Information Technology Reference
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Table 17. Examples of distributions and their link functions
Outcome
Distribution
Link function
Continuous
Normal
Identity
Binary
Binomial
Logit
Ordinal
Multinomial
Cumulative Logit
Binary
Poisson
Log
Continuous
Gamma
Reciprocal
-1
EY ugX u
Varu
(/)
=
(
b
+
)
()
=
G
1
2
1
2
VarY
(/)
uARA
=
where g
-1
() is the inverse link function and A is a diagonal matrix containing the variance functions.
Logistic regression is a special case of the generalized linear model. The link function specified is
the logit function. Besides the logit function, we use three different link functions: normal (which is
equal to the linear regression), negative binomial, and gamma. The density function for the negative
binomial is equal to
æ
ö
kr
k
+-
1
÷
÷
÷
÷
÷
ç
ç
ç
ç
ç
r
k
fkrp
(;,)
=
p
(
1
-
p
)
for k=0,1,2….
è
ø
where
ö
ö
æ
=-
æ
è
kr
k
+-
1
r
k
÷
÷
÷
÷
÷
=
G
kr
kr
(
)
!()
+
÷
÷
÷
÷
÷
ç
ç
ç
ç
ç
ç
ç
ç
ç
ç
k
()
1
G
è
ø
ø
and Γ(r)=(r-1)!
Figure 21 shows the shape of the distribution for different values of r; the distribution for r=1 and
r=2 is the closest to that of Figure 2 in Chapter 2.
Similarly, the density of the gamma function is defined by
-
x
e
q
=
-
k
1
fxk
(;,)
q
x
k
q
G
()
k
.
The distribution is shown in Figure 22.
In the case of the gamma function as well, the density function can take many different shapes de-
pending upon the parameters. We demonstrate using our example of pneumonia, septicemia, and immune
disorder. Table 18 shows the estimates of the coefficients for the three link functions.
The gamma link function indicates that having an infection such as septicemia results in a lower length
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