Download a Censoring

Chapter 16 -- Event Duration Models
This chapter covers models of elapsed duration.
 Customer Relationship Duration
 Loyalty Program Membership Duration
 Customer Retention Metrics
Mathematical
Marketing
This Section is 95% taken from
Helsen, Kristiaan and David C. Schmittlein (1993), "Analyzing Duration
Times in Marketing: Evidence for the Effectiveness of Hazard Rate
Models," Marketing Science, 12 (4), 395-414.
Slide 16.1
Hazard Rate Models
Module Sequence
The sequence of coverage





Mathematical
Marketing
Definitions
The Hazard Function
Truncation
Censoring
Parametric Models
Slide 16.2
Hazard Rate Models
Key Definitions
Define
Ti
as a random variable representing the duration for individual i. Then
F(t) = Pr(Ti < t)
is the probability function of duration failure times. The density, or unconditional
failure rate is
dF( t )
f(t) = F′(t) =
dt
Mathematical
Marketing
Slide 16.3
Hazard Rate Models
More On Survivorship and Failureship
The cumulative failure function can now be written as an integral
t

F(t) = Pr(Ti < t)  f (u )du
0
The survivorship function is the complement of the Failureship
distribution,

S(t) = 1 – F(t) = Pr(Ti > t) =
 f (u)du
t
Mathematical
Marketing
Slide 16.4
Hazard Rate Models
What Is a Hazard Function?
The hazard function, or conditional (age specific) failure rate is
h(t) 
Mathematical
Marketing
f (t)
f (t )

1  F( t ) S( t )
Slide 16.5
Hazard Rate Models
Elaboration on the Hazard Function
pr [failure at t]
f (t)
f (t )
h(t) 

1  F( t ) S( t )
pr [there has not been a failure up to t]
It is the instantaneous rate of failure given survival until now, or
the imminent failure risk
Mathematical
Marketing
Slide 16.6
Hazard Rate Models
The Shape of the Hazard Function
h(t) 
f (t)
f (t )

1  F( t ) S( t )
The hazard function can take on any shape:
Mathematical
Marketing
1.
h(t) increases – snowballing (product adoption)
2.
h(t) constant – no dynamics or memory
dh ( t )
0
dt
3.
h(t) decreases – inertia (interpurchase times)
dh ( t )
0
dt
Slide 16.7
Hazard Rate Models
Constant Hazard – No Memory
The exponential distribution
f(t) = e-t
implies
h(t) = 
and we have situation 2.
Mathematical
Marketing
Slide 16.8
Hazard Rate Models
The Two-Parameter Weibull
The Weibull distribution
 1  t 
f (t )  t e
implies
h(t) = t-1
and we can create any of the three situations.
Mathematical
Marketing
Slide 16.9
Hazard Rate Models
The Hazard Rate Impacts Average Retention
Since
f (t)
f (t )
h(t) 

1  F( t ) S( t )
then
f(t)  h(t)[1  F(t)]
So can we add independent variables to the model?
First, a digression on censoring.
Mathematical
Marketing
Slide 16.10
Hazard Rate Models
Customer Relationship Duration
Time
Ongoing Relationships
Are Right-Censored
Time of
Study
Mathematical
Marketing
Slide 16.11
Hazard Rate Models
Truncation and Censoring
Left
Truncation
Censoring
Mathematical
Marketing
Right
Ti is observed only if Ti < ai
Ti is observed only if Ti > ai
If Ti  ai, then Ti = ai
All values below a are observed as a
If Ti  ai, then Ti = ai
All values above a are observed as a
Slide 16.12
Hazard Rate Models
True Relationship of x and Duration
duration
Ti
Ti=0
Mathematical
Marketing
Some dependent variable
values will be reduced
due to censoring.
x
Slide 16.13
Hazard Rate Models
True Relationship of x and Duration
duration
Ti
Ti=0
Mathematical
Marketing
Some dependent variable
values will be reduced
due to censoring.
x
Slide 16.14
Hazard Rate Models
True Relationship of x and Duration
duration
Each dependent value
above the horizontal line
will be redefined as equal
to the line, i. e. y = a.
Ti=a
How will the bias work?
Ti=0
Mathematical
Marketing
x
Slide 16.15
Hazard Rate Models
Proportional Hazards
h(t) = h0(t) hx(t)
This part is a function of individual x values
It adjusts h0 up or down as a function of marketing
instruments
This part is constant for all individuals
Mathematical
Marketing
Slide 16.16
Hazard Rate Models
Proportional Hazard Models
We generally use this parametric approach:
h x (t)  e
Mathematical
Marketing
xiβ
Slide 16.17
Hazard Rate Models
Two Parametric Functional Forms
h(t) = h0(t) hx(t)
 λex β
i
 λγt γ 1e xiβ
Exponential distribution
Weibull distribution
Can you make the Exponential a special case of the Weibull?
Mathematical
Marketing
Slide 16.18
Hazard Rate Models
ML Estimation
Density function
Survivorship function Pr(Ti > t)
ln l   i ln f (Ti | β)   (1  i ) ln S(Ti | β)
i
i
with
Mathematical
Marketing
1
i  
0
for uncensored observations
for censored observations
Slide 16.19
Hazard Rate Models
SAS PROC LIFEREG ;
proc lifereg data=input-data-set;
model y *flag-var (1) = iv1 iv2 / distribution = weibull ;
class nominal-var ;
This var tracks whether the observation is right censored or not
If flag-var is equal to this value, the observation is censored.
Mathematical
Marketing
Slide 16.20
Hazard Rate Models