Questions and Answers of Categorical Data Analysis

Refer to Table 6.3. The survey also classified respondents by gender (G) and race (R). Table 7.1 shows the full data.a. Analyze these data using loglinear models, distinguishing between response and
Consider the following two-stage model for Table 6.8. The first stage is a logit model with S as the response, for the three-way G-L-S table. The second stage is a logit model with these three
Refer to the loglinear models for Table 6.8.a. Explain why the fitted odds ratios in Table 6.10 for model (GI, GL, GS, IL, IS, LS) suggest that the most likely case for injury is accidents for
Refer to Problem 3.1.a. Analyze these data using loglinear models. For the model you choose to describe the data, interpret parameter estimates.b. For the model in (a), fit the corresponding logit
Refer to Table 3.3.a. Fit the loglinear model of homogeneous association. Report the estimated conditional odds ratio between smoking and lung cancer. Obtain a 99% confidence interval for the true
Refer to Table 3.1. Let D= defendant's race, V = victims' race, and P = death penalty verdict. Fit model (DV, DP, PV).a. Using the fitted values, calculate the odds ratio between D and P at each
For the saturated model with Table 2.2, report {^} (a) using "sum to zero" constraints, (b) using a form of constraints for which only XY is nonzero. Show how to interpret these estimates using an
Fit the independence model to Table 2.2, and check the goodness of fit. Report {} (a) using a form of constraints for which a baseline estimate equals 0, (b) using "sum to zero" constraints.
5.6. Refer to Table 2.5. To investigate the association, one can use a logistic regres- sion model, but interchange roles of response and explanatory variables.a. Fit the model, treating party
Refer to the models discussed in Section 5.3.2 for the crab data. Show that the value of Ls, and hence G(Mo) and G(M1), depends on whether the data are entered as individual binary observations or
The logistic regression curve (5.1.1) has the shape of the cdf of a logistic distribution with mean -a/ and standard deviation = 1.814/(IBI). Show that for the ungrouped horseshoe crab data with
The slope of the line drawn tangent to the probit regression curve at a particu- lar x value equals (40) exp[-(a + Bx)2/2]. Show this is highest when x = -a/, where it equals 0.40. At this point,
Refer to the logistic regression model (5.1.1). Show that e" equals the odds of success when X = 0. Construct the odds of success when X = 1, X = 2, and X = 3. Use this to provide an interpretation
Refer to Table 7.10 in Chapter 7. Applying software for exact logistic re- gression, obtain exact P-values for testing the effects of each predictor in the multiple logistic regression model.
Refer to Table 5.9. Apply exact logistic regression to the model discussed in Section 5.7.3 for Cephalexin Use (C), Age (A), Length of Stay (L), and Diarrhea Outcome (D).a. Obtain an exact P-value
Refer to Table 2.7.a. Fit the logistic regression model using scores {0, .5, 1.5, 4, 7} for alcohol consumption. Check goodness of fit.b. Test independence, using the likelihood-ratio test and the
The horseshoe crab width values in Table 4.2 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 5.1.2, how large a
We expect two proportions to be about .20 and .30, and we want an 80% chance of detecting a difference between them using a 90% confidence interval. Assuming equal sample sizes, how large should they
About how large a sample is needed to test the hypothesis of equal proportions so that P(Type II error) = .05 when = .40 and 2 = .60, if the hypothesis is rejected when the P-value is less than .01?
Refer to Table 7.1, treating marijuana use as the response variable. Analyze these data using logit models. Interpret estimated effects.
Refer to Problem 6.12 and Table 6.16 in Chapter 6. Treating opinion about premarital sex as the response variable, use backward elimination to select a model. Interpret that model.
An alternative goodness-of-fit test when at least one predictor is continuous partitions values for the explanatory variables into a set of regions and adds a dummy variable to the model for each
For the horseshoe crab data, use backward elimination to select a good model, when weight, spine condition, and color are the predictors for the probability of a satellite.
Refer to the prediction equation logit() = -10.071-0.509c+0.458x, using width and quantitative color for the horseshoe crab data. Color has a mean of 2.44 and standard deviation of 0.80, and width
For the horseshoe crab data, fit a model using weight and width as predictors.a. Report the prediction equation. Interpret effects.b. Conduct a likelihood-ratio test of the hypothesis Ho B = B = 0
Refer to model (5.5.2). At width = 20 cm, find the predicted probabilities of a satellite for the colors medium dark and dark. Find the odds for each, and show that the odds ratio equals exp(1.106) =
Using models that treat color in a quantitative manner, repeat the analyses in (i) Problem 5.25, (ii) Problem 5.26c.
For the horseshoe crab data with color and width as predictors (Section 5.5.1), fit the logistic regression model permitting interaction.a. Report the prediction equations relating width to the
Refer to model (5.5.2). Explain how this model is analogous to the analysis of covariance model with no interaction for a continuous response, whereby the slope effect of a quantitative predictor is
A sample of subjects were asked their opinion about current laws legaliz- ing abortion (support, oppose). For the explanatory variables gender (female, male), religious affiliation (Protestant,
Refer to Problem 3.1. Analyze these data using logistic regression, treating death penalty as the response. Interpret results.
Refer to Table 3.6. Using logistic regression, conduct analogous analyses to those in Problem 3.10. Compare results.
Refer to Table 3.5. Analyze these data using logistic regression, with lung cancer as the response. Can one fit a linear probability model or a probit model to these data? Explain.
Refer to Table 3.3.a. Fit model (5.4.3). Conduct Pearson and likelihood-ratio tests of goodness of fit, and interpret. Since the tests provide tests of equality of odds ratios, compare results to
Refer to Table 3.1.a. Fit model (5.4.1). Interpret the parameter estimates using conditional odds ratios.b. Test goodness of fit. Interpret.c. Test the effect of defendant's race on the death
The U. S. National Collegiate Athletic Association (NCAA) conducted a study of graduation rates for student athletes who were freshmen during the 1984- 1985 academic year. Table 5.13 shows the data.
Fit model (5.4.1) to Table 5.5.a. For black veterans without immediate AZT use, obtain the predicted prob- abilities and fitted values for (yes, no) AIDS symptoms.b. Construct a 95% confidence
A recent article (D. J. Moritz and W. A. Satariano, J. Clin. Epidemiol., 46: 443-454 (1993)) used multiple logistic regression to predict whether the stage of breast cancer at diagnosis was advanced
For Table 4.2, fit a logistic regression model for the probability of a satellite, using color alone as the predictor.a. Treat color as qualitative (nominal). Conduct a likelihood-ratio test of the
Refer to the previous problem. By categorizing weight, test whether the model is adequate for predicting the presence of satellites. Obtain Pearson or adjusted residuals, and interpret. Obtain
Refer to Table 4.2. Using weight as the predictor, fit the logistic regression model for the probability of a satellite.a. Report and plot the ML fit, and find predicted probabilities at the values
5.5. Hastie and Tibshirani (1990, p. 282) described a study to determine risk factors for kyphosis, severe forward flexion of the spine following corrective spinal surgery. The age in months at the
Refer to Table 4.6. Use logistic regression to analyze these data.
Refer to Table 4.1. Using scores (0,2,4, 5} for levels of snoring, fit the logistic regression model. Interpret using fitted probabilities, linear approximations, and effects on the odds. Analyze
In the first nine decades of the twentieth century in baseball's National League, the percentage of times the starting pitcher pitched a complete game were: 72.7 (1900-1909), 63.4, 50.0, 44.3, 41.6,
5.1 For the horseshoe crab data (available at www.stat.ufl.edu/∼aa/intro-cda/appendix.html), fit a model using weight and width as predictors.a. Report the prediction equation.b. Conduct a
5.2 For the horseshoe crab data, use a stepwise procedure to select a model for the probability of a satellite when weight, spine condition, and color (nominal scale) are the predictors. Explain each
5.3 For the horseshoe crab data with width, color, and spine as predictors, suppose you start a backward elimination process with the most complex model possible.Denoted by C ∗ S ∗ W, it uses
5.4 Refer to Problem 4.16 on the four scales of the Myers–Briggs (MBTI) personality test. Table 5.10 shows the result of fitting a model using the four scales as predictors of whether a subject
5.5 Refer to the previous exercise. PROC LOGISTIC in SAS reports AIC values of 642.1 for the model with the four main effects and the six interaction terms, 637.5 for the model with only the four
5.6 Refer to the previous two exercises about MBTI and drinking.a. The sample proportion who reported drinking alcohol frequentlywas 0.092.When this is the cutpoint for forming a classification
5.7 From the same survey referred to in Problem 4.16, Table 5.11 cross-classifies whether a person smokes frequently with the four scales of the MBTI personality test. SAS reports model −2 log
5.8 Refer to the classification table in Table 5.3 with π0 = 0.50.a. Explain how this table was constructed.b. Estimate the sensitivity and specificity, and interpret.
5.9 Problem 4.1 with Table 4.8 used a labeling index (LI) to predict π = the probability of remission in cancer patients.a. When the data for the 27 subjects are 14 binomial observations (for the 14
5.10 For the horseshoe crab data, fit the logistic regression model with x = weight as the sole predictor of the presence of satellites.a. For a classification table using the sample proportion of
5.11 Here is an alternative to the Hosmer–Lemeshow goodness-of-fit test when at least one predictor is continuous: Partition values for the explanatory variables into a set of regions. Add these
5.13 Logistic regression is often applied to large financial databases. For example, credit scoring is a method of modeling the influence of predictors on the probability that a consumer is credit
5.14 Refer to the following artificial data:Denote by M0 the logistic model with only an intercept term and by M1 the model that also has x as a linear predictor. Denote the maximized log likelihood
5.15 According to the Independent newspaper (London, March 8, 1994), the Metropolitan Police in London reported 30,475 people as missing in the year ending March 1993. For those of age 13 or less, 33
5.16 In Chapter 4, exercises 4.29, 4.30, 4.31, and 4.32 asked for a data analysis and report. Select one of those analyses, and conduct a goodness-of-fit test for the model you used. Interpret.
5.17 Refer to Table 2.10 on death penalty decisions. Fit a logistic model with the two race predictors.a. Test the model goodness of fit. Interpret.b. Report the standardized residuals. Interpret.c.
5.18 Table 5.12 summarizes eight studies in China about smoking and lung cancer.a. Fit a logistic model with smoking and study as predictors. Interpret the smoking effect.b. Conduct a Pearson test of
5.19 Problem 7.9 shows a 2 × 2 × 6 table for Y = whether admitted to graduate school at the University of California, Berkeley.a. Set up indicator variables and specify the logit model that has
5.20 Refer to Table 2.7 on mother’s drinking and infant malformations.a. Fit the logistic regression model using scores {0, 0.5, 1.5, 4, 7} for alcohol consumption. Check goodness of fit.b. Test
5.21 In the previous exercise, the table has some small counts, and exact methods have greater validity than large-sample ones. Conduct an exact test of independence using the scores in (a).
5.22 For the example in Section 5.3.1, y = 0 at x = 10, 20, 30, 40, and y = 1 at x = 60, 70, 80, 90.a. Explain intuitively why ˆ β =∞for the model, logit(π) = α + βx.b. Report ˆ β and its SE
5.23 Table 5.13 refers to the effectiveness of immediately injected or 11 2 -hourdelayed penicillin in protecting rabbits against lethal injection withβ-hemolytic streptococci.a. LetX = delay, Y =
5.24 In the previous exercise, the small cell counts make large-sample analyses questionnable. Conduct small-sample inference, and interpret.
5.25 Table 5.14 is from a study of nonmetastatic osteosarcoma described in the LogXact 7 manual (Cytel Software, 2005, p. 171). The response is whether the subject achieved a three-year disease-free
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
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
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
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,
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
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 +
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
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
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
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
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)
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
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
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,
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
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
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
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
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
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.
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
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,
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
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,
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
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.
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
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
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;

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