An Introduction To Categorical Data Analysis 2nd Edition Alan Agresti - Solutions

5.26 Table 5.15 describes results from a study in which subjects received a drug and the outcome measures whether the subject became incontinent (y = 1, yes;y = 0, no). The three explanatory variables are lower urinary tract variables that represent drug-induced physiological changes.a. Report the
5.27 About howlarge a sample is needed to test the hypothesis of equal probabilities so thatP(type II error)=0.05 whenπ1 = 0.40 andπ2 = 0.60, if the hypothesis is rejected when the P-value is less than 0.01?
5.28 We expect two proportions to be about 0.20 and 0.30, and we want an 80% chance of detecting a difference between them using a 90% confidence interval.a. Assuming equal sample sizes, how large should they be?b. Compare the results with the sample sizes required for (i) a 90% interval with power
5.29 The horseshoe crab x = width values in Table 3.2 have a mean of 26.3 and standard deviation of 2.1. If the true relationship were similar to the fitted equation reported in Section 4.1.3, namely, ˆπ = exp(−12.351 + 0.497x)/[1 +exp(−12.351 + 0.497x)], how large a sample yields P(type II
5.30 The following are true–false questions.a. A model for a binary response has a continuous predictor. If the model truly holds, the deviance statistic for the model has an asymptotic chisquared distribution as the sample size increases. It can be used to test model goodness of fit.b. For the
6.1 A model fit predicting preference for President (Democrat, Republican, Independent)using x = annual income (in $10,000 dollars) is log(ˆπD/ˆπI ) =3.3 − 0.2x and log(ˆπR/ˆπI ) = 1.0 + 0.3x.a. State the prediction equation for log(ˆπR/ˆπD). Interpret its slope.b. Find the range of x
6.2 Refer to the alligator food choice example in Section 6.1.2.a. Using the model fit, estimate an odds ratio that describes the effect of length on primary food choice being either “invertebrate” or “other.”b. Estimate the probability that food choice is invertebrate, for an alligator of
6.3 Table 6.14 displays primary food choice for a sample of alligators, classified by length (≤2.3 meters, >2.3 meters) and by the lake in Florida in which they were caught.a. Fit a model to describe effects of length and lake on primary food choice.Report the prediction equations.b. Using the
6.4 Refer to the belief in afterlife example in Section 6.1.4.a. Estimate the probability of response “yes” for black females.b. Describe the gender effect by reporting and interpreting the estimated conditional odds ratio for the (i) “undecided” and “no” pair of response
6.5 For a recent General Social Survey, a prediction equation relating Y = job satisfaction (four ordered categories; 1 = the least satisfied) to the subject’s report of x1 = earnings compared with others with similar positions (four ordered categories; 1 = much less, 4 = much more), x2 = freedom
6.6 Does marital happiness depend on family income? For the 2002 General Social Survey, counts in the happiness categories (not, pretty, very) were (6, 43, 75)for below average income, (6, 113, 178) for average income, and (6, 57, 117)for above average income. Table 6.15 shows output for a
6.7 Refer to the previous exercise. Table 6.16 shows output for a cumulative logit model with scores {1, 2, 3} for the income categories.a. Explain why the output reports two intercepts but one income effect.b. Interpret the income effect.c. Report a test statistic and P-value for testing that
6.8 Table 6.17 results from a clinical trial for the treatment of small-cell lung cancer.Patients were randomly assigned to two treatment groups. The sequential therapy administered the same combination of chemotherapeutic agents in each treatment cycle. The alternating therapy used three different
6.9 A cumulative logit model is fitted to data from the 2004 General Social Survey, with Y = political ideology (extremely liberal or liberal, slightly liberal, moderate, slightly conservative, extremely conservative or conservative)and predictor religious preference (Protestant, Catholic, Jewish,
6.10 Refer to the interpretations in Section 6.2.6 for the mental health data. Summarize the SES effect by finding P(Y ≤ 2) for high SES and for low SES, at the mean life events of 4.3.
6.11 Refer to Table 6.12. Treating job satisfaction as the response, analyze the data using a cumulative logit model.a. Describe the effect of income, using scores {3, 10, 20, 35}.b. Compare the estimated income effect to the estimate obtained after combining categories “Very dissatisfied” and
6.12 Table 6.18 shows results from the 2000 General Social Survey relating happiness and religious attendance (1 = at most several times a year, 2 = once a month to several times a year, 3 = every week to several times a week).a. Fit a multinomial model. Conduct descriptive and inferential analyses
6.13 Fit an adjacent-categories logit model with main effects to the job satisfaction data in Table 6.12, using scores {1, 2, 3, 4} for income.a. Use proportional odds structure. Interpret the estimated effect of income.b. Fit the model allowing different effects for each logit, which is equivalent
6.14 Consider Table 6.4 on belief in an afterlife. Fit a model using (a) adjacentcategories logits, (b) alternative ordinal logits. In each case, prepare a one-page report, summarizing your analyses and interpreting results
6.15 Analyze the job satisfaction data of Table 6.12 using continuation-ratio logits.Prepare a one-page summary.
6.16 Table 6.19 refers to a study that randomly assigned subjects to a control group or a treatment group. Daily during the study, treatment subjects ate cereal containing psyllium. The purpose of the study was to analyze whether this resulted in lowering LDL cholesterol.a. Model the ending
6.17 Table 6.20 is an expanded version of a data set Section 7.2.6 presents about a sample of auto accidents, with predictors gender, location of accident, and whether the subject used a seat belt. The response categories are (1) not injured,(2) injured but not transported by emergency medical
6.18 A response scale has the categories (strongly agree, mildly agree, mildly disagree, strongly disagree, do not know). How might you model this response?(Hint: One approach handles the ordered categories in one model and combines them and models the “do not know” response in another model.)
6.19 The sample in Table 6.12 consists of 104 black Americans. A similar table relating income and job satisfaction for white subjects in the same General Social Survey had counts (by row) of (3, 10, 30, 27/7, 8, 45, 39/8, 7, 46, 51/4, 2, 28, 47) for females and (1, 4, 9, 9/1, 2, 37, 29/0, 10, 35,
6.20 For K = 1, the generalized CMH correlation statistic equals formula (2.10).When there truly is a trend, Section 2.5.3 noted that this test is more powerful than the X2 and G2 tests of Section 2.4.3. To illustrate, for Table 6.12 on job satisfaction and income, construct the marginal 4 × 4
6.21 For the 2000 GSS, counts in the happiness categories (not too, pretty, very) were (67, 650, 555) for those who were married and (65, 276, 93) for those who were divorced. Analyze these data, preparing a one-page report summarizing your descriptive and inferential analyses.
6.22 True, or false?a. One reason it is usually wise to treat an ordinal variable with methods that use the ordering is that in tests about effects, chi-squared statistics have smaller df values, so it is easier for them to be farther out in the tail and give small P-values; that is, the ordinal
7.1 For Table 2.1 on X = gender and Y = belief in an afterlife, Table 7.16 shows the results of fitting the independence loglinear model.a. Report and interpret results of a goodness-of-fit test.b. Report {ˆλY j}. Interpret ˆλY1− ˆλY2.
7.2 For the saturated model with Table 2.1, software reports the {ˆλXY ij} estimates:Show how to use these to estimate the odds ratio. Parameter gender*belief females yes gender*belief females no gender*belief males DF Estimate Std Error 1 0.1368 0.1507 0 0.0000 0.0000 yes 0 0.0000 0.0000
7.3 Table 7.17 is from a General Social Survey. White subjects in the sample were asked: (B) Do you favor busing (Negro/Black) and white school children from one school district to another?, (P) If your party nominated a (Negro/Black)for President, would you vote for him if he were qualified for
7.4 In a General Social Survey respondents were asked “Do you support or oppose the following measures to deal with AIDS? (1) Have the government pay all of the health care costs of AIDS patients; (2) develop a government information program to promote safe sex practices, such as the use of
7.5 Refer to Table 2.10 on death penalty verdicts. Let D = defendant’s race, V =victim’s race, and P = death penalty verdict. Table 7.20 shows output for fitting model (DV,DP,PV ). Estimates equal 0 at the second category for any variable.a. Report the estimated conditional odds ratio between D
7.6 Table 7.21 shows the result of cross classifying a sample of people from the MBTIStep II National Sample, collected and compiled byCPPInc., on the four scales of the Myers–Briggs personality test: Extroversion/Introversion (E/I), Sensing/iNtuitive (S/N), Thinking/Feeling (T/F) and
7.7 Refer to the previous exercise. Table 7.22 shows the fit of the model that assumes conditional independence between E/I and T/F and between E/I and J/P but has the other pairwise associations.a. Compare this to the fit of the model containing all the pairwise associations, which has deviance
7.8 Refer to the previous two exercises. PROC GENMOD in SAS reports the maximized log likelihood as 3475.19 for the model of mutual independence(df = 11), 3538.05 for the model of homogeneous association (df = 5), and 3539.58 for the model containing all the three-factor interaction terms.a. Write
7.9 Table 7.23 refers to applicants to graduate school at the University of California, Berkeley for the fall 1973 session. Admissions decisions are presented by gender of applicant, for the six largest graduate departments. Denote the three variables by A = whether admitted, G = gender, and D =
7.10 Table 7.24 is based on automobile accident records in 1988, supplied by the state of Florida Department of Highway Safety and Motor Vehicles. Subjects were classified by whether they were wearing a seat belt, whether ejected, and whether killed.a. Find a loglinear model that describes the data
7.11 Refer to the loglinear models in Section 7.2.6 for the auto accident injury data shown in Table 7.9. Explain why the fitted odds ratios in Table 7.11 for model(GI, GL, GS, IL, IS, LS) suggest that the most likely case for injury is accidents for females not wearing seat belts in rural
7.12 Consider the following two-stage model for Table 7.9. The first stage is a logistic model with S as the response, for the three-way G × L × S table. The second stage is a logistic model with these three variables as predictors for I in the four-way table. Explain why this composite model is
7.13 Table 7.25 is from a General Social Survey. Subjects were asked about government spending on the environment (E), health (H), assistance to big cities(C), and law enforcement (L). The common response scale was (1 = too little, 2 = about right, 3 = too much).a. Table 7.26 shows some results,
7.14 Table 7.27, from a General Social Survey, relates responses on R = religious service attendance (1 = at most a few times a year, 2 = at least several times a year), P = political views (1 = Liberal, 2 = Moderate, 3 = Conservative), B = birth control availability to teenagers between ages of 14
7.15 Refer to Table 7.13 in Section 7.4.5 on the substance use survey, which also classified students by gender (G) and race (R).a. Analyze these data using logistic models, treating marijuana use as the response variable. Select a model.b. Which loglinear model is equivalent to your choice of
7.16 For the Maine accident data modeled in Section 7.3.2:a. Verify that logistic model (7.9) follows from loglinear model (GLS, GI, LI, IS).b. Show that the conditional log odds ratio for the effect of S on I equalsβS 1− βS 2 in the logistic model and λIS 11+ λIS 22− λIS 12− λIS 21 in
7.17 For a multiway contingency table, when is a logistic model more appropriate than a loglinear model, and when is a loglinear model more appropriate?
7.18 For a three-way table, consider the independence graph,X——–Z Ya. Write the corresponding loglinear model.b. Which, if any, pairs of variables are conditionally independent?c. If Y is a binary response, what is the corresponding logistic model?d. Which pairs of variables have the same
7.19 Consider loglinear model (WXZ, WYZ).a. Draw its independence graph, and identify variables that are conditionally independent.b. Explain why this is the most general loglinear model for a four-way table for which X and Y are conditionally independent.
7.20 For a four-way table, are X and Y independent, given Z alone, for model(a) (WX, XZ, YZ,WZ), (b) (WX, XZ, YZ,WY)?
7.21 Refer to Problem 7.13 with Table 7.25.a. Showthat model (CE,CH,CL,EH,EL,HL) fits well. Showthat model(CEH,CEL,CHL,EHL) also fits well but does not provide a significant improvement. Beginning with (CE,CH,CL,EH,EL,HL), show that backward elimination yields (CE,CL,EH,HL). Interpret its fit.b.
7.22 Consider model (AC,AM,CM,AG,AR,GM,GR) for the drug use data in Section 7.4.5.a. Explain why the AM conditional odds ratio is unchanged by collapsing over race, but it is not unchanged by collapsing over gender.b. Suppose we remove the GM term. Construct the independence graph, and show that
7.23 Consider logit models for a four-way table in whichX1,X2, andX3 are predictors of Y . When the table is collapsed overX3, indicate whether the association between X1 and Y remains unchanged, for the model (a) that has main effects of all predictors, (b) that has main effects of X1 and X2 but
7.24 Table 7.28 is from a General Social Survey. Subjects were asked whether methods of birth control should be available to teenagers between the ages of 14 and 16, and how often they attend religious services.a. Fit the independence model, and use residuals to describe lack of fit.b. Using
7.25 Generalizations of the linear-by-linear model (7.11) analyze association between ordinal variables X and Y while controlling for a categorical variable that may be nominal or ordinal. The modelwith ordered scores {ui } and {vj } is a special case of model (XY, XZ, YZ) that replaces λXY ij by
7.26 For the linear-by-linear association model applied with column scores{vj = j }, show that the adjacent-category logits within row i have form (6.6), identifying αj with (λY j+1− λY j ) and the row scores {ui } with the levels of x.In fact, the two models are equivalent. The logit
7.27 True, or false?a. When there is a single categorical response variable, logistic models are more appropriate than loglinear models.b. When you want to model the association and interaction structure among several categorical response variables, logistic models are more appropriate than
8.1 Apply the McNemar test to Table 8.3. Interpret.
8.2 Arecent General Social Survey asked subjects whether they believed in heaven and whether they believed in hell. Table 8.10 shows the results.a. Test the hypothesis that the population proportions answering “yes” were identical for heaven and hell. Use a two-sided alternative.b. Find a 90%
8.3 Refer to the previous exercise. Estimate and interpret the odds ratio for a logistic model for the probability of a “yes” response as a function of the item(heaven or hell), using (a) the marginal model (8.3) and (b) the conditional model (8.4).
8.4 Explain the following analogy: The McNemar test is to binary data as the paired difference t test is to normally distributed data.
8.5 Section 8.1.1 gave the large-sample z or chi-squared McNemar test for comparing dependent proportions. The exact P-value, needed for small samples, uses the binomial distribution. For Table 8.1, consider Ha: π1+ > π+1, or equivalently, Ha: π12 > π21.a. The exact P-value is the binomial
8.6 For Table 7.19 on opinions about measures to deal with AIDS, treat the data as matched pairs on opinion, stratified by gender.a. For females, test the equality of the true proportions supporting government action for the two items.b. Refer to (a). Construct a 90% confidence interval for the
8.7 Refer to Table 8.1 on ways to help the environment. Suppose sample proportions of approval of 0.314 and 0.292 were based on independent samples of size 1144 each. Construct a 95% confidence interval for the true difference of proportions. Compare with the result in Section 8.1.2, and comment on
8.8 A crossover experiment with 100 subjects compares two treatments for migraine headaches. The response scale is success (+) or failure (−). Half the study subjects, randomly selected, used drug A the first time they got a migraine headache and drug B the next time. For them, six had
8.9 Estimate β in model (8.4) applied to Table 8.1 on helping the environment.Interpret.
8.10 A case–control study has eight pairs of subjects. The cases have colon cancer, and the controls are matched with the cases on gender and age. A possible explanatory variable is the extent of red meat in a subject’s diet, measured as“low” or “high.” For three pairs, both the case
8.11 For the subject-specific model (8.4) for matched pairs, logit[P(Yi1 = 1)] = αi + β, logit[P(Yi2 = 1)] = αi the estimated variance for the conditional ML estimate ˆ β = log(n12/n21) of βis (1/n12 + 1/n21). Find a 95% confidence interval for the odds ratio exp(β)for Table 8.1 on helping
8.12 For Table 7.3 on the student survey, viewing the table as matched triplets, you can compare the proportion of “yes” responses among alcohol, cigarettes, and marijuana.a. Construct the marginal distribution for each substance, and find the three sample proportions of “yes” responses.b.
8.13 Table 8.12, from the 2004 General Social Survey, reports subjects’ religious affiliation in 2004 and at age 16, for categories (1) Protestant, (2) Catholic,(3) Jewish, (4) None or Other.
a. The symmetry model has devianceG2 = 150.6 with df = 6. Use residuals for the model [see equation (8.9)] to analyze transition patterns between pairs of religions.b. The quasi-symmetry model has deviance G2 = 2.3 with df = 3. Interpret.c. Test marginal homogeneity by comparing fits in (a) and
8.14 Table 8.13, from the 2004 General Social Survey, reports respondents’ region of residence in 2004 and at age 16.a. Fit the symmetry and quasi-symmetry models. Interpret results.b. Test marginal homogeneity by comparing the fits of these models.
8.15 Table 8.14 is from a General Social Survey. Subjects were asked their opinion about amanand awoman having sexual relations before marriage and a married person having sexual relations with someone other than the marriage partner.The response categories are 1 = always wrong, 2 = almost always
8.16 Table 8.15 is from a General Social Survey. Subjects were asked “How often do you make a special effort to buy fruits and vegetables grown without pesticides or chemicals?” and “How often do you make a special effort to sort glass or cans or plastic or papers and so on for recycling?”
8.17 Table 8.16 is from the 2000 General Social Survey. Subjects were asked whether danger to the environmentwas caused by car pollution and/or by a rise in the world’s temperature caused by the “greenhouse effect.” The response categories are 1 = extremely dangerous, 2 = very dangerous, 3 =
8.18 Refer to Problem 6.16 with Table 6.19 on a study about whether cereal containing psyllium had a desirable effect in lowering LDL cholesterol. For both the control and treatment groups, use methods of this chapter to compare the beginning and ending cholesterol levels. Compare the changes in
8.19 Refer to Table 8.13 on regional mobility. Fit the independence model and the quasi-independence (QI) model. Explain why there is a dramatic improvement in fit with the QI model. (Hint: For the independence model, the standardized residuals are about 40 for the cells on the main diagonal; what
8.20 Table 8.17 displays diagnoses of multiple sclerosis for two neurologists. The categories are (1) Certain multiple sclerosis, (2) Probable multiple sclerosis,(3) Possible multiple sclerosis, and (4) Doubtful, unlikely, or definitely not multiple sclerosis.a. Use the independence model and
8.21 Refer to Table 8.5. Fit the quasi-independence model. Calculate the fitted odds ratio for the four cells in the first two rows and the last two columns. Interpret.Analyze the data from the perspective of describing agreement between choice of coffee at the two times.
8.22 In 1990, a sample of psychology graduate students at the University of Florida made blind, pairwise preference tests of three cola drinks. For 49 comparisons of Coke and Pepsi, Cokewas preferred 29 times. For 47 comparisons of Classic Coke and Pepsi, Classic Coke was preferred 19 times. For 50
8.23 Table 8.18 refers to journal citations among four statistical theory and methods journals (Biometrika, Communications in Statistics, Journal of the American Statistical Association, Journal of the Royal Statistical Society Series B) during 1987–1989. The more often that articles in a
8.24 Table 8.19 summarizes results of tennis matches for several women professional players between 2003 and 2005.a. Fit the Bradley–Terry model. Report the parameter estimates, and rank the players.b. Estimate the probability that Serena Williams beats Venus Williams.Compare the model estimate
8.25 Refer to the fit of the Bradley–Terry model to Table 8.9.a. Agassi did not play Henman in 2004–2005, but if they did play, show that the estimated probability of a Agassi victory is 0.78.b. The likelihood-ratio statistic for testingH0: β1 = · · · = β5 equals 26.3 with df = 4.
8.26 When the Bradley–Terry model holds, explain why it is not possible that A could be preferred to B (i.e., AB > 1 2 ) and B could be preferred to C, yet C could be preferred to A.
8.27 In loglinear model form, the quasi-symmetry (QS) model iswhere λij = λji for all i and j .a. For this model, by finding log(μij/μji ) show that the model implies a logit model of form (8.10), which isb. Show that the special case of QS with λX i = λY i for all i is the symmetry model in
8.28 For matched pairs, to obtain conditional ML { ˆ βj } for model (8.5) using software for ordinary logistic regression, letLet x ∗1i = x ∗1i1 − x ∗1i2, . . . , x ∗ki = x ∗ki1 − x ∗ki2. Fit the ordinary logistic model to y ∗ with predictors {x ∗1 , . . . , x ∗k }, forcing
9.1 Refer to Table 7.3 on high school students’ use of alcohol, cigarettes, and marijuana. View the table as matched triplets.a. Construct the marginal distribution for each substance. Find the sample proportions of students who used (i) alcohol, (ii) cigarettes, (iii) marijuana.b. Specify a
9.2 Refer to Table 7.13. Fit a marginal model to describe main effects of race, gender, and substance type (alcohol, cigarettes, marijuana) on whether a subject had used that substance. Summarize effects.
9.3 Refer to the previous exercise. Further study shows evidence of an interaction between gender and substance type. Using GEE with exchangeable working correlation, the estimated probability ˆπ of using a particular substance satisfies logit(πˆ ) = −0.57 + 1.93S1 + 0.86S2 + 0.38R − 0.20G
9.4 Refer to Table 9.1. Analyze the depression data (available at the text web site) using GEE assuming exchangeable correlation and with the time scores(1, 2, 4). Interpret model parameter estimates and compare substantive results to those in the text with scores (0, 1, 2).
9.5 Analyze Table 9.8 using a marginal logit model with age and maternal smoking as predictors. Report the prediction equation, and compare interpretations to the regressive logistic Markov model of Section 9.4.2.
9.6 Table 9.9 refers to a three-period crossover trial to compare placebo (treatment A) with a low-dose analgesic (treatment B) and high-dose analgesic(treatment C) for relief of primary dysmenorrhea. Subjects in the study were divided randomly into six groups, the possible sequences for
a. Assuming common treatment effects for each sequence and setting βA = 0, use GEE to obtain and interpret { ˆ βt } for the modelb. How would you order the drugs, taking significance into account? logit[P(Yi(k) = 1)] = k + t
9.7 Table 9.10 is from a Kansas State University survey of 262 pig farmers. For the question “What are your primary sources of veterinary information”?, the categories were (A) Professional Consultant, (B) Veterinarian, (C) State or Local Extension Service, (D) Magazines, and (E) Feed Companies
9.8 Table 10.4 in Chapter 10 shows General Social Survey responses on attitudes toward legalized abortion. For the response Yt about legalization (1 = support, 0 = oppose) for question t (t = 1, 2, 3) and for gender g (1 = female, 0 =male), consider the model logit[P(Yt = 1)] = α + γg + βt with
9.9 Refer to the clinical trials data inTable 10.8 available at the text web site, which are analyzed with random effects models in Section 10.3.2. Use GEE methods to analyze the data from the 41 centers, treating each center as a cluster.a. Specify a working correlation and fit a model.b. Explain
9.10 Refer to theGSSdata on sex inTable 8.14 in Exercise 8.15. UsingGEEmethods with cumulative logits, compare the two marginal distributions. Compare the results with those in Problem 8.15.
9.11 Analyze the data in the 3 × 3 × 3 × 3 table on government spending in Table 7.25 with a marginal cumulative logit model. Interpret the effects.
9.12 For the insomnia study summarized in Table 9.6, model (9.2) compared treatments while controlling for initial response of time to fall asleep.a. Add an interaction term to model (9.2). Summarize how the estimated treatment effect varies according to the initial responses by showing that the
9.13 Analyze Table 9.8 from Section 9.4.2 using a transitional model with two previous responses.a. Given that yt−1 is in the model, does yt−2 provide additional predictive power?b. How does the maternal smoking effect compare with the model using only yt−1 of the past responses?
9.14 Analyze the depression data in Table 9.1 using a Markov transitional model.Compare results and interpretations to those in this chapter using marginal models.
9.15 Table 9.13 is from a longitudinal study of coronary risk factors in school children. A sample of children aged 10–13 in 1977 were classified by gender and by relative weight (obese, not obese) in 1977, 1979, and 1981. Analyze these data, summarizing results in a one-page report.
9.16 Refer to the cereal diet and cholesterol study of Problem 6.16 (Table 6.19).Analyze these data with marginal models, summarizing results in a one-page report.

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An Introduction To Categorical Data Analysis 2nd Edition Alan Agresti - Solutions

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