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

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EXERCISES 345

duration ⫽ 16.155. Verify the McDonald–Moﬃtt decomposition of regressor

eﬀects for this application, as discussed in the text.

9.10 Means for the regressors used in Table 9.4 are child abuse index ⫽ 1.776, age ⫽

39.868, education ⫽ 4.786, nonwhite ⫽ .201, previously married ⫽ .334, never

married ⫽ .153, income ⫽ 4.031. Verify the McDonald–Moﬃtt decomposition

of regressor eﬀects for this application, as discussed in the text.

Exercises 9.11 to 9.13 involve the following (x,y) pairs: (1.5, 0), (2.2, 0), (1.8, 0),

(3.5, 1.25), (.82, 3.42), (3.85, 8.65). Also, tobit estimates for the regression of y on x

yield

⫽⫺

4.85827,

⫽

2.24803,

⫽

4.53832. Denote this information as Data One.

9.11 Using Data One, calculate Laitila’s R

9.12 Using Data One, calculate [corr(y, yˆ)]

, where yˆ is the predicted value for the

observed y’s, based on equation (9.23). Notice that this R

analog is for the

observed y’s, whereas R

is for Y*.

9.13 Give the McDonald–Moﬃtt decomposition for the impact of x on y in Data One.

Exercises 9.14 to 9.16 involve the following (x,y) pairs: (8, 0), (6.2, 0), (7, .4), (5.5,

.55), (5.8, .62), (2.3, .89). Also, tobit estimates for the regression of y on x yield

⫽

1.38552,

⫽⫺

.17864,

⫽

.26422. Denote this information as Data Two.

9.14 Using Data Two, calculate Laitila’s R

9.15 Using Data Two, calculate [corr(y, yˆ)]

, where yˆ is the predicted value for the

observed y’s, based on equation (9.23).

9.16 Give the McDonald–Moﬃtt decomposition for the impact of x on y in Data Two.

9.17 In the analyses reported in Table 9.7, PTSD was treated as an uncensored,

continuous response. Ignoring, for the moment, the problem of incidental

truncation, suppose that we treat PTSD as censored at its minimum value of

21, noting that 29.3% of women score at this value. A tobit analysis for

n ⫽ 330 women produces the following results:

Mean of X

Intercept 29.708

Minor physical assault ⫺3.390 .924

Severe physical assault 6.939** .455

Took time oﬀ due to assault 8.472*** .242

Child abuse index 1.244** 2.139

Rape with penetration 4.393* .333

Income ⫺1.275** 3.933

17.531

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346 TRUNCATED AND CENSORED REGRESSION MODELS

(a) Give the predicted observed PTSD score for a woman who has experi-

enced minor and severe physical assault, rape with penetration, and tak-

ing time oﬀ because of the physical assault, and who has average income

and child abuse scores.

(b) To what extent is the eﬀect of child abuse due to its eﬀect on elevating

average PTSD for those above the censoring threshold, as opposed to its

eﬀect on raising the risk of reporting any PTSD symptoms at all?

9.18 Suppose that y* ⫽ β

⫹ β

⫹ ε and z*⫽ γ

⫹ γ

⫹

⫹ γ

⫹ υ, where Cov(x

,ε) ⫽ Cov(x

,υ) ⫽ Cov(ε,υ) ⫽ 0, all vari-

ables are standardized, all x’s are positively correlated, and all parameters

except β

, β

, γ

,and γ

are positive, with the latter being negative. But you

estimate y*⫽ β

⫹ β

⫹ e and z*⫽ γ

⫹ γ

⫹ u. What is ρ

[Hint: Compute corr(e,u) ⫽ cov(e,u) using covariance algebra, and evaluate

the sign of the resulting expression.]

9.19 Suppose that y* ⫽ β

⫹ β

⫹ ε and z* ⫽ γ

⫹ γ

⫹

⫹ υ, where Cov(x

,ε) ⫽ Cov(x

,υ) ⫽ Cov(ε,υ) ⫽ 0, all variables are stan-

dardized, all x’s are positively correlated, and all parameter values are positive.

But you estimate y* ⫽ β

⫹ β

⫹ e and z* ⫽ γ

⫹ γ

⫹ u. What

is ρ

? (Hint: Follow procedure for Exercise 9.18.)

9.20 Refer to equation (9.33) and the accompanying discussion of sample-

selection bias. Suppose that the models for y*, z* are y* ⫽ 2 ⫹ 3.2x ⫹ ε if

z* ⬎ 0; y* is missing otherwise; z* ⫽ .8x ⫹ υ. Also, σ

⫽ 17.64, the squared

correlation of y* with x is .726, σ

⫽ 1.25, ρ ⫽ .66, and E(λ) ⫽ 3⫺1.5x. What

is the bias in b

as an estimate of β

if no correction is made for incidental

truncation, where bias(b

) is deﬁned in Exercise 9.21? Will b

over- or under-

estimate β

? [Hint: In general, in the simple linear regression of y on x,

⫽ Cov(x,y) Ⲑσ

9.21 Refer to equation (9.33) and the accompanying discussion of sample-selection

bias. Suppose that the models for y* and z* are: y* ⫽ 2 ⫹ 3.2x ⫹ ε if z* ⬎ 0; y*

is missing otherwise; z* ⫽ .8x ⫹ υ. Also, σ

⫽ 2.2, σ

⫽ 1.25, ρ ⫽ .66, and

Cov(x,λ) ⫽ .75. What is the bias in b

as an estimate of β

if no correction is

made for incidental truncation [where bias(b

) ⫽ plim b

⫺ β

]? Will b

over-

or underestimate β

? (Hint: See the hint for Exercise 9.20.)

9.22 Use the female subsample of the couples dataset to estimate a tobit model for

CESD as a function of FEDUC, IHTOT2, VIOLENT, UNSTABLE, DIS-

AGMT, and PRESCHDN. Then:

(a) Interpret the signiﬁcant eﬀects.

(b)Give R

for the underlying depressive symptomatology score.

(d) Give the predicted value of observed CESD at the mean of the regressors.

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9.23 In the students dataset, 15.7% of the 235 students are missing EXAM2 scores

because they dropped out of the course before the second exam. If the target

population is all students enrolling in introductory statistics at BGSU, a

regression of EXAM2 using only those with valid scores could be biased by

self-selection. Estimate a sample-selection model for EXAM2 regressed on

EXAM1, COLGPA, SCORE, and STATMOOD using either the ML or Heckit

procedure, along with uncorrected OLS. Use CLASSIF as the unique regres-

sor for the selection equation. Missing imputation: Substitute the parentheti-

cal values for missing data on each variable indicated: EXAM1 (76.9774766),

COLGPA (3.0827835), and SCORE (40.9358974). Evaluate the extent and

nature of any selection bias found.

9.24 Use the GSS98 dataset to estimate the regression of INCOME on EDUCAT,

RESPAGE, and MALE using OLS. In that 34.7% of the 2832 respondents are

missing on INCOME, correct for selection bias with either the ML or Heckit

procedure, employing CONSERV as the unique regressor for the selection

equation. Missing imputation: Substitute the parenthetical values for missing

data on each variable indicated: EDUCAT (13.251) and CONSERV (4.098).

Evaluate the extent and nature of any selection bias found.

9.25 Use the introductory sociology dataset to estimate the regression of T2 test anx-

iety (TESTANX2) on HSGPA, HSGRADE, COLLGPA, MALE, ACADSE1,

and TESTANX1. Because 36.5% of the 649 students are missing on TES-

TANX2, correct for selection bias with either the ML or Heckit procedure,

employing PAYOWN as the unique regressor for the selection equation.

Evaluate the extent and nature of any selection bias found.

EXERCISES 347

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348

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

By Alfred DeMaris

CHAPTER 10

Regression Models for

an Event Count

CHAPTER OVERVIEW

In this chapter we return to the study of discrete response variables, in particular,

count responses, variables that represent the number of occurrences of some event.

The number of events that occur to each case in a given observation period, or in a

given observation space, is assumed to be governed by an underlying rate of event

occurrence. In this chapter we focus on the outcome of this process, the count of

events over the period or the space. In Chapter 11 we focus on the waiting time from

the beginning of risk for an event until event occurrence, or the survival time in

the “nonevent” state. These chapters are therefore tied together by their common

reliance on an unobserved continuous rate of event occurrence. Examples of count

variables are the number of articles published during graduate school by male and

female biochemists (Long, 1990), the number of domestic violence incidents

reported for a given oﬀender in a 6- to 22-month follow-up of domestic violence

oﬀenders (Sherman et al., 1992), or the number of police contacts for a sample of

juveniles aged 8 to 26 (Land et al., 1996).

I begin by deﬁning count data and presenting probability distributions that are

commonly associated with count responses. I then consider why OLS is not opti-

mal for these types of data, and present instead the Poisson regression model. The

likelihood function for the model is illustrated, along with schemes for interpreting

the coeﬃcients, and analogs of R

. Truncated, censored, and sample-selected vari-

ants of the model are also discussed. Due to the restrictive property of Poisson

regression that the conditional mean and variance of the dependent variable be

identical, the Poisson model is rarely adequate for count data, which are frequently

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DENSITIES FOR COUNT RESPONSES 349

overdispersed; that is, the variance exceeds the mean. For this reason, I then con-

sider a few variations on the Poisson model. One is the negative binomial regres-

sion model, which is appropriate when data are generally overdispersed. I also

consider a version of this model designed for truncated count distributions. Finally,

for cases in which overdispersion is due to an excessive number of zero counts, I

consider zero-inﬂated Poisson and negative binomial models, as well as the hurdle

model.

DENSITIES FOR COUNT RESPONSES

An event count is the number of occurrences of an event within a ﬁxed domain of

observation (King, 1988). The domain can either be temporal or spatial. Hence, the

number of partners one has had sex with in the past month is an event count, where

the event is having a sex partner, and the domain is one month. From above, the

number of domestic violence incidents in a 6- to 22-month period is an event count

in which the event is the incident and the domain varies between 6 and 22 months.

The number of taverns per city in a given state is, similarly, an event count, with the

event being the occurrence of a tavern and the domain being city size, most likely

expressed as the number of inhabitants. An example of an event count from the

students dataset is the number of previous college-level math courses taken by each

student. In this case, the domain is assumed to be the student’s college career.

This variable, for 230 students, ranges from 0 to 7, has a sample mean of 1.274, and

a sample variance of 1.274. The sample distribution on the variable is shown in

Figure 10.1.

We notice right away that the distribution is right-skewed and that there is a siz-

able proportion of students, about 24%, who have had no previous math classes.

These features are fairly typical of count data. Often, our ﬁrst impulse with this type

of response might be to use linear regression with OLS estimation. For reasons to

be detailed below, this would be a poor choice of estimator. Another strategy might

be to collapse the variable into a dichotomy, contrasting taking one or fewer classes

with taking more than one class. One might then use logistic regression to analyze

this binary version of the variable. But this would be quite wasteful of information,

since we would be treating 0 or 1 class as the same answer, and we would also be

lumping two through seven classes into the same category. Fortunately, neither

strategy is necessary, as count variables are well represented by certain speciﬁc dis-

crete densities, which can then be used as the basis of a likelihood function for a

regression model.

Poisson Density

For a random variable, Y, the Poisson density is

f(y 冟µ) ⫽

ᎏ

⫺

ᎏ

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350 REGRESSION MODELS FOR AN EVENT COUNT

Figure 10.1 Distribution on number of previous math courses for 230 students in introductory statistics

(mean ⫽ 1.2739; variance ⫽ 1.2740).

for y ⫽ 0, 1, 2, . ... The parameter µ is both the mean and the variance of Y. Figure

10.2 shows a Poisson density with mean and variance equal to µ⫽ 1.274. Notice that

it resembles the sample distribution of number of previous math courses in Figure

10.1. The probability of a given value for Y is readily computed using this density.

For example, the probability that Y ⫽ 5, according to this density, is

f(5 冟µ ⫽ 1.274) ⫽

ᎏ

⫺1.27

.274

ᎏ

⫽ .0078.

The sample proportion of students having ﬁve previous math classes, on the

other hand, is .022. The Poisson density is constrained so that its variance equals its

mean, a property called equidispersion (Long, 1997). The next density relaxes that

restriction.

Negative Binomial Density

A second density that is very important for count data is the negative binomial

density. This density can be expressed in a variety of diﬀerent ways (see, e.g., Hoel

et al., 1971). Moreover, Cameron and Trivedi (1998) note that there are 13 distinct

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DENSITIES FOR COUNT RESPONSES 351

stochastic mechanisms for generating it. One such mechanism is as follows. Deﬁne

r to be an integer and p to be a number between 0 and 1. Then the negative binomial

density arises if we let y be the number of failures encountered before achieving the

rth success in a series of trials, where p is the probability of a success:

f(y 冟r, p) ⫽

ᎏ

(

⫹

⫺

)

ᎏ

(1 ⫺ p)

(10.1)

for y ⫽ 0, 1, 2, ...(Hoel et al., 1971). The negative binomial density is said to be

overdispersed, meaning that the variance exceeds the mean. This is evident, since for

the negative binomial density,

E(y) ⫽ r

ᎏ

1 ⫺

ᎏ

and V(y)⫽r

ᎏ

⫺

ᎏ

In that, for p between 0 and 1, p

⬍ p, the variance is larger than the mean.

Figure 10.3 shows a negative binomial distribution with parameters r ⫽ 2 and

p ⫽ .61089. As the reader can verify, the distribution again has a mean of 1.274, but

this time its variance is 2.0853. For this distribution, the probability that Y ⫽ 5 is

f(5 冟r ⫽ 2, p ⫽ .61089) ⫽

ᎏ

(

⫹

⫺

)

ᎏ

.61089

(1 ⫺ .61089)

⫽ .02.

Figure 10.2 Poisson distribution for a discrete variable with mean and variance equal to 1.2739.

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352 REGRESSION MODELS FOR AN EVENT COUNT

MODELING COUNT RESPONSES WITH POISSON REGRESSION

Problems with OLS

With Y as an event count, the linear regression model is Y ⫽ 冱β

⫹ ε. There are sev-

eral problems with this formulation, especially when using OLS estimation. First, the

usual assumption that the errors are normally distributed fails, since Y is typically

nonnormal, as is evident in Figure 10.1. With large samples, however, the parameter

estimates are approximately normally distributed, by the CLT, so inferences regard-

ing the regression parameters are still possible. Nevertheless, there are more serious

diﬃculties with using a linear function. The mean structure—the nature of the func-

tion relating the conditional mean to the linear predictor—is probably misspeciﬁed

with a linear model, given that a negative mean is not possible. Recall that as men-

tioned in Chapter 7, the right-hand side of the linear regression equation is not

constrained to be positive. If the mean structure is misspeciﬁed, OLS estimators

are inconsistent (Cameron and Trivedi, 1998). The OLS estimator also assumes a

homoscedastic error structure. This is problematic if the data are generated by a count

distribution such as the Poisson or negative binomial. In such distributions, the

Figure 10.3 Negative binomial distribution for a discrete variable with mean equal to 1.2739 and

variance equal to 2.0853.

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variance is a function of the mean, and as such, varies as the mean varies. If the errors

are really heteroscedastic, the standard error estimates produced by OLS are biased.

Poisson Regression Model

The Poisson regression model (PRM) assumes that the occurrence of events within

a given domain is governed by an unobserved, continuous process. Although the

process is unobserved, its end product—an event count over the domain of interest—is

observed. The process is encapsulated in µ

, the rate of event occurrence for the ith case.

The model makes two assumptions about this process (King, 1989). The independence

assumption is that the probability of a subsequent event is independent of the occur-

rence of a previous event. (Contagion is the situation in which the probability of a

subsequent event is enhanced by a previous event, whereas negative contagion obtains

when this probability is reduced by a previous event.) The homogeneity assumption is

that the rate of event occurrence is constant over the domain of interest, which is usu-

ally time. (In Chapter 11 we relax this particular assumption about event processes.)

The Poisson model assumes that the number of events for the ith case follows a

Poisson distribution with parameter µ

, and that µ

is an exponential function of the

covariates. Hence, the density for Y

f(y

冟x

ββ

) ⫽

ᎏ

⫺

ᎏ

for y

⫽ 0, 1, 2, . ..and the speciﬁcation for µ

is µ

⫽ exp(冱β

). The rate of event

occurrence, µ

, is conditional on the covariates for each case. Hence, µ

is both the

conditional mean and the conditional variance of Y

. The exponential speciﬁcation

ensures that the mean is always positive. Also, recall from Chapter 1 that in the gen-

eralized linear model, if the response has a Poisson distribution, the link function is

the log of the mean, which implies an exponential function for the mean itself [i.e.,

ln µ

⫽ 冱β

implies that µ

⫽ exp(冱β

)].

If the domain for the event process is not the same size for all cases, this needs to

be controlled, since a greater number of events would be expected the larger the

domain size. The domain size is the length of time that cases are at risk for events,

or the size of the geographical area over which events are counted, and so on. With

varying domain sizes, the solution is to model the expected count per domain size

(Cameron and Trivedi, 1998; King, 1988). If E

represents the exposure, or domain

size, for the ith case, the model becomes

ᎏ

⫽ exp

冢

冱

k⫽1

冣

which implies that

⫽ E

exp

冢

冱

k⫽1

冣

⫽ exp(ln E

)exp

冢

冱

k⫽1

冣

⫽exp

冤冢

冱

k⫽1

冣

⫹ ln E

冥

. (10.2)

MODELING COUNT RESPONSES WITH POISSON REGRESSION 353

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This suggests that the log of E

should be entered into the model with its coeﬃcient

constrained to 1. In some software (e.g., SAS) E

is referred to as an oﬀset when it

is to be entered into the model in this way. However, as others have noted (King,

1988), there is no harm in allowing the coeﬃcient for ln E

to be estimated. In fact,

in the author’s experience, the model is less likely to have convergence problems if

this parameter is estimated rather than constrained.

Estimation. The parameters of the model—the β’s—can be estimated via maximum

likelihood. The likelihood function follows from the joint density of the observed y

Letting y represent the vector, or set, of observed event counts, x represent the vector

of explanatory variables (including, possibly, ln E

), and

ββ

represent the vector of

parameters, we have

ββ

冟y,x) ⫽

兿

i⫽1

ᎏ

⫺

␮

ᎏ

which implies that the log-likelihood is

ln[L(

ββ

冟 y, x)]⫽

冱

i⫽1

ln µ

⫺ µ

⫺ ln y

!) ⫽

冱

i⫽1

冤

冢

冱

k⫽0

冣

⫺ exp

冢

冱

k⫽0

冣

⫺ ln y

冥

The MLEs of the betas are the b

that maximize this function. As with all MLEs, the

resulting estimates are consistent as well as asymptotically unbiased, eﬃcient, and

normally distributed.

Inferences in the Poisson Regression Model. As in logistic regression, the test for

the signiﬁcance of the model as a whole is performed with the model chi-squared

statistic. The null hypothesis is H

: β

⫽ β

⫽

...

⫽ β

⫽ 0. The alternative hypothe-

sis is that at least one of these betas is not zero. The test statistic is χ

⫽⫺2

ln L

⫺ (⫺2lnL

), where L

is the likelihood function for a model with only an

intercept and L

is the likelihood function for the hypothesized model, evaluated at

the MLEs of the parameters. Under the null hypothesis, this statistic has a chi-squared

distribution with K degrees of freedom. Comparison of nested models is achieved

with the nested chi-squared test. If model 2 is nested within model 1, the test for

whether the nesting constraints are valid is ∆χ

⫽⫺2lnL

⫺ (⫺2lnL

), with L

the

likelihood for the nested model, evaluated at its MLEs. Under the null hypothesis

that the constraints are valid, this statistic has the chi-squared distribution with

degrees of freedom equal to the number of constraints imposed to produce the nested

model. Finally, individual regression coeﬃcients can be tested with z-tests of the

form z ⫽ b

/σ

Interpretation of Regression Coefficients. As in logistic regression, the betas no

longer can be interpreted as the change in the mean for a unit increase in the pre-

dictors, due to the nonlinear functional form of the model. This is easily seen by

attempting to simplify the expression for the change in the mean when one of the

regressors, say X

, increases by 1 unit, holding all other (K ⫺ 1) regressors constant.

354 REGRESSION MODELS FOR AN EVENT COUNT

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