1. Event History Analysis 3
Sociology 8811 Lecture 17
Copyright © 2007 by Evan Schofer
Do not copy or distribute without permission

2. Announcements Topic: More Event History Analysis
Models, data structures, etc
Note: We have fallen a bit behind schedule
Lectures = slightly behind the reading assignments
I’ll try to catch up
But, you can adjust your reading efforts accordingly.

3. Review: EHA
In essence, EHA models a dependent variable that reflects both:
1. Whether or not a patient experiences mortality
2. When it occurs (like a OLS regression of duration
Dependent variable is best conceptualized as a rate of some occurrence
EHA involves both descriptive and parametric analysis of data

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

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

6. Hazard Plot: Marriage
Hazard Rate: Full Sample

7. Hazard Plot: Marriage
Smoothed Hazard Rate: Full Sample

8. From Plots to Tests to Models
It appears from the plots that women get married faster than men
Issue:
How do we test hypotheses about the difference in rates?
Can we be confident that the observed difference between men and women is not merely due to sampling variability?

9. Tests of Equality for Survivor Fns
Idea: Conduct a hypothesis test to see if survivor functions differ across groups
Like a t-test for difference in means…
Example: Log-Rank Test
Based on calculating the expected # failures at each point in time if there were no difference between groups
Then, compute difference between observed failures and expected value for each group
Analogous to a chi-square test of independence for a crosstab.

10. Log-rank Test Example: Do women marry earlier than men?
. sts test sex, logrank
failure _d: married == 1
analysis time _t: endtime
Log-rank test for equality of survivor functions
| Events Events
sex | observed expected
1 |
2 |
Total |
chi2(1) =
Pr>chi2 =
Significant Chi-square (p<.05) indicates that survivor plots differ

11. Tests of Equality for Survivor Fns
Stata offers a variety of tests
They mainly differ by how they weight cases
ex: some place greater weight on early failures
Tests available in Stata
Log rank, Wilcoxon, Tarone-Ware, Peto-Peto-Prentice
See Stata manual “Survivor Analysis & Epidemiological Tables” for advice about which to use
In many cases, the results are similar across tests
Also: Cox test
Based on a different principle
Can be used with weighted data (“pweights”).

12. EHA Models Strategy: Model the hazard rate as a function of covariates
Much like regression analysis
Determine coefficients
The extent to which change in independent variables results in a change in the hazard rate
Use information from sample to compute t-values (and p-values)
Test hypotheses about coefficients

13. EHA Models
Issue: In standard regression, we must choose a proper “functional form” relating X’s to Y’s
OLS is a “linear” model – assumes a liner relationship
e.g.: Y = a + b1X1 + b2X2 … + bnXn + e
Logistic regression for discrete dependent variables – assumes an ‘S-curve’ relationship between variables
When modeling the hazard rate h(t) over time, what relationship should we assume?
There are many options: assume a flat hazard, or various S-shaped, U-shaped, or J-shaped curves
We’ll discuss details later…

14. Constant Rate Models
The simplest parametric EHA model assumes that the base hazard rate is generally “flat” over time
Any observed changes are due to changed covariates
Called a “Constant Rate” or “Exponential” model
Note: assumption of constant rate isn’t always tenable
Formula:
Usually rewritten as:

15. Constant Rate Models
Question: Is the constant rate assumption tenable?

16. Constant Rate Models
Question: Is the constant rate assumption tenable?
Answer: Probably not
The hazard rate goes up and down over time
Not constant at all – even if smoothed
2. The change over time isn’t likely the result of changing covariates (X’s) in our model
However, if the change was merely the result of some independent variable, then the underlying (unobserved) rate might, in fact, be constant.

17. Constant Rate Models Let’s run an analysis anyway…
Ignore the violation of assumptions regarding the functional form of the hazard rate
Recall -- Constant rate model is:
In this case, we’ll only specify one X var:
DFEMALE – dummy variable indicating women
Coefficient reflects difference in hazard rate for women versus men.

18. Constant Rate Model: Marriage
A simple one-variable model comparing gender
. streg sex, dist(exponential) nohr
No. of subjects = Number of obs =
No. of failures =
Time at risk =
LR chi2(1) =
Log likelihood = Prob > chi2 =
_t | Coef. Std. Err z P>|z| [95% Conf. Interval]
Dfemale |
_cons |
The positive coefficient for DFemale indicates a higher hazard rate for women

19. Constant Rate Coefficients
Interpreting the EHA coefficient: b = .19
Coefficients reflect change in log of the hazard
Recall one of the ways to write the formula:
But – we aren’t interested in log rates
We’re interested in change in the actual rate
Solution: Exponentiate the coefficient
i.e., use “inverse-log” function on calculator
Result reflects the impact on the actual rate.

20. Constant Rate Coefficients
Exponentiate the coefficient to generate the “hazard ratio”
Multiplying by the hazard ratio indicates the increase in hazard rate for each unit increase in the independent variable
Multiplying by 1.21 results in a 21% increase
A hazard ratio of 2.00 = a 100% increase
A hazard ratio of .25 = a decreased rate by 75%.

21. Constant Rate Coefficients
The variable FEMALE is a dummy variable
Women = 1, Men = 0
Increase from 0 to 1 (men to women) reflects a 21% increase in the hazard rate
Continuous measures, however can change by many points (e.g., Firm size, age, etc.)
To determine effects of multiple point increases (e.g., firm size of 10 vs. 7) multiply repeatedly
Ex: Hazard Ratio = .95, increase = 3 units:
.95 x .95 x .95 = .86 – indicating a 14% decrease.

22. Hypothesis Tests: Marriage
Final issue: Is the 21% higher hazard rate for women significantly different than men?
Or is the observed difference likely due to chance?
Solution: Hazard rate models calculate standard errors for coefficient estimates
Allowing calculation of T-values, P-values
_t | Coef. Std. Err t P>|t|
Female |
_cons |

23. Types of EHA Models Two main types of proportional EHA Models
1. Parametric Models
specify a functional form of h(t)
Constant rate is one example
Also: Piecewise Exponential, Gompertz, Weibull,etc.
2. Cox Models
Doesn’t specify a particular form for h(t)
Each makes assumptions
Like OLS assumptions regarding functional form, error variance, normality, etc
If assumptions are violated, models can’t be trusted.

24. Parametric Models
These models make assumptions about the overall shape of the hazard rate over time
Much like OLS regression assumes a linear relationship between X and Y, logit assumes s-curve
Options: constant, Gompertz, Weibull
There is a piecewise exponential option, too
Note: They also make standard statistical assumptions:
Independent random sample
Properly specified model, etc, etc…

25. Cox Models The basic Cox model: Where h(t) is the hazard rate
h0(t) is some baseline hazard function (to be inferred from the data)
This obviates the need for building a specific functional form into the model
bX’s are coefficients and covariates

26. Cox Model Assumptions
Cox Models assume that independent variables don’t interact with time
At lease, not in ways you haven’t controlled for
i.e., that the hazard rate at different values of X are proportional (parallel) to each other over time
Example: Marriage rate – women vs. men
Women have a higher rate at all points in time
Question: Does the hazard rate for women diverge or converge with men over time?
If so, the proportion (or ratio) of the rate changes. The assumption is violated. Use a different model

27. Cox Model Assumptions:
Proportionality: Look for parallel h(t)’s for different sub-groups (values of X’s)
h(t)
time
Good
Women
Men
h(t)
Bad
Women
Men

28. Cox Model Assumptions:
Hazard rates are often too spiky to discern trends
Options:
1. Smooth the hazard plots OR
2. Check the integrated hazard rate
Look for differences in the overall shape of the curve
Note: divergence is OK on an integrated hazard

29. Cox Model: Example Marriage example:
No. of subjects = Number of obs = 29269
No. of failures = Time at risk =
LR chi2(1) =
Log likelihood = Prob > chi2 =
_t | Coef. Std. Err z P>|z|
Female |

30. Reading Discussion
Frank, David J., Ann M. Hironaka, and Evan Schofer “The Nation State and the Natural Environment, ” American Sociological Review, 65 (Feb):