Questions and Answers of Categorical Data Analysis

Table 8.8 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. Use a logit model to
The Human Suffering Index was created by the Population Crisis Committee (Washington, D.C.) to compare living conditions among countries. Table 8.11 shows results of this index in 1987 for countries
Table 8.12 refers to results of a clinical trial for the treatment of small-cell lung cancer. Patients were randomly assigned to two treatment groups. In the sequential therapy, the same combination
Table 8.13 refers to a study in which subjects were randomly assigned to a control group or a treatment group. Daily during the study, treatment subjects ate cereal containing psyllium. The purpose
Table 8.14 is an expanded version of the data set discussed in Section 6.4. The response categories are (1) not injured, (2) injured but not transported by emergency medical services, (3) injured and
8.6. Table 8.9 shows results of a survey of women, relating frequency of breast self-examination and age. Fit a proportional odds model. Analyze goodness of fit, and conduct descriptive and
8.7. For a sample of 40 subjects, Table 8.10 relates Y = mental impairment. (1 == None, 2 Mild, 3 Moderate, 4 Impaired) to socioeconomic status (SES 1, high; SES = 0, low) and a life events index,
Table 9.10 refers to a sample of juveniles convicted of a felony in Florida in 1987. Matched pairs were formed using criteria such as age and the number of prior offenses. For each pair, one subject
A crossover experiment with 100 subjects compares two drugs used to treat migraine headaches. The response scale is success (+) or failure (-). Half the study subjects, randomly selected, used drug A
Table 9.12, from the 1991 General Social Survey, reports subjects' religious affiliation in 1991 and at age 16, for categories (1) Protestant, (2) Catholic, (3) Jewish, (4) None or other.a. Test
Table 9.13, from the 1991 General Social Survey, reports respondents' current region of residence and region of residence at age 16. Analyze these data. Table 9.13 Residence in 1991. Residence at Age
Table 9.14 is from the 1989 General Social Survey. Subjects were asked their opinion on early teens (age 14-16) having sex relations and on a man and a woman having sex relations before marriage. The
Table 9.15 refers to a case-control study investigating a possible relationship between cataracts and the use of head coverings during the summer. Each casereporting to a clinic for care for a
Table 9.16 displays diagnoses of multiple sclerosis for two neurologists. The categories are (1) Certain multiple sclerosis, (2) Probable multiple sclerosis, (3) Possible multiple sclerosis, (4)
Table 9.17 refers to journal citations among four statistical theory and methods journals (Biometrika, Communications in Statistics, Journal of the American Statistical Association, Journal of the
Table 9.18 refers to tennis matches for several men players during 1989 and 1990.a. Fit the Bradley-Terry model. Analyze the lack of fit.b. Estimate the probability of Edberg beating Sampras. Compare
For the quasi-symmetry model, show thatThus, the main effect parameters determine departures from symmetry. Explain the connection with the Bradley-Terry model (9.6.1). log(ij/ji)=(x)-(x*x).
Explain THE PEARSON-YULE ASSOCIATION CONTROVERSY
Refer to the previous problem. Specify a logit model for the probability of rearrest, using court assignment as a predictor. Explain how to estimate and interpret the effect of court assignment.
When the Bradley-Terry model holds, explain why it is not possible that A could be preferred to B (i.e., IIAB > 1/2) and B could be preferred to C, yet C could be preferred to A.
Fit the Bradley-Terry model to Table 9.9.a. Estimate the probability of Graf beating Sanchez, and construct a 90% confidence interval for this probability. Interpret.b. By comparing the fit of this
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, Coke was preferred 29
Refer to Table 9.7. Based on the adjusted residuals, explain why the linear-by- linear association model (7.2.1) might fit these data well. Fit that model, and use the fit to describe the association
Refer to Table 9.5. 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
Refer to Table 9.13. Fit the independence model and the quasi-independence model. Describe lack of fit. What can you say about the numbers of people who moved from the Northeast to the South and from
Refer to Problem 8.18 with Table 8.13. For both the control and treatment groups, use methods of this chapter to compare the beginning and ending cholesterol levels. Compare the changes in
An alternative ordinal model for square tables, called the conditional symmetry model, has form log(ij/Mji) = , for all i < j. Show that symmetry is the special case = 0. Comparing the fit of this
Refer to all four items in Table 6.17.a. Generalize the symmetry model, so that the cell probability is identical for all cells with the same set of response outcomes; that is, hijk is identical for
Refer to Table 6.17. The two-way table relating health (as rows) and law enforcement (as columns) has cell counts, by row, (292, 117, 25/72,60,6/14, 12,9). Analyze these data.
Refer to Table 6.17. The two-way table relating responses for the environment (as rows) and cities (as columns) has cell counts, by row, (108, 179, 157/21, 55,49/5,6, 24).a. Fit the symmetry and
Refer to the previous problem. Using the ordinal quasi-symmetry model or a proportional odds model, estimate the marginal effect. Interpret.
Refer to Table 9.6. Using a generalized CMH procedure, test marginal homo- geneity.
Refer to Table 9.6. Fit the ordinal quasi-symmetry model, using u = 1 and u4 = 4 and picking u and u3 to represent sensible choices for distances be- tween categories that are unequally spaced.
Refer to Table 9.6. For the symmetry model, compute and interpret the adjusted residual for the pair of categories (1,4).
Refer to the previous problem. Using conditional ML, estimate the marginal parameters in logit model (9.2.2), and interpret using subject-specific odds ratios.
Refer to Table 6.3. Viewing the table as matched triplets, one can compare the proportion of "yes" responses among the three drugs.a. Construct the marginal distribution for each drug, and compute
The estimated variance for the conditional ML estimate = log(n12/n21) of in model (9.2.1) is (1/n12 + 1/n21). Show that a 95% confidence interval for the odds ratio exp() for Table 9.1 equals (1.34,
A case-control study has 8 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
Fitting a logistic regression model to difference scores, estimate in model (9.2.1) applied to Table 9.1. Interpret.
Refer to Table 9.1. Suppose sample proportions of approval of .59 and .55 were based on independent samples of size 1600 each. Construct a 95% confidence interval for the true difference of
Refer to Table 6.12, and treat the data as matched pairs on opinion, stratified by gender.a. For each gender, test equality of the true proportions supporting government action for the two items, and
Explain the following analogy: McNemar's test is to binary data as the paired difference / test is to normally distributed data.
Apply McNemar's test to Table 9.3. interpret
Suppose that a response scale has the categories (strongly agree, mildly agree, mildly disagree, strongly disagree, don't know). How might one model this re- sponse? (Hint: One approach uses two
Refer to Table 7.8. Treating opinion on birth control as the response variable, analyze these data.
Refer to the previous problem. Analyze the data using an alternative ordinal model. Interpret results.
Refer to Table 8.3. Treating belief in an afterlife as ordinal, fit a model using adjacent-category logits to these data. Test the goodness of fit, and interpret the estimated effects of gender and
Analyze Table 7.5 using continuation-ratio logits. Interpret.
Show that the homogeneous L XL model (7.3.1) with equally spaced scores for response Y corresponds to an adjacent categories logit model with a linear trend effect for X and a factor effect for Z.
Fit the adjacent-categories logit model (8.3.4) to Table 7.5, using scores {1,2,3,4} for income.a. Test goodness of fit, and interpret the estimated effects of income and gender.b. Perform a
Refer to Table 7.5. Treating job satisfaction as the response, analyze the data using a model with cumulative logits.a. Test the fit, and describe the effect of income.b. Conduct a likelihood-ratio
Refer to Problem 7.26 with Table 7.9.a. Analyze these data using a proportional odds model.b. Analyze the data using adjacent-category logits.c. Specify the loglinear model that corresponds to the
Show that the generalization of model (8.2.1) that has a separate slope Bj for each cumulative logit has a structural problem, cumulative probabilities possibly being misordered for some values of x.
8.5. Refer to Problem 2.21 with Table 2.15. Using modeling methods, describe and make inferences about the effect of mammography experience on women's attitudes.
Refer to Problem 6.7. Treating vote for President (P) as the response variable and B and D as explanatory variables, find a logit model that fits these data well. Interpret parameter estimates for
Refer to the example in Section 8.1.4.a. Using the fitted model, estimate the probability of response "no" for black females, (i) using parameter estimates, (ii) using fitted values.b. Fit the logit
Refer to the example in Section 8 .1 .2.a. Using the model fit, calculate an odds ratio that describes the estimated effect of length on primary food choice being either "invertebrate" or "other."b.
. 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,
. Silicon wafers for computer chips manufactured by a high-tech company have an average number of defects of 1.0 per wafer. If the number of defects has the Poisson distribution, find the probability
. 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. Specify the
. A balanced coin is flipped twice. Let Y = number of heads obtained.a. Specify the probabilities for the possible values for Y, and give the distri- bution's mean and variance.b. Calculate
. Refer to the previous problem.a. Calculate the binomial probabilities for N = 2 when the probability of a head for each flip equals (i) = .6, (ii) = .4.b. Suppose we observe Y = 1. Calculate and
. In his autobiography A Sort of Life, British author Graham Greene described a period of severe mental depression during which he played Russian Roulette. This "game" consists of putting a bullet in
. A sample of women suffering from dysmenorrhea have been taking an analgesic designed to diminish the effects. A new analgesic is claimed to provide greater relief. After trying the new analgesic,
. Refer to the previous problem. The researchers wanted a sufficiently large sample to be able to estimate the probability of preferring the new analgesic to within .08, with confidence .95. If the
. Newsweek magazine (March 27, 1989) reported results of a poll about religious beliefs, conducted by the Gallup Organization. Of 750 American adults, 24% believed in reincarnation. Treating this as
. A criminologist wants to estimate the proportion of U.S. citizens who live in a home in which firearms are available. The 1991 General Social Survey asked respondents, "Do you have in your home any
. If Y is a variate and c is a positive constant, then the standard deviation of the distribution of cY equals co (Y). Suppose Y is a binomial variate, and let p = Y/N. Show that (p)=(1-T)/N. Explain
. A variate has a Poisson distribution, with unknown parameter . The sole observation equals 0.a. Find and plot the likelihood function over the space of potential values for .b. What is the ML
. Using calculus, it is easier to derive the maximum of the log of the likelihood function, L = log 1, than the likelihood function / itself. Both functions have maximum at the same value, so it is
. Show that a value o for which the statistic z = (p- To)/(1-T)/N takes some fixed value zo is a solution to the equation (1 + 2/N) m +(-2p- /N)To+p = 0. Hence, using the formula x = (-b b2-4ac)/2a
A Swedish study considered the effect of low-dose aspirin on reducing the risk of stroke and heart attacks among people who have already suffered a stroke (Lancet 338: 1345-1349 (1991)). Of 1360
In the United States, the estimated annual probability that a woman over the age of 35 dies of lung cancer equals .001304 for current smokers and .000121 for nonsmokers (M. Pagano and K. Gauvreau,
The odds ratio between treatment (A, B) and response (death, survival) equals 2.0.a. Explain what is wrong with the interpretation, "The probability of death with treatment A is twice that with
An estimated odds ratio for adult females between the presence of squamous cell carcinoma (yes, no) and smoking behavior (smoker, non-smoker) equals 11.7 when the smoker category consists of subjects
Table 2.11 was taken from the 1991 General Social Survey.a. Identify each classification as a response or explanatory variable.b. Describe the association. Interpret the direction and strength of
A poll by Louis Harris and Associates of 1249 adult Americans in July 1994 indicated that 36% believe in ghosts and 37% believe in astrology. Can we compare the proportions using inferential methods
Table 2.12 is based on records of accidents in 1988 compiled by the Department of Highway Safety and Motor Vehicles in Florida. Compute and interpret the sample odds ratio, relative risk, and
In an article about crime in the United States, Newsweek magazine (Jan. 10, 1994) quoted FBI statistics stating that of all blacks slain in 1992, 94% were slain by blacks, and of all whites slain in
A 20-year study of British male physicians (R. Doll and R. Peto, British Med. J., 2: 1525-1536 (1976)) noted that the proportion who died from lung cancer was .00140 per year for cigarette smokers
Refer to Table 2.1.a. Construct a 90% confidence interval for the difference of proportions, and interpret.b. Construct a 90% confidence interval for the odds ratio, and interpret.c. Conduct a test
A large-sample confidence interval for the log of the relative risk isAntilogs of the endpoints yield an interval for the true relative risk. For Table 2.1, construct a 90% confidence interval. log
Refer to Table 2.3. Find the P-value for testing that the incidence of heart attacks is independent of aspirin intake, (a) using X2, (b) using G. Interpret results.
Refer to Table 2.4. Do these data provide evidence of an association between myocardial infarction and smoking? Use an inferential procedure, and interpret.
Table 2.13 was taken from the 1991 General Social Survey.a. Test the hypothesis of independence between party identification and race. Interpret.b. Use adjusted residuals to describe the evidence.c.
A recent article (D. J. Moritz and W. A. Satariano, J. Clin. Epidemiol., 46: 443-454 (1993)) investigated the relationship between stage of breast cancer at diagnosis (local or advanced) and a
Give examples of contingency tables for which a chi-squared test of inde- pendence using X or G should not be used, because of (a) sample size, (b) measurement scale.
Table 2.14 classifies a sample of psychiatric patients by their diagnosis and by whether their treatment prescribed drugs.a. Report the P-value for a test of independence, and interpret the result.b.
Refer to Table 7.5 (Chapter 7). Combine data for the two genders, yielding a single 4 X 4 table.a. Use X and G to test independence. Interpret.b. Partition G into three components for three 2 4
A study on educational aspirations of high school students (S. Crysdale, Int. J. Compar. Sociol., 16: 19-36 (1975)) measured aspirations using the scale (some high school, high school graduate, some
Table 2.15 refers to a study that assessed factors associated with women's attitudes toward mammography (Hosmer and Lemeshow, 1989, p. 220). The columns refer to their response to the question, "How
Refer to Table 8.12 (Chapter 8). Analyze these data using the methods of this chapter.
A study (B. Kristensen et al., J. Intern. Med., 232: 237-245 (1992)) con- sidered the effect of prednisolone on severe hypercalcaemia in women with metastatic breast cancer. Of 30 patients, 15 were
Refer to the previous problem. Compute the sample odds ratio. Using software, obtain a 95% "exact" confidence interval for the true odds ratio. Interpret, and note the effect of the zero cell count.
Table 2.16 contains results of a study comparing radiation therapy with surgery in treating cancer of the larynx. Use Fisher's exact test to test Ho: 0 = 1 against Ha: 0 > 1. Interpret results.
Refer to the previous problem.a. Obtain and interpret the one-sided mid-P value. Give advantages and dis- advantages of this type of P-value compared to the ordinary one.b. Obtain and interpret a
Suppose a researcher routinely conducts tests using a nominal probability of Type I error of .05, rejecting Ho if the P-value satisfies P < .05. Suppose an exact test using X2 has null distribution
Refer to Table 2.8.a. Construct the null distributions of the ordinary P-value and the mid P-value, for the one-sided alternative. Compute and compare their expected values.b. Repeat (a) for the test
Consider the 3 3 table having entries, by row, of (4,2,0/2,2,2/0,2,4).a. Using software, conduct an exact test of independence, using X. Interpret.b. Suppose the row and column classifications are

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