8/11/2019 GLMs - List of Standard Pearson Residual Results

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Pearson residuals

var( )

i i

i

y m

m

-

where var( )i is the variance of the distribution with any im s replaced by the

fitted results.

Useful for 2~ ( , )i iY N m s only as the Pearson residuals are (0,1)N .

non-normal distributions produce skewed (non-normal) Pearson residuals, which

makes them hard to interpret.

Poisson distribution ~ ( )i iY Poi

Pearson residual

var( )i i i i

i i

y y mm

- -

= = since var( )i iY =

Exponential distribution ~ ( ) ~ (1 )i i iY Exp Expl m

Pearson residual

var( )

i i i i

ii

y y m

mm

- -= = since

2

21var( )i

i iY lm= =

Gamma distribution ~ ( , ) ~ ( , )i i iY Ga Gaa l a a m

Pearson residual

var( )

i i i i

ii

y y m

am

- -= = since

2

2var( ) i

iiY

aal

= =

Normal distribution2~ ( , )i iY N m s

Pearson residual

var( )

i i i i

i

y y m

sm

- -= = since 2var( )iY s=

Binomial distribution

iZ

i nY =

where ~ ( , )i iZ Bin n m

Pearson residual ( )

var( ) (1 )

i i i i

i i i

y y

n

m

m m m

- -= =

-

since ( ) 2 2(1 ) (1 )1var( ) var var( )i i i i iZ ni in nn nY Z m m m- -

= = = =

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Poisson distribution ~ ( )i iY Poi with linear predictor, ih a= .

Pearson residual

var( )

i i i i i

i i

y y y e

e

a

a

m m

m m

- - -= = =

since var( )i iY m= and, using inv. link function, ii e eh am = =

Exponential distribution ~ ( ) ~ (1 )i i iY Exp Expl m with linear predictor, ih a= .

Pearson residual

1var( )

i i i ii

ii

y yy

ma

mm

- -= = = -

since2

21var( )i

i iY l= = and, using inv. link function, 1 1

ii ah

m = =

Gamma distribution ~ ( , ) ~ ( , )i i iY Ga Gaa l a a m with linear predictor, ih a=

.

Pearson residual

( 1)var( )

i i i ii

ii

y ya y

a

m ma

mm

- -= = = -

since2

2var( ) i

i

ai a

Y m

l= = and, using inv. link function, 1 1

ii ah

m = =

Normal distribution2~ ( , )i iY N m s with linear predictor, ih a= .

Pearson residual

var( )

i i i i i

i

y y ym a

s sm

- - -= = =

since var( )iY s= , and, using inv. link function, i ih a= =

Binomial distribution iZ

i nY = where ~ ( , )i iZ Bin n m with linear predictor,

ih a= .

Pearson residual

( ) (1 )

var( ) (1 )

i i i i i

i i i

y y y e e

n e n

a a

a

m m

m m m

- - + -= = =

-

since ( ) 2 2(1 ) (1 )1var( ) var var( )i i i i iZ ni in nn nY Z m m m- -

= = = =

and, using inv. link function,

11

ii

e ei

ee

h a

ahm

++

= =