Questions and Answers of Categorical Data Analysis

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
A diagnostic test is designed to detect whether subjects have a certain disease. A positive test outcome predicts that a subject has the disease. Given that the subject has the disease, the
For tests of independence, {j= n;+n+/n}. Show that {ij} have the same row and column totals as the observed data. For 2 x 2 tables, show that their odds ratio equals 1.0. Hence, they satisfy the null
The Pearson residual for a cell in a two-way table equals ej (niji).a. Show that they provide a decomposition of the Pearson chi-squared statistic, through X =b. Show that Pearson residuals are
Formula (2.4.3) has alternative expression X = n(Pij - Pi+P+1)/Pi+P+j For a particular set of {pij), X2 is directly proportional to n. Hence, X can be large when n is large, regardless of whether the
Let Z denote a standard normal variate. Then Z has a chi-squared distribution with df 1. A chi-squared variate with degrees of freedom equal to df has representation Z++Z, where Z.....Zaf are
In murder trials in 20 Florida counties during 1976 and 1977, the death penalty was given in 19 out of 151 cases in which a white killed a white, in 0 out of 9 cases in which a white killed a black,
For all trials in Florida involving homicides between 1976 and 1987, M. Radelet and G. Pierce (Florida Law Review, 43: 1-34 (1991)) reported the following results: The death penalty was given in 227
Smith and Jones are 1 seball players. Smith had a higher batting average than Jones in 1994 and 19 . Is it possible that for the combined data for these two years, Jones had the higher batting
Give a "real world" example of three variables X, Y, and Z, for which you expect X and Y to be marginally associated but conditionally independent, controlling for Z.
Based on 1987 murder rates in the United States, the Associated Press reported that the probability a newborn child has of eventually being a murder victim is 0.0263 for nonwhite males, 0.0049 for
Using graphs or tables to illustrate, explain what is meant by "no interaction" in modeling a response Y and explanatory variables X and Z, when (a) all variables are continuous (multiple
For three-way contingency tables, when any pair of variables is conditionally independent, explain why there is homogenous association. When there is not homogeneous association, explain why no pair
Table 3.5 refers to the effect of passive smoking on lung cancer. It summarizes results of case-control studies from three countries among nonsmoking women married to smokers. Test the hypothesis
Refer to the previous problem. Assume that the true odds ratio between passive smoking and lung cancer is the same for each study. Estimate its value, and use software to find a 95% confidence
Table 3.6 shows results of a three-center clinical trial designed to compare a drug to placebo for treating severe migraine headaches. At each center, subjects were randomly assigned to treatments.a.
Refer to Table 3.1. Treating this as a sample, analyze the data.
Refer to Problem 3.2. Test whether the odds ratios are the same at each level of victims' race. Interpret.
Refer to Table 3.7, which classifies police officers by rank, race, and promotion decisions made in 1988.a. Conduct an exact test of conditional independence of promotion and race, given rank.
Refer to Problem 3.10.a. Use an exact test to conduct this analysis. Compare results to the large- sample test.b. Conduct an exact test that the odds ratio is identical for all three centers. Compare
Table 3.8 refers to ratings of agricultural extension agents in North Carolina. In each of five districts, agents were classified by their race and by whether they qualified for a merit pay increase.
Describe the purpose of the link function of a GLM. Define the identity link, and explain why it is not often used with binomial or Poisson data.
Refer to Table 4.1. Refit the linear probability model or the logistic regression model using the scores (i) (0, 2, 4, 6), (ii) (0, 1, 2, 3), (iii). (1, 2, 3, 4). Compare the model parameter
Refer to Table 4.2. Let Y = 1 if a crab has at least one satellite, and Y = 0 otherwise. Using weight as the predictor, fit the linear probability model.a. Use ordinary least squares. Interpret the
Refer to Table 2.7 on alcohol consumption and infant malformation.a. Using scores {0, .5, 1.5, 4, 7}, fit a linear probability model. Interpret, and compare the sample proportions to the fitted
Table 4.6 refers to a sample of subjects randomly selected for an Italian study on the relation between income and whether one possesses a travel credit card (such as American Express or Diners
An experiment analyzes imperfection rates for two processes used to fabricate silicon wafers for computer chips. For treatment A applied to 10 wafers, the numbers of imperfections are 8, 7, 6, 6, 3,
Refer to the previous problem. Conduct the test of Ho A = B by using the fact that if X is Poisson with mean and Y is an independent Poisson variate with mean 2, then X given X + Y is binomial with
Refer to Problem 4.6. The wafers are also classified by thickness of silicon coating (z 0, low; z = 1, high). The first five imperfection counts reported for each treatment refer to z = 0 and the
Refer to Table 4.2.a. Using weight as the predictor and the number of satellites as the response, fit a Poisson loglinear model. Estimate the mean number of satellites for female crabs of average
Refer to the previous problem.a. Test goodness of fit by grouping levels of weight. Use residuals to describe lack of fit.b. Is there evidence of overdispersion? If necessary, adjust the standard
Refer to Table 4.2.a. Fit a Poisson loglinear model using both weight and color to predict the number of satellites. Assigning dummy variables, treat color as a nominal factor. Interpret the
In Section 4.3.2, refer to the Poisson regression model with identity link for the crab data. Explain why the fit differs from the least squares fit. (Hint: The least squares fit is the same as the
Refer to the injurious accident data in Section 4.3.4.a. Test the hypothesis of equal rates for men and women using a likelihood- ratio test. Compare results to the Wald test.b. White drivers had 348
Table 4.7 lists total attendance (in thousands) and the total number of arrests, in the 1987-1988 season for soccer teams in the Second Division of the British football league. (Thanks to Dr. P. M.
Table 4.8 shows the number of train miles (in millions) and the number of collisions involving British Rail passenger trains between 1970 and 1984. Is it plausible that the collision counts are
Table 4.9, based on a study with British doctors conducted by R. Doll and A. B. Hill, was analyzed by N. R. Breslow in A Celebration of Statistics, A. C. Atkinson and S. E. Fienberg, eds. (Berlin:
For rate data, the Poisson GLM with identity link isa. Since the model has form at +tx, argue that it is equivalent to a Poisson GLM for the response totals at the various levels of x, using identity
For the 23 space shuttle flights that occurred before the Challenger mission disaster in 1986, Table 5.10 shows the temperature (F) at the time of the flight and whether at least one primary O-ring
Table 5.11 contains results of a case-control study on the relationship between smoking and myocardial infarction (MI). The sample consisted of young and middle-aged women admitted to 30 coronary
Table 5.12 appeared in a national study of 15 and 16 year-old adolescents. The event of interest is ever having sexual intercourse. Analyze these data, including description and inference about the
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
In a study designed to evaluate whether an educational program makes sexually active adolescents more likely to obtain condoms, adolescents were randomly assigned to two experimental groups. The
Table 5.15 refers to results of a case-control study about effects of cigarette smoking and coffee drinking on myocardial infarction (MI) for a sample of men under 55 years of age.a. Fit a logit
Table 5.16 shows estimated effects for a fitted logistic regression model with squamous cell esophageal cancer (Y = 1, yes; Y = 0, no) as the response variable. Smoking status (S) equals 1 for at
In the 1988 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;
Table 6.13 refers to applicants to graduate school at the University of California at Berkeley for the fall 1973 session. Admissions decisions are presented by gender of applicant, for the six
Table 6.14 is from the 1991 General Social Survey. White subjects in the sample were asked: (B) Do you favor busing of (Negro/Black) and white school children from one school district to another?,
Table 6.15 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
Table 6.16, based on the 1991 General Social Survey, relates responses on four variables: How often you attend religious services (R = At most a few times a year, At least several times a year);
Table 6.17 is taken from the 1989 General Social Survey. Subjects were asked their opinions regarding government spending on the environment (E), health (H), assistance to big cities (C), and law
Table 7.8 is taken from the 1991 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
Table 7.9, taken from the 1991 General Social Survey, shows the relation between political party affiliation and political ideology, stratified by gender. Analyze these data. Table 7.9 Political
Consider Table 7.10, from a study of nonmetastatic osteosarcoma described in A. M. Goorin, J. Clinical Oncology, 5: 1178-1184 (1987) and the LogXact Turbo User Manual (1993, p. 5-22). The response is
Refer to Problem 6.12 with Table 6.16 and the loglinear model you selected for those data. Draw the association graph for the model. Remark on conditional independence patterns. For each pair of
A GLM has form g(u) = X, for a monotone function g. Explain what each symbol in this formula represents for fitting the ordinary linear regression model to n observations on a normally distributed
For Table 6.3, show the matrix representation of loglinear model (AC, AM, CM). Specify the parameter constraints used in your model ma- trix. Show the matrix representation of the corresponding logit
Refer to formula (7.5.3) applied to the independence model for a 2 2 table. Show that for the constraints A == 0, for which B = (A, A, ), X has rows (1,0,0), (1,0, 1), (1, 1, 0), (1, 1, 1).
Refer to {ijk = ni+kn+jk/n++k} for model (XZ, YZ). Show that these fitted values have the same X-Z and Y-Z marginal totals as the observed data. For 2 2 K tables, show that Oxy(k) = 1. Illustrate
For the model logit(T) = a +x, let (x, y) denote the x and y values for subject i, i = 1,..., N. Suppose y = 0 for all x below some point and y; = 1 for all x above that point. Explain intuitively
Suppose that all row and column marginal totals of a two-way table are positive, but some cells are empty. Show that all fitted values for the loglinear model of independence (6.1.1) are positive.
Show that a single cell containing any positive count makes a large contribution to X if the fitted value is close to 0. To illustrate, calculate the contribution of (a) a count of 1 in a cell having
Provide an example of contingency tables in which certain cells contain (a) structural zeroes, (b) sampling zeroes.
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
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
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
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
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,
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
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
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
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
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
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
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
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
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,
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
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
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
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
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.
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)
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
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
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
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

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