Questions and Answers of Categorical Data Analysis

Consider Table 9.13. Explain how one could analyze whether the hazard depends on time.Table 9.13: Number of Deaths from Lung Cancer Follow-up Histology“ Time III I II Interval Disease (months)
An article by W. A. Ray et al. (Amer. J. Epidemiol. 132: 873–884, 1992) dealt with motor vehicle accident rates for 16,262 subjects aged 65–84 years, with data on each for up to 4 years. In 17.3
A table at the text’s Web site (www.stat.ufl.edu/ ∼aa/cda/cda.html) shows the number of train miles (in millions) and the number of collisions involving British Rail passenger trains between
Table 9.18 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. Let Y =
Table 9.19 is based on a study with British doctors.a. For each age, find the sample coronary death rates per 1000 person-years for nonsmokers and smokers. To compare them, take their ratio and
In a 2 × 2 × K table, the true XY conditional odds ratios are identical, but different from the XY marginal odds ratio. Is there three-factor interaction? Is Z conditionally independent of X or Y?
Suppose that loglinear model (XY, XZ) holds.a. Find and log µij+. Show the loglinear model for the XY marginal table has the same association parameters as {λijXY} in (XY, XZ). Deduce that odds
For a four-way table, is the WX conditional association the same as the WX marginal association for the loglinear model (a) (WX, XYZ)? and (b) (WX, WZ, XY, YZ)? Why?
Loglinear model M0 is a special case of loglinear model M1.a. Haberman (1974a) showed that when {µ̂i} satisfy any model that is a special case of M0, ∑i µ̂1i log µ̂i = ∑i µ̂0i log µ̂i.
Consider the L × L model (9.6) with {υj = j} replaced by {υj = 2j}. Explain why β̂ is halved but {µ̂ij}, {θ̂ij}, and G2 are unchanged.
Lehmann (1966) defined (X, Y) to be positively likelihood-ratio dependent if their joint density satisfies f(x1, y1) f(x2, y2) ≥ f(x1, y2) f(x2, y1) whenever x1 < x2 and y1 < y2. Then, the
Yule (1906) defined a table to be isotropic if an ordering of rows and of columns exists such that the local log odds ratios are all nonnegative].a. Show that a table is isotropic if it
Consider the row effects model (9.8).a. Show that no loss of generality occurs in lettingb. Show that minimal sufficient statistics are {ni+}, {n+j}, and {ˆ‘j Ï…j nij, i = 1,...,
Refer to the homogeneous linear-by-linear association model (9.10).a. Show that the likelihood equations are, for all i, j, and k,b. Show that residual df = K(I €“ 1) (J €“ 1)
When model (XY, XZ, YZ) is inadequate and variables are ordinal, useful models are nested between it and (XYZ). For ordered scores {ui}, {Ï…j} and {wk}, considera. Show that log odds ratios
Construct a model having general XZ and YZ associations, but row effects for the XY association that are (a) Homogeneous, and (b) Heterogeneous across levels of Z. Interpret.
Refer to Section 9.7.3. Let T = ∑ti and W = ∑wi. Suppose that survival times have a negative exponential distribution with parameter λ.a. Using log likelihood (9.19), show that λ̂ = W/T.b.
Show that ML estimates do not exist for Table 9.15. [Haberman: If µÌ‚111= c > 0, then marginal constraints the model satisfy imply that µÌ‚222= €“
For a loglinear model, explain heuristically why the ML estimate of a parameter is infinite when its sufficient statistics takes its maximum or minimum possible value, for given values of other
Table 10.14 shows results when subjects were asked €œDo you think a person has the right to end his or her own life if this person has an incurable disease?€ and When a person
Refer to Table 8.16 and Problem 8.1. Treat the data as matched pairs on opinion, stratified by gender. Testing independence for the 2 Ã— 2 table using entries (6, 160) in row 1 and (11,
A crossover experiment with 100 subjects compares two drugs for treating migraine headaches. The response scale is success (1) or failure (0). Half the study subjects, randomly selected, used drug A
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
Each week Variety magazine summarizes reviews of new movies by critics in several cities. Each review is categorized as pro, con, or mixed, according to whether the overall evaluation is positive,
Refer to Table 10.8. Based on the reported standardized residuals, explain why the linear-by-linear association model (9.6) might fit well. Fit it and describe the association.Table 10.8: Diagnoses
Table 10.20 refers to journal citations among four statistics journals during 1987 €“ 1989. The more often articles in a particular journal are cited, the more prestige that journal
Table 10.21 refers to matches for several women tennis players during 1989 and 1990.a. Fit the Bradley€”Terry model. Interpret, and rank the players.b. Estimate the probability of Seles
Explain the following analogy: McNemar’s test is to binary data as the paired difference t test is to normally distributed data.
Refer to the subject-specific model (10.8) for binary matched pairs. a. Show that exp(Î²) is a conditional odds ratio between observation and outcome. Explain the distinction between
Consider marginal model (10.6) when Y1 and Y2 are independent and conditional model (10.8) when {αi} are identical. Explain why they are equivalent.
Let Î²Ì‚M= log (p+1p2+/p+2p1+) refer to marginal model (10.6) and Î²Ì‚C= log (n21/n12) to conditional model (10.8). Using the delta method, show that the
Give an example illustrating that when I > 2, marginal homogeneity does not imply symmetry.
Refer to the 34table on government spending in Table 8.19. Analyze these data with a marginal cumulative logit model. Interpret effects.Table 8.19: Cities 2 3 1. Law Environment Health Enforcement: 3
Refer to the air pollution data in Table 11.7. Using ML or GEE, fit marginal logit models that assume(a) Marginal homogeneity,(b) A linear effect of timeTable 11.7:
Refer to Table 11.2. Analyze the data using the scores (1, 2, 4) for the week number, using ML or GEE. Interpret estimates and compare substantive results to those in the text with scores (0, 1,
Table 11.11 is from a Kansas State University survey of 262 pig farmers. For the question €œWhat are your primary sources of veterinary information?,€ the categories were (A)
Refer to the matched-pairs data of Table 10.14 and Problem 10.1.a. Fit model (12.3). Interpret Î²Ì‚. If your software uses numerical integration, report
For marginal model (10.14) for Table 10.5 on premarital and extra marital sex, Table 12.12 shows results of fitting a corresponding random intercept model. Interpret Î²Ì‚.
For the insomnia example in Section 12.4.2, according to SAS the maximized log likelihood equals – 593.0, compared to – 621.0 for the simpler model forcing σ = 0. Compare models, using either a
Refer to Section 12.3.1. Using supplementary information improves predictions. Let qi denote the true proportion of votes for Clinton in state i in the 1992 election, conditional on voting for him or
For a binary response, consider the random effects modellogit[P(Yit = 1|ui)] = α + βt + ui, t = 1,..., T,where {ui} are independent N(0, σ2), and the marginal modellogit[P(Yt = 1)] = α + βt*, t
The GLMM for binary data using probit link function isΦ–1[P(Yit = 1 | ui)] = x’it β + z’it ui,where Φ is the N(0, 1) cdf and ui has N(0, ∑) pdf, f(ui; ∑).a. Show that the marginal mean
In the Rasch model, logit[P(Yit= 1)] = Î±i+ Î²t, Î±iis a fixed effect.a. Assuming independence of responses for different subjects and for different observations
Consider the matched-pairs random effects model (12.3). For given Î²0, let Î´0be such that µÌ‚12= n12+ Î´0and µÌ‚21=
Explain why the logistic-normal model is not helpful for capture–recapture experiments with only two captures.
For ordinal square I Ã— I tables of counts {nab}, model (12.3) for binary matched-pairs responses (Yi1, Yi2) for subject i extends tologit[P(Yit ‰¤ j|ui)] = Î±j +
Summarize advantages and disadvantages of using a GLMM approach compared to a marginal model approach. Describe conditions under which parameter estimators are consistent for (a) marginal models
For capture €“ recapture experiments, Coull and Agresti (1999) used a loglinear model with exchangeable association and no higher-order terms. Explain why the model expected frequencies
A data set on pregnancy rates among girls under 18 years of age in 13 north central Florida counties has information on a 3-year total for each county i on ni = number of births and yi = number of
In Problem 12.2 about Shaq O€™Neal€™s free-throw shooting, the simple binomial model, Ï€i= Î±, has lack of fit. Fit the beta-binomial model, or use the
For the train accidents in Problem 9.19, a negative binomial model assuming constant log rate over the 14-year period has estimate –4.177 (SE = 0.153) and estimated dispersion parameter 0.012.
One question in the 1990 General Social Survey asked subjects how many times they had sexual intercourse in the preceding month. Table 13.9 shows responses, classified by gender.a. The sample means
For the counts of horseshoe-crab satellites in Table 4.3, Table 13.10 shows the results of ML fitting of the negative binomial model using width as the predictor, with the identity link.a. State and
Refer to Table 13.6. For those with race classified as €œother,€ the sample counts for (0, 1, 2, 3, 4, 5, 6) homicides were (55, 5, 1, 0, 1, 0, 0). Fit an appropriate model
Let Y be a Poisson random variable with mean µ.a. For a constant c > 0, show thatE[log(Y + c)] = log µ + (c – 1/2)/µ + O(µ–2)(Note that log(Y + c) = log µ + log[1 + (Y + c – µ)/µ].)b.
Let p denote the sample proportion for n independent Bernoulli trials. Find the asymptotic distribution of the estimator [p(1 – p)]1/2 of the standard deviation. What happens when π = 0.5?
a. Refer to Problem 14.6. If Tn is Poisson, show √Tn has asymptotic variance 1/4.b. For a binomial sample with n trials and sample proportion p, show the asymptotic variance of sin-1(√p) is 1/4n.
For a multinomial (n, {πi}) distribution, show the correlation between pi and pj is –[πi πj/(1 – πi)(1 – πj)]1/2. What does this equal when πi = 1 – πj and πk = 0 for k ≠ i, j?
Consider the model for a 2 × 2 table. π11 = θ2, π12 = π21 = θ(1 – θ), π22 = (1 – θ)2, where θ is unknown (Problems 3.31 and 10.34).a. Find the matrix A in (14.14) for this model.b. Use
Suppose that {µij = nπij} satisfy the independence model (8.1).a. Show that λYa – λYb = log(π+a / π+b).b. Show that {all λYj = 0} is equivalent to π+j = 1/J for all j.
The book’s Web site (www.stat.ufl.edu/ ∼aa/cda/cda.html) has a 2 × 3 × 2 × 2 table relating responses on frequency of attending religious services, political views, opinion on making birth
For a multiway contingency table, when is a logit model more appropriate than a loglinear model? When is a loglinear model more appropriate?
Refer to the logit model in Problem 5.24. Let A = opinion on abortion.a. Give the symbol for the loglinear model that is equivalent to this logit model.b. Which logit model corresponds to loglinear
Refer to Table 8.19. Subjects were asked their opinions about government spending on the environment (E), health (H), assistance to big cities (C), and law enforcement (L).a. Table 8.20 shows some
Table 8.18 refers to automobile accident records in Florida in 1988.a. Find a loglinear model that describes the data well. Interpret associations.b. Treating whether killed as the response, fit an
Refer to Section 8.3.2. Explain why software for which parameters sum to zero across levels of each index reports λ̂11AC = λ̂22AC = 0.514 and λ̂12AC = λ̂21AC = – 0.514, with SE = 0.044 for
Refer to Table 8.17 from the 1991 General Social Survey. White subjects were asked: (B) €œDo you favor busing of (Negro/Black) and white school children from one school district to
The 1988 General Social Survey compiled by the National Opinion Research Center asked: €œDo you support or oppose the following measures to deal with AIDS9 (1) Have the government pay all
For the multinomial (n,{Ï€j}) distribution with c > 2, confidence limits for Ï€jare the solutions of a. Using the Bonferroni inequality, argue that these c intervals
Refer to Section 1.5.6. Using the likelihood function to obtain the information, find the approximate standard error of π̂.
In some situations, X2 and G2 take very similar values. Explain the joint influence on this event of (a) whether the model holds, (b) whether the sample size n is large, and (c) whether the number of
Consider the logistic-normal model (12.10) for the abortion opinion data, under the constraint σ = 0.a. Explain why the fit is the same as an ordinary logit model treating the three responses for
Refer to Table 4.8 on the free-throw shooting of Shaq O€™Neal. In game i, suppose that yi= number made out of niattempts is a bin(ni, Ï€i) variate and {yi} are independent.a. Fit
Gamblers A and B have a total of I dollars. They play games of pool repeatedly. Each game they each bet $1, and the winner takes the other’s dollar. The outcomes of the games are statistically
Suppose that loglinear model (Y0, Y1,...,YT) holds. Is this a Markov chain?
What is wrong with this statement?: “For a first-order Markov chain, Yt is independent of Yt–2.”
a. For a univariate response, how is quasi-likelihood (QL) inference different from ML inference? When are they equivalent?b. Explain the sense in which GEE methodology is a multivariate version of
Consider the model µi = β, i = 1,..., n, for independent Poisson observations. For β̂ = y̅, show that the model-based asymptotic variance estimate is y̅/n, whereas the robust estimate of the
Repeat Problem 11.23 assuming that υ(µi) = σ2 when actually var(Yi) = µi.Data from Problem 11.23:Consider the model µi = β, i = 1, ..., n, assuming that υ(µi) = µi. Suppose that actually
Consider the model µi = β, i = 1, ..., n, assuming that υ(µi) = µi. Suppose that actually var(Yi) = µi2. Using the univariate version of GEE described in section 11.4, show that u(β) = ∑i(yi
Refer to Problem 11.1. Suppose that we expressed the data with a 3 Ã— 2 partial table of drug-by-response for each subject, to use a generalized CMH procedure to test marginal
Refer to Problem 2.12.a. Fit the model with G and D main effects. Using it, estimate the AG conditional odds ratio. Compare to the marginal odds ratio, and explain why they are so different. Test its
For a sequence of s nested models M1,..., Ms, model Ms is the most complex. Let ν denote the difference in residual df between M1 and Ms.a. Explain why for j < k, G2(Mj | Mk) ≤
Prove that the Pearson residuals for the linear logit model applied to a I × 2 contingency table satisfy X2 = ∑1i = 1 e2i. Note that this holds for a binomial GL.M with any link.
Refer to Table 2.6. Let D = defendant€™s race, V = victims€™ race, and P = death penalty verdict. Fit the loglinear model (DV, DP, PV).a. Using the fitted values, estimate and
For model (AC, AM, CM) with Table 8.3, the standardized Pearson residual in each cell equals ± 0.63. Interpret, and explain why each one has the same absolute value. By contrast, model (AM,
Use odds ratios in Table 8.3 to illustrate the collapsibility conditions.a. For (A, C, M), all conditional odds ratios equal 1.0. Explain why all reported marginal odds ratios equal 1.0.b. For (AC,
Refer to Problem 5.1. Table 6.18 shows output for fitting a probit model. Interpret the parameter estimates (a) using characteristics of the normal cdf response curve, (b) finding the estimated rate
The horseshoe crab width values in Table 4.3 have xÌ… = 26.3 and sx= 2.1. If the true relationship were similar to the fitted equation in Section 5.1.3, about how large a sample yields
For the horseshoe crab data, fit a model using weight and width as predictors. Conduct (a) A likelihood-ratio test of H0: β1 = β2, = 0, (b) Separate tests for the partial effects. Why
Treatments A and B were compared on a binary response for 40 pairs of subjects matched on relevant covariates. For each pair, treatments were assigned to the subjects randomly. Twenty pairs of
For a sequence of independent Bernoulli trials, Y is the number of successes before the kth failure. Explain why its probability mass function is the negative binomial,[For it, E(Y) =
For the geometric distribution p(y) = πy(1– π), y = 0, 1, 2,... , show that the tail method for constructing a confidence interval [i.e., equating P(Y ≥ y) and P(Y ≤ y) to α/2] yields
For a diagnostic test of a certain disease, π1 denotes the probability that the diagnosis is positive given that a subject has the disease, and π2 denotes the probability that the diagnosis is
For a 2 × 2 table of counts {nij} show that the odds ratio is invariant to(a) Interchanging rows with columns, (b) Multiplication of cell counts within rows or within columns by c ≠ 0. Show
A 2 Ã— J table has ordinal response. Let Fj|i= Ï€1|i+ ..... + Ï€j|i. When Fj|2‰¤ Fj|1for j = 1,......, J, the conditional distribution in row 2 is stochastically higher than the one in
For binary data with sample proportion yi based on ni trials, we use quasi-likelihood to fit a model using variance function. Show that parameter estimates are the same as for the binomial GLM but
Suppose that Yi is Poisson with g(µi) = a + βxi, where xi = 1 for i = 1,..., nA from group A and xi = 0 for i = nA + 1,..., nA + nB from group B. Show that for any link function g, the likelihood
Refer to the tea-tasting data (Table 3.8). Construct the null distributions of the ordinary P-value and the mid-P-value for Fisher€™s exact test with HÎ±: 0 > 1. Find and
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,

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