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Statistical Methods for the Social Sciences

Alan Agresti, Barbara Finlay

Chapter 13

Combining Regression and ANOVA: Analysis of Covariance - all with Video Answers

Educators

Chapter Questions

13:46

Problem 1

The 1egression equation relating $Y=$ education (numbes of y'ears completed) to race ( $Z=$ 1 for whates. $Z=0$ for nonwhites) in a certan country is $E(Y)=11+2 Z$. The regression equarion relating educalion to race and to father's educalion $(X)$ is $E(Y)=3+.8 X-.6 \mathrm{Z}$. a) Find the mean education for whites. the mean education for nonwhites, and the difference between them (jgnoring father's education).
b) Plot the relationship between $X$ and the mean of $Y$ for whites and for nonuthites
c) Find the diflerence between the mean education of whites and nonwhites. controllung for father's education.
d) Find the mean education for whites and for nonuhites, when father's education equals 12 years

Heather Duong
Heather Duong
Numerade Educator
03:46

Problem 2

A regression analysis for the 100 th Congress, ending in 1988, predicted the proportion of each representalive's voles on dbortion issues that took the "pio-cheice" position (R. Tatalovich and D. Schier. Americom Politics Quanterd: Vol. 21, 1993, p. 125). The prediction equation was
$$
\hat{y}=.350+0111 \mathrm{D}+.094 \mathrm{REL}+.005 \mathrm{NH}+.005 \mathrm{INC}+.063 \mathrm{GEN}-.167 \mathrm{PAR}
$$
whese REl , $=$ religion $=1$ for non - Catholses, $\mathrm{GEN}=$ gender $=1$ for women, $\mathrm{PAR}=$ pary $=1$ for Democrats, $\mathrm{ID}=$ ideology is the member ADA score (ranging from 0 at most conser ative 10100 at most liberal). $N W=$ nonu hite is the percentage nonwhite of the member's district. and INC $=$ income is the median lamly income of the momber's district
a) Interpret the effect of percentage nonu bite.
b) Interpiet the effect of gender
c) Interpret the coefficient for party. Does this imply that. ignoring the controls. Democrats are less likely than Republicans to take a pro-choice position?
d) Using standardized vanables, the predicion equation is
$$
\hat{\bar{z}} 1=.83 \mathrm{HD}-.21 \mathrm{REL}+.38 \mathrm{NW}+.05 \mathrm{INC}+.03 \mathrm{GEN}-.18 \mathrm{PAR}
$$

Comiment on the relative sizes of the partial effects. Interpret the coefficient of ideology.

Lucas Finney
Lucas Finney
Numerade Educator

Problem 2

In the model $E(Y)=\alpha+\beta_1 X+\beta_2 Z$, where $Z$ is a dummy variable,
a) The qualutative predictor has two categories.
b) One line has slope $\beta_1$ and the other has slope $\beta_2$.
c) $\beta_2$ is the difference between the mean of $Y$ for the sccond and first categories of the qualitative variable.
d) $f_2$ is the difference between the adjusted mean of $Y$ (conlrolling for $X$ ) for the second and first categories of the qualitacive vanable

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Problem 3

Based on a national survey Table 13.14 shows resulls of a prediction cquation (ignonng some nonsignificant varables) 1eponted for the response viariable, $Y=$ alcohol consumption, measured as the number of alcolnolic drinks the subject drank during the past month (D. Umbeason and M Chen, Americon Sociolonical Reriew, Vol. 59. 1994, p 152)

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Problem 4

Refer to Table 9.4 in Clapter 9 Table 13.15 shows a printout for modeling $Y=$ selling price in terms of $X=$ size of home and $Z=$ whether the home is neu $(Z=1$, yes; $Z=0$, no

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Problem 5

Refer to the previous exercise Tablc 13.16 shows a SPSS primout from fitting the model allouing interaction. where NEWSIZE refers to the closs-product term.

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00:19

Problem 7

Refer to Table 13 I. Not reported there were obser vations for ten Asian Americans. Their $(X, Y)$ values follow:

Nick Johnson
Nick Johnson
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09:15

Problem 8

The printouts in Table 13.18 show results of using SAS to hit four models to data fiom a study of the relationship between $Y=$ percentage of adults voting, $X=$ percealage

Jameson Kuper
Jameson Kuper
Numerade Educator

Problem 9

Refer to the picvious exercisc The means of percentage registered Jor the threc categones are $\dot{X}_1=762, \bar{X}_2=49.5$. $\bar{X}_3=39.7$, with an overall mean of $\bar{X}=60.4$
a) Find the adjusted means on pescentage voting. and interpret.
b) Compare the adjusted mean for Auglos to the unadjusted nrean of 523 , and interpret
c) Shelch a plol of the no inleraction model for these dald. and identify on it the unadjusted and adjusted means
d) Using the appropriate model. tect the null hypothesis that the true unadjusted means on percentage voting are equal Report the test statistic and $P$-value, and interpret Compate the sesult of this test to the one in Problem I3.8(f).
e) Describe hou to expand the no interaction model to include the additional qualitative variable. location of peecinct, with categories urbun, suburban. small town, and rural.
f) Duscribe how to expand the no intcraction model to include the additional guantitatise explanalory variable. percentage of residents in precinct who are homeowners

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Problem 10

An andlysis of covariance model is fitted to annual income (thousands of dollars). using predicors age and mantal status Table 13,19 shows the sample mean incomes and the adjusted means for the model. How could the adjusted means be so differesi from the unadjusted means? Drew a sketeh to help explain

Shu Naito
Shu Naito
Numerade Educator
13:46

Problem 11

Refe1 to Problem 13.1 The overall mean of father's education is 12 years Find the adjusted mean educational lev els for whites and nonu híces. contrulling for father ${ }^{\circ}$ education, and compare them to the unadjusted means.

Heather Duong
Heather Duong
Numerade Educator

Problem 12

Refer to the WWW data set (Problem 1 7). Using computer sof(ware, conduct and interpret an analy'sis of covarnance using $Y=$ political ideology. Prcpare a report. presenting graphical. descriptive, and inferential analyses. using the predictors
a) religiosity and whether a regetarian
b) college GPA. number of times a wcek reading a newspaper. opinion about abortion, and whether a vegetarian.

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Problem 13

Repeat the previous excrcise. using $Y=$ college GPA with predictors ligh school GPA. gender. and religosity.

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Problem 14

Refer to the data file created in Problem 1.7. For variables chosen by your inctructor. use regression analysis as the basic of desenptive and inferential statistical analyses Sunmarize yout findings in a report in which you describe and interpret the fitted models and the related analyses

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Problem 15

Table J3.20 shows results of fittung a regression model to data from 1969 on sularies (in dollass) of about 35.000 college professors. Four predictors are qualitalive (binary). with dummy vanable defined in parentheses. The table shows estimates of parameters for each predictor. with standard enors in parentheses. Writc a short report. interpreting the ellects of the predictors.

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Problem 16

Table 1321 is a SPSS piuntout based on General Social Survey data combined for the years 1977, 1978, and 1980. The response vanable is an indcx of dttitudes toward premarital. extramarital. and homosexual sex. Higher scores represent more permissive atti-

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07:04

Problem 17

A researcher is interested in factors associated with fertility in a Latin American cily. Of particular interest is whether migrants from other cities or migrants from rural areas differfrom natives of the eity in their completed family sizes. The groups to be compared are urban natives, urban migrants. and rural mgrants. Fertility is defined to be a woman's total number of live births. Since fertility is negatively related to educational level. and since education might differ among the threc groups. it is decided to control that variable. Table 13.22 shows data for a random sample of marned women above age 45 Analyze these data. In your report, provide gaaphical presentations as well as interpretations for all your analyses, and present a summary of the main results.

Heather Duong
Heather Duong
Numerade Educator

Problem 18

Refer to Table 9 I , not including the observation for D C. Let $Z$ be a dummy vanable for whether a state is in the South. with $Z=1$ for AL, AR, FL. GA KY, LA, MD, MS, NC. OK, SC, TN, 7X, VA, W'V
a) Analyze the relationship between $Y=$ violent crime rate und the predictors $X=$ poverty rate and $Z$.
b) Add percentage white as a predictor. Find a model that describes the data well, and interpret,
c) Repeat the analysis with D.C in the data set. setting $Z=1$ for it. Do the conctusions change in any way?

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Problem 19

Refer to the previous exercise. Repeat using $Y=$ murder 1 ate

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Problem 20

Figure 134 exhibits at least one possibly influcntial oullier Remove the observation with the bighest income and reconduct the analyses. Did this one observation have any influence on the results?
2

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Problem 21

You have two groups, and you want to compare their regressions of $Y$ on $X$, in order to test the ly pothesis that the true slopes are identical for the two groups. Explain how you can do this using regression modeling.

Shu Naito
Shu Naito
Numerade Educator
02:14

Problem 22

Let $Y=$ dcath rate and $X=$ average age of residents. measured for each county in Massachusetts and in Florida. Draw a bypothelical scatuer diagram, identifying points for each state, when
a) The mean death rate is higher in Florida than in Massachusets when $X$ is ignored. but lower when it is controlled.
b) The mean death rate is higher in Flonda than in Massachusetts both when $X$ is ignored and when it is controlled.

Harsh Gadhiya
Harsh Gadhiya
Numerade Educator
01:46

Problem 23

Draw a scatter diagram of $X$ and $Y$ with seth of points representing two groups such that $H_0$. equal means on $\gamma$ would be rejected in a one-way ANOVA. but would not be rejected in an analysis of covanance.

Sriparna Bhattacharjee
Sriparna Bhattacharjee
Numerade Educator
01:56

Problem 24

Give an example of a situation in which you expect interaction between a quautitative variable and a qualitarive variable in their effects on a quantitative response variable.
In Problems 13.25-13.26, select the correct response(s)

Raymond Matshanda
Raymond Matshanda
Numerade Educator
01:02

Problem 26

In the United States, the mean annual income for blacks $\left(\mu_1\right)$ is smaller than for whites $\left(\mu_2\right)$. the mean number of years of education is smaller for blacks than for whites. and annual income is positively related to number of years of education. Assuming that there is no interaction, the difference in the mean annual income between whites and blacks, controlling for education, is
a) Less than $\mu_2-\mu_1$
b) Greater than $\mu_2-\mu_1$
c) Possibly equal to $\mu_2-\mu_1$

Jameson Kuper
Jameson Kuper
Numerade Educator
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Problem 27

Summarize the differences in purpose of the following:
a) A regression analysis for two quantitative vanables
b) A one-way analysis of vanance
c) A two-way analysis of vanance
d) An analysis of covanance

Rashmi Sinha
Rashmi Sinha
Numerade Educator

Problem 67

Refer to the previous problem. Figure 13.14 shows a SAS scalter diagram for the relationship between selling price and size of home. identifying the points by a I when the home is new and a 0 when it is not. Because of the crudeness of scale, many of the points are hidden in this plot, but note that the observation with the highest selling price is a new home that is somewhat removed fiom the general tread of points Table 13.17 shows a SAS printout for the intcraction inodel after removing this single observation. For thes adjusted data set. $R^2=.845$ for the no interaction model and $R^2=.848$ for the interaction model. Now, again answer the four parts of the previous exercise, and note what a large impact one obsen alion can have on the conclusions.

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