Alfred DeMaris - Regression with Social Data, Modeling Continuous and Limited Response Variables

Подождите немного. Документ загружается.

Allison (1984, 1995) and is relatively ﬂexible and easy to estimate. Its form is

(t) ⫽ h

(t ⫺ t

e⫺1

) exp(x

⬘

ββ

). (12.4)

The notation needs some clariﬁcation. The response, h

(t), is the hazard of the eth

event for the ith person. The baseline hazard function, h

(t ⫺ t

e⫺1

), is assumed to be

the same for each event. However survival time is denoted as “t ⫺ t

e⫺1

,” where t is

survival time until the eth event or censoring and t

e⫺1

is survival time until the pre-

vious event. The diﬀerence is the survival time for the current, or eth, event. In other

words, survival time for each spell begins at inception of risk for the current event,

which is normally the time of occurrence of the previous event (although there are

exceptions, e.g., the example of unemployment spells below). This follows the

assumption that cases are not at risk for a subsequent event until after the occurrence

of a prior event. [An alternative is to reckon survival time for all spells from the incep-

tion of risk for the ﬁrst event; see Hosmer and Lemeshow (1999) for a discussion of

alternative start times for recurrent event data.] The e superscript on x indicates that

covariate values may change with the risk periods for each event. Covariates may

also change over time within each risk period, as always. The recurrence of an event

terminates a given spell. Spells that are ongoing at the end of the study constitute

censored cases. Estimation is straightforward: One simply treats each spell for a

given case as a separate observation and then pools all spells over all n cases in the

sample to arrive at a total ﬁle of nE observations. One then estimates the Cox model

for “an event” in the usual fashion.

Example: Unemployment Spells

As an example, Goza and DeMaris (2003) examined transitions out of unemploy-

ment for a sample of 283 Brazilian immigrants residing in the United States and

Canada in 1990–1991. The authors’ study was primarily focused on testing predic-

tions from job search theory regarding the duration of unemployment for immi-

grants. Each respondent in the study experienced, on average, about 2.2 periods of

unemployment, ranging from under one month to as much as ﬁve years in duration.

Inception of risk for each spell was the date of loss of the previous job, rather than

the date on which the previous job began. Spells that ended in reemployment were

considered uncensored cases. Unemployment spells that were ongoing when data

collection ended were considered censored. When all spells were pooled over all

cases, a total of 620 unemployment spells constituted the analytic sample. Of these,

578 were uncensored and 42 were censored. A series of Cox models for the hazard

of reemployment was estimated using these data and is shown in Table 12.2.

Model 1 in Table 12.2 contains the eight focus variables in the study: age at the

start of unemployment, duration in months on the previous job, the respondent’s

number of relatives living in North America, cumulative time in years in North

America, the log of monthly income on the last job, education in years of schooling,

a dummy for being female, and a dummy for English proﬁciency being self-rated as

at least “good” when entering North America. All eﬀects are signiﬁcant at the .05

MULTIEPISODE MODELS 425

c12.qxd 8/27/2004 2:57 PM Page 425

level. Longer spells of unemployment are associated with being older at the start of

the unemployment spell, spending more time in the previous position, having been

in North America longer, being female, and surprisingly, having good English

proﬁciency. The last eﬀect is perhaps the result of those with better English skills

having higher expectations for good positions and therefore holding out longer for

them (Goza and DeMaris, in 2003). On the other hand, shorter unemployment spells

obtain for those with more relatives in North America and those with more human

capital in the form of attained educational level and income from the last job.

Nonindependence of Survival Times

Model 1 is the most restrictive possible model in the sense that it makes several

assumptions about the data that may or may not be tenable. To begin, it is assumed

that the 620 spells constitute a set of independent observations, given model covari-

ates. In all likelihood this assumption is violated since the same people contribute

more than one spell to the data set. Therefore any unmeasured factor that might ele-

vate (or diminish) the rate of reemployment for a given person will tend to shorten

(or lengthen) successive spells for the same person, resulting in correlated survival

times. One solution recommended by Allison (1984) is to add covariates that tap into

characteristics of the person’s prior event history, such as the number of events prior

to the current spell or the length of the previous spell. In that independence of obser-

vations is assumed to hold net of model covariates, including the proper covariates

might render the independence assumption tenable. As a result, model 2 includes

two additional factors pertaining to respondents’ job histories: The cumulative num-

ber of months of unemployment experienced and the cumulative number of jobs

426 MULTISTATE, MULTIEPISODE, AND INTERVAL-CENSORED MODELS

Table 12.2 Cox Multiepisode Models for Transitions Out of Unemployment

Predictor Model 1 Model 2 Model 3

Model 4

Age at unemployment ⫺.118* ⫺.124* ⫺.136* ⫺.113

Duration of previous job (months) ⫺.026** ⫺.028** ⫺.030** ⫺.026**

Number of relatives in North America .020* .019* .184 .021*

Cumulative time in North America ⫺.136*** ⫺.113* ⫺.152*** ⫺.136**

Log of monthly income last job .061*** .064*** .068*** .080***

Education .051*** .049*** .058*** .052***

Female ⫺.351** ⫺.349** ⫺.371** ⫺.369***

Good English proﬁciency ⫺.509** ⫺.509** ⫺.549* ⫺.477*

Cumulative months unemployed ⫺.018

Cumulative no. previous jobs .001

.128**

Note: n ⫽ 620 unemployment spells.

Shared-frailty model.

Stratiﬁed by job-spell number.

* p ⬍ .05. ** p ⬍ .01. *** p ⬍ .001.

c12.qxd 8/27/2004 2:57 PM Page 426

taken by the respondent. Neither additional factor is signiﬁcant. More important,

however, none of the other eﬀects are appreciably altered when these factors are con-

trolled.

A second approach to the dependence problem is to adjust the standard errors of

coeﬃcients to account for the dependence of survival times within a person’s event

history. Lin and Wei (1989) proposed a “sandwich” estimator for the covariance

matrix of parameter estimates from a potentially misspeciﬁed Cox model. The

resulting estimates of variances and covariances of Cox regression coeﬃcients are

robust to various types of misspeciﬁcation, including dependence among observations.

(As we will discover below, the problem of dependence can be cast in terms of an

omitted variable; hence such misspeciﬁcation is in the form of omitted-variable

bias.) The robust estimator of V(b) is available in mainstream software packages

such as SAS and STATA. Model 1 was reexamined employing the robust standard

errors of coefﬁcients (results not shown). However, again, results were not altered

appreciably. The largest change in standard error estimates was for the coeﬃcient of

good English proﬁciency: The unadjusted standard error was .182, while the robust

estimate was .192. The coeﬃcient remained signiﬁcant at p ⬍ .008 in either case.

A third approach to handling nonindependence of survival times is to employ the

ﬁxed-eﬀects partial likelihood (FEPL) approach (Allison, 1996). This strategy uti-

lizes a ﬁxed-eﬀects model for the hazard, which takes the form

(t) ⫽ h

(t ⫺ t

e⫺1

)α

exp(x

⬘

ββ

). (12.5)

The additional multiplicative term in the model, α

, represents a ﬁxed constant that

characterizes each person. These constants represent individual-level factors aﬀecting

the hazard, which are responsible for the correlated survival times for events per-

taining to the same person. Normally, these factors are unmeasured and are corre-

spondingly referred to as factors reﬂecting unmeasured heterogeneity in the hazard.

A convenient method of estimation absorbs the α

into the baseline hazard function,

thus converting equation (12.5) into

(t) ⫽ h

(t ⫺ t

e⫺1

) exp(x

⬘

ββ

). (12.6)

The reader should notice the resemblance of equation (12.6) to the stratiﬁed version

of the Cox model presented in Chapter 11. In fact, equation (12.6) is estimated sim-

ply by estimating the Cox model while stratifying on individuals. However, a major

drawback to this approach, as mentioned in Chapter 11, is that only the eﬀects of

covariates whose values vary across or within spells may be estimated. In other

words, individual characteristics such as gender or race that remain constant across

the spells for a given person cannot be assessed. Nevertheless, all such characteris-

tics are implicitly controlled during model estimation. In the unemployment exam-

ple the only covariates in model 1 that vary over unemployment spells for a given

immigrant are age at unemployment, duration of previous job, cumulative time in

North America, and log of monthly income. A FEPL model containing just these four

covariates was therefore estimated (results not shown). The only signiﬁcant eﬀect

MULTIEPISODE MODELS 427

c12.qxd 8/27/2004 2:57 PM Page 427

was for log of monthly income, and its value was .059, which tends to agree with the

covariate’s eﬀect in model 1.

A fourth approach to the dependence problem is similar in spirit to the unob-

served-heterogeneity approach just discussed. However, instead of being ﬁxed, the

heterogeneity term is modeled as a random variable and referred to as a frailty.

Frailty models account for a burgeoning literature within survival analysis (see, e.g.,

Aalen, 1994; Blossfeld and Hamerle, 1990; Blossfeld and Rohwer, 1995; Galler and

Poetter, 1990; Hosmer and Lemeshow, 1999; Klein and Moeschberger, 1997; Land

et al., 2001; McGilchrist and Aisbett, 1991) and are becoming more available in

mainstream software. When the frailty characterizes a group of observations, the

model is referred to as a shared-frailty model. In recurrent-event data, for example,

the shared-frailty model is applicable since each person contributes a group of obser-

vations to the data in the form of multiple episodes. The Cox shared-frailty model

for multiepisode data is (Klein and Moeschberger, 1997)

(t) ⫽ h

(t ⫺ t

e⫺1

)υ

exp(x

⬘

ββ

), (12.7)

where the frailty, υ

, is assumed to have some density with a mean of 1 and a vari-

ance of θ. The gamma density is often chosen for its mathematical tractability.

Frailties greater than 1 imply a greater hazard of event occurrence, net of covariates.

That is, these people are more “frail,” or susceptible to the event. Frailties less than

1 indicate greater resistance to the event, or longer survival times. The diﬀerence

between equations (12.5) and (12.7) is subtle but important. In equation (12.5), α

a ﬁxed eﬀect, whereas in equation (12.7), υ

is a random variable with some popu-

lation distribution. By “ﬁxed eﬀect” is meant that there is some ﬁnite set of α

in the

population that are constant values over repeated sampling. That is, each sample of

people is a sample from the same set of limited α

values, with the same value of α

characterizing potentially many people in the population. On the other hand, υ in

equation (12.7) is a random variable that is not ﬁxed over repeated sampling, and

may in fact be unique to each person. The set, assumed to be inﬁnite, of all possible

in the population is represented by some distribution function.

Model 3 in Table 12.2 is the Cox shared-frailty model for the unemployment data,

assuming a gamma distribution for the frailties. The model was estimated using

STATA. The estimate of the frailty variance is .128 and is signiﬁcantly diﬀerent from

zero according to a likelihood-ratio test. It is therefore important to take account of

individual frailties in the model. That said, however, results are, again, not radically

altered compared to model 1, except that in model 3 the number of relatives in North

America no longer has a signiﬁcant eﬀect on the hazard.

A word of caution is in order regarding unobserved heterogeneity. In the Cox

model, the shape of the hazard function is ignored. However, in parametric models

that specify some form for the hazard, care must be exercised in interpreting the

eﬀect of a hazard that appears to be declining over time. Unobserved heterogeneity,

if unaccounted for, can artiﬁcially generate a declining hazard. The reason is that

over time, the frailest people experience the event and drop out of the risk set. This

leaves a risk set that is composed increasingly of the most “resistant” individuals,

428 MULTISTATE, MULTIEPISODE, AND INTERVAL-CENSORED MODELS

c12.qxd 8/27/2004 2:57 PM Page 428

making it appear that the risk for the event is declining over time (Blossfeld and

Rohwer, 1995). As unobserved heterogeneity is rarely completely accounted for, a

declining hazard function should always be regarded with some skepticism.

Model Variation across Spells

An additional assumption of model 1 is that each successive spell of unemployment is

characterized by the same hazard function for reemployment. This, of course, may not

be the case. Many times the ﬁrst episode of any event has a diﬀerent survival trajec-

tory than later episodes. To investigate this possibility, I created a variable represent-

ing the sequencing of unemployment spells. This variable, job-spell number,was

coded 1 for the ﬁrst unemployment spell, 2 for the second unemployment spell, 3 for

spell numbers 3 or 4, and 4 for spell numbers 5 or higher. Later spells were collapsed

due to relatively few people having more than four unemployment episodes. I then

dummied up job-spell number and added the three dummies to model 1 (results not

shown), producing a base model for testing interaction. In the next step, I allowed the

job-spell number dummies to interact with time (results not shown) in order to capture

changes in baseline hazards across unemployment spells. The likelihood-ratio test for

the addition of the interaction terms produced a nonsigniﬁcant result (

∆χ

(3)

⫽ 2.2902,

p ⫽ .514

), suggesting that the baseline hazard function did not vary across unem-

ployment spells. Nevertheless, model 4 in Table 12.2 reestimates model 1 but stratiﬁes

by job-spell number. Again, the results are not appreciably altered, compared to model

1, except that age at unemployment in model 4 is not quite signiﬁcant.

A ﬁnal simplifying assumption made in model 1 is that the eﬀects of covariates

on the hazard of reemployment are the same for each successive unemployment

episode. Again, such an assumption may not be tenable. We might expect, for exam-

ple, that having many relatives to help in the job search would have a stronger eﬀect

in later unemployment periods than in earlier ones. During an initial episode of

unemployment, people may rely on only their own credentials to ﬁnd jobs. After

repeated unemployment spells, however, people may begin to call on their larger kin

network for additional aid in helping to secure a good position. A general test for the

interaction of model 1 covariates with job-spell number was eﬀected by creating the

cross-products of the three job-spell number dummies with the eight covariates in

model 1 and then adding these to the base model described above (results not

shown). Due to small cell sizes, however, the interaction of job-spell 4 with good

English proﬁciency could not be entered into the model. The test for interactions

resulted in a ∆χ

of 66.566, which, with 23 df is signiﬁcant at p ⬍ .00001. This sug-

gests that one or more regressors has diﬀerent eﬀects, depending on the sequencing

of the unemployment spell. Model 1 was therefore estimated separately for obser-

vations deﬁned by diﬀerent values of job-spell number (results not shown). As

expected, the eﬀect of having relatives in North America only becomes signiﬁcantly

positive in the later unemployment episodes—job-spell numbers 3 or 4. At the same

time, the eﬀects of gender and human capital (i.e., income and education) are only

signiﬁcant in earlier episodes—job-spell numbers 1 or 2—and decline to non-

signiﬁcance thereafter.

MULTIEPISODE MODELS 429

c12.qxd 8/27/2004 2:57 PM Page 429

MODELING INTERVAL-CENSORED DATA

As alluded to previously, survival time may be a discrete rather than a continuous

variable. Or, even if theoretically continuous, all that may be recorded is whether or

not events have been experienced in some interval of time. In either case, the data

are described as interval-censored and the appropriate model is a discrete-time haz-

ard model. However, a distinction should be made from the outset between a

discrete-time analysis vs. a discrete-time model. If survival time is truly discrete, the

model being estimated is a discrete-time model. If survival time, although interval-

censored, is truly continuous, on the other hand, then employing the discrete-time

approach results only in a discrete-time analysis. That is, the model for a discrete-

time process is being used to approximate the underlying continuous-time model.

Discrete-Time Hazard Model and Estimation

Suppose that survival time for the ith case, denoted again by T

, is now a discrete

variable representing time periods in which events occur. For example, in the study

of time to promotion among college professors by Long et al. (1993), T

was a dis-

crete variable measured in single years, since promotions take eﬀect at the beginning

of the academic year. Survival time was therefore measured in years from the date

of hire (for promotion to associate professor) or from the previous promotion (for

promotion to full professor) until the current promotion or censoring. If, in the union

disruption data, survival time were only measured in six-month intervals, then

⫽ 1,2,3,...,would capture the six-month interval in which separations or cen-

soring occurred. In either case, the discrete-time hazard, P

, is deﬁned as a proba-

bility

⫽ P(T

⫽ t 冟 T

ⱖ t). (12.8)

That is, P

is the conditional probability that an event occurs at time (or in time inter-

val) t to the ith case given that no event occurs before time t (Allison, 1982).

A popular model for the discrete-time hazard as a function of covariates is the logit

model, which employs the log-odds of event occurrence at time t as the response

ᎏ

1 ⫺

ᎏ

⫽ α(t) ⫹ x

⬘

ββ

, (12.9)

where α(t) represents some function of time, and x

⬘

ββ

, as in the Cox model, repre-

sents a weighted sum of covariates times parameters, excluding an intercept term.

The term α(t) captures the manner in which the log-odds of event occurrence

depends on time. As explained below, this term is quite ﬂexible and can take a

variety of forms. Also, note that when the probability of an event in any given time

interval is small, the log-odds of event occurrence is approximately equal to the log-

hazard of event occurrence. Why? Notice that when P

is small, say a value of .05

or less, then 1 ⫺ P

is approximately 1. For example, 1 ⫺ .05 ⫽ .95 ⬇ 1. Therefore,

/(1⫺ P

) ⬇ P

/1 ⫽ P

. Hence, equation (12.9) can be regarded as a model for the

430 MULTISTATE, MULTIEPISODE, AND INTERVAL-CENSORED MODELS

c12.qxd 8/27/2004 2:57 PM Page 430

log-hazard of event occurrence given small P

’s in each interval, and interpreted

accordingly.

Likelihood Function for Discrete-Time Data. Estimation of equation (12.9) is typ-

ically performed using a transformed data set. To provide the rationale for this trans-

formation, I now consider the likelihood function for model (12.9). This explication

is, admittedly, algebraically tedious. But it is key to the reader’s understanding of the

procedure employed to estimate discrete-time models. To continue, let δ

be a cen-

soring indicator for the ith case, coded 1 if an event occurs at time t

, and 0 if the case

is censored at time t

. The likelihood function for model (12.9) is

ββ

冟t, x

) ⫽

兿

i⫽1

[P(T

⫽ t

)]

[P(T

⬎ t

)]

1⫺

. (12.10)

Now, for those who experience the event at time t

, their contribution to the likeli-

hood is P(T

⫽ t

). For those who are censored at t

, the contribution is P(T

⬎t

Further, if P

is the probability that the ith case experiences the event at time t

, then

1 ⫺ P

is the probability that no event is experienced at t

. And by probability rules,

the probability of surviving to time t

without experiencing the event is simply the

product of (1 ⫺ P

) over all values of T

up to time t

. That is,

P(T

⬎ t

) ⫽

兿

j =1

(1 ⫺ P

) (12.11)

and the probability that the event does not occur before t

but then occurs at time t

is then P

times the probability of surviving to time t

⫺ 1:

P(T

⫽ t

) ⫽P

兿

⫺1

j⫽

(1 ⫺ P

(12.12)

Substituting expression (12.11) for P(T

⬎ t

) and expression (12.12) for P(T

⫽ t

)

into equation (12.10), the likelihood function can be expressed as

ββ

冟t,x

) ⫽

兿

i⫽1

冤

兿

⫺1

j⫽

(1 ⫺ P

)

冥

冤

兿

j=1

(1 ⫺ P

)

冥

1⫺δ

. (12.13)

Taking logs of both sides of equation (12.13), we have the log-likelihood for the

discrete-time model:

ln L(

ββ

冟t,x

) ⫽

冱

⫽

冦

冤

兿

⫺1

j⫽1

(1 ⫺ P

)

冥

⫹ ln

冤

兿

j⫽1

(1 ⫺ P

)

冥

1⫺δ

冧

⫽

冱

⫽

冤

ln P

⫹ δ

冱

⫺1

j⫽1

ln (1 ⫺ P

) ⫹ (1 ⫺ δ

)

冱

j⫽1

ln (1 ⫺ P

)

冥

⫽

冱

⫽

冤

ln P

⫹ δ

冱

⫺1

j⫽1

ln (1 ⫺ P

) ⫹

冱

j⫽1

ln (1 ⫺ P

) ⫺ δ

i 冱

j⫽1

ln (1 ⫺ P

)

冥

MODELING INTERVAL-CENSORED DATA 431

c12.qxd 8/27/2004 2:57 PM Page 431

⫽

冱

⫽

冤

ln P

⫹ δ

冱

⫺1

j⫽1

ln (1 ⫺ P

) ⫹

冱

j⫽1

ln (1 ⫺ P

) ⫺ δ

ln (1 ⫺ P

)

⫺ δ

冱

⫺1

j⫽1

(1 ⫺ P

)

冥

⫽

冱

⫽

冤

ᎏ

1⫺

ᎏ

⫹

冱

j⫽1

ln (1 ⫺ P

)

冥

ln L(β 冟t, x

) ⫽

冱

⫽

ᎏ

1⫺

ᎏ

⫹

冱

⫽

冱

j⫽1

ln (1 ⫺ P

(12.14)

Now, suppose that we deﬁne a dummy variable, y

, such that y

⫽ 1 if the ith case

experiences an event at time t

, and 0 otherwise. We can then rewrite equation

(12.14) as

ln L(

ββ

冟t,x

)

⫽

冱

⫽

冱

j⫽1

ᎏ

1⫺

ᎏ

⫹

冱

⫽

冱

j⫽1

ln (1 ⫺ P

(12.15)

Notice that the second sum in the ﬁrst term on the right-hand side of equation (12.15)

will be zero until time t

, at which point the ﬁrst term on the right-hand side of equa-

tion (12.15) will be identical to the ﬁrst term in equation (12.14). However, equation

(12.15) is now in a recognizable form. Consider the log-likelihood function for the

logistic regression model, which I will denote as ln L

[see equation (7.13)].

Letting π

⫽ exp(x

⬘

ββ

)/[1 ⫹ exp(x

⬘

ββ

)] in equation (7.13), we have

ln L

⫽

冱

⫽1

ln π

⫹ (1 ⫺ y

)ln(1⫺ π

)]

⫽

冱

⫽1

ln π

⫹ ln (1 ⫺ π

) ⫺ y

ln (1⫺π

)]

⫽

冱

⫽1

ᎏ

1 ⫺

ᎏ

⫹

冱

⫽1

ln (1 ⫺ π

(12.16)

What the reader should now recognize is that the log-likelihood for the discrete-time

hazard model in equation (12.15) is just the log-likelihood for a logistic regression

model [e.g., equation (12.16)] for which the units of analysis are time periods rather

than individuals. This means that model (12.9) is estimated by ﬁrst converting per-

son-level data, in which people are the analytical units, into person-period data, in

which people’s time periods are the analytical units. One then estimates equation

(12.9) as an ordinary logistic regression model using the person-period data (as

demonstrated below).

Approximating a Continuous-Time Process. If time is truly discrete, equation

(12.9) is an appropriate model. However, if time is truly continuous but the data are

432 MULTISTATE, MULTIEPISODE, AND INTERVAL-CENSORED MODELS

c12.qxd 8/27/2004 2:57 PM Page 432

interval-censored, the appropriate model is the interval-censored Cox model

(Allison, 1995; Hosmer and Lemeshow, 1999):

ln[⫺ln(1 ⫺ P

)] ⫽ α(t) ⫹ x

⬘

ββ

, (12.17)

which is simply the complementary log-log model applied to person-period data.

If survival times are generated by the Cox model and then grouped into intervals,

the corresponding model for P

is equation (12.17) (Allison, 1982, 1995). This

implies that the parameter vector,

ββ

, in equation (12.17) is identical to that in the

underlying Cox model and can therefore be given the same interpretation.

Moreover, the parameter vector in equation (12.17) is invariant to interval length,

which is not true of model (12.9). This doesn’t mean that sample estimates of

equation (12.17) are invariant to interval length, but rather only that the underly-

ing theoretical model is (Allison, 1982). At any rate, equation (12.17) is the cor-

rect model to use whenever time is theoretically continuous and the Cox model is

presumed to underlie the data.

Converting to Person-Period Data

The process of converting from person- to person-period data is straightforward. It

merely involves expanding individual observations by the number of time intervals

they contribute to the process under study. That is, a given case’s data record is repli-

cated (including the original record) as many times as its number of intervals of sur-

vival time. Covariate values are either duplicated in each replication if they are

time-invariant, or changed appropriately on successive replications if they are time-

varying. A binary indicator of event occurrence takes the value 0 in each replication

if the case is censored. If the case is uncensored, the event indicator is coded 0 in

each replication except the ﬁnal one, in which it is coded 1. A variable, T, repre-

senting the time-interval number, is also recorded with the values 1, 2, 3,...,τ,

where τ represents the last time interval observed for that case. Once the data are

converted, the event indicator can be regressed on a set of covariates, using model

(12.9) or (12.17). The term α(t) in the model utilizes some parameterization for T.

The most general parameterization is simply to represent T with a series of τ ⫺ 1

dummies in the model. This is equivalent to the unspeciﬁed function of time implicit

in the Cox model.

As an example, let’s consider the conversion of the data on union disruption for

the 1230 NSFH couples from Chapter 11. In this case, time is theoretically continu-

ous, which is why the Cox model was used for the analyses in Chapter 11. Suppose,

however, that we simulate the situation in which survival time is recorded only in

six-month intervals. We begin by recoding survival time for each couple according

to the number of six-month intervals it represents. For the moment I ignore left trun-

cation and treat inception of risk as beginning at the wave 1 survey. The maximum

time recorded from wave 1 until disruption or censoring was 86 months. I therefore

created a variable, INTERVAL, ranging from 1 to 15, representing survival time—

in six-month increments—for each couple. Those whose survival time was less than

MODELING INTERVAL-CENSORED DATA 433

c12.qxd 8/27/2004 2:57 PM Page 433

or equal to six months were coded 1, those surviving more than six months but no

more than a year were coded 2, and so on. I then replicated each couple’s data up to

a maximum of 15 times, depending on their value of INTERVAL. For example, if

INTERVAL was 7, the couple’s record was replicated seven times. If INTERVAL

was 15, it was replicated 15 times, and so on. In the resulting couple-period data set,

INTERVAL was relabeled TIME. The event indicator, named DISRUPTD, was

coded as 0 on each replicated record for censored cases. For uncensored cases, DIS-

RUPTD was coded 0 for each replicated record except the last, where it was coded

1. In all, 12,480 records were produced in the couple-period data set. Table 12.3 fur-

ther illustrates the conversion process.

Panel A in the table shows the couple-level data for three couples. The ﬁrst couple,

ID 18, is a continuously cohabiting couple who separated 33 months after wave 1.

They had been together nine months before the initial survey. The second couple was

cohabiting at wave 1 and had been living together three months before the start of

the survey. They then married 13 months after wave 1, and separated 38 months after

wave 1. The third couple had been married for 21 months prior to wave 1 and was

censored 67 months after wave 1 by the wave 2 survey. This couple had a birth to

the union 51 months after the initial survey.

Panel B shows the couple-period version of the data. To account for left-trunca-

tion, survival time prior to the start of the study was used to update the TIME vari-

able accordingly for each couple. Hence, as couple 18 was not observed until the

second six-month interval of their risk period, TIME begins with the value 2 for this

couple. Their total survival time was 9 ⫹ 33 ⫽ 42 months, which puts their survival

time in the seventh interval. Hence, the maximum TIME value for this couple is 7.

Notice that DISRUPTD is coded 0 for times 2 through 6 and then changes to 1 when

TIME ⫽ 7. The second couple, ID 630, had been together for only three months before

wave 1. As they are still coming under observation in the ﬁrst time interval, TIME

begins at 1 for them. Couple 630 got married 3 ⫹ 13 ⫽ 16 months after inception of

the union, which puts their marriage in the third interval. Thus, the time-varying

indicator of transition to marriage, COHTOMAR, changes from 0 to 1 at TIME ⫽ 3.

As their survival time is 41 months, they are last observed when TIME ⫽ 7, at which

time DISRUPTD again changes to 1. Finally, the married couple’s survival time is

21 ⫹ 67 ⫽ 88 months. However, as they are observed for only 67 months, they con-

tribute only 12 (since 67/6 ⫽ 11.17) records to the expanded data set. Nevertheless,

they do not come under observation until the fourth time interval, hence TIME

begins at 4 for this couple. Their birth took place 21 ⫹ 51 ⫽ 72 months after incep-

tion of the marriage, which puts their birth in the 12th interval. Thus, the time-vary-

ing covariate UNBIRTH is coded 0 until TIME ⫽ 12, at which point it changes to 1.

They were last observed at TIME ⫽ 15, where DISRUPTD remains at 0. Notice that

time-invariant covariates are duplicated as is on all records for all couples.

Discrete-Time Analysis: Examples

Union Disruption. Table 12.4 presents the results of estimating discrete-time mod-

els for the log-hazard of union disruption using the couple-period data for our 1230

NSFH couples. The “logit model” is equation (12.9), employing 14 dummies for the

434 MULTISTATE, MULTIEPISODE, AND INTERVAL-CENSORED MODELS

c12.qxd 8/27/2004 2:57 PM Page 434