1. Event History Analysis 7
Sociology 8811 Lecture 21
Copyright © 2007 by Evan Schofer
Do not copy or distribute without permission

2. Announcements Paper Assignment #2 Due April 26 Class topic:
Try to find a dataset soon
Class topic:
Parametric EHA models; diagnostics
Later (if time allows): AFT models, discrete time models

3. Parametric Proportional Hazard Models
Cox models do not specify a functional form for the hazard curve, h(t)
Rather, they examine effects of variables net of a baseline hazard trend (to be inferred from the data)
h(t) = h0(t)ebX = h0(t)exp(bX)
Parametric models specify the general shape of the hazard curve
Approach is more familiar – more like regression
We can model Y as a constant, a linear function, a logit function, a binomial function (poisson), etc
For instance, we could assume h(t) was a linear
Then solve for values of a hazard slope that best fit the data (plus effects of other covariates on hazard rate).

4. Parametric Proportional Hazard Models
Parametric models work best when you choose a curve that fits the data
Just like OLS regression – which works best when the relationship between two variables is roughly linear
If the actual relationship between two variables is non-linear, coefficient estimates may be incorrect
Though sometimes one can transform variables (e.g., logging them) to get a good fit…
Parametric models are more efficient than Cox models
They can generate more precise estimates for a given sample size
But, they can also be more wildly incorrect if you mis-specify h(t)!
Note: These are proportional hazard models – like Cox!
You must still check the proportional hazard assumption.

5. Exponential (Constant Rate) Model
Exponential models are simplest:
Note that there is no “t” in the equation… no coefficient that specifies time dependence of the hazard rate
Rather, there are just exponentiated BXs
PLUS: a, the constant
Note 2: Box-Steffensmeier & Jones: h(t)=e-(bX)
An exponential model solves for the constant value (a) that best fits the data…
Along with values of Bs, which reflect effects of X vars
In effect, the model assumes a constant hazard rate .

6. Exponential (Constant Rate) Model
Another way of looking at it: An exponential model is a lot like a cox model
But, with the assumption that the baseline hazard is a constant!
Exponential
Cox

7. Exponential (Constant Rate) Model
Basic Model. Constant reflects base rate
. streg gdp degradation education democracy ngo ingo, dist(exponential) nohr
Exponential regression -- log relative-hazard form
No. of subjects = Number of obs =
No. of failures =
Time at risk =
Wald chi2(6) =
Log pseudolikelihood = Prob > chi2 =
| Robust
_t | Coef. Std. Err z P>|z| [95% Conf. Interval]
gdp |
degradation |
education |
democracy |
ngo |
ingo |
_cons |
Constant shows base hazard rate estimated from data:
exp(-4.57) = .01

8. Exponential (Constant Rate) Model
Suppose we plotted the baseline hazard rate estimated from our exponential model
It would be a flat line: h(t) = .01
This is the estimated hazard if all X vars are zero
If we plotted the estimated hazard for some values of X (ex: democracy = 10), we would get a higher value
Since democracy has a positive effect, Democ = 10 would yield a higher hazard than democ = 0
But, again, the estimated hazard rate trend would be a flat line over time…

9. Exponential Model: Baseline Hazard
Ex: stcurve, hazard
See, the estimated baseline hazard really is flat!

10. Exponential Model: Estimated Hazard
stcurve, hazard at1(democ=1) at2(democ=10)
Here are estimated hazards for 2 groups
Other vars pegged at mean

11. Exponential Model: Baseline Hazard
Issue: Actual hazard is rising. A problem?
Is an exponential model appropriate?
Answer:
It can be, IF we have X variables that account for increasing hazard
If not, fit will be poor!

12. Exponential (Constant Rate) Model
Cleves et al. 2004, p. 216:
In the exponential model, h(t) being constant means that the failure rate is independent of time, and thus the failure process is said to lack memory.
You may be tempted to view exponential regression as suitable for use only in the simplest of cases. This would be unfair. There is another sense in which the exponential model is the basis for all other models.
The baseline hazard… is constant … the way in which the overall hazard varies is purely a function of bX. The overall hazard need not be constant with time; it is just that every bit of how the hazard varies must be specified in BX. If you fully understand a process, you should be able to do that.
When you do not understand a process, you are forced to assign a role to time, and in that way, you hope, put to the side your ignorance and still describe the part of the process that you do understand.
In addition, exponential models can be used to model the overall hazard as a function of time, if they include t or functions of t as covariates.

13. Exponential (Constant Rate) Model
The exponential model is extremely flexible…
You specify substantive covariates (X variables) to explain failures
It is probably not due to some inherent feature of time, but rather due to some variable that you hope to control for
If you do a great job, you will fully explain why hazard rate appears to go up (or down) over time
And, you can include functions of time as independent variables to address temporal variation
Independent (X) variable scan include time dummies, log time, linear time, time interactions, etc
That is, if you can’t explain time variation with substantive X variables, you can add time variables to model it
But, if you mis-specify your model, results will be biased
In that case, you might be better off with a Cox model…

14. Piecewise Exponential Model
If you have a lot of cases, you can estimate a piecewise model
Essentially a separate model for different chunks of time
Model will yield different coefficients and base rate (constant) for multiple chunks of time
Even if hazard is not constant over time, it may be more or less constant in each period
This allows you to effectively model any hazard trend
A related approach: Put in time-period dummies
This gives a single set of bX coefficient estimates
But, allows you to specify changes in the hazard rate over different periods
NOTE: Don’t forget to omit one of the time dummies!

15. Parametric Models Let’s try a more complex parametric model
Example: Let’s specify a linear time trend
Exponential
Linear
In this case, we estimate a constant (a) and slope (b0) which best summarize the time dependence of the hazard rate
Note: this isn’t common – we have better options…

16. Gompertz Models Another option: an exponentiated line
Rather than a linear function of time and exponentiated function of bX, we’ll exponentiate everything:
Exponentiated Linear: Gompertz
Slope coefficient is often represented by gamma: g
Note: Exponentiation alters the line… it isn’t a simple linear function anymore.
It is flat if gamma = 0
It is monotonically increasing if gamma > 0
It is monotonically decreasing if gamma < 0

17. Gompertz Models
Exponentiating a linear function generates a curve defined by the value of gamma (g)
Model estimates value of g that best fits the data
g = 0
g < 0
g > 0
g >> 0

18. Gompertz Model
Example: streg gdp degradation education democracy ngo ingo, robust nohr dist(gompertz)
Gompertz regression -- log relative-hazard form
No. of subjects = Number of obs =
No. of failures =
Time at risk =
Wald chi2(6) =
Log pseudolikelihood = Prob > chi2 =
_t | Coef. Std. Err z P>|z| [95% Conf. Interval]
gdp |
degradation |
education |
democracy |
ngo |
ingo |
_cons |
gamma |
Model estimates gamma to be positive, significant. Implies increasing baseline hazard

19. Gompertz Model: Estimated Hazard
stcurve, hazard at1(democ=1) at2(democ=10)
Estimated hazards for 2 groups
Other vars pegged at mean
Note: curves are actually proportional – hard to see because bottom curve is nearly zero…

20. Weibull Models Another option: the Weibull curve
Another curve that can fit monatonic hazards
Weibull
Model estimates p to best fit the model
Hazard is flat if p = 1
Hazard is monotonically increasing if p > 1
Hazard is monotonically decreasing if p < 1.

21. Weibull: Visually
The Weibull family: Monotonic increasing or decreasing, depending on p
Time
Hazard Rate
p = 1
p = 4
p = .5
p = 2

22. Weibull Model
Example: streg gdp degradation education democracy ngo ingo, robust nohr dist(weibull)
Weibull regression -- log relative-hazard form
No. of subjects = Number of obs =
No. of failures =
Time at risk =
LR chi2(6) =
Log likelihood = Prob > chi2 =
_t | Coef. Std. Err z P>|z| [95% Conf. Interval]
gdp |
degradation |
education |
democracy |
ngo |
ingo |
_cons |
/ln_p |
p |
1/p |

23. Ancillary Parameters
Gompertz & Weibull models have parameters that determine the shape of the curve
Gamma (g), p
Ex: Bigger g = greater increase of h(t) over time
You can actually specify covariate effects on those parameters
Effectively allowing a different curve shape across values of X variables
Ex: If you think that hazard increases more for men than women, you can look to see if Dmale affects g
streg male educ, dist(gompertz) ancillary(male)
Model estimates effect of male on hazard AND on gamma…

24. Parametric: Model Fit
Parametric models use maximum likelihood estimation (MLE)
Comparisons among nested models can be made using a likelihood ratio test (LR test)
Just like logit: Addition of groups of variables can be tested with lrtest
Some parametric models are themselves nested
Ex: A Weibull model simplifies to an exponential model if p = 1
Thus, exponential is nested within Wiebull
LR tests can be used to see if Weibull is preferable to exponential.

25. Parametric: Model Fit
Parametric models use maximum likelihood estimation (MLE)
Comparisons among nested models can be made using a likelihood ratio test (LR test)
Just like logit: Addition of groups of variables can be tested with lrtest
Some parametric models are themselves nested
Ex: A Weibull model simplifies to an exponential model if p = 1
Thus, exponential is nested within Wiebull
LR tests can be used to see if Weibull is preferable to exponential.

26. Parametric Model Fit: AIC
Non-nested parametric models can be compared via the Akaike Information Criterion
k = # independent variables in the model
c = # shape parameters in model (ex: p in Weibull)
Exponential has one parameter (a); Weibull has 2.
AIC compares likelihoods, but corrects for parameters in the model – rewarding simpler models…
Low values = better model fit
Even for negative values… is better than -50.

27. Frailty Two kinds of models: Shared Frailty – a “random effects” model
Useful for clustered data (non-independent cases)
Can be used with Cox & parametric models
We’ll discuss this in detail in coming weeks
Unshared Frailty
Models for “unobserved heterogeneity”
Only available for parametric models
Refers to individual-specific (unknown) characteristics that affect likelihood of failure.

28. Unobserved Heterogeneity
Unobserved heterogeneity = differences among cases in risk set that affect failure
Think of it as “omitted variable bias”
Example: Effect of drug on mortality
Question: What half of the patients are smokers but you didn’t know that?
An “unobserved” attribute that makes them different
Answer: The smokers and non-smokers might have very different hazard rates…
But, you wouldn’t know to control for this…

29. Unobserved Heterogeneity
The observed hazard rate is modeled w/o controlling for the cause of the drop off…
Observed h(t)
Visually:
Time (months)
Hazard Rate
Smokers die early… exhausting the sample. Then h(t) drops off
Smokers
Non-Smokers

30. Unobserved Heterogeneity
Result of unobserved heterogeneity:
1. Bias in the effects of covariates
Due to “uncontrolled antecedents” (Yamaguchi 1991)
2. Problems estimating duration effects
Because some leave the risk set early, resulting in a “depressed” rate later on
Evidence of decline in hazard rate may be misleading.

31. Unobserved Heterogeneity
Strategies:
1. Develop fully-specified models
The best solution
2. Specify the form of the heterogeneity (frailty)
Approach: assume unobserved alpha (a) – case-specific factor that makes events more (or less) likely
Frailty Model
Where h(t) is some familiar model (ex: Weibull)
Requires functional form assumptions to estimate
Ex: Assume a is gamma (or inv gaussian) distributed…

32. PH Assumption & Outliers
Models discussed today are proportional hazard models…
Require the same assumption as Cox models
But, most of the “tests” of proportionality are only available in Cox models
But: You can still use piecewise models and interaction terms to check the assumption
Cumulative Cox-Snell residuals can be used to identify outliers
Use “predict”: predict ccs, ccsnell
Then, plot residuals by case ID, time, etc.

33. Parametric Models: Outliers
Cumulative Cox-Snell residuals vs case ID
Note that Scandinavia has highest residuals
Probably not outliers, but interesting nevertheless

34. Accelerated Failure Time Models
An alternative approach: model log time
Using parametric approach like exponential or Weibull
Focus is time rather than hazard rate
But, models are similar to hazard rate models – just in a different “metric”
Where last term “e” is assumed to have a distribution that defines the model (e.g., making it Weibull)
AFT models aren’t very common in sociology
But, don’t be intimidated by them… they are similar to parametric proportional hazard models…
But some software presents coefficient signs that are opposite!

35. Discrete Time EHA Models
Another completely different approach to EHA
Described in Yamaguchi reading
Break time into discrete chunks (ex: months, years)
Model dichotomous outcome (event vs. non-event) for all chunks of time
Allows use of simple model, like logit
Other common discrete time models: Probit, complementary log log models (“cloglog”)
Data structure is similar to what we did for time-varying covariates, but…
All records must cover the same length of time
Logit models don’t weight cases based on start/end time
Instead, time in analysis is represented simply by the number of cases.

36. Choosing a Hazard Model
A Cox model is a good starting point
Less problems due to accidental mis-specification of the time-dependence of the hazard rate
Box-Steffensmeier & Jones point to cites: Cox models are 95% as efficient as parametric models under many circumstances
Cox models treat time dependence as a “nuisance”, put the focus on substantive covariates
Which is often desirable.

37. Choosing a Hazard Model
Parametric models are good when
1. You have strong theoretical expectations about the hazard rate
2. You are confident that you can fit the time dependence well with a parametric model
3. You need the most efficient estimates possible
AGAIN: Substantive model specification is typically more important
Biases due to omitted variables are often greater than biases due to poor model choice (e.g., Cox vs. Weibull)
Also: In small samples, outliers are likely to be more important.