This document discusses using the probit model as a continuous-time survival model. It defines survival analysis as modeling the time until an event occurs. The probit model is presented as a way to specify survival probabilities over time in a continuous-time survival model, where the hazard rate characterizes the instantaneous rate of the event occurring. The document provides mathematical definitions and equations to describe how the probit model can be used to specify both survival probabilities and the hazard rate in a continuous-time survival analysis framework.

1. 1
The
Probit
Model
as
a
Continuous-­‐time
Survival
Model
Eddie
Cruz,
Dylan
Lojac,
Deniz
Öncü,
Becky
Turlip
Math-­‐UA
251
Introduction
to
Mathematical
Modelling
May
9,
2016
Abstract.
That
the
Probit
model
can
be
used
as
a
survival
model
in
discrete
time
is
well
known.
We
show
that
the
Probit
model
can
be
used
as
a
survival
model
also
in
continuous
time.
We
then
compare
and
contrast
the
predictions
of
the
Probit-­‐based
survival
model
with
the
predictions
of
a
version
of
the
commonly
employed
Cox-­‐based
survival
model
on
a
simple
set
of
simulated
data
in
discrete
as
well
as
in
continuous
time.
1.
Introduction
Although
it
appears
that
the
term
“survival
analysis”
originated
from
statistical
studies
in
medicine
where
the
event
of
interest
was
“death”,
the
survival
analysis
can
be
used
for
studying
any
process
for
which
the
outcome
variable
of
interest
is
the
time
until
an
event
occurs.
Other
examples
include
the
arrival
time
of
equipment
failures,
traffic
accidents,
stock
market
crashes,
default
of
borrowers,
unemployment
of
individuals
and
other
such
events.
A
survival
model
can
be
characterized
by
a
stochastic
process
𝑍 𝑡
which
takes
values
from
a
set
of
two
states,
say,
1,2
over
a
period
𝒯 = 0, 𝑇
or
𝒯 = 0, 𝑇
with
𝑇 ≤ ∞.
We
assume
that
the
transition
probabilities
depend
on
a
time
varying
covariate
vector
𝑋 𝑡
and
that
the
first
entry
of
𝑋 𝑡
is
1.
We
assume
further
that
the
process
is
Markov
so
that
the
transition
probabilities
from
state
𝑖
at
time
𝑠
to
state
𝑗
at
time
𝑡
where
𝑠 < 𝑡
depend
only
on
𝑍 𝑠
and
𝑋 𝑠 ,
and
are
independent
from
their
history
prior
to
time
𝑠.
That
is,
𝑃45 𝑠, 𝑡 = 𝑃𝑟𝑜𝑏 𝑍 𝑡 = 𝑗 𝑍 𝑠 = 𝑖, 𝑋(𝑠)
(1)

2. 2
Since
we
are
interested
in
survival
models,
we
assume
that
the
state
1
is
the
“survival”
state
and
the
state
2
is
the
“death”
state
so
that
𝑃;< 𝑠, 𝑡 = 0
and
𝑃;; 𝑠, 𝑡 = 1.
The
survival
probability
from
time
𝑠
to
time
𝑡
is
then
𝑃<< 𝑠, 𝑡 = 𝑆(𝑠, 𝑡)
so
that
𝑃<; 𝑠, 𝑡 =
1 − 𝑆(𝑠, 𝑡).
The
function
𝑆 𝑡 = 𝑆(0, 𝑡)
is
called
the
survival
function.
The
above
can
be
summarized
into
the
transition
probabilities
matrix
𝑷 𝑠, 𝑡
as
𝑷 𝑠, 𝑡 =
𝑃<<(𝑠, 𝑡) 𝑃<;(𝑠, 𝑡)
0 1
=
𝑆(𝑠, 𝑡) 1 − 𝑆(𝑠, 𝑡)
0 1
(2)
and,
since
the
process
is
Markov,
we
have
𝑷 𝑠, 𝑢 =
𝑷 𝑠, 𝑡 𝑷 𝑡, 𝑢 ,
𝑠 < 𝑡 < 𝑢 ∈
𝒯.
(3)
To
complete
the
model,
it
remains
to
specify
the
survival
probabilities
𝑆 𝑠, 𝑡
or
the
survival
function
𝑆 𝑡
in
some
fashion.
We
do
this
after
the
next
section.
2.
The
hazard
function
An
alternative
characterization
of
the
model
is
given
by
the
hazard
function,
or
instantaneous
rate
of
occurrence
of
the
event,
defined
in
our
formalization
as
𝜆 𝑡 = lim∆G→I
𝑃<; 𝑡, 𝑡 + ∆𝑡
∆𝑡
.
(4)
It
then
follows
from
the
ongoing
definitions,
and
the
equations
(2)
and
(3)
that
𝑃<< 0, 𝑡 𝑃<; 𝑡, 𝑡 + ∆𝑡 = 𝑃<; 0, 𝑡 + ∆𝑡 − 𝑃<; 0, 𝑡 ,
(5)
so
that
𝑃<; 𝑡, 𝑡 + ∆𝑡 = −
𝑆 𝑡 + ∆𝑡 − 𝑆 𝑡
𝑆 𝑡
.
(6)
Combining
the
equations
(4)
and
(6),
we
get
𝜆 𝑡 = −
𝑑
𝑑𝑡
ln 𝑆 𝑡
(7)

3. 3
We
are
almost
ready
to
discuss
some
alternative
specifications
of
the
survival
function.
As
a
first
step
in
this
direction,
let
us
solve
the
equation
(7)
with
the
natural
initial
condition1
𝑆 0 = 1
to
get
𝑆 𝑡 = exp − 𝜆 𝑠 𝑑𝑠
G
I
.
(8)
It
should
be
mentioned
that
under
our
assumptions
𝜆 𝑠
may
depend
on
𝑋 𝑠
but
we
have
been
supressing
this
dependence
for
convenience
all
along.
We
close
this
section
by
noting
that
an
alternative
way
of
characterizing
the
survival
models
is
to
look
at
the
probability
distribution
function
of
a
continuous
random
variable
𝜏 ∈ 𝒯
with
the
probability
density
function
𝑓(𝑡)
and
the
cumulative
distribution
function
𝐹(𝑡)
which
gives
the
probability
that
the
event
has
occurred
by
time
𝑡 > 𝜏,
𝑡 ∈ 𝒯.
It
is
evident
that
𝑆 𝑡 = 1 − 𝐹(𝑡)
and
it
follows
from
the
equation
(7)
that
𝑓 𝑡 = 𝜆 𝑡 𝑆 𝑡 .
(9)
3.
Two
alternatives
In
view
of
Sections
1
and
2,
we
have
two
alternatives.
We
can
specify
either
the
hazard
function
𝜆 𝑡
or
the
survival
function
𝑆(𝑡).
The
Cox
model
is
one
of
the
commonly
employed
approaches
for
specifying
the
hazard
function.
Since
we
are
interested
in
the
influence
of
a
time
varying
covariate
vector
𝑋 𝑡
on
the
survival
probabilities,
we
adopt
the
following
version
of
the
Cox
model:
𝜆 𝑡 = exp
{ 𝛼′𝑋(𝑡)},
(10)
where
𝛼
is
a
vector
of
parameters.
The
second
alternative
is
specifying
the
survival
function
directly.
Let
Φ ∙
be
the
cumulative
distribution
function
of
some
distribution
whose
probability
distribution
function
is
𝜙 ∙ .
Although
any
distribution
might
do,
we
assume
that
the
Φ ∙
is
the
cumulative
distribution
function
of
the
standard
normal
distribution
and
specify
the
survival
function
as
𝑆 𝑡 = Φ 𝜇 𝑡 ,
(11)
where
𝜇 𝑡
is
a
cut-­‐off
function.
This
is
a
Probit
specification.
1
Otherwise,
there
is
nothing
to
observe.

4. 4
Our
main
reason
for
choosing
Probit
is
that
it
provides
a
simple
tool
for
the
joint
estimation
of
death
and
the
covariates,
as
we
describe
on
a
simple
example
in
the
Appendix.
For
other
specifications,
the
joint
estimation
of
death
and
the
covariates
might
be
very
difficult,
if
not
impossible.
Why
this
is
important
is
explained
in
the
Appendix
as
well.
Note
that
given
the
properties
of
the
survival
function
𝑆(𝑡),
the
cut-­‐off
function
𝜇 𝑡
must
be
decreasing
and
we
must
have
lim
G→I
𝜇 𝑡 = ∞.
From
the
equations
(7)
and
(11),
we
see
also
that
𝜆 𝑡 = −
𝜙 𝜇 𝑡 𝜇′ 𝑡
Φ 𝜇(𝑡)
(12)
which
verifies
that
𝜇 𝑡
must
be
decreasing.
We
defer
the
specification
of
the
cut-­‐off
function
to
the
next
section.
In
closing
this
section,
we
note
that
the
survival
probability
from
time
𝑠
to
time
𝑡
should
take
the
form
𝑆 𝑠, 𝑡 = Φ 𝜇b(𝑡 − 𝑠)
(13)
where
𝜇b(∙)
is
the
cut-­‐off
function
associated
with
time
s.
Of
course,
𝜇b ∙
must
have
the
same
properties
𝜇 ∙
has.
4.
The
setup
In
what
follows,
we
will
consider
a
set
of
𝑀
“entities”
such
as
patients
or
firms
or
machines
and
the
like,
and
study
their
failure
probabilities.
In
the
case
of
patients,
the
failure
can
be
death;
in
the
case
of
firms,
it
can
be
default
on
debt
and
so
on
so
forth.
For
simplicity
in
discussion,
we
will
refer
to
all
of
these
as
patients
and
our
failure
of
interest
will
be
death.
We
will
begin
with
a
single
patient
and
extend
our
results
to
𝑀
patients
later.
4.1.
A
single
patient
In
this
subsection,
we
will
look
at
a
single
patient
and
assume
that
the
covariates
𝑋(𝑡)
are
sampled
with
intervals
of
equal
length
∆𝑡.2
Suppose
now
that
the
observations
were
made
2
Note
that
although
the
assumption
that
the
covariates
sampled
with
intervals
of
equal
length
is
not
necessary,
it
will
simplify
the
formulation.
Our
results
can
easily
be
extended
to
the
case
of
unequal
length
sampling
intervals
after
minor
modifications.

5. 5
in
the
interval
[0, 𝑇]
where
𝑇 = 𝑁∆𝑡
and
𝑁
is
the
number
of
intervals.
Let
us
set
𝑡4 =
𝑖∆𝑡, 𝑖 = 0,1, … , 𝑁.
Since
the
sampling
is
done
discretely,
further
suppose
that
𝑋 𝑡 =
𝑋(𝑡4i<)
=𝑋Gjkl
for
any
𝑡 ∈ [𝑡4i<, 𝑡4),
𝑖 = 1, … , 𝑁.
With
this
assumption,
we
have
also
that
𝜆 𝑡 = 𝜆 𝑡4i< =
𝜆Gjkl
for
any
𝑡 ∈ [𝑡4i<, 𝑡4),
𝑖 = 1, … , 𝑁.
Let
us
now
assume
that
either
at
some
time
𝜏 ∈ [𝑡mi<, 𝑡m)
for
some
𝑘,
0 < 𝑘 ≤ 𝑁,
that
is,
in
the
𝑘Go
period,
the
death
occurred
or
that
the
patient
survived
in
the
observation
interval
0, 𝑇 .
If
the
survival
occurred
in
0, 𝑇 ,
set
𝜏 = 𝑡m
and
𝑘 = 𝑁.
Lastly,
define
the
indicator
variables
𝑌5 𝑡 = 1 𝑍 𝑡 = 𝑗 , 𝑗 = 1,2.
This
means
that
if
𝑌<(𝑡) =1
at
time
𝑡 ∈
[0, 𝑇]
,
the
patient
survived
until
time
𝑡.
Otherwise,
𝑌;(𝑡) =1
and
the
patient
is
dead
at
time
𝑡.
Under
these
assumptions,
from
the
equation
(8)
we
have
𝑆 𝜏 = 𝑆 𝑡mi<, 𝜏 𝑆 𝑡I, 𝑡mi<
,
(14)
where
𝑆 𝑡4i<, 𝑡4 = exp −
𝜆Gjkl
Δ 𝑡 , 𝑖 = 1,2, … , 𝑘 − 1,
(15)
𝑆 𝑡mi<, 𝜏 = exp −𝜆Grkl
(τ − 𝑡mi<) ,
(16)
and
𝑆 𝑡I, 𝑡mi<
= 𝑆 𝑡4i<, 𝑡4
mi<
4t<
,
(17)
Given
these,
we
have
two
alternatives.
4.1.1.
The
discrete
case
The
first
alternative
is
to
ignore
that
the
death
occurred
in
the
interior
of
the
interval
[𝑡mi<, 𝑡m),
set
𝜏 = 𝑡m.
Then
the
associated
likelihood
function
for
this
patient
is
ℒv 𝜃 = 𝑌< 𝑡m 𝑆 𝑡mi<, 𝑡m + 𝑌; 𝑡m [1 − 𝑆 𝑡mi<, 𝑡m ] 𝑆 𝑡I, 𝑡mi< ,
(17)
where
𝜃
is
the
parameter
vector
to
be
estimated
once
the
modelling
approach
is
chosen.

6. 6
If
we
choose
to
model
the
𝜆Gj
as
in
the
version
of
Cox
model
given
by
the
equation
(10),
then
𝜃 = 𝛼
and
we
are
done.
If,
instead,
we
choose
to
model
the
survival
probabilities
𝑆 𝑡4i<, 𝑡4 , 𝑖 = 1,2, … , 𝑘,
then
we
can
proceed
as
follows.
Set
𝜇Gjkl
= 𝜇Gjkl
𝑡4 − 𝑡4i< , 𝑖 = 1,2, … , 𝑘
and
model
the
𝜇Gjkl
as
𝜇Gjkl
= 𝛽y
𝑋Gjkl
.
(18)
In
this
case,
the
parameter
vector
𝜃 = 𝛽
and
the
survival
probabilities
are
𝑆 𝑡4i<, 𝑡4 = Φ 𝜇Gjkl
.
(19)
We
can
now
rewrite
the
likelihood
function
for
this
patient
as
ℒv 𝜃 = 𝑌< 𝑡m Φ 𝜇Grkl
+ 𝑌; 𝑡m [1 − Φ 𝜇Grkl
] Φ 𝜇Gjkl
mi<
4t<
,
(20)
which
is
the
usual
likelihood
function
of
the
Probit
model
in
discrete
time.
4.1.2.
The
continuous
case
If
we
do
not
ignore
that
the
death
occurred
in
the
interior
of
the
interval
[𝑡mi<, 𝑡m)
and
recall
from
the
equation
(9)
that
𝑓 𝜏 = 𝜆 𝜏 𝑆(𝜏),
then
from
the
ongoing
development
the
likelihood
function
for
this
patient
is
ℒz 𝜃 = {𝑌< 𝜏 + 𝑌; 𝜏 𝜆Grkl
}𝑆 𝑡mi<, 𝜏 𝑆 𝑡I, 𝑡mi< .
(21)
If
we
choose
to
model
the
hazard
function
as
in
the
above
Cox-­‐based
formulation,
we
are
done
already.
The
equations
(10),
(15)
and
(16)
complete
the
model
and
the
parameter
vector
to
be
estimated
is
𝜃 = 𝛼.
Let
us
now
proceed
to
the
Probit
formulation
and
observe
from
the
ongoing
development
that
exp −
𝜆Grkl
Δ 𝑡 = 𝑆 𝑡mi<, 𝑡m = Φ 𝜇Grkl
.
(22)
Then,
𝜆Grkl
=
1
Δ𝑡
ln
1
Φ 𝜇Grkl
,
(23)
and
𝑆 𝑡mi<, 𝜏 = Φ 𝜇Grkl
{iGrkl
|G .
(24)

7. 7
Combining
all
of
the
above
gives
us
the
Probit
version
of
the
continuous
time
likelihood
function
for
this
patient.
It
is
ℒz 𝜃 = 𝑌< 𝜏 + 𝑌; 𝜏 ln
1
Φ 𝜇Grkl
Φ 𝜇Grkl
{iGrkl
|G Φ 𝜇Gjkl
mi<
4t<
.
(25)
Now,
the
parameter
vector
to
be
estimated
is
𝜃 = 𝛽.
4.2.
Many
patients
Extension
to
the
case
of
multiple
patients
is
trivial.
Index
all
of
the
relevant
variables
with
𝑚 = 1,2, …
, 𝑀
and
denote
by
ℒ~ 𝜃
a
generic
for
all
of
the
likelihood
functions
written
above
for
the
𝑚Go
patient.
Then
the
likelihood
function
ℒ 𝜃
associated
with
our
patient
sample
is
ℒ 𝜃 = ℒ~ 𝜃
•
~t<
.
(26)
And
we
are
done.
5.
Numerical
Experiments
In
this
section,
we
compare
and
contrast
the
predictions
of
the
Probit
and
Cox
models
for
four
simulated
data
sets
of
10,000
firm-­‐periods.
We
set
the
data
sampling
period
length
as
∆𝑡=1
for
convenience,
suppose
there
is
an
external
variable
𝑋
that
drives
the
defaults,
and
draw
10,000
values
for
𝑋
assuming
that
it
is
identically
and
independently
distributed
standard
normal.
Finally,
we
generate
our
data
from
the
following
three
models
that
give
the
survival
probabilities
as
follows.
a)
Probit:
𝑃<<(0,1) = Φ 1.45 + 𝑋 ,
b)
Cox:
𝑃<<(0,1) = exp −exp
(2.17 − 𝑋) ,
c)
Chi
Squared:
𝑃<<(0,1) = Χ;
exp(1 + 𝑋) ,
In
the
above,
Φ ∙
and
Χ;
∙
and
are
the
cumulative
distribution
functions
of
the
standard
normal
and
Chi
Squared
with
unit
degree
of
freedom
distributions.
We
chose
the
parameter

8. 8
values
of
the
models
in
such
a
way
that
the
resulting
in-­‐sample
unconditional
survival
probabilities
in
the
data
sets
are
about
85%.
Next,
we
estimate
the
models
for
each
of
the
data
sets
under
the
assumption
that
the
observations
are
made
in
discrete-­‐time.
Table
1
summarizes
the
estimation
results.
All
parameter
estimates
and
models
are
statistically
significant
at
better
than
1%.
Although
a
comparison
of
models
based
on
the
likelihood
ratios
is
not
a
proper
comparison
for
non-­‐
nested
models,
we
nevertheless
see
from
this
comparison
in
Panels
A
and
B
that
when
Probit
is
the
data-­‐generating
model,
Probit
appears
to
fit
the
data
better
than
Cox,
whereas
when
Cox
is
the
data-­‐generating
model,
Cox
appears
to
fit
the
data
better
than
Probit.
These
results
are
expected
and
included
only
as
a
check
on
our
results.
Table
1.
Comparison
of
Probit
and
Cox
Models
in
Discrete-­‐time
This
table
presents
the
estimation
results
of
Probit
and
Cox
models
for
three
simulated
data
sets
of
10,000
patient-­‐periods.
All
parameter
estimates
and
models
are
statistically
significant
at
better
than
1%.
Panel
A.
Data
Generating
Model:
Probit
Model
Probit
Cox
Constant
1.473
-­‐2.554
Slope
0.993
-­‐1.427
Loglikelihood
Null
-­‐4185.23
-­‐4185.23
Model
-­‐2975.39
-­‐3009.39
Likelihood
Ratio
2419.68
2351.68
Panel
B.
Data
Generating
Model:
Cox
Model
Probit
Cox
Constant
1.232
-­‐2.214
Slope
0.608
-­‐0.980
Loglikelihood
Null
-­‐4155.31
-­‐4155.31
Model
-­‐3548.29
-­‐3541.40
Likelihood
Ratio
1,214.06
1,227.82
Panel
C.
Data
Generating
Model:
Chi
Squared
Model
Probit
Cox
Constant
1.355
-­‐2.339
Slope
0.783
-­‐1.101
Loglikelihood

9. 9
Null
-­‐4114.45
-­‐4114.45
Model
-­‐3258.21
-­‐3320.03
Likelihood
Ratio
1712.48
1,587.54
In
this
Panel
C,
the
data
generated
from
the
Chi
Squared
represent
“real
world”
data
whose
distribution
is
“unknown”.
Although
from
Panel
C
–
again
based
on
a
comparison
of
the
likelihood
ratios
–
we
see
that
when
the
data-­‐generating
model
is
Chi
Squared,
the
Probit
model
appears
to
fit
the
data
better
than
the
Cox
model,
one
should
not
conclude
that
the
Probit
is
better
than
the
Cox
model
based
on
a
single
simulated
data
set.
Figure
1
depicts
the
predicted
one-­‐period
survival
probabilities
from
the
Probit
and
Cox
models
as
functions
of
the
underlying
external
variable
𝑋,
and
compares
their
predictions
with
the
“true”
survival
probabilities
for
the
data
sets
where
the
unconditional
survival
probability
is
about
85%.
Figure
1.
Comparison
of
Probit
and
Cox
Models
in
Discrete-­‐time
–
In
sample
unconditional
survival
probability
is
about
85%
The
figures
plot
one-­‐period
survival
probabilities
as
estimated
by
discrete-­‐time
Probit,
Cox
and
data-­‐generating
models
as
functions
of
the
simulated
patient-­‐period
variable
X
for
the
simulated
data
of
10,000
patient-­‐periods.
In
Figures
1.a
and
1.b,
only
the
estimated
models
are
included
because
the
data-­‐generating
model
and
its
estimation
produce
indistinguishable
graphs.
a)
Data
Generating
Model:
Probit
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
-­‐2.5 -­‐1.5 -­‐0.5 0.5 1.5 2.5
Survival
Probability
X
Probit Cox │Probit
-­‐ Cox
│

10. 10
b)
Data
Generating
Model:
Cox
c)
Data
Generating
Model:
Chi
Squared
In
Figures
1.a
and
1.b,
we
compare
the
Probit
and
Cox
models
when
the
data-­‐
generating
model
is
one
of
them,
and
see
that
although
the
models
agree
in
their
predictions
for
the
most
part,
the
deviations
at
the
left
extreme
–
where
the
predicted
survival
probabilities
are
small
–
can
be
as
large
as
7%
for
our
simulated
data
sets.
In
Figure
1.c,
we
compare
the
Probit
and
Cox
models
when
the
data-­‐generating
process
is
“unknown”
(that
is,
when
its
Chi
Squared).
We
see
from
the
figure
that
not
only
the
Probit
and
Cox
models
generally
agree
with
each
other,
but
also
they
do
not
deviate
from
the
“true”
model
significantly
except
at
the
extremes.
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
-­‐2.5 -­‐2 -­‐1.5 -­‐1 -­‐0.5 0 0.5 1 1.5 2 2.5
Survival
probability
X
Probit Cox │Probit
-­‐ Cox
│
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
-­‐2.5 -­‐2 -­‐1.5 -­‐1 -­‐0.5 0 0.5 1 1.5 2 2.5
Survival
Probability
X
Chi
Squared Probit Cox │Probit
-­‐ Cox
│

11. 11
Lastly,
we
turn
our
attention
to
continuous-­‐time,
and
focus
on
the
last
data-­‐
generating
model,
that
is,
Chi
Squared.
We
keep
our
original
simulated
data
set
generated
by
this
model,
but
randomly
assign
to
the
deaths
a
default
time
between
zero
and
one
using
the
uniform
distribution.
The
results
of
our
estimations
based
on
these
data
are
reported
in
Table
2.
All
of
the
parameter
estimates
and
models
reported
in
Table
2
are
statistically
significant
at
better
than
1%.
As
before,
we
note
that
although
a
comparison
between
two
non-­‐nested
models
based
on
likelihood
ratios
is
not
appropriate,
our
results
show
that
our
Probit
formulation
appears
to
fit
the
data
better
than
the
Cox
model
based
on
this
criterion
in
our
simulated
sample.
Table
2.
Comparison
of
Probit
and
Cox
Models
in
Continuous-­‐time
This
table
presents
the
estimation
results
of
Probit
and
Cox
models
for
a
simulated
data
set
of
10,000
patient-­‐periods.
All
parameter
estimates
and
models
are
statistically
significant
at
better
than
1%.
Data
Generating
Model:
Chi
Squared
with
Randomized
Death
Times
Model
Probit
Cox
Constant
1.354
-­‐2.333
Slope
0.764
-­‐1.070
Loglikelihood
Null
-­‐4116.62
-­‐4116.62
Model
-­‐3272.58
-­‐3336.73
Likelihood
Ratio
1688.08
1659.78
Since
in
this
set
of
simulated
data
the
“true”
data
generating
processes
is
“truly
unknown”,
we
compare
predictions
of
the
Probit
and
Cox
models
without
any
reference
to
any
“true”
default
probabilities
in
Figure
2.
From
Figure
2
we
see
that
the
agreement
between
the
Probit
and
Cox
models
is
much
better
in
their
continuous
versions.
This
can
be
observed
also
from
Table
2,
if
their
likelihood
ratios
are
compared.
Figure
2.
Comparison
of
Probit
and
Cox
Models
in
Continuous-­‐time
–
In
sample
unconditional
survival
probability
is
about
85%
The
figure
plots
one-­‐period
survival
probabilities
as
estimated
by
continuous-­‐time
Probit
and
Cox
Models
as
functions
of
the
simulated
pazient-­‐period
variable
X
for
the
simulated
data
of
10,000
patient-­‐periods.
The
data-­‐generating
models
is
Chi
Squared
with
randomized
death

12. 12
times.
Let
us
close
this
section
by
summarizing
our
observations
from
the
numerical
experiments
of
this
section
as
follows.
1)
As
long
as
the
underlying
data
generating
process
for
the
deaths
is
not
known,
it
is
not
possible
to
decide
whether
the
Probit
or
the
Cox
model
is
the
better
suited
to
the
task;
2)
No
matter
which
class
of
models
is
chosen,
it
is
worthwhile
to
estimate
the
models
in
continuous-­‐time,
for
otherwise,
death
probabilities
may
be
exaggerated.
6.
Conclusions
We
showed
that
the
Probit
model
is
equivalent
to
the
commonly
employed
Cox
model
in
continuous-­‐time,
indicating
that
survival
probabilities
could
be
modelled
using
the
Probit
model
in
addition
to
the
Cox
model.
We
then
constructed
the
likelihood
function
of
the
Probit
model
for
the
estimation
of
deaths
in
continuous-­‐time
and
presented
a
simpler
discrete-­‐time
version.
Furthermore,
since
patient
specific
variables
are
expected
to
be
dependent,
as
we
explain
in
the
Appendix,
the
patient
specific
covariates
and
the
death
processes
can
be
jointly
estimated
in
our
framework.
Lastly,
we
compared
and
contrasted
the
Probit
and
Cox
models
through
numerical
experiments
and
demonstrated
that
none
of
the
alternatives
necessarily
dominates
the
other.
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
-­‐2.5 -­‐2 -­‐1.5 -­‐1 -­‐0.5 0 0.5 1 1.5 2 2.5
Survival
probability
X
Probit Cox │Probit
-­‐ Cox
│

13. 13

14. 14
Appendix
As
we
mentioned
in
Section
3,
one
advantage
of
our
Probit
formulation
is
that
it
provides
a
simple
tool
for
the
joint
estimation
of
death
and
the
covariates.
To
explain
why
the
joint
estimation
of
death
and
the
covariates
is
important,
consider
a
set
of
patients
and
suppose
that
the
covariates
are
some
patient
related
variables
that
are
recorded
during
doctor
visits.
For
convenience,
focus
on
a
single
patient
and
suppose
that
there
is
one
patient
specific
variable,
say,
𝐵 𝑡 .
Recall
that
state
1
is
the
survival
state
and
state
2
is
the
death
state,
and
focus
on
two
observations
in
discrete-­‐time,
at
times
𝑡 = 𝑡I
and
𝑡 = 𝑡< > 𝑡I.
The
process
𝑍(𝑡)
is
initialized
at
𝑡 = 𝑡I
as
𝑍 𝑡I = 1
and
the
latent
process
𝑍∗
(𝑡)
determines
the
next
state
according
as
𝑍 𝑡< = 1,
𝑖 𝑓
𝑍∗
𝑡< ≤ 𝑎
and
𝑍 𝑡< = 2,
𝑖 𝑓
𝑍∗
𝑡< > 𝑎
( 𝐴. 1)
For
further
simplicity,
consider
the
following
processes:
𝐵 𝑡< = 𝑏 + 𝜂
( 𝐴. 2)
𝑍∗
𝑡< = 𝜖
( 𝐴. 3)
where
𝑏
is
some
constant,
and
𝜂
and
𝜖
are
some
random
variables.
Since
both
𝐵(𝑡<)
and
𝑍∗
𝑡<
are
variables
associated
with
the
patient,
it
may
be
desirable
to
consider
the
possibility
that
they
may
be
dependent.
This
possibility
can
be
addressed
in
our
framework
with
no
essential
difficulty,
as
we
demonstrate
below.
To
this
end,
let
us
suppose
that
𝜂
and
𝜖
are
jointly
normally
distributed
with
mean
zero
and
the
covariance
matrix
Σ =
𝜎;
𝜌𝜎
𝜌𝜎 1
In
this
case,
the
joint
probability
distribution
is
given
by
𝜙; 𝜖, 𝜂 =
1
2𝜋 1 − 𝜌; 𝜎;
𝑒𝑥𝑝 −
𝜂;
− 2𝜌𝜎𝜂𝜖 + 𝜎;
𝜖;
2 1 − 𝜌; 𝜎;
(𝐴. 4)
Suppose
now
that
the
patient
did
not
die
at
time
𝑡<.
Then
the
likelihood
of
this
observation
is
ℒ 𝑎, 𝑏 =
1
2𝜋𝜎;
𝑒𝑥𝑝 −
(𝐵(𝑡<) − 𝑏);
2𝜎;
Φ
𝑎
(1 − 𝜌;
−
𝜌(𝐵(𝑡<) − 𝑏)
1 − 𝜌; 𝜎;
𝐴. 5
whereas
if
the
death
occurred
at
time
𝑡<,
then
the
likelihood
of
this
observation
is

15. 15
ℒ 𝑎, 𝑏 =
1
2𝜋𝜎;
𝑒𝑥𝑝 −
(𝐵(𝑡<) − 𝑏);
2𝜎;
1 − Φ
𝑎
(1 − 𝜌;
−
𝜌(𝐵(𝑡<) − 𝑏)
1 − 𝜌; 𝜎;
𝐴. 6
where
Φ ∙
is
the
standard
normal
cumulative
distribution
function.
Since
in
most
applications
one
works
with
a
handful
of
patient
specific
variables,
the
above
can
be
extended
to
several
patient
specific
variables
without
much
difficulty.

16. 16