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

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COUNT-DATA MODELS THAT ALLOW FOR OVERDISPERSION 375

days. That is, let p

be the sample proportion exhibiting a count of y depression days

and pˆ

be the mean predicted probability of a count of y according to a count model,

where M is, variously, PRM, NBRM, ZIP, or ZINB. Then each line in Figure 10.6 is

a plot of p

⫺ pˆ

over the values of Y ⫽ number of depression days. The ideal model

is the one whose plot is closest to the zero line—the horizontal line in the middle of

the graph. The PRM is clearly a poor-ﬁtting model, particularly for 0, 1, or 2 depres-

sion days. The NBRM is somewhat better but still has a noticeable overprediction

for a count of one day. Both the ZIP and the ZINB models appear to have similarly

good ﬁts to the data, with ZINB being slightly superior to ZIP.

Hurdle Models

The hurdle model was introduced by Mullahy (1986) in the context of a discussion

of modiﬁed count models. In contrast to the approach taken by the zero-inﬂated

model, the hurdle model assumes that all cases are at risk for events and that the

probability of a zero count is governed by a stochastic process. However, the hurdle

model allows model regressors to have diﬀerent eﬀects on the likelihood of experi-

encing at least one event, as opposed to the likelihood of a subsequent event given

that at least one event has occurred. The idea is that experiencing at least one event

is a “hurdle” that requires a diﬀerent parameterization than is the case for experi-

encing subsequent events. As an example, the NVAW survey asked women respon-

dents to report the number of physical assaults they had ever experienced from any

type of oﬀender. For the 1343 minority women in the sample, the range was 0 to 124

assaults, with a mean of 2.081, a standard deviation of 10.035, and 71.7% reporting

no assaults. For this variable, it is not reasonable to suppose that there is a popula-

tion that is not at risk for assault. As this type of event is not under individual con-

trol, everyone is presumed at risk for an assault. However, the process generating

subsequent assaults may well be diﬀerent than the process aﬀecting the likelihood of

at least one assault. We will explore this using the Poisson hurdle model.

Parameterization of the Poisson Hurdle Model. In the Poisson hurdle model,

allowance is made for the rate of event occurrence to be diﬀerent for zero, as

opposed to positive, counts. The model for a zero count is, therefore,

f(0 冟 x

γγ

) ⫽

ᎏ

⫺µ

ᎏ

⫽ e

⫺µ

where µ

⫽ exp(冱γ

). The model for positive counts is then a truncated PRM

based on a diﬀerent parameter set, adjusted for the probability of a nonzero count

(Cameron and Trivedi, 1998). That is, for Y ⬎ 0,

f(y

冟 x

ββ

γγ

) ⫽ [1 ⫺ f(0 冟 x

γγ

)][f(y

冟 y

⬎ 0,x

ββ

]

⫽(1⫺e

⫺µ

)

ᎏ

⫺

)

ᎏ

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where µ

⫽ exp(冱β

). Now in the event that

γγ

⫽

ββ

, we have that µ

⫽ µ

, and the

hurdle model reduces to the ordinary PRM, since the speciﬁcation is now

f(0 冟 x

ββ

) ⫽

ᎏ

⫺µ

ᎏ

⫽ e

⫺µ

f(y

冟 x

ββ

) ⫽ (1⫺e

⫺µ

)

ᎏ

⫺

)

ᎏ

⫽

ᎏ

⫺µ

ᎏ

for y

⫽ 1,2,3,....

This means that the PRM is nested inside the Poisson hurdle model (King, 1989) and

a nested chi-squared test can be used to compare these models. The test is also a test

for the constraint that

γγ

⫽

ββ

, that is, that the covariates have the same eﬀects on the

probability of at least one count as they do on the probability of each additional

count. The test has K ⫹ 1 degrees of freedom.

Estimation. The likelihood function for the hurdle model is worth considering in

some detail, as it suggests how the model can be estimated, even in the absence of

specialized software. Let 0 represent the zero counts in the sample and y ⬎ 0 repre-

sent the positive counts. Then the likelihood function is of the form

ββ

,γ 冟y,x) ⫽

兿

y∈0

P(y ⫽ 0)

兿

y∈y⬎0

P(y ⬎ 0)P(y 冟 y ⬎ 0)

⫽

兿

y∈0

⫺µ

兿

y∈y⬎0

(1 ⫺ e

⫺µ

)

ᎏ

⫺

1)y!

ᎏ

which, in terms of the parameters is

ββ

,γ 冟y,x) ⫽

兿

y∈0

exp

冤

⫺exp

冢

冱

冣冥

兿

y∈y⬎0

冦

1 ⫺ exp

冤

⫺exp

冢

冱

冣冥冧

(10.14)

⫻

(10.15)

Of importance is that this partitions into two separate likelihoods that can be maxi-

mized independently. The reader should recognize expression (10.14) as the likeli-

hood for a complementary log-log model of the probability that Y is greater than zero

(see Chapter 7), whereas expression (10.15) is the likelihood for the truncated PRM.

Therefore, the Poisson hurdle model can be estimated in two steps: a complemen-

tary log-log model for the probability of a positive count, followed by a truncated

PRM for cases with positive counts.

Example: Number of Lifetime Physical Assaults. Table 10.5 presents estimates of

the regular PRM and of the Poisson hurdle model for the number of lifetime physi-

cal assaults for the 1343 minority women in the minority women dataset.

Explanatory variables are education, income, the child abuse index (as described

previously), plus dummy variables for employment status (other employment,

兿

y∈y⬎0

exp冢y冱β

冣

ᎏᎏ

冢冦

exp

冤

exp

冢

冱β

冣冥冧

⫺1

冣

376 REGRESSION MODELS FOR AN EVENT COUNT

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COUNT-DATA MODELS THAT ALLOW FOR OVERDISPERSION 377

unemployment, with full- or part-time employment as the reference group), dummy

variables for marital status (formerly married, never married, with currently married

as the reference group), and a dummy for having been raped (either with or without

penetration). The variable exposure is the log of the woman’s age in years, the ration-

ale being that the older a woman is, the longer she is at risk for assault. Exposure

should be positively related to the number of assaults. Moreover, I would expect

prior experience of victimization (e.g., having been abused as a child or having been

raped) to raise the expected number of assaults. Research suggests that prior victim-

ization may increase women’s tolerance of abusive situations, including those that

eventuate in physical aggression (Kalmuss, 1984).

The PRM suggests that several factors in addition to exposure aﬀect the mean num-

ber of assaults: child abuse, employment status, marital status, and the experience of

having been raped. However, signiﬁcance tests for the PRM will be too liberal if the

data are overdispersed, as these data appear to be, judging from the mean and standard

deviation reported above. The next two columns report the Poisson hurdle results. Of

interest is whether the hurdle model is an improvement over the PRM. The test is ⫺2

ln L

⫺ (⫺2lnL

), where model 1 is the PRM and model 2 is the Poisson hurdle model.

The log-likelihood for the hurdle model is simply the sum of the log-likelihoods for the

complementary log-log part and the truncated PRM part of the model. The test statistic

is therefore ⫺2[⫺6312.701 ⫺ (⫺3554.224 ⫺ 738.603)] ⫽ 4039.749. This statistic has

Table 10.5 Unstandardized PRM vs. Poisson Hurdle Estimates for the Regression of

Number of Lifetime Physical Assaults Experienced by Minority Women in the NVAW

Survey

Hurdle Model

PRM P(Pos. Count) Truncated PRM

Regressor bb b

Intercept ⫺.953*** ⫺1.306 ⫺4.285

Education .006 .037 ⫺.411

Income ⫺.018 .027 ⫺.236

Abused as a child .282*** .179*** 1.493***

Other employment

⫺.154** ⫺.117 ⫺.243

Unemployment

⫺.169* .202 ⫺1.259

Formerly married

.647*** .469*** 3.134

Never married

⫺.257*** .125 ⫺1.607

Raped .299*** .408*** .631

Exposure .283*** ⫺.149 3.109

Model χ

2423.127*** 122.890*** 882.762***

.161 .077 .110

for hurdle 4039.749***

Note: n ⫽ 1343.

Employed either full- or part-time is the reference category.

Currently married is the reference category.

* p ⬍ .05. ** p ⬍ .01. *** p ⬍ .001.

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378 REGRESSION MODELS FOR AN EVENT COUNT

the chi-squared distribution with 10 degrees of freedom under the null hypothesis that

γγ

⫽

ββ

(i.e., that the hurdle oﬀers no improvement over the PRM). It is highly signiﬁ-

cant (p ⬍ .0001), suggesting that the hurdle model is to be preferred. According to this

model, the probability of being assaulted is elevated by having been abused as a child,

having been raped, or having been married before. The last eﬀect is probably due to

the fact that an overwhelming proportion of physical assaults on women are perpe-

trated by intimate partners. Given that a woman has been assaulted, however, only

having been abused as a child increases the expected number of assaults. This ﬁnding

is consistent with others’ work (see, e.g., Kalmuss, 1984).

EXERCISES

10.1 Recall that for a discrete variable, Y, the sum of f( y) over the values of Y

equals 1.0, where f( y) is the density of Y. Show that the truncated Poisson

density has this property.

10.2 Show that the ZIP density has the property mentioned in Exercise 10.1.

10.3 Suppose that Y has a Poisson density with mean equal to 3.2. Find

(a) P(Y ⫽ 0); (b) P(Y ⫽ 3); (c) P(Y ⫽ 10).

10.4 Suppose that Y has a Poisson density with mean equal to 1.5. Find

(a) P(Y ⫽ 0); (b) P(Y ⫽ 3); (c) P(Y ⫽ 10).

10.5 Suppose that Y has a negative binomial density with parameters r ⫽ 4,

p ⫽ .1. Find (a) P(Y ⫽ 0); (b) P(Y ⫽ 3); (c) P(Y ⫽ 10).

10.6 Suppose that Y has a negative binomial density with parameters r ⫽ 1,

p ⫽ .45. Find (a) P(Y ⫽ 0); (b) P(Y ⫽ 3); (c) P(Y ⫽ 10).

10.7 Suppose that Y has a negative binomial density and let r ⫽ 1. (a) Find p such

that E(Y) ⫽ 3.2. Then ﬁnd (b) P(Y ⫽ 0); (c) P(Y ⫽ 3); (d) P(Y ⫽ 10).

10.8 For a PRM with covariates X

, X

, ..., X

, show that the multiplicative

change in E(Y ) for a c-unit increase in X

is exp(β

c).

10.9 Give the log of the likelihood function for the truncated PRM.

10.10 Prove that 1 ⫺ µe

⫺µ

/(e

⫺ 1) is always less than 1 for µ ⬎ 0. (Hint: Start by

assuming that the assertion is true, and manipulate the inequality until a

more obviously correct assertion appears.)

10.11 Greene (1998) gives the following recursion formula for the probabilities

of various Y-values under the PRM: p

⫽ exp(⫺µ); p

⫽ (µ/j)(p

j ⫺ 1

) for j ⫽ 1,

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2, .... Show that this recursion is equivalent to the formula e

⫺µ

/y! for

y ⫽ 0, 1, 2, 3, 4.

10.12 Use Greene’s recursion from Exercise 10.11 to ﬁnd f

( y 冟 x,β

) for y ⫽ 0, 1,

2, 3, 4 if µ

⫽ 1.75.

10.13 Greene (1998) gives the following recursion formula for probabilities under

the NBRM:

⫽

冢

ᎏ

⫺

⫹

ᎏ

冣

⫺1

, p

⫽

ᎏ

⫺1

⫹

j⫺1

ᎏ

⫺1

⫹ µ

ᎏ

j⫺1

If α

⫽ 3.5806 and µ

⫽ 1.75, give the estimated probabilities for y ⫽ 0, 1, 2, 3,

4, based on the NBRM, using this recursion.

10.14 Sherman et al. (1992) randomly assigned 1200 domestic violence oﬀenders

to arrest versus no arrest treatments and then followed them for 6 to 18

months posttreatment. The response of interest was the number of subse-

quent police reports to a local women’s shelter for the same oﬀender.

Employing the NBRM, their estimated model was

ln µ

⫽ β

⫹ gⴕx ⫹ .198 arrest ⫹ .261 employed ⫹ .026 married

⫺ .434 arrest * employed ⫺ .3 arrest * married,

where arrest, employed, and married are all dummy variables for these

respective statuses, and gⴕx represents the other terms in the model. Ln L for

this model was ⫺1218.77, while ln L for the model without the two cross-

product terms was ⫺1222.29.

(a) Test whether the interactions of married and employed with arrest are

signiﬁcant as a block.

(b) Interpret the interaction eﬀects with respect to the conditional mean of Y.

10.15 King (1988) employed the PRM to examine several predictors of the num-

ber of members of the U.S. House of Representatives who switched political

parties in a given year for the years 1802–1876. The exposure variable was

the log of the number of members of the House in each year. His estimated

model was ln µ

⫽ β

⫹ gⴕx ⫹ 3.49 exposure.

(a) Let N ⫽ the number of members of the House of Representatives in a

given year, and let the estimated equation, in general, be ln µ

⫽ β

⫹

gⴕx ⫹ d exposure, and derive an expression for the proportionate change

in the rate of event occurrence for each additional member in the House,

net of the other covariates.

(b) Let N ⫽ 250, and estimate the proportionate change in the expected

number of switchers for each additional member of the House of

Representatives, using King’s equation.

EXERCISES 379

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10.16 Show that

冱

P(Y ⫽ j) ⫽ 1 for j⫽ 1,2,...,in the truncated negative binomial

density.

10.17 For the 416 couples in the couples dataset, a PRM for the number of chil-

dren under 18 in the household produces the following results:

⫽ exp[1.9939 ⫺ .5976 cohabiting ⫺ .0428 male’s age at union

⫺ .0389 union duration ⫺ .0054 household income],

where cohabiting is a dummy for cohabiting as opposed to being married,

male’s age at union and union duration are measured in years, and house-

hold income is in thousands of dollars. All eﬀects are signiﬁcant at p ⬍ .05.

(a) Interpret the cohabiting eﬀect.

(b) Give the predicted probability of zero, one, and two children for a cou-

ple married for 12 years with a household income of $150,000, in which

the male was 18 at inception of the marriage.

(c)LnL⫽⫺620.6443 for the intercept-only model versus ⫺555.5196 for

the hypothesized model. Test the null hypothesis that all regression

coeﬃcients equal zero, and give R

10.18 Cameron and Trivedi’s (1990) regression-based test suggests that the res-

ponse in exercise 10.17 is overdispersed. An NBRM for the same data

produces the following estimates:

⫽ exp[2.1295 ⫺ .6295 cohabiting ⫺ .0457 male’s age at union

⫺ .0427 union duration ⫺ .0056 household income],

⫽ .1812, σ

⫽ .0987, and ln L ⫽⫺552.5279.

(a) Do both Wald and likelihood-ratio chi-squared tests for equidispersion

versus overdispersion. What do you conclude?

(b) Repeat Exercise 10.17(b) based on the NBRM model.

10.19 A ZIP model for the data in Exercises 10.17 and 10.18 produces the follow-

ing results:

ᎏ

1⫺

ψˆ

ᎏ

⫽⫺12.4046 ⫹ 2.1386 cohabiting ⫹ .2064 male’s age at union

⫹ .2716 union duration ⫹ .0054 household income;

⫽ exp[.7758⫺ .3345 cohabiting ⫺ .0085 male’s age at union

⫺ .0225 union duration ⫺ .0085 household income],

and V ⫽ 10.0016 for ZIP/PRM.

(a) Interpret the model with respect both to the probability of being in the

structural-zero group and to the expected count given that one is in the

at-risk group.

(b) Which model is to be preferred: the ZIP or the PRM?

for the ZIP. [Hint: Use

additional information from Exercise 10.17(c).]

380 REGRESSION MODELS FOR AN EVENT COUNT

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10.20 Repeat Exercise 10.17(b) based on the ZIP results.

For Exercises 10.21 to 10.25, use the 1485 cases in the inmates’ dataset to model the

number of class III tickets (NUM3TIX) received by inmates as a function of LOG-

TIME (the exposure factor), ETHNIC, EDUCCL, FIRSTARR, CENAGE,

CENAGESQ, SMALL, MEDIUM. [Also, you need software (e.g., LIMDEP, STATA)

for count models.]

10.21 Estimate the PRM for the number of class III tickets.

(a) Show the regression estimates and their signiﬁcance levels.

(b) Perform Cameron and Trivedi’s (1990) regression-based test for equidis-

persion and give your conclusion.

, R

, and/or R

10.22 Estimate the NBRM for these data.

(a) Show the regression estimates and their signiﬁcance levels.

(b) Test the null hypothesis of equidispersion versus overdispersion via both

Wald and likelihood-ratio tests.

10.23 Estimate the ZIP model for these data.

(a) Show estimates for ln [ψ/(1 ⫺ ψ)] and for µ.

(b) Which is to be preferred, ZIP or PRM?

10.24 Estimate the Poisson hurdle model for these data.

(a) Show coeﬃcient estimates both for the probability of a positive count

and for the truncated PRM.

(b) Test the hurdle model against the PRM.

or the hurdle?

10.25 Construct a graph after the fashion of Figure 10.6 to compare the ﬁt of PRM,

NBRM, and ZIP to the observed sample proportions for the ﬁrst 11 values

of NUM3TIX (i.e., values 0 to 10).

EXERCISES 381

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382

Regression with Social Data: Modeling Continuous and Limited Response Variables,

By Alfred DeMaris

CHAPTER 11

Introduction to Survival Analysis

CHAPTER OVERVIEW

In Chapter 10 the response of interest was the number of events occurring in some

ﬁxed period of time. This chapter is concerned, instead, with the waiting, or survival,

time in the nonevent state until some event occurs. For example, rather than focus-

ing on the number of subsequent domestic violence oﬀenses committed in a given

follow-up period by a sample of those arrested for domestic assault (Sherman et al.,

1992), we might instead wish to model the time from release from arrest to the ﬁrst

incidence of recidivism. Or, equivalently, we may be interested in modeling the risk

of recidivating at any given time, conditional on having survived up to that moment

without recidivating. The corpus of tools for analyzing such data is typically col-

lected under the rubric of survival analysis, and the risk of an event occurring at any

given time is referred to as the hazard of the event. These terms stem from the bio-

medical literature, where many of the techniques were ﬁrst developed and where the

event of interest was often death. In the social sciences, another name for this body

of techniques is event history analysis. These techniques have been used in the social

science arena to study the occurrence of such events as the formation of the ﬁrst mar-

ital or cohabiting union (Lamb et al., 2003), transitions out of cohabiting unions

(DeMaris, 2001; Sanchez et al., 1998), the role of premarital cohabitation in marital

disruption (Bennett et al., 1988), and gender diﬀerences in the promotion process in

academic settings (Long et al., 1993), to cite a few examples.

Because event history data present some unique challenges for the data analyst, I

begin this chapter by deﬁning terms and acquainting the reader with the nature of sur-

vival data. I then illustrate the major concepts of survival analysis, such as survival

and hazard functions, using the life table technique, and employ as an application the

event of disruption from a marital or cohabiting union. I then consider regression

models for event history data, beginning with a parametric model but moving quickly

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to the much more commonly employed Cox semiparametric regression model. Again,

the event of union disruption is used to illustrate the major ideas. The Cox model is

explored in some detail, along with strategies for dealing with time-varying covari-

ates, nonproportional eﬀects of covariates, and left truncation. Plots of survival curves

and hazard functions along with examples from published research are used through-

out to help the reader assimilate the techniques as they are used in practice.

NATURE OF SURVIVAL DATA

Key Concepts in Survival Analysis

Survival, or event history, data consist of observations on when events occur to a sam-

ple of people over time. An event is a qualitative change in state. In the study of exist-

ing marriages, for example, the event of divorce is a change from the married to the

unmarried state. In the study of academic promotions, the event of promotion is a

change in state from, say, associate professor to full professor, and so on. In the ideal

scenario we would follow a collection of people from the time when they ﬁrst become

susceptible to a change in state until they all eventually experience that change. Some

events lend themselves readily to this protocol. A sample of patients diagnosed with

a terminal illness, for example, may be followed from diagnosis with the illness until

death, an event that is sure to occur to all patients if the study continues for suﬃcient

time. Hence survival times can usually be observed for everyone in the sample.

However, many events in the social sciences, such as ﬁrst marriage, divorce, or

pregnancy, are not inevitable; a number of people fail to experience the event regard-

less of study length. In these cases the survival time is said to be right-censored: All

that is known is that survival time is greater than the last recorded time for a partic-

ular person. For example, if a sample of marriages is followed for 20 years and a

given marriage is still intact at the end of the study, that marriage’s survival time is

censored at 20 years. This means that the couple’s survival time is not exactly

observed but is known only to be greater than 20 years. A critical assumption is that

censoring is noninformative; that is, the process leading to being censored is inde-

pendent of the hazard of event occurrence. If this assumption holds, censored cases

are representative of all other persons surviving up until the same time, controlling

for relevant covariates (Collett, 1994). In studies of marital dissolution, for example,

couples’ survival times are typically considered censored upon one partner becom-

ing widowed. As the probability of dying is not normally presumed to be aﬀected by

marital instability, this type of censoring is typically considered noninformative.

I have been referring to “following” people or cases over time as though survival

data are exclusively prospective. This is not at all the case. Much of the data on event

histories in the social sciences are collected retrospectively. For example, the NSFH

interview schedule contains long sections exploring respondents’ marital and cohab-

itation histories, job histories, fertility histories, and so on. As long as signiﬁcant

events can be associated with speciﬁc dates in a case’s biography, event histories can

be constructed after the fact.

NATURE OF SURVIVAL DATA 383

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384 INTRODUCTION TO SURVIVAL ANALYSIS

A word is in order about the use of the term time. In event history analysis, time

typically refers to the duration in a given state rather than calendar time. Nevertheless,

sometimes “time” refers to calendar time. In fact, a critical concept in survival analy-

sis is the calendar time at which a case is ﬁrst exposed to the risk of an event. I refer

to this time as the inception of risk. Once the inception of risk is given, survival time

is then calculated as the calendar time at which the event of interest occurs minus the

inception of risk. If cases are lost to follow-up or the study ends before an event

occurs, the time at which the case was last observed minus the inception of risk gives

the censored survival time for that case.

In this chapter I moreover assume that time is a continuous variable, which gives

rise to continuous-time survival models. In practice, time is never measured so pre-

cisely as to be truly continuous, but it can be treated as such if measured ﬁnely

enough—say in days, weeks, or months. Use of a continuous-time model is predi-

cated, however, on the notion that an event can occur at any given time or after any

given amount of time has elapsed. However, some events can only occur at particu-

lar calendar times. For example, promotion in academic rank typically occurs only

at the beginning of the school year. In this type of process, survival time is a discrete

variable and the appropriate analytic technique is a discrete-time survival model.

Even if the true event-generating process is a continuous one, however, a discrete-

time analysis may still be appropriate if events are only known to occur in some

interval of time. Discrete-time analyses are considered in Chapter 12.

Nature of Event Histories

To further tease out critical concepts in survival analysis, I direct the reader to Figure

11.1, which provides a schematic of various event histories in a given study. For sim-

plicity, it is assumed in the ﬁgure that the event of interest is a single, nonrepeatable

event. Examples would be the ﬁrst instance of sexual intercourse or the ﬁrst marriage.

Although either of these events may be repeated several times, the ﬁrst instance of

either is a one-time aﬀair. The horizontal axis represents survival time. Each line stands

for a person’s event history and represents the time period from inception of risk until

either censoring or termination in the event of interest. These periods are also referred

to as episodes or spells. Each line moves from left to right in the ﬁgure, with vertical

bars on the left indicating known inceptions of risk and vertical bars on the right indi-

cating right-censoring times. An arrowhead at the end of the line indicates that the per-

son experienced the event of interest at a given time. Solid lines represent durations

observed in a given study, while dotted lines reﬂect survival time that is outside the

study. Individuals a, b, c, and e experience inception of risk at t

, which is also the

beginning of observation, or start time for the study. I consider these people ﬁrst, as

they are the most common kinds of observations. Case a represents an observation that

is right-censored by the ending date of the study, denoted t

. Case b, in contrast, is

right-censored by virtue of either being lost to follow-up, or by experiencing a diﬀerent

type of event that removes him or her from the risk for the event of interest (e.g., the

death of a spouse removes couples from the risk for divorce). Case c experiences the

event of interest during the observation period and is referred to as being uncensored.

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