GLMs

Question 1

Advantages to GLMs

Answer 1

Adjusts for correlation, with less restrictive assumptions to classical linear models

Question 2

Disadvantages to GLMs

Answer 2

Often difficult to explain results

Question 3

Components of GLMs

Answer 3

Random component

Systematic component

Question 4

Random component of GLM

Answer 4

Each y_i assumed to be independent and come from exponential family of distributions

Mean = µ_i

Variance = [ØV(µ_i)]/w_i

Question 5

Variance within a GLM, formula

Answer 5

Variance = [ØV(µi)]/w_i

Question 6

Ø

Answer 6

Dispersion/Scale factor

Same for all observations

Question 7

V(µ)

Answer 7

Variance function, given for selected distribution type

Describes relationship between variance and mean

Same distribution type must be used for all observations

Question 8

Systematic component of GLM

Answer 8

g(µ_i) = ß₀ + ß₁x_i1 +… + ß_px_ip + offset

Allows you to manually specify estimates of variables

Question 9

Link functions, g(x)

Answer 9

Allows for transformations of linear predictor

Question 10

Link function often used for binary target variables

Answer 10

Logit link:

g(µ) = ln [µ / (1 - µ)]

Question 11

Link function used in rating plans

Answer 11

g(µ) = ln(µ)

Allows transformation of linear predictor into multiplicative structure

Question 12

Advantages of multiplicative rating plans

Answer 12

Simple/practical to implement

Guarantees positive premiums

Impact of risk characteristics more intuitive

Question 13

Variance functions for exponential families

Answer 13

Question 14

When to use weights in a GLM

Answer 14

When a single observation contains grouped information

Different observations represent different time periods

(If neither apply, weights are all 1)

Weights are usually the denominator of the modeled quantity

Question 15

Severity model distributions

Answer 15

Tend to be right-skewed

Lower bound at zero

Gamma and inverse Gaussian distributions exhibit these properties

Question 16

Gamma vs. inverse Gaussian distributions

Answer 16

Gamma most commonly used for severity

Inverse Gaussian has sharper peak, wider tail (more appropriate for more skewed distributions)

Question 17

Frequency model distributions

Answer 17

Poisson (technically, ODP) - most common, Ø can be > 1

Negative binomial - Ø = 1 but there is a K in variance function

Question 18

Pure premium distributions

Answer 18

Large point mass at zero (most policies have no claims)

Right-skewness due to severity distribution

Tweedie distribution most commonly used

Question 19

Tweedie distribution

Answer 19

p = “Power parameter”

1 < p < 2: Poisson frequency and Gamma severity

Assumes frequency and severity move in same direction (not realistic)

Question 20

Mean, Tweedie

Answer 20

Poisson mean x Gamma mean

= λ x αθ

Question 21

Variance, Tweedie

Answer 21

ص^p

Question 22

p, Tweedie

Answer 22

p = (α + 2) / (α + 1)

Question 23

Dispersion factor, Tweedie

Answer 23

Question 24

Probability model distributions

Answer 24

Binomial distribution used

Typically use logit function

µ / (1 - µ) known as odds

Question 25

Odds, logit function

Answer 25

µ / (1 - µ) For each unit increase in a given predictor variable iwth coefficient ß increases the odds by e^ß - 1 in percentage terms

Question 26

Offset term

Answer 26

Incorporates pre-determined values so GLM takes as given

Question 27

Continuous predictor variables

Answer 27

If log link function is used, continuous variables should be logged (flexibility in fitting curve shapes) Exceptions: year variables for trends, variables containing values of 0

Question 28

Categorical predictor variables

Answer 28

Have 2 or more levels, converted to binary variables Level with highest number of observations usually deemed base levels Using a different level results in wider CIs

Question 29

Matrix notation of GLM

Answer 29

g(**µ**) = **Xß** **µ** is the vector of µ_i values **ß** is the vector of ß parameters (coefficients) X is the design matrix (rows are results ignoring coefficients)

Question 30

Degrees of freedom

Answer 30

Number of parameters that need to be estimated for the model

Question 31

If a variable is not significant in a GLM

Answer 31

Should be removed, grouped with the base level

Question 32

Options for dealing with very high correlation in GLMs

Answer 32

Question 33

Multicollinearity

Answer 33

Near-perfect linear dependency among 3 or more predictor variables Ex: x₁ + x₂ ~ x₃ Detected with variance inflation factor statistic (VIF) \> 10 is considered high

Question 34

Aliased variables

Answer 34

Perfect linear dependency among predictor variables GLM will not converge, but most will detect and remove one of the variables from the model

Question 35

GLM limitations

Answer 35

1. GLMs give full credibility (partially adressed by p-values, SEs) 2. GLMs assume randomness of outcomes are uncorrelated (violated if dataset has several renewals of same policy, or by weather events)

Question 36

Model-building process

Answer 36

1. Setting goals and objectives 2. Communication (IT, legal, UWs) 3. Collecting/processing data 4. Exploratory data analysis 5. Specifying the form of the model 6. Evaluating model output 7. Validation 8. Translation into a product 9. Maintenance and rebuild

Question 37

Splitting data for testing

Answer 37

Training set and test (holdout) set

Question 38

Model testing strategies

Answer 38

Train and test Train, validate, test Cross-validation

Question 39

Train and test

Answer 39

Split into single training and single test sets (60/40 or 70/30) Can split randomly or on time (if not done by time, could lead to over-optimistic validation results)

Question 40

Train, validate, and test

Answer 40

Validation set can be used to refine model and make tweaks Test set should be left until model is final Typically 40/30/30

Question 41

Cross-validation

Answer 41

Less common in insurance (hand-picked variables) Most common is *k*-fold: 1. Pick a *k* and split into *k* folds 2. For each fold, train the model using the other *k* - 1 folds, and test using *k*^th fold Superior (more data for training and testing) but more time-consuming (models built completely separately)

Question 42

When to use test dataset

Answer 42

Only when model is complete If too many decisions are made based on test set, it is effectively a training set (leads to overfitting)

Question 43

Advantages of modeling freq/sev over PP

Answer 43

Gain more insight and intuition about each Each is more stable separately PP modeling can lead to overfitting if predictor variable only impacts freq or sev but not both, since randomness of other component may be considered signal Tweedie assumes both freq and sev move in same direction

Question 44

Handling perils in a GLM

Answer 44

1. Run each peril model separately 2. Aggregate expected losses 3. Run model using all-peril LC as target variable and union of all predictor variables as predictors (focus on dataset more reflective of future mix of business)

Question 45

Criteria for variable inclusion in GLM

Answer 45

p-values Cost-effectiveness of collecting data ASOPs/legal requirements IT constraints

Question 46

Partial residuals for predictor variables

Answer 46

Plot r against x and see if the points match y = ßx

Question 47

Transformations if residuals do not match line

Answer 47

Binning (increases DOF, variation within bins is ignored) Adding polynomial terms (loses interpretability without a grap) Add piecewise linear function: hinge function max (0, x - c) at each break point c

Question 48

Interaction term combinations

Answer 48

Two categorical: 1/0 Continuous and categorical: f(x)/0 Two continuous: product of the two

Question 49

Log-likelihood

Answer 49

Log of the product of the likelihood for all observations using the model

Question 50

Deviance

Answer 50

Question 51

Comparing models using LL and deviance

Answer 51

Only valid if datasets used are identical Comparisons of deviance only valid if assumed distribution and dispersion are the same

Question 52

Nested models (F-test)

Answer 52

Question 53

Akaike Information Criterion (AIC)

Answer 53

AIC = -2LL + 2p

Question 54

Bayesian Information Criterion (BIC)

Answer 54

BIC = -2LL + p ln (n) Less reasonable for insurance data (large n, large BIC)

Question 55

Q-Q plot

Answer 55

Sort deviance residuals in ascending order (y-axis) Ø^-1 [(i - 0.5) / n] for x-coordinates If model is well-fit, points will appear on a straight line

Question 56

Model stability measures

Answer 56

Cook's distance Cross-validation Bootstrapping

Question 57

Model selection methods

Answer 57

Plotting actual vs. predicted Simple quantile plots Double lift charts Loss ratio charts Gini Index

Question 58

Lift

Answer 58

Economic value of the model (ability to prevent adverse selection)

Question 59

Creating simple quantile plot

Answer 59

Sort holdout datased based on predicted LC Bucket into quantiles by exposure Calculate average predicted LC and average actual LC from each bucked and plot (divide both values by overall average predicted LC)

Question 60

Winning model, simple quantile plots

Answer 60

1. Predictive accuracy 2. Monotonicity 3. Vertical distance of actual LC between first and last quantiles

Question 61

Double lift chart

Answer 61

1. Calculate sort ratio 2. Sort by sort ratio 3. Bucket into quantiles by exposure 4. Calculate average predicted LC for each model and average actual LC for each bucket, divide by overall average LC

Question 62

Loss ratio chart

Answer 62

1. Sort holdout dataset based on predicted LC 2. Bucket into quantiles by exposure 3. Calculate actual loss ratio (using current rating plan) Greater distance between lowest and highest, greater model does identifying further segmentation opportunites

Question 63

Gini Index

Answer 63

1. Sort holdout dataset based on predicted LC 2. Plot the graph with x-axis as % cumulative exposures, y-axis as cumulative % of actual losses

Question 64

True positive

Answer 64

Correct prediction that the event occurs

Question 65

False positive

Answer 65

Prediction that the event occurs, but it does not

Question 66

False negative

Answer 66

Prediction that the event does not occur, but it does

Question 67

True negative

Answer 67

Correct prediction that the event does not occur

Question 68

Sensitivity of a model

Answer 68

Ratio of true positives to total event occurrences Sometimes called true positive or hit rate

Question 69

Specificity

Answer 69

Ratio of true negatives to total event non-occurrences

Question 70

Receiver Operating Characteristic (ROC) Curve

Answer 70

Plots sensitivity as a function of 1 - specificity

Question 71

Ensemble models

Answer 71

If two or more models perform roughly equally well, can combine those models; only really works when model errors are as uncorrelated as possible