The Clayton & Hills book Mean blood pressure in patients ...publicifsv.sund.ku.dk/~nk/epiE12/PKA...
26
Ph.d. course in Epidemiology: Fall 2012 Regression models Clayton & Hills, Ch. 22-26 23, 30 October, 6 November 2012 www.biostat.ku.dk/~nk/epiE12 Per Kragh Andersen and Henrik Ravn 1 The Clayton & Hills book 1. Introductory concepts and methods, ch. 1-11. 2. Simple methods for cohort and case-control studies, ch. 13-19. 3. Regression models, ch. 22-27. In the first part a general method to estimate parameters in statistical models is introduced: LIKELIHOOD In some examples we may be able to come up with a suggestion to the parameter estimate without this general tool: • Mean blood pressure in patients with heart disease, • probability of heads in coin tossing, • risk of death in cohort study, • cancer rate among asbestos workers 2 But: 1. What about confidence intervals and significance tests? 2. What about examples like: • Effect of drug on blood pressure for heart patients adjusted for sex, age and weight? • Risk of death in cohort study and its dependence on age, cholesterol, stage of disease? • Cancer rate adjusted for age, degree of exposure, duration of exposure? In these examples we are not able to suggest a simple, obvious method and we have to rely on our great tool! 3 Regression models. - Outcome variable is related to - Explanatory variables. In epidemiology: binary outcome ⎧ ⎨ ⎩ rate Poisson, Cox odds or probability logistic In other fields, some times quantitative outcome: mean value Gaussian C & H, Ch. 34 4
Transcript of The Clayton & Hills book Mean blood pressure in patients ...publicifsv.sund.ku.dk/~nk/epiE12/PKA...
Ph.d
.co
urs
ein
Epid
em
iolo
gy:
Fall
2012
Regre
ssio
nm
odels
Cla
yto
n&
Hills
,C
h.
22-2
6
23,30
Oct
ober,
6N
ovem
ber
2012
www.biostat.ku.dk/~nk/epiE12
Per
Kra
ghA
nder
sen
and
Hen
rik
Rav
n
1
The
Cla
yto
n&
Hills
book
1.In
troduct
ory
conce
pts
and
met
hods,
ch.
1-1
1.
2.Sim
ple
met
hods
for
cohort
and
case
-contr
olst
udie
s,ch
.13
-19.
3.R
egre
ssio
nm
odel
s,ch
.22
-27.
Inth
efirs
tpar
ta
gener
alm
ethod
toes
tim
ate
para
met
ers
in
stat
isti
calm
odel
sis
intr
oduce
d:
LIK
ELIH
OO
D
Inso
me
exam
ple
sw
em
aybe
able
toco
me
up
wit
ha
sugges
tion
to
the
para
met
eres
tim
ate
wit
hout
this
gen
eralto
ol:
•M
ean
blo
od
pre
ssure
inpati
ents
wit
hhea
rtdis
ease
,
•pro
bability
ofhea
ds
inco
into
ssin
g,
•ri
skof
dea
thin
cohort
study,
•ca
nce
rra
team
ong
asbes
tos
wor
ker
s
2
But:
1.W
hat
abou
tco
nfiden
cein
terv
als
and
sign
ifica
nce
test
s?
2.W
hat
abou
tex
ample
slike:
•E
ffec
tof
dru
gon
blo
od
pre
ssure
for
hea
rtpat
ients
adju
sted
for
sex,age
and
wei
ght?
•R
isk
ofdea
thin
cohor
tst
udy
and
its
dep
enden
ceon
age,
chol
este
rol,
stag
eof
dis
ease
?
•C
ance
rra
tead
just
edfo
rag
e,deg
ree
ofex
pos
ure
,dura
tion
of
expos
ure
?
Inth
ese
exam
ple
sw
ear
enot
able
tosu
gges
ta
sim
ple
,ob
vio
us
met
hod
and
we
hav
eto
rely
on
our
grea
tto
ol!
3
Regre
ssio
nm
odels
.
-O
utc
om
eva
riable
isre
late
dto
-E
xpla
nato
ryva
riable
s.
Inepid
em
iolo
gy:
bin
ary
outc
om
e
⎧ ⎨ ⎩ra
tePois
son,C
ox
odds
or
pro
bability
logis
tic
Inot
her
fiel
ds,
som
eti
mes
quan
tita
tive
outc
ome:
mea
nva
lue
Gauss
ian
C&
H,C
h.
34
4
Regre
ssio
nm
odels
.
Inepid
em
iolo
gy:
expla
nat
ory
vari
able
s
⎧ ⎨ ⎩ex
pos
ure
s
confo
under
s
tech
nic
ally
trea
ted
iden
tica
lly
inre
gres
sion
model
s(a
sop
pos
edto
stra
tified
analy
sis)
:
WE
AK
NE
SS/ST
RE
NG
TH
?
5
Exam
ple
:A
ge
stra
tified
com
pari
son
oftw
ora
tes
Age
-ban
d,
t=
0,1,
2:
λt 1
=θλ
t 0
λt 1∼
expos
edλ
t 0∼
not
expose
d.
Expos
ure
isas
sum
edto
hav
eth
esa
me
mult
iplica
tive
effec
tin
all
age-
ban
ds;
θ(=
θ 1)
isth
iseff
ect
(Table
22.1
):
Expos
ure
Exposu
re
Age
01
Age
01
0λ
0 0λ
0 0θ 1
0λ
0 0λ
0 0θ 1
1λ
1 0λ
1 0θ 1
1λ
0 0ϕ
1λ
0 0ϕ
1θ 1
2λ
2 0λ
2 0θ 1
2λ
0 0ϕ
2λ
0 0ϕ
2θ 1
Sim
ilar
ly,th
eag
eeff
ect
isdes
crib
edusi
ng
mult
iplica
tive
effec
ts
rela
tive
toa
bas
elin
ele
vel
(=A
ge
0)
(∼Table
22.2
).
6
Exam
ple
:A
ge
stra
tified
com
pari
son
oftw
ora
tes
Fin
ally:
anew
nam
eλ
Cin
stea
dof
λ0 0
for
the
rate
inth
ere
fere
nce
gro
up
(Tab
le22.3
).
Expos
ure
Age
01
0λ
Cλ
Cθ 1
1λ
Cϕ
1λ
Cϕ
1θ 1
2λ
Cϕ
2λ
Cϕ
2θ 1
Exerc
ise
22.1
,p.
219
7
Exerc
ise
22.1
.
Exposu
re
Age
01
0λ
C=
5.0
λC
θ 1=
15.0
1λ
Cϕ
1=
12.0
λC
ϕ1
θ 1=
36.
0
2λ
Cϕ
2=
30.0
λC
ϕ2
θ 1=
90.
0
What
are
the
valu
esofθ 1
,ϕ1,ϕ
2,λ
C?
8
Gre
ekle
tter
sar
esu
itab
lefo
rdoi
ng
the
mat
h,not
asnam
esin
com
pute
rpro
gram
s
Inst
ead,use
nam
eslike
–A
ge
–E
xposu
re
Expos
ure
Age
01
0C
orner
Cor
ner
×E
xpos
ure
(1)
1C
orner
×A
ge(1
)C
orner
×A
ge(1
)×
Expos
ure
(1)
2C
orner
×A
ge(2
)C
orner
×A
ge(2
)×
Expos
ure
(1)
Rat
e=
Cor
ner
×E
xpos
ure
×A
ge
Mor
eof
ten
the
par
amet
ers
are
wri
tten
ona
log-
scal
e.
log(
Rat
e)=
Cor
ner
+E
xpos
ure
+A
ge
9
An
exam
ple
:
Table
22.6
.E
ner
gyin
take
and
IHD
inci
den
ceper
1000
per
son-y
ears
Expose
dU
nex
pose
d
Curr
ent
(<2750
kca
l)(≥
2750
kca
l)
age
Case
sP
-yrs
.R
ate
Case
sP
-yrs
.R
ate
RR
40–49
2311.9
6.4
14
607.9
6.5
80.9
7
50–59
12
878.1
13.6
75
1271.1
3.9
33.4
8
60–69
14
667.5
20.9
78
888.9
9.0
02.3
3
Tota
l28
1857.5
15.0
717
2768.9
6.1
42.4
5
RA
TE
=C
OR
NE
R×
EX
PO
SU
RE
×A
GE
The
regre
ssio
nm
odel
claim
sth
era
tera
tios
tobe
const
ant
over
age
bands.
10
Est
imate
s:
Table
22.7
.E
stim
ated
valu
esof
the
par
amet
ers
for
the
IHD
dat
a
Par
amet
erE
stim
ate
Cor
ner
0.00
444
Expos
ure
(1)
×2.
39
Age
(1)
×1.
14
Age
(2)
×2.
00
obta
ined
from
the
likel
ihood
funct
ion.
Com
par
epre
dic
ted
rate
sw
ith
obse
rved
rate
s:
Exerc
ise
22.3
11
Exerc
ise
22.3
:so
luti
on.
Table
22.6
.E
ner
gyin
take
and
IHD
inci
den
ceper
1000
per
son-y
ears
Expose
dU
nex
pose
d
Curr
ent
(<2750
kca
l)(≥
2750
kca
l)
age
Case
sP
-yrs
.R
ate
Case
sP
-yrs
.R
ate
40–49
2311.9
6.4
1(1
0.6
1)
4607.9
6.5
8(4
.44)
50–59
12
878.1
13.6
7(1
2.1
0)
51271.1
3.9
3(5
.06)
60–69
14
667.5
20.9
7(2
1.2
2)
8888.9
9.0
0(8
.88)
12
The
esti
mat
esar
efr
equen
tly
given
ona
log
scale
:
log(R
AT
E)=
CO
RN
ER
+E
XP
OSU
RE
+A
GE
Table
22.8
.E
stim
ated
par
amet
ers
and
SD
son
alo
gsc
ale
Par
amet
erE
stim
ate
(M)
SD
(S)
Cor
ner
-5.4
180
0.44
20
Expos
ure
(1)
0.8697
0.3
080
Age
(1)
0.12
900.
4753
Age
(2)
0.69
200.
4614
Her
e,SD
’sar
eob
tain
edby
Gauss
ian
appro
xim
ations
toth
elo
gpro
file
likelihood.
13
Appro
xim
ate
90%
confiden
cein
terv
alfo
reff
ect
ofex
posu
re:
exp(0
.8697)/
×ex
p(1
.645·0
.3080)
Fro
m2.3
9/1.6
6=
1.4
4to
2.3
9·1.6
6=
3.9
6
Exerc
ise
22.4
14
Exerc
ise
22.4
:so
luti
on.
Appro
xim
ate
90%
confiden
cein
terv
alfo
reff
ect
ofex
pos
ure
:
exp(0
.869
7)/×
exp(1
.645
·0.3
080)
Fro
m2.3
9/1.6
6=
1.4
4to
2.3
9·1.6
6=
3.9
6
–an
dfo
rth
efirs
tag
eeff
ect:
exp(0
.129
0)/×
exp(1
.645
·0.4
753)
.
Fro
m1.1
4/2.1
9=
0.5
2to
1.1
4·2.1
9=
2.4
9
15
Pois
son
regre
ssio
n
Table
22.6
again
Pois
son
log
likelihood
for
one
“ce
ll”
Dlo
g(λ
)−
Y×
λ
e.g.
,ex
pos
ed=
level
1,ag
eban
d=
level
2
14
log(λ
2 1)−
667.5×
λ2 1
=14{C
OR
NE
R+
EX
PO
SU
RE
(1)
+A
GE
(2)}
–667.5
exp{C
OR
NE
R+
EX
PO
SU
RE
(1)
+A
GE
(2)}
16
Pois
son
regre
ssio
n
Tot
allo
glikel
ihood
can
be
com
pute
dfr
omth
efr
equen
cyre
cord
s:
Table
23.1
.T
he
IHD
dat
aas
freq
uen
cyre
cord
s
Cas
esPer
son-y
ears
Age
Expos
ure
4607.9
00
2311.9
01
51272.1
10
12
878.1
11
8888.9
20
14
667.5
21
and
max
imis
ed.
17
SA
Spro
gra
m
/*
*****DATASTEP******/
dataihd;
inputekspalderpyrscases;
lpyrs=log(pyrs);
datalines;/*or,alternatively,readfromwwworfromfile*/
02
311.92
01
878.112
00
667.514
12
607.94
11
1272.15
10
888.98
; run;
/*
*****PROCedureSTEP
******/
procgenmoddata=ihd;
classekspalder;
modelcases=ekspalder/dist=poioffset=lpyrstype3;
run;
18
SA
Soutp
ut
(edit
ed)
TheGENMODProcedure
ModelInformation
DataSet
WORK.IHD
Distribution
Poisson
LinkFunction
Log
DependentVariable
cases
OffsetVariable
lpyrs
ObservationsUsed
6
ClassLevelInformation
Class
Levels
Values
eksp
20
1
alder
30
12
19
CriteriaForAssessingGoodnessOfFit
Criterion
DF
Value
Value/DF
Deviance
21.6727
0.8364
ScaledDeviance
21.6727
0.8364
PearsonChi-Square
21.6516
0.8258
ScaledPearsonX2
21.6516
0.8258
LogLikelihood
52.5435
20
AnalysisOfParameterEstimates
Standard
Wald95%
Chi-
Parameter
DF
Estimate
Error
ConfidenceLimits
Square
Intercept
1-5.4177
0.4421
-6.2841
-4.5513
150.20
eksp
01
0.8697
0.3080
0.2659
1.4734
7.97
eksp
10
0.0000
0.0000
0.0000
0.0000
.
alder
01
0.6920
0.4614
-0.2123
1.5964
2.25
alder
11
0.1290
0.4754
-0.8027
1.0607
0.07
alder
20
0.0000
0.0000
0.0000
0.0000
.
Parameter
Pr
>ChiSq
Intercept
<.0001
eksp
00.0048
eksp
1.
alder
00.1337
alder
10.7861
alder
2.
21
LR
StatisticsForType3
Analysis
Chi-
Source
DF
Square
Pr>
ChiSq
eksp
18.30
0.0040
alder
24.02
0.1342
22
Data
asin
div
idualre
cord
s:ex
ample
Per
son
dat
eof
dat
eof
dat
eof
exit
ener
gyag
eat
age
at
no.
bir
then
try
exit
statu
sin
take
entr
yex
it
···
0104
2701
0171
0107
931
2600
43.7
566
.25
� Age
�
40
�
50
�
60
�
70
�
�
�
�
Fig
.23.1
.Splitt
ing
the
follow
-up
reco
rd.
23
Contr
ibuti
on
tolo
glikel
ihood:
0×
log(λ
0 1)
-6.2
5×
λ0 1
+0×
log(λ
1 1)
-10
×λ
1 1
+1×
log(λ
2 1)
-6.2
5×
λ2 1
Sum
min
gov
erin
div
iduals
giv
esto
tallo
glikel
ihood
24
SA
Spro
gra
m-
usi
ng
the
Lexis
MA
CR
Oto
the
die
tdata
.
data
ihdindiv;
filename
dietdata
url
‘‘http://www.biostat.ku.dk/~pka/epidata/diet.txt’’;
infile
dietdata
firstobs=2;
input
id
doe
dox
chd
dob
job
month
energy
height
weight
fat
fibre;
informat
doe
dox
dob
mmddyy10.;
format
doe
dox
dob
entry
exit
ddmmyy10.;
exposure=energy<2.75;
fail=chd;
drop
job
month
energy
height
weight
fat
fibre;
run;
/*
We
include
the
MACRO
*/
filename
lexispr
url
‘‘http://www.biostat.ku.dk/~pka/epidata/Lexis.sas’’;
%inc
lexispr;
25
/*
We
compute
cases
and
person-years
in
10
year
age-intervals
using
the
MACRO
*/
%lexis(data=ihdindiv,out=ald,entry=doe,exit=dox,fail=fail,
breaks=40
to
70
by
10,
origin=dob,
scale=365.25,
left=ageinterval);
/*
Finally,
weapply
GENMOD
to
the
individual
data
*/
proc
genmod
data=ald;
class
exposure
ageinterval;
model
fail=exposure
ageinterval/dist=poi
offset=lrisk
type3;
run;
26
Logis
tic
regre
ssio
n
Table
23.2
.C
ases
ofle
pro
syan
dco
ntr
ols
by
age
and
BC
Gsc
ar
Lep
rosy
Hea
lthy
Odds
case
spopula
tion
rati
o
BC
G−
+−
+es
tim
ate
Age
0–4
11
7593
11719
0.6
5
Age
5–9
11
14
7143
10184
0.8
9
Age
10–14
28
22
5611
7561
0.5
8
Age
15–19
16
28
2208
8117
0.4
8
Age
20–24
20
19
2438
5588
0.4
1
Age
25–29
36
11
4356
1625
0.8
2
Age
30–34
47
65245
1234
0.5
4
Tota
l159
101
34594
46028
0.4
8
27
ω=
odds
that
aper
son
(in
the
study)
isa
case
esti
mat
edby
case
/con
trol
rati
os:
Table
23.3
.C
ase/
contr
olra
tio
(×103)
by
age
and
BC
Gsc
ar
BC
Gsc
ar
Age
Abse
nt
Pre
sent
OR
0-4
0.1
30.0
80.6
5
5-9
1.5
41.3
70.8
9
10-1
44.9
92.9
10.5
8
15-1
97.2
53.4
50.4
8
20-2
48.2
03.4
00.4
1
25-2
98.2
66.7
70.8
2
30-3
48.9
64.8
60.5
8
28
Ber
noullilo
glikel
ihood
for
one
“ce
ll”
D×
log(ω
)−
N×
log(1
+ω)
=D
log(ω
)−
(D+
H)lo
g(1
+ω)
Model
:
log(o
dds)
=C
OR
NE
R+
BC
G+
AG
E
e.g.
,Sca
r+,A
ge10
-14:
22×
log(ω
)−
(22
+75
61)×
log(1
+ω)
=22×
{CO
RN
ER
+B
CG
(1)
+A
GE
(2)}
−75
83×
log(1
+ex
p{C
OR
NE
R+
BC
G(1
)+
AG
E(2
)})
29
Data
as
frequency
reco
rds:
Table
23.4
.T
he
BC
Gdata
as
freq
uen
cyre
cord
s
Case
sTota
lScar
Age
17594
00
111720
10
11
7154
01
14
10198
11
28
5639
02
22
7583
12
16
2224
03
28
8145
13
20
2458
04
19
5607
14
36
4392
05
11
1636
15
47
5292
06
61240
16
30
Maxim
um
likelihood
est
imate
s
ofpar
amet
ers
inth
em
odel
:
log(o
dds)
=C
OR
NE
R+
AG
E+
BC
G
Table
23.5
Para
mete
rE
stim
ate
SD
Corn
er
-8.8
80
0.7
093
Age(1
)2.6
24
0.7
340
Age(2
)3.5
83
0.7
203
Age(3
)3.8
24
0.7
228
Age(4
)3.9
00
0.7
244
Age(5
)4.1
56
0.7
224
Age(6
)4.1
58
0.7
213
BC
G(1
)-0
.547
0.1
409
Inte
rpre
tati
on:
Exerc
ise
23.1
31
Exerc
ise
23.1
:so
luti
on.
Table
23.3
.C
ase/
contr
olra
tio
(×103)
by
age
and
BC
Gsc
ar
BC
Gsc
ar
Age
Abse
nt
Pre
sent
OR
raw
OR
ML
E=
exp(−
0.5
47)
OR
MH
0-4
0.13
0.08
0.6
50.5
79
0.5
87
5-9
1.54
1.37
0.8
90.5
79
0.5
87
10-1
44.
992.
910.5
80.5
79
0.5
87
15-1
97.
253.
450.4
80.5
79
0.5
87
20-2
48.
203.
400.4
10.5
79
0.5
87
25-2
98.
266.
770.8
20.5
79
0.5
87
30-3
48.
964.
860.5
80.5
79
0.5
87
32
We
know
that
if
π=
risk
offa
ilure
inth
est
udy
base
then
ω=
K×
π
1−
π
wher
e
K=
Pro
b(a
“fai
lure
”is
incl
uded
asca
se)
Pro
b(a
“su
rviv
or”
isin
cluded
asco
ntr
ol)
33
Then
log
π
1−
π=
Cor
ner
+A
ge+
BC
G
=⇒
log(ω
)=
log(K
)+
Cor
ner
+A
ge+
BC
G
sam
eodds
rati
osw
hen
Kdoes
not
dep
end
on
Age
and
BC
G.
But
the
esti
mat
edC
orner
par
amet
erca
nnot
be
inte
rpre
ted.
34
SA
Spro
gra
m
databcgdata;
filenamebcgfileurl’http://www.biostat.ku.dk/~pka/epidata/bcgalldata.txt’;
infilebcgfilefirstobs=2;
inputagescarstatus$
n;
run;
procgenmoddata=bcgdata;
wherestatus=’case’orstatus=’conall’;
classagescar;
modelstatus=agescar/dist=binlink=logittype3;
weightn;
run;
35
TheGENMODProcedure
ModelInformation
DataSet
WORK.BCGDATA
Distribution
Binomial
LinkFunction
Logit
DependentVariable
status
ScaleWeightVariable
n
Numberof
ObservationsRead
28
Numberof
ObservationsUsed
28
SumofWeights
80882
Numberof
Events
14
Numberof
Trials
28
ClassLevelInformation
Class
Levels
Values
age
71
234567
scar
20
1
36
ResponseProfile
Ordered
Total
Value
status
Frequency
1case
260
2conall
80622
PROCGENMODismodelingtheprobabilitythatstatus=’case’.Oneway
tochangethistomodeltheprobabilitythatstatus=’conall’is
to
specifytheDESCENDINGoptioninthePROCstatement.
CriteriaForAssessingGoodnessOfFit
Criterion
DF
Value
Value/DF
Deviance
20
3288.0412
164.4021
ScaledDeviance
20
3288.0412
164.4021
PearsonChi-Square
20
81842.1402
4092.1070
ScaledPearsonX2
20
81842.1402
4092.1070
LogLikelihood
-1644.0206
37
AnalysisOfParameterEstimates
Standard
Wald95%
Chi-
Parameter
DF
Estimate
Error
ConfidenceLimits
Square
Intercept
1-5.2695
0.1855
-5.6330
-4.9060
807.26
age
11
-4.1576
0.7222
-5.5731
-2.7422
33.14
age
21
-1.5341
0.2475
-2.0193
-1.0489
38.40
age
31
-0.5745
0.2028
-0.9720
-0.1771
8.03
age
41
-0.3335
0.2193
-0.7633
0.0963
2.31
age
51
-0.2575
0.2210
-0.6906
0.1756
1.36
age
61
-0.0020
0.2014
-0.3967
0.3927
0.00
age
70
0.0000
0.0000
0.0000
0.0000
.
scar
01
0.5471
0.1409
0.2709
0.8232
15.07
scar
10
0.0000
0.0000
0.0000
0.0000
.
Scale
01.0000
0.0000
1.0000
1.0000
38
AnalysisOf
Parameter
Estimates
Parameter
Pr
>ChiSq
Intercept
<.0001
age
1<.0001
age
2<.0001
age
30.0046
age
40.1283
age
50.2439
age
60.9920
age
7.
scar
00.0001
scar
1.
Scale
NOTE:Thescaleparameterwasheldfixed.
39
LR
StatisticsForType3
Analysis
Chi-
Source
DF
Square
Pr>
ChiSq
age
6181.18
<.0001
scar
115.30
<.0001
40
Data
as
indiv
idualre
cord
s:
Log
likel
ihood
contr
ibuti
on
for
one
per
son
d×
log(ω
)−
1×
log(1
+ω)
Sum
min
gov
erper
sons
gives
tota
llo
glikel
ihood.
41
SA
Spro
gra
m-
mela
nom
adata
.
data
mel;
filename
melfile
url
‘‘http://www.biostat.ku.dk/~pka/epidata/melanom.txt’’;
infile
melfile
firstobs=2;
input
casecon
sex
brevald
agr
hudfarve
hair
eyes
fregner
akutrea
kronrea
nvsmall
nvlarge
nvtot
ant15;
run;
proc
genmod
data=mel
descending;
class
hudfarve;
model
casecon
=hudfarve/
dist=bin
type3;
run;
42
Age
matc
hin
g(g
roup
matc
hin
g)
Table
23.6
.A
sim
ula
ted
grou
p-m
atch
edst
udy
Cas
esC
ontr
ols
BC
G−
+−
+
Age
0–4
11
35
5–9
1114
4852
10–1
428
2267
133
15–1
916
2846
130
20–2
420
1950
106
25–2
936
1112
662
30–3
447
617
438
Her
e,K
does
dep
end
onA
ge!
43
Case
/co
ntr
olra
tios:
Abse
nt
Pre
sent
0-4
0.3
30.2
0
5-9
0.2
30.2
7
10-1
40.4
20.1
7
15-1
90.3
50.2
2
20-2
40.4
00.1
8
25-2
90.2
90.1
8
30-3
40.2
70.1
6
44
We
know
that
we
shou
ldco
rrec
tfo
rag
eth
ough
the
Age
esti
mat
esin
the
model
:
log(ω
)=
Cor
ner
+A
ge+
BC
G
cannot
be
inte
rpre
ted
(rel
ated
toth
est
udy
bas
e)
Case
sC
ontr
ols
Odds
Str
atu
m+
−+
−ra
tio
189
1180
202.
0
267
3350
502.
0
333
6720
802.
0
Tot
al18
911
115
015
01.
7
Table
18.4
.B
ias
due
toig
nori
ng
matc
hin
g
45
Matc
hed
and
unm
atc
hed
analy
sis.
Unm
atch
edA
ge-m
atch
ed
Par
amet
erE
stim
ate
SD
Est
imat
eSD
Cor
ner
-8.8
800.7
093
-1.0
670
0.8
00
Age
(1)
2.62
40.7
340
-0.0
421
0.8
27
Age
(2)
3.58
30.7
203
0.0
119
0.8
12
Age
(3)
3.82
40.7
228
0.0
713
0.8
14
Age
(4)
3.90
00.7
244
0.0
244
0.8
16
Age
(5)
4.15
60.7
224
-0.1
628
0.8
14
Age
(6)
4.15
80.7
213
-0.2
380
0.8
13
BC
G(1
)-0
.547
0.1
409
-0.5
721
0.1
55
46
Can
we
ever
inte
rpre
tth
eC
orner
par
amet
er?
Yes
,in
cum
ula
tive
inci
den
ceor
pre
vale
nce
studie
s
wher
eth
eabso
lute
risk
/pre
vale
nce
can
be
esti
mat
ed.
47
Hypoth
esi
ste
sts,
ch.
24.
Wald
test
for
asi
ngle
para
met
er:
( M−
0
S
) 2∼
χ2 1
(chi-sq
uare
)
dir
ectl
ybas
edon
com
pute
routp
ut:
Table
24.1
.P
rogra
moutp
ut
for
the
die
tdata
Par
amet
erE
stim
ate
SD
W
Cor
ner
-5.4
180
0.4
420
Expos
ure
(1)
0.8
697
0.3
080
7.9
7
Age
(1)
0.1
290
0.4
753
0.0
7
Age
(2)
0.6
920
0.4
614
2.2
5
48
Lik
elihood
ratio
test
:co
mpare
max.
log
likel
ihoods
“under
”and
“outs
ide”
the
hypot
hes
is.
Tes
tst
atis
tic
=-2
×diff
eren
cebet
wee
nm
ax.
log
likel
ihoods
Model
Max.
log
likel
ihood
#para
met
ers
Cor
ner
+A
ge+
Expos
ure
-247
.03
4
Cor
ner
+E
xpos
ure
-249
.04
2
Cor
ner
+A
ge-2
51.1
83
#par
amet
ers
rem
oved
=#
d.f.
inχ
2
Expos
ure
:8.
30,1
d.f.
Age
:4.
02,2
d.f.
Model
shav
eto
be
”nes
ted”
-w
eca
nnot
com
pare
the
last
two
model
s
inth
eta
ble
.
49
Max.
log
likelihood:
mea
sure
ofgoodnes
sof
fit
ofa
model
:
larg
erlo
glikel
ihood
=⇒
bet
ter
fit
Inte
rpre
tati
onof
–247
.03
?N
o!
Som
eti
mes
,th
edevia
nce
isin
troduce
das
asu
pple
men
tto
the
max.
log
likel
ihood.
50
Inte
ract
ion,se
ct.
24.3
We
hav
eas
sum
edth
atth
eeffect
ofexposu
reis
const
ant
over
age
bands
(and
vic
ever
sa).
Isth
atre
ason
able
?
Or
isth
ere
inte
raction
bet
wee
nag
ean
dex
pos
ure
?
log(R
ate
)=C
orn
er+
Exposu
re+
Age
+E
xpos
ure·A
ge
Not
eth
ere
lati
onsh
ipw
ith
the
Bre
slow
-Day
test
for
hom
ogen
eity
over
age
stra
ta.
How
ever
,w
enow
:
•ge
ta
quan
tifica
tion
ofhet
erog
enei
ty
•are
able
toad
just
for
oth
erex
pla
nato
ryva
riable
sw
hen
exam
inin
gin
tera
ctio
n
51
Table
24.5
.E
stim
ates
ofpar
amet
ers
inth
em
odel
wit
hin
tera
ctio
n
Par
amet
erE
stim
ate
SD
Cor
ner
-5.0
237
0.50
0
Expos
ure
(1)
-0.0
258
0.8
66
Age
(1)
-0.5
153
0.6
71
Age
(2)
0.3
132
0.6
12
Age
(1)·E
xposu
re(1
)1.2
720
1.0
20
Age
(2)·E
xposu
re(1
)0.8
719
0.9
73
Test
for
no
inte
raction:
Max.
log
likel
ihood
for
Corn
er+
Age
+E
xposu
re+
Age.
Exposu
re
is-2
46.1
9le
adin
gto
the
LR
test
1.6
7(2
d.f.)
52
Illu
stra
tive
exam
ple
without
inte
raction
Table
22.4
Expos
ure
Age
01
05.
015.0
112.0
36.0
230.0
90.0
05.
05.0
×3.0
112.0
12.0
×3.0
230.0
30.0
×3.0
05.
05.0
×3.0
15.
0×
2.4
5.0×
2.4
×3.0
25.
0×
6.0
5.0×
6.0
×3.0
Cor
ner
=5.
0A
ge(1
)=
2.4
Expos
ure
(1)
=3.0
Age(
2)
=6.0
53
Exam
ple
:Illu
stra
tive
valu
es
ofra
tes
wit
hin
tera
ctio
n
Table
24.2
.D
efinit
ion
ofin
tera
ctio
ns
inte
rms
ofexposu
re
Exposu
re
Age
01
05.
015.0
112.0
42.0
230.0
135.0
05.
05.0
×3.0
112.0
12.0
×3.5
230.0
30.0
×4.5
05.
05.0
×3.0
112.0
12.0
×3.0
×1.1
67
230.0
30.0
×3.0
×1.5
inte
ract
ion
para
met
ers
54
Exam
ple
:Illu
stra
tive
valu
es
ofra
tes
wit
hin
tera
ctio
n
Table
24.3
.D
efinit
ion
ofin
tera
ctio
ns
inte
rms
ofage
Expos
ure
Age
01
05.
015.0
112.0
42.0
230.0
135.0
05.
015.0
15.
0×
2.4
15.0
×2.8
25.
0×
6.0
15.0
×9.0
05.
015.0
15.
0×
2.4
15.0
×2.4
×1.1
67
25.
0×
6.0
15.0
×6.0
×1.5
inte
ract
ion
par
amet
ers
55
Table
24.4
.D
efinit
ion
ofin
tera
ctio
ns
inte
rms
ofexposu
reand
age
Exposu
re
Age
01
05.
05.0
×3.0
15.
0×
2.4
5.0
×3.0
×2.4
×1.1
67
25.
0×
6.0
5.0
×3.0
×6.0
×1.5
Exerc
ise
24.4
.
56
Exerc
ise
24.4
:so
luti
on.
Par
amet
erE
stim
ate
SD
Cor
ner
-5.0
237
0.50
0
Expos
ure
(1)
-0.0
258
0.8
66
Age
(1)
-0.5
153
0.67
1
Age
(2)
0.31
320.
612
Age
(1)·E
xpos
ure
(1)
1.2720
1.0
20
Age
(2)·E
xpos
ure
(1)
0.8719
0.9
73
log
4
607.9
=−
5.02
37,l
og
2311.9
4607.9
=−
0.02
58
(exce
pt
for
roundin
ger
rors
)
57
SA
Spro
gra
m
dataihd;
inputekspalderpyrscases;
lpyrs=log(pyrs);
datalines;/*or,alternatively,readfromwww*/
02
311.92
01
878.112
00
667.514
12
607.94
11
1272.15
10
888.98
; run;
procgenmoddata=ihd;
classekspalder;
modelcases=ekspaldereksp*alder/dist=poioffset=lpyrs
type3;
run;
58
TheGENMODProcedure
ModelInformation
DataSet
WORK.IHD
Distribution
Poisson
LinkFunction
Log
DependentVariable
cases
OffsetVariable
lpyrs
ObservationsUsed
6
ClassLevelInformation
Class
Levels
Values
eksp
20
1
alder
30
12
59
CriteriaForAssessingGoodnessOfFit
Criterion
DF
Value
Value/DF
Deviance
00.0000
.
ScaledDeviance
00.0000
.
PearsonChi-Square
00.0000
.
ScaledPearsonX2
00.0000
.
LogLikelihood
53.3799
60
AnalysisOfParameterEstimates
Standard
Wald95%
Chi-
Parameter
DF
Estimate
Error
ConfidenceLimits
Square
Intercept
1-5.0237
0.5000
-6.0037
-4.0437
100.95
eksp
01
-0.0258
0.8660
-1.7232
1.6716
0.00
eksp
10
0.0000
0.0000
0.0000
0.0000
.
alder
01
0.3132
0.6124
-0.8871
1.5134
0.26
alder
11
-0.5153
0.6708
-1.8301
0.7995
0.59
alder
20
0.0000
0.0000
0.0000
0.0000
.
eksp*alder
00
10.8719
0.9728
-1.0349
2.7786
0.80
eksp*alder
01
11.2720
1.0165
-0.7204
3.2643
1.57
61
AnalysisOf
Parameter
Estimates
Parameter
Pr>
ChiSq
Intercept
<.0001
eksp
00.9762
eksp
1.
alder
00.6091
alder
10.4424
alder
2.
eksp*alder
00
0.3701
eksp*alder
01
0.2108
62
LR
StatisticsForType3
Analysis
Chi-
Source
DF
Square
Pr>
ChiSq
eksp
13.09
0.0790
alder
24.37
0.1125
eksp*alder
21.67
0.4333
63
Table
24.5
.R
epor
ting
esti
mate
sfr
om
the
model
wit
hin
tera
ctio
n:
Rep
aram
etri
zein
tose
par
ate
effec
tsofE
xposu
rew
ithin
each
Age
band.
Par
amet
erE
stim
ate
SD
RR
Cor
ner
-5.0
237
0.50
0
Expos
ure
(1)·
Age(
0)
-0.0
258
0.8
66
0.9
7
Expos
ure
(1)·
Age(
1)
1.2
461
0.5
32
3.4
8
Expos
ure
(1)·
Age(
2)
0.8
461
0.4
43
2.3
3
Age
(1)
-0.5
153
0.6
71
0.6
0
Age
(2)
0.3
132
0.6
12
1.3
7
64
Inte
ract
ions:
whic
hto
study?
When
the
model
conta
ins
pco
vari
ates
ther
ear
ep(p
−1)
/2poss
ible
two-fact
or
inte
ract
ions
(e.g
.,45
for
p=
10).
Itis
out
ofth
eques
tion
tost
udy
them
all,
soa
gener
al
reco
mm
endat
ion
isto
rest
rict
atte
nti
onto
thos
eth
atw
ere
pre
-spec
ified
inth
ere
sear
chpro
toco
l:
“D
on’t
ask
aques
tion
ifyou
are
not
inte
rest
edin
the
reply
!”
Ther
ew
illal
sobe
aty
pe
Ier
ror
pro
ble
m:
“ifyou
ask
too
man
y
ques
tion
syou
willge
tto
om
any
wro
ng
answ
ers”
.
65
Inte
ract
ion
issc
ale
dependent.
Tab
leof
dis
ease
rate
s:
Fac
tor
AFac
tor
B
Abse
nt
Pre
sent
Abse
nt
0.1
0.2
Pre
sent
0.3
λ
Ifλ
=0.
6th
enth
era
tera
tio
asso
ciat
edw
ith
the
pre
sence
offa
ctor
A
is3
bot
hw
hen
fact
orB
isabse
nt
or
pre
sent;
and
the
rate
ratio
asso
ciat
edw
ith
the
pre
sence
offa
ctor
Bis
2both
when
fact
or
Ais
abse
nt
orpre
sent.
How
ever
,th
era
tediff
ere
nce
asso
ciat
edw
ith
the
pre
sence
offa
ctor
A
is0.
2w
hen
fact
orB
isabse
nt
and
0.4
ifit
ispre
sent
and
the
rate
diff
ere
nce
asso
ciat
edw
ith
the
pre
sence
offa
ctor
Bis
0.1
when
fact
or
Ais
abse
nt
and
0.3
ifit
ispre
sent 6
6
Fac
tor
AFac
tor
B
Abse
nt
Pre
sent
Abse
nt
0.1
0.2
Pre
sent
0.3
λ
Ifλ
=0.
4th
enth
era
tediff
ere
nce
asso
ciat
edw
ith
the
pre
sence
of
fact
or
Ais
0.2
bot
hw
hen
fact
orB
isab
sent
orpre
sent;
the
rate
diff
ere
nce
asso
ciat
edw
ith
the
pre
sence
offa
ctor
Bis
0.1
both
when
fact
or
Ais
abse
nt
orpre
sent.
How
ever
,th
era
tera
tio
asso
ciat
edw
ith
the
pre
sence
offa
ctor
Ais
3
when
fact
orB
isabse
nt
and
2if
itis
pre
sent
and
the
rate
ratio
asso
ciat
edw
ith
the
pre
sence
offa
ctor
Bis
2w
hen
fact
orA
isab
sent
and
1.33
ifit
ispre
sent
67
Inte
ract
ion
betw
een
2exposu
res
Table
24.6
.C
ases
(con
trol
s)fo
rora
lcancer
study.
Alc
ohol(o
z/day,1
dri
nk∼
0.3
oz/day).
Tobacco
01
23
(cig
s/day)
00.1
-0.3
0.4
-1.5
1.6
+
0(0
)10
(38)
7(2
7)
4(1
2)
5(8
)
1(1
-19)
11
(26)
16
(35)
18
(16)
21
(20)
2(2
0-3
9)
13
(36)
50
(60)
60
(49)
125
(52)
3(4
0+
)9
(8)
16
(19)
27
(14)
91
(27)
Table
24.7
.C
ase/
contr
olra
tios
for
the
ora
lcancer
data
.
Alc
ohol
Tobacco
01
23
00.2
60.2
60.3
30.6
3
10.4
20.4
61.1
31.0
5
20.3
60.8
31.2
22.4
0
31.1
20.8
41.9
33.3
7
68
Isth
eeff
ect
ofto
bac
coth
esa
me
for
allle
vel
sof
alco
hol
consu
mpti
on?
SY
NER
GIS
M?
=IN
TER
AC
TIO
N
ButC
OR
RELA
TIO
Nis
som
ethin
gco
mple
tely
diff
eren
t
69
Fig
.24.2
.N
est
ing
ofm
odels
.
5.
Corner+
Alc
ohol+
Tobacco+
Alc
ohol.Tobacco
4.
Corner+
Alc
ohol+
Tobacco
2.
Corner+
Alc
ohol
3.
Corner+
Tobacco
1.
Corner
��
�� ��
��
�� ��
��
�� ��
��
�� �� �
Exerc
ise
24.6
70
Exerc
ise
24.6
:Log-lik
elihoods
5.
-577.6
5
4.
-580.9
9
2.
-596.6
23.
-608.5
9
1.
-643.9
3
��
�� ��
��
�� ��
��
�� ��
��
�� �� � 71
Dose
-resp
onse
models
Expla
nato
ryva
riable
sw
ith
ord
ere
dcate
gori
es.
Table
25.1
.A
lcoholand
tobacco
trea
ted
asca
tego
rica
lva
riab
les
Par
amet
erE
stim
ate
SD
Cor
ner
-1.6
090
0.2
654
Alc
ohol(
1)
0.2
897
0.2
327
Alc
ohol(
2)
0.8
437
0.2
383
Alc
ohol(
3)
1.3
780
0.2
256
Tob
acco
(1)
0.5
887
0.2
844
Tob
acco
(2)
1.0
260
0.2
544
Tob
acco
(3)
1.4
090
0.2
823
72
Alt
ern
ati
ve:
monoto
ne
effect
ofto
bacc
oFig
.20.1
.Log
-lin
ear
tren
d
�
�Log(o
dds)
��
��
01
23
Dose
,z
�
�
�
β
β
73
Look
atsu
ccess
ive
diff
ere
nces
bet
wee
neff
ects
:
Tob
acco
(1),
Tob
acco
(2)-
Tobacc
o(1
),Tobacc
o(3
)-Tobacc
o(2
)
Exerc
ise
25.1
Intr
oduce
avari
able
takin
gva
lues
0,1,
2or
3an
dden
ote
its
effec
tby
[Tob
acco
]
Model
:lo
g(O
dds)
=C
orn
er+
Alc
ohol+
[Tobacc
o]
74
Exerc
ise
25.1
:so
luti
on.
Table
25.1
.A
lcoholand
tobacco
trea
ted
asca
tego
rica
lva
riab
les
Par
amet
erE
stim
ate
SD
Succ
.diff
.
Cor
ner
-1.6
090
0.26
54
Alc
ohol(
1)
0.2
897
0.2
327
0.2
897
Alc
ohol(
2)
0.8
437
0.2
383
0.5
540
Alc
ohol(
3)
1.3
780
0.2
256
0.5
543
Tob
acco
(1)
0.58
870.
2844
0.58
87
Tob
acco
(2)
1.02
600.
2544
0.43
73
Tob
acco
(3)
1.40
900.
2823
0.38
30
75
Model:
log(O
dds)
=C
orn
er
+A
lcohol+
[Tobacc
o]
Table
25.2
.T
he
linear
effect
ofto
bacco
consu
mpti
on
Alc
ohol
Tobacco
log(O
dds)=
Corner
+...
00
-
01
1×
[Tobacco]
02
2×
[Tobacco]
03
3×
[Tobacco]
10
Alc
ohol(
1)
11
Alc
ohol(
1)+
1×
[Tobacco]
12
Alc
ohol(
1)+
2×
[Tobacco]
13
Alc
ohol(
1)+
3×
[Tobacco]
20
Alc
ohol(
2)
21
Alc
ohol(
2)+
1×
[Tobacco]
22
Alc
ohol(
2)+
2×
[Tobacco]
23
Alc
ohol(
2)+
3×
[Tobacco]
30
Alc
ohol(
3)
31
Alc
ohol(
3)+
1×
[Tobacco]
32
Alc
ohol(
3)+
2×
[Tobacco]
33
Alc
ohol(
3)+
3×
[Tobacco] 76
Table
25.3
.Lin
ear
effect
ofto
bacco
per
level
Par
amet
erE
stim
ate
SD
Cor
ner
–1.5
250
0.21
9
Alc
ohol(
1)
0.3
020
0.2
32
Alc
ohol(
2)
0.8
579
0.2
37
Alc
ohol(
3)
1.3
880
0.2
25
[Tob
acco
]0.
4541
0.08
3
77
Sim
ilarl
yw
ith
alc
oholconsu
mption:
intr
oduce
vari
able
wit
hva
lues
=0,1,2
or
3
and
den
ote
its
effec
t[A
lcoh
ol]
Table
25.4
.Lin
ear
effects
ofalc
oholand
tobacco
per
level
Par
amet
erE
stim
ate
SD
Cor
ner
–1.6
290
0.1
860
[Alc
ohol]
0.4
901
0.0
676
[Tob
acco
]0.4
517
0.0
833
Exerc
ise
25.3
78
Exerc
ise
25.3
:so
luti
on.
Tob
acc
o(3
)+A
lcohol(
3)=
2.7
870
3×
[Tob
acco
]+
3×
[Alc
ohol]
=2.
8254
79
Alt
ern
ati
ve
ways
ofsc
ori
ng
Tob
acco
:ci
gare
ttes
/day
(0:
0,1-1
9:
10,20-3
9:
30,40+
:50)
Alc
ohol
:ou
nce
s/day
(0.0
:0,0.1
-0.3
:0.2
,0.4
-1.5
:1.0
,1.6
+:
2.0
)
Table
25.5
.A
lcoh
olin
ounces/
day
and
tobac
coin
cig
are
ttes/
day
Par
amet
erE
stim
ate
SD
Cor
ner
–1.2
657
0.1
539
[Alc
ohol]
0.6
484
0.0
881
[Tob
acco
]0.0
253
0.0
046
80
Test
ing
–Tes
tfo
rlinea
rity
:
1)C
ompar
ing
the
“nes
ted”
model
s:
log(
Odds)
=C
orner
+A
lcoh
ol+
Tob
acco
and
log(
Odds)
=C
orner
+A
lcoh
ol+
[Tob
acco
],
her
e:LR
test
=0.
38,2.
d.f.,
or 2)
elim
inati
ng
[Tob
sq](=
0,1,4,9)
from
log(
Odds)
=C
orner
+A
lcoh
ol+
[Tob
acco
]+
[Tob
sq],
her
eLR
test
=0.
02.
81
–Tre
nd
test
:(1
.d.f.)
Elim
inat
ing
[Tobac
co]fr
omth
em
odel
:
log(
Odds)
=C
orn
er+
Alc
ohol+
[Tobacc
o],
her
eLR
test
=30
.88.
Why
not
use
indiv
idualle
vel
s,th
at
is,a
truly
quanti
tati
ve
cova
riate
and
no
cate
gori
zati
on
atall?
Pro
san
dco
ns
•In
form
atio
nis
lost
by
cate
gori
zati
on
•C
ateg
orie
sm
aybe
more
robust
(e.g
.,sm
okin
g)
•Few
outl
iers
may
hav
ela
rge
influen
ce(“
Casa
nov
aeff
ect”
!)
•M
odel
wit
ha
linea
reff
ect
isno
longer
“nes
ted”
inca
tegori
cal
model
⇒al
tern
ativ
ealt
ernati
ves
are
nee
ded
when
test
ing
linea
rity
82
Indic
ato
rvari
able
s
The
way
inw
hic
hth
eca
tego
rica
lco
vari
ates
are
ente
red
into
the
regr
essi
onm
odel
.
Table
25.8
.In
dic
ator
vari
able
sfo
rth
eth
ree
alco
hol
par
amet
ers
A1
A2
A3
Lev
ello
g(O
dds)
=C
orner
+···
00
00
–
10
01
Alc
ohol(
1)
01
02
Alc
ohol(
2)
00
13
Alc
ohol(
3)
83
The
use
ofin
dic
ator
vari
able
sen
able
sth
epro
gra
mm
erto
choose
his
/her
pre
ferr
edre
fere
nce
level.
Inte
raction
term
sare
sim
ple
pro
ducts
ofin
dic
ato
rva
riabel
s.
Table
25.1
0.
Indic
ato
rva
riable
sfo
rin
tera
ctio
npara
met
ers
A1
A2
A3
TA
1·T
A2·T
A3·T
00
00
00
0
00
01
00
0
10
00
00
0
10
01
10
0
01
00
00
0
01
01
01
0
00
10
00
0
00
11
00
1
NB
:Tob
acco
isher
eon
2le
vel
sonly
84
Tre
ati
ng
the
zero
leveldiff
ere
ntl
yFig
.25.1
.Sep
arat
ing
zero
expos
ure
from
the
dos
e-re
spon
se.
�
�Log(o
dds)
��
��
01
23
Dose
,z
�
�
�
�
Corn
er
Corn
er+
Non-s
moker
Non-s
moker
��
��
85
Corr
esponds
toaddin
ga
new
vari
able
[Non-s
moker
]
Table
25.1
1.
Sep
arat
ing
zero
exposu
refr
om
the
dose
-res
ponse
Tob
acco
Non
-sm
oker
log(
Odds)
=C
orner
+···
01
[Non
-sm
oker
]
10
1×
[Tob
acco
]
20
2×
[Tob
acco
]
30
3×
[Tob
acco
]
(Alt
ernat
ive:
incl
ude
[Sm
oker
]=
1-
[Non-s
moker
])
86
Indic
ator
vari
able
sm
aybe
chos
enin
sever
alw
ays.
E.g
.,to
model
success
ive
diff
ere
nces
Table
25.1
2.
Indic
ator
sto
com
par
eea
chle
vel
wit
hth
eon
ebef
ore
Tob
acco
D1
D2
D3
00
00
11
00
21
10
31
11
Her
e,D
1=
indic
ator
for
Tob
acco
≥1
D2
=in
dic
ator
for
Tob
acco
≥2
D3
=in
dic
ator
for
Tob
acco
≥3
87
Tru
lyquanti
tati
ve
covari
ate
s,x
Ina
model
like
log(
Rat
e)=
Corn
er+
Exposu
re+
[x]
the
effec
tof
xis
assu
med
tobe
linea
r,i.e.
[x]ex
pre
sses
the
change
in
log(
Rat
e)per
1unit
chan
ge
ofx.
To
test
for
linea
rity
,on
em
ayadd
[xsq
]to
the
model
wher
exsq
=x
2.
An
alt
ernati
ve
alte
rnati
ve
isa
linea
rsp
line.
88
Lin
ear
splines
An
alte
rnat
ive
toa
stra
ight
line
isa
bro
ken
line.
Intr
oduce
bre
ak
poi
nts
for
x,e.
g.,
a1,a
2,a
3and
add
the
thre
elinea
r
splines
I 1×
[x−
a1],
I 2×
[x−
a2],
I 3×
[x−
a3]
to[x
]:
Her
e,I 1
=in
dic
ator
for
x≥
a1
I 2=
indic
ator
for
x≥
a2
I 3=
indic
ator
for
x≥
a3
The
par
amet
erfo
rth
esp
line
I 1×
[x−
a1]gi
ves
the
change
inslope
at
the
bre
ak
poi
nt
a1.
Sim
ilarl
yfo
ra2,a
3.
Splines
are
easy
topro
gram
and
par
amet
ers
are
easi
erto
inte
rpre
t
than
for
quadra
tic
term
s(q
uadra
tic
and
cubic
splines
als
oex
ist)
.
89
24
68
10
−2.0 −1.5 −1.0 −0.5 0.0 0.5
x
Linear Predictor
24
68
10
−2.0 −1.5 −1.0 −0.5 0.0 0.5
x
Linear Predictor
24
68
10
−2.0 −1.5 −1.0 −0.5 0.0 0.5
x
Linear predictor
24
68
10
−2.0 −1.5 −1.0 −0.5 0.0 0.5
x
Linear Predictor
90
Inte
ract
ion:
Are
searc
her
’satt
itude
tow
ards
inte
ract
ion
dep
ends
on
the
kin
dof
vari
able
sin
vol
ved
:
–a)
2ex
pos
ure
s
–b)
2co
nfo
under
s
–c)
1ex
pos
ure
and
1co
nfo
under
a)For
exam
ple
the
effec
tof
alc
oholand
tobacco
onora
l
cancer
b)
Pre
vale
nce
ofm
onoclo
nalgam
mapath
yby
occ
upat
ion
91
Table
26.1
.P
reva
lence
ofm
onoclo
nalgam
mapath
y
Agri
cult
ura
l(0
)N
on-a
gri
cult
ura
l(1
)
Age
Mal
e(0
)Fem
ale
(1)
Male
(0)
Fem
ale
(1)
<40
(0)
1/15
901/1926
2/1527
0/712
40-4
9(1
)12
/234
57/2677
3/854
0/401
50-5
9(2
)24
/278
715/2902
5/675
4/312
60-6
9(3
)53
/248
938/3145
3/184
1/80
70+
(4)
95/2
381
63/2918
2/75
0/20
How
toco
ntr
olfo
rse
xan
dage
when
studyin
gth
eeff
ect
ofw
ork
?
92
Str
atification
(Man
tel-H
aensz
el)∼
logis
tic
regre
ssio
nm
odel:
log(
Odds)
=C
orner
+A
ge+
Sex
+A
ge·Sex
+W
ork
Work
:–0
.134
(SD
=0.
244)
(10
par
amet
ers
+W
ork)
Sim
ple
rm
odel:
log(
Odds)
=C
orner
+A
ge+
Sex
+W
ork
(i.e
.,no
Age·Sex
inte
raction):
Work
:–0.1
36
(SD
=0.
243)
Lik
elihood
ratio
test
for
no
inte
raction:
is0.
878
wit
h4
deg
rees
offr
eedom
.
When
the
confo
under
shav
em
any
level
sth
ere
willbe
man
yd.f.’s.
93
c)in
tera
ctio
nbet
wee
nw
ork
and
sex
orw
ork
and
age
Table
26.2
.Tes
ting
for
inte
ract
ion
Model
−lo
gL
Cor
ner
+A
ge+
Sex
+W
ork
36.4
0
Cor
ner
+A
ge+
Sex
+W
ork
+W
ork·A
ge
35.4
8
Cor
ner
+A
ge+
Sex
+W
ork
+W
ork·S
ex36.2
0
Effect-
modifi
cation:
Exerc
ise
26.2
94
Exerc
ise
26.2
:so
luti
on.
LR
test
for
no
Wor
k·A
ge
inte
ract
ion:
1.84,4
d.f.
LR
test
for
no
Wor
k·S
exin
tera
ctio
n:
0.41,1
d.f.
95
Tre
ati
ng
Age
as
aquanti
tati
ve
vari
able
�
�Log(o
dds)
������
-8-7-6-5-4-3
��
��
40
50
60
70
Age
�
�
�
�
�
Fig
.26.1
.Log
pre
vale
nce
odds
by
age:
male
s,agri
cult
ura
lw
ork
ers
Exerc
ise
26.3
96
Incl
udin
g[A
ge](3
5,45
,55
,65
,75
)gi
ves
:
Work
=–0.1
86
Incl
udin
g[A
ge]an
d[A
gesq
](3
5)2,(4
5)2,(5
5)2,(6
5)2,(7
5)2,gi
ves
:
Table
26.3
.A
quadra
tic
rela
tionsh
ipw
ith
age
Par
amet
erE
stim
ate
SD
Cor
ner
-6.6
820.
344
Wor
k(1
)-0
.148
0.2
43
[Age
]1.
204
0.26
4
[Age
sq]
-0.0
840.
049
Sex
(1)
-0.5
830.
115
Tes
tfo
r[A
ges
q]:
3.13
(1d.f.)
The
esti
mat
edeff
ect
ofex
pos
ure
(Wor
k)
may
dep
end
onhow
the
confo
under
(Age
)is
model
led.
97
Tes
tfo
rA
ge-W
ork
inte
raction
usi
ng
[Age]
:
Table
26.4
.In
tera
ctio
nbet
wee
nag
e(q
uan
tita
tive)
and
wor
k
Par
amet
erE
stim
ate
SD
Cor
ner
–6.2
11
0.2
01
Wor
k(1
)–0.2
99
0.4
71
[Age
]0.7
63
0.0
58
Sex
(1)
–0.5
84
0.1
15
[Age
]·W
ork
(1)
0.0
53
0.1
88
Exerc
ises
26.4
-5
98
Exerc
ise
26.4
:so
luti
on.
Wal
dte
stfo
rno
Work·[A
ge]in
tera
ctio
n:
(0.5
3/0.
188)
2=
0.0
79,
1d.f.
99
Table
26.5
.In
tera
ctio
nbet
wee
n[A
ge]and
Work
Par
amet
erE
stim
ate
SD
Cor
ner
–7.0
64
0.5
53
Age
(1)
1.6
66
0.5
67
Age
(2)
2.3
94
0.5
62
Age
(3)
3.2
39
0.5
62
Age
(4)
3.8
60
0.5
59
Sex
(1)
–0.5
85
0.1
15
Wor
k(1
)0.0
46
0.5
44
[Age
]·W
ork
(1)
–0.0
83
0.2
20
Wald
test
for
inte
ract
ion:
( −0.0
83
0.2
20
) 2=
0.14
Lik
elihood
ratio
test
:0.1
4
100
Concl
usi
ons:
For
the
exam
ple
:no
evid
ence
what
soev
erof
any
effec
tof
wor
kon
the
outc
ome
Ingenera
l:th
ere
isoft
ena
LA
RG
E
num
ber
ofpos
sibilit
ies
avai
lable
when
anal
ysi
ng
regr
essi
on
model
s:
–w
hic
hva
riable
sto
incl
ude
–how
toin
clude
them
(dos
e-re
spon
se,in
tera
ctio
ns)
Kee
pth
isin
min
dw
hen
read
ing
the
publish
edlite
ratu
re!
Kee
pin
min
dth
epurp
ose
ofth
est
udy
when
analy
sing
the
data
!
101
