EHA: Terminology and basic
non-parametric graphs
Sociology 229 Advanced Regression
Class 4
Copyright © 2010 by Evan Schofer
Do not copy or distribute without permission
Announcements
• Assignment 2 Due
• Assignment 3 handed out
• Agenda:
• Event history analysis – basic issues.
Review: Why we need EHA
• Example: Drug dosage and mortality
• Question: What are the limits of using OLS
regression to model time-to-mortality?
– Answer:
• Censoring: some patients don’t die
• Violation of normality assumptions: outcome variable
is not normal
– This also causes issues for “censored normal regression”
– Question: What about Logistic Regression?
• Answer: Fails to utilize information on timing.
Motivation
• Event history analysis is more than just a “fix”
for censoring and violations of normality…
– EHA concepts and data structures put “dynamic”
processes at the foreground
• In short, EHA helps us think about how time matters.
EHA: Overview and Terminology
• EHA is referred to as “dynamic” modeling
• i.e., addresses the timing of outcomes: rates
• Dependent variable is best conceptualized
as a rate of some occurrence
• Not a “level” or “amount” as in OLS regression
• Think: “How fast?” “How often?”
• The “occurrence” may be something that can
occur only once for each case: e.g., mortality
• Or, it may be repeatable: e.g., marriages, strategic
alliances.
EHA: Types of Questions
• Some types of questions EHA can address:
• 1. Mortality: Does drug dosage reduce rates?
• Does “rate” decrease with larger doses?
• Also: control for race, gender, treatment options, etc
• 2. Life stage transitions: timing of marriage
• Is rate affected by gender, class, religion?
• 3. Organizational mortality
• Is rate affected by size, historical era, competition?
• 4. Inter-state war
• Is rate affected by economic, political factors?
EHA: Overview
• EHA involves both descriptive and
parametric analysis of data
• Just like regression:
• Scatterplots, partialplots = descriptive
• OLS model/hypothesis tests = parametric
– Descriptive analyses/plots
• Allow description of the overall rate of some outcome
• For all cases, or for various subgroups
– Parametric Models
• Allow hypothesis testing about variables that affect
rate (and can include control variables).
EHA Terminology: States & Events
• EHA has evolved its own terminology:
• “State” = the “state of being” of a case
• Conceptualized in terms of discrete phenomena
• e.g., alive vs. dead
• “State space” = the set of all possible states
• Can be complex: Single, married, divorced, widowed
• “Event” = Occurrence of the outcome
• Also called “transition”, “failure”
• Shift from “alive” to “dead”, “single” to “married”
• Occurs at a specific, known point in time
Terminology: Risk & Spells
• “Risk Set” = the set of all cases capable of
experiencing the event
• e.g., those “at risk” of experiencing mortality
• Note: the risk often changes over time
– Shrinks as cases experience events
– Or grows, if new cases enter the study
• “Spell” = A chunk of time that a case
experiences, bounded by: events, and/or the
start or end of the study
• As in “I’m gonna sit here for a spell…”
• Sometimes called a “duration”.
States, Spells, & Events: Visually
• If we assign numeric values to states, it is
easy to graph cases over time
• As they experience 1 or more spells
• Example: drug & mortality study
• States:
• Alive = 0
• Dead = 1
• Time = measured in months
• Starting at zero, when the study begins
• Ending at 60 months, when study ends (5 years).
States, Spells, & Events: Visually
• Example of mortality at month 33
Event
End of
Study
State
Spell #2
1
Spell #1
0
0
10
20
30
40
Time (Months)
50
60
• Note: It takes 2 spells to describe this case
– But, we may only be interested in the first spell. (Because there is no
possibility of change after transition to state = 1)
States, Spells, & Events: Visually
• Example of a patient who is cured
– Doesn’t experience mortality during study
State
End of
Study
1
Spell #1
0
0
10
20
30
40
Time (Months)
50
60
• Note: Only 1 spell is needed
– The spell indicates a consistent state (0), for the
period of time in which we have information
More Terminology: Censoring
• Note: In both cases, data runs out after
month 60
• Even if the patient is still alive
• In temporal analysis, we rarely have data for
all relevant time for all cases
• “Censored” = indicates the absence of data
before or after a certain point in time
• As in: “data on cases is censored at 60 months”
• “Right Censored” = no data after a time point
• “Left Censored” = no data before a time point
States, Spells, & Events: Visually
• A more complex state space: marital status
• 0 = single, 1 = married, 2 = divorced, 3 = widowed
• Individual history:
• Married at 20, divorced at 27, remarried at 33
3 Spell #1
Spell #2
Spell #3
Spell #4
State
2
Right
Censored
at 45
1
0
16
20
24
28
Age (Years)
32
36
40
44
Measuring States and Times
• EHA, in short, is the analysis of spells
• It takes into account the duration of spells, and
whether or not there was a change of state at the end
• States at start and end of spell are measured
by assigning pre-defined values to a variable
• Much like logit/probit or multinomial logit
• Times at the start and end of spell must also
be measured
• Time Unit = The time metric in the study
• e.g., minutes, hours, days, months, years, etc
Time Clock
• Time Clock = time reference of the analysis
• Possibilities:
• Duration since start of study
• Chronological age of case (person, firm, country)
• Duration since end of last spell
– i.e., clock is set to zero at start of each spell
• Historical time – the actual calendar date
• The choice of time-clock can radically
change the analysis and meaning of results
• It is crucial to choose a clock that makes sense for the
hypotheses you wish to test
Time Clocks Visually: Age
3 Spell #1
Spell #2
Spell #3
Spell #4
End of
Study
State
2
1
0
16
20
24
28
Age (Years)
32
36
40
44
• EHA examines rate of transitions as a function of
a person’s age
Time Clocks Visually: Duration
Single from 16-20 (4 years), married from 20-27 (7 years),
divorced from 27-33 (6 yrs), remarried at 33-45 (12 yrs)
Spell #3
Spell #2
Spell #4
3 Spell #1
State
2
1
0
0
4
6
12
Duration (Years)
18
22
• EHA examines rate of transitions as a function
of a person’s duration in their current state
Time Clocks: General Advice
• Different time-clocks have different strengths
• We’ll discuss this more…
• Chronological Age = good for processes
clearly linked to age
• Biological things: fertility, mortality
• Liability of newness
• Historical time = useful for examining the
impact of historical change on ongoing
phenomena
• E.g., effects of changing regulatory regimes on rates
of strategic alliances
Moving Toward Analyses: Example
• Example: Employee retention
• How long after hiring before employees quit?
• Data: Sample of 12 employees at McDonalds
• Time-Clock/Time Unit: duration of employment
from time of hiring (measured in days)
• 2 Possible states:
• Employed & No longer employed
• We are uninterested in subsequent hires
• Therefore, we focus on initial spell, ending in quitting.
Example: Employee Retention
• Visually – red line indicates length of
employment spell for each case:
Right
Cases
Censored
0
20
40
60
80
Time (days)
100
120
Simple EHA Descriptives
• Question: What simple things can we do to
describe this sample of 12 employees?
• 1. Average duration of employment
• Only works if all (or nearly all) have quit
• Many censored cases make “average” meaningless
– This is a fairly useful summary statistic
• Gives a sense of overall speed of events
• Especially useful when broken down by sub-groups
• e.g., average by gender or compensation plan.
Descriptives: Average Duration
• Simply calculate the mean time-to-quitting
Average =
33.4 days
Cases
Right
Censored
0
20
40
60
80
Time (days)
100
120
Simple EHA Descriptives
• Question: What simple things can we do to
describe this sample of 12 employees?
• 2. Compute “Half Life” of employee tenure
– i.e., median failure time… a better option than “mean”
• Determine time at which attrition equals 50%
• Also highlights the overall turnover rate
• Note: Exact value is calculable, even if there are
censored cases
• Again, computing for sub-groups is useful
Descriptives: Half Life
• Determine time when ½ of sample has had
event
Right
Censored
Cases
Half Life = 23 days
0
20
40
60
80
Time (days)
100
120
Simple EHA Descriptives
• Question: What simple things can we do to
describe this sample of 12 employees?
• 3. Tabulate (or plot) quitters in different timeperiods: e.g., 1-20 days, 21-40 days, etc.
• Absolute numbers of “quitters” or “stayers”
– or
• Numbers of quitters as a proportion of “stayers”
• Or look at number (or proportion) who have “survived”
(i.e., not quit)
Descriptives: Tables
• For each period, determine number or
proportion quitting/staying
20-40
40-60
60-80 80-100
Cases
Day 1-20
0
20
40
60
80
Time (days)
100
120
EHA Descriptives: Tables
Time
Range
1 Day 1-20
2 Day 21-40
3 Day 41-60
4 Day 61-80
Quitters:
Total #, %
5 quit, 42% of all,
42% of remaining
2 quit, 16% of all
29% of remaining
1 quit, 8% of all
20% of remaining
1 quit, 8% of all
25% of remaining
# staying
7 left, 58 % of all
5 left, 42% of all
4 left, 33 % of all
3 left, 25% of all
EHA Descriptives: Tables
• Remarks on EHA tables:
• 1. Results of tables change depending on
time-ranges chosen (like a histogram)
• E.g., comparing 20-day ranges vs. 10-day ranges
• 2. % quitters vs. % quitters as a proportion
of those still employed
• Absolute % can be misleading since the number of
people left in the risk set tends to decrease
• A low # of quitters can actually correspond to a very
high rate of quitting for those remaining in the firm
• Typically, these ratios are more socially meaningful
than raw percentages.
EHA Descriptives: Plots
• We can also plot tabular information:
100
% Quit (of Remaining)
% Remaining
90
80
Percent
70
60
50
40
30
20
10
0
0
1
2
3
Time Period
4
5
The Survivor Function: S(t)
• A more sophisticated version of % remaining
• Calculated based on continuous time (calculus), rather
than based on some arbitrary interval (e.g., day 1-20)
• Survivor Function – S(t): The probability (at
time = t) of not having the event prior to time t.
• Always equal to 1 at time = 0 (when no events can have
happened yet
• Decreases as more cases experience the event
• When graphed, it is typically a decreasing curve
• Looks a lot like % remaining
Survivor Function: S(t)
• McDonald’s Example:
Survivor Function: McDonalds Employees
1
0.9
Steep decreases
indicate lots of
quitting at
around 20 days
0.8
0.7
S(t)
0.6
0.5
0.4
0.3
0.2
0.1
0
0
20
40
60
Time
80
100
120
Survivor Function: S(t)
• Interpretation: The survivor function reflects
the probability of surviving beyond time t
• A monotone, non-increasing function of time
• Always starts at 1, decreases as cases experience
events
• Let’s try to draw some possible survivor
functions
•
•
•
•
For human mortality
For the failure of a computer hard-drive
For having a (first) baby
For large US cities having major protests in the civil
rights movement.
Survivor Ex: First Marriage
• Compare survivor for women, men:
Kaplan-Meier survival estimates, by dfem
1.00
Survivor plot
for Men
(declines later)
0.75
Survivor plot
for Women
(declines earlier)
0.50
0.25
dfem 0
dfem 1
0.00
0
50
analysis time
100
The Hazard Function: h(t)
• A more sophisticated version of # events
divided by # remaining
• Hazard Function – h(t) = The probability of
an event occurring at a given point in time,
given that it hasn’t already occurred
• Formula:
P(t  t  T  t T  t )
h(t )  lim
t 0
t
• Think of it as: the rate of events occurring for
those at risk of experiencing the event
The Hazard Function
• Example:
McDonalds Employees: Hazard Rate
0.12
High (and wide)
peaks indicate
lots of quitting
0.10
h(t)
0.08
0.06
0.04
0.02
0.00
0.00
10.00
20.00
30.00
40.00
Time
50.00
60.00
70.00
80.00
The Hazard Function: h(t)
• Interpretation: The hazard function reflects
the rate of events at a given point in time
• For cases that made it that far…
• It reflects the “rate that risk is accumulating”
• Let’s draw some hazard functions
•
•
•
•
For human mortality
For the failure of a computer hard-drive
For having a (first) baby
For large US cities having major protests in the civil
rights movement.
Hazard Plot: First Marriage
• Hazard Rate: Full Sample
Estimated Hazard Rate
Figure 3. Estimated hazard rate
of entry into first marriage for entire sample
12
20
30
40
50
60
70
80
.2
.2
.15
.15
.1
.1
.05
.05
0
0
12
20
30
40
50
Age in Years
60
70
80
Cumulative Hazard Function: H(t)
• The “cumulative” or “integrated” hazard
• Use calculus to “integrate” the hazard function
• Recall – An integral represents the area under the
curve of another function between 0 and t
– Hazard is a rate, like “60 miles per hour”
• Integrated hazard is total distance driven…
• In three hours, it would be 180 miles
– Integrated hazard functions always increase
(opposite of the survivor function).
• Big increases indicates that the hazard is high
Cumulative Hazard Function: H(t)
• Example:
McDonalds Employees: Integrated Hazard
“Flat” areas
indicate low
hazard rate
1.8
1.6
Integrated Hazard
1.4
1.2
1
0.8
Steep increases
indicate peaks in
hazard rate
0.6
0.4
0.2
0
0
20
40
60
Time
80
100
The Cumulative Hazard: H(t)
• Interpretation: The cumulative hazard
function reflects the total amount of risk that
has accumulated at a given point in time…
• Let’s draw some integrated hazard functions
•
•
•
•
For human mortality
For the failure of a computer hard-drive
For having a (first) baby
For large US cities having major protests in the civil
rights movement.
Integrated Hazard: First Marriage
• Compare Integrated Hazard for women, men:
Nelson-Aalen cumulative hazard estimates, by dfem
3.00
df em 1
2.00
df em 0
Integrated Hazard for
men increases slower
(and remains lower)
than women
1.00
0.00
0
50
analysis time
100
Cumulative Hazard Example
• Ex: Edelman et al. 1999: EEOC Grievance procedures
EHA Plots: Remarks
• Plotting EHA data is extremely useful
•
•
•
•
Helps you understand your data
Helps you figure out the correct time-clock
Helps you to develop arguments about dynamics
Allows you to compare different groups
– We’ll pick this up in the future.