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
  15. Link functions
    • 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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A.2. Anderson - GLMs
2015-09-02 23:30:10
Anderson GLM

Anderson GLM
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