An Introduction to Event History Analysis

Dawn L. Teeledawn.teele@yale.edu

April 2008

Abstract

This paper is meant to be a guide to using event history modeling.Topics include the formulation of the hazard rate and the survival func-tion, parameterization of the model, and common problems such as rightand left censoring and ties. The appendices contain commands for settingup event history models in Stata.

1 Event History Modeling

Event history models, also known as survival-time analysis, duration models, orhazard models take their origin in the natural sciences. These models estimatewhat is known as a hazard rate, the probability of an event occurring given thatit has not already occurred. In other words, the conditional probability of anevent.Embedded in the hazard is the notion of a failure rate and a survival func-

tion. In the natural sciences the hazard was used to understand how medicaltreatments a¤ected the probability of death of patients with terminal illnesses(the failure rate) given a distribution of longevity (the survival function).In the social sciences we might also be concerned with the conditional prob-

ability of an event occurring, and with some data � especially those that havevariation in observable characteristics over time � an event history model mightbe appropriate.

1.1 Why event history?

When the estimate of interest is the conditional probability of an event occur-ring, a linear regression speci�cation will not su¢ ce. If left in a linear model, theexplanatory variables will yield predictions for the coe¢ cients that are di¢ cultto compute and interpret (Allison 1984), and little information will be shed onthe conditions that lead to an �event�occurring.There are two statistical techniques that can be used to understand the

e¤ects of the covariates on the response variable: The logit or log-odds of the

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event variable can be used in an OLS regression,1 or survival time regressionscan be performed.Logit regression has been a traditional method for analyzing events within

social science. However, if the length of time leading up to an event is also ofinterest, then duration models hold a unique advantage over traditional logitmodels. For example, some subjects will have time-varying observable charac-teristics that need to be controlled for, such as GDP per capita, trade and evenan indicator of how democratic a country is. Event history models performmuch better than traditional OLS regression with these types of independentvariable (Allison 1984).2

Issues with event history data such as truncation, late entry, and censoringare also better dealt with in survival analysis packages. In many situations,researchers won�t have data for subjects prior to when their sample begins (es-sentially an incomplete life history). This is called �forward censoring�and canbe handled quite well by survival time packages. Another type of censoring,�right censoring�occurs when a subject has not experienced the event by thetime the sample ends. Mathematically, right censoring does not pose problemsfor estimating a hazard because subjects for which an event has not happenedcontribute to the survival function S(t) but not to the unconditional failurefunction f(t) (Box-Ste¤ensmeier and Jones, 2004).To add to the laundry list of concerns that event history packages handle

well, some subjects have what is known as �delayed risk�meaning that they arenot part of the relevant sample at the beginning of the history. Concretely, ifwe want to know the probability of a relapse into civil war � and the durationof time between those events � countries that have not yet had one civil war atthe beginning, but who have at least one by the end of the history, have delayedrisk.Finally, �failure ties�occur when more than one subject experiences an event

in the same time period. Ties are a statistical problem because it is impossibleto determine which subject failed �rst hence we cannot specify the conditionsthat were instrumental at the time of failure. Ties are less of an issue withdiscrete than continuous data, and �lters can be applied to deal with theseissues (Box-Ste¤ensmeier and Jones, 2004).

1.2 An example to work with

In the following discussion I will make reference to work that I conducted for mysenior thesis to provide a practical example of the way in which event historyanalysis can be used (Teele 2006). The basic question I asked is, �what causescountries to ratify the child labor convention with the International Labor Or-ganization?� taking rati�cation as the �event� to be studied. Bear in mind

1Formally, the logit or "log-odds" transformation is constructed by: logit_Depvar=ln Depvar(1�Depvar ) : The transformation is necessary to have standard errors that are normally

distributed.2 If all of the covariates used in the regression are constant over a country�s event history,

cross-sectional analysis of the dependent variable in the year of rati�cation would su¢ ce.

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Figure 1: Rati�cation of the ILO Child Labor Convention by Region, 1976-2004

that I use this example here to demonstrate the steps that one should take toimplement an event history model, and am not purporting to have uncoveredun-falsi�able truths in this demonstration.The sample contains 150 countries from 1976-2004, the periods are in years

(discrete time) and the independent variables will be both time-varying andtime-invariant. To give you a sense of the data, Figure 1 shows the rati�cationpatterns by region. It is clear that there was a surge in rati�cation during thelater part of the sample (1997-2003), and so there must be a story to tell aboutthe conditions leading up to this time period. For this reason the event historymethodology seems, conceptually, to �t the data.

1.3 Duration variable (the dependent variable)

Event history analysis uses the duration of time � either discrete or continuoustime � before an event occurred as the dependent variable (though not thedependent variable in the conventional sense). In our example, Convention 138was introduced to the ILO assembly in 1976, so the event history begins in thisyear and ends in 2004, the last year that data for this sample are available.However, there are 47 countries that were not members of the ILO in 1976,

meaning that they have �delayed risk� in the sample. Practically speaking, asubject cannot experience an event if it is not part of the relevant sample (wecannot understand the e¤ect that a blood pressure drug has on increasing thetime between heart attacks if a subject has not in fact had a heart attack), andso we must be careful when constructing the dataset to deal with these cases

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(read: get the Stata commands correct).3

In this study the duration variable will not begin for each country until theyare members. Thus for all countries the duration variable begins either in 1976or in the year that a country joined the ILO, and ends in the year that theyratify C138. The late-entries are classi�ed as �left-censored�. On the oppositeend of the sample period we have many countries that have not rati�ed theILO convention. In event history models these cases are classi�ed as �rightcensored�.

1.4 Estimating the hazard rate

Event history models estimate a �hazard rate�, or the conditional probabilitythat an event will occur given that it hasn�t already. The basic formulationof the hazard rate H(t), is similar to a cumulative distribution function ofthe probability of an event. The hazard rate is constructed by estimating themaximum likelihood that a subject will experience an event.Mathematically, the hazard rate has two components: the probability of

failure, f(t) and the survival function, S(t). As I mentioned earlier, the hazardrate originally hails from techniques used in the natural sciences to predictthe probability that a subject will die, hence �survival�and �failure�functions(Allison 1984).Following Box-Ste¤ensmeier and Jones (2004), let T be a discrete random

variable indicating the timing of an event. In our example the probability thata country will ratify C138 at time tj , is found in the failure function where jrepresents the year in which a country rati�es C138:

f(t) = Pr(T = tj) (1)

Because rati�cation (or �failure�) can only occur once for each country, thisis a �single spell�analysis. However, there may be more than one failure at atime if countries ratify in the same year; the Breslow method of estimation for�failure ties�is accounted for in Stata.The second component of the hazard, the survival function is de�ned as:

S(t) = Pr(T � tj) =Xj=i

f(tj) (2)

The basic hazard rate for discrete-time is simply the ratio of the probabilityof failure to the probability of survival, or the conditional probability of survivalgiven that failure has not already occurred. This relationship can be expressedin two ways:

h(tj) =f(t)

S(t)(3)

3 I have run across some literature (speci�cally, Bockmann 2001) in which authors disregardthe observations that are not in the entire sample. This seems to be a waste, both because itlowers the n but also because it devalues any general conclusions that may be inferred fromthe results.

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h(tj) = Pr(T = tj jT � tj) (4)

Equation 4 gives the rate at which countries ratify C138 conditional on theirsurvival until j. The hazard can be augmented to include the e¤ect of a vectorof time-varying explanatory variables xij :

h(tj jxij) =f(tj jxit)S(tj jxit)

(5)

In this speci�cation it is assumed that the only thing that a¤ects the haz-ard over time are elements of xij , the vector of time-varying covariates, butqualitative variables can be included in the model just as easily.Implicit in the hazard estimate is what is known as the �baseline hazard�,

h0(t), which is a parameter representing the distribution of the data with respectto time. The idea is that the hazard is a function of h0(t), an intercept, and avector of explanatory variables, xij .The e¤ect of the baseline h0(t) on the hazard ratio depends on the restric-

tions that are placed on this parameter. Using a Cox �proportional hazards�speci�cation is a way to allow the hazard to be estimated without placing anyparametric assumptions on the baseline. Meaning that the regression remainsagnostic as to how time in�uences the probability of the event.Proportional hazards, especially the Cox model, has become increasingly

more popular in he social science literature due to its �exibility.4 However, ifthe distribution is known (or suspected), a functional form for h0(t) can, andshould, be speci�ed.So long as the hazard is a multiplicative of the covariates, the model is classi-

�ed as proportional hazards, and most parametric models can be written in thisnotation. There are substantive di¤erences in the estimates when other distrib-utions are assumed, and alternatives to the Cox model include the exponential,Gompertz and Weibull distributions.

1.5 Survivor and Hazard Functions

Two preliminary steps in event history analysis entail looking at the componentsof the hazard function for di¤erent sub-groups, and testing to see whether thedi¤erences are statistically signi�cant. Given that the division between countriesin the OECD and those who are not members generally corresponds to the levelof development, I will compare these two sub-groups, but others (such as regionof the world) could just as easily be chosen.The cumulative hazard estimates for OECD and non-OECD countries can

be seen in Figure 2. No parametric assumptions were placed on the estimates of

4Estimating the hazard rate was �rst done within a conditional logit model, and is at-tributed to Cox (1972). Over time, the �Cox Proportional Hazard� model has becomemore prevalent in empirical social science. For a discussion of the bene�ts of the propor-tional hazards assumption as opposed to other functional forms of the hazard model seeBox-Ste¤ensmeier and Jones (2004) and Yamaguchi (1991).

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Figure 2: Cumulative Hazard Estimates for OECD and non-OECD Countries

the cumulative hazard functions, and at �rst glance the OECD countries appearto have a higher hazard ratio based on the higher intercept and steeper functiontoward the end of the sample.Another common way to look at sub-groups within the event history panel

is to calculate a Kaplan-Meier estimate of the survivor function, and then graphthese results for di¤erent qualitative sub-groups. The Kaplan-Meier estimatoralso has no parametric assumptions and is given by:

S(t) =Yjjtj�t

nj � rjnj

(6)

where nj represents the number of countries who are at risk of rati�cationat time tj , and rj represents the number of countries that ratify C138 in periodtj . In other words, the Kaplan-Meier estimate is the product of the percent ofcountries in the sample that survive past each period. This estimator is non-parametric in that it doesn�t place any restriction on the shape of the survivorfunction takes. The Kaplan-Meier survival estimates, often referred to as the�Cumulative Survival�estimate for the rati�cation of C138 can be seen in Figure3.The probability that countries within each group, here OECD and non-

OECD countries, will make it to the next year without ratifying C138 is shownon the y-axis, whereas the x-axis shows the time that has passed since C138 wasintroduced in 1976. Figure 3 displays survival functions that look almost iden-tical in shape (as opposed to the cumulative hazard graph) and it is clear thatfor both OECD and non-OECD countries, the survival function is monotoni-

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Figure 3: Kaplan-Meier Survival Estimates for OECD and non-OECD Countries

cally decreasing over the sample. Though the intercept for OECD countries isvisibly lower than the intercept for non-OECD countries, a statistical test �the �log rank�test for the equality of survivor functions � demonstrates thatthe di¤erence is not statistically signi�cant.Taken together, the Cumulative Hazard and Kaplan-Meier graphs illustrate

that though the hazard functions display some variation across groups, thesurvivor functions move very closely to one another. Solely looking at the cu-mulative hazard functions (or not looking at the log-rank tests) could lead to amisspeci�cation of the model. Appendix A shows the output for the log-ranktest in the scenario above, and lists a few other statistical tests that can beused to test the null hypothesis that the survivor functions for two (or more)sub-groups are the same.In the literature on ILO convention rati�cation, the Cox proportional haz-

ards model is the predominant speci�cation, although little rationale other thatits �exibility is given (see Boockmann 2001, Chau and Kanbur 2001, and AbuSharkh 2002). The Cox model may have advantages when the shape of thebaseline hazard is unknown, however for this data the survivor functions aredecreasing monotonically over time, which leads to a suspicion that the risk ofrati�cation as represented by the hazard is increasing over time.

1.6 The Weibull Regression Model

A parametric regression model that provides e¢ cient coe¢ cient estimates formonotonic functions is theWeibull speci�cation (Allison 1984, Box-Ste¤ensmeierand Jones 2004). The baseline hazard function that the Weibull estimates is

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given by:

h0(t) = ptp�1exp(a) (7)

where p is a shape parameter that remains constant, and a is a scale para-meter that is estimated by the covariates. When the hazard is monotonicallyincreasing with respect to time, p > 1. Given a set of covariates xij the Weibullequation is:

h(tj jxij) = h0(t)exp(xit�x) (8)

= ptp�1exp(�0 + xit�x)

As can be seen from Equation 8, the covariates that in�uence the hazard doso as a multiple of the baseline hazard. For this reason the Weibull still fallsinto the category of a proportional hazards model, but is di¤erentiated fromthe Cox model based on the ancillary parameters a and p. The �tted Weibullmodel estimates a, p and �0x (Cleves, Gould & Gutierrez 2002).

2 Estimation Results

Table 1 lists the coe¢ cients and standard errors for Weibull estimates of therati�cation of Convention 138. The signs of the coe¢ cients indicate the directionof the e¤ect that the variable has on the probability that a country will ratifyC138 given that it has not already. The coe¢ cients themselves do not havemeaningful interpretations, but because the Weibull model still �ts into thefamily of proportional hazard models, the e¤ect that each covariate has on thehazard ratio can be found by taking the exponential the coe¢ cients.5

The �ve regressions in Table 1 are not nested models, but have been includedto demonstrate two things: �rst, as can be seen in regressions (1) and (3), per-capita GDP and its quadratic term do not have a statistically signi�cant e¤ecton the hazard ratio. This result is consistent with the results found by Chauand Kanbur (2001), and is intuitive given that many of the richest countriesrati�ed C138 very late into the sample, if at all.The e¤ect of international trade on the propensity to ratify C138 is posi-

tive and statistically signi�cant in all tested variations of the model (for othercombinations of covariates see Teele (2006)). The negative coe¢ cient on trade�squadratic term indicates that trade has a positive e¤ect on rati�cation but ata decreasing rate. This result predicts that as trade increases, the probabilityof rati�cation increases as well. This result could be motivated by a desire to

5For hazard ratios the null hypothesis tested is that coe¢ cient has no e¤ect on the hazardsuch that exp(Bx) = 1. This is equivalent to testing whether the non-exponentiated coe¢ -cients are equal to zero, as with standard linear regression. For this reason, traditional waysof assessing signi�cance apply in this analysis.

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Table 1: Weibull Regressions on Child Labor Convention(1) (2) (3) (4) (5)

ln GDP per cap 0.393 -0.274(0.639) (0.689)

GDP squared -0.0285 0.00956(0.0419) (0.0446)

Region -0.00287 -0.0262 0.0494 0.128(0.0749) (0.0536) (0.0795) (0.0893)

Trade, % GDP 2.208� 2.422� 3.138�� 3.525��

(0.969) (1.023) (1.120) (1.159)

Trade squared -0.923 -0.974� -1.131� -1.255�

(0.477) (0.489) (0.532) (0.542)

Aid per capita -0.00494 -0.00485(0.00296) (0.00292)

Child Labor % 1.436 2.440�

(0.797) (1.042)

Constant -7.427�� -7.009��� -5.909� -7.908��� -8.793���

(2.455) (0.744) (2.555) (0.874) (1.082)ln_pConstant 0.581��� 0.570��� 0.574��� 0.616��� 0.640���

(0.0979) (0.0978) (0.0978) (0.106) (0.105)Observations 3646 3559 3479 2779 2779

Standard errors in parentheses�p < 0:05, ��p < 0:01, ���p < 0:001

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�signal� to trade partners that the country has a similar set of values and re-strictions placed on its labor standards � a move that is meant to facilitatetrade negotiations.Per-capita international aid is statistically signi�cant at the 10 percent level

and is found to have a negative e¤ect on the hazard. This means that countriesthat receive more international aid per person are less likely to ratify the childlabor convention. This result is contrary to sociological theories that �pressure�from international aid organizations and foreign governments a¤ects rati�cationpatterns for aid recipients.6 However, it should be noted that countries that havehigher per-capita aid are probably some of the poorest, and this variable maypicking up on other correlates of poverty.The other result of interest in Table 1, found in Regression (5) is that child

labor has a positive e¤ect on the hazard ratio, and is statistically signi�cant.7

This result is particularly intriguing given that C138, which is technically the�minimum age to work�convention speci�es that a minimum age of 15 be re-quired for economic participation. The regression shows that for higher levelsof child labor � speci�cally de�ned as the percent of children 10�14 who areeconomically active � countries are more likely to ratify the child labor con-vention.Regression (5) includes all of the covariates that have a statistically signi�-

cant e¤ect on the hazard. Hazard ratios have been calculated for these covariatesand can be found in Table 2. For estimates of the hazard ratio that are largerthan one, the covariate has a positive e¤ect on the probability of rati�cation.Covariates whose hazard estimates are less than one have a negative e¤ect onthe probability of rati�cation. The hazards reported in Table 2 mirror the previ-ous discussion. Both child labor and trade have very large e¤ects on the hazardratio.In the next section I will construct a graph holding child labor constant at the

saple mean and varying the level of trade in order to give a visual interpretationof the results above.

2.1 Fitted Values of Survival Analysis

Because there are so many steps that go into the data collection even before theregression command can be hit, it can be tempting to let the regressions speakfor themselves. However, regression coe¢ cients or, in our case above, hazardrates, are not always easy to understand. The saying goes that pictures areworth a thousand words, so for the next page or so I will show you a picturethat can help us interpret the results from above.First, it is important to look at the data to see what a probable range for the

6 It could be argued that it is not aid per-capita but the overall reliance on internationalaid that allows countries to be pressured into rati�cation. For this reason, international aidas a percent of government expenditures was also tested for its e¤ect on the hazard but wasnot statistically signi�cant.

7Without the inclusion of regional dummies, child labor is signi�cant at the 10 percentlevel.

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Table 2: Hazard Estimates for Model (5)(1)

Trade, % GDP 33.96(3.04)��

Trade squared 0.285(-2.32)�

Aid per capita 0.995(-1.66)

Child Labor % 11.48(2.34)�

Region 1.137(1.44)

Observations 2779

Exponentiated coe¢ cients; t statistics in parentheses�p < 0:05, ��p < 0:01, ���p < 0:001

variables of interest are. Table ?? below shows means and standard deviationsfor trade and child labor, the two variables that were statistically signi�cantin the section above. Figure 4 presents the �tted values for Regression (5)evaluated at the average level of child labor, 17 percent, for di¤erent levels oftrade. The groups with lower trade (one standard deviation below the mean)as a percent of GDP are less likely to ratify the convention than those withabove-average trade (one standard deviation above the mean).The slopes of the �tted value curves show that for higher levels of trade, the

hazard rate is much higher. The direct interpretation of this is: holding childlabor constant, countries with higher levels of trade as a percent of GDP aremuch more likely to ratify the ILO convention banning child labor.This result supports the hypothesis that countries may sign conventions to

enhance their �reputational capital� within the global market place. If ques-tions of labor standards arise in trade negotiations, countries can point to theconvention as a law that they uphold, while knowing full well that the ILO hasvery little coercive power to punish their actions should they be discovered.

3 Conclusion

Finally we come to the end of this foray into event history modeling. TheAppendices that follow are meant to help you get started with setting up yourevent history panel in Stata, and I have also included the commands that were

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Table 3: Summary Statistics(1)

Trade, % GDP 0.729(0.406)

Child Labor % 0.136(0.158)

Observations 3825mean coe¢ cients; sd in parentheses

Figure 4: Fitted Values of Regression (5) Evaluated at Mean Level of ChildLabor with Varying Levels of Trade

used to make each graph in this paper.

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A ST commands used on this data

As a brief introduction to some of the survival time commands that are necessaryto set up an event history panel:

snapspan panelid year dcnv138, gen(date0) replace/* Convert snapshot data to time-span data */rename year date1

stset date1, id(panelid) failure(dcnv138) origin(ismember==1) exit(iloflake==1)

stdes /* describes st set */stvary

note: obtain K-M survival estimatests generate kmS=slabel var kmS "K-M"

note: obtained N-A cumulative hazard estimatests generate naH=na\qquadlabel var naH "N-A"

note: calculate N-A survivor estimateg naS=exp(-naH)label var naS "N-A"

note: calcualte K_M cumulative hazard estimateg kmH=-log(kmS)label var kmH "K-M"

Cox regression -- Breslow method for tiesstcox lpcGDP lpcGDPsq, basesurv(s) basehc(h)

sts test OECD, logrank

/* failure _d: dcnv138analysis time _t: (date1-origin)

origin: ismember==1exit on or before: iloflake==1

id: panelid

Log-rank test for equality of survivor functions

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| Events EventsOECD | observed expected------+-------------------------0 | 91 89.331 | 18 19.67------+-------------------------Total | 109 109.00

chi2(1) = 0.18Pr>chi2 = 0.6710

*or can use any of the following to test for the equality of survivor functions:sts test OECD, wilcoxon /* Wilcoxon-Breslow Test */sts test OECD, tware\qquad /* Tarone-Ware Test */sts test OECD, peto /* Peto-Peto Test */

sts test OECD, logrank strata(legalsys) detail /* stratified long-rank tests */

B Graph commands used in this paper

The following are the stata commands used to construct the graphs and tablesabove.For the bar chart in Figure 1:

graph bar (asis) Rat138 if Year!=1975,over(Year, label(angle(forty_five)labsize(small))) stackytitle(, size(small)) ylabel(, angle(horizontal) labsize(small))title(Ratification of ILO Convention 138)subtitle(by region and year)note(Source: International Labour Organization)

For the hazard rate estimates in Figure 2:

sts graph, hazard by(OECD)ylabel(,angle(horizontal))ytitle(Conditional Probability of Ratification)xtitle(Years since C138 introduced, margin(medium))legend(order(1 "non-OECD" 2 "OECD") size(small))xlabel(0 "1976" 10 "1986" 20 "1996" 30 "2006", valuelabel)title(Cumulative Hazard Estimates)subtitle(OECD and non-OECD countries)note(Source: International Labour Organization)

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For the Kaplan-Meier survival estimates in Figure 3:

sts graph, by(OECD)ylabel(,angle(horizontal))ytitle(Conditional Probability of Ratification)xtitle(Years since C138 introduced, margin(medium))legend(order(1 "non-OECD" 2 "OECD") size(small))xlabel(0 "1976" 10 "1986" 20 "1996" 30 "2006", valuelabel)title(Kaplan-Meier Survival Estimates)subtitle(OECD and non-OECD countries)note(Source: International Labour Organization)

Finally, for the hazard graphs of �tted values evaluated at the mean level ofchild labor for di¤erent levels of trade in Figure 4:

stcurve, cumhazat1( trade=.701 clabWDI=.15 )at2( trade=1.07 clabWDI=.15 )at3( trade=.295 clabWDI=.15 )ylabel(,angle(horizontal))xtitle(Year)xlabel(0 "1976" 10 "1986" 20 "1996" 30 "2006", valuelabel)note(Source: ILO & WDI)legend(order (1 "Average Trade (70%)"2 "One standard deviation above"3 "One standard deviation below"))title(Fitted values of Cumulative Hazard Function)subtitle(evaluated at average level of child labor)

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