1 p. j. safˇ ´arik university, ko ˇ antoch/robust10/prednasky/... · pdf...

Logistic, multinomial, and ordinal regression
I. Zezula1
1Faculty of ScienceP. J. Safarik University, Kosice
Robust 201031.1. 5.2.2010, Kralky

Logistic regressionMultinomial regression
Ordinal regression
Obsah
1 Logistic regressionIntroductionBasic modelMore general predictorsGeneral modelTests of association
2 Multinomial regressionIntroductionGeneral modelGoodness-of-fit and overdispersionTests
3 Ordinal regressionIntroductionProportional odds modelOther models
I. Zezula Robust 2010

Ordinal regression
IntroductionBasic modelMore general predictorsGeneral modelTests of association
1) Logistic regression
IntroductionTesting a risk factor: we want to check whether a certain factor addsto the probability of outbreak of a disease.
This corresponds to the following contingency table:
risk factordisease status
sumill not ill
exposed n11 n12 n10unexposed n21 n22 n20
sum n01 n02 n

Ordinal regression
IntroductionTesting a risk factor: we want to check whether a certain factor addsto the probability of outbreak of a disease.
This corresponds to the following contingency table:
risk factordisease status
sumill not ill
exposed n11 n12 n10unexposed n21 n22 n20
sum n01 n02 n
Odds of the outbreak for both groups are:
o1 =n11
n10 n11=
n11n10
1 n11n10=
p11 p1
, o2 =p2
1 p2

Ordinal regression
As a measure of the risk, we can form odds ratio :
OR =o1o2
=p1
1 p1
1 p2p2
o1 = OR o2

Ordinal regression
OR =o1o2
=p1
1 p1
1 p2p2
o1 = OR o2
Then, it holds
log (o1) = log (o2) + log (OR) ,

Ordinal regression
OR =o1o2
=p1
1 p1
1 p2p2
o1 = OR o2
Then, it holds
log (o1) = log (o2) + log (OR) ,
ory = log (o2) + log (OR) x ,
where x {0, 1} and y {log (o1) , log (o2)}.

Ordinal regression
This is the basic logistic model. Formally it is a regression model
y = 0 + 1x
with baseline 0 = log (o2) and slope 1 = log (OR) effect ofthe exposure.

Ordinal regression
y = 0 + 1x
with baseline 0 = log (o2) and slope 1 = log (OR) effect ofthe exposure.If we denote the probability of the event (outbreak of the disease)by p, then
log(
p1 p
)
= 0 + 1x
in either group.

Ordinal regression
y = 0 + 1x
with baseline 0 = log (o2) and slope 1 = log (OR) effect ofthe exposure.If we denote the probability of the event (outbreak of the disease)by p, then
log(
p1 p
)
= 0 + 1x
in either group.It follows
p =exp (0 + 1x)
1 + exp (0 + 1x)=
11 + exp (0 1x)

Ordinal regression
Estimation is done via ML-method. Likelihood of observedfrequencies for given s is
L() =(
n10n11
)
pn111 (1 p1)n10n11
(
n20n21
)
pn212 (1 p2)n20n21

Ordinal regression
L() =(
n10n11
)
pn111 (1 p1)n10n11
(
n20n21
)
pn212 (1 p2)n20n21
Then
() = log L() = n11 log (p1) + n12 log (1 p1)+
+ n21 log (p2) + n22 log (1 p2)

Ordinal regression
L() =(
n10n11
)
pn111 (1 p1)n10n11
(
n20n21
)
pn212 (1 p2)n20n21
Then
() = log L() = n11 log (p1) + n12 log (1 p1)+
+ n21 log (p2) + n22 log (1 p2)
It is easy to derive that ML-equations are
1= n11 n10p1 = 0 ,
0= n11 n10p1 + n21 n20p2 = 0

Ordinal regression
In this case we already know the solution, p1 =n11n10
and p2 =n21n20
(and corresponding s).

Ordinal regression
and p2 =n21n20
Asymptotic variance matrix of s can also be established. In thiscase it is
var(
01
)
= J1 =1
ab
(
a aa a + b
)
,
where a = n10p1 (1 p1), b = n20p2 (1 p2).

Ordinal regression
and p2 =n21n20
Asymptotic variance matrix of s can also be established. In thiscase it is
var(
01
)
= J1 =1
ab
(
a aa a + b
)
,
where a = n10p1 (1 p1), b = n20p2 (1 p2).
J =(
a + b aa a
)
is the information matrix.

Ordinal regression
Example: Baystate Medical Center in Springfield, MA, studiedfactors influencing low birth weights of babies. Let us take as arisk factor smoking during pregnancy. We get the followingcontingency table:
smokinglow birth weight
sumyes no
exposed 30 44 74unexposed 29 86 115
sum 59 130 159

Ordinal regression
Example: Baystate Medical Center in Springfield, MA, studiedfactors influencing low birth weights of babies. Let us take as arisk factor smoking during pregnancy. We get the followingcontingency table:
smokinglow birth weight
sumyes no
exposed 30 44 74unexposed 29 86 115
sum 59 130 159
Odds for the unexposed is o1 = 30/44 = 0.681818, for theexposed o2 = 29/86 = 0.337209, OR = 2.021644. Sakodacoefficient S = 0.225344 indicates moderate association, butstatistically significant (2 = 4.92 > 3.84 = 21(0.05)).

Ordinal regression
As log(2.021644) = 0.7040592, log(0.337209) = 1.087051,we get the model
y = 1.087 + 0.704 x .

Ordinal regression
As log(2.021644) = 0.7040592, log(0.337209) = 1.087051,we get the model
y = 1.087 + 0.704 x .
Software gets
coefficient std. error z P > |z| 95% conf. intervalsmoking 0.7040592 0.319639 2.20 0.028 0.077579 1.330539constant -1.087051 0.21473 -5.06 0.000 -1.50791 -0.66619

Ordinal regression
Int

1 p. j. safˇ ´arik university, ko ˇ antoch/robust10/prednasky/... · pdf...

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