Chapter 18 Generalised Linear Modelling

Question 1

What is an explanatory variable?

Answer 1

• input into a model that is expected to influence the response variable.
• i.e. rating factor

-it is important that explanatory variables make intuitive sense.

Question 2

What is a response variable?

Answer 2

• output variable from a model is likely to be influenced by an explanatory variable.
• ie price

Question 3

What is a categorical variable?

Answer 3

• These are explanatory variables which are discrete and distinct, often cannot be given any natural ordering score.
• Eg gender

Question 4

What is non-categorical variable?

Answer 4

-can take numerical values eg age.

Question 5

What is an interaction term?

Answer 5

-Used where the pattern in response variable is better modelled by including an extra parameter for each combination of two or more factors.

Question 6

One-way analysis merits

Answer 6

• prior to use of GLMs the effect of frequency and severity of each rating factor was considered separately.
• This one-way analysis ignores correlations and interaction effects between variables and so may underestimate or double count the effects of variables.

Question 7

Uses of GLMs

Answer 7

• A GLM can be used to model the behaviour of a random variable that is believed to depend on the values of several other characteristics eg age, sex, chronic condition.
• It is a generalisation of the normal model for multiple linear regression.

Question 8

What are the drawbacks for the normal model for multiple linear regression?

Answer 8

• it assumes the response variable has a normal distribution
• the normal distribution has a constant variance which may not be appropriate
• it adds together the effects of different explanatory variables, but is often not what is observed
• it becomes long-winded with more than two explanatory variables.

Question 9

Assumptions of classical linear models

Answer 9

• error term are independent and come from a normal distribution
• the mean is a linear combination of the explanatory variables
• the error terms have constant variance (or homoscedasticity)

Question 10

What are the two properties of any member of the exponential family?

Answer 10

• the distribution is completely specified in terms of its mean and variance.
• the variance is a function of its mean

Question 11

What is the link function?

Answer 11

• the link function acts to remove the assumption that the effects of different variables must simply be added together.
• it must be both differentiable and monotonic.
• include:log, logit & identity functions.

Question 12

Steps for obtaining predicted values from a single GLM

Answer 12

• Specify design matrix X and the vector of parameters Beta
• Choose a distribution for the response variable and the link function.
• Identify the log-likelihood function
• Take logarithm to convert the product of many terms into a sum
• Maximise the logarithm of the likelihood function by taking partial derivatives with respect to each parameter.
• Compute predicted values.

Question 13

What techniques are used to analyse significance of explanatory variables?

Answer 13

• chi-squared test
• the F-statistic - models need to be nested for this to work.
• Akaike Criterion Information - appropriate where models are not nested.
• other methods

Question 14

Define degrees of freedom

Answer 14

-number of observations - number of parameters

Question 15

AIC formula

Answer 15

AIC = -2 * log likelihood + 2* number of parameters

• the lower the AIC the better the model.
• fewer parameters is better/parsimonious model.

Question 16

Measuring uncertainty in the estimators of the model parameters

Answer 16

• The cramer-rao lower bound is used.
• the maximum likelihood estimator theta-hat is distributed N(theta,CRLB).
• standard errors in a GLM will be found using the Hessian matrix.
• this is a matrix of 2nd derivatives.

Question 17

What other ways can be used to test significance?

Answer 17

• Comparisons with time

- Consistency checks with other factors

Question 18

Comparisons with time

Question 19

Consistency checks with other factors

Answer 19

• time is not the only factor that can be used as a consistency check.
• eg an explanatory variable like age would be expected to show the same pattern regardless of geopraphical region.

Question 20

Testing the appropriateness of models

Answer 20

• The hat matrix is one of the outputs of the model-fitting process.
• It is the matrix H such that y-Hat = Hy
• For Normal multiple linear regression model.
• The diagonal entries, h(i,i) of the matrix are called leverages. h(i,i) in interval (0,1).
• Leverages measure the influence that each observed value has on the fitted value for that observation.
• Data points with high leverages or residuals may distort the outcome and accuracy of a model.

Question 21

Deviance residuals

Answer 21

• This is the measure of the distance between the actual observation and the fitted value.
• deviance corrects the skewness of the distribution.

Question 22

Standardised Pearson residuals

Answer 22

• A standardised residual is the difference between the observed response and the predicted value, adjusted for the standard deviation of the predicted value and the leverage of the observed response.
• These adjustments make it possible to compare Standardised Pearson residuals even where observations have different means.

Question 23

Residual Plots

Answer 23

-For a particular method if the distribution chosen for the response variable is appropriate then the residuals chart should produce residuals that:

• are symmetrical about the x-axis
• have an average residual of zero
• are fairly constant across the width of the fitted values

Question 24

Cook’s distance and leverage

Answer 24

• Cook’s distance is used to estimate the influence of a data point on the model results.
• Data points with a Cook’s distance of 1 or more are considered to merit closer examination in the analysis.
• As a result of the investigation into any data points with a high Cook’s distance, decision might be made to remove the observations altogether.

Question 25

Models may be refined using (4)?

Answer 25

• Interactions
• Aliasing
• Resctrictions
• Smoothing

Question 26

Model refinement: Interactions

Answer 26

• After choosing a structure of the model and checked that it is appropriate for the factors chosen the model can be refined further.
• The may be complete or marginal.

Question 27

Model refinement: Aliasing

Answer 27

• Aliasing occurs when there is a linear dependency among the observed covariates X1,….,Xp.
• That is, 1 covariate can be expressed as a linear combination of other covariates
• eg X3= 5 +2X1+ 3X2

Question 28

There are two types of aliasing

Answer 28

• Intrinsic

- Extrinsic

Question 29

Intrinsic aliasing

Answer 29

• Occurs because of the dependencies inherent within the definition of covariates.
• arise mostly when categorical variables are included
• This is dealt by modelling software.

Question 30

Extrinsic aliasing

Answer 30

-Occurs when two or more factors contain levels that are perfectly correlated.

Question 31

Near aliasing

Answer 31

• When modelling in practice, a common problem occurs when two or more factors contain levels that are almost, but not quite, perfectly correlated.
• In order to understand problems where model suggest very large negative/positive parameters 2 way tables of exposure and claim counts.
• From this it should be possible to identify combinations that cause near aliasing.
• The issue can then be resolved either deleting or excluding those rogue records or reclassifying the rogue records into another, more appropriate, factor.

Question 32

Parameter smoothing

Answer 32

• A GLM can be improved by smoothing the parameter values. This can be achieved by grouping level factors.
• The granularity of data can be kept in modeling since softwares can use this granularity & patterns to better group the different levels of variables.

Question 33

Factors can be simplified by?

Answer 33

• Grouping and summarising data prior to loading. Requires knowledge of expected patterns.
• Grouping in the modelling package. eg grouping age into two age bands.

Question 34

Restrictions when pricing: GLM restrictions

Answer 34

• Legal or commercial considerations may impose rigid restrictions on the way particular factors are used in practice.
• eg legal: restricting use of age and gender in pricing for medical scheme.
• When the use of certain factors is restricted the model will be able to compensate for this to an extent for this artificial restriction by adjusting the fitted relativities for correlated factors.
• this is achieved by the offset term.

Question 35

Restrictions when pricing: Further restrictions

Answer 35

• theoretical risk premium results from a GLM claims analysis will differ from the rates implemented in practice since consideration needs to be given to price demand elasticity and competitive situation.
• price elasticity demand is a measure of how the demand changes to a change in price.
• The competitive situation is relevant because there may be pockets of business where theoretical risk premium rates would produce a market premium that are much higher or lower than those of your competitors.
• eg if your premiums are much lower than those of you competitors then there may be an opportunity to increase your rates and still be cheapest in the market, thereby increase profits without decreasing volumes.