Comparison of Outcomes Among Cancer Patients with Multiple Covariates: Using Logistic Regression and Cox Regression

1.
MedicReS Good Biosta/s/cal & Publica/on Prac/ce
in Cancer Research with "Real World Data"
February 13 -­‐ 14, 2017 VIENNA
Comparison
of
Outcomes
Among
Cancer
Pa4ents
with
Mul4ple
Covariates:
Using
Logis4c
Regression
and
Cox
Regression
Nicholas
P.
Jewell,
PhD
Professor
of
Biosta4s4cs
and
Sta4s4cs.
University
of
California
Berkeley

2. Natural History Schema/c Illness-­‐Death Model
Time%
Origin%
(t"="0)%
Disease%
Ini2a2on%
Disease%
Expression%
Death%
Can be many stages between Initiation and Expression
Many similar schematics in different fields

3. Disease Ini/a/on/Expression
• Disease
ini4a4on
studies:
epidemiological
inves4ga4on
of
risk
factors
for
cancer
incidence
• Cohort
or
case-­‐control
observa4onal
studies
of
cumula4ve
incidence
• Case-­‐control
studies
require
use
of
odds
ra4os
(not
rela4ve
risks)
• Ignores
dynamic
of
popula4on
and
disease
incidence
within
risk
interval
• Disease
expression
studies
• Randomized
clinical
trials
• OPen
(right)
censoring
• Use
4me-­‐to-­‐event
models
(e.g.
propor4onal
hazards
models)

4. Cumula/ve Incidence Propor/ons
Interval
at
risk
• Need
4me
origin
and
endpoint
• Need
4me
scale
(age,
exposure
4me,
4me
since
diagnosis,
no.
of
contacts,
etc)
• Need
defini4on
of
incidence
(incidence
preferable
to
prevalence)
or
endpoint
(e.g.
death)
during
interval
• Need
defini4on
of
target
popula4on
at
risk
at
the
origin
(last
part
arbitrary
but
conven4onal)
• Incidence
propor4on
is
frac4on
of
at
risk
popula4on
who
experience
D
during
risk
interval
I
or
P(D)
• I(t)
is
incident
propor4on
over
interval
[0,
t]
;
S(t)
=
1-­‐I(t)
Survival
Frac9on
(Func9on)
Time
=
0
Time
=
T
D

5. Mortality Risk Calcula/ons for Cervical Cancer
Wider
Racial
Gap
Found
in
Cervical
Cancer
Deaths
By
JAN
HOFFMAN
JAN.
23,
2017
(New
York
Times)
In
the
new
analysis,
the
mortality
rate
for
black
women
was
10.1
per
100,000.
For
white
women,
it
is
4.7
per
100,000.
Previous
studies
had
put
those
figures
at
5.7
and
3.2.
The
new
rates
do
not
reflect
a
rise
in
the
number
of
deaths,
which
recent
es4mates
put
at
more
than
4,000
a
year
in
the
United
States.
Instead,
the
figures
come
from
a
re-­‐examina4on
of
exis4ng
numbers,
in
an
adjusted
context.
Typically,
death
rates
for
cervical
cancer
are
calculated
by
assessing
the
number
of
women
who
die
from
a
disease
against
the
general
popula4on
at
risk
for
it.
But
these
epidemiologists,
who
looked
at
health
data
from
2000
to
2012,
also
excluded
women
who
had
had
hysterectomies
from
that
larger
popula4on.
A
hysterectomy
almost
always
removes
the
cervix,
and
thus
the
possibility
that
a
woman
will
develop
cervical
cancer.
“We
don’t
include
men
in
our
calcula4on
because
they
are
not
at
risk
for
cervical
cancer
and
by
the
same
measure,
we
shouldn’t
include
women
who
don’t
have
a
cervix,”
said
Anne
F.
Rositch,
the
lead
author
and
an
assistant
professor
of
epidemiology
at
the
Johns
Hopkins
Bloomberg
School
of
Public
Health.
“If
we
want
to
look
at
how
well
our
programs
are
doing,
we
have
to
look
at
the
women
we’re
targe4ng.”

6. Incidence Rates
• Interval
at
risk
• People
entering
and
leaving
during
the
period
of
risk
–
not
observed
for
en4re
interval
• (Average)
Incidence
Rate
(over
4me
interval)
=
#D/cum.
9me
at
risk
• What
if
incidence
changes
substan4ally
over
4me
interval,
and/or
observa4on
window
changes
• Divide
interval
into
2,
then
4,
then
8,
then
.
.
.
intervals
etc.
• Plot
of
incidence
rate
against
midpoint
of
interval
hazard
func4on
h(t)
Time
=
0
Time
=
T

7. Survival
Func4on
(1-­‐I(t))
for
Caucasian
Males
in
California
in
1980

8. Hazard
Func4on
for
Caucasian
Males
in
California
in
1980

9. Measures of Associa/on : Outcomes with
Exposure or Treatment
• Rela4ve
Risk
• Odds
Ra4o
• OR
and
RR
are
very
similar
if
risks
P(D|E),
etc.
are
small
• OR
is
symmetric
in
roles
of
D
and
E
)|(
)|(
EnotDP
EDP
RR =
OR =
P(D | E)
P(notD | E)
P(D | notE)
P(notD | notE)

10. Rela/ve Hazard
• Rela4ve
Hazard
(Hazard
Ra4o)
• If
HR
does
not
depend
on
t
propor4onal
hazards
• Also
similar
to
RR
and
OR
with
propor4onal
hazards
HR =
hE (t)
hE (t)
RH
RR
OR

11. 2 x 2 Table Nota/on
Disease/Outcome
Status
D
Not
D
Exposure/
Treatment
E
a
b
a+b
Not
E
c
d
c+d
a+c
b+d
n
bc
ad
dcd
dcc
bab
baa
RO =
!
"
#
$
%
&
+
+
!
"
#
$
%
&
+
+
=
)/(
)/(
)/(
)/(
ˆ

12. Case-­‐Control Study of Pancrea/c Cancer
Pancrea2c
Cancer
Incidence
Cases
Controls
Coffee
Drinking
Yes
347
555
902
No
20
88
108
367
643
1010
75.2
20555
88347ˆ =
×
×
=RO

13. Sampling Distribu/ons of Odds Ra/os
Not
Normal-­‐-­‐skewed
log
scale–
not
skewed
Use
log
scale
for
inference

14. Case-­‐Control Study of Pancrea/c Cancer
Pancrea2c
Cancer
Incidence
Cases
Controls
Coffee
Drinking
Yes
347
555
902
No
20
88
108
367
643
1010
75.2
20555
88347ˆ =
×
×
=RO log(O ˆR) = log(2.75) =1.01
vˆar log(O ˆR)( )=
1
347
+
1
555
+
1
20
+
1
88
= 0.066
95% CI for log(OR): 1.01±1.96 0.066 = (0.508, 1.516)
95% CI for OR: e1.01±1.96 0.066
= (e0.508
,e1.516
) = (1.66, 4.55)

15. Confounding/Adjustment
C
E
D
?
Condi4ons
for
confounding
• C
must
cause
D
• C
must
caused
E
Stra4fy
by
levels
of
C
• Assume
OR
is
same
at
each
level
(no
interac4on
or
effect
modifica4on)

16. Logis/c Regression
log
px
1− px
"
#
$
%
&
' = log odds for D | X = x( )= a + bx
px =
1
1+e−(a+bx)
≡
ea+bx
1+ea+bx
eb1
=
OR
associated
with
unit
increase
in
X1,
holding
X2
fixed
(think
stra4fica4on)
Think,
e.g.,
D
=
breast
cancer,
X1
=
age
at
menarche,
X2
=
parity
log
p0,K,0
1− p0,K,0
!
"
##
$
%
&& = a
log
px1+1,x2,K,xk
/ (1− px1+1,x2,K,xk
)
px1,x2,K,xk
/ (1− px1,x2,K,xk
)
!
"
##
$
%
&& = log px1+1,x2,K,xk
/ (1− px1+1,x2,K,xk
)( )− log px1,x2,K,xk
/ (1− px1,x2,K,xk
)( )
= a + b1(x1 +1)+ b2 x2 +L+ bk xk[ ]− a + b1x1 + b2 x2 +L+ bk xk[ ]= b1

17. Mul/ple Logis/c Regression
eb
=
OR
associated
with
unit
increase
in
X
(i.e
exposure
E)
log
px1,K,xk
1− px1,K,xk
"
#
$$
%
&
'' = log odds for D | X1 = x1,K, Xk = xk( )
= a + b1x1 + b2 x2 +L+ bk xk
px1,K,xk
=
1
1+e−(a+b1x1+b2x2+L+bk xk )
≡
ea+b1x1+b2x2+L+bk xk
1+ea+b1x1+b2x2+L+bk xk

18. Pancrea/c Cancer example: Logis/c
Regression Models
X =
1 Coffee drinker ( ≥1 cups/day)
0 Coffee abstainer (0 cups/day)
"
#
$
%$
Y =
1 Female
0 Male
!
"
#
Model
Parameter
Es2mate
SD
OR
P-­‐value
b
1.012
0.257
2.751
<
0.001
b
0.957
0.258
2.603
<
0.001
c
-­‐0.406
0.133
0.667
0.002
bxapp +=- )1/log(
cybxa
pp
++
=- )1/log(

19. Time to Event Analysis—the Propor/onal
Hazards (Cox) Model
Kaplan-Meier survival estimate
analysis time
0 1000 2000 3000 4000
.880806
1
Western
Collabora4ve
Group
Study
of
CHD
in
men
Es4mate
of
Survival
Frac4on
(Func4on)
• S(t)
=
1-­‐I(t)
Handles
different
follow-­‐up
periods
straigthforwardly—right-­‐censoring
(can
also
handle
delayed
entry)

20. Comparing Survival Func/ons
Kaplan-Meier survival estimates, by dibpat0
analysis time
0 1000 2000 3000 4000
.835189
1
dibpat0 0
dibpat0 1
Type
B
Type
A

21. Propor/onal Hazards
RH(t) =
h1(t)
h0 (t)
= K
Logis4c
Regression
log(OR : X1vs X0 ) = b(X1 − X0 )
Cox
Propor4onal
Hazards
Model
log(RH : X1vs X0 ) = c(X1 − X0 )
h(T | X) = h0 (t)ecX
ec
is
the
Hazard
Ra4o
associated
with
unit
increase
in
X

22. WCGS: Logis/c Regression Fit
.
logit
chd69
diage
disbp
dismoke
dichol
dibpat0
------------------------------------------------------------------------------
chd69
|
Coef.
Std.
Err.
z
P>|z|
[95%
Conf.
Interval]
diage
|
.5370678
.1367423
3.93
0.000
.2690577
.8050778
disbp
|
.733672
.1464785
5.01
0.000
.4465794
1.020765
dismoke
|
.5510715
.1372644
4.01
0.000
.2820382
.8201049
dichol
|
.9433532
.1495272
6.31
0.000
.6502853
1.236421
dibpat0
|
.7612871
.1430818
5.32
0.000
.4808519
1.041722
_cons
|
-­‐4.402261
.2058241
-­‐21.39
0.000
-­‐4.805668
-­‐3.998853
------------------------------------------------------------------------------ logit chd69 diage disbp dismoke dichol dibpat0, or
------------------------------------------------------------------------------
chd69 | Odds Ratio Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
diage | 1.710982 .2339637 3.93 0.000 1.308731 2.23687
disbp | 2.082714 .305073 5.01 0.000 1.562957 2.775316
dismoke | 1.735111 .2381691 4.01 0.000 1.325829 2.270738
dichol | 2.56858 .3840724 6.31 0.000 1.916087 3.443268
dibpat0 | 2.14103 .3063425 5.32 0.000 1.617452 2.834094
------------------------------------------------------------------------------
In
terms
of
Odds
Ra4os
Dichotomous
measure
of
age,
blood
pressure,
smoking,
Serum
cholesterol,
behavior
type

23. WCGS: Propor/onal Hazards Model Fit
In
terms
of
Rela4ve
Hazards
Dichotomous
measure
of
age,
blood
pressure,
smoking,
Serum
cholesterol,
behavior
type
stcox
diage
disbp
dismoke
dichol
dibpat0,
nohr
------------------------------------------------------------------------------
Coef.
Std.
Err.
z
P>|z|
[95%
Conf.
Interval]
diage
|
.5273534
.1268302
4.16
0.000
.2787708
.775936
disbp
|
.6822427
.1391327
4.90
0.000
.4095476
.9549377
dismoke
|
.5282569
.1287685
4.10
0.000
.2758752
.7806386
dichol
|
.9072686
.1429656
6.35
0.000
.6270613
1.187476
dibpat0
|
.737248
.135597
5.44
0.000
.4714828
1.003013
-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐
stcox diage disbp dismoke dichol dibpat0
----------------------------------------------------------------------------
Haz. Ratio Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
diage | 1.694442 .2149064 4.16 0.000 1.321504 2.172625
disbp | 1.978309 .2752475 4.90 0.000 1.506136 2.598509
dismoke | 1.695974 .218388 4.10 0.000 1.317683 2.182866
dichol | 2.477546 .3542038 6.35 0.000 1.872101 3.278795
dibpat0 | 2.090175 .2834214 5.44 0.000 1.602368 2.726485
------------------------------------------------------------------------------

24. Comparison of Logis/c and Propor/onal
Hazards Model
------------------------------------------------------------------------------
chd69
|
Odds
Ra4o
Std.
Err.
z
P>|z|
[95%
Conf.
Interval]
-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐+-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐
diage
|
1.710982
.2339637
3.93
0.000
1.308731
2.23687
disbp
|
2.082714
.305073
5.01
0.000
1.562957
2.775316
dismoke
|
1.735111
.2381691
4.01
0.000
1.325829
2.270738
dichol
|
2.56858
.3840724
6.31
0.000
1.916087
3.443268
dibpat0
|
2.14103
.3063425
5.32
0.000
1.617452
2.834094
-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐
------------------------------------------------------------------------------
chd69 | Odds Ratio Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
diage | 1.710982 .2339637 3.93 0.000 1.308731 2.23687
disbp | 2.082714 .305073 5.01 0.000 1.562957 2.775316
dismoke | 1.735111 .2381691 4.01 0.000 1.325829 2.270738
dichol | 2.56858 .3840724 6.31 0.000 1.916087 3.443268
dibpat0 | 2.14103 .3063425 5.32 0.000 1.617452 2.834094
------------------------------------------------------------------------------
Logis4c
Regression
Propor4onal
Hazards

25. When is it Important to use a PH Model?
• Time-­‐dependent
exposures/treatments
• Varying
loss
to
follow
ups
(i.e
right
censoring)
• Par4cularly
when
follow-­‐up
payerns
vary
across
exposure/treatment
groups
(i.e.
differen4al
loss
to
follow
up)
• Logis4c
regression
can
be
badly
biased
in
either
situa4on

26.
• Nicholas
P.
Jewell,
“Natural
history
of
diseases:
Sta4s4cal
designs
and
issues,”
Clinical
Pharmacology
&
Therapeu9cs,
100,
2016,
353-­‐361.
• Nicholas
P.
Jewell,
Sta9s9cs
for
Epidemiology,
2004,
Chapman
&
Hall/CRC
Press.
• Nicholas
P.
Jewell,
“Risk
interpreta4on,
percep4on,
and
communica4on,”
American
Journal
of
Opthalmology,
148,
2009,
636-­‐638.
• Nicholas
P.
Jewell,
“Risk
comparisons,”
American
Journal
of
Opthalmology,
148,
2009,
484-­‐486.
• J.
C.
Schroeder
et
al.,
“The
North
Carolina-­‐Louisiana
prostate
cancer
project
(PCaP):
Methods
and
design
of
a
mul4disciplinary
popula4on-­‐based
cohort
study
of
racial
differences
in
prostate
cancer
outcomes,”
The
Prostate,
66,
2006,
1162-­‐1176.
• R.
C.
Millikan
et
al.,
“Epidemiology
of
basal-­‐like
breast
cancer,”
Breast
Cancer
Res.
Treat.,
109,
2008,
123-­‐139.
• K.
A.
Cronin
et
al.,
“Case-­‐control
studies
of
cancer
screening,
JNCI,
90,
1998,
498-­‐504.
• IARC
Handbooks
of
Cancer
Preven4on,
Cervix
Cancer
Screening,
2005
-­‐-­‐
Chapter
5:
Effec4veness
of
screening
in
popula4ons.