PL-2
An Introduction to
GLM Theory
2006 CAS Seminar on
Ratemaking
Claudine Modlin, FCAS
Watson Wyatt Worldwide
WWW.WATSONWYATT.COM
Generalized linear model
benefits

Consider all factors simultaneously

Allow for nature of random process

Provide diagnostics

Robust and transparent
Example of GLM output
0.25
22%
180
0.2
160
0.15
10%
0.1
7%
6%
140
0.05
0%
Log of multiplier
-4%
-5%
-0.05
100
-0.1
-15%
-16%
-0.15
-19%
-17%
80
-19%
-20%
-0.2
60
-0.25
40
-0.3
-0.35
20
-0.4
-0.45
0
1
2
3
4
5
6
Factor
Exposure
Oneway relativities
Approx 2 SE from estimate
GLM estimate
7
Exposure (policy years)
120
0
Agenda



Formularization of GLMs
–
linear predictor, link function, offset
–
error term, scale parameter, prior weights
–
typical model forms
Model testing
–
use only variables which are predictive
–
make sure model is reasonable
Aliasing
Agenda



Formularization of GLMs
–
linear predictor, link function, offset
–
error term, scale parameter, prior weights
–
typical model forms
Model testing
–
use only variables which are predictive
–
make sure model is reasonable
Aliasing
Linear models

Linear model Yi =  i + error

 i based on linear combination of measured factors

Which factors, and how they are best combined is to be
derived
i =  + .agei + .agei2 + .heighti.agei
i =  + .agei + .(sexi=female)
i = ( + .agei) * exp(.heighti.agei)
Linear models - formularization
E[Yi] = i = SXijj
Var[Yi] =
2
s
Yi ~ N(i,
2
s)
What is


SXijj?
X defines the explanatory variables to be
included in the model
–
could be continuous variables - "variates"
–
could be categorical variables - "factors"
 contains the parameter estimates which relate
to the factors / variates defined by the structure
of X
–
"the answer"
What is X. ?

Write SXijj as X.

Consider 3 rating factors
- age of driver ("age")
- sex of driver ("sex")
- age of vehicle ("car")

Represent  by , , , , ...
What is X. ?

Suppose we wanted a model of the form:
 =  + .age + .age2 + .car27.age52½

X. would need to be defined as:
1 age1 age12 car127.age152½




1 age2 age22 car227.age252½
1 age3 age32 car327.age352½
1 age4 age42 car427.age452½
1 age5 age52 car527.age552½
................................
................................

What is X. ?

Suppose we wanted a model of the form:
 = + 1 if age < 30
+ 2 if age 30 - 40
+ 3 if age > 40
+ 1 if sex male
+ 2 if sex female
What is X. ?
1
2
3
4
5
Age
Sex
<30 30-40 >40
M F
1 010
10
1 100
10
1 100
01
1 001
10
1 010
01
..........................
..........................


1
2
3
1

2
What is X. ?

Suppose we wanted a model of the form:
 = + 1 if age < 30
+ 2 if age 30 - 40
+ 3 if age > 40
+ 1 if sex male
+ 2 if sex female
What is X. ?

Suppose we wanted a model of the form:
 = + 1 if age < 30
+ 2 if age 30 - 40
"Base levels"
+ 3 if age > 40
+ 1 if sex male
+ 2 if sex female
X. having adjusted for base levels
1
2
3
4
5
Age
Sex
<30 30-40 >40
M F
1 010
10
1 100
10
1 100
01
1 001
10
1 010
01
..........................
..........................


1
2
3
1

2
Linear models - formularization
E[Yi] = i = SXijj
Var[Yi] =
2
s
Yi ~ N(i,
2
s)
Generalized linear models
i = f(  + .agei + .agei2 + .heighti.agei )
i = f(  + .agei + .(sexi=female) )
Generalized linear models
i = g-1(  + .agei + .agei2 + .heighti.agei )
i = g-1(  + .agei + .(sexi=female) )
Generalized linear models
Linear Models
Generalized Linear Models
E[Yi] = i = SXijj
E[Yi] = i = g-1(SXijj + xi)
Var[Yi] = s2
Var[Yi] = fV(i)/wi
Y from
Y from a distribution from the
Normal distribution
exponential family
Generalized linear models

Each observation i from distribution with mean i
0.045
0.04
0.035
0.03
0.025
0.02
0.015
0.01
0.005
0
Women
Men
Generalized linear models
E[Yi] = i =
-1
g (SX

ij j
j
+ xi )
Var[Yi] = f.V(i)/wi
Generalized linear models
E[Y] =  = g ( X + x )
-1
Var[Y] = fV() / w
Generalized linear models
E[Y] =  = g ( X
-1
Some function
(user defined)
Observed thing
(data)
)
Parameters to be
estimated
(the answer!)
Some matrix based on data
(user defined)
as per linear models
What is g (X.) ?
-1
-1
Y = g (X.) + error
Assuming a model with three categorical factors,
each observation can be expressed as:
Yijk= g (  + i +  j +  k ) + error
-1
age is in group i

 = =0
=
2
1
3
sex is in group j
car is in group k
What is g (X.) ?
-1


g(x) = x
 Yijk =  + i + j + k + error
( + i + j + k)
g(x) = ln(x)  Yijk = e
= A.Bi.Cj.Dk
+ error
+ error
where Bi = ei etc
 Multiplicative form common for frequency and amounts
Multiplicative model
$ 207.10 x
Age
17
18
19
20
21-23
24-26
27-30
31-35
36-40
41-45
46-50
50-60
60+
Factor
2.52
2.05
1.97
1.85
1.75
1.54
1.42
1.20
1.00
0.93
0.84
0.76
0.78
Group
1
2
3
4
5
6
7
8
9
10
11
12
13
Factor
0.54
0.65
0.73
0.85
0.92
0.96
1.00
1.08
1.19
1.26
1.36
1.43
1.56
Sex
Male
Female
Factor
1.00
1.25
Area Factor
A
0.95
B
1.00
C
1.09
D
1.15
E
1.18
F
1.27
G
1.36
H
1.44
E(losses) = $ 207.10 x 1.42 x 0.92 x 1.00 x 1.15 = $ 311.14
Generalized linear models
E[Y] =  = g ( X + x )
-1
"Offset"
Eg Y = claim numbers
Smith:
Male, 30, Ford, 1 years, 2 claims
Jones:
Female, 40, VW, ½ year, 1 claim
What is x ?

g(x) = ln(x)

xijk = ln(exposureijk)

( + i + j + k + xijk)
E[Yijk] = e
(ln(exposureijk))
= A.Bi.Cj.Dk. e
= A.Bi.Cj.Dk. exposureijk
Restricted models
E[Y] =  = g ( X + x )
-1
Offset

Constrain model (eg increased limits, territory,
amount of insurance, discounts)

Other factors adjusted to compensate
Generalized linear models
E[Y] =  = g ( X + x )
-1
Var[Y] = fV() / w
Generalized linear models
Var[Y] = fV() / w
Normal: f = s2, V(x) = 1  Var[Y] = s2.1
Poisson: f = 1, V(x) = x  Var[Y] = 
Gamma: f = k, V(x) = x  Var[Y] = k
2
2
The scale parameter
Var[Y] = fV() / w
Normal: f = s2, V(x) = 1  Var[Y] = s2.1
Poisson: f = 1, V(x) = x  Var[Y] = 
Gamma: f = k, V(x) = x  Var[Y] = k
2
2
Example of effect of
changing assumed error - 1
10
8
6
4
2
0
1
2
Data
Normal
3
Example of effect of
changing assumed error - 1
10
8
6
4
2
0
1
2
Data
Normal Poisson
3
Example of effect of
changing assumed error - 1
10
8
6
4
2
0
1
2
Data
Normal Poisson Gamma
3
Example of effect of
changing assumed error - 2

Example portfolio with five rating factors, each
with five levels A, B, C, D, E

Typical correlations between those rating factors

Assumed true effect of factors

Claims randomly generated (with Gamma)

Random experience analyzed by three models
Example of effect of
changing assumed error - 2
0.6
0.5
Log of multiplier
0.4
0.3
0.2
0.1
0
A
B
C
-0.1
-0.2
True effect
D
E
Example of effect of
changing assumed error - 2
0.6
0.5
Log of multiplier
0.4
0.3
0.2
0.1
0
A
B
C
D
-0.1
-0.2
True effect
One way
E
Example of effect of
changing assumed error - 2
0.6
0.5
Log of multiplier
0.4
0.3
0.2
0.1
0
A
B
C
D
-0.1
-0.2
True effect
One way
GLM / Normal
E
Example of effect of
changing assumed error - 2
0.6
0.5
Log of multiplier
0.4
0.3
0.2
0.1
0
A
B
C
D
-0.1
-0.2
True effect
One way
GLM / Normal
GLM / Gamma
E
Example of effect of
changing assumed error - 2
0.6
0.5
Log of multiplier
0.4
0.3
0.2
0.1
0
A
B
C
D
-0.1
-0.2
True effect
GLM / Gamma
+ 2 SE
- 2 SE
E
Prior weights
Var[Y] = fV() / w

Exposure

Other credibility
Eg Y = claim frequency
Smith:
Male, 30, Ford, 1 years, 2 claims, 100%
Jones:
Female, 40, VW, ½ year, 1 claim, 100%
Typical model forms
Y
Claim
frequency
Claim
number
Average
claim
amount
g(x)
ln(x)
ln(x)
ln(x)
ln(x/(1-x))
Error
Poisson
Poisson
Gamma
Binomial
f
V(x)
1
x
1
x
estimate
x2
1
x(1-x)
w
exposure
1
# claims
1
x
0
ln(exposure)
0
0
Probability
(eg lapses)
Tweedie distributions
0.2

Incurred losses have a
point mass at zero and
then a continuous
distribution

Poisson and gamma
not suited to this

Tweedie distribution has point mass and parameters
which can alter the shape to be like Poisson and gamma
above zero

f Y ( y;  ,  ,  )  
n 1
(w )
0
  (1 / y )
. exp w[ 0 y    ( 0 )] for y  0
(n )n! y
1
p(Y  0)  exp w ( 0 )
n
Generalized linear models
Var[Y] = fV() / w
Normal: f = s , V(x) = 1  Var[Y] = s .1
2
2
Poisson:f = 1, V(x) = x  Var[Y] = 
Gamma:f = k, V(x) = x  Var[Y] = k
2
2
Tweedie:f = k, V(x) = x  Var[Y] = k
p
p
Tweedie distributions
Tweedie:f = k, V(x) = x  Var[Y] = k
p
p

Defines a valid distribution for p<0, 1<p<2, p>2

Can be considered as Poisson/gamma process
for 1<p<2

Typical values of p for insurance incurred claims
around, or just under, 1.5
Tweedie distributions

Helpful when important to fit to pure premium

Often similar results to traditional approach but
differences may occur if numbers and amounts
models have effects which are both large and
insignificant

No information about whether frequencies or
amounts are driving result
Offset x
Link function g(x)
Error Structure V()
Linear Predictor Form
X. = i + j + k + l
Scale Parameter
f
Data
Y
Prior Weights
w
Numerical MLE
Parameter Estimates
, , , 
Diagnostics
Maximum likelihood estimation

Seek
parameters
which give
highest
likelihood
function
given data
Parameter 1
Likelihood
Parameter 2
Newton-Raphson


In one dimension: xn+1 = xn – f '(xn) / f ''(xn)
In n dimensions: n+1= n – H-1.s
where  is the vector of the parameter estimates (with p elements), s is the
vector of the first derivatives of the log-likelihood and H is the (p*p) matrix
containing the second derivatives of the log-likelihood
Agenda



Formularization of GLMs
–
linear predictor, link function, offset
–
error term, scale parameter, prior weights
–
typical model forms
Model testing
–
use only variables which are predictive
–
make sure model is reasonable
Aliasing
Model testing
 Use
only those variables which are
predictive
–
–
–
standard errors of parameter estimates
F tests / c2 tests on deviances
consistency with time
 Make sure the model is reasonable
– histogram of deviance residuals
– residual vs fitted value
Standard errors

Roughly speaking, for a parameter p: SE = -1 / (2/ p2 Likelihood)
Likelihood
Parameter 1
Parameter 2
GLM output (significant factor)
1.2
200000
180000
1
154%
138%
0.8
160000
105%
140000
84%
73%
0.6
120000
72%
58%
100000
45%
0.4
39%
80000
31%
Exposure (years)
Log of multiplier
93%
60000
0.2
5%
40000
0%
0
20000
-0.2
0
1
2
3
4
5
6
7
8
9
10
11
12
13
Vehicle symbol
P value = 0.0%
Onew ay relativities
Approx 95% confidence interval
Parameter estimate
GLM output (insignificant factor)
0.2
200000
180000
0.15
160000
0.1
7%
3%
3%
0%
120000
0%
-1%
0
-5%
-1%
-1%
-1%
-5%
100000
80000
-0.05
60000
-0.1
40000
-0.15
20000
-0.2
0
1
2
3
4
5
6
7
8
9
10
11
12
13
Vehicle symbol
P value = 52.5%
Onew ay relativities
Approx 95% confidence interval
Parameter estimate
Exposure (years)
4%
0.05
Log of multiplier
140000
4%
Awkward cases
2
1.8
1.6
1.4
200%
1.2
159%
Log of multiplier
1
123%
0.8
92%
101%
65%
0.6
42%
0.4
16%
11%
0.2
0
-10%
0%
-5%
-10%
-14%
-22%
-0.2
-0.4
-0.6
-0.8
-1
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Deviances

Single figure measure of goodness of fit

Try model with & without a factor

Statistical tests show the theoretical significance
given the extra parameters
Deviances
Age
Sex
Vehicle
Model A
Zone
Fitted
value
Deviance = 9585
df = 109954
Mileage
?
Claims
Age
Sex
Vehicle
Model B
Mileage
Claims
Fitted
value
Deviance = 9604
df = 109965
Deviances

If f known, scaled deviance S output
n
S=
S
u=1
Yu
2 wu / f  (Yu - ) / V() d


S1 - S2 ~ c

u
2
d1-d2
If f unknown, unscaled deviance D = f.S output
(D1 - D2)
(d1 - d2) D3 / d3
~ Fd -d , d
1
2
3
Consistency over time
1.2
100000
185%
1
150%
149%
140%
80000
118%
104%
Log of multiplier
97%
81%
73%
0.6
0.4
33%
27%
45%
38%
34%
51%
44%
40%
78%
76%
64%
63%
56%
81%
108%
105%
104%
89%
86%
60000
69%
65%
40000
Exposure (years)
0.8
0.2
0
4%
0%
-4%
10%
5%
2%
20000
-0.2
0
1
2
3
4
5
6
7
8
9
10
11
12
13
Vehicle symbol.Year of exposure
Approx 95% confidence interval, Year of exposure: 2000
Approx 95% confidence interval, Year of exposure: 2001
Approx 95% confidence interval, Year of exposure: 2002
Parameter estimate, Year of exposure: 2000
Parameter estimate, Year of exposure: 2001
Parameter estimate, Year of exposure: 2002
Consistency over time
0.8
95%
0.7
100000
0.6
0.5
49%
Log of multiplier
29%
0.3
0.2
0.1
14%
2%
30%
26%
16%
60000
4%
3%
0%
2%
0
-9%
-13%
-0.1
-0.2
-21%
-23%
-15%
Exposure (years)
80000
0.4
40000
-15%
-20%
-22%
-25%
20000
-0.3
-0.4
-0.5
0
1
2
3
4
5
6
7
Territory.Year of exposure
Approx 95% confidence interval, Year of exposure: 2000
Approx 95% confidence interval, Year of exposure: 2001
Approx 95% confidence interval, Year of exposure: 2002
Smoothed estimate, Year of exposure: 2000
Smoothed estimate, Year of exposure: 2001
Smoothed estimate, Year of exposure: 2002
Model testing
 Use
only those variables which are
predictive
–
–
–
standard errors of parameter estimates
F tests / c2 tests on deviances
interaction with time
 Make sure the model is reasonable
– histogram of deviance residuals
– residual vs fitted value
Residuals
4.5
4
3.5
3
2.5
2
1.5
0
1
2
3
4
Residuals Fitted values
5
Data
6
7
Residuals

Several forms, eg
– standardized deviance
sign (Yu-u) / ( f(1-hu) ) ½
Yu
2 wu ( Yu- ) / V()d


u
–
standardized Pearson
Yu -  u
( f.V( u ).(1-h u ) / wu )
½

Standardized deviance - Normal (0,1)

Numbers/frequency residuals problematical
Residuals
Histogramof Deviance Residuals
Run 12 (Final models with analysis) Model 8 (AD amounts)
6000
5000
Frequency
4000
3000
2000
1000
Pretium 08/01/2004 12:24
5.25
5
4.5
4.75
4
4.25
3.5
3.75
3
3.25
2.5
2.75
2
2.25
1.5
Size of deviance residuals
1.75
1
1.25
0.5
0.75
0
0.25
-0.25
-0.5
-0.75
-1
-1.25
-1.5
-1.75
-2
-2.25
-2.5
-2.75
-3
-3.25
-3.5
-3.75
-4
-4.25
-4.5
0
Residuals
4
3
2
Residual
1
0
-1
-2
-3
-4
5.6
5.7
5.8
Pretium 04/05/2004 11:03
5.9
6.0
6.1
6.2
6.3
6.4
6.5
6.6
6.7
Log of fitted value
6.8
6.9
7.0
7.1
7.2
7.3
7.4
7.5
Residuals
Gamma data, Gamma error
Plot of deviance residual against fitted value
Run 12 (All claim types, final models, N&A) Model 6 (Own damage, Amounts)
2
Deviance Residual
1
0
-1
-2
-3
500
600
700
800
900
1000
1100
1200
1300
1400
Fitted Value
Pretium 08/01/2004 12:32
1500
1600
1700
1800
1900
2000
2100
2200
Gamma data, Normal error
Plot of deviance residual against fitted value
Run 12 (All claim types, final models, N&A) Model 7 (Own damage, Amounts)
6000
5000
4000
Deviance Residual
3000
2000
1000
0
-1000
-2000
500
600
700
800
900
1000
1100
1200
1300
1400
1500
Fitted Value
Pretium 08/01/2004 12:32
1600
1700
1800
1900
2000
2100
2200
2300
Agenda



Formularization of GLMs
–
linear predictor, link function, offset
–
error term, scale parameter, prior weights
–
typical model forms
Model testing
–
use only variables which are predictive
–
make sure model is reasonable
Aliasing
Aliasing and "near aliasing"

Aliasing
–

Intrinsic aliasing
–

the removal of unwanted redundant parameters
occurs by the design of the model
Extrinsic aliasing
–
occurs "accidentally" as a result of the data
Intrinsic aliasing
X. = + 1 if age 20 - 29
+ 2 if age 30 - 39
+ 3 if age 40 +
+ 1 if sex male
+ 2 if sex female

"Base levels"
Intrinsic aliasing
Example job
Run 16 Model 3 - Small interaction - Third party material damage, Numbers
1.2
250000
155%
138%
200000
Log of multiplier
0.8
0.6
63%
150000
0.4
28%
24%
100000
0.2
0%
-6%
0
-11%
50000
-19%
-0.2
-0.4
0
17-21
22-24
25-29
30-34
35-39
40-49
50-59
60-69
Age of driver
Onew ay relativities
Approx 95% confidence interval
Unsmoothed estimate
Smoothed estimate
70+
Exposure (years)
1
Extrinsic aliasing

If a perfect correlation exists, one factor can alias levels
of another

Eg if doors declared first:
Exposure: # Doors  2
Color 
13,234
Selected base Red
Further aliasing

Selected base
3
4
5 Unknown
12,343
13,432
13,432
0
Green
4,543
4,543
13,243
2,345
0
Blue
6,544
5,443
15,654
4,565
0
Black
4,643
1,235
14,565
4,545
0
Unknown
0
0
0
0
3,242
This is the only reason the order of declaration can
matter (fitted values are unaffected)
Extrinsic aliasing
Example job
Run 16 Model 3 - Small interaction - Third party material damage, Numbers
1
135%
0.8
600000
103%
500000
91%
72%
Log of multiplier
0.6
70%
400000
57%
45%
0.4
39%
300000
30%
0.2
200000
0%
0%
0
100000
-0.2
0
50
55-59
60-64
65-69
70-74
75-79
80-84
85-89
90-94
95-99
No claim discount
Onew ay relativities
Approx 95% confidence interval
Unsmoothed estimate
Smoothed estimate
100
Malus
Exposure (years)
82%
"Near aliasing"

If two factors are almost perfectly, but not quite aliased,
convergence problems can result as a result of low exposures
(even though one-ways look fine), and/or results can become hard
to interpret
Selected base
Exposure: # Doors  2
Color 
Selected base Red
13,234

3
4
5 Unknown
12,343
13,432
13,432
0
Green
4,543
4,543
13,243
2,345
0
Blue
6,544
5,443
15,654
4,565
0
Black
4,643
1,235
14,565
4,545
2
Unknown
0
0
0
0
3,242
Eg if the 2 black, unknown doors policies had no claims, GLM would
try to estimate a very large negative number for unknown doors,
and a very large positive number for unknown color
"A Practitioner's Guide to
Generalized Linear Models"

CAS 2004 Discussion
Paper Program

CAS Exam 9 syllabus as of
2006

Copies available at
www.watsonwyatt.com/glm
PL-2
An Introduction to
GLM Theory
2006 CAS Seminar on
Ratemaking
Claudine Modlin, FCAS
Watson Wyatt Worldwide
WWW.WATSONWYATT.COM