Questions and Answers of Categorical Data Analysis

Show that (11.10) is equivalent to the formula for the large-sample covariance of the ML estimator in a GLM, estimated by (4.28).
Formulate a model using adjacent-categories logits that is analogous to model (12.14) for cumulative logits. Interpret parameters.
For the Poisson GLMM (13.14), use the normal mgf to show that for t ≠ s,cov(Yit, Yis) = exp[(x’it + x’is)β][exp(σ2)(exp(σ2)–1)]Hence, find corr(Yit, Yis).
Let {Yi} be independent Poisson random variables. Show by the delta method that the estimated asymptotic variance of ∑ai log(Yi) is ∑a2i/yi. [This formula applies to ML estimators of parameters
Cell counts {Yi} are independent Poisson random variables, with µi = E(Yi). Consider the Poisson loglinear modellog µ = Xa θa, where µ = (µ1,..., µN).Using arguments similar to those in
Use the delta method, with derivatives (14.22), to derive the asymptotic covariance matrix in (14.23) for residuals. Show that this matrix is idempotent.
Show X and θ in multinomial representation (14.27) for the independence model for an I × J table. By contrast, show Xa for the corresponding Poisson loglinear model (14.29).
Refer to Problem 15.2. Using these data, describe the differences between (a) WLS and ML, and (b) WLS and GEE methods for marginal models with multivariate categorical response data.Data from Problem
Consider marginal homogeneity for an I × I table.a. Letting F(π) = Aπ, explain how (i) F(π) = 0, where A has I – 1 rows, and (ii) F(π) = Xβ, where A has 2(I – 1) rows and β has I – 1
For the Dirichlet prior for multinomial probabilities, show the posterior expected value of πi is (15.3). Derive the expression for this Bayes estimator as a weighted average of pi and E(πi).
For Bayes estimator (15.4), show that the total mean squared error is[K/(n + K)]2[∑(πi – γi)2] + [n/(n + K)]2[1 – ∑πi2].Show that (15.5) is the value of K that minimizes this.
Refer to Problem 15.6. For marginal homogeneity, explain why the minimum modified chi-squared estimates are identical to WLS estimates.Data from Problem 15.6:Consider marginal homogeneity for an I ×
Show that the kernel estimator (15.9) is the same as the Bayes estimator (15.3) for the Dirichlet prior with {βi = αn/(1 – α)N}. Using this result, suggest a way of letting the data determine
Refer to correlation model (9.16) (Goodman 1985, 1986).a. Show that λ is the correlation between the scores.b. If this model holds, show that ∑i µi (πij/π+j) = λvj and ∑j υj (πij/πi+) =
For the general canonical correlation model, show that ∑λ2k = ∑i ∑j (πij – πi+ π+j)2/πi+ π+j. Thus, the squared correlations partition a dependence measure that is the noncentrality
Refer to model (9.18). Given the times at risk {tij}, show that sufficient statistics are {ni+} and {n+j}.
Refer to Table 10.5. Fit the ordinal quasi-symmetry model using u1 = 1 and u4 = 4 and picking u2 and u3 that are unequally spaced but represent sensible choices. Compare results and interpretations
For a 2 × 2 table, derive cov(p+1, p1+), and show that var[√n(p+1 – p1+)] equals (10.1).
Consider the conditional symmetry (CS) model (10.28). a. Show that it has the loglinear representation log µab = λmin(a, b), max(a, b) + τI( a where I(·) is an indicator. b. Show that the
Refer to model (10.32). a. Construct a more general model having home-team parameters {βHi} and away-team parameters {βAi}, such that the probability team i beats team j when i is the home team is
Analyze Table 11.9 using a marginal logit model with age and maternal smoking as predictors. Compare interpretations to the Markov model of Section 11.5.5. Table 11.9:
Refer to the clinical trials data in Table 12.5, analyzed with random effects models in Section 12.3.4. Use GEE methods to analyze them, treating each center as a correlated cluster. Table 12.5:
Refer to Table 10.5. Using GEE methods with cumulative logits, compare the two marginal distributions. Compare results to those using ML in Section 10.3.2. Table 10.5:
Analyze Table 11.9 using a transitional model with two previous responses. Does it fit better than the first-order model of Section 11.5.5? Interpret.Section 11.5.5:Table 11.9 is also from the
Analyze Table 11.2 using a first-order transitional model. Compare interpretations to those in this chapter using marginal models. Table 11.2:
In Section 13.5.1 and Problem 13.42 we saw that for Poisson GLMMs, the marginal effects are the same as the cluster-specific effects. This does not imply that ML estimates of effects are the same for
Liang and Hanfelt (1994) described a teratology study comparing control and treatment groups in which the ML estimate of the treatment effect in a beta-binomial model differs by a factor of 2
Refer to Problem 13.14. a. Fit a negative binomial model with log link. Interpret. Plot the counts against width and indicate which link seems more appropriate b. Fit a Poisson GLMM with log link,
Refer to the crossover study in Problem 12.7. Kenward and Jones (1991) reported results using the ordinal response scale (none, moderate, complete) for relief. Explain how to formulate an ordinal
Generalize model (12.16) to apply simultaneously to Tables 12.8 and 12.15, using a gender main effect but the same membership effect and the same attitude effect for each gender. Fit the model. Use
For Problem 12.7, compare estimates of βB – βA and βC – βA and SE values to those using (a) a marginal model (Problem 11.6), and (b) conditional logistic regression (Section 10.2), treating
For I × 2 contingency tables, explain why the linear-by-linear association model is equivalent to the linear logit model (5.5).
The book’s Web site (www.stat.ufl.edu/ ∼aa/cda/cda.html) has a 4 × 4 × 5 table that cross-classifies assessment of cognitive impairment, Alzheimer’s disease, and age. Analyze these data,
Refer to the log-likelihood function for the baseline-category logit model (Section 7.14). Denote the sufficient statistics by npj = ∑i yij and Sjk = ∑i xik yij, j = 1,... J – 1,...,p. Let S =
For cumulative link model (7.7), show that for 1 ≤ j < k ≤ J – 1, P(Y ≤ k | x) = P(Y ≤ j | x*), where x* is obtained by increasing the ith component of x by (αk – αj)/βi. Interpret.
Consider the model Link [ωj(x)] = αj + βj’x, where ωj(x) is (7.14).a. Explain why this model can be fitted separately for j = 1,..., J – 1.b. For the complementary log-log link, show that
A café has four entrees: chicken, beef, fish, vegetarian. Specify a model of form (7.22) for the selection of an entrée using x = gender (1 = female, 0 = male) and µ = cost of entrée, which is a
Refer to the independence model, µij = µαi βj. For the corresponding loglinear model (8.1): a. Show that one can constrain ∑λiX = ∑λjY = 0 by setting b. Show that one can constrain λ1X =
Write the log likelihood L for model (XZ, YZ). Calculate ∂L/∂λ and show that it implies µ̂+++ = n. Show that ∂L/∂λiX = ni++ – µi++. Similarly, differentiate with respect to each
Verify that loglinear model (GLS, GI, LI, IS) implies logit model (8.16). Show that the conditional log odds ratio for the effect of S on I equals β1S – β2S in the logit model and λ11IS + λ22IS
Consider model (10.12) for a study with matched sets of T observations rather than matched pairs. Explain how (10.13) generalizes and construct the form of the conditional likelihood.
For the quasi-symmetry model (10.19), let λa = λaX – λaY. Show that one can express it equivalently as log µab = λ + λa + λ*ab, with λ*ab = λ*ba. Hence, one needs only one set of
Derive the covariance matrix (10.16) for the difference vector d.
For independent binomial sampling, construct the log likelihood and identify the sufficient statistics to be conditioned out to perform exact inference about β in model (6.4).
Refer to logit model (6.4) for a 2 × 2 × K contingency table {nijk}.a. Using dummy variables, write the log-likelihood function. Identify the sufficient statistics for the various parameters.
Refer to Table 6.16. Apply conditional logistic regression to the model discussed in Section 6.7.8. a. Obtain an exact P-value for testing no C effect against the alternative of a positive effect.
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
Table 6.17, refers to the effectiveness of immediately injected or 1 ½ hour-de1ayed penicillin in protecting rabbits against lethal injection with β-hemolytic streptococci. a. Let X = delay, Y =
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, and (b) separate tests for the partial effects. Why does
Consider the linear logit model (5.5) for an I × 2 table, with yi a bin(ni, πi) variate. a. Show that the log likelihood is b. Show that the sufficient statistic for β is ∑iyi xi, and explain
Find the likelihood equations for model (5.10). Show that they imply the fitted values and that the sample values are identical in the marginal two-way tables.
The book’s Web site (www.swr.ufl.edu/ aa/cda/cda.htrnl) contains a five-way table relating occupational aspirations (high, low) to gender, residence, IQ, and socioeconomic status. Analyze these
Find the form of the deviance residual (4.35) for an observation in a (a) binomial GLM, and (b) Poisson GL.M. Illustrate part (b) for a cell count in a two-way contingency table for the model of
In Problem 4.30, when u is also a parameter, show that it satisfies the exponential dispersion family (4.14).Data from Prob. 4.30:Show the normal distribution N(µ, σ2) with fixed σ satisfies
Show representation (4.18) for the binomial distribution.
For known k, show that the negative binomial distribution (4.12) has exponential family form (4.1) with natural parameter log[µ/(µ + k)].
Show that the sample value of the uncertainty coefficient (2.13) satisfies Û = –G2 / 2n(∑p+j). [Haberman (1982) gave its standard error.]
Explain why {n+j} are sufficient for {π+j} in (3.17).
Motivate partitioning (3.14) by showing that the multiple hypergeometric distribution (3.19) for {nij}factors as the product of hypergeometric distributions for the separate component tables
For a 2 × 2 table, show that:a. The four Pearson residuals may take different values.b. All four standardized Pearson residuals have the same absolute value. (This is sensible, since df = 1.)c. The
Refer to Problem 2.23. For multinomial sampling, show how to obtain a confidence interval for AR by first finding one for log(1 – AR) (Fleiss 1981, p. 76). Data from Prob. 2.23: For two
Refer to Problem 15.14. For N = 2 groups with n1 and n2 independent observations, find the minimum modified chi-squared estimator of π. Compare it to the ML estimator.Data from Problem 15.14:Let yi
Let yi be a bin(ni=, πi) variate for group i, i = 1,..., N, with {yi} independent. Consider the model that π1 = ... = πN. Denote that common value by π.a. Show that the ML estimator of π is p =
Consider the Bayes estimator of a binomial parameter π using a beta prior distribution.a. Does any beta prior distribution produce a Bayes estimator that coincides with the ML estimator?b. Show that
The response functions F(p) have asymptotic covariance matrix VF. Derive the asymptotic covariance matrix of the WLS model parameter estimator b and the predicted values F̂ = Xb.
With WLS, show that [F(p) – Xβ]’V̂F–1[F(p) – Xβ] is minimized by β = (X’ V̂F–1 X)–1 X’ V̂F–1 F(p).
For WLS with F(π) = C[log(Aπ)], show that Q = C[diag(Aπ)]–1A.
14.27 Let \(\zeta\) denote a generic measure of association. For \(K\) independent multinomial samples of sizes \(\left\{n_{k}\right\}\), suppose that
Consider the ML estimator π̂ij = pi+ p+j of πij for the independence model, when that model does not hold. Show that E(pi+ p+j) = πi+ π+j(n–1)/n + πij/n. To what does π̂ij converge as n
For a given set of parameter constraints, show that weak identifiability conditions hold for the independence loglinear model for a two-way table; that is, when two values for θ give the same π,
Justify the use of estimated asymptotic covariance matrices. For instance, for large samples, why is Â’Â close to A’A?
For loglinear model (XY, XZ, YZ), ML estimates of {?ijk} and hence the X2 and G2 statistics are not direct. Alternative approaches may yield direct analyses. For 2 ? 2 ? 2 tables, find a statistic
In an I × J contingency table, let θij denote local odds ratio (2.10), and let θ̂ij denote its sample value.a. show that asymp. cov(√n log θ̂ij, √n log θ̂i+1, j) = –[π–1i+1, j +
Refer to Problem 3.27. The sample size may need to be quite large for the sampling distribution of γ̂ to be approximately normal, especially if |γ| is large. The Fisher-type transform ξ̂ = 1/2
Assuming two independent binomial samples, derive the asymptotic standard error of the log relative risk.
An animal population has N species, with population proportion ?i of species i. Simpson?s index of ecological diversity (Simpson 1949) is I(?) = 1 ? ??i2. [Rao (1982) surveyed diversity measures.] a.
Suppose that Tn has a Poisson distribution with mean λ = nµ, for fixed µ > 0. For large n, show that the distribution of log Tn is approximately normal with mean log(λ) and variance λ–1.
a. Use Tchebychev’s inequality to show that if E(Xn) = µn and var(Xn) = σ2n < ∞, then (Xn – µn) = Op(σn).b. Suppose that Y1,..., Yn are independent with E(Yi) = µ and var(Yi) = σ2 for
If X2 has an asymptotic chi-squared distribution with fixed df as n → ∞, then explain why X2/n = op(1).
Explain why:a. If c > 0, n–c = o(1) as n → ∞.b. If c ≠ 0, czn has the same order as zn; that is, o(czn) is equivalent to o(zn) and O(czn) is equivalent to O(zn).c. o(yn) o(zn) = o(yn zn),
Consider the loglinear random effects model log[E(Yit | ui)] = x?it ? + z?it ui, where {?i} are independent N(0, ?). Show that this implies the marginal loglinear model with the same fixed effects
The negative binomial distribution is unimodal with a mode at the integer part of µ(k – 1)/k. Show that the mode is 0 when µ ≤ 1, and that when µ > 1 the mode is still 0 if k < µ/(µ
Suppose that given u, Y is Poisson with E(Y | u) = u µ, where µ may depend on predictors. Suppose that u is a positive random variable with E(u) = 1 and var(u) = τ. Show that E(Y) = µ and var(Y)
Altham (1978) introduced the discrete distribution where c(?, ?) is a normalizing constant. Show that this is in the exponential family. Show that the binomial occurs when ? = 0. [Altham noted that
Consider the logistic-normal model, logit(πi) = α + x’i β + ui. For small σ, show that it corresponds approximately to a mixture model for which the mixture distribution has var(πi) = [µi(1
Express the numerator of the beta density in terms of µ and θ. Using this, show that it is (a unimodal when θ < min(µ, 1 – µ), and (b) the uniform density when µ = θ = ½.)
Refer to Problem 12.7. Let μk(a, b, c) denote the expected frequency of outcomes (a, b, c) for treatments (A, B, C) under treatment sequence k, where outcome 1 = relief and 0 = nonrelief. With a
Let ? denote an I ? J matrix of cell probabilities for the joint distribution of X and Y. Suppose that there exist I ? 1 column vectors ?1k and J ? 1 column vectors ?2k of probabilities, k = 1,...,q,
Refer to Problem 13.14. Using an appropriate subset of width, weight, color, and spine condition as predictors, find and interpret a reasonable model for predicting the number of satellites. Data
Conduct a latent class analysis of the data in Espeland and Handel man (1989).
With data at the book’s Web site (www.stat.ufl.edu/∼aa/cda/cda.html), use methods of this chapter to analyze how the countywide vote for the Reform Party candidate Pat Buchanan in the 2000
Conduct the analyses of Problem 4.6 on defects in the fabrication of computer chips, but use a negative binomial GLM. Compare results to those for the Poisson GLM. Indicate why results are
Use a quasi-likelihood approach to analyze Table 13.6 on counts of murder victims. Table 13.6: Number of Victims of Murder Known in Past Year, by Race, with Fit of Poisson and Negative Binomial
Table 13.8 reports the results of a study of fish hatching under three environments. Eggs from seven clutches were randomly assigned to three treatments, and the response was whether an egg hatched
For the toxicity study of Table 12.9, collapsing to a binary response, consider linear logit models for the probability a fetus is normal. a. Does the ordinary binomial model show evidence of
Analyze Table 8.19 on government spending using latent class models. Table 8.19: Cities 1 2 3 Law Environment Health Enforcement: 3 3 62 17 90 42 3 74 31 11 22 18 19 14 3 11 1 3 3 1 3 3 21 13 2 20
Analyze Table 8.3 using a latent class model with q = 2. a. For a subject in the first latent class, estimate the probability of having used (i) marijuana, (ii) alcohol, (iii) cigarettes, (iv) all
Analyze Table 11.9 with age and maternal smoking as predictors using a (a) logistic-normal model, (b) marginal model, and (c) transitional model. Explain how the interpretation of the maternal
Refer to Problem 11.12 for Table 8.19 on government spending. Analyze these data using a cumulative logit model with random effects. Interpret. Compare results to those with a marginal model (Problem
For the student survey in Table 9.1, (a) analyze using GLMMs, and (b) compare results and interpretations to those with marginal models in Problem 11.2. Data from Problem 11.2: Refer to Table 9.1.

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