Foundations of Linear and Generalized Linear Models 1st edition solutions

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Foundations of Linear and Generalized Linear Models 1st edition Solutions Manual

Name: Foundations of Linear and Generalized Linear Models
Rating: 4.7 (6 reviews)
Author: CrazyForStudy
ISBN: 9781118730034

(Advanced Math)

Edition: 1st edition

Author: Crazy for study

ISBN: 9781118730034

4.7 Rating

6 Reviews

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Chapter

Section

Question

Question : 1E

Exercise
- Suppose that yi has aN(??i, ??2) distribution, i = 1,
Question : 2E

Exercise
- Link function of a GLM: a. Describe the purpose of the link function g. b. The identity link is the standard one with normal responses but is not often used with binary or count responses. Why do you think this is?
Question : 3E

Exercise
- What do you think are the advantages and disadvantages of treating an ordinal explanatory variable as (a) quantitative, (b) qualitative?
Question : 4E

Exercise
- Extend the model in Section 1.2.1 relating income to racial
Question : 5E

Exercise
- Suppose you standardize the response and explanatory variables before fitting a linear model (i.e., subtract the means and divide by the standard deviations). Explain how to interpret the resulting standardized regression coefficients.
Question : 6E

Exercise
- When X has full rank p, explain why the null space of X consists only of the 0 vector.
Question : 7E

Exercise
- For any linear model ?? = X??, is the origin 0 in the model space C(X)? Why or why not?
Question : 8E

Exercise
- A model M has model matrix X. A simpler model M0 results from removing the final term in M, and hence has model matrix X0 that deletes the final column from X. From the definition of a column space, explain why C(X0) is contained in C(X)
Question : 9E

Exercise
- For the normal linear model, explain why the expression yi = ?p j=1 ??jxij + ??i with ??i ? N(0, ??2) is equivalent to yi ? N( ?p j=1 ??jxij, ??2).
Question : 10E

Exercise
- GLMs normally use a hierarchical structure by which the presence of a higher-order term implies also including the lower-order terms. Explain why this is sensible, by showing that (a) a model that includes an x2 explanatory variable but not x makes a strong assumption about where the maximum or minimum of E(y) occurs, (b) a model that includes x1x2 but not x1 makes a strong assumption about the effect of x1 when x2 = 0.
Question : 11E

Exercise
- Show the form of X?? for the linear model for the one-way layout, E(yij) = ??0 + ??i, using a full-rank model matrix X by employing the constraint ? i ??i = 0 to make parameters identifiable
Question : 12E

Exercise
- Consider the model for the two-way layout for qualitative factors A and B, E(yijk) = ??0 + ??i + ??j, for i = 1,
Question : 13E

Exercise
- Consider the model for the two-way layout shown in the previous exercise. Suppose r = 2, c = 3, and n = 2. a. Show the form of a full-rank model matrix X and corresponding parameter vector ?? for the model, constraining ??1 = ??1 = 0 to make ?? identifiable. Explain how to interpret the elements of ??. b. Show the form of a full-rank model matrix and corresponding parameter vector ?? when you constrain ? i ??i = 0 and ? j ??j = 0 to make ?? identifiable. Explain how to interpret the elements of ??. c. In the full-rank case, what is the rank of X?
Question : 14E

Exercise
- For the model in the previous exercise with constraints ??1 = ??1 = 0, generalize the model by adding an interaction term ??ij. a. Show the new full-rank model matrix. Specify the constraints that {??ij} satisfy. Indicate how many parameters the ??ij term represents in ??. b. Show how to write the linear predictor using indicator variables for the factor categories, with the model parameters as coefficients of those indicators and the interaction parameters as coefficients of products of indicators.
Question : 15E

Exercise
- Refer to Exercise 1.12. Now suppose r = 2 and c = 4, but observations for the first two levels of B occur only at the first level of A, and observations for the last two levels of B occur only at the second level of A. In the corresponding model, E(yijk) = ??0 + ??i + ??j(i), B is said to be nested within A. Specify a full-rank model matrix X, and indicate its rank.
Question : 16E

Exercise
- Explain why the vector space of p
Question : 17E

Exercise
- If A is a nonsingular matrix, show that C(X) = C(XA). (If two full-rank model matrices correspond to equivalent models, then one model matrix is the other multiplied by a nonsingular matrix.)
Question : 18E

Exercise
- For the linear model for the one-way layout, Section 1.4.1 showed the model matrix that makes parameters identifiable by setting ??1 = 0. Call this model matrix X1. a. Suppose we instead obtain identifiability by imposing the constraint ??c = 0. Show the model matrix, say Xc. b. Show how to obtain X1 as a linear transformation of Xc.
Question : 19E

Exercise
- Consider the analysis of covariance model without interaction, denoted by 1 + X + A. a. Write the formula for the model in such a way that the parameters are not identifiable. Show the corresponding model matrix. b. For the model parameters in (a), give an example of a characteristic that is (i) estimable, (ii) not estimable. c. Now express the model so that the parameters are identifiable. Explain how to interpret them. Show the model matrix when A has three groups, each containing two observations.
Question : 20E

Exercise
- Show the first five rows of the model matrix for (a) the linear model for the horseshoe crabs in Section 1.5.2, (b) the model for a one-way layout in Section 1.5.3, (c) the model containing both weight and color predictors.
Question : 21E

Exercise
- Littell et al. (2000) described a pharmaceutical clinical trial in which 24 patients were randomly assigned to each of three treatment groups (drug A, drug B, placebo) and compared on a measure of respiratory ability (FEV1 = forced expiratory volume in 1 second, in liters). The data file8FEV.dat at www.stat.ufl.edu/~aa/glm/data has the form shown in Table 1.4. Here, we let y be the response after 1 hour of treatment (variable fev1 in the data file), x1 = the baseline measurement prior to administering the drug (variable base in the data file), and x2 = drug (qualitative with labels a, b, p in the data file). Download the data and fit the linear model for y with explanatory variables (a) x1, (b) x2, (c) both x1 and x2. Interpret model parameter estimates in each case.
Question : 22E

Exercise
- Refer to the analyses in Section 1.5.3 for the horseshoe crab satellites. a. With color alone as a predictor, why are standard errors much smaller for a Poisson model than for a normal model? Out of these two very imperfect models, which do you trust more for judging significance of the estimates of the color effects? Why? b. Download the data (file Crabs.dat) from www.stat.ufl.edu/~ aa/glm/data. When weight is also a predictor, identify an outlying observation. Refit the model with color and weight predictors without that observation. Compare results, to investigate the sensitivity of the results to this outlier.
Question : 23E

Exercise
- Another horseshoe crab dataset9 (Crabs2.dat at www.stat.ufl.edu/~ aa/glm/data) comes from a study of factors that affect sperm traits of male crabs. A response variable, SpermTotal, is measured as the log of the total number of sperm in an ejaculate. It has mean 19.3 and standard deviation 2.0. Two explanatory variables are the crab
Question : 24E

Exercise
- For 72 young girls suffering from anorexia, the Anorexia.dat file at the text website shows their weights before and after an experimental period. Table 1.5 shows the format of the data. The girls were randomly assigned to receive one of three therapies during this period. A control group received the standard therapy, which was compared to family therapy and cognitive behavioral therapy. Download the data and fit a linear model relating the weight after the experimental period to the initial weight and the therapy. Interpret estimates.

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A valuable overview of the most important ideas and results in statistical modeling Written by a highly-experienced author, foundations of linear and generalized linear models is a clear and comprehensive guide to the key concepts and results of linearstatistical models. The book presents a broad, in-depth overview of the most commonly usedstatistical models by discussing the theory underlying the models, R software applications,and examples with crafted models to elucidate key ideas and promote practical modelbuilding. The book begins by illustrating the fundamentals of linear models, such as how the model-fitting projects the data onto a model vector subspace and how orthogonal decompositions of the data yield information about the effects of explanatory variables. Subsequently, the book covers the most popular generalized linear models, which include binomial and multinomial logistic regression for categorical data, and Poisson and negative binomial loglinear models for count data. Focusing on the theoretical underpinnings of these models, Foundations ofLinear and Generalized Linear Models also features: An introduction to quasi-likelihood methods that require weaker distributional assumptions, such as generalized estimating equation methods An overview of linear mixed models and generalized linear mixed models with random effects for clustered correlated data, Bayesian modeling, and extensions to handle problematic cases such as high dimensional problems Numerous examples that use R software for all text data analyses More than 400 exercises for readers to practice and extend the theory, methods, and data analysis A supplementary website with datasets for the examples and exercises An invaluable textbook for upper-undergraduate and graduate-level students in statistics and biostatistics courses, foundations of linear and generalized linear models is also an excellent reference for practicing statisticians and biostatisticians, as well as anyone who is interested in learning about the most important statistical models for analyzing data.

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