Actuarial Notes

Generalized Linear Models: Study Note for Exam S

By Michael Larsen · Casualty Actuarial Society · 2015

Study guide: Exam MAS-I

A three-page CAS study note adding six topics to the textbook treatment of generalized linear models, from quasi-likelihood to exponential-family formulas. Written by Michael Larsen as a study note for Exam S and marked "Revised 6/29/2016", it works alongside Dobson & Barnett and Hogg, McKean and Craig, pointing to the sections of each that it supplements.

On the syllabus

  • Exam MAS-I — objectives C1–C9; the whole note (cited as December 2015, revised June 2016; an online publication).

Quasi-Likelihood

  • Quasi-likelihood solves for the parameters of a Generalized Linear Model when the assumption that the distribution belongs to the exponential family is not supported: it assumes a relationship between the mean and the variance, but the underlying distribution need not be known.
  • Its equations start from a variation on the derivative of the likelihood equations, which gives the method its name.
  • The example is over dispersion — variance larger than the mean — in Poisson regression (Dobson & Barnett §§3.2 and 9.8) and in the binomial (§7.7): quasi-likelihood solves for a constant of proportionality between the mean and the variance at the same time as the betas.

Regression Through the Origin

  • Regression through the origin leaves the intercept out of the linear equation, which a modeler may choose when the test of significance shows the intercept is not significantly different from zero.
  • Dropping it can give non-intuitive results: in a general linear model it implies the dependent variable is zero whenever every explanatory variable is zero.

QQ and Box Plots to test distribution assumptions

  • The Normal probability plot the Dobson text uses to test Normality of the residuals is an example of a QQ plot, built from order statistics; the algorithm is in Section 4.4.1 of Hogg, McKean and Craig (7th edition).
  • Box plots, described in the same section, use order statistics to describe how the observations behave, as an aid to evaluating how well a model fits the data.

Tweedie Distributions

  • The Tweedie distributions are a family whose variance is a power function of the mean, \text{Var}[Y] = a\,[E(Y)]^{p}: Normal p = 0, Poisson p = 1, compound Poisson–gamma 1 < p < 2, gamma p = 2, inverse Gaussian p = 3.
  • The Tweedie is useful for a data set that combines zero observations of the dependent variable with continuous non-zero values, viewed as a compound Poisson–gamma distribution.
  • Some software packages accept a Tweedie specification and find the best value of p while solving for the betas.

Canonical Link Function

  • The inverse of the link function transforms the linear formula for the mean back to the scale of the original data; the canonical form of the link varies with the distribution selected.
  • The note states that with the canonical link the GLM's estimates are unbiased — for example the log link with the Poisson — and that a non-canonical link does not give an unbiased estimate of the mean.
  • Common canonical links: Normal — identity; binomial — logit; Poisson — natural logarithm; gamma — inverse.

Exponential Family Formulas

  • A random variable Y whose distribution depends on a single parameter \theta (all other parameters treated as known) is a member of the exponential family if its density can be written in the form below.
  • A table gives \theta, a(y), b(\theta), c(\theta) and d(y) for the binomial, Normal, Poisson, exponential, gamma, inverse Gaussian and negative binomial distributions, supplementing the table in Section 3.2 of Dobson & Barnett.

f(y;\theta) = \exp\left[a(y)\,b(\theta) + c(\theta) + d(y)\right]

Related readings

  • An Introduction to Generalized Linear Models (Dobson - 2018) — the text the note supplements, citing its Sections 3.2, 7.7 and 9.8 and the Chapter 3 exercises
  • Introduction to Mathematical Statistics (Hogg et al. - 2018) — cited (in its 7th edition, Section 4.4.1) for the QQ-plot algorithm and box plots

Sources