Generalised Linear Models

Question 1

A member of exponential family can be written in form

Answer 1

Question 2

Θ in exponential family

Answer 2

Natural or canonical parameter

Question 3

Φ in exponential family

Answer 3

Nuisance parameter (if unknown)

Question 4

Score of exponential family

Answer 4

Question 5

Hessian of exponential family

Answer 5

Question 6

Fisher information matrix of exponential family

Answer 6

Question 7

Expectation of score of exponential family

Answer 7

Question 8

E[Y] for exponential family? Why?

Answer 8

Question 9

Variance of score of exponential family

Answer 9

Question 10

Variance of Y for exponential family

Answer 10

Question 11

μ for exponential family

Answer 11

b’(θ)

Question 12

Variance function for exponential family

Answer 12

Question 13

a(φ) in exponential family

Answer 13

= (σ^2)/w where σ^2 is called the dispersion/scale parameter and w the weight

Question 14

For normal distribution;
Θ =

Answer 14

μ

Question 15

For normal distribution;
b(θ)

Answer 15

(Θ^2)/2

Question 16

For normal distribution;
a(φ)

Answer 16

σ^2

Question 17

For normal distribution;
c(y,φ)

Answer 17

Question 18

For normal distribution;
E(Y)

Answer 18

Question 19

For normal distribution;
Var(Y)

Answer 19

Question 20

For normal distribution;
V(μ)

Answer 20

1

Question 21

For poison distribution;
Distribution?

Answer 21

Question 22

For normal distribution;
Distribution?

Answer 22

Question 23

For poison distribution;
Θ?

Answer 23

log(λ)

Question 24

For poison distribution;
b(θ)?

Answer 24

exp(θ)

Question 25

For poison distribution; a(φ)?

Answer 25

1

Question 26

For poison distribution; c(y, φ)?

Answer 26

-log(y!)

Question 27

For poison distribution; E(Y)

Answer 27

Question 28

For poison distribution; Var(Y)

Answer 28

Question 29

For poison distribution; V(μ)

Answer 29

μ

Question 30

For Bernoulli distribution; Distribution

Answer 30

Question 31

For Bernoulli distribution; Θ

Answer 31

log(p/(1-p))

Question 32

For Bernoulli distribution; b(θ)

Answer 32

log(1+exp(θ))

Question 33

For Bernoulli distribution; a(φ)

Answer 33

1

Question 34

For Bernoulli distribution; c(y, φ)

Answer 34

0

Question 35

For Bernoulli distribution; E(Y)

Answer 35

Question 36

For Bernoulli distribution; Var(Y)

Answer 36

Question 37

For Bernoulli distribution; V(μ)

Answer 37

μ(1-μ)

Question 38

For binomial distribution; Distribution

Answer 38

Question 39

For binomial distribution; Θ

Answer 39

log(p/(1-p))

Question 40

For binomial distribution; b(θ)

Answer 40

log(1+exp(θ))

Question 41

For binomial distribution; a(φ)

Answer 41

1/n

Question 42

For binomial distribution; c(y, φ)

Answer 42

Question 43

For binomial distribution; E(Y)

Answer 43

Question 44

For binomial distribution; Var(Y)

Answer 44

Question 45

For binomial distribution; Var(μ)

Answer 45

μ(1-μ)

Question 46

Random component of general linear model; Parameters

Answer 46

For each observation, given the fitted distribution, functions a,b and c (and usually) scale parameter φ are the same for all observations, only θ changes

Question 47

Random component of general linear model; Joint density

Answer 47

Question 48

Random component of general linear model; Vector y , observed responses

Answer 48

Is likelihood function for θ and φ

Question 49

Systematic/Structural component of general linear model; Linear predictor

Answer 49

Distribution of response

Question 50

Systematic/Structural component of general linear model; Design matrix

Answer 50

Question 51

Link function does?

Answer 51

Describes relationships between E(Y) and linear predictor

Question 52

Link function must

Answer 52

Any function g that is one to one, monotonic and differentiable (limitations May apply due to distribution) (eg poisson must have μ_i >0)

Question 53

How to pick link function

Answer 53

-normally choose so that range is entire real line

Question 54

Get θ_i from generalised linear model given that

Answer 54

Question 55

Canonical link function

Answer 55

Question 56

Canonical link function normal

Answer 56

Question 57

Canonical link function poisson

Answer 57

Question 58

Canonical link function Bernoulli/binomial

Answer 58

Question 59

Normal linear model; linear predictor

Answer 59

Question 60

Normal linear model; link between

Answer 60

Through the

Question 61

Objective of binary regression

Answer 61

Model success probability p as a function of the covariates

Question 62

Binary regression; Θ when using canonical link

Answer 62

(Logit)

Question 63

CDF of logistic dist

Answer 63

Question 64

Binary regression; Probit link

Answer 64

Using CDF of standard normal to model p(**x**)

Question 65

Binary regression; Probit link has g(μ)=

Answer 65

Where Φ is CDF of standard normal dist

Question 66

Binary regression; CDF of log-Weibull

Answer 66

Question 67

Verify that log-Weibull CDF does in fact define a CDF

Answer 67

Question 68

Binary regression; Difference between logit, Probit and log-log link

Answer 68

Logit and Probit CDFs are symmetric about 1/2. Log-log link isn’t, hence this should be used when asymmetry as a function of the linear predictor is suspected

Question 69

When to use Logit

Answer 69

Heavier tailed than standard normal dist, hence use when outliers are suspected in linear predictor