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KCL Statistical Modelling > Generalised Linear Models > Flashcards

Generalised Linear Models Flashcards

1
Q

A member of exponential family can be written in form

A
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2
Q

Θ in exponential family

A

Natural or canonical parameter

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3
Q

Φ in exponential family

A

Nuisance parameter (if unknown)

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4
Q

Score of exponential family

A
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5
Q

Hessian of exponential family

A
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6
Q

Fisher information matrix of exponential family

A
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7
Q

Expectation of score of exponential family

A
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8
Q

E[Y] for exponential family? Why?

A
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9
Q

Variance of score of exponential family

A
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10
Q

Variance of Y for exponential family

A
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11
Q

μ for exponential family

A

b’(θ)

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12
Q

Variance function for exponential family

A
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13
Q

a(φ) in exponential family

A

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

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14
Q

For normal distribution;
Θ =

A

μ

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15
Q

For normal distribution;
b(θ)

A

(Θ^2)/2

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16
Q

For normal distribution;
a(φ)

A

σ^2

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17
Q

For normal distribution;
c(y,φ)

A
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18
Q

For normal distribution;
E(Y)

A
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19
Q

For normal distribution;
Var(Y)

A
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20
Q

For normal distribution;
V(μ)

A

1

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21
Q

For poison distribution;
Distribution?

A
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22
Q

For normal distribution;
Distribution?

A
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23
Q

For poison distribution;
Θ?

A

log(λ)

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24
Q

For poison distribution;
b(θ)?

A

exp(θ)

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25
Q

For poison distribution;
a(φ)?

A

1

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26
Q

For poison distribution;
c(y, φ)?

A

-log(y!)

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27
Q

For poison distribution;
E(Y)

A
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28
Q

For poison distribution;
Var(Y)

A
29
Q

For poison distribution;
V(μ)

A

μ

30
Q

For Bernoulli distribution;
Distribution

A
31
Q

For Bernoulli distribution;
Θ

A

log(p/(1-p))

32
Q

For Bernoulli distribution;
b(θ)

A

log(1+exp(θ))

33
Q

For Bernoulli distribution;
a(φ)

A

1

34
Q

For Bernoulli distribution;
c(y, φ)

A

0

35
Q

For Bernoulli distribution;
E(Y)

A
36
Q

For Bernoulli distribution;
Var(Y)

A
37
Q

For Bernoulli distribution;
V(μ)

A

μ(1-μ)

38
Q

For binomial distribution;
Distribution

A
39
Q

For binomial distribution;
Θ

A

log(p/(1-p))

40
Q

For binomial distribution;
b(θ)

A

log(1+exp(θ))

41
Q

For binomial distribution;
a(φ)

A

1/n

42
Q

For binomial distribution;
c(y, φ)

A
43
Q

For binomial distribution;
E(Y)

A
44
Q

For binomial distribution;
Var(Y)

A
45
Q

For binomial distribution;
Var(μ)

A

μ(1-μ)

46
Q

Random component of general linear model;
Parameters

A

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

47
Q

Random component of general linear model;
Joint density

A
48
Q

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

A

Is likelihood function for θ and φ

49
Q

Systematic/Structural component of general linear model;
Linear predictor

A

Distribution of response

50
Q

Systematic/Structural component of general linear model;
Design matrix

A
51
Q

Link function does?

A

Describes relationships between E(Y) and linear predictor

52
Q

Link function must

A

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

53
Q

How to pick link function

A

-normally choose so that range is entire real line

54
Q

Get θ_i from generalised linear model given that

A
55
Q

Canonical link function

A
56
Q

Canonical link function normal

A
57
Q

Canonical link function poisson

A
58
Q

Canonical link function Bernoulli/binomial

A
59
Q

Normal linear model; linear predictor

A
60
Q

Normal linear model; link between

A

Through the

61
Q

Objective of binary regression

A

Model success probability p as a function of the covariates

62
Q

Binary regression;
Θ when using canonical link

A

(Logit)

63
Q

CDF of logistic dist

A
64
Q

Binary regression;
Probit link

A

Using CDF of standard normal to model p(x)

65
Q

Binary regression;
Probit link has g(μ)=

A

Where Φ is CDF of standard normal dist

66
Q

Binary regression;
CDF of log-Weibull

A
67
Q

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

A
68
Q

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

A

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

69
Q

When to use Logit

A

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

KCL Statistical Modelling (9 decks)

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