This presentation shows financial managers how to predict how long accounts will likely stay open. It is based on a sophisticated statistical probability model.

1. Core Deposits
Sensitivity and Survival
Analysis
Laura Roberts
Hugh Blaxall
Brian Velligan
Sept. 13, 2010

2. Research question 1a
• Question 1a.
How can we visually summarize account duration?

3. Research question 1b
• Question 1b. How can we predict the length of time a
person will keep a core account open (account duration)?
We cannot simply compute an average of account
durations because we do not know how far into the future
current accounts will “survive.” Simple means will
produce a negatively biased estimate.
• Perhaps we can revise our question to read, “What is the
probability a person will keep an account open for a
specific period of time?” This new question allows us to
use survival analysis, hazard probabilities, and risk
functions to get a detailed picture of account duration.

4. Question 1b (continued)
• Can we create a model using time and other
indictors (e.g. interest rate or change in the interest
rate on the account) as predictors of account
duration? This is a more sophisticated question for
another time…food for thought for now…

5. Question 1c
• 1c – How can we summarize typical account
duration with a single index? Remember means and
other simple average indices will not do the trick
because we do not know how long accounts will stay
open…

6. What is the best statistical tool for
answering each question?
• Question 1a – to visually summarize duration use a
histogram of the frequency of duration for censored
and uncensored accounts. I’ll show you how to do
this.
• Question 1b - To predict duration, use survival
analysis.
• Question 1c – for a single index, we can use median
lifetime survival probability…more on this…

7. Background for Study
• 1. Use a multi-cohort analysis such as accounts
opened between 1972 and 1977 and studied until
1984.
• 2. Measure duration of each account.
• 3. Predict length of time until a given event, in this
case, closing of the account.
• 4. Some people will not close the account within the
time period of observation. These people (accounts)
are considered to be censored.

8. © Willett, Harvard University
Graduate School of Education,
03/19/14
S052/II.2(b) – Slide 8
Dataset acctdur.txt
Overview Discrete-time person-level dataset on the duration of
accounts opened between 1972 and 1977, and which were
followed uninterruptedly until 1984.
Source Bank records.
Sample size 3941 accounts.
More Info Singer & Willett, 2003
Let’s examine an example …Let’s examine an example …
Note on the labeling of the discrete time “bins.” We regarded an account’s first year as their zeroth
year. If they then are closed sometime during
the following year, they were classified as having a duration of one year and having been closed in “year one.”
Note on the labeling of the discrete time “bins.” We regarded an account’s first year as their zeroth
year. If they then are closed sometime during
the following year, they were classified as having a duration of one year and having been closed in “year one.”
S052/II.2(a1): Introducing The Central Concepts In Classical Survival Analysis
Introducing A Dataset On Account Duration
S052/II.2(a1): Introducing The Central Concepts In Classical Survival Analysis
Introducing A Dataset On Account Duration
““Multiple Cohort” Sample DesignMultiple Cohort” Sample Design
Be aware that multiple annual cohorts of
accounts are pooled together into this
single sample:
• Cohorts entered the sample
sequentially between the 1972 and
1977.*
• All cohorts were followed until the
end of 1984.
““Multiple Cohort” Sample DesignMultiple Cohort” Sample Design
Be aware that multiple annual cohorts of
accounts are pooled together into this
single sample:
• Cohorts entered the sample
sequentially between the 1972 and
1977.*
• All cohorts were followed until the
end of 1984.
Important Distinction YouImportant Distinction You
Must Keep In MindMust Keep In Mind
The two “modern” approaches
to survival analysis are distinct
in the way that they require
duration to be measured:
• In discrete-time survival
analysis, time is measured in
discrete units, such as
semesters, years, etc.
• In continuous-time survival
analysis, time can be
measured to any level of
precision.
Important Distinction YouImportant Distinction You
Must Keep In MindMust Keep In Mind
The two “modern” approaches
to survival analysis are distinct
in the way that they require
duration to be measured:
• In discrete-time survival
analysis, time is measured in
discrete units, such as
semesters, years, etc.
• In continuous-time survival
analysis, time can be
measured to any level of
precision.
Research QuestionResearch Question
Whether, and if so when,
accounts are closed?
Research QuestionResearch Question
Whether, and if so when,
accounts are closed?

9. © Willett, Harvard University
Graduate School of Education,
03/19/14
S052/II.2(b) – Slide 9
The dataset is straightforward, containing IDs and length of account, with one small hitch …The dataset is straightforward, containing IDs and length of account, with one small hitch …
Structure of Dataset
Col
#
Var Name Variable Description Variable Metric/Labels
1 ID Customer identification code. Integer
2 acctopen
Number of years that the account
remained open, or until the
account was censored in 1984 by
the end of the study.
Integer
3 CENSOR
Dummy variable to indicateto indicate
whether an account was
censored by the end of data
collection in 1984.
Dichotomous variable:
0 = not censored,
1 = censored.
S052/II.2(a1): Introducing The Central Concepts In Classical Survival Analysis
The Difficult Problem of Censoring!!!
S052/II.2(a1): Introducing The Central Concepts In Classical Survival Analysis
The Difficult Problem of Censoring!!!
There is a problem that is intrinsic to survival data,
and is illustrated in this dataset:
 The event of importance in the study isis
“closing an account.”
 But not every customer (account) actually
experiences this event while being observed by
researchers.
 We say that they are “censored” by the end of
the data-collection.
There is a problem that is intrinsic to survival data,
and is illustrated in this dataset:
 The event of importance in the study isis
“closing an account.”
 But not every customer (account) actually
experiences this event while being observed by
researchers.
 We say that they are “censored” by the end of
the data-collection.
And, of course, some of the censored accounts will
eventually experience the event of interest, but not
while the researchers are watching!
 Ignoring this can seriously impact estimates
of time-to-event.
 And, given that time-to-event is the focus of
our research question, we need to figure out
how to deal with this!
And, of course, some of the censored accounts will
eventually experience the event of interest, but not
while the researchers are watching!
 Ignoring this can seriously impact estimates
of time-to-event.
 And, given that time-to-event is the focus of
our research question, we need to figure out
how to deal with this!

10. © Willett, Harvard University
Graduate School of Education,
03/19/14
S052/II.2(b) – Slide 10
One sensible thing you can do is display the frequency with which each account length
occurs, in a vertical histogram that includes all the accounts in the sample, both censored
and un-censored.
One sensible thing you can do is display the frequency with which each account length
occurs, in a vertical histogram that includes all the accounts in the sample, both censored
and un-censored.
I created this vertical
histogram by typing the
frequencies of each account
length into an EXCEL
spreadsheet. You can
create similar vertical
histograms in SAS too, but
I created this vertical
histogram by typing the
frequencies of each account
length into an EXCEL
spreadsheet. You can
create similar vertical
histograms in SAS too, but
Note the impact of the
multi-cohort research
design – any account
that was opened after
1977 and remained open
longer than 6 years is a
censored case.
Note the impact of the
multi-cohort research
design – any account
that was opened after
1977 and remained open
longer than 6 years is a
censored case.
S052/II.2(a1): Introducing The Central Concepts In Classical Survival Analysis
Exploring the Account Data ANSWER TO RESEARCH QUESTION 1a
S052/II.2(a1): Introducing The Central Concepts In Classical Survival Analysis
Exploring the Account Data ANSWER TO RESEARCH QUESTION 1a

11. © Willett, Harvard University
Graduate School of Education,
03/19/14
S052/II.2(b) – Slide 11
S052/II.2(a1): Introducing The Central Concepts In Classical Survival Analysis
Exploring the Data
S052/II.2(a1): Introducing The Central Concepts In Classical Survival Analysis
Exploring the Data
Here are two hopeless strategies for dealing with censoring,
while summarizing account duration length …
Here are two hopeless strategies for dealing with censoring,
while summarizing account duration length … If we set the duration
lengths of the censored
accounts to their
longest observed career
length, the mean
account duration for all
accounts is 6.31 years.
This too is a negatively
biased estimate of true
duration even if onlyonly
one account has lastedone account has lasted
longer than thelonger than the
censored durationcensored duration.
If we set the duration
lengths of the censored
accounts to their
longest observed career
length, the mean
account duration for all
accounts is 6.31 years.
This too is a negatively
biased estimate of true
duration even if onlyonly
one account has lastedone account has lasted
longer than thelonger than the
censored durationcensored duration.
If you take the average of the duration
lengths of only the uncensored accounts,
their mean account duration is 3.73 years,
which is a negatively biased estimate of
the average population account
If you take the average of the duration
lengths of only the uncensored accounts,
their mean account duration is 3.73 years,
which is a negatively biased estimate of
the average population account
duration.

12. © Willett, Harvard University
Graduate School of Education,
03/19/14
S052/II.2(b) – Slide 12
Dataset Acct dur_PP.txt
Overview Person-period dataset containing the same information
as the Acctdur.txt person dataset, on the career duration
of accounts who began between 1972 and 1977, and
who were followed uninterruptedly until 1984.
Source Bank records.
Sample size 24875 annual person-period records.
More Info Singer & Willett, 2003
You can resolve these problems by working with your
data in a different format:
 Re-format the data into a person-period format.
 In a person-period dataset, you can estimate a
different class of summary statistics that address the
“whether” and “when” questions.
 Hazard probability.
 Survival probability.
 Median lifetime.
You can resolve these problems by working with your
data in a different format:
 Re-format the data into a person-period format.
 In a person-period dataset, you can estimate a
different class of summary statistics that address the
“whether” and “when” questions.
 Hazard probability.
 Survival probability.
 Median lifetime.
S052/II.2(a1): Introducing The Central Concepts In Classical Survival Analysis
Resolving The Problem Of Censoring By Working In A Person-Period Dataset
S052/II.2(a1): Introducing The Central Concepts In Classical Survival Analysis
Resolving The Problem Of Censoring By Working In A Person-Period Dataset
Notice that the name of the
dataset is different
Here’s a clue to the
difference between the
person-level and the person-

13. © Willett, Harvard University
Graduate School of Education,
03/19/14
S052/II.2(b) – Slide 13
Person-Level Dataset
ID acctopen CENSOR
1 1 Not censored
2 2 Not censored
3 1 Not censored
4 1 Not censored
5 12 Censored
6 1 Not censored
7 12 Censored
8 1 Not censored
9 2 Not censored
10 2 Not censored
12 7 Not censored
13 12 Censored
14 1 Not censored
15 12 Censored
16 12 Censored
Etc.
Person-Level Dataset
ID acctopen CENSOR
1 1 Not censored
2 2 Not censored
3 1 Not censored
4 1 Not censored
5 12 Censored
6 1 Not censored
7 12 Censored
8 1 Not censored
9 2 Not censored
10 2 Not censored
12 7 Not censored
13 12 Censored
14 1 Not censored
15 12 Censored
16 12 Censored
Etc.
Person-Period
Dataset
ID PERIOD EVENT
1 1 1
2 1 0
2 2 1
3 1 1
4 1 1
5 1 0
5 2 0
5 3 0
5 4 0
5 5 0
5 6 0
5 7 0
5 8 0
5 9 0
5 10 0
5 11 0
5 12 0
6 1 1
7 1 0
7 2 0
7 3 0
7 4 0
7 5 0
7 6 0
7 7 0
7 8 0
7 9 0
7 10 0
7 11 0
7 12 0
Etc.
Person-Period
Dataset
ID PERIOD EVENT
1 1 1
2 1 0
2 2 1
3 1 1
4 1 1
5 1 0
5 2 0
5 3 0
5 4 0
5 5 0
5 6 0
5 7 0
5 8 0
5 9 0
5 10 0
5 11 0
5 12 0
6 1 1
7 1 0
7 2 0
7 3 0
7 4 0
7 5 0
7 6 0
7 7 0
7 8 0
7 9 0
7 10 0
7 11 0
7 12 0
Etc.
In a person-period dataset:
• Each person has one row of data for
each time-period,
• Their data record continues until
the time-period in which they
either experience the event of
interest, or they are censored.
In a person-period dataset:
• Each person has one row of data for
each time-period,
• Their data record continues until
the time-period in which they
either experience the event of
interest, or they are censored.
S052/II.2(a1): Introducing The Central Concepts In Classical Survival Analysis
Inspecting the Person-Period Dataset
S052/II.2(a1): Introducing The Central Concepts In Classical Survival Analysis
Inspecting the Person-Period Dataset
account #2 is not censored
and so it experiences the event
of interest (i.e. closes
account ) in the 2nd
year.
account #2 is not censored
and so it experiences the event
of interest (i.e. closes
account ) in the 2nd
year.
account #7 is censored – it
never experiences the event of
interest (i.e. never closes
account ) in all the 12 years
during which accounts are
observed.
account #7 is censored – it
never experiences the event of
interest (i.e. never closes
account ) in all the 12 years
during which accounts are
observed.

14. © Willett, Harvard University
Graduate School of Education,
03/19/14
S052/II.2(b) – Slide 14
EVENT(Did Customer close Account in this Time Period?)
Frequency‚
Col Pct
‚ 1‚ 2‚ 3‚ 4‚ 5‚ 6‚
ƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒ
ˆ
No close ‚ 3485 ‚ 3101 ‚ 2742 ‚ 2447 ‚ 2229 ‚ 2045
‚
‚ 88.43 ‚ 88.98 ‚ 88.42 ‚ 89.24 ‚ 91.09 ‚ 91.75
‚
ƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒ
ˆ
close ‚ 456 ‚ 384 ‚ 359 ‚ 295 ‚ 218 ‚ 184
‚
‚ 11.57 ‚ 11.02 ‚ 11.58 ‚ 10.76 ‚ 8.91 ‚ 8.25
‚
ƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒ
ˆ Total 3941 3485 3101 2742 2447
2229
EVENT(Did Customer close Account in this Time Period?)
Frequency‚
Col Pct
‚ 1‚ 2‚ 3‚ 4‚ 5‚ 6‚
ƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒ
ˆ
No close ‚ 3485 ‚ 3101 ‚ 2742 ‚ 2447 ‚ 2229 ‚ 2045
‚
‚ 88.43 ‚ 88.98 ‚ 88.42 ‚ 89.24 ‚ 91.09 ‚ 91.75
‚
ƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒ
ˆ
close ‚ 456 ‚ 384 ‚ 359 ‚ 295 ‚ 218 ‚ 184
‚
‚ 11.57 ‚ 11.02 ‚ 11.58 ‚ 10.76 ‚ 8.91 ‚ 8.25
‚
ƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒ
ˆ Total 3941 3485 3101 2742 2447
2229
PERIOD(Current Time Period)
‚ 7‚ 8‚ 9‚ 10‚ 11‚ 12‚ Tota
l
ˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆ
‚ 1922 ‚ 1563 ‚ 1203 ‚ 913 ‚ 632 ‚ 386
‚ 22668
‚ 93.99 ‚ 95.19 ‚ 95.78 ‚ 96.31 ‚ 97.53 ‚ 98.72 ‚
ˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆ
‚ 123 ‚ 79 ‚ 53 ‚ 35 ‚ 16 ‚ 5
‚ 2207
‚ 6.01 ‚ 4.81 ‚ 4.22 ‚ 3.69 ‚ 2.47 ‚ 1.28 ‚
ˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆ
2045 1642 1256 948 648 391
24875
PERIOD(Current Time Period)
‚ 7‚ 8‚ 9‚ 10‚ 11‚ 12‚ Tota
l
ˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆ
‚ 1922 ‚ 1563 ‚ 1203 ‚ 913 ‚ 632 ‚ 386
‚ 22668
‚ 93.99 ‚ 95.19 ‚ 95.78 ‚ 96.31 ‚ 97.53 ‚ 98.72 ‚
ˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆ
‚ 123 ‚ 79 ‚ 53 ‚ 35 ‚ 16 ‚ 5
‚ 2207
‚ 6.01 ‚ 4.81 ‚ 4.22 ‚ 3.69 ‚ 2.47 ‚ 1.28 ‚
ˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆ
2045 1642 1256 948 648 391
24875
Here’s the Life Table – a Two-Way Contingency Table Analysis of EVENT by PERIOD …Here’s the Life Table – a Two-Way Contingency Table Analysis of EVENT by PERIOD …
S052/II.2(a1): Introducing The Central Concepts In Classical Survival Analysis
Beginning Of The Life Table Analysis – Estimating The Sample Hazard Probability
S052/II.2(a1): Introducing The Central Concepts In Classical Survival Analysis
Beginning Of The Life Table Analysis – Estimating The Sample Hazard Probability
We can use these frequencies to estimate a hazard probability that
describes the “risk of closing” in each time-period.
Hazard probability is the (conditional) probability that an account will
experience the event of importance (i.e., close) in a particular time-period,
given that it has “survived” up until this period.
We can use these frequencies to estimate a hazard probability that
describes the “risk of closing” in each time-period.
Hazard probability is the (conditional) probability that an account will
experience the event of importance (i.e., close) in a particular time-period,
given that it has “survived” up until this period.
In discrete time period #1, for instance:
 There are 3941 accounts “at risk of closing.” Of this “risk set of accounts,” 456 were
observed to close. Hence, the probability that an account will close in this period, given that it
entered it, is (456/3941), or 0.1157. So, the sample hazard probability in discrete time-
period #1 is
In discrete time period #1, for instance:
 There are 3941 accounts “at risk of closing.” Of this “risk set of accounts,” 456 were
observed to close. Hence, the probability that an account will close in this period, given that it
entered it, is (456/3941), or 0.1157. So, the sample hazard probability in discrete time-
period #1 is ˆh t( ) = 0.1157

15. © Willett, Harvard University
Graduate School of Education,
03/19/14
S052/II.2(b) – Slide 15
EVENT(Was account closed in this Time Period?)
Frequency‚
Col Pct
‚ 1‚ 2‚ 3‚ 4‚ 5‚ 6‚
ƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒ
ˆ
No ‚ 3485 ‚ 3101 ‚ 2742 ‚ 2447 ‚ 2229 ‚ 2045 ‚
‚ 88.43 ‚ 88.98 ‚ 88.42 ‚ 89.24 ‚ 91.09 ‚ 91.75
‚
ƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒ
ˆ
Yes ‚ 456 ‚ 384 ‚ 359 ‚ 295 ‚ 218 ‚ 184 ‚
‚ 11.57 ‚ 11.02 ‚ 11.58 ‚ 10.76 ‚ 8.91 ‚
8.25 ‚
ƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒ
ˆ Total 3941 3485 3101 2742 2447
2229
EVENT(Was account closed in this Time Period?)
Frequency‚
Col Pct
‚ 1‚ 2‚ 3‚ 4‚ 5‚ 6‚
ƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒ
ˆ
No ‚ 3485 ‚ 3101 ‚ 2742 ‚ 2447 ‚ 2229 ‚ 2045 ‚
‚ 88.43 ‚ 88.98 ‚ 88.42 ‚ 89.24 ‚ 91.09 ‚ 91.75
‚
ƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒ
ˆ
Yes ‚ 456 ‚ 384 ‚ 359 ‚ 295 ‚ 218 ‚ 184 ‚
‚ 11.57 ‚ 11.02 ‚ 11.58 ‚ 10.76 ‚ 8.91 ‚
8.25 ‚
ƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒ
ˆ Total 3941 3485 3101 2742 2447
2229
PERIOD(Current Time Period)
‚ 7‚ 8‚ 9‚ 10‚ 11‚ 12‚ Tota
l
ˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆ
‚ 1922 ‚ 1563 ‚ 1203 ‚ 913 ‚ 632 ‚ 386
‚ 22668
‚ 93.99 ‚ 95.19 ‚ 95.78 ‚ 96.31 ‚ 97.53 ‚ 98.72 ‚
ˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆ
‚ 123 ‚ 79 ‚ 53 ‚ 35 ‚ 16 ‚ 5
‚ 2207
‚ 6.01 ‚ 4.81 ‚ 4.22 ‚ 3.69 ‚ 2.47 ‚ 1.28 ‚
ˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆ
2045 1642 1256 948 648 391
24875
PERIOD(Current Time Period)
‚ 7‚ 8‚ 9‚ 10‚ 11‚ 12‚ Tota
l
ˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆ
‚ 1922 ‚ 1563 ‚ 1203 ‚ 913 ‚ 632 ‚ 386
‚ 22668
‚ 93.99 ‚ 95.19 ‚ 95.78 ‚ 96.31 ‚ 97.53 ‚ 98.72 ‚
ˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆ
‚ 123 ‚ 79 ‚ 53 ‚ 35 ‚ 16 ‚ 5
‚ 2207
‚ 6.01 ‚ 4.81 ‚ 4.22 ‚ 3.69 ‚ 2.47 ‚ 1.28 ‚
ˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆ
2045 1642 1256 948 648 391
24875
And the sample hazard probabilities for discrete time-periods #4, #5, #6 and #7…And the sample hazard probabilities for discrete time-periods #4, #5, #6 and #7…
S052/II.2(a1): Introducing The Central Concepts In Classical Survival Analysis
Continuing The Life Table Analysis – Estimating Sample Hazard Probability
S052/II.2(a1): Introducing The Central Concepts In Classical Survival Analysis
Continuing The Life Table Analysis – Estimating Sample Hazard Probability
Something different is
happening here in the Life
Table?
What is it?
Why is it occurring?
Is it a problem?
2229

16. © Willett, Harvard University
Graduate School of Education,
03/19/14
S052/II.2(b) – Slide 16
Conclusion? The hazard probability
provides the risk of closing at each
year after an account is open.
Conclusion? The hazard probability
provides the risk of closing at each
year after an account is open.
Collect the sample hazard probabilities together and plot them as a sample hazard function …Collect the sample hazard probabilities together and plot them as a sample hazard function …
S052/II.2(a1): Introducing The Central Concepts In Classical Survival Analysis
Plotting the Sample Hazard Function
S052/II.2(a1): Introducing The Central Concepts In Classical Survival Analysis
Plotting the Sample Hazard Function

17. © Willett, Harvard University
Graduate School of Education,
03/19/14
S052/II.2(b) – Slide 17
Time
Period
Sample
Hazard
Probability
h(t)
Sample
Survival
Probability
S(t)
0 1.0000
1 0.1157 0.8843
2 0.1102 0.7869
3 0.1158 0.6958
4 0.1076 0.6209
5 0.0891 0.5656
6 0.0825 0.5189
7 0.0601 0.4877
8 0.0481 0.4642
9 0.0422 0.4446
10 0.0369 0.4282
11 0.0247 0.4177
12 0.0128 0.4123
Once you have the sample hazard probabilities, you can cumulate them to get sample
survival probabilities …
Once you have the sample hazard probabilities, you can cumulate them to get sample
survival probabilities …
Sample Survival
Probability
Survival probability in
any time period is the
probability of “surviving”
beyond that period (ie, the
probability of not
experiencing the event of
interest until after the
period).
Sample Survival
Probability
Survival probability in
any time period is the
probability of “surviving”
beyond that period (ie, the
probability of not
experiencing the event of
interest until after the
period).
S052/II.2(a1): Introducing The Central Concepts In Classical Survival Analysis
Continuing The Life Table Analysis – Estimating Sample Survival Probability
S052/II.2(a1): Introducing The Central Concepts In Classical Survival Analysis
Continuing The Life Table Analysis – Estimating Sample Survival Probability
Here, all accounts survived the 0th
time period, so the estimated sample
survival probability in the 0th
period
is 1.000.
Here, all accounts survived the 0th
time period, so the estimated sample
survival probability in the 0th
period
is 1.000.
The estimated hazard
probability suggests that a
proportion of 0.1157 of
accounts in the 1st
period
risk set will “die” in the 1st
period (i.e., close).
The estimated hazard
probability suggests that a
proportion of 0.1157 of
accounts in the 1st
period
risk set will “die” in the 1st
period (i.e., close).
 Because a proportion of 0.1157
of the risk set will “die” in the 1st
period, we know that (1 - 0.1157)
or 0.8843 of the 1st
period risk set
will survive.
 In other words, 0.8843 of the
entering “1.0000” will remain
“alive” beyond the 1st
time-
period (and will therefore be
potentially available to close at
some later time).
 The sample survival probability in
the 1st
time period is therefore
0.8843 × 1.000, or:
 Because a proportion of 0.1157
of the risk set will “die” in the 1st
period, we know that (1 - 0.1157)
or 0.8843 of the 1st
period risk set
will survive.
 In other words, 0.8843 of the
entering “1.0000” will remain
“alive” beyond the 1st
time-
period (and will therefore be
potentially available to close at
some later time).
 The sample survival probability in
the 1st
time period is therefore
0.8843 × 1.000, or:
8843.0)(ˆ
1 =tS

18. © Willett, Harvard University
Graduate School of Education,
03/19/14
S052/II.2(b) – Slide 18
Time
Period
Sample
Hazard
Probability
h(t)
Sample
Survival
Probability
S(t)
0 1.0000
1 0.1157 0.8843
2 0.1102 0.7869
3 0.1158 0.6958
4 0.1076 0.6209
5 0.0891 0.5656
6 0.0825 0.5189
7 0.0601 0.4877
8 0.0481 0.4642
9 0.0422 0.4446
10 0.0369 0.4282
11 0.0247 0.4177
12 0.0128 0.4123
And, the estimated survival probability in discrete time
period #2…
And, the estimated survival probability in discrete time
period #2…
S052/II.2(a1): Introducing The Central Concepts In Classical Survival Analysis
Continuing The Life Table Analysis – Estimating Sample Survival Probability
S052/II.2(a1): Introducing The Central Concepts In Classical Survival Analysis
Continuing The Life Table Analysis – Estimating Sample Survival Probability
Here, according to the
estimated sample survival
probability, a proportion of
0.8843 of the accounts survived
the 1th
time period.
Here, according to the
estimated sample survival
probability, a proportion of
0.8843 of the accounts survived
the 1th
time period.
The estimated hazard
probability suggests that a
proportion of 0.1102 of
accounts in the 2nd
period
risk set will “die” in the 2nd
period (i.e., close).
The estimated hazard
probability suggests that a
proportion of 0.1102 of
accounts in the 2nd
period
risk set will “die” in the 2nd
period (i.e., close).
 Because a proportion of 0.1102
of the risk set will “die” in the 2nd
period, we know that (1 -
0.1102), or 0.8898, of the 2nd
period risk set will survive.
 In other words, a proportion of
0.8898 of the entering “0.8843”
will remain “alive” beyond the
2nd
time period (and be
potentially available to close
later).
 The sample survival probability
in the 2nd
time period is therefore
0.8898 × 0.8843, or:
 Because a proportion of 0.1102
of the risk set will “die” in the 2nd
period, we know that (1 -
0.1102), or 0.8898, of the 2nd
period risk set will survive.
 In other words, a proportion of
0.8898 of the entering “0.8843”
will remain “alive” beyond the
2nd
time period (and be
potentially available to close
later).
 The sample survival probability
in the 2nd
time period is therefore
0.8898 × 0.8843, or:
7869.0)(ˆ
2 =tS

19. © Willett, Harvard University
Graduate School of Education,
03/19/14
S052/II.2(b) – Slide 19
Time
Period
Sample
Hazard
Probability
h(t)
Sample
Survival
Probability
S(t)
0 1.0000
1 0.1157 0.8843
2 0.1102 0.7869
3 0.1158 0.6958
4 0.1076 0.6209
5 0.0891 0.5656
6 0.0825 0.5189
7 0.0601 0.4877
8 0.0481 0.4642
9 0.0422 0.4446
10 0.0369 0.4282
11 0.0247 0.4177
12 0.0128 0.4123
And, the estimated survival probability in discrete time period #3 … etcAnd, the estimated survival probability in discrete time period #3 … etc
S052/II.2(a1): Introducing The Central Concepts In Classical Survival Analysis
Continuing The Life Table Analysis – Estimating Sample Survival Probability
S052/II.2(a1): Introducing The Central Concepts In Classical Survival Analysis
Continuing The Life Table Analysis – Estimating Sample Survival Probability
Here, according to the
estimated sample survival
probability, a proportion of
0.7869 of the accounts
survived the 2nd
time period.
Here, according to the
estimated sample survival
probability, a proportion of
0.7869 of the accounts
survived the 2nd
time period.
The estimated hazard
probability suggests that a
proportion of 0.1158 of
accounts in the 3rd
period
risk set will “die” in the 3rd
period (i.e., close).
The estimated hazard
probability suggests that a
proportion of 0.1158 of
accounts in the 3rd
period
risk set will “die” in the 3rd
period (i.e., close).
 Because a proportion of 0.1158
of the risk set will “die” in the 3rd
period, we know that (1 -
0.1158), or 0.8842, of the 3rd
period risk set will survive.
 In other words, a proportion of
0.8842 of the entering “0.7869”
will remain “alive” beyond the
3rd
time period (and be
potentially available to close
later).
 The sample survival probability
in the 3rd time period is
therefore 0.8842 × 0.7869, or:
 Because a proportion of 0.1158
of the risk set will “die” in the 3rd
period, we know that (1 -
0.1158), or 0.8842, of the 3rd
period risk set will survive.
 In other words, a proportion of
0.8842 of the entering “0.7869”
will remain “alive” beyond the
3rd
time period (and be
potentially available to close
later).
 The sample survival probability
in the 3rd time period is
therefore 0.8842 × 0.7869, or:
6958.0)(ˆ
3 =tS

20. © Willett, Harvard University
Graduate School of Education,
03/19/14
S052/II.2(b) – Slide 20
Time
Period
Sample
Hazard
Probability
h(t)
Sample
Survival
Probability
S(t)
jt )(ˆ
jth )(ˆ
jtS
1−jt )(ˆ
1−jtS
As a general principle, the
estimated survivor probability
in any time period j can be
found by substituting into a
simple little rule …
As a general principle, the
estimated survivor probability
in any time period j can be
found by substituting into a
simple little rule …
S052/II.2(a1): Introducing The Central Concepts In Classical Survival Analysis
Simple Rule For Estimating Sample Survival Probability
S052/II.2(a1): Introducing The Central Concepts In Classical Survival Analysis
Simple Rule For Estimating Sample Survival Probability
So, in general, in any time period j ..So, in general, in any time period j ..
)(ˆ)](ˆ1[)(ˆ
1−−= jjj tSthtS

21. © Willett, Harvard University
Graduate School of Education,
03/19/14
S052/II.2(b) – Slide 21
Plotting the sample survival probabilities against time period provides the sample survivor function.Plotting the sample survival probabilities against time period provides the sample survivor function.
Typical monotonically
decreasing survivor
function …Median
lifetime survival
probability is 6.6, point at
which half of accounts are
“still alive.”
Typical monotonically
decreasing survivor
function …Median
lifetime survival
probability is 6.6, point at
which half of accounts are
“still alive.”
S052/II.2(a1): Introducing The Central Concepts In Classical Survival Analysis
Plotting the Sample Survivor Function And Estimating Median Lifetime Survivor Probability
S052/II.2(a1): Introducing The Central Concepts In Classical Survival Analysis
Plotting the Sample Survivor Function And Estimating Median Lifetime Survivor Probability

22. Research Question 2
for Next Time…
• Question 2. How can we predict core deposit interest
rates?
• A. from prime interest rate?
• B. from market interest rate?
• 1. Can we predict core deposit interest rate from 3 month
LIBOR (one index of market interest rate)?
• 2. from lagged LIBOR indices?
• 3. Are there other market interest rate indices we want to
include to predict core deposit interest rate?