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

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violence” by a factor of exp(.064) ⫽ 1.066. Or each unit increase in positive commu-

nication lowers the odds of “intense male violence” by exp(⫺.443) ⫽ .642, whereas it

lowers the odds of “any violence” by exp(⫺.441) ⫽ .643, a virtually indistinguishable

diﬀerence in eﬀects. If the eﬀects of predictors are invariant to the cutpoint, a more par-

simonious speciﬁcation of equation (8.3) is possible. This is

log O

ⱕj

⫽ β

⫹ β

⫹

...

⫹ β

. (8.4)

This is the ordered logit or proportional odds model (Agresti, 2002). In this model, the

eﬀects of predictors are the same, regardless of the cutpoint for the odds. That is, each

unit increase in a given predictor, say X

, multiplies the odds by a proportionality con-

stant of exp(β

), regardless of the cutpoint chosen. The results of estimating this model

(using procedure LOGISTIC in SAS) are shown in the last column of Table 8.7. Notice

that the intercept is allowed to depend on the cutpoint, so there are two intercepts in

the equation. (In fact, there are two diﬀerent equations, but the coeﬃcients are being

constrained to be the same in each.) In that predictors are assumed to be invariant to

the cutpoint, there is only one set of regression coeﬃcients. Eﬀects are interpreted just

as in binary logistic regression, except that the response is the log odds of “more

severe” versus “less severe” violence, rather than, as in Table 8.4, violence per se.

Thus, cohabiting is seen to raise the odds of “more severe” violence by exp(.909) ⫽

2.482, or about 148%, whereas each unit increase in positive communication lowers

the odds of “more severe” violence by about 35%.

Test of Invariance. In the ﬁrst two columns of Table 8.7, where eﬀects are allowed

to depend on the cutpoint, some predictors appear to have diﬀerent eﬀects on the

odds of “intense male violence” compared to violence per se. For example, the eﬀect

of cohabiting on the odds of “intense male violence” is exp(1.202) ⫽ 3.327, whereas

its eﬀect on the odds of “any violence” is only exp(.867) ⫽ 2.380. Moreover, other

regressors, for example, female’s age at union or economic disadvantage,have

signiﬁcant eﬀects on only one of the odds. Are these variations signiﬁcantly diﬀerent

or just the result of sampling error? This can be tested using the score test for the

proportional odds assumption (provided automatically in SAS). The test statistic

tests the null hypothesis that regressor eﬀects are the same across all J ⫺ 1 possible

cutpoints. That is, H

is that for each of the K regressors in the model, β

⫽ β

for

j ⫽ 1, 2, ..., J ⫺ 1. Under the null hypothesis, the score statistic is asymptotically

distributed as chi-squared with degrees of freedom equal to K(J ⫺ 2). This is the

diﬀerence in the number of parameters required to estimate the model in equation

(8.3) versus equation (8.4): K(J ⫺ 1) ⫺ K ⫽ K(J ⫺ 1 ⫺ 1) ⫽ K(J ⫺ 2). As shown in

Table 8.7 for the current example, its value is 12.457, which, with 11 degrees of free-

dom, is not signiﬁcant. This suggests that there is insuﬃcient evidence to reject the

proportional odds assumption. Apparently, the more parsimonious proportional-odds

model appears reasonable for the data.

Estimating Probabilities with the Proportional Odds Model. In the event that esti-

mates of P(Y ⫽ j) for j ⫽ 1, 2,..., J, based on the proportional odds model, are of

MULTINOMIAL MODELS 305

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interest, they are relatively straightforward to calculate. We draw on the probability

rule that for an integer-valued random variable, Y,P(Y ⫽ j) ⫽ P(Y ⱕ j) ⫺ P(Y ⱕ j ⫺ 1).

Now, note that, according to the proportional odds model,

log O

ⱕj

⫽ log

ᎏ

(

⬎

ⱕ

)

ᎏ

⫽ β

⫹ x

⬘

ββ

, (8.5)

where, in this case, x

⬘

ββ

⫽ β

⫹ β

⫹

...

⫹ β

. Thus, equation (8.5) implies

that

⫽⫽

⫽

ᎏ

P(Y

ⱕ

(

)

⬎

⬎ j)

ᎏ

⫽ P(Y ⱕ j).

Therefore, for j ⫽ 2,3,...,J ⫺ 1, the formula for P(Y ⫽ j) is

P(Y ⫽ j) ⫽

ᎏ

1⫹

(

g O

ⱕj

ⱕ

)

ᎏ

⫺

ᎏ

1⫹

(

g O

ⱕj

ⱕ

⫺

)

ᎏ

. (8.6)

However, for j ⫽ 1, the lowest ordered value of Y, the formula for P(Y ⫽ 1) is

P(Y ⫽ 1) ⫽

ᎏ

1⫹

(

g O

ⱕ1

ⱕ

)

ᎏ

(8.7)

since P(Y ⱕ 0) ⫽ 0. And for j ⫽ J, the highest-ordered value of Y, the formula for

P(Y ⫽ J), is

P(Y ⫽ J) ⫽ 1 ⫺ P(Y ⱕ J ⫺ 1) ⫽ 1 ⫺

ᎏ

1⫹

(

g O

ⱕJ

ⱕ

⫺

)

ᎏ

(8.8)

As an example, suppose that Y is a four-category ordered response, with values

j ⫽ 1, 2, 3, 4, and the model is

log O

ⱕ1

⫽⫺3.2 ⫹ .45X,

log O

ⱕ2

⫽⫺2.8 ⫹ .45X,

log O

ⱕ3

⫽⫺1.25 ⫹ .45X.

For X ⫽ 1.75, the four probabilities are

P(Y ⫽ 1) ⫽⫽.082,

exp[⫺3.2 ⫹ .45(1.75)]

ᎏᎏᎏ

1 ⫹ exp[⫺3.2 ⫹ .45(1.75)]

P(Y ⱕ j)/P(Y ⬎ j)

ᎏᎏᎏᎏ

[P(Y ⬎ j) ⫹ P(Y ⱕ j)]/P(Y ⬎ j)

P(Y ⱕ j)/P(Y ⬎ j)

ᎏᎏᎏ

1 ⫹ P(Y ⱕ j)/P(Y ⬎ j)

exp(logO

ⱕj

)

ᎏᎏ

1⫹exp(logO

ⱕj

)

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P(Y ⫽ 2) ⫽⫺

⫽ .118 ⫺ .082 ⫽ .036,

P(Y ⫽ 3) ⫽⫺

⫽ .386 ⫺ .118 ⫽ .268,

P(Y ⫽ 4) ⫽ 1 ⫺ .386 ⫽ .614.

It is easily seen that the four probabilities sum to 1, and that P(Y ⱕ j) ⱕ P(Y ⱕ j ⫹ 1)

for j ⫽ 1, 2, 3.

Alternatives to the Proportional Odds Model. If the score test for the proportional

odds assumption is signiﬁcant, so that the proportional odds assumption is unten-

able, what should one do? In this case the researcher has several options. First, he or

she can use the proportional odds model anyway, especially if noninvariant eﬀects

are only peripheral to the study. As an example, if I were interested primarily in how

open disagreement and positive communication aﬀect more severe violence, net of

other factors, it is clear that the proportional odds model in the last column of Table

8.7 summarizes these variables’ eﬀects on the cumulative logits in an elegant fash-

ion. Particularly when there are several categories of the response variable, equation

(8.4) is a substantially more parsimonious description of the data than any of the

other multinomial approaches. Nevertheless, it is frequently desirable to use a

diﬀerent modeling strategy when invariance is rejected. So one alternative is to

choose the most informative bifurcation of the response variable and proceed with

binary logistic regression. For example, either the ﬁrst or second column in Table 8.7

could be a legitimate model to estimate. Or, if it is especially important to preserve

the distinctions among diﬀerent response categories, the multinomial analysis pre-

sented in Table 8.5 is a workable strategy. Even if the response variable is ordinal,

use of the multinomial approach is always legitimate. At worst, we are only wasting

some information by not taking advantage of the ordering information in the values

of the response. Of course, if the response variable has at least ﬁve levels, its sam-

ple distribution is not too skewed, and the sample is large, the researcher may just

want to treat it as continuous and employ OLS.

Empirical Consistency and Discriminatory Power in Multinomial Models. As indi-

cated above, measures of empirical consistency (or goodness of ﬁt) are not as well

developed for multinomial models as they are for, say, binary logistic regression.

Therefore, one means of assessing ﬁt for the multinomial logistic regression of

unordered responses has already been suggested above. This is to use the Hosmer–

Lemeshow χ

statistic along with the LIML estimation technique to evaluate the

empirical consistency of each separate equation. For ordered outcomes, Lipsitz et al.

(1996) have recently developed a goodness-of-ﬁt test that is an extension of the

Hosmer–Lemeshow statistic to an ordinal response. The test is readily conducted using

existing software for the proportional odds model (e.g., SAS’s PROC LOGISTIC).

exp[⫺2.8 ⫹ .45(1.75)]

ᎏᎏᎏ

1 ⫹ exp[⫺2.8 ⫹ .45(1.75)]

exp[⫺1.25 ⫹ .45(1.75)]

ᎏᎏᎏ

1 ⫹ exp[⫺1.25 ⫹ .45(1.75)]

exp[⫺3.2 ⫹ .45(1.75)]

ᎏᎏᎏ

1 ⫹ exp[⫺3.2 ⫹ .45(1.75)]

exp[⫺2.8 ⫹ .45(1.75)]

ᎏᎏᎏ

1 ⫹ exp[⫺2.8 ⫹ .45(1.75)]

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308 ADVANCED TOPICS IN LOGISTIC REGRESSION

The reader is referred to the authors’ article for further details. Discriminatory power

is easier to handle. Any of the measures such as R

, R

, or R

GSC

, which are based on

the log-likelihood, can be used with multinomial models. In Tables 8.5 and 8.7, I

have relied on R

to indicate the predictive eﬃcacy of the models. According to this

statistic, discriminatory power for the multinomial models of couple violence proﬁle

is modest, at best.

EXERCISES

8.1 Let P

⫽ the model-estimated probability, for a given case, of “intense male

violence,” P

⫽ the probability of “physical aggression,” and P

⫽ the proba-

bility of “nonviolence.” Using equation group (8.2), show that these three

probabilities sum to 1 for any given case.

8.2 Show how equation group (8.2) generates each probability deﬁned in

Exercise 8.1. That is, show that equation group (8.2) is the correct formula for

generating each probability. [Hint: Substitute ln (P

) for U and ln (P

)

for V in equation group (8.2) to recover the probabilities.]

8.3 Based on model 1 in Table 8.2, give the odds ratio for those with vs. without

alcohol or drug problems at 1 standard deviation below,

ᎏ

standard deviation

above, and 1

ᎏ

standard deviations above mean economic disadvantage

(SD ⫽ 5.1283). In particular, in each case, by what factor is the impact of

alcohol/drug problem reduced/inﬂated as a function of economic disadvan-

tage?

8.4 Based on model 1 in Table 8.3, give the estimated probability of violence at

each duration quintile for cohabiting, minority couples, with an alcohol or

drug problem, who are 1 standard deviation above the mean on male’s isola-

tion (SD = 6.2958) and economic disadvantage (SD ⫽ 5.1283) and 1 standard

deviation below the mean on female’s age at union (SD ⫽ 7.0807).

8.5 Based on model 3 in Table 8.3, give the eﬀect (i.e., partial derivative) of rela-

tionship duration on the log odds of violence at .5, 1.5, and 2 standard devi-

ations (SD ⫽ 12.8226) above mean relationship duration.

8.6 Based on the FIML estimates in Table 8.5, give the equation for the log odds

of “intense male violence” versus “physical aggression.”

8.7 Using the FIML model in Table 8.5, verify the probabilities in the last row of

Table 8.6, within rounding error.

8.8 Give the eﬀect of alcohol/drug problem on the odds of “intense male violence”

versus “nonviolence,” at mean economic disadvantage, 1 standard deviation

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below mean economic disadvantage, and 1 standard deviation above mean

economic disadvantage, based on the FIML estimates in Table 8.5. Use .8672

for the coeﬃcient of alcohol/drug problem, .0479 for the coeﬃcient for the

interaction term, and 5.1283 as the SD of economic disadvantage.

8.9 Estimate the same eﬀects as in Exercise 8.8, but this time using the probabil-

ities in Table 8.6, showing that these eﬀects agree with those in Exercise 8.8,

within rounding error.

8.10 Based on the proportional odds model in Table 8.7, give the odds of “intense

male violence” versus any other response, and the odds of “violence” versus

“nonviolence,” for cohabiting, minority couples, with alcohol or drug prob-

lems, at mean levels of the continuous regressors.

8.11 Using the principle, for a discrete variable, Y, that P(Y ⫽ j) ⫽ P(Y ⱕ j) ⫺

P(Y ⱕ j ⫺ 1), along with the formulas in equations (8.6) to (8.8), give the esti-

mated probabilities of “intense male violence,” “physical aggression,” and

“nonviolence,” for the couples of Exercise 8.10, based also on the proportional

odds model in Table 8.7.

8.12 Using the students dataset, regress MISSG (a dummy for having missing data

on EXAM2 that reﬂects dropping the class before the second exam) on the

EXAM1 score and its square, student classiﬁcation (CLASSIF), and STAT-

MOOD, using logistic regression. Missing imputation: Substitute 76.9774766

for missing data on EXAM1. Then:

(a) Interpret all coeﬃcients, including the intercept and the main eﬀect of

EXAM1.

(b) Center EXAM1 and form EXAM1

from the centered term. Rerun the

regression and interpret the eﬀect of EXAM1 in this model.

tive) of EXAM1 on the log odds of being missing on EXAM2 at 1 stan-

dard deviation below the mean of EXAM1.

(d) Compute the standard error for the test in part (c) using the formula for

V(b ⫹ 2gx) given in the chapter, and compare it to the standard error in

part (c). For V(b), V(g), and Cov(b,g), use the variance–covariance matrix

of parameter estimates based on the centered (not target-centered) ver-

sion of the model for EXAM1.

8.13 Using the students dataset, estimate the proportional odds model for FIRS-

TEX, the ordinal version of EXAM1, as a function of COLGPA, SCORE,

STATMOOD, and RATIO. Missing imputation: Follow the instructions for

Exercise 6.20. Then:

(a) Test whether the proportional odds hypothesis is supported.

(b) Interpret the eﬀects of all regressors (excluding the intercept) on the odds

of a higher, as opposed to a lower, grade.

EXERCISES 309

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8.14 Based on the estimates from Exercise 8.13:

(a) Give and interpret the estimated cumulative logit for each possible bifur-

cation of FIRSTEX for someone with a college GPA of 3.2, a math diag-

nostic score of 39, a STATMOOD of 3, and a RATIO of 25.

(b) Give the estimated probabilities of an A, B, C, D, and F, for the student

in part (a).

Use the following results for Exercises 8.15 to 8.19. The variable PORNLAW in the

GSS98 dataset was recoded as PORN18 as follows: 1 ⫽ “no prohibition of any

kind,” 2 ⫽ “prohibit its distribution to all ages,” and 3 ⫽ “prohibit its distribution to

persons under 18.” A multinomial logistic regression of PORN18 on FREQBARS

(frequency of going to bars, coded 1 to 6), AGE (in years), MALE (a dummy for

being male), RELOSITY (religiosity score; the sum of standardized items with the

high score being most religious), and CONSERV (coded 1 to 7, with the high score

being most politically conservative) produced the following results for n ⫽ 879 cases

using PROC CATMOD in SAS. The ﬁrst parameter estimate listed for a regressor is

for the log odds of PORN18 ⫽ 1 versus PORN18 ⫽ 3; the second estimate is for the

log odds of PORN18 ⫽ 2 vs. PORN18 ⫽ 3:

Intercept-only model: ⫺2lnL⫽ 1446.8468.

Wald χ

(2)

for global eﬀects of each regressor:

Hypothesized model: ⫺2lnL⫽ 1264.5865.

Eﬀects in hypothesized model:

Eﬀect Parameter Estimate Std. Error

Intercept 1 ⫺4.2337 .8784

2 ⫺1.8312 .3925

FREQBARS 3 .0266 .1014

4 ⫺.1507 .0514

AGE 5 .0222 .0105

6 .0286 .00473

MALE 7 ⫺.3028 .3715

8 ⫺.4796 .1589

RELOSITY 9 ⫺.2397 .0944

10 .2915 .0495

CONSERV 11 .1376 .1335

12 .1661 .0600

Intercept 39.15

FREQBARS 9.03

AGE 37.41

MALE 9.23

RELOSITY 46.16

CONSERV 8.04

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8.15 Give the model χ

for the hypothesized model and assess its discriminatory

power using an appropriate measure. Is the model signiﬁcant?

8.16 Construct a table of regression estimates along the lines of the ﬁrst two

columns of Table 8.5, starring the signiﬁcant coeﬃcients, and noting which

regressors have signiﬁcant global eﬀects on both log odds.

8.17 Interpret the eﬀects of the regressors (excluding the intercept) on each odds.

Does it appear, from the nature of the regressor eﬀects, that PORN18 might

be treated as ordinal and estimated with the proportional odds model? Why or

why not?

8.18 Give the estimated equation for the log odds of “no prohibition” versus “pro-

hibition for all,” based on the results for PORN18. Interpret all regressor

(excluding the intercept) eﬀects in terms of odds.

8.19 Give the predicted probabilities associated with each response to PORN18 for

a 21-year-old man of average religiosity, with FREQBARS ⫽ 3 and CON-

SERV ⫽ 4.

Use the following results for Exercises 8.20 to 8.23. The variable DIVLAW (should

divorce be easier or more diﬃcult to obtain?) in the GSS98 dataset is coded

1 ⫽ “make divorce easier to obtain,” 2 ⫽ “make divorce more diﬃcult to obtain,”

3 ⫽ “keep it as is.” A multinomial logistic regression of DIVLAW on AGE (in years),

EDUCAT (education in years of schooling), MALE (a dummy for being male),

RELOSITY (religiosity score; the sum of standardized items with the high score

being most religious), BLACK (a dummy for being black, with white as the contrast

group), and OTHRACE (a dummy for being other than black or white, with white as

the contrast group) produced the following results, for n ⫽ 879 cases, using PROC

CATMOD in SAS. The ﬁrst parameter estimate listed for a regressor is for the log

odds of DIVLAW ⫽ 1 vs. DIVLAW ⫽ 3; the second parameter estimate is for the log

odds of DIVLAW ⫽ 2 versus DIVLAW ⫽ 3:

Intercept-only model: ⫺2lnL⫽ 1734.7917.

Wald χ

(2)

for global eﬀects of each regressor:

Intercept 30.31

AGE 17.13

EDUCAT 16.17

MALE 2.88

RELOSITY 52.07

BLACK 55.61

OTHRACE 16.78

EXERCISES 311

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Hypothesized model: ⫺2lnL⫽ 1575.1477.

Eﬀects in hypothesized model:

8.20 Construct a table of regression estimates along the lines of the ﬁrst two

columns of Table 8.5, starring the signiﬁcant coeﬃcients, and noting which

regressors have signiﬁcant global eﬀects on both log odds. At the bottom of

the table, include the model χ

, its df, and a measure of discriminatory power.

8.21 Interpret the eﬀects of the regressors (excluding the intercept) on each odds. Does

it appear from the nature of the regressor eﬀects that DIVLAW might be treated

as ordinal and estimated with the proportional odds model? Why or why not?

8.22 Give the estimated equation for the log odds of “making divorce easier” ver-

sus “making it more diﬃcult.” Interpret the eﬀects in terms of odds. Also,

what is the odds ratio for blacks versus those of other races?

8.23 Give the predicted probabilities associated with each response to DIVLAW

for a 45-year-old black female with 12 years of education and a religiosity

score of 5.

8.24 Use the kids dataset to estimate a binary logistic regression equation for MUL-

TIPLE (whether or not the focal child has had more than one sex partner in the

last month) as a function of FCAGE2, MALE, NONINTAC, MSEXATT,

FSEXATT, MSTYLE1, MSTYLE2, FSTYLE1, FSTYLE2, PERMISIV, and

the interaction of PERMISIV with MALE. Center PERMISIV to guard against

collinearity problems.

(a) Interpret the interaction eﬀect by examining how the impact of PERMI-

SIV diﬀers for male and female focal children, and by examining whether

Eﬀect Parameter Estimate Std. Error

Intercept 1 3.2549 .7097

2 3.1548 .5994

AGE 3 ⫺.0283 .00688

4 ⫺.0112 .00552

EDUCAT 5 ⫺.1535 .0404

6 ⫺.1124 .0334

MALE 7 ⫺.3396 .2253

8 ⫺.0574 .1879

RELOSITY 9 .00313 .0609

10 .3241 .0536

BLACK 11 1.3872 .3197

12 ⫺.3424 .3164

OTHRACE 13 1.1421 .4322

14 ⫺.0789 .4301

312 ADVANCED TOPICS IN LOGISTIC REGRESSION

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PERMISIV has a signiﬁcant eﬀect on having multiple partners, sepa-

rately within each gender.

(b) Perform Allison’s (1999) Chow test analog to discern whether the model

in part (a), minus the interaction term, diﬀers on the whole between male

and female children.

in part (a).

8.25 Use the students dataset for the following exercise. Missing imputation: Do

not impute any missing values for this problem; use listwise deletion.

(a) ORDEXAM1 is a trichotomous version of EXAM1 with three values: A

(⬎89), B (⬎79), and C or worse (ⱕ79). Estimate a multinomial logistic

regression model for this variable, with C as the baseline category, as a

function of SCORE, MALE, STATMOOD, COLGPA, RMAJOR (dum-

mied up, with “sociology majors” as the reference group), and PRE-

VMATH. Interpret the coeﬃcient estimates in terms of odds.

(b) Give the model χ

for this model.

(d) Test the collapsibility of the A and B categories.

(e) Estimate the ordered logit model for this response, interpret the coeﬃcients

and their signiﬁcance levels, and assess whether the hypothesis of ordinal-

ity (proportional odds) is supported.

EXERCISES 313

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314

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

By Alfred DeMaris

CHAPTER 9

Truncated and Censored

Regression Models

CHAPTER OVERVIEW

In this chapter we once again focus on continuous (or, at least, approximately inter-

val-level) response variables, but take up the subject of regression models for trun-

cated or censored data. Truncated and censored responses arise when the data

observed on a continuous response are not fully representative of the range of values

of the response in the target population. I begin the chapter by deﬁning truncation,

censoring, and a special variant of truncation, known as incidental truncation, that

gives rise to sample-selection bias. I then introduce the truncated regression model

and show how it is estimated via maximum likelihood. The discussion then moves

to the more commonly used censored regression, or tobit model. For this model I

discuss estimation via maximum likelihood, various ways of interpreting model

coeﬃcients, and an analog of the R

used in OLS. An alternative to the tobit model

is also presented that relaxes one of its major constraints. Finally, I take up sample-

selected regression, presenting both two-step and maximum-likelihood estimation

techniques for this model. Substantive applications are illustrated throughout the

chapter to enhance understanding of each technique.

TRUNCATION AND CENSORING DEFINED

Truncation

In contrast to approaches in other chapters, I begin this one with a more abstract set

of examples. First, consider a normally distributed variable, X, with a mean of 5 and

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