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

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冟 x⫺1

. Then a scatterplot of e

y 冟 x⫺1

against e

冟 x⫺1

is the partial plot for X

. In that

OLS residuals are always uncorrelated with all regressors in the same equation,

y 冟 x⫺1

and e

冟 x⫺1

represent versions of Y and X

, respectively, that are uncorrelated

with all other model regressors. The same procedure is used to generate plots for the

other K ⫺ 1 regressors.

Figures 3.10 to 3.14 present the partial plots for the regression of exam score in

introductory statistics. The plots are for model 3 in Table 3.1. Shown, in order, are

the plots for math diagnostic score, college GPA, attitude toward statistics, class

hours in the current semester, and number of previous math courses. The ﬁtted line

on each plot is the OLS line for the simple linear regression of each e

y 冟 x⫺k

each e

冟 x⫺k

. None of the relationships depicted in the plots suggest noticeable non-

linear associations between e

y 冟 x⫺k

and e

冟 x⫺k

. This conﬁrms the acceptability of

modeling each of the predictors as having a linear relationship to exam scores. Also,

note that the simple correlation between e

y 冟 x⫺k

and e

冟 x⫺k

is called the partial cor-

relation coeﬃcient between Y and X

, controlling for all other regressors, and is

denoted pr

in this book. This represents the correlation between Y and X

after

removing from each variable the linear association with all other regressors in the

model. The squared partial correlation, or pr

, can be used as a measure of the

EVALUATING EMPIRICAL CONSISTENCY 115

Figure 3.10 Partial regression leverage plot of residualized exam 1 score (ey-x1) on residualized score

on the math diagnostic (ex1).

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Figure 3.11 Partial regression leverage plot of residualized exam 1 score (ey-x2) on residualized col-

lege GPA (ex2).

Figure 3.12 Partial regression leverage plot of residualized exam 1 score (ey-x3) on residualized atti-

tude toward statistics score (ex3).

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Figure 3.13 Partial regression leverage plot of residualized exam 1 score (ey-x4) on residualized num-

ber of current semester hours (ex4).

Figure 3.14 Partial regression leverage plot of residualized exam 1 score (ey-x5) on residualized num-

ber of previous math courses (ex5).

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unique variance in Y accounted for by X

after the inﬂuence of all other variables has

been removed (McClelland and Judd, 1993).

Using this measure of eﬀect size—the size of the eﬀect of a given regressor on

Y—the most unique variance in exam scores is accounted for by college GPA. Its

squared partial correlation with exam score is .21. The second largest eﬀect size is

for math diagnostic score, with a squared partial correlation of .196. The least

inﬂuential factor, net of the others, is the number of previous math courses taken.

This variable accounts for only 1.4% of unique variance in exam scores. Unlike the

standardized regression coeﬃcient, which can fall outside the range [⫺1,1] in the

presence of strong collinearity among the X’s, the partial correlation coeﬃcient is

conﬁned within this range under all conditions. A simple (although not trivial to

prove) formula for pr

is (from Greene, 2003)

⫽

ᎏ

⫹

dfE

ᎏ

where t

is the t test for X

in the corresponding MULR model, and dfE is the error

degrees of freedom from this regression, which is n ⫺ K ⫺ 1, assuming a total of K

regressors in the model. Using this formula, we assign to pr

the sign of the

coeﬃcient for X

in the MULR. As an example of this calculation, the t-value for

math diagnostic score (variable 1) in model 3 in Table 3.1 is 7.127. The squared

partial correlation of exam score with math diagnostic score, controlling for college

GPA, attitude toward statistics, class hours in the current semester, and number of

previous math courses is then equal to

⫽

ᎏ

7.12

⫹

208

ᎏ

⫽ .196.

EXERCISES

3.1 In the sample regression equation yˆ ⫽ b

⫹ b

, show how holding X

constant at the respective values 3, 6, and 9 changes the intercept for the

regression of Y on X

, but not the slope.

3.2 In the sample regression equation yˆ ⫽ b

⫹ b

⫹ gX

, show how

holding X

constant at the respective values 3, 6, and 9 changes the intercept

and slope for the regression of Y on X

3.3 The squared partial correlation between Y and X

, controlling for X

, is pr

⫽

Y 冟 X

⫺ R

Y 冟X

)/(1 ⫺ R

Y 冟X

). That is, pr

is the additional variation in Y accounted

for by X

, over and above the variance accounted for by X

, as a proportion

of the variance unaccounted for by X

(a) Using this formula for pr

, ﬁnd the partial correlation between college

GPA and exam 1 score, while controlling for math diagnostic score, for

the 214 students represented in Table 3.1.

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(b) Also, ﬁnd the same partial correlation using Greene’s (2003) formula if

the t statistic for the eﬀect of college GPA in model 2 is 7.397.

3.4 The regression of CMODERN (couple modernism) on MEDUC (male’s edu-

cation) and FEDUC ( female’s education) for the 416 couples in the couples

dataset produces the following results:

Predictor bt

Intercept 22.6853

FEDUC .2497 5.139

MEDUC .0898 2.069

RSS 322.58233

SSE 2237.94588

Standard deviations:

CMODERN 2.4839

FEDUC 2.8

MEDUC 3.1

(a)Give R

and R

adj

(b) Give the F-value and signiﬁcance for the test of the model’s utility.

(d) Give the partial correlation between CMODERN and each predictor, con-

trolling for the other predictor.

(e) Interpret the intercept and the unstandardized coeﬃcients.

3.5 To the model in Exercise 3.4 are added FCHATT, DURYRS, MCHATT,

MALEAGE, and FEMAGE, producing the following result: R

⫽ .16672.

Test whether the addition of these ﬁve variables produces an improvement in

the model (i.e., test the null hypothesis that the ﬁve additional coeﬃcients are

zero in the population).

3.6 Regard model 3 in Table 3.1. The regression of each predictor on the other

four predictors in the model produces the following R

’s: .1007 for math diag-

nostic score, .0953 for college GPA, .2102 for attitude toward statistics, .0942

for class hours in the current semester, and .1291 for number of previous

math courses. Give the VIF’s for each coeﬃcient in the model. Does multi-

collinearity appear to be a problem?

3.7 The regression of STATMOOD (attitude toward statistics) on COLGPA (col-

lege GPA), SCORE (score on the math diagnostic), HOURS (class hours in

EXERCISES 119

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the current semester), and PREVMATH (number of previous math courses),

for 214 students in introductory statistics, produces the following sample

equation (standard errors of coeﬃcients are in parentheses underneath the

coeﬃcients) and results:

yˆ ⫽⫺6.748 ⫹ 1.597 colgpa ⫹ .296 score ⫺ .490 hours ⫹ 1.516 prevmath.

(4.981) (.604) (.110) (.145) (.344)

TSS ⫽ 6505.32710, R

adj

⫽ .195

(a) Test the null hypothesis that all regression coeﬃcients (except the inter-

cept) in the population equal zero. (Hint: Since

adj

⫽ 1 ⫺

ᎏ

⫺

n ⫺

K ⫺

ᎏ

use this to solve for MSE, then compute the F statistic.)

(b) Give the value of R

(d) Regardless of your answer to part (a), test the signiﬁcance of each regres-

sion coeﬃcient in the equation.

(e) Give the predicted STATMOOD score for someone with a GPA of 3.15,

a diagnostic score of 40, who’s taking 12 hours this semester, and who has

had two previous math courses.

(f) Is the intercept meaningful? Why or why not?

3.8 Using the students dataset, verify that statistical control for the variables

SCORE and COLGPA when looking at the eﬀect of STATMOOD on

EXAM1 is accomplished by partialing out of STATMOOD its association

with a linear combination of SCORE and COLGPA. (Hint: You need to

regress STATMOOD on SCORE and COLGPA among only those with valid

EXAM1 scores and save the residuals. Then regress EXAM1 on these resid-

uals and verify that the regression coeﬃcient is the same as the coeﬃcient for

STATMOOD in the MULR of EXAM1 on STATMOOD, SCORE, and COL-

GPA.) Missing imputation: Substitute 40.9358974 for missing data on

SCORE and 3.0827835 for missing data on COLGPA.

3.9 Using the students dataset, use residual and partial regression leverage plots

to judge the empirical consistency of the MULR of STATMOOD on COL-

GPA, SCORE, PREVMATH, HOURS, STUDYHRS, and TVHRS for the

235 students. Missing imputation: Substitute the parenthetical values for

missing data on each variable indicated: COLGPA (3.0827835), SCORE

(40.9358974), PREVMATH (1), STUDYHRS (16.7844828), TVHRS

(9.5128755). Also, recode values greater than 4 on PREVMATH to the value

4. What do you conclude?

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3.10 The regression of STATMOOD on COLGPA, SCORE, PREVMATH, HOURS,

STUDYHRS, and TVHRS for 235 students in the students dataset produces the

following results:

Calculate both the standardized coeﬃcients and the elasticities for all regres-

sors. Is there any diﬀerence in relative importance of the predictors when the

’s are used to judge this rather than the b

’s? Interpret the elasticities for

SCORE and for HOURS.

3.11 The regression for STATMOOD in Exercise 3.10 has an R

of .2256 and a

TSS of 7056.65532.

(a) Give the value of R

adj

. [Hint: See the hint for Exercise 3.7(a).]

(b) Test whether the model is of any utility in predicting STATMOOD.

the MULR.

(d) Give the estimate of σ

3.12 Suppose that the true model for Y is .5X ⫺ 1.2Z ⫹ ε, but you estimate Y ⫽

βX ⫹ ε⬘. Assuming that the correlation between X and Z is ⫺.75 and σ

⫽ 1.75

while σ

⫽ 2.25, what parameter value is your sample regression coeﬃcient

consistent for? Repeat this computation assuming that the true coeﬃcients for

X and Z are ⫺2, ⫺4.148, and again if they are ⫺2, 2.75. In each case, describe

the nature of the bias in estimating the true eﬀect of X on Y.

3.13 For the 416 couples in the couples dataset, the regression of CONFLICT

(couple verbal conﬂict) on MCHATT, FCHATT (male and female church

attendance), MALEAGE, FEMAGE (male and female age), MEDUC,

FEDUC (male and female education), IHTOT2 (total household income), and

DURYRS (duration of the relationship in years) produces the following

results:

⫽ 2.842 ⫺ .001667 mchatt ⫺ .007571 fchatt ⫺ .006181 maleage ⫺ .001743

femage ⫺ .002062 meduc ⫺ .007608 feduc ⫺ .003551 ihtot2 ⫺ .000858

duryrs.

Variable Variable

Variable Coeﬃcient Mean Standard Deviation

STATMOOD 5.0383 5.4915

INTERCEPT ⫺6.7106

COLGPA 1.1721 3.0828 .5012

SCORE .3222 40.9359 3.1133

PREVMATH 1.4474 1.2383 1.0308

HOURS ⫺.5075 14.8426 2.3362

STUDYHRS .0662 16.7845 10.1542

TVHRS ⫺.0444 9.5129 7.5350

EXERCISES 121

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Partial variance–covariance matrix of parameter estimates:

MCHATT FCHATT MALEAGE FEMAGE

MCHATT .000165 ⫺.000103 MALEAGE .000015 ⫺.000010

FCHATT ⫺.000103 .000162 FEMAGE ⫺.000010 .000017

MEDUC FEDUC

MEDUC .000113 ⫺.000056

FEDUC ⫺.000056 .000131

Use t tests to test the diﬀerences between the eﬀects of male and female char-

acteristics on couple conﬂict.

3.14 Use the GSS98 dataset to run the following: regress ABORTION (attitude

toward abortion, the sum of the seven abortion-attitude items coded 1 ⫽ “yes,”

0 ⫽ “no,” where a high score indicates a liberal attitude toward obtaining

abortions) on EDUCAT, INCOME, PAED, MAED, and RESPAGE for the

1868 respondents who are nonmissing on ABORTION. Then, add the variables

RELOSITY, PARTNRS5, and CONSERV to the model. Missing imputation:

Substitute the parenthetical values for missing data on each variable indicated:

EDUCAT (13.2824919), INCOME (13.2024490), CONSERV (4.0702152).

(a) Test whether the addition of the last three variables results in a signiﬁcant

improvement in the model’s utility.

(b) Test whether the eﬀects of PAED and MAED in the full model are signi-

ﬁcantly diﬀerent, using the t test for b

PAED

⫺ b

MAED

3.15 Using the kids dataset, estimate the model for the regression of ADVENTRE

on PERMISIV, MSEXATT, FSEXATT, MVALUES, FVALUES, MSTYLE1,

FSTYLE1, MSTYLE2, and FSTYLE2. Then conduct an omnibus test for the

equality of eﬀects of male versus female parents’ sex attitudes, values, and

parenting styles on the focal child’s sexual adventurism (ADVENTRE).

3.16 Using the students dataset, estimate the following models of STATMOOD:

(a) The regression of STATMOOD on COLGPA, PREVMATH, HOURS,

STUDYHRS, and TVHOURS. Evaluate the hypothesis that the number

of previous math courses elevates attitude toward statistics because it

improves math proﬁciency as measured by the math diagnostic score.

That is, add SCORE to the model and test whether this results in a

signiﬁcant reduction in the eﬀect of PREVMATH.

(b) Finally, add to the model interactions between SCORE and COLGPA and

between HOURS and COLGPA. Test whether this block of two interac-

tion terms is signiﬁcant. Regardless of signiﬁcance, interpret the eﬀect of

COLGPA in this model.

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Missing imputation: Follow the instructions for Exercise 3.9. Also, center

COLGPA, SCORE, and HOURS in all models, and be sure to create the

cross-product terms from the centered variables.

3.17 In the model E(Y) ⫽ β

⫹ β

, the OLS estimate of β

is (Hanushek

and Jackson, 1977)

⫽ .

Use this formula to demonstrate why:

(a) b

is no diﬀerent from the value for b in the SLR of Y on X

alone if

⫽ 0.

(b) Multicollinearity tends to inﬂate the magnitudes of MULR coeﬃcients.

3.18 Demonstrate the mathematical equivalence of the two forms of the nested F

test shown in the text.

3.19 Show that in the interaction model

E(Y) ⫽ β

⫹ β

⫹

...

⫹ β

⫹

...

⫹ β

⫹ γX

a unit increase in X

results in a change in E(Y) of β

⫹ γX

3.20 Verify that ∂E(Y )/∂X

in the interaction model in Exercise 3.3 is β

⫹ γX

3.21 Based on the results for model 2 in Table 3.4, compute the estimated partial

slope for years at the university for those with 2 years below, 2 years above,

and 9 years above mean years in rank. Then give the estimated mean salary

for these faculty members, assuming that they have been at the university 3

years more than average, that they have 3 years of prior experience, and that

their marketability factor is 1.1.

3.22 Although R

is conﬁned to the range [0,1], is it also true that R

adj

is so con-

ﬁed? That is, can R

adj

ever be ⬍ 0 or ⬎ 1? If so, under what conditions? (Hint:

Set R

adj

⬍ 0 and R

adj

⬎ 1 and see what these inequalities imply.)

3.23 In the kids dataset, a MULR of oﬀspring’s sexual adventurism (ADVEN-

TRE) on CPERMISS (a centered version of oﬀspring’s sexual permissive-

ness), FCAGE2 (oﬀspring’s age at time 2), MONITOR ( parental monitoring

at time 1), and the interaction of CPERMISS with MONITOR for 357 cases

produces the following sample equation:

⫽ .0197 ⫹ .1625 cpermiss ⫹ .0001 fcage2 ⫹ .0155 monitor

⫹ .024 cpermiss * monitor.

ᎏ

⫺ r

ᎏ

1 ⫺ r

EXERCISES 123

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Signiﬁcant coeﬃcients are in boldface type. MONITOR is a scale consisting of

the sum of six standardized items and therefore has a mean of zero. Also, let-

ting a represent the coeﬃcient for CPERMISS and d the coeﬃcient for CPER-

MISS * MONITOR, we have v(a)⫽ .00157, v(d) ⫽ .00014, and cov(a,d) ⫽

.000016.

(a) Interpret the main eﬀect of CPERMISS in the model.

(b) Show the partial slope for CPERMISS in the model, and give its value at

1 standard deviation, 3.4 units, above mean parental monitoring.

standard deviation above mean parental monitoring, as well as at 2 stan-

dard deviations below mean parental monitoring.

3.24 Using the students dataset, test whether the following model for EXAM1

score is any diﬀerent for males versus females in the population, using the

Chow test (use only the 214 cases with valid scores on EXAM1):

exam1 ⫽ β

⫹ β

colgpa ⫹ β

score ⫹ β

prevmath ⫹ β

hours

⫹ β

studyhrs ⫹ β

statmood ⫹ ε.

Missing imputation: Follow the instructions for Exercise 3.9. What do you

conclude?

3.25 For the students dataset, a regression of STATMOOD on COLGPA, SCORE,

PREVMATH, HOURS, STUDYHRS, and TVHRS, for 235 students, gives

SSE ⫽ 5464.69347, while separate regressions of STATMOOD on the same

predictors give SSE ⫽ 3415.49428 for the 163 female students and SSE ⫽

1451.03996 for the 72 male students. Conduct a Chow test to discern whether

there is any diﬀerence in the population between the models for males versus

females.

3.26 Refer to Exercise 3.25. A variable called MALE is coded 1 for males and 0 for

females. This type of binary variable is called a dummy, design, or indicator

variable. If it is added to the model for STATMOOD for the combined sample

of 235 males and females in Exercise 3.25, it allows the intercept for males to

be diﬀerent from the intercept for females, even though the coeﬃcients of

regressors for males versus females are constrained to be the same.

(a) Why does adding MALE allow the intercepts to diﬀer for males and

females?

(b) If SSE for this model in the combined sample is now 5018.55625, test

whether just the regressor eﬀects (but not the intercepts) are diﬀerent for

males versus females, using the Chow test. (Hint: The degrees of freedom

in the numerator of the test is no longer simply p ⫽ the number of param-

eters in the model. Rather, the df is the diﬀerence in the number of param-

eters estimated in the combined-sample model versus the total number of

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