Chapter 18 - GLM

Question 1

What are explanatory variables

Answer 1

They are inputs into a model that are expected to influence the response variable.

Question 2

What are response variables

Answer 2

They are outputs from the model that are likely to be affected by the explanatory variables

Question 3

What are categorical values and non-categorical values

Answer 3

They are explanatory variables that are used for modelling where the values of each level are distinct, and often cannot be given any natural ordering or score. Eg. gender which takes on the value male or female.

Non-categorical values can take numerical values, eg age

Question 4

What are the drawbacks of using a normal model for linear regression

Answer 4

CLAN

• The normal distribution has a CONSTANT variance, which may not be appropriate for the variable being modelled
• More than 2 explanatory variables makes the time LONG to compute
• The normal model ADDS together the effects of the different explanatory variables, but this is seldom what is observed in practice
• It assumes that the response variable, Y, has a NORMAL distribution , which may not be appropriate for the variable being modelled

Question 5

How does GLMs address the problems for the normal model for linear regression

Answer 5

• The response variable can take any distribution from the exponential family - it no longer has to take the normal distribution
• A link function is introduced - this acts to remove the assumption that the effects of different variables must simply be added together

Question 6

What is the purpose of the link function

Answer 6

• The link function acts to remove the assumption that the effects of different variables must simply be added together

Question 7

What does the deviance of a model compare

Answer 7

It compares the observed value Yi to the fitted value Ui

In essence, the deviance is a measure of how much the fitted values differ from the obervations

Question 8

What do the chi-squared statistic measure

Answer 8

This measures whether the inclusion of one or more additional explanatory variables in a model improves the fit significantly

Question 9

What can be used to measure the uncertainty in the parameter estimators used in GLM

Answer 9

The Cramer-rao lower bound

Question 10

What are deviance residuals

Answer 10

Is a measure of the distance between the actual observation and the fitted values

Question 11

What are standardised pearson residuals

Answer 11

It 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

Question 12

What is Cook’s distance used for

Answer 12

It is used to estimate the influence of a data point on the model results

Cook’s distance of 1 or more is considered to merit closer examination in the analysis

Question 13

What is Aliasing

Answer 13

Aliasing occurs when there is dependency among the observed covariates. i.e one covariate may be identical to some linear combination of other covariates

Question 14

What is intrinsic aliasing

Answer 14

This occurs because of dependencies inherent in the definition of the covariates.

These intrinsic dependencies arise most commonly whenever categorical variables are included in the model.

Question 15

What is Extrinsic aliasing

Answer 15

It arises from a dependency among the covariates. It arises when the dependency results from the nature of the data itself, rather than as a result of the inherent properties of the covariates.

Question 16

What is an interaction term

Answer 16

It is used where the pattern in the response variable is better modelled by including extra parameters for each combination of two or more factors

An interaction exists when the effect of one factor varies depending on the value of another factor

Question 17

What are the assumptions of the classical normal model for linear regression

Answer 17

• The mean is a linear combination of the explanatory variables
• The error terms are independent and come from a normal distribution
• The error terms have constant variance

Question 18

What 2 properties do the members of the exponential family have

Answer 18

• The distribution is completely specified in terms of its mean and variance
• The variance of Yi is a function of its mean

Question 19

What is special about the Tweedie distribution

Answer 19

• The Tweedie distribution is a special member of the exponential family that has a variance function proportional to \mu^{p}, with p being an additional parameter
• In the case of 1<p<2, the Tweedie distribution has a point mass at zero and corresponds to the compound distribution of a Poisson claim number process and a gamma claim size distribution
• The distribution can be Poisson-like (as p tends to 1) or gamma-like (as p tends to 2)

Question 20

How is the number of degrees of freedom calculated

Answer 20

It is the number of observations less the number of parameters

Question 21

What are nested models

Answer 21

Two models are nested if one model contains explanatory variables that are a subset of the explanatory variables in the other model

Question 22

What is the equation of the Akaike Information Criteria and what does it measure

Answer 22

AIC = -2xlog likelihood + 2x number of parameters

It looks at the trade-off of the likelihood of a model against the number of parameters: the lower the AIC, the better the model

Question 23

What does the deviance residual measure

Answer 23

It measures the distance between the actual observation and fitted value

Question 24

What is the difference between correlations and interactions

Answer 24

• Correlations occur when there is a relationship between the distribution of exposure between levels of two or more factors
• GLMs automatically take account of correlations (unlike one-way tables)
• Interactions relate to the effect that factors have upon the risk
• To define the risk accurately, an interaction would be necessary where the effect of two (or more) factors depend on each other. GLMs can be specified to include interactions.

Question 25

What are GLMs used for

Answer 25

- They are used to assess and quantify the relationship between a response variable and a set of possible explanatory variables

Question 26

What does the leverage measure

Answer 26

It measures how much influence each observed value has on the fitted value for that observation