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

Подождите немного. Документ загружается.

average couple conﬂict?

4.3 For the students dataset, CLASSF is recoded into the categories “sophomore,”

“junior,” “senior,” and “postgraduate/graduate” (there are no freshmen). The

means (n’s) on the ﬁrst exam for these groups are “sophomore” 81.595 (21);

“junior” 74.681 (70); “senior” 74.884 (88); “postgraduate/graduate” 84.064 (35).

(a) Show how the categories of CLASSF should be eﬀect coded, naming the

eﬀect variables SOPH, JUN, and POST (with “senior” as the reference

group).

(b) Using only hand calculations, give the sample regression equation for

regressing EXAM1 on SOPH, JUN, and POST.

4.4 Suppose that a continuous variable Y is regressed on two categorical vari-

ables: Z

with categories A and B, and Z

with categories C, D, and E. If Z

is dummied with B as the interest category and Z

is dummied with C and D

as the interest categories, the sample regression equation is

yˆ  4  6B  2C  D  4BC  2BD.

Fill in the following table with each of the cell means, based on this equation:

4.5 Using the GSS98 dataset, perform a two-way ANOVA for the eﬀect of gen-

der and marital status on attitude toward cohabitation (COHABTN) using

dummy-variable regression. In particular:

(a) Test whether marital status aﬀects cohabitation attitude, controlling for

gender.

(b) Test whether there is a signiﬁcant interaction eﬀect between gender and

marital status.

for the cross-classiﬁcation of gender with marital status.

4.6 In the two-way ANOVA model without interaction in Exercise 4.5, use

Bonferroni–Holm to test all 10 mean contrasts in cohabitation attitude by

marital-status categories.

4.7 In DeMaris (2002b) I present several regression analyses of the frequency

of sexual activity in the past month for 2997 couples in the NSFH. A regression

EXERCISES 155

c04.qxd 8/27/2004 2:49 PM Page 155

of sexual frequency on a dummy variable reﬂecting cohabiting status (1 

“cohabiting unmarried,” 0  “married”) produces the following equation:

yˆ  8.171  3.314 cohabiting. A second analysis adds the continuous vari-

ables female age in years (mean  33.43) and relationship duration in months

(mean  92.63) to the model. The second equation is

yˆ  13.969  2.322 cohabiting  .144 female age

 .01 relationship duration.

(a) Give the unadjusted mean sexual frequency for cohabiting vs. married

couples.

(b) Give the adjusted mean sexual frequency for cohabiting vs. married cou-

ples, after adjusting for female age and relationship duration. Comment

on the diﬀerences between the unadjusted and adjusted means.

4.8 In the analyses in Exercise 4.7, MSE for the ﬁrst equation is 43.183, MSE for

the second equation is 40.631, and the standard error of the cohabiting eﬀect

is .201 in each equation. Test whether female age and relationship duration

account for a signiﬁcant part of the cohabitation eﬀect on sexual frequency,

using the Clogg et al. (1995) test discussed in Chapter 3.

4.9 For the couples dataset, the variable MPERCENT is the percent of total

weekly housework hours contributed by the male partner, calculated as

100

冢冣

This was regressed on MPARTIME and MUNEMP, dummies for male part-

ners who work “part time” or are “unemployed,” respectively (working “full

time” is the reference category); FPARTIME and FUNEMP, dummies for

female partners who work “part time” or are “unemployed,” respectively

(working “full time” is the reference category); and MINCOME and FIN-

COME (male and female income, respectively). The sample equation is

yˆ  23.766  2.104 MPARTIME  .177 MUNEMP  6.043 FPARTIME

 4.738 FUNEMP  .020 MINCOME  .286 FINCOME.

(a) Interpret the intercept and each of the dummy coeﬃcients.

(b) Give the adjusted mean percentages of housework done by male partners

in each of the nine cross-classiﬁcations of male employment with female

employment. (Note: Mean MINCOME  23.5, mean FINCOME 

9.04.)

4.10 For the couples dataset, the regression of couple happiness (the mean of

HUSHAP and WIFHAP) on PRESCHDN (a dummy representing the presence

MMHOURS



MMHOURSFFHOURS

156 MULTIPLE REGRESSION WITH CATEGORICAL PREDICTORS

c04.qxd 8/27/2004 2:49 PM Page 156

of children in the household), CONFLICT (couple conﬂict), and their inter-

action (PRESCONF) produces the following equation:

yˆ  6.997  .185 PRESCHDN  .479 CONFLICT  .157 PRESCONF.

Interpret the interaction eﬀect with, alternately, presence of children and

conﬂict as the focus variable. Note that the couple conﬂict scale ranges from

1 (“no conﬂict”) to 6 (“maximum conﬂict”).

4.11 For the 230 main-respondent females in the couples dataset, the following

three regression equations were produced, where Y  the Center for Epidemi-

ological Studies Depression Scale [the 12-item version (CESD)], and the

regressors are some combination of OWNKID, STEPKID, and CONFLICT

(signiﬁcant coeﬃcients are shown in boldface type):

(1) yyˆ  13.2178  .6332 OWNKID  3.0679 STEPKID; R

 .0043.

(2) yˆ 6.9947  11.8270 CONFLICT; R

 .2062.

(3) yˆ 7.0448  5.0846 OWNKID  .4777 STEPKID

 13.0701 CONFLICT; R

 .2279.

(a) Test whether child type is a signiﬁcant predictor of women’s depressive

symptomatology after controlling for couple conﬂict.

(b) Interpret the coeﬃcient for OWNKID in the last equation.

equation 1 to equation 3.

4.12 Refer to Exercise 4.3. A regression of EXAM1 on SCORE, COLGPA, and

PREVMATH for the 214 students with valid grades produces an R

of .4312.

When SOPH, JUN, and POST (coded as eﬀect variables) are added to the

model, the R

goes up slightly, to .4383.

(a) Test whether student classiﬁcation is a signiﬁcant predictor of exam scores

once SCORE, COLGPA, and PREVMATH have been accounted for.

(b) If the sample equation for the more complete model is

yˆ 55.122  1.728 SOPH  1.429 JUN  1.603 POST

 2.2 SCORE  13.139 COLGPA  1.805 PREVMATH

and the means for SCORE, COLGPA, and PREVMATH are, respectively,

40.925, 3.091, and 1.257, give the estimated average EXAM1 scores for

sophomores, juniors, seniors, and postgraduate/graduate students with aver-

age values of SCORE, COLGPA, and PREVMATH.

4.13 Refer to Exercise 4.12. A regression of EXAM1 on SCORE, COLGPA, and

PREVMATH and dummy variables representing student classiﬁcation and

EXERCISES 157

c04.qxd 8/27/2004 2:49 PM Page 157

named SOPHM (“sophomore”), JUNR (“junior”), and POSTG (“postgrad/

grad”), with seniors as the contrast group, produced the following sample

equation:

yˆ 57.025  3.63 SOPHM  .474 JUNR  3.506 POSTG

 2.2 SCORE  13.139 COLGPA  1.805 PREVMATH.

Give the adjusted mean EXAM1 scores for sophomores, juniors, seniors, and

postgrad/grads. Compare your answer to part (b) of Exercise 4.12.

4.14 One analysis of faculty salary for the 725 faculty members at BGSU looks at

the interaction of gender with college (“ﬁrelands” is the group omitted here)

in their eﬀects on faculty salary. The estimated equation, including several

controls (contained in the vector CONTROLS), is

yˆ  36959  702.006 FEMALE  1201.846 ARTSCI  7738.588 BUSINESS

 499.644 EDUCATN  2900.191 OTHER

 1301.887 FEMALE * ARTSCI 4817.316 FEMALE * BUSINESS

 1388.864 FEMALE * EDUCATN 1058.814 FEMALE * OTHER

 gⴕ CONTROLS.

There are a total of 20 regressors in this model, and the R

is .8382. The model

without the interaction terms has an R

of .8363.

(a) Test whether the interaction is signiﬁcant.

(b) Interpret the intercept and all of the main eﬀects of gender and college in

the model.

gender and college alternating as the focus variables.

4.15 Using the faculty salary dataset, estimate the regression of AYSALARY on

rank (captured by the dummies R1 for “full professor,” R2 for “associate pro-

fessor,” and R4 for “instructor/lecturer,” with “assistant professors” as the ref-

erence group), centered versions of PRIOREX, YRBG, YRRANK, and

SALFAC, plus the dummies TERMDEG, GRAD, ADMIN, and FIRELAND.

(a) Intepret the rank eﬀects.

(b) Give the adjusted mean AYSALARY for those in each rank in the base-

line group (i.e., no terminal degree, not on graduate faculty, not in an

administrative position, on the main campus).

4.16 Using the GSS98 dataset, conduct a one-way ANOVA via dummy-variable

regression, where Y  ABORTION and X  religious aﬃliation (“Protestant,”

“Catholic,” “Jewish,” “None,” “Other”). Use Bonferroni–Holm to test all 10

158 MULTIPLE REGRESSION WITH CATEGORICAL PREDICTORS

c04.qxd 8/27/2004 2:49 PM Page 158

mean contrasts. Then add the covariates EDUCAT and CONSERV to the

model and once again use Bonferroni–Holm to test the 10 religious-group con-

trasts on abortion attitude. Missing imputation: Follow the instructions for

Exercise 3.14.

4.17 For the 416 couples in the couples dataset, an analysis was conducted to

replicate DeMaris (1997). That article examined the impact of coital fre-

quency on women’s depressive symptomatology and how this might be mod-

erated by the male’s physical aggression against the female partner. The

analysis regressed the CESD on coital frequency (COITFREQ), dummies for

male and female violence (MALEHIT and FEMAHIT, respectively), the

interaction between COITFREQ and each violence dummy (MHITCOIT and

FHITCOIT, respectively), and a vector of control variables (CONTROLS).

The model has 15 parameters. The results for all 416 couples produced an

SSE  81596.74241. The model was then estimated separately for the 186

couples in which the main respondent was male, and the 230 couples in which

the main respondent was female. For the male sample, SSE  39246.83241,

MSE  229.51364, df

 171; for the female sample, SSE  39607.02799,

MSE  184.21873, df

 215.

The females’ equation was

yˆ  .3344  gⴕ CONTROLS  .0183 COITFREQ  28.4758 MALEHIT

3.0606 FEMAHIT  1.2076 MHITCOIT  .2841 FHITCOIT.

(a) Test whether the error variances in the equations for males and females

are equal.

(b) Regardless of your answer to part (a), test whether the same model

(including the intercept) holds in the population of partnered males ver-

sus partnered females.

by partner violence.

4.18 Using the GSS98 dataset, estimate the regression of ABORTION on MALE,

RELOSITY, CONSERV, and EDUCAT for the 1868 respondents with

valid ABORTION scores. Then perform the Chow test with both constrained

and unconstrained intercepts to test model invariance across categories of

RACE (“White,” “Black,” “Other”). First, use Bartlett’s test to test the homo-

geneity of error variance across the racial groups; but do the Chow test

regardless of this outcome. Missing imputation: Follow the instructions for

Exercise 3.14.

4.19 Retest the model’s equivalence across racial groups in Exercise 4.18 using

the cross-product method. Again, use both constrained- and unconstrained-

intercepts approaches. You should demonstrate that this produces the same

EXERCISES 159

c04.qxd 8/27/2004 2:49 PM Page 159

results as in Exercise 4.18. Missing imputation: Follow the instructions for

Exercise 3.14.

The following information is to be used for Exercises 4.20 and 4.21.

• For the 725 faculty members in the faculty salary dataset, the ANCOVA

model in Exercise 4.15, when estimated using the entire sample, has:

RSS  112828737391, SSE  28911032834, MSE  40548433.147, and

 713.

• If a dummy for gender (FEMALE) is added, the results are RSS 

112913052400, SSE  28826717824, MSE  40486963.236, and df



712.

• If the cross-product of FEMALE with all 11 other covariates is then added

to the model, the results are RSS  113515775771, SSE  28223994454,

MSE  40262474.256, and df

 701.

• If the model in Exercise 4.15 is estimated for the 511 male faculty mem-

bers, the results are RSS  72214388564, SSE  23630713768, MSE 

47356139.816, and df

 499.

• If the model in Exercise 4.15 is estimated for the 214 female faculty mem-

bers, the results are RSS  18209728848, SSE  4593280685.4, MSE 

22739013.294, and df

 202.

4.20 Test the model equivalence of the model in Exercise 4.15 for males versus

females using the Chow test. Perform the test with both constrained- and

unconstrained-intercept versions. First, test homogeneity of error variance in

each group; then do the Chow test regardless of this outcome.

4.21 Redo Exercise 4.20 using the cross-product approach.

4.22 For the 214 students with valid EXAM1 scores in the students dataset, EXAM1

was regressed on COLGPA, STATMOOD, and SCORE. The results were

RSS  26652.37864, SSE  34995.71120, MSE  166.64624, and df

 210.

• For the 69 “sociology” majors, the same model produced RSS 

12969.83816, SSE  7162.26656, MSE  110.18872, and df

 65.

• For the 43 “other social science” majors, the same model produced

RSS  3070.59860, SSE  5257.61617, MSE  134.81067, and df

 39.

• For the 102 “other ﬁelds” majors, the same model produced RSS 

11694.08078, SSE  21442.49456, MSE  218.80096, and df

 98.

Use Barlett’s test to test for homogeneity of error variance across the three

groups of majors. Regardless of the outcome, conduct a Chow test for model

equivalence across the three groups of majors, constraining the intercept to be

the same across groups.

160 MULTIPLE REGRESSION WITH CATEGORICAL PREDICTORS

c04.qxd 8/27/2004 2:49 PM Page 160

4.23 Suppose that a model for Y is estimated as a function of a categorical variable,

Z, with categories A, B, C, and D (the reference category), and a continuous

variable X (range: 10 to 10). Suppose that the sample equation is

yˆ  2  .5A  .8B  .2C  1.5X  .05AX  .15BX  .25CX.

Determine whether the interaction eﬀect in X and in Z is ordinal or disordi-

nal, with, alternately, Z and then X as the focus variables.

4.24 Suppose that a model for Y is estimated as a function of a categorical variable,

Z, with categories A, B, C, and D (the reference category), and a continuous

variable X (range: 10 to 10). Suppose that the sample equation is

yˆ  3.5  A  4B  2C  .5X  .5AX  .25BX  .75CX.

Determine whether the interaction eﬀect in X and in Z is ordinal or disordi-

nal, with, alternately, Z and then X as the focus variables.

4.25 Prove that the t test for d (the dummy coeﬃcient) in the SLR of Y on a dummy

variable X is equivalent to the two-sample t test. Note that for two independ-

ently sampled groups, denoted group 0 and group 1, with sample sizes n

and

, respectively, and means y

苶

and y

苶

, respectively, the two-sample t test is

n

2)





兹

苶



)

苶



苶

(

苶

)

苶



where σ

is the pooled estimate of the common population variance of Y, with

the formula

 .

(a) First, prove that σ

in the two-sample t test equals MSE in the dummy-

variable SLR. [Hint: Start with MSE  冱(y  yˆ)

/(n  2); then use the fact

that yˆ  y

苶

for X  0, and yˆ  y

苶

for X  1, plus the fact that n

 n

 n.]

(b) Then prove that the t tests are equivalent. [Hint: Start with σ

苶

y

苶

, which

equals σ

((1/n

) (1/n

)). Then use the fact that (1/n

)  (1/n

) 

 n

)/n

; n

 n(1  π

), n

 nπ

; and 冱(x  x

苶

)

 nπ

(1  π

).]

1)s

 (n

1)s



 n

2

EXERCISES 161

c04.qxd 8/27/2004 2:49 PM Page 161

162

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

By Alfred DeMaris

CHAPTER 5

Modeling Nonlinearity

CHAPTER OVERVIEW

Until now the models under discussion have been strictly linear. By this is meant that

(1) Y is held to be a linear, additive function of the explanatory variables, or, if inter-

action is present, it is of a linear nature; and (2) the model is linear in the parameters

(i.e., it is a weighted sum of regressors times parameters). (More formal deﬁnitions

of linearity in terms of the explanatory variables or the model as a whole are given

below.) In this chapter I introduce models in which Y is held to be a nonlinear func-

tion of either the explanatory variables or the model parameters. I begin by deﬁning

these concepts and giving examples of models illustrating various nonlinear rela-

tionships between Y and the structural component of the model. I then illustrate

some of the more common nonlinear functions of explanatory variables that may be

helpful in modeling the response variable. This discussion segues into a lengthy

illustration of the use and interpretation of the quadratic model, perhaps the most

commonly employed nonlinear function in social data analysis. Here I also deﬁne

and discuss nonlinear interaction eﬀects. Finally, I introduce the reader to nonlinear

regression, the technique employed when one’s model is nonlinear in the parameters

and cannot be converted to a convenient linear form. Readers unfamiliar with deriv-

atives from calculus are advised to peruse Section IV of Appendix A before tackling

this chapter. Because derivatives and partial derivatives are central to an understand-

ing of nonlinearity, they ﬁgure prominently in the subject matter that follows.

NONLINEARITY DEFINED

I begin by making a distinction between nonlinearity in the functional form of the

relationship between Y and X, versus nonlinearity of the model for Y. Nonlinearity

c05.qxd 8/27/2004 2:52 PM Page 162

in the functional form of the relationship between Y and X is determined by the ﬁrst

partial derivative of the model with respect to X. In any model in which Y is a func-

tion of X, if the ﬁrst partial derivative of Y with respect to X is a function of X, the

model is nonlinear in X; otherwise, the model is linear in X. For example, suppose

that model A is

Y  α  βX  γZ  ε.

Since the ﬁrst partial derivative of Y with respect to X, or ∂Y/∂X, is β, which is not a

function of X, the model is linear in X. However, in model B,

Y  α  βX  δX

 γZ  ε,

the ﬁrst partial derivative is ∂Y/∂X  β  2δX. Since this is a function of X, the model

is nonlinear in X. In particular, this model, called a quadratic model, or curvilinear

model, in X, describes a parabolic curve (or part of a parabolic curve) relating Y to

X at any given value of Z. We can also deﬁne nonlinearity in the functional form

relating Y to X using the second partial derivative. If the second partial derivative of

Y with respect to X is not zero, the model for Y is nonlinear in X; if it is zero, the

model is linear in X. In model A, ∂

Y/∂X

 ∂(β)/∂X  0 showing again that the

model is linear in X. On the other hand, in model B, ∂

Y/∂X

 2δ, which is nonzero

provided that δ is not zero. This once again reveals that model B is nonlinear in X.

Intuitively, the ﬁrst derivative measures the change in Y with change in X at the point

x. As long as this is not a function of X, that change is constant over levels of X. This

condition means that the relationship between Y and X can be represented by a

straight line, which is characterized by a constant slope (see Section I.P of Appendix

A). If, on the other hand, the ﬁrst derivative is a function of X, this means that the

rate at which Y changes with change in X is itself changing with levels of X, describ-

ing some type of curve instead of a straight line. Moreover, if the ﬁrst partial deriv-

ative of Y with respect to X is not a function of X, the second partial derivative of Y

with respect to X is necessarily zero. Note that conventional interaction models such

as model C,

Y  α  βX  δZ  γXZ  ε,

are not nonlinear in X, since ∂Y/∂X  β  γZ is not a function of X.

The linearity or nonlinearity of the model as a whole is determined by the ﬁrst

partial derivative of Y with respect to the model’s parameters. Denote each of the P

parameters of any model by θ

, for p  1,2,...,P (e.g., in linear regression we typ-

ically have P  K  1 parameters). If the ﬁrst partial derivative of Y with respect to

at least one of the θ

is a function of any of the model parameters, the model for Y is

nonlinear (Ratkowsky, 1990). Model B, which is nonlinear in X, is nevertheless not

a nonlinear model, since the ﬁrst partial derivatives of Y with respect to, alternately,

α, β, δ, and γ are 1, X, X

, and Z. Notice that none of these terms involves any of the

model parameters. On the other hand, consider model D: Y  α  X

 ε. Now,

NONLINEARITY DEFINED 163

c05.qxd 8/27/2004 2:52 PM Page 163

ﬁnding the ﬁrst partial derivative of Y with respect to β involves ﬁnding the ﬁrst

derivative of X

with respect to β. To do this we write X

as exp(log X

). Then



[exp(log X

)] 



[exp (β log X )]  exp(β log X )log X  X

log X.

Since this expression involves β, model D is nonlinear for Y. Notice that it is also

nonlinear in X, since ∂Y/∂X  βX

β1

is also a function of X. (Most models that are

nonlinear for Y are also nonlinear in X.) Some other examples of nonlinear models

are the exponential model with additive error term (Neter et al., 1985),

Y  γ

 ε,

the exponential model with multiplicative error term (Fox, 1997),

Y  γ

the logistic population-growth model (Fox, 1997),

Y 



1exp(

β



 ε,

and the gravity model of migration (Fox, 1997),

 α



where Y

is the number of migrants moving from city i to city j, D

is the distance

between cities i and j, and P

and P

are the respective population sizes of cities i and j.

Employing the terminology of Fox (1997) and Neter et al. (1985), I make a fur-

ther distinction among three types of models. Linear models are those that are linear

in the parameters, such as models A through C above. Intrinsically linear models

(Neter et al., 1985) are those that are linear in the parameters after applying some

kind of transformation to the response and/or explanatory variables. An example is

the exponential model with multiplicative errors. If we transform Y by taking its nat-

ural logarithm, the model becomes log Y  log γ

 γ

X  ε. Deﬁning α as log γ

, the

model is log Y  α  γ

X  ε, which is now linear in the parameters, and therefore a

linear model. Similarly, logging Y

in the gravity model of migration produces

log Y

 αβ log P

 γ log P

 δ(log D

)  log ε

where αlog α. This model is, again, linear in the parameters. Provided that the

errors in each model (ε and log ε

, respectively) are independent and normally dis-

tributed with zero mean and constant variance, these models can be estimated using

ordinary least squares. Essentially nonlinear models (Fox, 1997) are nonlinear mod-

els that cannot be made linear by any transformations of the response or explanatory

164 MODELING NONLINEARITY

c05.qxd 8/27/2004 2:52 PM Page 164