Workshop 10.4: Generalized linear models Murray Logan 16 Aug 2016 - - PowerPoint PPT Presentation

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Workshop 10.4: Generalized linear models Murray Logan 16 Aug 2016 Linear models Homogeneity of variance 2 . 0 0 . . 2 0 . 2 ) y i = 0 + 1 x i + i i N ( 0 , . V =

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Workshop 10.4: Generalized linear models

Murray Logan 16 Aug 2016

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Linear models

yi = β0 +β1 ×xi

Linearity

+εi εi ∼ N (0,. σ2)

Normality

. V = cov =      . σ2 ··· σ2 ··· . . . . . . ··· σ2 . . . . ··· ··· σ2      . Homogeneity of variance . Zero covariance (=independence) .

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Other data types

Binary - only 0 and 1 (dead/alive)

(present/absent)

Proportional abundance - range from 0 to

100

Count data - min of zero

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Linear models

● ● ● ● ● ● ● ●
● ● ● ● ● ●

Absent Present 0.0 0.2 0.4 0.6 0.8 1.0 Predicted probability

f presence

a) X Frequency 0.0 0.4 0.8 2 4 6 8 10 12 b)

expected values outside logical bounds
response not normally distributed

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Logistic models

● ● ● ● ● ● ● ●
● ● ● ● ● ●

Absent Present 0.0 0.2 0.4 0.6 0.8 1.0 b) X Frequency 0.0 0.4 0.8 2 4 6 8 10 12 b)

expected values outside logical bounds
response not normally distributed

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Section 1 Exponential family distributions

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Gaussian distribution

Virtually unbound measurements (weight, lengths etc)

Probability density function

µ = 25, σ2 = 5 µ = 25, σ2 = 2 µ = 10, σ2 = 2 5 10 15 20 25 30 35 40

Cumulative density function

5 10 15 20 25 30 35 40

f(x | µ, σ2) =

1 √ 2σ2π e− (x−µ)2

2σ2

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Binomial distribution

Presence/absence and data bound to the range [0,1]

Probability density function

n = 50 n = 20 n = 3 5 10 15 20 25 30 35 40

Cumulative density function

5 10 15 20 25 30 35 40

f(k | n, p) =

)

pk(1 − p)n−k

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Poisson distribution

Count data (or count derivatives - like low densities)

Probability density function

λ = 25 λ = 15 λ = 3 5 10 15 20 25 30 35 40

Cumulative density function

5 10 15 20 25 30 35 40

f(x | λ) = e−λλx

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Negative Binomial

Count data (or count derivatives - like low densities)

Probability density function

n = 25 n = 10 n = 1.5 5 10 15 20 25 30 35 40

Cumulative density function

5 10 15 20 25 30 35 40

f(x | µ, ω) = Γ(x+ω)

Γ(ω)x! × µxωω (µ+ω)µ+ω

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General linear models

yi = β0 +β1 ×xi

Linearity

+εi εi ∼ N (0,. σ2)

Normality

. V = cov =      . σ2 ··· σ2 ··· . . . . . . ··· σ2 . . . . ··· ··· σ2      . Homogeneity of variance . Zero covariance (=independence) .

E(Y)

Link function

= β0 + β1x1 + ... + βpxp

Systematic

+ε, ε ∼ Dist(...)

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General linear models

E(Y)

Link function

= β0 + β1x1 + ... + βpxp

Systematic

+ e

Random
Random component.

E(Yi) ∼ N(µi, σ2) A nominated distribution (Gaussian, Poisson, Binomial, Gamma, Beta,฀)

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General linear models

E(Y)

Link function

= β0 + β1x1 + ... + βpxp

Systematic

+ e

Random
Random component.
Systematic component

β0 + β1x1 + ... + βpxp

Link function

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Generalized linear models

Response variable Probability Distribu- tion Link function Model name Continuous measurements Gaussian identiy:

Linear regression Binary,proportions Binomial logit: log

(

π 1−π

)

Logistic regression probit: 1 √ 2π

∫ α+β.X

−∞ exp

( − 1

2 Z2) dZ Probit regression complimentary: log (−log(1 − π)) Logistic regression Quasi-binomial logit: log

(

π 1−π

)

Logistic regression Counts Poisson log: log µ Poisson regression / log- linear model Negative binomial log

(

µ µ−θ

)

Negative binomial regression Quasi- poisson log: logµ Poisson regression

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OLS

6 8 10 12 14 Sum of squares

µ=10

Parameter estimates

8 10 12 14

Parameter estimates

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Maximum Likelihood

f(x | µ, σ2) =

1 √ 2σ2π e− (x−µ)2

2σ2

lnL(µ, σ2) = − n

2ln(2π) − n 2lnσ2 − 1 2σ2

∑2

i=1(xi − µ)2

Maximum likelihood estimates:

ˆ µ = ¯

x = 1

∑n

i=1 xi

ˆ σ2 = 1

∑n

i=1(xi − ¯

x)2

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Maximum Likelihood

6 8 10 12 14 Log−likelihood

µ=10

Parameter estimates

8 10 12 14 6 8 10 12 14 6 8 10 12 14 Parameter estimates