Foundations Of Linear And Generalized Linear Models 1st Edition Alan Agresti - Solutions

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11.2 Suppose that (Yjk,xjk) are observations on the kth subject in cluster k (with j = 1,...,J;k = 1,...,K) and the goal is to fit a “regression through the origin” model E(Yjk) = βxjk,
11.1 The measurement of left ventricular volume of the heart is important for studies of cardiac physiology and clinical management of patients with heart disease. An indirect way of measuring the volume, y, involves a mea-surement called parallel conductance volume, x. Boltwood et al. (1989)found
10.6 The data in Table 10.5 are survival times, in months, of 44 patients with chronic active hepatitis. They participated in a randomized controlled trial of prednisolone compared with no treatment. There were 22 patients in each group. One patient was lost to follow-up and several in each group
10.5 As the survivor function S(y) is the probability of surviving beyond time y, the odds of survival past time y are O(y) = S(y)1−S(y).For proportional odds models the explanatory variables for subject i, ηi, act multiplicatively on the odds of survival beyond time y
10.4 For proportional hazards models the explanatory variables for subject i, ηi, act multiplicatively on the hazard function. If ηi = e xT iβ, then the hazard function for subject i is hi(y) = e xT iβh0(
10.3 For accelerated failure time models the explanatory variables for subject
10.2 The log-logistic distribution with the probability density function f(y) = eθ λyλ−1(1+eθ yλ )2 is sometimes used for modelling survival times.(a) Find the survivor function S(y), the hazard function h(y) and the cumulative hazard function H(y).(b) Show that the median survival time is
10.1 The data in Table 10.4 are survival times, in weeks, for leukemia patients.There is no censoring. There are two covariates, white blood cell count(WBC) and the results of a test (AG positive and AG negative). The data set is from Feigl and Zelen (1965) and the data for the 17 patients with AG
9.7 Mittlbock and Heinzl (2001) compare Poisson and logistic regression models for data in which the event rate is small so that the Poisson distribution provides a reasonable approximation to the Binomial distribution. An example is the number of deaths from coronary heart disease among British
9.6 Consider a 2×K contingency table (Table 9.14) in which the column totals y.k are fixed for k = 1,...,K.
9.5 Use log-linear models to examine the housing satisfaction data in Table 8.5. The numbers of people surveyed in each type of housing can be regarded as fixed.a. First, analyze the associations between level of satisfaction (treated as a nominal categorical variable) and contact with other
9.4 For a 2×2 contingency table, the maximal log-linear model can be written asη11 = µ +α +β + (αβ), η12 = µ +α −β −(αβ),η21 = µ −α +β −(αβ), η22 = µ −α −β + (αβ), where ηjk = logE(Yjk) = log(nθ jk) and n = ∑∑Yjk.Show that the interaction term (αβ) is
9.3 This question relates to the flu vaccine trial data in Table 9.6.a. Using a conventional chi-squared test and an appropriate log-linear model, test the hypothesis that the distribution of responses is the same for the placebo and vaccine groups.b. For the model corresponding to the hypothesis
9.2 The data in Table 9.13 are numbers of insurance policies, n, and numbers of claims, y, for cars in various insurance categories, CAR, tabulated by age of policy holder, AGE, and district where the policy holder lived (DIST =1, for London and other major cities, and DIST = 0, otherwise). The
9.1 Let Y1,...,YN be independent random variables with Yi ∼ Po(µi) and log µi = β1 +∑J j=2 xi jβj, i = 1,...,N.a. Show that the score statistic for β1 is U1 = ∑N i=1(Yi − µi).b. Hence, show that for maximum likelihood estimates µbi, ∑µbi = ∑yi.c. Deduce that the expression for
8.4 Consider ordinal response categories which can be interpreted in terms of continuous latent variable as shown in Figure 8.2. Suppose the distribution of this underlying variable is Normal. Show that the probit is the natural link function in this situation (Hint: See Section 7.3).
8.3 The data in Table 8.6 show tumor responses of male and female patients receiving treatment for small-cell lung cancer. There were two treatment regimes. For the sequential treatment, the same combination of chemotherapeutic agents was administered at each treatment cycle. For the alternating
8.2 The data in Table 8.5 are from an investigation into satisfaction with housing conditions in Copenhagen (derived from Example W in Cox and Snell, 1981, from original data from Madsen, 1971). Residents in selected areas living in rented homes built between 1960 and 1968 were questioned about
8.1 If there are only J = 2 response categories, show that models (8.4), (8.13),(8.15) and (8.16) all reduce to the logistic regression model for binary data.
7.5 Let Yi be the number of successes in ni trials with Yi ∼ Bin(ni,πi), where the probabilities πi have a Beta distributionπi ∼ Be(α,β).The probability density function for the Beta distribution is f(x;α,β) =xα−1(1 − x)(β−1)/B(α,β) for x in [0, 1], α > 0,β > 0 and the beta
7.4 Let l(bmin) denote the maximum value of the log-likelihood function for the minimal model with linear predictor x T β = β1, and let l(b) be the corresponding value for a more general model x T β = β1 + β2x1 + ... +βpxp−1.a. Show that the likelihood ratio chi-squared statistic is C =
7.3 Tables 7.16 and 7.17 show the survival 50 years after graduation of men and women who graduated each year from 1938 to 1947 from various faculties of the University of Adelaide (data compiled by J.A. Keats). The columns labelled S contain the number of graduates who survived and the columns
7.2 Odds ratios. Consider a 2×2 contingency table from a prospective study in which people who were or were not exposed to some pollutant are followed up and, after several years, categorized according to the presence or absence of a disease. Table 7.15 shows the probabilities for each cell. The
7.1 The number of deaths from leukemia and other cancers among survivors of the Hiroshima atom bomb are shown in Table 7.14, classified by the radiation dose received. The data refer to deaths during the period 1950–1959 among survivors who were aged 25 to 64 years in 1950 (from data set 13 of
6.9 Examine if there is a non-linear association between age and cholesterol using the fractional polynomial approach for the data in Table 6.24.
6.8 Table 6.27 shows the data from a fictitious two-factor experiment.a. Test the hypothesis that there are no interaction effects.b. Test the hypothesis that there is no effect due to Factor A(i) by comparing the models E(Yjkl) = µ +αj +βk and E(Yjkl) = µ +βk;(ii) by comparing the models
6.7 For the balanced data in Table 6.12, the analyses in Section 6.4.2 showed that the hypothesis tests were independent. An alternative specification of the design matrix for the saturated Model (6.9) with the corner point ?
6.6 The weights (in grams) of machine components of a standard size made by four different workers on two different days are shown in Table 6.26;five components were chosen randomly from the output of each worker on each day. Perform an analysis of variance to test for differences among workers,
6.5 Table 6.25 shows plasma inorganic phosphate levels (mg/dl) one hour after a standard glucose tolerance test for obese subjects, with or without hyperinsulinemia, and controls (data from Jones, 1987).
6.4 It is well known that the concentration of cholesterol in blood serum increases with age, but it is less clear whether cholesterol level is also associated with body weight. Table 6.24 shows for thirty women serum cholesterol (millimoles per liter), age (years) and body mass index (weight
6.3 Analyze the carbohydrate data in Table 6.3 using appropriate software (or, preferably, repeat the analyses using several different regression programs and compare the results).a. Plot the responses y against each of the explanatory variables x1,x2 and x3 to see if y appears to be linearly
6.2 Table 6.23 shows response of a grass and legume pasture system to various quantities of phosphorus fertilizer (data from D. F. Sinclair; the results were reported in Sinclair and Probert, 1986). The total yield, of grass and legume together, and amount of phosphorus (K) are both given in
6.1 Table 6.22 shows the average apparent per capita consumption of sugar(in kg per year) in Australia, as refined sugar and in manufactured foods(from Australian Bureau of Statistics, 1998).Table 6.22 Australian sugar consumption.Period Refined Sugar in sugar manufactured food 1936–39 32.0 16.3
5.4 For the leukemia survival data in Exercise 4.2:a. Use the Wald statistic to obtain an approximate 95% confidence interval for the parameter β1.b. By comparing the deviances for two appropriate models, test the null hypothesis β2 = 0 against the alternative hypothesis β2 =6 0. What can you
5.3 Suppose Y1,...,YN are independent identically distributed random variables with the Pareto distribution with parameter θ.a. Find the maximum likelihood estimator θb of θ.b. Find the Wald statistic for making inferences about θ (Hint: Use the results from Exercise 3.10).c. Use the Wald
5.2 Consider a random sample Y1,...,YN with the exponential distribution f(yi;θi) = θi exp(−yiθi).Derive the deviance by comparing the maximal model with different values of θi for each Yi and the model with θi = θ for all i.
5.1 Consider the single response variable Y with Y ∼ Bin(n,π).a. Find the Wald statistic (πb − π)T I(πb − π), where πb is the maximum likelihood estimator of π and I is the information.b. Verify that the Wald statistic is the same as the score statistic U T I−1U in this case (see
4.3 Let Y1,...,YN be a random sample from the Normal distribution Yi ∼N(logβ,σ2) where σ2 is known. Find the maximum likelihood estimator of β from first principles. Also verify Equations (4.18) and (4.25) in this case.
4.2 The data in Table 4.6 are times to death, yi, in weeks from diagnosis and log10 (initial white blood cell count), xi, for seventeen patients suffering from leukemia. (This is Example U from Cox and Snell, 1981.)a. Plot yi against xi. Do the data show any trend?
4.1 The data in Table 4.5 show the numbers of cases of AIDS in Australia by date of diagnosis for successive 3-month periods from 1984 to 1988.(Data from National Centre for HIV Epidemiology and Clinical Research, 1994.)
3.12 See some more relationships between distributions in Figure 3.3.
3.11 For the Pareto distribution, find the score statistics U and the information I = var(U). Verify that E(U) = 0.
3.10 Let Y1,...,YN be independent random variables with E(Yi) = µi = β0 +log(β1 +β2xi); Yi ∼ N(µ,σ2)for all i = 1,...,N. Is this a generalized linear model? Give reasons for your answer.
3.9 Suppose Y1,...,YN are independent random variables each with the Pareto distribution and E(Yi) = (β0 +β1xi)2.Is this a generalized linear model? Give reasons for your answer.
3.8 Is the extreme value (Gumbel) distribution, with probability density function f(y;θ) = 1φexp(y−θ)φ−exp(y−θ)φ(where φ > 0 is regarded as a nuisance parameter) a member of the exponential family?
3.7 Consider N independent binary random variables Y1,...,YN with P(Yi = 1) = πi and P(Yi = 0) = 1−πi.The probability function of Yi, the Bernoulli distribution B(π), can be written asπyi i(1−πi)1−yi, where yi = 0 or 1.a. Show that this probability function belongs to the exponential
3.6 Do you consider the model suggested in Example 3.5.3 to be adequate for the data shown in Figure 3.2? Justify your answer. Use simple linear regression (with suitable transformations of the variables) to obtain a model for the change of death rates with age. How well does the model fit the
3.5a. For a Negative Binomial distribution Y ∼ NBin(r,θ), find E(Y) and var(Y ).b. Notice that for the Poisson distribution E(Y) = var(Y ), for the Binomial distribution E(Y) > var(Y) and for the Negative Binomial distribution E(Y) < var(Y ). How might these results affect your choice of a model?
3.4 Use results (3.9) and (3.12) to verify the following results:a. For Y ∼ Po(θ), E(Y) = var(Y ) = θ.b. For Y ∼ N(µ,σ2), E(Y) = µ and var(Y ) = σ2.c. For Y ∼ Bin(n,π), E(Y) = nπ and var(Y) = nπ(1−π).
3.3 Show that the following probability density functions belong to the exponential family:a. Pareto distribution f(y;θ) = θy−θ−1.b. Exponential distribution f(y;θ) = θe−yθ
3.2 If the random variable Y has the Gamma distribution with a scale parameter β, which is the parameter of interest, and a known shape parameterα, then its probability density function is f(y;β) = βαΓ(α)yα−1 e−yβ.Show that this distribution belongs to the exponential family and find
3.1 The following associations can be described by generalized linear models.For each one, identify the response variable and the explanatory variables, select a probability distribution for the response (justifying your choice)and write down the linear component.a. The effect of age, sex, height,
2.5 The model for two-factor analysis of variance with two levels of one factor, three levels of the other and no replication is E(Yjk) = µjk = µ +αj +βk; Yjk ∼ N(µjk,σ2), where j = 1,2; k = 1,2,3 and, using the sum-to-zero constraints, α1 +α2 = 0,β1 + β2 + β3 = 0. Also the Yjk’s are
2.4 Suppose you have the following data x: 1.0 1.2 1.4 1.6 1.8 2.0 y: 3.15 4.85 6.50 7.20 8.25 16.50 and you want to fit a model with E(Y) = ln(β0 +β1x+β2x 2).Write this model in the form of (2.13) specifying the vectors y and β and the matrix X.
2.3 For Model (2.7) for the data on birthweight and gestational age, using methods similar to those for Exercise 1.4, show Sb1 =J∑j=1 K∑k=1(Yjk −aj −bjxjk)2=J∑j=1 K∑k=1(Yjk −(αj +βjxjk)2 −K J∑j=1(Y j −αj −βjx j)2−J∑j=1(bj −βj)2(K∑k=1 x2 jk −Kx 2j
2.2 The weights, in kilograms, of twenty men before and after participation in a “waist loss” program are shown in Table 2.8 (Egger et al. 1999). We want to know if, on average, they retain a weight loss twelve months after the program?
2.1 Genetically similar seeds are randomly assigned to be raised in either a nutritionally enriched environment (treatment group) or standard conditions (control group) using a completely randomized experimental design.After a predetermined time all plants are harvested, dried and weighed.The
1.6 The data in Table 1.4 are the numbers of females and males in the progeny of 16 female light brown apple moths in Muswellbrook, New South Wales, Australia (from Lewis, 1987).a. Calculate the proportion of females in each of the 16 groups of progeny.b. Let Yi denote the number of females and ni
1.5 This exercise is a continuation of the example in Section 1.6.2 in which Y1,...,Yn are independent Poisson random variables with the parameter θ.a. Show that E(Yi) = θ for i = 1,...,n.b. Suppose θ = eβ. Find the maximum likelihood estimator of β.c. Minimize S = ∑Yi −eβ2 to obtain a
1.4 Let Y1,...,Yn be independent random variables each with the distribution N(µ,σ2). Let Y =1 nn∑i=1 Yi and S 2 =1 n−1 n∑i=1(Yi −Y)2.a. What is the distribution of Y?b. Show that S 2 =1 n−1∑n i=1(Yi − µ)2 −n(Y − µ)2.c. From (b) it follows that ∑(Yi−µ)2/σ2 = (n−1)S
1.3 Let the joint distribution of Y1 and Y2 be MVN(µ,V) withµ =2 3and V =4 1 1 9 .a. Obtain an expression for (y−µ)TV−1(y−µ). What is its distribution?b. Obtain an expression for y TV−1y. What is its distribution?
1.2 Let Y1 and Y2 be independent random variables with Y1 ∼ N(0,1) and Y2 ∼N(3,4).a. What is the distribution of Y 21?b. If y =Y1(Y2 −3)/2, obtain an expression for y T y. What is its distribution?c. If y =Y1 Y2and its distribution is y ∼ MVN(µ,V), obtain an expression for y TV−1y.
1.1 Let Y1 and Y2 be independent random variables with Y1 ∼ N(1,3) and Y2 ∼ N(2,5). If W1 = Y1 +2Y2 and W2 = 4Y1 −Y2, what is the joint distribution of W1 and W2?
For the anorexia study of Exercise 1.24, write a report in which you pose a research question and then summarize your analyses, including graphical description, interpretation of a model fit and its inferences, and checks of assumptions.
For the Student survey.dat data file at the text website, model how political ideology relates to number of times per week of newspaper reading and religiosity. Prepare a report, posing a research question, and then summarizing your graphical analyses, models and interpretations, inferences, checks
Refer to the study for comparing instruction methods mentioned in Exercise 2.45. Write a short report summarizing inference for the model fitted there, interpreting results and attaching edited software output as an appendix.
In a study11 at Iowa State University, a large field was partitioned into 20 equal-size plots. Each plot was planted with the same amount of seed corn, using a fixed spacing pattern between the seeds. The goal was to study how the yield of corn later harvested from the plots depended on the levels
For the house selling price data, fit the model with size of home as the sole explanatory variable. Find a 95% confidence interval for E( y) and a 95%prediction interval for y, at the sample mean size. Interpret.
Using the house selling price data at the text website, describe the predictive power of various models by finding adjusted R2 when (i) size is the sole predictor, (ii) size and new are main-effect predictors, (iii) size, new, and taxes are main-effect predictors, (iv) case (iii) with the addition
For the house selling price data of Section 3.4, when we include size, new, and taxes as explanatory variables, we obtain---------------------------------------------------------------> summary(lm(price ~ size + new + taxes))Estimate Std. Error t value Pr(>|t|)(Intercept) -21.3538 13.3115 -1.604
Section 3.4.1 used x1 = size of house and x2 = whether new to predict y = selling price. Suppose we instead use a GLM, log(????i) = ????0 + ????1 log(xi1) +????2xi2.a. For this GLM, interpret ????1 and ????2. (Hint: Adjusting for the other variable, find multiplicative effects on ????i of (i)
Refer to Exercise 2.47 on carapace width of attached male horseshoe crabs.Extend your analysis of that exercise by conducting statistical inference, and interpret.
For the horseshoe crab dataset Crabs.dat at the text website, analyze inferentially the effect of color on the mean number of satellites, treating the data as a random sample from a conceptual population of female crabs. Fit the normal one-way ANOVA model using color as a qualitative factor. Report
Refer to Exercise 1.21 on a study comparing forced expiratory volume (y =fev1 in the data file) for three drugs (x2), adjusting for a baseline measurement(x1).a. Fit the normal linear model using both x1 and x2 and their interaction.Interpret model parameter estimates.b. Test to see whether the
Suppose the relationship between y = college GPA and x = high school GPA satisfies yi ∼ N(1.80 + 0.40xi, 0.302). Simulate and construct a scatterplot for n = 1000 independent observations taken from this model when xi has a uniform distribution (a) over (2.0, 4.0), (b) over (3.5, 4.0). In each
Construct a Q–Q plot for the model for the house selling prices that uses size, new, and their interaction as the predictors, and interpret. To get a sense of how such a plot with a finite sample size may differ from its expected pattern when the model holds, randomly generate 100 standard normal
In the previous exercise, suppose truncation instead occurs on x. Would you expect this to affect (a) E(????̂1)? (b) inference about ????1? Why?
Selection bias: Suppose the normal linear model ????i = ????0 + ????1xi holds with????1 > 0, but the responses are truncated and we observe yi only when yi > L(or perhaps only when yi < L) for some threshold L.a. Describe a practical scenario for which this could happen. How would you expect the
In the one-way layout with c groups and a fixed common sample size n, consider simultaneous confidence intervals for pairwise comparisons of means, using family-wise error probability ???? = 0.05. Using software such as R, analyze how the ratio of margins of error for the Tukey method to the
An analyst plans to construct family-wise confidence intervals for normal linear model parameters {????(1),…, ????(g)} in estimating an effect as part of a metaanalysis with g independent studies. Explain why constructing each interval with confidence level (1 − ????)1∕g provides exactly the
For the normal linear model for the r × c two-way layout with n observations per cell, explain how to use the Tukey method for family-wise comparisons of all pairs of the r row means with confidence level 95%.
Based on the expression for a squared partial correlation in Section 3.4.4, show how it relates to a partial SS for the full model and SSE for the model without that predictor.
Consider the null model, for simplicity with known ????2. After estimating ???? =E(y) by ȳ, you plan to predict a future y from the N(????, ????2) distribution. State the formula for a 95% prediction interval for this model. Suppose, unknown to you, ȳ = ???? + zo????∕√n for some particular
When there are no explanatory variables, show how the confidence interval in Section 3.3.2 simplifies to a confidence interval for the marginal E( y).
Mimicking the derivation in Section 3.3.2, derive a confidence interval for the linear combination ????????. Explain how it simplifies for the case ????j − ????k.
Suppose a one-way layout has ordered levels for the c groups, such as dose levels in a dose–response assessment. The model E( yij) = ????0 + ????i treats the groups as a qualitative factor. The model E( yij) = ????0 + ????xi has a quantitative predictor that assumes monotone group scores {xi}.a.
Explain how to use the F test for the general linear hypothesis H0: ???????? = c to invert a test of H0: ???? = ????0 to form a confidence ellipsoid for ????. For p = 2, describe how this could give you information beyond what you would learn from separate intervals for ????1 and ????2.
For a normal linear model with p parameters and n observations, explain how to test H0: ????j = ????k in the context of the (a) general linear hypothesis and (b)F test comparing two nested linear models.
For the linear model E( yij) = ????0 + ????i for the one-way layout, explain how H0:????1 = ⋯ = ????c is a special case of the general linear hypothesis.
Using the F formula for comparing models in the previous exercise, show that adjusted R2 being larger for the more complex model is equivalent to F > 1.
a. Show that the F statistic in Section 3.2.4 for testing that all effects equal 0 has expression in terms of the R2 value as F = R2∕(p − 1)(1 − R2)∕(n − p)b. Show that the F statistic (3.1) for comparing nested models has expression in terms of the R2 values for the models as F = (R2 1
Refer to the previous exercise. Now consider the model permitting interaction.Table 3.4 shows the resulting ANOVA table.a. Argue intuitively and in analogy with results for one-way ANOVA that the SS values for factor A, factor B, and residual are as shown in the ANOVA table.b. Based on the results
For the balanced two-way r × c layout with n observations {yijk} in each cell, denote the sample means by {ȳij.} in the cells, ȳi.. in level i of A, ȳ.j. in level j of B, and ȳ overall for all N = nrc observations. Consider the model that assumes a lack of interaction.a. Construct the
Section 2.3.4 considered the projection matrices and ANOVA table for the two-way layout with one observation per cell. For testing each main effect in that model, show how to construct test statistics and explain how to obtain their null distributions, based on theory in this chapter.
Refer to the previous exercise. Based on inverting significance tests with nonzero null values, show how to construct a confidence interval for ????1 − ????2.
Using principles from this chapter, inferentially compare ????1 and ????2 from N(????1, ????2) and N(????2, ????2) populations, based on independent random samples of sizes n1 and n2.a. Put the analysis in a normal linear model context, showing a model matrix and explaining how to interpret the
Based on the formula s2(XTX)−1 for the estimated var(????̂), explain why the standard errors of {????̂j} tend to decrease as n increases.
A one-way ANOVA uses ni observations from group i, i = 1,…, c.a. Verify the noncentrality parameter for the scaled between-groups sum of squares.b. Suppose c = 3, with ????1 − ????2 = ????2 − ????3 = ????∕2. Evaluate the noncentrality, and use it to find the power of a F test with size ????
Consider the normal linear model for the one-way layout (Section 3.2.1).a. Explain why the F statistic used to test H0: ????1 = ⋯ = ????c has, under H0, an F distribution.b. Why is the test is called analysis of variance when H0 deals with means?(Hint: See Section 3.2.5.)
For y1,…, yn independent from N(????, ????2), apply Cochran’s theorem to construct a F test of H0: ???? = ????0 against H1: ???? ≠ ????0 by applying the SS decomposition with the projection matrix for the null model shown in Section 2.3.1 to the adjusted observations {yi − ????0}. State the

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Foundations Of Linear And Generalized Linear Models 1st Edition Alan Agresti - Solutions

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