Statistical Methods for the Social Sciences

Alan Agresti, Barbara Finlay

Chapter 10 Introduction to Multivariate Relationships - all with Video Answers

Educators

Chapter Questions

Problem 1

For cach of the three criteria for a caucal telationship. describe a rclationship between two varables that is not causal hecause that criterion would be violated

Maxime Rossetti

Numerade Educator

01:42

An association exists berween college GPA and whether one has ever used marijuana
a) Explain how the direction of a causal arrow might go is either direction.
b) Explain how a third vanable might be responsible for the association.

Tyler Moulton

Numerade Educator

02:17

Problem 3

Give an example of two variables for which the time order for a causal connection is ambiguous, and explain why.

Prabhakar Kumar

Numerade Educator

00:48

Problem 4

Explain clearly what it means to control for a vanable. Describe a situation in which an association would change dramatically after controlling a certain vanable

Neel Faucher

Numerade Educator

06:26

Problem 6

Explain what is meant by a spurious association.
a) Illustrate using the variables $X_1=$ sloc size. $X_2=$ age, and $Y=$ number of books one has ever ıead. for a random samplc of children froun scliools in Winnipeg, Canada.
b) Illustrate using the vanables $X_1=$ shoe size. $X_2=$ gender, and $Y=$ annual income, for a random sample of adults. Draw a scatter diagram to illustrate this case if. overall, men tend ro have larger shoes and higher income, on the average, than females.
f. Refer to Problem 9.17 (Table 9 13). Those data exhibit a strong negative corrclation between birth rate and per capita television ownersbip. While there could potentially be a causal relationshijn, there are other plausible explanauons for this association. Using another varnable from that exercisc. provide an explanation for whch this association is spunous.

Sherrie Fenner

Numerade Educator

00:49

Problem 7

Table 10.9 Ielales occupational level (white collar, biue collar) and political party choice. controlling for incomc.
a) Construct the bivariate table betweec occupational level and political party, ignoring income Is therc an association? If so. describe it.
b) Do the partial tables displas an association? Interprel them.
c) Using the nature of the association between income and each of the other variables. explain why the bitanate table has sucly different association than the partial tables
d) Do the pantial tables exhibit statistical interdction? Explain.
e) Construcı a chain diagram that might explain the relationships for these data. identifying the intervening variable.
f) Shou that the dara are also consistent with a spurious association, and draw the corresponding diagram.
g) Refel to parts (e) and (f) Which diagrain seems more appropriate? Why?

Maxime Rossetti

Numerade Educator

03:13

Problem 8

In ınurder tnals in 20 Florida counties during 1976 and 1977, the death penalty was given in 19 out of 151 cases in which a whitc killed a white, in 0 out of 9 cases in which a white killed a black. in 11 out of 63 cases in which a black killed a white, and in 6 out of 103 cases in which a black killed a black (M Radelet, American Sociological Reriew, Vol. 46. 1981. pp. 918-927).
a) Construct partial tables to study the association between defendant's race and the death penalty verdict, controlling tor victin's riwe.
b) Describe the association in the partial tables, using the diffecence of proportions on the odds ratio.
c) Construct the bivariate table, ignoring rather than controlling victim's race. Describe the association, and compare to (b)
d) Simpson's paradox states that the associations in partial tables can all have a different durection than the association in the bivariate table. Show that these data satisfy Simpson's paradox.

Hossam Mohamed

Numerade Educator

01:01

Problem 9

The news media report that a study has found that children who eat breakfast get better math giades than those who do not eat breakast This result was based on a simple bivariate association. with $X=$ whether cat breakiast (yes; no) and $Y=$ grade in last math course taken. How might this result be spurious. and how could you check for that possibility?

Nick Johnson

Numerade Educator

Problem 10

For a particular Big Ten universily, the mean incone for male faculty is $$\$ 8000$$ higher than the mean income for female faculy. Explain how this difference could disappear.
a) Controlling for number of years since received bighcst degee. if inale professors tend to be older and more expenenced, relatively few female professors having been hired until recent years.
b) Controlling for college of employment. if ıelatively more femalc faculty are in lowsalary colleges (Arts and Science, Education) and relatively more nalc facully are in highsalary colleges (Medical School. Law. Engineering)
c) In (a) and (b), explajn how the variables mught satisfy a chain relationship

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01:32

Problem 11

Reler to the vanables in Table 9.4. The number of bedrooms has a noderately strong positive corıclation with selling pnce ( $r=59$ ). Contuolling for size of home, howcver, this association diminislres greally Explain how this could happer, illustrating with a diagram showing potential direct and indirect effects of number of bedrooms on selling pricc.

Shu Naito

Numerade Educator

01:22

Problem 12

Refer to the variables in Table 9.16 Perhaps surpnsingly, a moderale positive correlation exists between crime rate and percent who are at least high school graduates $(r=$ .468). Percentage living in metropolitan areas is also strongly positively correlated both with cnme rate ( $r=.678$ ) and with high school graduation rate $(r=.791)$.
a) Explain why the association between crime rate and high school graduarion rate could disappear. or even change direction, when we control for percenage living in melropol itan areas.
b) Which type of relationship is more plausible for these vartables. a spurious relationship or a chain relationsbip? Explain.

Tyler Moulton

Numerade Educator

06:26

Problem 13

Give an example of three vanables for which the effect of $X_1$ on $Y$ would be
a) Spurious, disappcaring when $X_2$ is controlled.
b) Part of a chain relatronship, disappearing when an intervcning variable $X_2$ is controlled.
c) Weakened, but not elimnated, when $X_2$ is controlled
d) Unaflected by conırolling $X_2$.
e) Different at diffcrent levels of $X_2$ (i.e.. showing interaction).

Sherrie Fenner

Numerade Educator

Problem 14

Opposition to the legal availabiling of abortion is strongel among the very religious than the nonreligious. and it is also stronger among those with conservative sexual attitudes than those with more permissive attitudes.
a) Draw a unee-variable diagran of how these variables might be iclated, treating abortion attitude as the Iesponse vanable. (Note: Morc than one diaglam is plausible.)
b) Explain how you would test your diagram, showing porential sample data thal are consistent with it.

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03:25

Problem 15

Table 1010 lists the mean salary, in thousands of dnllars, of lull-time instructional faculty on nine-month contracts in United States institutions of higher education in 1993-1994, by gender and ıank.
a) Suppose that gender is the cxplanatory variable. Identify the response variable and the control varjable.
b) Describe the bivariate relationship between gender and salary.
c) Describe the relationship between gender and salary. controlling for academic rank.

Sheryl Ezze

Numerade Educator

03:13

Problem 16

Refer to Table 10.11 For in the overall table. summing over social class Compare the proportions of Catholics and Protestanis favoring the death penalty, both ignoning and controlling social class. Whal lype of relationship do these data satisfy?

Gaurav Kalra

Numerade Educator

03:13

Problem 17

Rcfer to Table 15.4 on death penalty verdict, defendant" race, and victims race.
a) Construcl the bivanaue table relating defendant's race to death penalty verdicl. ignoring victims race Calculate the conditional distrbution on the death penalty verdict, for each race of defendanl.
b) Construct partial tabies relating death penalty verdict to defendani's race treatung victims' 1ace as a control variable. For edch partial table, describe the association by computing the difference between the pioportions of white and black defendants who reccived the death penalty. Interprel.
c) Notice that the association changes direction when we control for victims race; that is, in the bivariate table relatively more whites received the death penalty, but in each partial table relatively more blacks received the death penalty. (This is Simpson's paradox; see Pıoblem 108 d ) By describing how victims' race is associated with each of these vanables, explain why the partial association differs as it does from the bivariate association. d) For these variables, indicate whether each of the following diagrams seems to provide a reasonable model. Give your reasoning. (Ilere. $P=$ death penally, $D=$ defendanl's race. $V=$ yicrims' race )
e) Is it plausible that tictims' race may be a suppressor variable? Why or why not ${ }^{\text {? }}$

Hossam Mohamed

Numerade Educator

05:12

Problem 18

For the cell counts in Table 10.12 uclating $Y=$ exam score $(1=$ below median. $2=$ above median) to gender. controlling for subject of exam (Math. Verbal).
a) Is subject of exam a suppressor variable? Explan.
b) Is there interaction? Explain.

Heather Duong

Numerade Educator

07:24

Problem 19

Reler to Pioblem $8.4^2$ in Chapter 8 .
a) Treat expectation of voting as the control sariable, and construct the partial tables relating political party affiliation to present voting intention
b) Show how to combine the partial tables to obtain the bivanate table constructed in Problem 8.43.
c) From constructing condilional distributions in the parual tables in part (a). indicate whether sample evidence of statistical interaction seems to exist.

Saeeda Aman

Numerade Educator

01:03

Problem 20

Refer to Table 15.17 on AZT use and AIDS symptoms.
a) Using the difference of proportions or the odds rapio. describe the association for (i) whites. (ii) blacks.
b) Does there secm to be strong evidence of statistical interaction? Explain.

Dominador Tan

Numerade Educator

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Problem 21

. A study of the association between whether a smoker (yes. nor and whether have had some form of cancer (yes, no) has odds ratio 11 for subjects of age less than 30.2 . 4 for subjecte of age 30 to 50 , and 4.3 for subjects of agc over 50 .
a) Identify the response variable, explanatory variable. and conuol variable. and interpret the associarions.
b) Does the study show evidence of interaction? Explain.

Shu Naito

Numerade Educator

01:43

Problem 22

Refer to Table 8.29 Treal the four groups as a cross-classification of the two variables, race and gender. Treating race as a control variable, is there evidence of severe statistical interaction for these data? Explain.

James Kiss

Numerade Educator

Problem 23

A study of students at Oregon State University found an association between frequency of church attendance and favorability toward the legalizanion of marijuana. Both variables were measured in ordered categorres When gender of student was controlled, the resulting gamma measures for the two partial tables were as follows:
Males gamnia $=-287$. standard error $=081$
Females: gamma $=-.581$. standard error $=.091$
a) loterpret these results
b) These results show u slight degree of $\qquad$ . since the asqociation is somewhat stonger for females than males.

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04:43

Problem 24

A study of the relationship betw'een student s high school GPA and mother's employment (yes, no) suspects an interaction with gender of student. Controlling gender of student, Table 10.13 conlains the mean GPA for each conbination of gender and mother's employment.
a) Descnbe the relationship between mother's employment and GPA for females and for males Does this sample slıow evidence of statistical interaction? Explain.

Jeremiah Mbaria

Numerade Educator

Problem 25

Refer to the IVYW data set (Pıoblem 1 7). Construct partial tables selating opinion about abortion to opinion about life after death, controlling for attendance at religious services, ineasured using the iwo categoties, (Never or occasionally. Most weeks or every week). Prepare a report (a) posing and interpreting a possible arrou diagram, bcfore you analyze the data, for relationships among the variables. (b) interpreting the sample associations in the bivariate table and the partial tables. (c) revising. if necessaly, your arrow diagram based on the evidence in the sample data.

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Problem 26

Refer to Problein 1.7. Repeat the previous problem for three variables chosen by your instructor

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04:04

Problem 27

. Refer to Problem 1.7. Are there any pairs of these variables for whicl you expect the assuciatiou to disappear under control tor a third variable? Explaim.

Khoobchandra Agrawal

Numerade Educator

03:36

Problem 28

Using the most recent General Social Survey, construct a contingency table relatung gender and party affiliation ("PARTYID"). Is there still a gender gap" Control for political ideology ("POLVIEWS") by forming partial tables for the most conservative and the most liberal subjects. Does the assocjation seem to persisl for these subjects?

Nick Johnson

Numerade Educator

Problem 29

. Table 10.14 is a contingency table that shows counts on self-esteem, race, and cumulative GPA, for a sample of females. Analyze the type of multivariate relationship that these variables scem to satisfy. Prepare a report, explaining the analyses you conducted to reach your conclusion and providing interprelations.

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Problem 30

Table 10.15 shows the relation between allitude loward a nuclear freeze and church attendance, controlling for educational levels of the respondents. Assessing association by the difference of proportions, analyze these data using methods of this chapter, and explain why it is misleading to report only the results from the bivanate lable.

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06:01

Problem 31

In 1980. SAT total scores (verbal plus math) had a mcan of 890 . In 1984 the mean increased to 897. for an increase of 7 . Table 10.16 shows the means and thein changes, controlling for race. For each year; the table also shows the pcrcentage representation in the sample of each race.
a) In each year, using the means tor each racc and the percentage representations, verify the overall means.
b) For the overall sample, the change in the mean from 1980 to $1984(+7)$ uas smaller than the change for either whites ( $$\$ 8$$ ) or nonwhites ( +15 ). Explain this apparent paradox.

Bryan Luo

Numerade Educator

Problem 33

Table 10.18 show's the mean number of children in Canadian faunilies, classilied by whether the family was English speaking or French speaking and by whether the family was in Quebec or in another province. Let $Y=$ number of children in fanuly. $X_1=$ primary language of family, and $X_2=$ province (Quebec, others).
a) Descnbe the as sociatuon between $Y$ and $X_1$, based on the overall means in this table.
b) Describe the association between $Y$ and $X_1$, controlling for $X_2$.
c) Explain how it is possible that for each level of province the mean is higher for Fiench speaking families, yet overall the mean is ligher for English speaking families (This illustrates Simpson's paradox: see Problem 10.8d)

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Problem 34

Relgions affiljation is associated with attitudes about racial intermaniage. with white Catholics tending to be less racially prejudiced than Protestints, members of theologically liberal Protestant groups tending to be less prejudiced than those from the more conservative denominations. and Jews and the nonreligious bcing least prcjudiced (see Ptoblemy 8.35). Describe some variables than you miglat control to sce if they can explain. at least partially, this association

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22:02

Problem 35

A research study funded by Wobcgon Springs Mineral Wate1, Inc., discovers that the probability that a newborn child has a birth defect is lower for fanilies that regularly buy bottled water than for families that do not. Is this assocjation likely to reflect a causal link between drinking botred water and a reduction in bith defects? Wlyy or why not?

Evelyn Cunningham

Numerade Educator

03:23

Problem 36

The peicentage of women who get breast cancer is hugher now than at the beguning of this century. Suppose that cancer incidence tends to increase with age. and suppose diat women tend to live longer lives now than earlier in this century. Explain why a comparisou of breast cancer rates now with the beginning of thus century could show differeut results from these if ue control for the age of the woman.

Rabeya Zahid

Numerade Educator

Problem 37

Refer to Table 9 1. A regression analy sis reveals a moderately strong negauive correlation ( -.677 ) bctween percent of white residcnts and violent crime rate. Another vanable measured in Table 91 is percent of families headed by a single parent. Thus is positively correlated with violent crime 1 ate (.839) and negatively correlated with percent white recidents $(-.657)$.
a) Controlling for single-parent rate, would you expect the association between percentage of white residents and violent crime rate to be stronger, or weaker, or the samc as in the bivariate relationshup. Explain.
b) Explain how these variables could plausibly have a chain relationship.

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00:45

Problem 38

In the L'nited States median age of residents is lowest in Utah At each age level. the death rate from heart disease is higher in Utah than in Colorado; yet overall. the death rate from heart disease is lower in Utah than Colorado. Are there any contradictions here, or is this possible? Explain.

Adnan Gill

Numerade Educator

Problem 39

For lower-level managerial employees of a fast-food chain. the prediction equation relatung $Y=$ amual income (thousands of dollars) to $X_1=$ number of years experience on the job equals $\dot{Y}=14.2+1.1 X_1$ fo: $X_2=$ males and $\hat{Y}=14.2+.4 X_1$ for $X_2=$ females.
These equations show evidence of (select one. and explain)
a) Spurious relationship
b) Chain relationship
c) Statistical interaction

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Problem 40

Statistical interaction lefers to which of the following?
a) Association exists between two vanables
b) The degree of association between Iwo variables varies gıcatly over the partial levels of a control varable.
c) The partial associauon is the same at each level of the control variable, hut it is different from the overall bivariate associatun. ignoring the control vanable.
d) For a collection of three valiables, each pair of vanables is associated.
e) All of the above

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Problem 41

Consider the relationslip betweell $Y=$ political parly preference (Democrat. Republican) and $X_1=1$ ace (Black, White) and $X_2=$ gender. There is an association betwreen $Y$ and both $X_1$ and $X_2$, with the Democrat preference being more likely for blacks than whites and for women than ment Select the conecl responsets)
a) $X_1$ and $X_2$ are probably independent causes of $Y$
b) The association between $Y$ and $X_1$ is probably spurious, controlling for $X_2$.
c) Since botl variables affect $Y$, there is probably interaction.
d) The variables probably satisfy a chain relationship.
e) Race is probably a suppiessor tarlable
f) None of the above.

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05:18

Problem 42

Table 10.17 rcports the median family income (in dollars) in 1992 for white and black families The median for white families is $$\$ 17.748$$ higher than the median for black families. For each of the three lypes of families, however, the median for white families is about $$\$ 8000$$ higher than the median for black faunilics Explain what could cause this discrepancy.

Ryan Mcalister

Numerade Educator