Questions and Answers of Categorical Data Analysis

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
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
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.
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
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
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
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
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
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
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 =
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
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
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
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
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
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
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
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
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
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
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
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
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
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,
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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.
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
8.28 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
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
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
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
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
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
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
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?
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
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,
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
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
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
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)
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.
9.17 What is wrong with this statement: “For a first-order Markov chain, Yt is independent of Yt−2”?
9.18 True, or false? With repeated measures data having multiple observations per subject, one can treat the observations as independent and still get valid estimates, but the standard errors based
10.1 Refer back to Table 8.10 from a recent General Social Survey that asked subjects whether they believe in heaven and whether they believe in hell.a. Fit model (10.3). If your software uses
10.2 You plan to apply the matched-pairs model (10.3) to a data set for which yi1 is whether the subject agrees that abortion should be legal if the woman cannot afford the child (1 = yes, 0 = no),
10.3 A dataset on pregnancy rates among girls in 13 north central Florida counties has information on the total in 2005 for each county i on Ti = number of births and yi = number of those for which
10.4 Table 10.9 shows the free-throw shooting, by game, of Shaq O’Neal of the Los Angeles Lakers during the 2000 NBA (basketball) playoffs. In game i, let yi = number made out of Ti attempts.a. Fit
10.5 For 10 coins, let πi denote the probability of a head for coin i. You flip each coin five times. The sample numbers of heads are {2, 4, 1, 3, 3, 5, 4, 2, 3, 1}.a. Report the sample proportion
10.6 For Table 7.3 from the survey of high school students, let yit = 1 when subject i used substance t (t = 1, cigarettes; t = 2, alcohol; t = 3, marijuana).Table 10.10 shows output for the
10.7 Refer to the previous exercise.a. Compare { ˆ βt } to the estimates for the marginal model in Problem 9.1.Why are they so different?b. How is the focus different for the model in the previous
10.8 For the student survey summarized by Table 7.13, (a) analyze using GLMMs,(b) compare results and interpretations to those with marginal models in Problem 9.2.
10.9 For the crossover study summarized by Table 9.9 (Problem 9.6), fit the model logit[P(Yi(k)t = 1)] = ui(k) + αk + βt where {ui(k)} are independent N(0, σ). Interpret { ˆ βt } and ˆσ .
10.10 For the previous exercise, compare estimates of βB − βA and βC − βA and their SE values to those using the corresponding marginal model of Problem 9.6.
10.11 Refer to Table 5.5 on admissions decisions for Florida graduate school applicants.For a subject in department i of gender g (1 = females, 0 = males), let yig = 1 denote being admitted.a. For
10.12 Consider Table 8.14 on premarital and extramarital sex. Table 10.11 shows the results of fitting a cumulative logit model with a random intercept.a. Interpret ˆ β.b. What does the relatively
10.13 Refer to the previous exercise. Analyze these data with a corresponding cumulative logit marginal model.a. Interpret ˆ β.b. Compare ˆ β to its value in the GLMM. Why are they so different?
10.14 Refer to Problem 9.11 for Table 7.25 on government spending.a. Analyze these data using a cumulative logit model with random effects.Interpret.b. Compare the results with those with a marginal
10.15 Refer to Table 4.16 and Problem 4.20, about an eight-center clinical trial comparing a drug with placebo for curing an infection. Model the data in a way that allows the odds ratio to vary by
10.16 See http://bmj.com/cgi/content/full/317/7153/235 for a meta analysis of studies about whether administering albumin to critically ill patients increases or decreases mortality. Analyze the data
10.17 Refer to the insomnia example in Section 10.3.1.a. From results in Table 10.7 for the GLMM, explain how to get the interpretation quoted in the text that “The response distributions are
10.18 Analyze Table 9.8 with age and maternal smoking as predictors using a (a)logistic-normal model, (b) marginal model, (c) transitional model. Summarize your analyses in a two-page report,
10.19 Refer to the toxicity study with data summarized in Table 10.12. Collapsing the ordinal response to binary in terms of whether with data summarized in the outcome is normal, consider logistic
10.20 Refer to the previous exercise. Analyze the data with a marginal model and with a GLMM, both of cumulative logit form, for the ordinal response.Summarize analyses in a two-page report.
10.21 Table 10.13 reports results of a study of fish hatching under three environments.Eggs from seven clutches were randomly assigned to three treatments, and the response was whether an egg hatched
10.22 Problem 3.15 analyzed data from a General Social Survey on responses of 1308 subjects to the question, “Within the past 12 months, how many people have you known personally that were victims
10.23 A crossover study compares two drugs on a binary response variable. The study classifies subjects by age as under 60 or over 60. In a GLMM, these two age groups have the same conditional effect
In the following examples, identify the response variable and the explanatory variables.a. Attitude toward gun control (favor, oppose), Gender (female, male), Mother’s education (high school,
Which scale of measurement is most appropriate for the following variables –nominal, or ordinal?a. Political party affiliation (Democrat, Republican, unaffiliated).b. Highest degree obtained (none,
Each of 100 multiple-choice questions on an exam has four possible answers but one correct response. For each question, a student randomly selects one response as the answer.a. Specify the
A coin is flipped twice. Let Y = number of heads obtained, when the probability of a head for a flip equals π.a. Assuming π = 0.50, specify the probabilities for the possible values for Y , and
Refer to the previous exercise. Suppose y = 0 in 2 flips. Find the ML estimate of π. Does this estimate seem “reasonable”? Why? [The Bayesian estimator is an alternative one that combines the
Genotypes AA, Aa, and aa occur with probabilities (π1, π2, π3). For n = 3 independent observations, the observed frequencies are (n1, n2, n3).a. Explain how you can determine n3 from knowing n1

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