1. 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

2. Announcements Assignment 2 Due Agenda: Assignment 3 handed out
Event history analysis – basic issues.

3. 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.

4. 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.

5. 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.

6. 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?

7. 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).

8. 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

9. 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”.

10. 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).

11. States, Spells, & Events: Visually
Example of mortality at month 33
End of Study
Event
Spell #2 1
Time (Months)
State
Spell #1
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)

12. States, Spells, & Events: Visually
Example of a patient who is cured
Doesn’t experience mortality during study
End of Study 1
Time (Months)
State
Spell #1
Note: Only 1 spell is needed
The spell indicates a consistent state (0), for the period of time in which we have information

13. 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

14. 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
Right Censored at 45
Spell #4 3 2 1
Age (Years)
State
Spell #1
Spell #2
Spell #3

15. 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

16. 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

17. Time Clocks Visually: Age3 2 1
Age (Years)
State
Spell #1
End of Study
Spell #4
Spell #2
Spell #3
EHA examines rate of transitions as a function of a person’s age

18. Time Clocks Visually: Duration
Single from (4 years), married from (7 years), divorced from (6 yrs), remarried at (12 yrs)
Spell #3
Spell #2
Spell #4 3 2 1
Duration (Years)
State
Spell #1
EHA examines rate of transitions as a function of a person’s duration in their current state

19. 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

20. 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.

21. Example: Employee Retention
Visually – red line indicates length of employment spell for each case:
Time (days)
Cases
Right Censored

22. 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.

23. Descriptives: Average Duration
Simply calculate the mean time-to-quitting
Average = 33.4 days
Time (days)
Cases
Right Censored

24. 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

25. Descriptives: Half Life
Determine time when ½ of sample has had event
Time (days)
Cases
Right Censored
Half Life = 23 days

26. Simple EHA Descriptives
Question: What simple things can we do to describe this sample of 12 employees?
3. Tabulate (or plot) quitters in different time-periods: e.g., 1-20 days, 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)

27. Descriptives: Tables
For each period, determine number or proportion quitting/staying
Day 1-20
20-40
40-60
60-80
80-100
Time (days)
Cases

28. EHA Descriptives: Tables
Time Range
Quitters:
Total #, %
# staying 1
Day 1-20
5 quit, 42% of all,
42% of remaining
7 left, 58 % of all 2
Day 21-40
2 quit, 16% of all
29% of remaining
5 left, 42% of all 3
Day 41-60
1 quit, 8% of all
20% of remaining
4 left, 33 % of all 4
Day 61-80
25% of remaining
3 left, 25% of all

29. 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.

30. EHA Descriptives: Plots
We can also plot tabular information:

31. 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

32. Survivor Function: S(t)
McDonald’s Example:
Steep decreases indicate lots of quitting at around 20 days

33. 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.

34. Survivor Ex: First Marriage
Compare survivor for women, men:
Survivor plot for Men (declines later)
Survivor plot for Women (declines earlier)

35. 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:
Think of it as: the rate of events occurring for those at risk of experiencing the event

36. High (and wide) peaks indicate lots of quitting
The Hazard Function
Example:
High (and wide) peaks indicate lots of quitting

37. 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.

38. Hazard Plot: First Marriage
Hazard Rate: Full Sample

39. 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

40. Cumulative Hazard Function: H(t)
Example:
“Flat” areas indicate low hazard rate
Steep increases indicate peaks in hazard rate

41. 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.

42. Integrated Hazard: First Marriage
Compare Integrated Hazard for women, men:
Integrated Hazard for men increases slower (and remains lower) than women

43. Cumulative Hazard Example
Ex: Edelman et al. 1999: EEOC Grievance procedures

44. 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.