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Chapter 5. Logistic Regression
Deyuan Li
School of Management, Fudan University
Feb. 28, 2011
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Outline
•
5.1 Interpreting Parameters in Logistic Regression
•
5.2 Inference for Logistic Regression
•
5.3 Logit Models with Categorical Predictors
•
5.4 Multiple Logistic Regression
•
5.5 Fitting Logistic Regression Models
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5.1 Interpreting Parameters in Logistic Regression
For a binary response variable
Y
and an explanatory variable
X
,let
π
(
x
)=
P
(
Y
=1

X
=
x
)=1
−
P
(
Y
=0

X
=
x
).
The logistic regression model is
π
(
x
exp(
α
+
β
x
)
1+exp(
α
+
β
x
)
.
Equivalently, the log odds, called the
logit
, has the linear
relationship
{
Insert .
.....
}
This equates the logit link function to the linear predictor.
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5.1.1 Interpreting
β
: odds, probabilities, and linear
approximations
The sign of
β
determines whether
π
(
x
) is increasing or decreasing
as
x
increases.
The

β

determines the rate of climb or descent.
β
→
0:
the curve ﬂattens to a horizontal straight line.
β
:
Y
is independent of
X
.
For quantitative
x
with
β>
0, the curve for
π
(
x
)hastheshapeo
f
the cdf of the logistic distribution (recall Section 4.2.5).
Since the logistic density is symmetric,
π
(
x
) approaches 1 at the
same rate that it approaches 0.
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and odds
Exponentiating both sides of the logit equation shows that the
odds are an exponential function of
x
:
odds =
π
(
x
)
/
[1
−
π
(
x
)] = exp(
α
+
β
x
)=
e
α
e
β
x
.
Interpretation for the magnitude of
β
:
•
the odds increase multiplicatively by
e
β
for every 1unit
increase in
x
;
•
e
β
is an odds ratio, i.e.,
odds at
X
=
x
+1
odds at
X
=
x
=
π
(
x
+1)
/
[1
−
π
(
x
+1)]
π
(
x
)
/
[1
−
π
(
x
)]
=
e
α
e
β
(
x
+1)
e
α
e
β
x
=
e
β
.
β
and linear approximation
Since
π
(
x
) changes with
x
and
∂π
(
x
)
/∂
x
=
βπ
(
x
)[1
−
π
(
x
)], the
rate of change in
π
(
x
) per unit change in
x
varies.
This expression also shows that the logistic density is symmetric,
i.e.,
π
(
x
) approaches a value
v
at the same rate that it approaches
(1
−
v
).
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A straight line drawn tangent to the logistic regression curve at a
particular
x
value describes the rate of change at that point (see
Figure 5.1).
FIGURE 5.1
Linear approximation to logistic regression curve.
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Example
π
(
x
)s
l
o
p
e
=
(
x
)[1
−
π
(
x
)]
1
/
2
β/
4
0
.
9o
r0
.
10
.
09
β
→
1o
→
0
The steepest slope occurs when
π
(
x
)=1
/
2
⇒
odds = 1
⇒
logit = 0
⇒
α
+
β
x
=0
⇒
x
=
−
α/β.
This
x
value (
−
α/β
) is sometimes called the
median efective level
and denoted by EL
50
.
In toxicology studies it is called LD
50
(LD = lethal dose), i.e., the
dose with a 50% chance of a lethal result.
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From this linear approximation, near
x
where
π
(
x
/
2, a
change in
x
of 1
/β
corresponds to change in
π
(
x
) of roughly
(1
/β
)(
4) = 1
/
4;
that is, 1
/β
approximates the distance between
x
values where
π
(
x
)=0
.
25 or 0
.
75 (in reality, 0
.
27 and 0
.
73) and where
π
(
x
.
50.
The linear approximation works better for smaller changes in
x
.
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 Spring '11
 DeyuanLi

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