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

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CENSORED REGRESSION MODEL 325

where, without loss of generality, c is typically taken to be zero. [See Long (1997)

for a discussion of the case in which censoring involves an upper threshold, and for

a more general development in which the threshold can be any value. Regardless of

the value of c, though, the response can always be rescaled so that the censoring

threshold is zero, as was done above in creating the simulated data.] Thus, the

observed response, Y

, is deﬁned as

⫽

冦

0ifY

ⱕ 0, (9.18)

if Y

⬎ 0. (9.19)

In other words, what we observe are only zero or positive responses, whereas the

model of interest, in equation (9.17), pertains to an outcome that can, in theory at

least, take on a wider range of values. The task, as with truncated data, is to estimate

the parameters of equation (9.17) using incomplete information on the response. A

naive approach might be either to attempt to estimate equation (9.17) with all of the

observations using OLS, or to estimate equation (9.17) using OLS after limiting the

sample to those with positive scores on the response. As we will see below, either

approach will result in biased and inconsistent estimates of the conditional mean

function.

Social Science Applications

The social science literature is replete with examples of response variables that are

treated as censored. An interesting use of the tobit model is Fair’s (1978) analysis of

the impact of various factors on leisure time spent in extramarital aﬀairs among a

sample of married women. In one of his analyses, the dependent variable was a

measure of the rate of extramarital sex per length of marriage, and constructed as

[(number of diﬀerent extramarital partners ⫻ frequency of sexual relations with

each)/number of years married]. Those reporting no aﬀairs had values of zero and

were treated as censored cases. In this instance, we might deﬁne Y* as the desired

rate of extramarital sex, which is observed in the form of an actual rate only when

some threshold value is crossed. Another example is Walton and Ragin’s (1990)

study of protest severity in response to austerity programs imposed in debtor nations

to stabilize the economy. In their sample of 60 debtor nations, 26 had positive val-

ues on the dependent variable of protest severity, an index based largely on journal-

istic accounts of protest activities in each country. Countries with no recorded

austerity protests were assigned a value of 0 on the response. These cases were those

whose protests were “. . . not severe enough to be recorded in international media”

(Walton and Ragin, 1990, p. 884).

Mean Functions

Given the model in equations (9.17) to (9.19), let’s derive the functions for the con-

ditional mean of Y, given Y

⬎ 0, and for the conditional mean of Y among all obser-

vations.

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326 TRUNCATED AND CENSORED REGRESSION MODELS

Regression Function for E(Y

冟冟

⬎⬎

0, x

). The regression function for the condi-

tional mean of the positive responses is as follows. The model implies that Y is pos-

itive only if Y

⫽ x

⬘

ββ

⫹ ε

⬎ 0, which implies that ε

⬎⫺x

⬘β. Thus:

E(Y

冟Y

⬎ 0, x

) ⫽ E(x

⬘

ββ

⫹ ε

冟ε

⬎⫺x

⬘

ββ

)

⫽ x

⬘

ββ

⫹ E(ε

冟ε

⬎⫺x

⬘

ββ

) ⫽ x

⬘

ββ

⫹ σλ (α

). (9.20)

This result follows from applying equation (9.15), where c ⫽ 0. Notice that this is

just the truncated regression model again, since we are conditioning on Y* being

greater than zero. In this case, however, α

simpliﬁes to

⫽

ᎏ

0 ⫺

⬘

ββ

ᎏ

⫽

ᎏ

⫺x

⬘

ββ

ᎏ

, (9.21)

and therefore

λ(α

) ⫽

ᎏ

1 ⫺

φ(

⫺

(

⫺

⬘

ββ

⬘

ββ

)

/σ)

ᎏ

⫽

ᎏ

φ(

(

⬘

ββ

)

ᎏ

, (9.22)

with the last expression resulting from the symmetry of the normal distribution.

Regression Function for E(Y

冟冟

). The regression function for Y among all obser-

vations can be obtained by realizing that equation (9.17) implies that Y* is normally

distributed with mean x

⬘

ββ

and variance σ

. We can then apply equation (9.7) with

“X” equal to Y* and “c” ⫽ 0 to arrive at (Greene, 2003)

E(Y

冟x

) ⫽ Φ

冢

ᎏ

⬘

ββ

ᎏ

冣

⬘

ββ

⫹ σλ (α

)]. (9.23)

This is the censored regression, or tobit model. Here, again, the symmetry of the nor-

mal distribution allows us to write

1 ⫺ Φ

冢

ᎏ

⫺x

⬘

ββ

ᎏ

冣

冢

ᎏ

⬘

ββ

ᎏ

冣

It should now be clear that attempts to estimate regression models for Y using OLS with

either all of the data or only the cases in which Y is positive will result in biased and

inconsistent estimators due to the additional terms shown in equations (9.20) and (9.23).

Estimation

Unlike the case in truncated regression, censored regression models involve measures

of the explanatory variables for all cases. With this information it would actually be

possible to estimate λ(α

) for each case based on a probit analysis of the probability that

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CENSORED REGRESSION MODEL 327

Y* is uncensored. This estimate could then be substituted for λ(α

) in equation (9.20)

so that the model could be estimated with OLS [see Breen (1996) for further details].

This two-step technique is not necessary in this case, however, as the log-likelihood

function for the tobit model is quite well-behaved. That function is (Greene, 2003)

ln L(

ββ

,σ 冟 y

) ⫽

冱

⬎ 0

⫺

ᎏ

冤

ln (2π) ⫹ ln σ

⫹

冢

ᎏ

⫺

⬘

ββ

ᎏ

冣

冥

⫹

冱

⫽0

冤

1⫺Φ

冢

ᎏ

⬘

ββ

ᎏ

冣冥

In this expression, the ﬁrst sum represents the contribution of the uncensored obser-

vations to the log-likelihood, whereas the second sum represents the contribution of

the censored observations. Maximizing this function with respect to the parameters

(

ββ

and σ) of the model results in estimators with the usual asymptotic properties of

MLEs (Greene, 2003). It should be mentioned that tobit estimates based on the nor-

mal distribution may be inconsistent if the equation disturbance is nonnormal.

Estimation under diﬀerent distributional assumptions for the error term is possible

in programs such as LIMDEP or PROC LIFEREG in SAS.

Interpretation of Parameters

The eﬀects of individual regressors can be interpreted with respect to three diﬀerent

mean functions in the tobit model (McDonald and Moﬃtt, 1980). The ﬁrst is for the

mean of Y*, the underlying response; the second, shown in expression (9.20), is for

the mean of the positive observations; and the third, shown in equation (9.23), is for

the mean of all of the observed responses, the Y

Based on equation (9.17), the eﬀect of the jth regressor on the mean of Y* is

ᎏ

∂

ᎏ

E(Y

冟x

) ⫽ β

;

that is, the regression coeﬃcients have the usual interpretation as the change in the

mean of Y* for a unit increment in X

, net of the other regressors.

For the second function, the marginal, or partial, eﬀect of X

ᎏ

∂

ᎏ

Ε(Y

冟Y

⬎ 0, x

) ⫽ β

{1 ⫺ z

λ(z

) ⫺ [λ(z

)]

}, (9.24)

where

⫽

ᎏ

⬘

ββ

ᎏ

and λ(z

) ⫽ λ(α

) is deﬁned in equation (9.22).

The marginal eﬀects for the third function are of the form

ᎏ

∂

ᎏ

E(Y

冟x

) ⫽ Φ(z

)β

. (9.25)

The marginal eﬀects of independent variables shown in equations (9.24) and (9.25)

are no longer constant, but depend on the values of all variables in the models,

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including X

. The reason for this is that they are functions of z

, which changes across

covariate patterns. Most often, however, these eﬀects would be evaluated at the

means of the regressors. That is, for z

we substitute

苶苶

⫽

ᎏ

苶

⬘⬘

ββ

ᎏ

, (9.26)

with x

苶

being the vector of regressor means.

McDonald and Moffitt’s Decomposition. McDonald and Moﬃtt (1980) suggested

a useful decomposition for the eﬀect of the jth regressor on E(Y

冟x

). They showed

that equation (9.25) could be expressed as

ᎏ

∂

ᎏ

E(Y

冟x

) ⫽ Φ(z

)

ᎏ

∂

ᎏ

E(Y

冟Y

⬎ 0,x

) ⫹ E(Y

冟Y

⬎ 0,x

)

ᎏ

∂

ᎏ

Φ(z

). (9.27)

Note that Φ(z

) is the probability of being uncensored for the ith case. Thus, equation

(9.27) shows that the eﬀect of X

on the mean of the observed Y

is a weighted sum of

its eﬀect on the mean of the positive observations, weighted by the probability of being

uncensored, and its eﬀect on the probability of being uncensored, weighted by the mean

of the positive observations. It is therefore possible to partition the eﬀect of X

into its

constituent parts. McDonald and Moﬃtt further simpliﬁed this partition by showing

that the proportion of X

’s eﬀect due to its eﬀect on the mean of the positive observa-

tions is simply 1 ⫺ z

λ(z

) ⫺ [λ(z

)]

. Once this is estimated, the proportion of X

’s eﬀect

due to its inﬂuence on being uncensored is just 1 minus this value. Again, this partition

is usually accomplished at the mean of the regressors by substituting z

苶苶

,deﬁned in equa-

tion (9.26), for z

in this expression. Examples are given in the applications below.

Analog of R

A useful measure of discriminatory power for the tobit model has been proposed by

Laitila (1993). His pseudo-R

, denoted R

, is essentially identical to R

, presented

in Chapter 7 in connection with logistic regression. As in that situation, the measure

captures the variance in the underlying continuous variable, Y*, accounted for by the

structural part of the model. Denoting the tobit coeﬃcient estimates by b

, we have

⫽

ᎏ

冢

冱

冢

冱

冣

⫹

冣

ᎏ

where the numerator is just the variance of the linear predictor, and the denominator

is an estimate of the variance of Y* and consists of the variance of the linear predic-

tor plus the estimated variance of the conditional errors. Laitila (1993) comments

that this measure is bounded by 0 and 1 and has an explained-variance interpretation

with respect to Y*. However, it may decrease with the addition of regressors to the

model, particularly when they are irrelevant to the response. The calculation of R

demonstrated in the applications below.

328 TRUNCATED AND CENSORED REGRESSION MODELS

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Alternative Speciﬁcation

The tobit model imposes a constraint that may or may not always be reasonable.

Consider the probability that a case is uncensored. In the tobit model, that probabil-

ity is

P(Y

⬎ 0

冟x

) ⫽ Φ

冢

ᎏ

⬘

ββ

ᎏ

冣

whereas the mean of the positive values, from expression (9.20), is

E(Y

冟Y

⬎ 0, x

) ⫽ x

⬘

ββ

⫹ σλ (α

). (9.28)

We see that because

ββ

is the same in both functions, the coeﬃcients governing

whether a case is uncensored are constrained to be proportional to those governing

the mean of Y, given that Y is uncensored. That is,

ββ

is proportional to

ββ

/σ, with 1/σ

being the proportionality constant. This precludes some interesting possibilities. For

example, suppose that we are investigating the relationship between the age of an

oﬀender and the length of sentence given, for a sample of criminal oﬀenders. In that

one has to get caught in order to be sentenced, it may be that older oﬀenders are more

experienced at crime and therefore less likely to get caught. However, given that they

are caught, they may get longer sentences because they are perceived to be more incor-

rigible than younger oﬀenders. In other words, age may have opposite eﬀects on the

likelihood of being sentenced vs. the length of sentence given. The tobit model, on

the other hand, forces age to have the same kind of eﬀect on each component of the

process.

To free this constraint, Cragg (1971) proposed an alternative model, in which the

probability of being uncensored and the mean of the uncensored observations are

allowed to be governed by diﬀerent parameters. In this model, the probability of

being uncensored is assumed to follow a probit model with parameter vector

δδ

P(Y

⬎ 0 冟 x

) ⫽ Φ(x

⬘

δδ

where since σ and

δδ

are not separately identiﬁable in probit, σ is assumed to equal 1.

The mean function for the positive responses is then characterized by the truncated

regression model in equation (9.28), employing a diﬀerent parameter vector,

ββ

. If the

constraint is imposed that

δδ

⫽

ββ

/σ, the Cragg speciﬁcation reduces to the tobit model

(Greene, 2003; Smith and Brame, 2003). To estimate the model we simply use a pro-

bit model to estimate whether a case is uncensored, using all of the observations. We

then estimate a truncated regression model for the conditional mean of the response,

using only the uncensored observations. A test statistic for whether the tobit model is

valid, compared to the Cragg model, that is, a test statistic for the null hypothesis that

δδ

⫽

ββ

/σ, is constructed as

(∆df )

⫽⫺2lnL

tobit

⫺ [⫺2lnL

probit

⫹ (⫺2lnL

truncated

)],

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330 TRUNCATED AND CENSORED REGRESSION MODELS

where ∆df is the diﬀerence in the number of parameters estimated between the Cragg

and tobit models. If this result is signiﬁcant, the Cragg model is to be preferred. An

example is given below.

Simulated Data Example

As was the case with the truncated regression reported above for the simulated data,

the underlying model is Y*⫽⫺3.859 ⫹ 1.5X ⫹ ε. The last two columns of Table 9.1

show the results of estimating the underlying model using OLS on all of the obser-

vations, compared to using the tobit model. (The tobit model was estimated using

PROC LIFEREG in SAS.) As is evident, the OLS estimates are all too low, whereas

the tobit estimates, like those for the truncated regession model, are close to the true

parameter values. We can apply the McDonald–Moﬃtt decomposition to partition

the eﬀect of X into its eﬀect on the probability of being uncensored and its eﬀect on

the conditional mean of the positive responses. For these data, x

苶

⫽ 3.024, so z

苶苶

⫽

[⫺ 3.717 ⫹ 1.465(3.024)]/2.018 ⫽ .353. Now φ(.353) ⫽ .375, whereas Φ(.353) ⫽ .638.

Hence, λ(.353) ⫽ .375/.638 ⫽ .588. The proportion of X’s eﬀect that is due to its

eﬀect on the conditional mean of the positive Y’s is, therefore, 1 ⫺ .353(.588) ⫺

.588

⫽ .447. So about 45% of X’s eﬀect is due to its impact on the conditional mean,

while 55% is due to its impact on the probability of being uncensored. Finally, the

OLS estimate of P

is also an underestimate of the parameter. Laitila’s pseudo-R

,on

the other hand, at .478, is fairly close to the true value of .496.

In this particular case, because the data were truly generated by the tobit model,

the Cragg speciﬁcation should be no improvement over tobit. Or, to put it diﬀerently,

the tobit model should ﬁt no worse than the Cragg model. Minus twice the log-like-

lihood for the tobit model is 3018.49, while ⫺2 ln L for the probit model of whether

a case is uncensored is 974.923, and ⫺2 ln L for the truncated regression model

(shown in the column “MLE: truncated sample” in Table 9.1) is 2043.234. The test sta-

tistic is therefore a chi-squared variate equal to 3018.49 ⫺ (974.923 ⫹ 2043.234) ⫽

.333. The degrees of freedom for the test are ﬁgured as follows. We estimate two

parameters for the probit model (intercept and slope) and three parameters for the

truncated regression model (intercept, slope, and σ), whereas we estimate just three

parameters in tobit (intercept, slope, and σ). The diﬀerence in parameters estimated

is 5 ⫺ 3 ⫽ 2. As a χ

equal to .333 with 2 df is quite insigniﬁcant, the hypothesis that

δδ

⫽

ββ

/σ would not, in this case, be rejected—a correct decision.

Applications of the Tobit Model

Two response variables that can exhibit considerable censoring are depressive sympto-

matology and the severity of physical assaults. In both cases, paper-and-pencil measures

may simply not be sensitive enough to capture scores at the lower end of the construct.

Therefore, both variables will typically exhibit a large number of zero scores.

Depressive Symptomatology. Table 9.3 presents the results of regressing depressive

symptomatology—tapped by the Center for Epidemiological Studies Depression Scale

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CENSORED REGRESSION MODEL 331

(CESD) (Mirowsky and Ross, 1984)—on gender and characteristics of the relation-

ship for the couples in the couples dataset. Of the 416 couples, 64, or 15.4%, had

scores of zero on the scale and are considered censored cases. The CESD was meas-

ured for the main respondent, who could have been either the man or the woman. The

purpose of the current analysis was to examine whether relationship conﬂict has the

same eﬀect on depression for women as it does for men. I expect that open disagree-

ment will have a stronger eﬀect on depression for women, because they are typically

more sensitive than men are to the tenor of the relationship. Tobit estimates for a

model with gender, open disagreement, and their interaction—plus controls (a

dummy for being in an unstable relationship and a continuous measure of relation-

ship duration)—are shown in the column labeled “model 1.” OLS estimates are also

shown, for comparison purposes. As is evident, gender, open disagreement, and their

interaction are all signiﬁcant. The tobit eﬀects are somewhat larger than those esti-

mated by OLS. As expected, open disagreement has a stronger eﬀect on depressive

symptomatology for women than for men. According to Tobit model 1, for men the

eﬀect is simply the main eﬀect of open disagreement, 6.753. For women, the eﬀect is

6.753 ⫹ 6.494 ⫽ 13.247, a substantially larger value.

As open disagreement is centered, the main eﬀect of “female” suggests that at aver-

age levels of disagreement, women have mean depressive symptomatology that is about

3.4 points higher than men’s. But is there a gender diﬀerence in depression when there

is little open disagreement? This question is easily answered by employing targeted cen-

tering. One standard deviation of open disagreement is .601. Let’s calculate the gender

diﬀerence in depressive symptomatology at 1 standard deviation below mean disagree-

ment, or at a value of ⫺.601 for centered open disagreement. Subtracting ⫺.601 from

Table 9.3 Unstandardized OLS and ML Estimates for Censored Regression

Models of Depressive Symptomatology

OLS

Tobit (ML) Estimates

Regressor Estimates Model 1 Model 2

Intercept 9.842*** 7.667*** 3.608

Female 2.685 3.371* ⫺.532

Open disagreement

6.202** 6.753**

Female ⫻ open disagreement 6.157* 6.494*

Unstable relationship .807 1.668 1.668

Relationship duration .083 .077 .077

(Open disagreement ⫹ 1 SD) 6.753**

Female ⫻ (open disagreement ⫹ 1 SD) 6.494*

14.542 16.305 16.305

.148

.145 .145

Note: n ⫽ 416.

Centered variable.

* p ⬍ .05. ** p ⬍ .01. *** p ⬍ .001.

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332 TRUNCATED AND CENSORED REGRESSION MODELS

open disagreement is equivalent to adding .601 to open disagreement, hence the new

variable is shown as “open disagreement ⫹ 1 SD” in Table 9.3. We then create the cross-

product of female ⫻ (open disagreement ⫹ 1 SD) and include these variables in the

model in place of open disagreement and female ⫻ open disagreement. The results are

shown in model 2. The main eﬀect of “female” is now the eﬀect of being female among

couples who are 1 SD below the mean of open disagreement. The eﬀect is ⫺.532 and

is no longer signiﬁcant. Apparently, when there is little conﬂict in the relationship, there

is no gender diﬀerence in depressive symptomatology.

Additional Issues.

Laitila’s R

is easily calculated, since the variance of the linear

predictor (not shown) is 45.024, and the estimated error variance is 16.305

⫽ 265.853.

is thus 45.024/(45.024 ⫹ 265.853) ⫽ .145. Using McDonald and Moﬃtt’s decom-

position of eﬀects (calculations are left as an exercise for the reader), we ﬁnd that 46%

of regressor eﬀects are due to their impact on the probability of exhibiting any depres-

sive symptomatology, while 54% are due to their inﬂuence on mean depressive symp-

tomatology for those above the threshold. In other words, slightly more than half of the

eﬀects of the explanatory variables is due to their inﬂuence on the conditional mean.

The test of tobit against the Cragg speciﬁcation (not shown) in this case is highly

signiﬁcant ( p ⬍ .0001), suggesting that the tobit constraint is not justiﬁed for these

data. Inspection of the Cragg coeﬃcients (not shown) suggests that they are all sub-

stantially larger than those in Table 9.3; however, only the eﬀect of open disagreement

even approaches signiﬁcance (p ⬍ .07) in the Cragg model.

Physical Assault Severity. Table 9.4 presents the results of regressions of physical

assault severity, measured using the conﬂict tactics scale (CTS) (Straus, 1979), for

Table 9.4 Unstandardized OLS and ML Estimates for

Censored Regression Models of Physical Assault Severity

OLS Tobit (ML)

Regressor Estimates Estimates

Intercept 91.552*** 60.581*

Child abuse index 11.405*** 15.013***

Age ⫺.143 ⫺.329

Education ⫺9.290** ⫺12.715**

Nonwhite ⫺16.642* ⫺12.670

Previously married

6.355 12.341

Never married

⫺16.465 ⫺22.328

Income ⫺3.046* ⫺2.158

138.559 173.329

.048

.049

Note: n ⫽ 1779.

Currently married is the reference category.

* p ⬍ .05. ** p ⬍ .01. *** p ⬍ .001.

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1779 women in the victims dataset (NVAWS) who had been the victims of either

physical assault, sexual assault, stalking, or threats. Predictors include the child

abuse index (interval variable coded 1 to 9, with higher scores indicating more severe

abuse experienced as a child), age (in years), education (interval-level measure

coded from 1 ⫽ “no schooling” to 7 ⫽ “postgraduate”), nonwhite (dummy for being

nonwhite, as opposed to white), previously married, never married (currently mar-

ried is the reference category for both dummies), and annual income (interval vari-

able coded from 1 ⫽ “less than $5,000” to 10 ⫽ “over $100,000”). In this example,

30.7% of the cases are censored with values of zero for physical assault severity.

Both OLS and tobit estimates are shown.

Some eﬀects (e.g., those for child abuse and for education) are estimated to be

larger in magnitude in tobit, while others (e.g., for being nonwhite and for income)

are larger using OLS. The tobit results suggest that child abuse enhances physical

assault severity, whereas education reduces it. Aside from these eﬀects, none others

are signiﬁcant. McDonald and Moﬃtt’s decomposition (calculations are again left as

an exercise) suggests that 63.5% of regressor eﬀects are due to their inﬂuence on

reporting any violence, while the other 36.5% are due to their impact on the aver-

age level of violence among those reporting it. In this particular case, the Cragg

speciﬁcation could not be estimated, because the truncated part of the model would

not converge, even with the tobit estimates as start values. According to Wooldridge

(2000), however, one way to evaluate the appropriateness of the tobit speciﬁcation

informally is to compare the δ

from the probit analysis to β

/ σ

from tobit. If these

are not too diﬀerent from each other, the tobit model might be justiﬁed. Examining

the signiﬁcant eﬀects in this example, we compare eﬀects for child abuse (.087 in

tobit; .045 in probit) and for education (⫺.073 in tobit; ⫺.04 in probit). As these are

almost twice as large in tobit than in probit, we conclude, again that the tobit

speciﬁcation may not be justiﬁed.

SAMPLE-SELECTION MODELS

The sample-selection model consists of two equations. The substantive equation

describes the response of interest:

⫽ x

⬘

ββ

⫹ ε

, (9.29)

where ε

is assumed to be normally distributed with mean 0 and variance σ

However, whether Y

is actually observed or not depends on a second variable, Z

the selection propensity, whose equation is

⫽ w

⬘

γγ

⫹ u

, (9.30)

where u

is normally distributed with mean 0 and variance σ

, and w includes x as a

proper subset; that is, all elements of x are in w, but w may contain elements that are

not in x (Wooldridge, 2000). Now Y

is observed only if Z

is greater than zero. That

SAMPLE-SELECTION MODELS 333

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334 TRUNCATED AND CENSORED REGRESSION MODELS

is, the observed variable, Y

, is such that

is missing if Z

ⱕ 0,

⫽ Y

if Z

⬎ 0.

Further, the joint distribution of ε and u is bivariate normal, with correlation ρ. The

expected value of Y

is, therefore (Greene, 2003),

E(Y

冟Z

⬎ 0) ⫽ E(Y

冟u

⬎⫺w

⬘

γγ

)

⫽ x

⬘

ββ

⫹ E(ε

冟u

⬎⫺w

⬘

γγ

) ⫽ x

⬘

ββ

⫹ ρσ

λ(α

E(Y

冟Z

⬎ 0) ⫽ x

⬘

ββ

+θλ(α

), (9.31)

where

θ ⫽ ρσ

⫽

ᎏ

⫺

⬘

γγ

ᎏ

and

λ (α

) ⫽

ᎏ

φ(

(

⬘

γγ

)

ᎏ

Equation (9.31) follows from (9.9) by letting “y” equal ε,“z” equal u, and “a” equal

⫺w

⬘

γγ

, and recognizing that µ

is zero here. As in previous models, attempts to esti-

mate the mean function for the observed Y using OLS and the variables in x result

in omitted-variable bias, due to omission of the term λ(α

Conceptual Framework

The sample-selection model is employed to adjust for nonrandom selection into the

current sample. Self-selection into the sample is a pervasive condition in the social

sciences (Berk, 1983; Breen, 1996). Individuals selectively choose to respond to sur-

veys, to respond to particular questions in surveys, to be followed up in subsequent

waves of a survey, and so on. Such selectivity is a problem to the extent that unob-

served elements that determine selection into the sample also aﬀect the substantive

outcome, Y

(more on this below). As an example, DeMaris et al. (2003) tested an

integrated model of domestic violence using a sample of couples remaining intact

over both waves of the NSFH. In that the authors only included couples with non-

missing responses on the questions about intimate violence, their ﬁnal sample was

aﬀected by three diﬀerent sources of selectivity: panel attrition, couple attrition

(through divorce or separation), and item nonresponse. A total of 45% of couples were

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