# A.2. Anderson - GLMs

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1. Modeling techniques
• one-way analysis: simple to calculate, intuitive, but hides impact of correlated variables (age vs vehicle age) and inter-dependencies between variables in the way they impact modeled variable (male-female vs age)
• minimum bias method: adjusts for correlated variables, but unable to test significance of variables, get a range for the parameters, or assess quality of the model
• classical linear models: adjusts for correlated variables, has statistical basis but has restrictive assumptions about distribution of dependent variables
• generalized linear models: same as linear models but with less restrictions, but still more complex and difficult to explain results
2. Linear models
• express the relationship between observed response variable Y and covariates X
• Y = µ + ɛ, where µ = X β, ɛ is normally distributed (0, σ2)
3. Components of a linear model
• (LM 1) random component: each component of Y is independent and normally distributed with common variance σ2
• (LM 2) systematic component: the p covariates are combined to give the linear predictor η = X β
• (LM 3) link function: relationship between random & systematic component (1 for linear)
4. Limitations of linear models
• difficult to assert normality and constant variance for response variables
• values for the response variable may be restricted to be > 0 (can't be normally distributed)
• if response variable is > 0 then variance of Y → 0 as E(Y) → 0
• model can't work with multiplicative predictors
5. Define GLM
Statistical method to measure the relationship that a function of a linear combination of one or more explanatory variables has on a single dependent random variable that is assumed to come from the exponential family of distributions.
6. Benefits of GLMs
• statistical framework allows for explicit assumptions about the nature of the data and its relationship with predictive variables
• method of solving GLMs is more technically efficient than iterative methods
• GLMs provide statistical diagnostics which aid in selecting only significant variables and validating model assumptions
• adjusts for correlations between variables and allows for interaction effects
7. Components of a generalized linear model
• from linear models, remove assumptions of normality, constant variance, additivity
• (GLM 1) random component: each component of Y is independent and from one of the exponential family of distributions
• (GLM 2) systematic component: the p covariates are combined to give the linear predictor η = X β
• (GLM 3) link function: relationship between random and systematic components is specified via link function g that’s differentiable and monotonic such that E[Y] = µ = g-1(η)
8. Properties of members of the exponential families
• distribution is completely specified by mean and variance
• the variance of Yi is a function of its mean, with Var(Yi) = 𝜙V(µi) / ωi
• ω is the prior weight, usually exposure so the model is more responsive to credible data
9. Common Exponential Family Variance Function
• Normal: V(x) = 1
• Poisson: V(x) = x
• Gamma: V(x) = x2
• Inverse Gaussian: V(x) = x3
• Binomial: V(x) = x (1-x)
• Normal assumes each observation is attracted to the original data point with equal weight (all observations have same weight)
• Poisson, Gamma variable function assumes the variance increases with the expected value (gives less weight to observations with high expected values)
10. Tweedie distribution
• special case of V(x) = (1/λ) xp, where p<0 or 1<p<2 or p>2
• good for pure premium, allows a large point mass at zero (many policies have no claims)
• for 1<p<2, ≈ compound distribution of Poisson (frequency) and Gamma (severity)
11. Solving a CLM
• write the general equation with Y, β, X, ɛ
• write actual equations for each i
• solve each equation for ɛi
• define SSE equation = ∑ ɛi2
• minimize SSE (set derivative with respect to βi equal to 0)
• solve for values of β
12. Solving a GLM
• identify the likelihood function
• take the log to turn the product of several items into a sum
• maximize the log of the likelihood function and set the partial derivatives for each parameter to zero
• solve the resulting system of equations
• compute the predicted values
13. The scale parameter ϕ
• for Poisson, ϕ = 1
• for other distributions, ϕ not known (estimated from data)
• it is not necessary to know ϕ to estimate β’s, but it’s used for statistical assessments (SSE)
14. Methods to evaluate 𝜙
• maximum likelihood: not feasible in practice (no explicit formula)
• Pearson 𝝌2 statistic: 𝜙(hat) = 1/(n - p) ∑ [ ω(Yi - μi)2 / V(μi) ]
• total deviance estimator: 𝜙(hat) = D / (n - p), D given or equal to SSE in classical LM
• LM / GLM requires Y / g(Y) to be additive
• link function must be differentiable and monotonic (strictly increasing or decreasing)
• Identity g(x) = g-1(x) = x. Used for classical linear model
• Log g(x) = ln(x); g-1(x) = ex. Common as it makes everything multiplicative
• Logit g(x) = ln(x/(1-x)); g-1(x) = ex/(1+ex); Used for retention or 0 < probability < 1
• Reciprocal g(x) = g-1(x) = 1/x
16. Offset term
• used to fix the impact of an explanatory variable
• offset term is known (not estimated), and different for each observation
• common use is when fitting a multiplicative GLM to an observed number/count, use ƹi = ln(xi)
• η = X β + ƹ
17. Typical GLM models for insurance
• frequency: multiplicative Poisson (Log link function, Poisson error term); invariant to time, i.e. frequency by month/year is the same. ωi is usually set as the exposure (or offset = log of exposure for claim count)
• severity: multiplicative Gamma (Log link function, Gamma error term); invariant to currency
• pure premium: Tweedie (compound of Poisson and Gamma)
• probability (e.g. retention): Logistic (Logit link and binomial error term)
18. GLM Summary
• μi = E[Yi] = g-1( ∑Xij βj + ƹi)
• Var[Yi] = 𝜙V(μi) / ωi
• Yi is the vector of responses
• g(x) is the link function; relates expected response to linear combination of observed factors
• Xij is the design matrix produced from the factors
• βi is a vector of model parameters (estimated)
• ƹi is a vector of known effects or "offsets"
• 𝜙 is a parameter to scale the variance function V(x)
• ωi is the prior weight assigning a credibility to each observation
19. Aliasing
• aliasing: linear dependency among observed covariates resulting in a model that's not uniquely defined
• intrinsic: dependencies inherent in the Xi definition (e.g. can only be in 1 group at a time)
• extrinsic: dependency resulting from the nature of the data (e.g. same unknowns)
• near aliasing: occurs when factors are almost but not quite correlated
20. Correcting for aliasing
• aliasing: remove covariates that create a linear dependency until there are fewer parameters than unique levels of the variables
• near-aliasing: re-classify or delete problematic observations
21. Model diagnostics (testing variables in the model)
• standard error / size of confidence interval: visually analyzing speed with which log-likelihood falls from the maximum given a change in parameter
• deviance / type III test: create 2 models, with and without parameters to be tested
• 𝝌2 test: D1 - D2 compared to the 𝝌2 distribution with df1 - df2 d.f. (larger ✓)
• F-test: (D1 - D2) / [(df1 - df2)D2 / df2] compare to F with df1 - df2, df2 d.f. (larger ✓)
• testing parameter consistency over time
• intuition: whether we expect a factor to impact the results

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 Author: EExam8 ID: 304885 Filename: A.2. Anderson - GLMs Updated: 2015-09-02 23:30:10 Tags: Anderson GLM Folders: Description: Anderson GLM Show Answers:

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