Generalized Linear Models 1
April 18, 2019

GLMs
I
We have so far studied linear models. We have
n
observations on a response variable
y
1
, . . . ,
y
n
and on
each of
p
explanatory variables
x
ij
for
i
=
1
, . . . ,
n
and
j
=
1
, . . . ,
p
.

GLMs
I
We have so far studied linear models. We have
n
observations on a response variable
y
1
, . . . ,
y
n
and on
each of
p
explanatory variables
x
ij
for
i
=
1
, . . . ,
n
and
j
=
1
, . . . ,
p
.
I
The linear model that we have seen models
μ
i
:=
E
y
i
=
β
0
+
β
1
x
i
1
+
· · ·
+
β
p
x
ip
.

GLMs
I
We have so far studied linear models. We have
n
observations on a response variable
y
1
, . . . ,
y
n
and on
each of
p
explanatory variables
x
ij
for
i
=
1
, . . . ,
n
and
j
=
1
, . . . ,
p
.
I
The linear model that we have seen models
μ
i
:=
E
y
i
=
β
0
+
β
1
x
i
1
+
· · ·
+
β
p
x
ip
.
I
What this model implies is that when there is a unit
increase in the explanatory variable
x
j
, the mean of the
response variable changes by the amount
β
j
.

GLMs
I
We have so far studied linear models. We have
n
observations on a response variable
y
1
, . . . ,
y
n
and on
each of
p
explanatory variables
x
ij
for
i
=
1
, . . . ,
n
and
j
=
1
, . . . ,
p
.
I
The linear model that we have seen models
μ
i
:=
E
y
i
=
β
0
+
β
1
x
i
1
+
· · ·
+
β
p
x
ip
.
I
What this model implies is that when there is a unit
increase in the explanatory variable
x
j
, the mean of the
response variable changes by the amount
β
j
.
I
This may not always be a reasonable assumption.

GLMs
I
For example, if the response
y
i
is a binary variable, then its
mean
μ
i
is a probability which is always constrained to stay
between 0 and 1.

GLMs
I
For example, if the response
y
i
is a binary variable, then its
mean
μ
i
is a probability which is always constrained to stay
between 0 and 1.
I
Therefore, the amount by which
μ
i
changes per unit
change in
x
j
would now depend on the value of
μ
i
(for
example, the change when
μ
i
=
0
.
9 may not be the same
as when
μ
i
=
0
.
5).

GLMs
I
For example, if the response
y
i
is a binary variable, then its
mean
μ
i
is a probability which is always constrained to stay
between 0 and 1.
I
Therefore, the amount by which
μ
i
changes per unit
change in
x
j
would now depend on the value of
μ
i
(for
example, the change when
μ
i
=
0
.
9 may not be the same
as when
μ
i
=
0
.
5).
I
Therefore, modeling
μ
i
as a linear combination of
x
1
, . . . ,
x
p
may not be the best idea always.

GLMs
I
For example, if the response
y
i
is a binary variable, then its
mean
μ
i
is a probability which is always constrained to stay
between 0 and 1.
I
Therefore, the amount by which
μ
i
changes per unit
change in
x
j
would now depend on the value of
μ
i
(for
example, the change when
μ
i
=
0
.
9 may not be the same
as when
μ
i
=
0
.
5).
I
Therefore, modeling
μ
i
as a linear combination of
x
1
, . . . ,
x
p
may not be the best idea always.
I
A more general model might be
g
(
μ
i
) :=
β
0
+
β
1
x
i
1
+
· · ·
+
β
p
x
ip
(1)
for a function
g
that is not necessarily the identity function.

I
Another feature of the linear model that people do not
always like is that some aspects of the theory are tied to
the normal distribution.

I
Another feature of the linear model that people do not