Lecture 6

Generalized Linear Models

Olivier MISSA, om502@york.ac.uk

Advanced Research Skills

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Outline

Continue exploring options available when

assumptions of classical linear models are untenable.

In this lecture:

What can we do when observations are not

continuous

and the residuals are not normally distributed nor

identically distributed ?

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Defined by three assumptions:

(1) the response variable is continuous.

(2) the residuals (ε) are normally distributed and ...

(3) ... independently (3a) and identically distributed (3b).

Today, we will consider a range of options available

when assumptions (1) (2) and/or (3b) are not verified.

Classical Linear Models

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Many situations exist:

The response variable could be

(1) a count (number of individuals in a population)(number of species in a community)

(2) a proportion (proportion "cured" after treatment) (proportion of threatened species)

(3) a categorical variable (breeding/non-breeding)

(different phenotypes)

(4) a strictly positive value (esp. time to success) (or time to failure)

( ... ) and so forth

Non-continuous response variable

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These types of non-continuous variables also tend to deviate from the assumptions of

Normality (assumption #2) and Homoscedasticity (assumption #3b)

(1) A count variable often follows a Poisson distribution (where the variance increases linearly with the mean)

(2) A proportion often follows a Binomial distribution (where the variance reaches a maximum for intermediate values

and a minimum at either end: 0% or 100%)

Added difficulties

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These types of non-continuous variables also tend to deviate from the assumptions of

Normality (assumption #2) and Homoscedasticity (assumption #3b).

(3) A categorical variable tends to follow a Binomial distribution

(when the variable has only two levels) or a Multinomial

distribution (when the variable has more than two levels)

(4) Time to success/failure can follow an exponential distribution or

an inverse Gaussian distribution (the latter having a variance

increasing more quickly than the mean).

Added difficulties

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Many of these situations can be unified under a central framework.

Since all these distributions (and a few more) belong to the exponential family of distributions.

Fortunately

),(

)(

)(exp,

yc

a

byyf

Probability density function (if y is continuous)

Probability mass function (if y is discrete)

Canonical (location) parameter

Dispersion parameter

Canonical form

bEY

abY var

mean

variance

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The Normal distribution

2

2

2exp

2

1,

y

yfProbability

density function

Canonical form

)2log(

2

12/exp 2

2

2

2

2

yy

Canonical (location) parameter

Dispersion parameter

2

bEY

2var abY

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The Poisson distribution

!

,y

eyf

y

Probability mass

function

Canonical form

!lnlnexp yy

= 1

Canonical (location) parameter

Dispersion parameter

ln1

bEY

abYvar

)exp()( b

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The Binomial distribution

yny

y

nyf

1,

Probability mass

function

Canonical form

y

nyny ln1lnlnexp

= 1

Canonical (location) parameter

Dispersion parameter

1

ln

1

nbEY

)1(var nabY

)exp1log()1ln()( nnb

y

nny ln1ln

1lnexp

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Why is that remotely useful ?1) A single algorithm (maximum likelihood)

will cope with all these situations.

2) Different types of Variance can be accommodated

When Var is constant -> Normal (Gaussian)

When Var increases linearly with the mean -> Poisson

When Var has a humped back shape -> Binomial

When Var increases as the square of the mean -> Gamma(means the coefficient of variation remains constant)

When Var increases as the cube of the mean -> inverse Gaussian

3) Most types of data are thus effectively covered

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Two ways to cope with non-independent observations

When design is balanced ("equal sample size")

We can use factors to partition our observations in different "groups" and analyse them as an ANOVA or ANCOVA.

We already know how to do that (when factors are "crossed")

We just need to figure out how to cope with nested factors.

When design is unbalanced ("uneven sample size")

Mixed effect models are then called for.

Non-independent Observations

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How does it work ?1) You need to specify the family of distribution to use

2) You need to specify the link function

ppxxx 22110 iyg

linear predictorlink function

For each type of variable the "natural" link function to use is indicated by the canonical parameter

Link

Normal Identity

Poisson Log

Binomial Logit

Gamma Inverse

Inv.Gaussian Inverse square

1

ln

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Count variableThis type of response variable often follows a

Poisson distribution with Variance increasing in direct relation with the Mean.

The family to use is Poisson and the canonical link is log.

Example: What are the environmental variables associated with plant diversity on the Galapagos ?

> library(faraway)> data(gala)> names(gala)[1] "Species" "Endemics" "Area" "Elevation" "Nearest" [6] "Scruz" "Adjacent"> attach(gala)

Beware some missing data in the original dataset have been filed for convenience.

Johnson, M.P. & Raven, P.H. (1973) Science 179(4076): 893-895.

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Count variable> summary(gala) Species Endemics Area Min. : 2.00 Min. : 0.00 Min. : 0.0100 1st Qu.: 13.00 1st Qu.: 7.25 1st Qu.: 0.2575 Median : 42.00 Median :18.00 Median : 2.5900 Mean : 85.23 Mean :26.10 Mean : 261.7087 3rd Qu.: 96.00 3rd Qu.:32.25 3rd Qu.: 59.2375 Max. :444.00 Max. :95.00 Max. :4669.3200

Elevation Nearest Scruz Adjacent Min. : 25.00 Min. : 0.20 Min. : 0.00 Min. : 0.03 1st Qu.: 97.75 1st Qu.: 0.80 1st Qu.: 11.03 1st Qu.: 0.52 Median : 192.00 Median : 3.05 Median : 46.65 Median : 2.59 Mean : 368.03 Mean :10.06 Mean : 56.98 Mean : 261.10 3rd Qu.: 435.25 3rd Qu.:10.03 3rd Qu.: 81.08 3rd Qu.: 59.24 Max. :1707.00 Max. :47.40 Max. :290.20 Max. :4669.32> gala <- gala[,-2] ## removing variable "Endemics" > modp <- glm(Species ~ ., family=poisson, data=gala)

by default the link for a Poisson is log

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Count variable> summary(modp) Coefficients: Estimate Std. Error z value Pr(>|z|) (Intercept) 3.155e+00 5.175e-02 60.963 < 2e-16 ***Area -5.799e-04 2.627e-05 -22.074 < 2e-16 ***Elevation 3.541e-03 8.741e-05 40.507 < 2e-16 ***Nearest 8.826e-03 1.821e-03 4.846 1.26e-06 ***Scruz -5.709e-03 6.256e-04 -9.126 < 2e-16 ***Adjacent -6.630e-04 2.933e-05 -22.608 < 2e-16 ***---

(Dispersion parameter for poisson family taken to be 1)

Null deviance: 3510.73 on 29 degrees of freedomResidual deviance: 716.85 on 24 degrees of freedomAIC: 889.68

Number of Fisher Scoring iterations: 5

Only valid if the Response variable is indeed following a Poisson

n

iiiiii yyyD

1

)ˆ()ˆln(2

Need to be broadly similar

also called G-statistic

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Count variable

> (dp <- sum(residuals(modp, type="pearson")^2)/modp$df.res)[1] 31.74914

Pearson's residuals

This dispersion parameter () must be calculated.

pn

y

pni iii

ˆˆˆ22

Residual degrees of freedom

Suggests that the Variance is 31.8 times the Mean.

In statistical terms this is called Overdispersion.

In biological terms, it suggests that the counts are not independent from each other but instead are Aggregated(i.e. Clumped).

Typically Overdispersed count data follow a Negative Binomial distribution, which is not part of the Exponential families of distribution.

It won't be covered here, but it can be approximated as a quasi-Poisson (family="quasipoisson").

If you need it in your future work, you can also try glm.nb (in MASS package)

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Count variable

> summary(modp, dispersion=dp)Coefficients: Estimate Std. Error z value Pr(>|z|) (Intercept) 3.1548079 0.2915897 10.819 < 2e-16 ***Area -0.0005799 0.0001480 -3.918 8.95e-05 ***Elevation 0.0035406 0.0004925 7.189 6.53e-13 ***Nearest 0.0088256 0.0102621 0.860 0.390 Scruz -0.0057094 0.0035251 -1.620 0.105 Adjacent -0.0006630 0.0001653 -4.012 6.01e-05 ***---(Dispersion parameter for poisson family taken to be 31.74914)

Null deviance: 3510.73 on 29 degrees of freedomResidual deviance: 716.85 on 24 degrees of freedomAIC: 889.68

The summary table can be adjusted with the dispersion parameter

These Values can now be taken at face value

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Count variable

> drop1(modp, test="F") When you have overdispersed data Model:Species ~ Area + Elevation + Nearest + Scruz + Adjacent Df Deviance AIC F value Pr(F) <none> 716.85 889.68 Area 1 1204.35 1375.18 16.3217 0.0004762 ***Elevation 1 2389.57 2560.40 56.0028 1.007e-07 ***Nearest 1 739.41 910.24 0.7555 0.3933572 Scruz 1 813.62 984.45 3.2400 0.0844448 . Adjacent 1 1341.45 1512.29 20.9119 0.0001230 ***---Warning message:In drop1.glm(modp, test = "F") : F test assumes 'quasipoisson' family

The drop1 function can be used to simplify the model

AIC values dodgy when quasipoisson is used

"Nearest" should probably be removed from the model.

Safer to use the F-values

and their p-values

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Count variable> modp2 <- update(modp, ~. - Nearest)> (dp2 <- sum(residuals(modp2, type="pearson")^2)/

modp2$df.res) [1] 29.53501> summary(modp2, dispersion=dp2)Coefficients: Estimate Std. Error z value Pr(>|z|) (Intercept) 3.1599640 0.2805140 11.265 < 2e-16 ***Area -0.0005978 0.0001396 -4.283 1.85e-05 ***Elevation 0.0035769 0.0004675 7.651 1.99e-14 ***Scruz -0.0038565 0.0025216 -1.529 0.126 Adjacent -0.0007030 0.0001521 -4.621 3.82e-06 ***---(Dispersion parameter for poisson family taken to be 29.53501)

Null deviance: 3510.73 on 29 degrees of freedomResidual deviance: 739.41 on 25 degrees of freedomAIC: 910.24

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> drop1(modp2, test="F")Model:Species ~ Area + Elevation + Scruz + Adjacent Df Deviance AIC F value Pr(F) <none> 739.41 910.24 Area 1 1290.08 1458.91 18.6184 0.0002200 ***Elevation 1 2525.09 2693.92 60.3749 3.981e-08 ***Scruz 1 818.74 987.57 2.6822 0.1140018 Adjacent 1 1570.87 1739.70 28.1123 1.709e-05 ***---Warning message:In drop1.glm(modp, test = "F") : F test assumes 'quasipoisson' family> modp3 <- update(modp2, ~. – Scruz)> (dp3 <- sum(residuals(modp3, type="pearson")^2)/

modp3$df.res) [1] 30.08155

Count variable

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> summary(modp3, dispersion=dp3)

Coefficients: Estimate Std. Error z value Pr(>|z|) (Intercept) 2.9613109 0.2617588 11.313 < 2e-16 ***Area -0.0005704 0.0001381 -4.129 3.64e-05 ***Elevation 0.0035891 0.0004721 7.602 2.91e-14 ***Adjacent -0.0007508 0.0001524 -4.928 8.32e-07 ***---(Dispersion parameter for poisson family taken to be 30.08155)

Null deviance: 3510.73 on 29 degrees of freedomResidual deviance: 818.74 on 26 degrees of freedomAIC: 987.57

Count variable

How good is the model ? 1 – (Res. Dev. / Null Dev.)

= 76.68 %

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> plot(residuals(modp3) ~ predict(modp3, type="response"), xlab=expression(hat(mu)), ylab="Deviance residuals")

Count variable Checking the Model

Plotting residuals vs fitted values (Several options)

by default the Deviance version In the Original Response Scale

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Count variable Checking the Model

Plotting residuals vs fitted values (Several options)

> plot(residuals(modp3) ~ predict(modp3, type="link"), xlab=expression(hat(eta)), ylab="Deviance residuals")

both in the linked scale (Log for Poisson)

Clearest to inspect

"Good Spread"

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> plot(residuals(modp3, type="response") ~ predict(modp3, type="response"), xlab=expression(hat(mu)), ylab="Response residuals")

Count variable Checking the Model

Plotting residuals vs fitted values (Several options)

both in the original response scale

Harder to read

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> shapiro.test(residuals(modp3, type="deviance"))

Shapiro-Wilk normality test

data: residuals(modp3, type = "deviance") W = 0.9811, p-value = 0.854

Count variable Checking the Model

Do the residuals have the right distribution ?