event history analysis 7 sociology 8811 lecture 21 copyright © 2007 by evan schofer do not copy or...
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Event History Analysis 7
Sociology 8811 Lecture 21
Copyright © 2007 by Evan SchoferDo not copy or distribute without permission
Announcements
• Paper Assignment #2 Due April 26• Try to find a dataset soon
• Class topic: • Parametric EHA models; diagnostics• Later (if time allows): AFT models, discrete time
models
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)eX = h0(t)exp(X)
• 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).
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.
Exponential (Constant Rate) Model
• Exponential models are simplest:)()( 2211)( βXaXbXbXba eeth nn
• 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-(X)
• 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 .
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!
)(0 )()( βXethth
Cox
)()()( βXaXa eeeth Exponential
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 = 92 Number of obs = 1938No. of failures = 77Time at risk = 1938 Wald chi2(6) = 94.29Log pseudolikelihood = 282.11796 Prob > chi2 = 0.0000
------------------------------------------------------------------------------ | Robust _t | Coef. Std. Err. z P>|z| [95% Conf. Interval]-------------+---------------------------------------------------------------- gdp | -.044568 .1842564 -0.24 0.809 -.4057039 .3165679 degradation | -.4766958 .1044108 -4.57 0.000 -.6813372 -.2720543 education | .0377531 .0130314 2.90 0.004 .0122121 .0632942 democracy | .2295392 .0959669 2.39 0.017 .0414475 .417631 ngo | .4258148 .1576803 2.70 0.007 .1167671 .7348624 ingo | .3114173 .365112 0.85 0.394 -.4041891 1.027024 _cons | -4.565513 1.864396 -2.45 0.014 -8.219663 -.9113642------------------------------------------------------------------------------
Constant shows base hazard rate estimated from data:
exp(-4.57) = .01
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…
Exponential Model: Baseline Hazard• Ex: stcurve, hazard
-.96
9705
91.
030
294
Ha
zard
func
tion
1970 1980 1990 2000analysis time
Exponential regression
See, the estimated baseline hazard really is flat!
Exponential Model: Estimated Hazard• stcurve, hazard at1(democ=1) at2(democ=10)
.05
.1.1
5.2
.25
.3H
aza
rd fu
nctio
n
1970 1980 1990 2000analysis time
democracy=1 democracy=10
Exponential regression
Here are estimated hazards for 2 groups
Other vars pegged at mean
Exponential Model: Baseline Hazard• Issue: Actual hazard is rising. A problem?
0.0
2.0
4.0
6.0
8.1
1970 1980 1990 2000analysis time
Smoothed hazard estimateIs an exponential model appropriate?
Answer:
It can be, IF we have X variables that account for increasing hazard
If not, fit will be poor!
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 X. 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.
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…
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!
Parametric Models
• Let’s try a more complex parametric model• Example: Let’s specify a linear time trend
)(0 )()( βXetβath
Linear
)()()( βXaXa eeeth Exponential
• In this case, we estimate a constant (a) and slope (0) which best summarize the time dependence of the hazard rate
• Note: this isn’t common – we have better options…
Gompertz Models
• Another option: an exponentiated line• Rather than a linear function of time and exponentiated
function of X, we’ll exponentiate everything:
• Slope coefficient is often represented by gamma: • 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
)()()( 0)( βXtaβXtβa eeeth Exponentiated Linear: Gompertz
Gompertz Models• Exponentiating a linear function generates a
curve defined by the value of gamma () • Model estimates value of that best fits the data
= 0
< 0
> 0
>> 0
Gompertz Model• Example: streg gdp degradation education democracy ngo
ingo, robust nohr dist(gompertz)Gompertz regression -- log relative-hazard form
No. of subjects = 92 Number of obs = 1938No. of failures = 77Time at risk = 1938 Wald chi2(6) = 46.48Log pseudolikelihood = 307.64758 Prob > chi2 = 0.0000
_t | Coef. Std. Err. z P>|z| [95% Conf. Interval]-------------+---------------------------------------------------------------- gdp | .4633559 .2104244 2.20 0.028 .0509316 .8757802 degradation | -.4394712 .1434178 -3.06 0.002 -.720565 -.1583775 education | .0026837 .0145341 0.18 0.854 -.0258026 .03117 democracy | .2890106 .092612 3.12 0.002 .1074943 .4705268 ngo | .2522894 .1658275 1.52 0.128 -.0727265 .5773054 ingo | .0037688 .2275176 0.02 0.987 -.4421575 .4496952 _cons | -253.035 45.28363 -5.59 0.000 -341.7892 -164.2807-------------+---------------------------------------------------------------- gamma | .124117 .0224506 5.53 0.000 .0801146 .1681195------------------------------------------------------------------------------
Model estimates gamma to be positive, significant. Implies increasing baseline hazard
Gompertz Model: Estimated Hazard• stcurve, hazard at1(democ=1) at2(democ=10)
Estimated hazards for 2 groups
Other vars pegged at mean
01
23
4H
aza
rd fu
nctio
n
1970 1980 1990 2000analysis time
democracy=1 democracy=10
Gompertz regression
Note: curves are actually proportional – hard to see because bottom curve is nearly zero…
Weibull Models
• Another option: the Weibull curve• Another curve that can fit monatonic hazards
• 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.
)(1)( βXap eptth Weibull
Weibull: Visually
• The Weibull family: Monotonic increasing or decreasing, depending on p
Time
Haz
ard
Rat
e
p = 1
p = 4
p = .5
p = 2
Weibull Model• Example: streg gdp degradation education democracy ngo ingo, robust nohr dist(weibull)
Weibull regression -- log relative-hazard form
No. of subjects = 92 Number of obs = 1938No. of failures = 77Time at risk = 1938 LR chi2(6) = 23.71Log likelihood = 307.6045 Prob > chi2 = 0.0006
_t | Coef. Std. Err. z P>|z| [95% Conf. Interval]-------------+---------------------------------------------------------------- gdp | .4631871 .2360589 1.96 0.050 .0005202 .9258541 degradation | -.4396978 .1486662 -2.96 0.003 -.7310781 -.1483175 education | .0027319 .0141652 0.19 0.847 -.0250314 .0304953 democracy | .288927 .0913855 3.16 0.002 .1098147 .4680394 ngo | .2522595 .1610192 1.57 0.117 -.0633324 .5678514 ingo | .004058 .1835743 0.02 0.982 -.355741 .363857 _cons | -1884.071 280.0398 -6.73 0.000 -2432.939 -1335.203-------------+---------------------------------------------------------------- /ln_p | 5.511481 .1486542 37.08 0.000 5.220124 5.802837-------------+---------------------------------------------------------------- p | 247.5173 36.79449 184.9571 331.2381 1/p | .0040401 .0006006 .003019 .0054067------------------------------------------------------------------------------
Ancillary Parameters
• Gompertz & Weibull models have parameters that determine the shape of the curve
• Gamma (), p• Ex: Bigger = 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
– streg male educ, dist(gompertz) ancillary(male) – Model estimates effect of male on hazard AND on gamma…
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.
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.
Parametric Model Fit: AIC
• Non-nested parametric models can be compared via the Akaike Information Criterion
)(2)ln(2 ckLAIC • 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… -100 is better than -50.
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.
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…
Unobserved Heterogeneity
• Visually:
Time (months)0 10 20 30 40 50 60 70
Haz
ard
Rat
e
Non-Smokers
Smokers die early… exhausting the sample.
Then h(t) drops offSmokers
The observed hazard rate is modeled w/o
controlling for the cause of the drop off…
Observed h(t)
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.
Unobserved Heterogeneity
• Strategies:– 1. Develop fully-specified models
• The best solution
– 2. Specify the form of the heterogeneity (frailty)• Approach: assume unobserved alpha () – case-
specific factor that makes events more (or less) likely
)()|( thth Frailty Model
• Where h(t) is some familiar model (ex: Weibull)• Requires functional form assumptions to estimate
– Ex: Assume is gamma (or inv gaussian) distributed…
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.
Parametric Models: Outliers• Cumulative Cox-Snell residuals vs case ID
LATVIAMACEDONIA
SLOVAKIA
SLOVENIA
ALGERIA
ANGOLA
BENIN
BUR-FASO
BURUNDICAMEROONCHAD
COMOROSCONGO
EGYPTETHIOPIA w eGAMBIA
GHANA
GUINEA
IVORY-CO
KENYA
MADAGASCMALAWIMALIMAURITAN
MAURITIUS
MOROCCO
MOZAMBIQNIGERNIGERIARWANDA
SENEGAL
SIERRA-L
SO-AFRICA
TANZANIA
TOGO
UGANDA
ZAMBIA
ZIMBABWE
CANADA
COSTA-RI
DOM-REP
EL-SALVA
GUATEMAHONDURAS
JAMAICAMEXICONICARAGPANAMA
TRIN&TOB
USA
ARGENTIN
BOLIVIA
BRAZIL
CHILE
COLOMBIA
ECUADOR
GUYANA
PARAGUAY
PERU
URUGUAY
BANGLAD
KAMPUCH
INDIA
INDONES
IRAN
ISRAEL
JAPAN
JORDANKOREA-R(S
LEBANON
MALAYSIA
NEPAL
PAKISTAN
PHILIPPI
SINGAPORSRI-LAN
SYRIA
THAILAND
TURKEY
BELGIUM
DENMARKFINLAND
ICELAND
IRELAND
LUXEMB
NETHERL
NORWAY
PORTUGAL
SWEDEN
SWITZERL
AUSTRAL
NEW-ZEAL
01
23
cum
. C
ox-
Sne
ll re
sidu
al
0 1000 2000 3000caseid
Note that Scandinavia has highest residuals
Probably not outliers, but interesting nevertheless
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”
Xt )ln(• 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!
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.
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.
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.