odds ratios)

Linear predictors

• The linear predictor is a linear combination of the explanatory variables and their associated coefficients, representing the systematic component of the model
• It captures the relationship between the explanatory variables and the transformed expected value of the response variable
• The coefficients in the linear predictor quantify the effect of each explanatory variable on the response variable, while controlling for the other variables in the model
• The linear predictor can include main effects, interactions, and polynomial terms, allowing for flexible modeling of complex relationships

Model fitting and estimation

• Model fitting and estimation involve determining the values of the model coefficients that best describe the relationship between the explanatory variables and the response variable, based on the observed data
• The process of fitting a GLM typically involves maximum likelihood estimation, iterative weighted least squares, and assessing the model's goodness of fit using deviance and other measures

Maximum likelihood estimation

• Maximum likelihood estimation (MLE) is a statistical method used to estimate the model coefficients by maximizing the likelihood function, which measures the probability of observing the data given the model parameters
• MLE finds the set of coefficients that make the observed data most likely, assuming that the data are generated from the specified exponential family distribution and link function
• The likelihood function is constructed based on the chosen distribution and link function, and optimization algorithms (such as Newton-Raphson or Fisher scoring) are used to find the maximum likelihood estimates
• MLE provides asymptotically unbiased, consistent, and efficient estimates of the model coefficients, under certain regularity conditions

Iterative weighted least squares

• Iterative weighted least squares (IWLS) is an algorithm used to solve the maximum likelihood estimation problem for GLMs, by iteratively updating the coefficient estimates and the weights assigned to each observation
• IWLS transforms the GLM into a weighted least squares problem, where the weights are determined by the current estimates of the mean and variance of the response variable
• At each iteration, the algorithm computes the working response and weights based on the current coefficient estimates, fits a weighted least squares regression, and updates the coefficients using the results
• The process is repeated until the coefficient estimates converge, typically requiring a small number of iterations to achieve a satisfactory level of accuracy

Deviance and goodness of fit

• Deviance is a measure of the discrepancy between the fitted model and the saturated model (a model with a separate parameter for each observation), used to assess the goodness of fit of a GLM
• It compares the log-likelihood of the fitted model to that of the saturated model, with smaller deviance indicating a better fit to the data
• The deviance follows an approximate chi-square distribution with degrees of freedom equal to the difference in the number of parameters between the saturated and fitted models
• Other goodness-of-fit measures for GLMs include the Pearson chi-square statistic, the Akaike information criterion (AIC), and the Bayesian information criterion (BIC), which balance the model's fit and complexity

Types of generalized linear models

• GLMs encompass a wide range of models that can be used to analyze different types of response variables, including continuous, binary, and count data
• The choice of the specific GLM depends on the nature of the response variable and the research question of interest, with linear regression, logistic regression, and Poisson regression being among the most commonly used types in actuarial practice

Linear regression models

• Linear regression models are used when the response variable is continuous and normally distributed, assuming a linear relationship between the explanatory variables and the response
• The identity link function is used, equating the expected value of the response variable to the linear predictor
• The coefficients in a linear regression model represent the change in the expected value of the response variable for a one-unit change in the corresponding explanatory variable, holding other variables constant
• Linear regression models are widely used in actuarial applications, such as modeling claim severity or loss reserves, where the response variable is a continuous monetary amount

Logistic regression models

• Logistic regression models are used when the response variable is binary or categorical, such as the occurrence or non-occurrence of an event (claim, policy lapse, etc.)
• The logit link function is used, relating the log-odds of the event to the linear predictor
• The coefficients in a logistic regression model represent the change in the log-odds of the event for a one-unit change in the corresponding explanatory variable, holding other variables constant
• Logistic regression models are commonly used in actuarial practice for modeling claim frequency, policy retention, or risk classification, where the focus is on predicting the probability of an event occurring

Poisson regression models

• Poisson regression models are used when the response variable is a count, such as the number of claims or the frequency of events over a fixed period
• The log link function is used, relating the logarithm of the expected count to the linear predictor
• The coefficients in a Poisson regression model represent the change in the log of the expected count for a one-unit change in the corresponding explanatory variable, holding other variables constant
• Poisson regression models are frequently used in actuarial applications, such as modeling claim frequency or the number of policy renewals, where the response variable is a non-negative integer

Interpretation of model coefficients

• Interpreting the coefficients of a GLM is crucial for understanding the relationship between the explanatory variables and the response variable, as well as for making inferences and predictions based on the model
• The interpretation of coefficients depends on the type of GLM, the link function, and the scale of the explanatory variables, and can be facilitated by significance testing, confidence intervals, and exponentiation

Significance testing

• Significance testing is used to assess the statistical significance of individual coefficients in a GLM, determining whether the observed relationship between an explanatory variable and the response variable is likely to have occurred by chance
• Hypothesis tests, such as the Wald test or the likelihood ratio test, are used to compare the fitted model to a reduced model without the coefficient of interest
• The p-value associated with each coefficient indicates the probability of observing a relationship as strong as the one in the sample data, assuming that there is no true relationship in the population
• Coefficients with p-values below a chosen significance level (e.g., 0.05) are considered statistically significant, providing evidence against the null hypothesis of no relationship

Confidence intervals

• Confidence intervals provide a range of plausible values for each coefficient in a GLM, quantifying the uncertainty associated with the point estimates
• A confidence interval is typically constructed using the point estimate and its standard error, based on the asymptotic normality of the maximum likelihood estimator
• For example, a 95% confidence interval indicates that if the model fitting process were repeated many times, 95% of the resulting intervals would contain the true value of the coefficient
• Confidence intervals that do not include zero suggest that the corresponding explanatory variable has a significant relationship with the response variable, consistent with the results of hypothesis testing

Exponentiated coefficients

• Exponentiated coefficients, also known as odds ratios or rate ratios, provide a more intuitive interpretation of the coefficients in GLMs with non-identity link functions, such as logistic or Poisson regression
• For a logistic regression model, the exponentiated coefficient represents the multiplicative change in the odds of the event for a one-unit increase in the corresponding explanatory variable, holding other variables constant
• For a Poisson regression model, the exponentiated coefficient represents the multiplicative change in the expected count for a one-unit increase in the corresponding explanatory variable, holding other variables constant
• Exponentiated coefficients are easier to interpret than the raw coefficients, as they express the relationship between the explanatory variables and the response variable on the original scale of the data

Rating factors in insurance pricing

• Rating factors are the explanatory variables used in GLMs for insurance pricing and ratemaking, capturing the characteristics of policyholders, insured objects, or coverage that are associated with the risk of claims or losses
• The selection and inclusion of rating factors in a GLM are guided by actuarial judgment, regulatory constraints, and statistical considerations, with the goal of developing fair, accurate, and competitive prices

Categorical vs continuous factors

• Rating factors can be either categorical or continuous, depending on the nature of the underlying variable and the granularity of the available data
• Categorical factors, such as gender, occupation, or vehicle type, take on a finite number of distinct values or levels, and are typically represented using dummy variables in the GLM
• Continuous factors, such as age, driving experience, or sum insured, can take on any value within a given range and are directly included in the linear predictor
• The choice between treating a factor as categorical or continuous depends on the relationship between the factor and the response variable, the sample size, and the desired interpretability of the model

Interactions between factors

• Interactions between rating factors occur when the effect of one factor on the response variable depends on the level of another factor
• Including interaction terms in a GLM allows for a more flexible and accurate representation of the complex relationships between the explanatory variables and the response variable
• For example, the interaction between age and gender might be significant in a model for life insurance pricing, as the effect of age on mortality risk may differ for males and females
• Interactions can be specified as products of the corresponding main effects, and their coefficients represent the additional effect of the interaction over and above the main effects

Relativities and factor levels

• Relativities are the exponentiated coefficients associated with the levels of a categorical rating factor, representing the relative impact of each level on the response variable, compared to a chosen reference level
• For example, in a GLM for auto insurance pricing, the relativities for different vehicle types (e.g., sports car, sedan, SUV) would indicate the expected claim frequency or severity for each type, relative to a base vehicle type
• Factor levels are the specific values or categories of a rating factor, and their definition and granularity can have a significant impact on the model's fit and interpretability
• The choice of factor levels involves balancing the need for detailed risk differentiation with the availability of data and the simplicity of the rating structure

Model selection and validation

• Model selection and validation are essential steps in the development of GLMs for actuarial applications, ensuring that the chosen model is parsimonious, accurate, and generalizable to new data
• The process involves comparing alternative model specifications, assessing their relative performance, and testing their predictive ability using techniques such as stepwise selection, cross-validation, and information criteria

Stepwise selection procedures

• Stepwise selection procedures are algorithmic approaches to model selection that iteratively add or remove explanatory variables from the GLM based on their statistical significance or contribution to the model's fit
• Forward selection starts with an empty model and sequentially adds the most significant variable at each step until no further improvement can be achieved
• Backward elimination starts with a full model containing all potential explanatory variables and sequentially removes the least significant variable at each step until all remaining variables are significant
• Stepwise selection combines forward selection and backward elimination, allowing for both the addition and removal of variables at each step, based on a set of predefined criteria (e.g., p-value thresholds, AIC, or BIC)

Cross-validation techniques

• Cross-validation is a model validation technique that assesses the performance and generalizability of a GLM by repeatedly fitting the model to subsets of the available data and evaluating its predictive accuracy on the remaining observations
• K-fold cross-validation divides the data into K equally sized subsets, and iteratively uses each subset as a validation set while fitting the model to the remaining K-1 subsets
• Leave-one-out cross-validation (LOOCV) is a special case of K-fold cross-validation, where K is equal to the number of observations, and each observation is used as a validation set in turn
• The cross-validation error, computed as the average prediction error across all validation sets, provides an estimate of the model's performance on new, unseen data and can be used to compare different model specifications

Akaike and Bayesian information criteria

• The Akaike information criterion (AIC) and the Bayesian information criterion (BIC) are model selection criteria that balance the goodness of fit of a GLM with its complexity, penalizing models with a larger number of parameters
• AIC is defined as -2 times the log-likelihood of the model plus 2 times the number of parameters, while BIC is defined as -2 times the log-likelihood plus the number of parameters times the logarithm of the sample size
• Models with lower AIC or BIC values are preferred, as they indicate a better trade-off between fit and parsimony
• AIC and BIC can be used to compare non-nested models, such as GLMs with different link functions or distributions, and to select the most appropriate model for a given application

Assumptions and limitations

• GLMs, like all statistical models, rely on a set of assumptions about the data generating process and the relationship between the explanatory variables and the response variable
• Violating these assumptions can lead to biased or inefficient coefficient estimates, incorrect inferences, and poor model performance, making it crucial to assess and address potential issues through residual diagnostics and model refinements

Independence of observations

• GLMs assume that the observations in the data are independent, meaning that the value of the response variable for one observation is not influenced by the values of other observations
• Violation of the independence assumption, such as in the presence of clustered or longitudinal data, can lead to underestimated standard errors and overstated significance of the coefficients
• Techniques for handling non-independence include using clustered standard errors, random effects models, or generalized estimating equations (GEEs)

Overdispersion and underdispersion

• Overdispersion occurs when the variance of the response variable is greater than what is expected under the assumed distribution, while underdispersion occurs when the variance is smaller than expected
• In the context of GLMs, overdispersion is commonly encountered in Poisson regression models, where the variance of the count response may exceed the mean
• Ignoring overdispersion can lead to underestimated standard errors and overstated significance of the coefficients, while ignoring underdispersion can lead to overestimated standard errors and understated significance
• Strategies for handling overdispersion include using a quasi-Poisson or negative binomial distribution, or incorporating random effects to account for unobserved heterogeneity

Residual diagnostics

• Residual diagnostics are used to assess the adequacy of a GLM and to identify potential violations of the model assumptions, such as non-linearity, heteroscedasticity, or outliers
• Residuals are the differences between the observed values of the response variable and the values predicted by the model, and can be standardized or deviance-based to facilitate comparison across observations
• Plotting the residuals against the fitted values, the explanatory variables, or the observation order can reveal patterns that suggest model misspecification or assumption violations
• Residual diagnostics can also be used to identify influential observations or leverage points that have a disproportionate impact on the model estimates, and to guide model refinements or data preprocessing steps

Applications in actuarial practice

• GLMs have become an essential tool in actuarial practice, providing a flexible and powerful framework for modeling and analyzing insurance data
• The applications of GLMs span a wide range of actuarial tasks, from pricing and ratemaking to reserving and capital modeling, enabling actuaries to make data-driven decisions and to communicate the results to stakeholders

Pricing and ratemaking

• GLMs are widely used in insurance pricing and ratemaking to estimate the expected claim frequency and severity for individual policyholders or risk classes, based on their characteristics and exposure
• By fitting separate GLMs for frequency and severity, actuaries can develop a granular and accurate rating structure that reflects the underlying risk factors and ensures fairness and competitiveness
• GLMs allow for the inclusion of a wide range of rating factors, such as demographic, geographic, and behavioral variables, as well as interactions and non-linear effects, providing a high level of flexibility and customization
• The coefficients of the GLMs can be directly translated into relativities or base rates, which form the basis for the premium calculation and the communication of the pricing decisions to regulators, agents, and policyholders

Claim frequency and severity modeling

• GLMs are used to model claim frequency and severity separately, as these two components of the total claim cost often have different distributions and are influenced by different risk factors
• For claim frequency modeling, Poisson or negative binomial regression models are commonly used, with the log link function relating the expected number of claims to the linear predictor
• For claim