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

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Explain why replacing in model (7.3.1) by B gives a heterogeneous L XL model. The fit is equivalent to fitting the L XL association model for two-way tables separately for each gender. Fit the heterogeneous model to Table 7.5, and test whether it fits significantly better than the homogeneous model
Refer to the fit of the homogeneous L XL model in Section 7.3.4. Describe the conditional association using the fitted odds ratio for the corner cells at the extreme categories of income and job satisfaction. Interpret, and compare to the local odds ratios. Show that the estimated odds ratio for
For K = 1, the generalized CMH correlation statistic equals (2.5.1). When there truly is a trend, Section 2.5.3 noted that this test is more powerful than the X and G tests of Section 2.4. To illustrate, for Table 7.5, construct the marginal 44 table relating job satisfaction and income. Show that
Refer to the data in Problem 7.22.a. Test conditional independence against the alternative that the mean job satisfaction varies by level of income, controlling for gender. Report and interpret a relevant test statistic.b. Conduct the test and interpret results for the alternative that mean income
Refer to the data in the previous problem. Test the hypothesis of conditional independence, (a) using ordinary loglinear models, (b) using the category orderings. Interpret, and compare results.
The sample in Table 7.5 consists of 104 black Americans. A similar table relating income and job satisfaction for white subjects in the 1991 General So- cial 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, 39/7, 14,69,
Refer to Table 7.5. Perform a sensitivity analysis to check the dependence of the analyses in Section 7.3 on the choice of scores.
Refer to Problem 6.6 with Table 6.13.a. Fit model (AD, AG, DG). Using the fitted values, estimate the A-G condi- tional odds ratio and the A-G marginal odds ratio. Explain why they give different indications of the A-G association.b. Is it proper to use G((AD, DG) | (AD, AG, DG)] to test
Refer to Table 3.3 in Section 3.2. Using a model-based procedure, test con- ditional independence and estimate the strength of conditional association between smoking and lung cancer. Compare results with the CMH approach to testing and estimation.
Refer to Problem 3.10 with Table 3.6. Test conditional independence of group and response, given center, by comparing two logit or loglinear models. Using a relevant model, estimate the conditional odds ratio between group and response, given center. Interpret.
Political ideology is often measured using the categories (Liberal, Mod- erate, Conservative). For the 1991 General Social Survey, Democrats had counts (161, 171,96) in these categories, whereas Republicans had counts (76, 148, 183). Analyze these data.
For Table 7.3, fit the linear-by-linear association model using scores {1,2,4, 5}. Explain why the fitted local log odds ratio using levels 2 and 3 of each classi- fication equals four times the fitted local log odds ratio using levels 1 and 2 of each or 3 and 4 of each. What is the relation
Refer to Problem 2.21 with Table 2.15. Test the hypothesis of independence by testing the fit of the independence model. Using an ordinal model, describe the association and conduct an alternative test that may be more powerful. Interpret.
Refer to Table 2.5. Treat party identification as ordinal by fitting the linear-by- linear association model. Conduct a likelihood-ratio test of independence for that model. Interpret, and compare results to the ordinary likelihood-ratio test of independence.
Consider logit models for a four-way table in which X1, X2, and X3 are predictors of Y. When the table is collapsed over X3, indicate whether the association between X and Y remains unchanged, for the model (i) that has main effects of all predictors, (ii) that has main effects of X, and X2 but
Show that when any variable is independent of all others, collapsing over that variable does not affect other model terms. Illustrate using model (W,XY, XZ, YZ), showing that associations among X, Y, and Z are the same as in model (XY, XZ, YZ).
Consider loglinear model (WX, WY, WZ, XZ, YZ). Though X and Y are condi- tionally independent, explain why they may be marginally dependent, collaps- ing over W and Z. Explain why the W-X and X-Z associations are unchanged when one collapses over Y, and the W-Y and Y-Z associations are unchanged
Show that model (WX, WY, WZ, XY, XZ, YZ) or any more complex model for four variables has the same association graph. Explain why each pair of vari- ables may have differing partial and marginal associations.
Consider model (WX, YZ). Applying the collapsibility conditions in Section 7.1.4 with sets A {W.X), C (Y,Z), and with B empty, explain why the W-X partial association is the same as the W-X marginal association, collapsed over Y and Z.
Fit the model to Table 7.1 having association graph portrayed in Section 7.1.5.a. Explain why the A-M conditional odds ratio is unchanged by collapsing over race, but it is not unchanged by collapsing over gender.b. Examine and interpret the associations among the drug-use variables and the effects
Refer to Problem 6.13 with Table 6.17.a. Show that model (CE, CH, CL, EH, EL, HL) fits well. Show that 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 back- ward elimination yields (CE, CL, EH, HL).
Refer to the clinical trial in Problem 3.10 with Table 3.6 a. Fit logit model (7.1.1). Using the fitted values, compute the estimated odds ratio between group and response (i) for each center, (ii) for the marginal table. Why do they differ?b. Fit the simpler logit model deleting the center
Refer to Problem 6.3 with Table 3.1. Show the association graph for model (DV, PV), and fit the model. Using the fitted values, compute the estimated P-V odds ratios at the two levels of D, and compute the marginal P-V odds ratio. Compare the values. Are the collapsibility conditions satisfied?
For a four-way table, are X and Y independent, given Z alone, for model (i) (WX, XZ, YZ, WZ), (ii) (WX, XZ, YZ, WY)?
Draw the association graph for loglinear model (WXZ, WYZ). Which, if any, variables are conditionally independent in this model?
For four categorical variables W, X, Y, Z, explain why (WXZ, WYZ) is the most general loglinear model for which X and Y are conditionally independent.
For a multiway contingency table, when is a logit model more appropriate than a loglinear model? When is a loglinear model more appropriate than a logit model?
Verify that logit model (6.5.4) follows from loglinear model (GLS, GI, LI, IS). Show that the conditional log odds ratio for the effect of S on / equals - B in the logit model and A+ A A-A in the loglinear model.
Refer to the logit model in Problem 5.23. Let A denote the response variable, opinion on abortion.a. Give the symbol for the loglinear model that is equivalent to this logit model. Which logit model corresponds to loglinear model (AR, AP, GRP)?b. State the equivalent loglinear and logit models for
Refer to Problem 3.2. Analyze these data with loglinear models, using residuals to describe lack of fit. Show how logit models for the death penalty response provide the same results.
Refer to Problem 6.3. Using P as the response, fit logit models that give the same results as these two loglinear models. Show the relation between parameter estimates for the loglinear and logit models.
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 explanatory variables. Summarize your main conclusions based on studying the relevant estimated
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 variables as predictors for / in the four- way table. Explain why this composite model is sensible, fit
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 females not wearing seat belts in rural locations.b. Fit model (GLS, GI, IL, IS). Show, using model
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 model, treating death penalty as the response. Show the correspondence between parameter estimates for
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 odds ratio. Interpret.b. Test goodness of fit of the model. Interpret.c. Consider the simpler model of
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 level of V. Note the "common odds ratio" property of this model, and interpret the value.b. Calculate
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 odds ratio.
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. Interpret - .
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 identification as nominal. Note that it is satu- rated. (The likelihood-ratio test for an absence of a
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 summary binomial counts. However, Lo and Li, and hence G(Mo | M), do not depend on this choice.
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 width as a predictor, the curve has the shape of a logistic cdf with mean 24.8 and standard deviation
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, (x)=. For the fit of the probit model to the crab data using width as a predictor, where does the
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 of . Generalize these results to the multiple logistic regression model (5.5.1).
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. Interpret results.
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 for testing the hypothesis of no C effect against the alternative of a positive effect, and construct
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 Wald test for the model in (a). (Note: The trend test of Section 2.5 is the "efficient score" test for
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 sample yields P(Type II error) = .10 in an a = .05-level test of independence against the alternative
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 be? Compare results to the sample sizes required for a 90% interval with power 90%, a 95% interval
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 region. The test statistic compares the fit of this model to the simpler one, testing that the extra
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 has a mean of 26.30 and standard deviation of 2.11. For standardized versions of the predictors,
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 that neither variable affects the response. Interpret.c. Conduct separate tests for the partial
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) = 3.02. When each probability is close to zero, the odds ratio is similar to the ratio of
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 probability of a satel- lite, for each of the four colors. Plot them, and interpret.b. Test for a
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 the same at all levels of a qualitative factor. Fit the model, using weight instead of width as x.
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, Catholic, Jewish), and political party affiliation (Democrat, Republican, Independent), the model for the
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 those given by the Breslow-Day test of Section 3.2.4.b. Check Pearson or adjusted residuals to analyze
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 penalty, controlling for victims' race. Interpret.
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. Analyze and interpret.
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 interval for the conditional odds ratio between AZT use and the development of symptoms.c. Describe and
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 or was local for a sample of 444 middle-aged and elderly women. A table referring to a particular
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 hypothesis that color has no effect. Compare the result to the test for color conducted in Section
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 influence diagnostics, and interpret.
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 1.20, 2.44, and 5.20 kg of weight, which are the sample minimum, mean, and maximum.b. Find the weight
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 time of the operation for the 18 subjects for whom kyphosis was present were 12, 15, 42, 52, 59,
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 goodness of fit.
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, 32.8, 27.2, 22.5, 13.3 (1980-1989) (Source: George Will, Newsweek, April 10, 1989).a. Use a logistic
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 likelihood-ratio test of H0: β1 = β2 = 0. Interpret.c. Conduct separate likelihood-ratio tests for the
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 step of the process.
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 main effects for each term as well as the three two-factor interactions and the three-factor
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 drinks alcohol frequently.a. Conduct a model goodness-of-fit test, and interpret.b. If you were to
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 binary main effect terms, 644.0 for the model with only TF as a predictor, and 648.8 for the model with
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 table, sensitivity =0.53 and specificity = 0.66. Explain what these mean.b. Using (a), show that the
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 likelihood values of 1130.23 with only an intercept term, 1124.86 with also the main effect predictors,
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 distinct levels of LI), the deviance for this model is 15.7 with df = 12. Is it appropriate to use
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 0.642 as the cutoff, report the sensitivity and specificity. Interpret.b. Form a ROC curve, and report
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 regions as a predictor in the model by setting up dummy variables for the regions. The test statistic
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 worthy. The data archive found under the index at www.stat.uni-muenchen.de for a textbook by L.
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 values by L0 for M0, L1 for M1, and Ls for the saturated model. Recall that G2(Mi ) = −2(Li − Ls
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 of 3271 missing males and 38 of 2486 missing females were still missing a year later. For ages
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. Interpret the parameter estimates.
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 goodness of fit. Interpret.c. Check residuals to analyze further the quality of fit. Interpret.
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 department as a predictor (with no gender effect) for Y = whether admitted (1 = yes, 0 = no).b. For the
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 independence using the likelihood-ratio test for the model in (a). (The trend test of Section 2.5.1 is
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 for the software you use.c. Add two observations at x = 50, y = 1 for one and y = 0 for the
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 = whether cured, andZ = penicillin level. Fit the model, logit[P(Y = 1)] = βx + βZ k , deleting an
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 interval.a. Show that each predictor has a significant effect when it is used individually without

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