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Statistical Methods for the Social Sciences

Alan Agresti, Barbara Finlay

Chapter 8

Analyzing Association Between Categorical Variables - all with Video Answers

Educators

Chapter Questions

07:05

Problem 1

A Gallup Poll in 1995 suggested that in the Unted States. about $15 \%$ of males and $15 \%$ of females believe that abortion should be illegal in all circuinstances (The Gallup Poll Monthh, No 354. Princeton, NJ: The Gallup Poll, March 1995, p. 30).
a) For males and for fenales, report the conditional distnbution on whether abortion should be illegal in all circumstances. using calegorics (yes, no).
b) Based on these resulls, docs statistical independence seem plausible between gender and opinion about whether abortion should be illegal in all circumstdnces? Explain.

Saeeda Aman
Saeeda Aman
Numerade Educator
02:30

Problem 2

Whether a wuman becomes pregnant in the next year is a categoncal variable with categones (ves. no), and whcther she and her partner use contraceptives is another categoncal vanable with categones (yes, no). Would you expect these variables to be statustically independent, or associated? Explain.

Lucas Finney
Lucas Finney
Numerade Educator
04:00

Problem 3

For a sample of 100.000 subjects, Table 8.22 cross-classifies whether a subject is HIV+ by whether a diagnostic lest for HIV+ status is positive (i.e.. indicates that the subject is $\mathrm{HIV}+2$
a) Construct the conditional distnbutions for the test result, given the true discase status.
b) For those u ho truly are HIV+, what percentage are diagnosed as HIV+ by the diagnostic test? For those who are not HIV+. what percenlage gel a negative test ıesult? Does the diagnostic test appear to be a good one?
c) Consiruct the conditional distribution on disease starus, for those who have a posituve diagnostic test result Of those subjects, what percentage trulv are HIV+? Comunent. (For discussion of the possibility of a high false posilive rate even for a seemingly good diagnostic test, see Chutterjce et al.. 1995. pp, 37-40).

Sajin Shajee
Sajin Shajee
Numerade Educator

Problem 4

In an article about cnme in the Uniled States, Newsweek magazone (January 10, 1994) quored FBI statistics stating that of all blacks slain in 1992, $94 \%$ vere slain by blacks, and of all whites slain in $1992,83 \%$ were slain by whites. Let $Y$ denote race of victim and $X$ denote race of murderer.
a) Which conditional distributions do these statistics refer to. 山hose of $Y$ at given levels of $X$, or those of $X$ at given levels of $Y$ ? Set up a table with race of murderes as rows and race of viclim as columns, showing these conditional distributions.
b) Are $X$ and $Y$ indcpendent or dependent? Explain.

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03:11

Problem 5

A young chuld wonders uhat caluses uomen to have babics. For each woman who lives on her block, she observes whether her hain is gray and whether she bas young clildren. with the results shown in Table 8.23.
a) Construct the $2 \times 2$ contingcncy table that cross-classifies "gray hair" (yes, no) with "young chuldren" (yes. no) For these nine women
b) Treating "young children" as the response vanable, obtain the conditional distributions Jor those women who have gray hair and for those who do not. Does there seem to be an association?
c) Noticing this association, the child concludes that not having gray hair is what causes women to have childen. Use this example to explain why associabon does not necessarily imply causation (Chapter 10 discusses the connection berween association and caasation.)

Lynn Larson
Lynn Larson
Numerade Educator

Problem 6

How large a $y^2-$-\alue puovides is $P$-value of .05 for testing independence tor the follou ing table dinensions?
a) $2 \times 2$
b) $3 \times 3$
c) $2 \times 5$
d) $5 \times 5$
c) $3 \times 9$

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01:40

Problem 7

Shou that the contingency table in Table 8.24 las four degrees of freedoun, by filling in the missing cell counts.

Sheryl Ezze
Sheryl Ezze
Numerade Educator
06:09

Problem 8

Table 825 tefers to a survey conducted in 1992 of senior high school students in Dayton. Ohio
a) Construct condirional distributions that treal cigarette smoking as the response variable. Interpret.
b) Construct conditional distributions that tuedt alcohol use as the response variablc. Interpiet.
c) Test whether cigarette use and alcohol use are statistically independent. Report the $P$ value. and interpret.

Saeeda Aman
Saeeda Aman
Numerade Educator

Problem 9

Table 826 shows an SPSS pintout for some analy scs for a data set taken from the 1991 General Social Sur ey, with saisbles race and party identification.
a) Form the conditonal distribution on party identification, for each race. Interprel
b) Report the expected fiequency for the first cell, and show hou SPSS obtained it
c) Tcst the hypothesis of independence between party identificaton and race. Report the tesi stanstic. $P$-value, and interpreL.
d) Use the adjusted residuals to describe the evidence about association
e) Summarize the associauon between race and whether one is a Democrat or a Republican. by computing the odds ratio or by companing percentages. Interprel.

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05:55

Problem 10

Table 8.27 is from the 1991 Gencral Social Survey Subjects were asked their opinion about a man and woman having sex relarions before marriage. They were also asked whether methods of birth control should be made available to teenagers between the ages of 14 and 16
a) Software reports that $\chi^2=128.7$, based on $d f=9$. Interpret.
b) Table 827 also shows. in pareatheses. the adjusted residuals Use these to describe how the dara deviate from independence in the four corner cells. Interpret.

Saeeda Aman
Saeeda Aman
Numerade Educator
19:12

Problem 11

Refer to Problem 8.8 .
a) Describe the nature of the association using adjusted residuals.
b) Describe the strength of ussociation using the difference between users and nonusers of alcohol in the proportions who have used cigarettes. Interpret.
c) Describe the strength of association using the difference between users and nonusers of cigarettes in the proportions who have used alcohol. Interpret
d) Describe the strength of association using the odds ratio. Jnterpret

Heather Duong
Heather Duong
Numerade Educator
06:23

Problem 12

Table 8.28 refers to 68,694 passengers in autos and light truchs involved in accideuts in the state of Maine in 1991. The table classifies passengers by whether they werc weanug a seat belt and by whether they were injured or killed. Test the hypothesis of independence. Interpret the $P$-value.

Jeremiah Mbaria
Jeremiah Mbaria
Numerade Educator

Problem 13

Refer to the previous problem, Show how to follow up the rest descriptivel) with
a) An analysis using adjusted ıesiduals
b) The difference between two proportions.
c) The odds ratio.

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05:55

Problem 14

Table 8.29 refers to a nationdl study of 15 - and 16 -yeat-old adolescents. The groups are four combinations of gender and race The event of interest is ever having sexual intercourse Is there an association berween group and the response? Interpret the results in the computer printout supplied with the table.
(TABLE CAN'T COPY)

Saeeda Aman
Saeeda Aman
Numerade Educator
01:52

Problem 15

Refer to the previous exercise.
a) Describe the naturc of the association, using conditional distributions.
b) Describe the nature of the association for females, using adjusted residuals for the Yes category in the first two rous.
c) Describe the strength of association. using the difference between the proportions of blachs and whites who have expenenced sexual intercourse. separately for females and for males.
d) Descnbe the strength of association, using the difference between the proportions of males and temales who have expenenced sexual incercourse, separately for blacks and for whites.
e) Desclibe the strength of association using the odds atio between gender and response, sepsrately for each race.
f) Describe the strength of association using the odds ratio belween race and response. separately for each gender.
g) Summarize what you analyses have revealed about these data

Maxime Rossetti
Maxime Rossetti
Numerade Educator
02:27

Problem 16

Refer to the paevious two exercises
a) Analyze the association between race and response. given gender, by conductung two chi-cquated rests, one for females using the first two row's of the table and one for males usiug the last two rows of the table. Interpre..
b) Analyze the overall association betwcen gender and response by combining the first two rows. combining the last two row s. and conducting a chi-squared test for the resulting $2 \times 2$ table. Interpret. (Note: The constuction in (a) and (b) shows a way of analyzing the rable with three scparate chi-squared statistics, each having $d f=1$, rather than a single statistic having $d f=3$ The sum of the three separate statistics having $d f=1$ is approximatcly equal to the overall staristic lhaving $d f=3$. For discussion of ways of "partitioning" chj-squared, see Agresti. 1996. Scc 2.4 6.)

Maxime Rossetti
Maxime Rossetti
Numerade Educator
12:23

Problem 17

Table 830 is from the 1991 General Social Survey. White subjects wetc asked "If your party nominated a (Negno/Black) for President, would you vote for him if he were qualified fot the job?" and "During the last few years, has anyone in yout fannily brought a Inend who was a (Negro/Black) home for dinner?
a) Find the statistical significance of the association using the chi-squared test. Interptet the $P$-value.
b) Somc expected It equencies are small, so the chi-squared test may not be valid. If you have software available. conduct a snall-sample exact test, and interpret the $P$-value.
c) Follow up the rest, describing the nature of the association

Evelyn Cunningham
Evelyn Cunningham
Numerade Educator

Problem 18

Refer to the previous exercise. Using the yes and no caregories of each response, estimate the odds ratio. Interpret.

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03:32

Problem 19

A neu spaper article preceding the 1994 World Cup semifinal match between Italy and Bulgaria stated that "Italy is fayored $10-11$ to beat Bulgaria. which is rated at $10-3$ to reach the final." Suppose this means that the odds that Ital) wins are $11 / 10=1.1$ and the odds that Bulgand wins are $3 / 10=3$. Find the probability that each team wins, and comment.

Harsh Gadhiya
Harsh Gadhiya
Numerade Educator
01:33

Problem 20

According to data from the 1993 Substance Abuse and Mental Health Administration national houschold since, on drug abuse, for Americans aged $26-34,59 \%$ had used marijudnd at least once in their lifetime, and $26 \%$ had used cocaine at least once
a) Find the odds of having used marijuana. Interpret
b) Find the odds of traving used cocaine. Interpret
c) Find the odds ratio comparing ınarijuana use to cocaine use (i.e., this odds ratio refens to the $2 \times 2$ table having rows (marijuana. cocaine) and colunins (used at least once, never used) Inierpret.

Nick Johnson
Nick Johnson
Numerade Educator
04:28

Problem 21

When asked by the National Opinon Research Center "Is there any area right around here-that is. within a mule-where you would be afrand to walk alone at nght' $\gamma$, in 1973 $60 \%$ of fennales answered yes and $20 \%$ of males answered yes, whereas in $199460 \%$ of females answered yes and 305 of males answered yes.
a) Calculate the odds that a female responded yes in (i) 1973. (ii) 1994. Interpret.
b) Calculate the odds that a male responded yes (i) in 1973, (ii) in 1994.
c) Calculate the odds ratio between gender and response, in 1973 and in 1994. In winch year is the association stronger?

Tatiana Graham
Tatiana Graham
Numerade Educator

Problem 22

For mulders in the the United States in 1993 having a single victim and single offender, Table 831 cross classifies the sex of the victim by the sex of the offender. Calculate and interpret the odds ratio.

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Problem 23

According to the U.S Deparment of Justice, in 1994 the incarceration rale in the nation's pnsons was 646 per 100.000 male residenis. 45 per 100,000 female residents, 1471 per 100,000 black residents, and 207 per 100,000 white residents (Bureau of Justice Statistics Bulletin: Prisoners in 1994).
a) Find the odds ratio betwecn gender and whether incarcerated Interprel.
b) Find the odds ratio between race and whether incarcerated. Interpret.
c) According to the odds ratio. which has the stronger association with whether incarcerated, gender or race?

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00:57

Problem 24

For college freshmen in 1994. the odds ratio belween gender (male, female) and opimon abour whether homosexual relacions should be legally prohibited ( ycs , no) equaled 2.6 (Higher Education Research Institute, University of California at Los Angeles)
a) Explain what is wrong with the interpretation. "The probability of a yes response for males is 2.6 times the probability of a yes response for females." Give the correct interprctation.
b) The odds of a yes response equaled .82 for males. What is the probability of a yes response for males?
c) Based on the odds of .82 for males and the odds ratio of 2.6 , find the prohability of a yes response for females.

Bryan Meares
Bryan Meares
Numerade Educator
01:02

Problem 25

Consider the $3 \times 3$ table having entries, by row, of
(4.2.1/2,2,2/1,2,4).
a) Is the chi-squared test of independence appropriate for these data? Explain.
b) Using software, conduct a valid test of independence. Interpret.

Sheryl Ezze
Sheryl Ezze
Numerade Educator

Problem 26

Refer to the previous exerc.se. Suppose the variables are ordinal. with levels (high, modium, low) for each classification.
a) Compute the nunnbers of concordant and discordant pairs
b) Compule gamma, and interpret.
c) Shou hou to express gamma as a difference belwcen two proporions.
d) Using software. conduct a test of independencc using gamma, and coinpare the result to the previous cxercise hhy is the $P$-value so much smaller here?

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Problem 27

Table 8.32 refers to a study that assessed factors as sociated with wonnen's artitudes to ard mammography. The columns refer to their response to the question, "Hou likely is it that a mammogram could find a new case of breast cancer $\gamma^{\circ}$ Use gamma to summarize the strength of associalion. Interpret the sign and magnitude

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13:46

Problem 28

A study on educational aspirations of high school sludents (S. Crysdale, Internarional Journal of Comparative Sociolog1; Vol. 16. 1975, pp. 19-36) measured aspirations using the scale (some high school. high school graduate. some college. college graduaue). For students whose family income was luu, the counts in thcse categories were (9, 44. 13.10); when family income was uniddle, the counts were (11.52.23,22), when family income was high, the counts were ( $9,41.12,27$ ). Sof(ware provides the results shown in Table 8.33
a) Report the talue of an ordinal measure of association and use it to summarize the association.
b) Test independence of educalional aspirations and fanily income using the chi-squared test. Interpret, and explain the deficiency of this lest for these data.
c) Find a $90 \%$ confidence interval for an ordinal measure. Interpret
d) Conduct an alternative test of indcpendence that takes caregory ordering into account. Interpret results. and compale to results of the chi-squared test.

Heather Duong
Heather Duong
Numerade Educator

Problem 29

Table 8.34 comes from as study of econonic conditions and political attitudes of Cuban workers toward the Cuban revolution. A comnputet analysis reports:
$$
\begin{array}{lcc}
\text { Stat1stic } & \text { Value } & \text { ASE } \\
\hline \text { Gamma } & -0.481 & 0.117 \\
\text { Kendall's Tau-b } & -0.238 & 0.059
\end{array}
$$
a) Interprel one of these sample statistucs.
b) Obtain a $95 \%$ confidence interval for the population value. Interpret.
c) Conduct an ordinal test of independence. Interpret.

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Problem 30

Refer to Problem 8.10. The analy sis there does not take into account the ordinality of the vanables. Using software, summarize the strength of association by finding and interpreting garma. Construct and interpred a $95 \%$. confidence inlerval for the population value of gamma.

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Problem 31

Suppose $\hat{\gamma}=.30$ for the relationsbip between two ordinal variables.
a) Of the pairs that are concordant or discordant, what proportion are concordant? Discordant ${ }^2$
b) Is this a stonger or a weaker dssociation than one having $\hat{\gamma}=-, 70$ ? Explain

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Problem 32

Conduct the test of independencc for Table 8.17 using Kendall's tau- $b$ rather thun gamma. Compare results to those obrained using gammd.

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07:05

Problem 33

Refer to the WWW data sel (Problem 1,7). Using computer software. create and analyze descriptivel, and inferentially the contingency table relating
a) Political affiliation and opirion about abortion,
b) Religiosity and opinion about abortion.
c) Gender and belicf in life alter death

Saeeda Aman
Saeeda Aman
Numerade Educator

Problem 34

Refer to the data bile you created in Pioblem 1.7 For varrables chosen by your instructoi. conduct descriptive and inferential statustical analyses. Luterpret and summarnze yout findings.

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Problem 35

Several sociologists have reported that racial prejudice varies according to religious group Examine this using Table 8.35, for white respondents to the 1980 General Social Survey. Religious preference has four categories: (i) Liberal Protestant, encompassing the theologically morc hiberal groups such as Presbyterians. Episcopalians. Methodists, and Congregatuonalists; (ii) Conserv ative Protestant, consisting of conservative and fundamentalist denominations and sects such as Baptists and Pentecostals: (iii) Catholic; and (i) None, including those with no religious preference. (Jews and other religious preferences are not shown because of the small number of cases in the sample.) The indicalor of racial prejudice is the survey question, "Do you thunk there should be laws aganst marnages between blacks and whites Analyze these data. Prepare a report. descnbing y our dnalyses and providing interpretations of the data.

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Problem 36

Analyzc the data in Table 8.36, relating political ideology to political party Prepare a report of your analyses, emphasizing interpretations of the results.

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Problem 37

Table 8.3 ? classifies a sample of psychiatric patients by their diagnosis and by whether their treatment prescribed drugs. Analyze these data. providung interprelations of all your analyses.

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Problem 38

Table 8.38 , fion the 1991 General Social Sutney, refers to the association between belief in the afterlife and gender. Analyze these data.

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Problem 39

The printour in Table 8.39 refer to a sample of voters from a recent presidential primary in Wisconsin. Democrats and Republicans weıe classified on political ideology. Explain how to interpret thesc analyses

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Problem 40

The sample in Table 8.17 consists of 104 black Americans. A similar table relating income and job satisfaction for white subjects in the 1991 General Social Survel has counts (18. 39.36$)$ in row $1,(43,163,158)$ in row 2 , and $(27,97,156)$ in row 3. Analyze these data.

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05:42

Problem 41

Refer to Problem 8.8. A similar table relating alcohol use as rows to manjuana use as columns has cell counts $(322,5)$ in row I and $(994,955)$ in row 2 . Analyze these dala:

Carson Merrill
Carson Merrill
Numerade Educator
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Problem 42

In a survey conducted in 1994 by Market Segment Research and Consulting. Inc., as part of the 1994 Ethnic Market Report, ahout 5000 respondents were asked "Which of the following issues facing your community today do you feel is the most important?" Table 8.40 shows the data. Describe the patterns of association that this table displays.

Rashmi Sinha
Rashmi Sinha
Numerade Educator
07:24

Problem 43

Shortly before a gubematural election, a randomi sample of 50 potential voters are asked the following questions:
Do you consider yourself to be a Democrat (D), a Republican (R), or Independent (1)?
If you were to vote today, would you vote for the Democratic candidate (D), the Republican (R). or would you be undecided (U) about how to vote?
Do you plan on voting in the election? Yes $(\mathrm{Y})$ or mo $(\mathrm{N})$ ?
For each person interviewed. the answers to the three questions are entered in a data file. For example. the entry (D. U, M) represents a registered Democrat who is undecided and who does not expect io vote Table 8.41 summarizes results of the 50 interviews. Using computer software, create a data file and conduct the foliowing analyses:
a) Construct the $3 \times 3$ conungency table relating political party affiluation to untended vote (the first two variables listed). Report the conditunal distributions on intcaded vote for each of the threc polutocal party affiliations. Are they very different?
b) Report the result of the test of the hypothesis that intended vote is independent of political party affiliation: proride the test statistic, the $P$-value, and interpret the result.
c) Construct the $2 \times 3$ contingency table relating expectation of voting to political party affiliation (the first and third variables listed). Report the conditional distnbutoons on expectation of voting for each of the poltical party affiliations, and interpret. Test the null hypothesis of independence, and interprel.
d) Supplement the analyses in (a)-(c) to investigate the association more fully. Interpret.

Saeeda Aman
Saeeda Aman
Numerade Educator
00:19

Problem 44

Give an example of a contingency table for which the chi-squared test of independence should not be used. bocause of
a) Sample size
b) Measurement scale
c) The rous of the table are dependeat samples

Sheryl Ezze
Sheryl Ezze
Numerade Educator
01:03

Problem 45

True or false? Interchanging two rows in a contingency table has no effect on the chlsquared statistic.

Willis James
Willis James
Numerade Educator
00:30

Problem 46

True or false? The null hypothesis for the test of indepondence between two categorical variables is $H_0: \chi^2=0$

Hossam Mohamed
Hossam Mohamed
Numerade Educator

Problem 47

a) When the sample size is very large, we have not necessarily established an imporant result when we show a statustically significant assocjation. Explain
b) The remark in Section 84 about small $P$-t alues not necessarily referring to an important eftect applies for any significance test. Explain why. illustratung with the observation in Example 67 that. when $n$ is large, a small $P$-value may occur in testing $H_0 . \mu=\mu_0$ when $\bar{Y}$ is not very different from $\mu_0$ in practical terms.

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01:15

Problem 48

True or false? If $\gamma=0$ for two varables, then the variables are statistically independent Explain.

Kaylee Mcclellan
Kaylee Mcclellan
Numerade Educator

Problem 49

True or false? Interchanging two rows in a conungency table has no eflect on gamma.

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02:24

Problem 50

The correct answer in Problem 8.45 imples that if the chi-squared statistic is used for a contingency table having ordered categones in both duections. then (select the corroct response(s)):
a) The statistic actually treats the variables as nominal.
b) Information about the ordering is ignored.
c) The test is usually not as powerful for detecting associarion as a lest statisuc based on numbers of concordant and discordant pairs
d) The statistic cannot differentiate befu een positive and negative associations

Sheryl Ezze
Sheryl Ezze
Numerade Educator
02:09

Problem 51

Show how to gei the chi-squared values for $d f=1$ in Table C from $z$-scores in the standard normal table (Table A). Illustrate for the chi-squared value that has $P$-value 01 .

Maxime Rossetti
Maxime Rossetti
Numerade Educator
00:59

Problem 52

Each subject in a sample of 100 men and 100 women is asked to indicate which of the following factors (one or more) are responsible for increases in crimc conmitted by leenagers. A - the increasing gap in income between the nch and poor, B - the increase in the perceutage of single-parent families, C - insulficient time that parents spend w ith their children, D - crmmual penaltics given by courts are too lenient, E - increasing problems with drugs in sociery: $F$ - increasing levels of violence shown on TV. To analyzc uhether responses differ by gender of respondent. u c cross-classify the responses by gender, as Ta ble 842 shows.
a) Is it valid to apply the chi-squared test of independence to these data? Explain.
b) Explain how this table actually pruvides information needed to cıoss-classily gender with each of six variables. Construct the contingency table relating gender to opinion about whether the increasing gap in income is responsible for increases in tecnage crime, and analyze.

Maxime Rossetti
Maxime Rossetti
Numerade Educator

Problem 53

Tahle 8.43 exhibits the maximum possiblc association betu een two binary variables for a sample of size $n$. Show that $\chi^2=n$ for this table and, hence, thal the maxamum value of $\chi^2$ for $2 \times 2$ tables is $n$.

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02:39

Problem 54

Refer to the previous exercise. The pli-squared measure of association for $2 \times 2$ contingency tables has sample value
$$
\hat{\phi}^2=\frac{x^2}{n}
$$
Explain why this measwe falls between 0 and 1 , with a population value of 0 corresponding to independence. (It is a special case, for $2 \times 2$ tables, of the Goodman and Kruskal tau micasure introduced in Problem 8.58 and of the $r^2$ measure introduced in the next chapter.)

Carolyn Behr-Jerome
Carolyn Behr-Jerome
Numerade Educator

Problem 55

For $2 \times 2$ tables. gamma simplifies to a measure first proposed about 1900 by the statistician G. Udny Yule, who ulso introduced the odds ratio. In that special case, gamma is called Yule's $\mathbf{Q}$.
a) Show that for a genenc table $u$ ith counts $(a, b)$ in row 1 and $(c . d)$ in row 2 , the number of concordant pairs equals $a d$, the number of discordant paurs equals $b c$, and $Q=(a d-b c) /(a d+b c)$.
b) Show that the absolute value of garmma equals I for any $2 \times 2$ table in which one of the cell frequencies is 0 .

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Problem 56

You have data on two ordinal variables, degree of political conservatism (low, niedium, high) and willingness to raise taxes for uelfare programs (low, medium, high). If you compute gamma for these daa, would you cxpect its value to be positive, or uould yon expect it to be negative? Explain.

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00:59

Problem 57

Construcl a $3 \times 3$ table for cach of the following conditions:
a) Gamma equals 1.
b) Gamma equals -1 .
c) Gamma equals 0 .

Raj Bala
Raj Bala
Numerade Educator

Problem 58

Goodman and K'ruskal's tani, al so called the concentration measure, is a PRE measurc of association for nominal vanables. We illusirate it for Table 8.35. Rule 1 for predicting artitude without using religious preference is based on the marginal distribution for that response variable. To illustrate. since the proportion $381 / 1166=.327$ of the sample is classilied Favor, we gucss a classification of Favor for that same proportion of the sample. We predict Oppose for the remaining 785 people. For rule 2. in predicting attirudes. we nou do it for the conditional distnbution of the response vanable within each level of religious affiliation. For example, we predict Favor lor a random sdimple of 103 of 290 Liberal Protestants and predict Oppose for the other 187 Liberal Protestants.
a) Show that the number of errors expected in predicting atutude using rule $\downarrow$ is $E_1=381(785 / 1166)+785(381 / 1166)=513.0$
b) Show that the expected number of errors in predictung attitude using rule 2 is $E_2=4904$
c) Show that tau, the proportional reduction in error, equals .044. (Properties of this measure includc (i) $0 \leq \tau \leq 1$, (ii) $\tau=0$ corresponds to independence, (iii) $\tau=1$ when for each category of the predictor variable, all the observations occur in one category of the response vanable.)

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04:03

Problem 59

Refer to the previous problem. A contingency table relating race (the rows) to opinion (the columns) has counts $(490,410)$ in row I and $(10,90)$ in row 2 . A contingency table relating gender (the rows) to opinion has counts ( 350.150 ) in row 1 and ( 150.350 ) in row 2.
a) Compute the difference of proportions in each table. Which table seems to have the stronger assuciation?
b) Repeat (a) using the odds ratio.
c) Now compare the tables using Goodman and Kruskal's tau. This reveals the dependence of this incasure on the margins it is easier to get a larger value for the second lable. for which row totals are similar for each gender, than for the first table, for which row Iolals are quite different for the races. One should use tau to comparc tables only if they have similar marginals.
d) For the first table. suppose the counts equal $(50,450)$ instead of $(10,90)$ in row 2 . Show that the conditional distnbutions do not change, the difference of proportions docs not change, the odds ratio does not change, but tau changes.

Sheryl Ezze
Sheryl Ezze
Numerade Educator

Problem 60

Another PRE measure of association for nomínal variables is called lambda. For this measure, rule 1 predicts for each observation the modal category for the marginal distribution of the response variable. For Table 835 , for example. rule 1 predicts Oppose always. Rule 2 predicts the modal category for the conditional distribution of the response variable, within cach level of the prcdictor variable For each row in Table 8.35, rule 2 predicts Oppose.
a) Show that $E_1=381, E_2=103+182+80+16=381$, and $\hat{\imath}=\left(E_1-E_2\right) / E_1=0$. b) Explain why lambda equals 0 when the modal categor; of the response variable is the same for each level of the explanator, variablc. (The main difference between lambda and tau is that lambda can equal 0 even when the variables are statustically dependent.)

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Problem 61

${ }^7$ In cross-classifications of ordinal vanables, not all pairs are concordant or discordant. Subjects in the same category of a vatiable arc tied on that variable. Tho observalions that fall in the same row are ted on the row variable, and two obser vations that fall in the same column are tied on the column tranable Let $T_2$ denote the number of pairs tied on $X$, and ler $T_1$ denote the number of pairs tied on $Y$. The sample value of Kendall's tan- $b$ is
$$
\hat{\tau}_b=\frac{C-D}{\sqrt{\left[n(n-1) / 2-T_\lambda\right]\left[n(n-1) / 2-T_r\right]}}
$$
Show that $\hat{\gamma}=\hat{t}_b$ when $\tau_x=T_Y=0$. (The common valuc is then known as Kendall's tau)

Rashmi Sinha
Rashmi Sinha
Numerade Educator