Advanced Linear Modeling 3rd Edition Ronald Christensen - Solutions

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Show that for a 3×3 table with n11 = n13, n31 = n33, and n·1 = n·3 that the ˆm’s for the independence model are also the ˆm’s for the uniform association model.
Waite (1911) reports data on classifications of general intelligence made for students from a secondary school in London. Classifications were made after two different school terms and each student was classified by two instructors. Classifications were based on Pearson’s criteria as explained in
Use the saturated model and Theorem 10.2.1 to find the large sample distribution of a multinomial sample in terms of its category probabilities.
The Delta Method.Let vN be a sequence of q × 1 random vectors and suppose that√N(vN − θ) L→ N0, Σ(θ).Suppose that F(·) is a differentiable function taking q vectors into r vectors.Let dF be the r × q matrix of partial derivatives of F. Then√NF(vN ) − F(θ)L→ N0, dF Σ(θ)
Use the delta method of the previous exercise to show that if√N(vN − θ) L→ N0, Σ(θ), then for any r × q matrix A,√N(AvN − Aθ) L→ N0, AΣ(θ)A.Show that if Cov(√NvN ) = Σ(θ), then Cov(√NAvN ) = AΣ(θ)A.
Testing Marginal Homogeneity in a Square Table.For an I × I table, the hypothesis of marginal homogeneity is H0 : pi· = p·i , i = 1,...,I. A natural statistic for testing this hypothesis is the vector d = (n1· − n·1,...,nI· − n·I ) Clearly, E(di) = pi· − p·i .Use the results of
Analyze the data of Exercise 8.4.1 as a logistic regression with nodal involvement as the response. Include the investigation of higher-order interactions in your analysis. The original investigator was particularly interested in whether acid was a valuable predictor of nodal involvement.
Asymptotic Inference for the LD(50).In Exercise 4.8.9, models and methods for estimating the LD(50) were discussed. Use the delta method of Exercise 10.8.4 to obtain an asymptotic standard error for the LD(50). Using the data of Exercise 4.8.11, give a 99% confidence interval for the LD(50).
Fieller’s Method for the LD(50).Fieller’s method is an alternative to the delta method for obtaining an asymptotic confidence interval for the LD(50), cf. Exercise 11.8.2.Fieller’s method is thought to be less sensitive to the high correlation that is typically present between ˆα and βˆ.
Show that the conditional likelihood given by equation(11.7.1) and display (11.7.2) does not depend on the αi’s or the intercept terms γh1. Here, xi = (1, xi2,...,xik) and γ = (γh1,...,γhk).
Use the delta method of Exercise 10.8.4, the logistic transform, and (11.1.3) to show thatpˆij − pij pˆij (1 − pˆij )ˆaii/Ni∼ N(0, 1), where Ni = ni1 + ni2.
In a Newsweek article on “The Wisdom of Animals”(May 23, 1988), one of the key issues considered was whether animals (other than humans) understand relationships between symbols. Some animals can associate symbols with objects; the question is whether they can tell the difference between
Consider a 2×2 table of multinomial probabilities that models how subjects respond on two separate occasions.First Trial Second Trial A B A p11 p12 B p21 p22 Show that Pr(A Second Trial |B First Trial ) = Pr(B Second Trial |A First Trial )if and only if the event that a change occurs between the
Weisberg (1975) reports the following data on the number of boys among the first seven children born to a collection of 1,334 Swedish ministers.Number of Boys 0 1 2 3 4 5 6 7 Frequency 6 57 206 362 365 256 69 13 Assume that the number of boys has a Bin(7, .5) distribution. Compute the probabilities
The data given in the previous problem may be 1,334 independent observations from a Bin(7, p) distribution. If so, use the defining assumptions of the binomial distribution to show that this is the same as one observation from a Bin(1334 × 7, p) distribution. Estimate p with pˆ = Total number of
Let y1, y2, y3, and y4 be random variables and let a1, a2, a3, and a4 be real numbers. Show that the following relationships hold for finite discrete distributions.(a) E(a1y1 + a2y2 + a3) = a1E(y1) + a2E(y2) + a3.(b) Var(a1y1 + a2y2 + a3) = a2 1Var(y1) + a2 2Var(y2) for y1 and y2 independent.(c)
Assume that(n1,...,nq) ∼ Mult(N, p1,...,pq)and let t be an integer less than q. Define y = n1 + ··· + nt and ˜p =p1 + ··· + pt. Show that(y, nt+1,...,nq) ∼ Mult(N, p, p ˜ t+1,...,pq)so that E(y) = Np˜and Var(y) = Np˜(1 − p˜).
Suppose y ∼ Bin(N, p). Let ˆp = y/N. Show that E(ˆp) =p and that Var(ˆp) = p(1 − p)/N.
The data in Table 2.2 are on graduate admissions by sex at the University of California, Berkeley, and are given by Bickel et al. (1975) and Freedman et al. (1978). Test for independence, examine the
Cram´er (1946) presents data on the distribution of birth dates for males and females born in Sweden in 1935. The data given in Table 2.3 presume a natural ordering for the months of the year that Cram´er does not specify. Analyze the data. Is it better to think of this as one multinomial sample
Gilby (1911) presents data on the relationships among instructor’s evaluation of general intelligence, quality of clothing, and school standard. General intelligence was classified using a system of Karl Pearson’s that was reported in Waite (1911). Briefly, the Intelligence classifications are
Partitioning Tables.The examination of odds ratios and residuals provide two ways to investigate lack of independence in a two-way table. The partitioning methods of Irwin (1949) and Lancaster (1949) provide another. Christensen (1996a, Section 8.6) gives extensive examples of the application of
Fisher’s Exact Test.Consider the problem of testing whether the probability of success is the same for two independent binomials. Let yi ∼ Bin(Ni, pi), i = 1, 2. Write the 2 × 2 table as TABLE 2.8. Partitioned Tables Reduced Table Religion A B C D E F G H White Baptist 43 78 64 135 135 57 86
Yule’s Q.For 2×2 tables, a measure of association similar to a correlation coefficient is Yule’s Q, which is defined as Q = p11p22 − p12p21 p11p22 + p12p21.Find Q in terms of the odds ratio. Show that Q lies between −1 and 1.
Freeman-Tukey Residuals.Freeman and Tukey (1950) suggest a variance stabilizing transformation for Poisson data that leads to using the quantities√nij + nij + 1 −4 ˆm(0)ij + 1 as residuals, cf. Bishop, Fienberg, and Holland (1975, Section 4.4). Reexamine the data of Example 2.3.1 using the
Power Divergence Statistics.Cressie and Read (1984) and Read and Cressie (1988) have introduced the power divergence family of test statistics 2Iλ = 2λ(λ + 1)ij nij⎡⎣nij mˆ (0)ij λ− 1⎤⎦, where for λ = −1, 0 the statistics are defined by taking limits. They establish that for any
Compute the power divergence test statistics 2I−1/2 and 2I1/2 for the knee injury data of Example 2.3.1. Compare the results to G2 and X2. What conclusions can be reached about knee injuries?
Testing for Symmetry.Consider a multinomial sample arranged in an I ×I table. In square tables with similar categories for the two factors, it is sometimes of interest to test H0 : pij = pji for all i and j.(a) Give a procedure for testing this hypothesis based on testing equality of probabilities
Correlated Data.There are actually 410 observations in Exercise 2.7.10 and Table 2.9. There are 205 men and 205 women. Why was Table 2.9 set up as a 3 × 3 table with only 205 observations rather than as Table 2.10, a 2 × 3, sex versus height table with 410 observations?TABLE 2.10. Heights of
McNemar’s Test.McNemar (1947) proposes a method of testing for homogeneity of proportions among two binary populations when the data are correlated. (A binary population is one in which all members fall into one of two categories. Homogeneity means that the proportions in each category are the
Suppose the random variables nij , i = 1, 2, j =1,...,Ni, are independent Poisson(µi) random variables. Find the maximum likelihood estimates for µ1 and µ2 and find the generalized likelihood ratio test statistic for H0 : µ1 = µ2.
Yule’s Q (cf. Exercise 2.7.6.) is one of many measures of association that have been proposed for 2 × 2 tables. Agresti (1984, Chapter 9) has a substantial discussion of measures of association. It has been suggested that measures of association for 2 × 2 multinomial tables should depend solely
Complete an analysis similar to that of Example 3.4.2 for the classroom behavior data of Example 3.0.1.
Complete an analysis similar to that of Example 3.4.2 for the auto accident data of Example 3.2.4.
Radelet (1981) gives data on the relationship between race and the imposition of the death penalty. The data are given in Table 3.2. Analyze the data.TABLE 3.2. Race and the Death Penalty Defendant’s Victim’s Death Penalty Race Race Yes No Black Black 6 97 White 11 52 White Black 0 9 White 19
The data on graduate admissions at Berkeley given in Exercise 2.6.1 was actually collapsed over the six largest departments within the university. The possibility exists that the data may display Simpson’s paradox. The full data are given in Table 3.3. Analyze the three-dimensional table and
Discuss Simpson’s paradox in terms of the following probability inequalities.Pr(A|B and C) < Pr(A| not B and C),TABLE 3.3. Graduate Admissions at Berkeley Male Female Dept. Admitted Rejected Admitted Rejected A 512 313 89 19 B 353 207 17 8 C 120 205 202 391 D 138 279 131 244 E 53 138 94 299 F 22
Reevaluate your analysis of the data discussed in Exercise 2.6.3 in light of Simpson’s paradox. Are there other factors that need to be accounted for in a correct analysis of these data?
For the data of Example 3.2.4, do the first step of the iterative proportional fitting algorithm for ˆm(7)ijk using a hand calculator.Use starting values of ˆm[0]ijk = 1. Compare the results after one step to the fully iterated estimates.
Consider the model log(mijk) = u + u1(i) + u2(j) +u3(k) + u12(ij).(a) Show that the maximum likelihood estimate of u3(1) − u3(2) is log(n··1) − log(n··2).(b) Show that maximum likelihood estimation gives loguˆ12(11)uˆ12(22)uˆ12(12)uˆ12(21) = log(n11·) − log(n12·) − log(n21·) +
The Mantel-Haenszel Statistic.In biological and medical applications, it is not uncommon to be confronted with a series of 2 × 2 tables that examine the same effect under different conditions. If there are K such tables, the data can be combined to form a 2 × 2 × K table. Because each 2 × 2
Using the data of Table 3.1 fit the all main effects model, the all two-factor effects model, and the all three-factor effects model. Perform all of the tests possible among these three models. Discuss your results.
With regard to Section 3, show that the ˆm[3t+2]ijk ’s and mˆ [3(t+1)]ijk ’s also satisfy M(7).
As can be seen from the iterative proportional fitting algorithm, the ˆm’s for the model of no three-factor interaction depend only on the 3 two-dimensional marginal tables. Discuss how this fact can be used to develop a more complete analysis for the Gilby data of Exercise 2.7.3.What
The auto accident data of Example 3.2.4 was actually a subset of a four-dimensional table. The complete data are given in Table 4.15. Analyze the data treating severity of injury as a response variable.What conclusions can you reach from examining the ˆmhijk’s, the odds, and odds ratios?TABLE
Breslow and Day (1980) present data on the occurrence of esophageal cancer in Frenchmen. Explanatory factors are age and alcohol consumption. High consumption was taken to be anything over the equivalent of one liter of wine per day. The data are given in Table 4.16. Analyze the data as a logit
The data in the previous experiment is a series of 2×2 tables collected under five different age conditions. This is the same situation as the Mantel-Haenszel setup of Exercise 3.8.9.The Mantel-Haenszel test is one of conditional independence given age. It assumes that the model of no three-factor
Haberman (1978) reports data from the National Opinion Research Center on attitudes toward abortion (cf. Table 4.17). The data TABLE 4.16. Occurrence of Esophageal Cancer Alcohol Cancer Age Consumption Yes No 25-34 High 1 9 Low 0 106 35-44 High 4 26 Low 5 164 45-54 High 25 29 Low 21 138 55-64 High
Feigl and Zelen (1965), Cook and Weisberg (1982), and Johnson (1985) give data on survival of 33 leukemia patients as a function of their white blood cell count and the existence of a certain morphological characteristic in the cells. The characteristic is referred to as either AG positive or AG
Finney (1941) and Pregibon (1981) present data on the occurrence of vasoconstriction in the skin of the fingers as a function of the rate and volume of air breathed. The data are reproduced in Table 4.19.A constriction value of 1 indicates that constriction occurred. Analyze the data.
Mosteller and Tukey (1977) reported data on verbal test scores for sixth graders. They used a sample of 20 Mid-Atlantic and New England schools taken from The Coleman Report. The dependent variable y was the mean verbal test score for each school. The predictor variables were x1 — staff salaries
The Logistic Distribution.Show that F(x) = ex/(1 + ex) satisfies the properties of a cumulative distribution function (cdf). Any random variable with this cdf is said to have a logistic distribution.
Stimulus–Response Studies.The effects of a drug or other stimulus are often studied by choosing r doses of the drug (levels of the stimulus), say x1,...,xr, and giving the dose xj to each of Nj subjects. The data consist of the number yj who exhibit some predetermined response. Often this
Probit Analysis.An alternative to the logistic analysis of dose-response data is probit analysis. In probit analysis, the model isΦ−1(pj ) = α + β log(xj )where Φ(·) is the cumulative distribution function of a standard normal distribution. Graph Φ(·) and the logistic cdf and compare the
Woodward et al. (1941) report several data sets, one of which examines the relationship between exposure to chloracetic acid and the death of mice. Ten mice were exposed at each dose level. The data are given in Table 4.21. Doses are measured in grams per kilogram of body weight. Fit the logistic
Consider a sample of j = 1,...,r independent binomials yj ∼ Bin(Nj , pj ), each with a covariate xj . Suppose that for some cumulative distribution function F(·), pj = F(xj ).Show that for some transformation zj = g(xj ) and parameters α and β, this logistic regression model holds:log pj 1
The data of Exercise 4.8.2 are actually a retrospective study. A sample of cancer patients was compared to a sample of men drawn from the electoral lists of the department of Ille-et-Vilaine in Brittany.Reanalyze the data in light of this knowledge.
Any multinomial response model can be viewed as the model for an I × J table. Assume product-multinomial sampling from J independent multinomials each with I categories. Defineπij = pij Ih=i phj so that the continuation ratios introduced in Section 4.2 areπij 1 − πij= pijI h=i+1 phj.Let rij
Give the log-linear model corresponding to (4.1.1).
Analyze the trauma data that are described in Example 13.2.2.
Using the methods of Section 5.1, discuss the independence relationships for all of the models given below.(a) [123][24][456](b) [12][13][23][24][456](c) [123][124][456](d) [123][24][456][15](e) [123][24][456][15][36]
In the saturated log-linear model for a four-dimensional table, let u34 = 0 and let all of the corresponding higher-order terms also be zero, e.g., u134 = 0.(a) Based on this model, find a formula for ˆmhijk without using graphical methods.(b) Use graphical methods to find ˆmhijk.
The vertices for a five-factor model are given below. Connect the dots to give a graphical representation of the model[123][135][34][24]. Use the illustration to show that [123][135][34][24][25] is not a graphical model.
Which of the models given below are graphical? Graph them. Which of these are decomposable? Discuss the independence relationships for all of the models. For each model, what marginal tables will provide valid inferences?(a) [123][24][456](b) [12][13][23][24][456](c) [123][124][456](d)
Consider all of the graphs in Example 5.4.2. Classify each as equivalent or not equivalent to a decomposable log-linear model. For those that are equivalent, prove the equivalence of the probability models.
Exercise 1.13.1. Rescale x3 to make its values lie between 0 and 1. Plot the data.Using least squares, fit models with p = 10 using polynomials and cosines. Plot the regression lines along with the data. Which family works better on these data, cosines or polynomials?
Exercise 1.13.2. Using s = 8, fit the Coleman Report data using Haar wavelets.How well does this do compared to the cosine and polynomial fits?
Exercise 1.13.3. Based on the s = 10 polynomial and cosine models fitted in
Exercise 1.13.1, useCp to determine a best submodel for each fit. Plot the regression lines for the best submodels. Which family works better on these data, cosines or polynomials? Use Cp to determine the highest frequency needed when fitting sines and cosines.
Exercise 1.13.4. Investigate whether there is a need to consider heteroscedastic variances with the Coleman Report data. If appropriate, refit the data.
Exercise 1.13.5. Fit a cubic spline nonparametric regression to the Coleman Report data.
Exercise 1.13.6. Fit a regression tree to the Coleman Report data using just variable x4.
Exercise 1.13.7. In Sect. 1.7 we set up the interpolating cubic spline problem as one of fitting a saturated linear model that is forced be continuous, have continuous first and second derivatives, and have 0 as the second derivative on the boundaries.In our discussion, the dots are only connected
Exercise 1.13.8. Fit a tree model to the battery data and compare the results to fitting Haar wavelets.
Exercise 6.9.1. Show that the following functions are nonnegative definite.(a)σ (k) =⎧⎪⎨⎪⎩1 ifk = 0 14 27 if k = ±1 427 if k = ±2 0 other k .(b)σ (k) =⎧⎪⎨⎪⎩1 ifk = 0ρ if k = ±1ρ2 if k = ±2 0 other k .The function σ (k) is nonnegative definite if, for any n, the n×n
Exercise 6.9.4. Consider the simple linear regression model yt = α +β t +et , where et is a white noise process. Let wt be the symmetric moving average of order 5 introduced in Example 6.6.1 as applied to the yt process. Find E(wt ) and Cov(wt ,wt+k). Is the wt process stationary? Why or why not?
Exercise 6.9.5. Shumway (1988) reports data from Waldmeier (1960-1978, 1961) on the number of sunspots from 1748 to 1978. The data are collected monthly, and a symmetric moving average of length 12 has been applied. The data in Table 6.7 are the moving averages corresponding to June and December
Exercise 6.9.6. Box, Jenkins, and Reinsel (1994, p. 597) report data on international air travel. The values in Table 6.8 are the number of passengers in thousands.Do a frequency analysis of these data including appropriate transformations of the data, confidence intervals, and plots.
Exercise 6.9.7. Let et be a white noise process. The spectral density of et is fe(ν) =σ 2; see Exercise 6.2. Let |φ1| < 1 and |θ1| < 1.(a) Find the spectral density of yt = et −θ1et−1 .Sketch the graph of the spectral density for θ1 = .5 and θ1 = 1. Take σ 2 = 1. Which frequencies are
Exercise 6.9.8. Let et and εt be uncorrelated white noise processes. Let yt =φ1yt−1+et and wt =φ1wt−1+yt +εt .Find the spectral density of wt .
Exercise 6.9.9. Let et be second-order stationary and define yt =∞Σs=−∞aset−s and wt =∞Σr=−∞bryt−r, where Σ∞s=−∞ |as| < ∞ and Σ∞r =−∞ |br| < ∞.(a) Show that fw(ν) = |A(ν )|2|B(ν )|2 fe(ν ).(b) Show that wt =∞Σs=−∞cset−s, where cs
Exercise 6.9.10. Let yt and wt be two second-order stationary processes and suppose that fw(ν ) ≤ fy(ν ) for all ν ∈ [−π,π]. Show thatΣyy−Σww is nonnegative definite, where Σyy and Σww are the covariance matrices of(y1, · · · , yn) and (w1, · · · ,wn), respectively.
Exercise 6.9.11. Show that the spectral density f (ν) =π −|ν |π2 determines the covariance functionσ (k) =σ 2 if k = 0 4σ 2/(π|k|)2 if k is odd 0 otherwise.
Exercise 6.9.12. If yt and wt are independent second-order stationary processes with spectral densities fy and fw, find the spectral distribution of the stationary process zt = yt +wt .
Exercise 6.9.13. Suppose yt is second-order stationary and fy(ν ) is nonnegative, bounded, and fy(1/2) = 0. Let wt = yt −yt−1.Find fw(ν ) in terms of fy(ν ).
Exercise 6.9.14. For ν ∈ [0,π], define fy(ν) =50 ν ∈ [14−.01, 14+.01]0 otherwise and fy(−ν) = fy(ν ). Find σ (0), σ (1), σ (2).
Exercise 6.9.15.(a) Use the relationships sin(a+b) = sin(a) cos(b)+cos(a) sin(b)and References 245 cos(a+b) = cos(a) cos(b)−sin(a) sin(b)to show the following.cos(a) cos(b) = 1 2 {cos(a+b)+cos(a−b)}.cos(a) sin(b) = 1 2 {sin(a+b)−sin(a−b)}.sin(a) sin(b) = 1 2 {cos(a−b)−cos(a+b)}.(b)
Exercise 6.9.16. Apply the relationship in Exercise 6.9.15a on the sine of a sum to the process in Exercise 6.9.2 to show that the process can be rewritten as yt = Asin(2πνt +φ ), where A is the random amplitude and φ is the random phase of the sine curve.Find the relationship between (A,φ )
Exercise 7.9.1. Which of the following processes are stationary? Which are invertible?(a) (1−1.5B+0.54B2)yt = (1−0.5B)et .(b) (1− 58 B)yt = (1+0.1B−0.56B2)et .(c) (1−1.6B+0.55B2)yt = (1−0.6B)et .(d) (1−B+0.2475B2)yt = (1−B− 7 36B2)et .(e) (1−0.7B−0.66B2+0.432B3)yt =
Exercise 7.9.2. Consider the AR(1) model yt = 10−0.7yt−1+et , σ 2 = 3.(a) Find μ, σ (0), σ (1), σ (2), and σ (3).(b) If y4 = 12, will y5 tend to be less than or greater than μ?(c) Is the process stationary?(d) Give ˆE (y4+k|Y) and the prediction variance for k = 1,2,3 and Y
Exercise 7.9.3. Consider the MA(1) model yt = 10+et −0.4et−1, σ 2 = 3.(a) Find μ, σ (0), σ (1), σ (2), and σ (3).(b) If y4 = 12, will y5 tend to be less than or greater than μ?(c) Is the process stationary?(d) Give ˆE(y4+k|Y∞) and the prediction variance for k = 1,2,3 and Y =
Exercise 7.9.4. Consider the ARMA(1,1) model yt = 10−0.7yt−1+et −0.4et−1, σ 2 = 3(a) Find μ, σ (0), σ (1), σ (2), and σ (3).(b) If y4 = 12, will y5 tend to be less than or greater than μ?(c) Is the process stationary?(d) Give ˆE(y4+k|Y∞) and the prediction variance for k = 1,2,3
Exercise 7.9.5. Do a time domain analysis of the sunspot data in Exercise 6.9.5.This should include estimation, model fitting, and checking of assumptions.Exercise 7.9.6. Use a multiplicative seasonal model to analyze the international air passenger data of Exercise 6.9.6. This should include
Exercise 7.9.7. For an MA(1) process, find an estimate of θ1 in terms of ρˆ (1).Are any restrictions on ρˆ (1) needed?
Exercise 7.9.8. For an AR(2) process, use the Yule–Walker equations, with σˆy(·)replacing σy(·) to obtain estimatesφˆ1 =ρˆ (1) (1−ρˆ (1))1−ρˆ (1)2 andφˆ2 =ρˆ (2)−ρˆ (1)2 1−ρˆ (1)2 .
Exercise 7.9.9. Show that the variance for predicting k steps ahead in an AR(1)process isσ 2 1−φ 2k 11−φ 2 1.
Exercise 7.9.10. Show, for an MA(1) process with parameter θ1, thatˆE(yn+1|Y∞) = −∞Σs=0θ s 1yn−s
Exercise 7.9.11. Let et be a second-order stationary process. Show that yt = et −θ1et−1 and wt = et − 1θ1 et−1 have the same correlation function.
Exercise 7.9.12. What are the largest possible values of |ρ(1)| and |ρ(2)| for an MA(2) process?
Exercise 7.9.13. Find the ψis, μ, and σ (k) for the following processes.(a) The AR(3) process(1−0.9B)(1+0.8B)(1−0.6B)yt = et .(b) The ARMA(2,1) process(1−0.9B)(1−0.6B)yt = (1−0.5)et .

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Advanced Linear Modeling 3rd Edition Ronald Christensen - Solutions

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