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

Let yi be a bin(1, ????i) variate, i = 1,…,N. For the model logit(????i) = ????0 + ????1xi, show that the deviance depends on ̂????i but not yi. Hence, it is not useful for checking model fit. (This exercise and the previous one show that goodnessof-fit statistics are uninformative for ungrouped
A study has ni independent binary observations {yi1,…, yini} at xi, i =1,…,N, with n = ∑i ni. Consider the model logit(????i) = ????0 + ????1xi, where????i = P(yij = 1).a. Show that the kernel of the likelihood function is the same if treating the data as n Bernoulli observations or N
Use the following toy data to illustrate comments in Section 5.5 about grouped versus ungrouped binary data in the effect on the deviance:--------------------------------------------------------x Number of trials Number of successes 0 4 1 1 4 2 2 4 4
Refer to the deviance comparison statistic G2(M0 ∣ M1) introduced in Section 4.4.3. For a sequence of s nested binary response models M1,…, Ms, model Ms is the most complex. Let v denote the difference in residual df between M1 and Ms.a. Explain why for j < k, G2(Mj ∣ Mk) ≤ G2(Mj ∣ Ms).b.
In a football league, for matches involving teams a andb, let ????ab be the probability that a defeatsb. Suppose ????ab + ????ba = 1 (i.e., ties cannot occur).Bradley and Terry (1952) proposed the model log(????ab∕????ba) = ????a − ????b.For a
Let yi, i = 1,…,N, denote N independent binary random variables.a. Derive the log-likelihood for the probit model Φ−1[????(xi)] = ∑j ????jxij.b. Show that the likelihood equations for the logistic and probit regression models are∑N i=1(yi − ̂????i)zixij = 0, j = 1,…, p, where zi = 1
An alternative latent variable model results from early applications of binary response models to toxicology studies (such as Table 5.4) of the effect of dosage of a toxin on whether a subject dies, with an unobserved tolerance distribution. For a randomly selected subject, let xi denote the dosage
Consider the choice between two options, such as two product brands. Let Uy denote the utility of outcome y, for y = 0 and y = 1. Suppose Uy = ????y0 +????y1x + ????y, using a scale such that ????y has some standardized distribution. A subject selects y = 1 if U1 > U0 for that subject.a. If ????0
When Φ−1(????i) = ????0 + ????1xi, explain why the response curve for ????i [or for 1 − ????i, when ????1 < 0] has the appearance of a normal cdf with mean ???? =−????0∕????1 and standard deviation ???? = 1∕|????1|. By comparison, explain why the logistic regression curve for ????i has
Consider binary GLM F−1(????i) = ????0 + ????1xi, where F is a cdf corresponding to a pdf f that is symmetric around 0. Show that xi at which ????i = 0.50 is xi = −????0∕????1. Show that the rate of change in ????i when ????i = 0.50 is ????1f(0), and find this for the logit and probit links.
For the model log[− log(1 − ????i)] = ????0 + ????1xi, find xi at which ????i = 1 2 . Show that the greatest rate of change of ???? occurs at x = −????0∕????1, and find ???? at that point. Give the corresponding result for the model with log–log link, and compare with the logistic and
In a study of the presence of tumors in animals, suppose {yi} are independent counts that satisfy a Poisson loglinear model, log(????i) = ∑j ????jxij. However, the observed response merely indicates whether each yi is positive, zi = I(yi >0), for the indicator function I. Show that {zi} satisfy a
Suppose y = 0 at x = 10, 20, 30, 40 and y = 1 at x = 60, 70, 80, 90. Using software, what do you get for estimates and standard errors when you fit the logistic regression model (a) to these data? (b) to these eight observations and two observations at x = 50, one with y = 1 and one with y = 0? (c)
For the logistic model (5.7) for a 2 × 2 table, give an example of cell counts corresponding to (a) complete separation and ????̂1 = ∞, (b) quasi-complete separation and ????̂1 = ∞, (c) non-existence of ????̂1.
You plan to study the relation between x = age and y= whether belong to a social network such as Facebook (1 = yes). A priori, you predict that P(y = 1)is currently between about 0.80 and 0.90 at x = 18 and between about 0.20 and 0.30 at x = 65. If the logistic regression model describes this
In one of the first studies of the link between lung cancer and smoking7, Richard Doll and Austin Bradford Hill collected data from 20 hospitals in London, England. Each patient admitted with lung cancer in the preceding year was queried about their smoking behavior. For each of the 709 patients
To illustrate Fisher’s exact test, Fisher (1935) described the following experiment: a colleague of his claimed that, when drinking tea, she could distinguish whether milk or tea was added to the cup first (she preferred milk first). To test her claim, Fisher asked her to taste eight cups of tea,
For the horseshoe crab dataset (Crabs.dat at the text website) introduced in Section 4.4.3, let y = 1 if a female crab has at least one satellite, and let y = 0 if a female crab does not have any satellites. Fit a main-effects logistic model using color and weight as explanatory variables.
The dataset Crabs2.dat at the text website collects several variables that may be associated with y = whether a female horseshoe crab is monandrous(eggs fertilized by a single male crab) or polyandrous (eggs fertilized by multiple males). A probit model that uses as explanatory variables Fcolor
Refer to the previous exercise. Download the file from the text website.Using year of observation, Fcolor, Fsurf, FCW = female’s carapace width, AMCW = attached male’s carapace width, AMcolor = attached male’s color, and AMsurf = attached male’s surface condition, conduct a logistic
The New York Times reported results of a study on the effects of AZT in slowing the development of AIDS symptoms (February 15, 1991). Veterans whose immune symptoms were beginning to falter after infection with HIV were randomly assigned to receive AZT immediately or wait until their T cells showed
Download the data for the example in Section 5.7.1. Fit the main effects model.What does your software report for ????̂1 and its SE? How could you surmise from the output that actually ????̂1 = ∞?
Refer to the previous exercise. For these data, what, if anything, can you learn about potential interactions for pairs of the explanatory variables? Conduct the likelihood-ratio test of the hypothesis that all three interaction terms are 0.
Table 5.5 shows data, the file SoreThroat.dat at the text website, from a study about y = whether a patient having surgery experienced a sore throat on waking (1 = yes, 0 = no) as a function of d = duration of the surgery (in minutes) and t = type of device used to secure the airway (1 = tracheal
Show that the multinomial variate y = (y1,…, yc−1)T (with yj = 1 if outcome j occurred and 0 otherwise) for a single trial with parameters (????1,…, ????c−1)has distribution in the (c − 1)-parameter exponential dispersion family, with baseline-category logits as natural parameters.
For the baseline-category logit model without constraints on parameters,????ij = exp(xi????j)∑c h=1 exp(xi????h), show that dividing numerator and denominator by exp(xi????c) yields new parameters ????∗j = ????j − ????c that satisfy ????∗c = 0. Thus, without loss of generality, we can take
Derive Equation (6.3) for the rate of change. Show how the equation for binary models is a special case.
With three outcome categories and a single explanatory variable, suppose????ij = exp(????j0 + ????jxi)∕[1 + exp(????10 + ????1xi) + exp(????20 + ????2xi)], j = 1, 2. Show that ????i3 is (a) decreasing in xi if ????1 > 0 and ????2 > 0, (b) increasing in xi if ????1 < 0 and ????2 < 0, and (c)
Derive the deviance expression in Equation (6.5) by deriving the corresponding likelihood-ratio test.
For a multinomial response, let uij denote the utility of response outcome j for subject i. Suppose that uij = xi????j + ????ij, and the response outcome for subject i is the value of j having maximum utility. When {????ij} are assumed to be iid standard normal, this model is the simplest form of
Derive the likelihood equations and the information matrix for the discretechoice model (6.6).
Consider the baseline-category logit model (6.1).a. Suppose we impose the structure ????j = j????, for j = 1,…, c − 1. Does this model treat the response as ordinal or nominal? Explain.b. Show that the model in (a) has proportional odds structure when the c − 1 logits are formed using pairs
Section 5.3.4 introduced Fisher’s exact test for 2 × 2 contingency tables. For testing independence in a r × c table in which the data are c independent multinomials, derive a conditional distribution that does not depend on unknown parameters. Explain a way to use it to conduct a small-sample
Does it make sense to use the cumulative logit model of proportional odds form with a nominal-scale response variable? Why or why not? Is the model a special case of a baseline-category logit model? Explain.
Show how to express the cumulative logit model of proportional odds form as a multivariate GLM (6.4).
For a binary explanatory variable, explain why the cumulative logit model with proportional odds structure is unlikely to fit well if, for an underlying latent response, the two groups have similar location but very different dispersion.
Consider the cumulative logit model, logit[P(yi ≤ j)] = ????j + ????jxi.a. With continuous xi taking values over the real line, show that the model is improper, in that cumulative probabilities are misordered for a range of xi values.b. When xi is a binary indicator, explain why the model is
For the cumulative link model, G−1[P(yi ≤ j)] = ????j + xi????, show that for 1 ≤ j < k ≤ c − 1, P(yi ≤ k) equals P(yi ≤ j) at x∗, where x∗ is obtained by increasing component h of xi by (????k − ????j)∕????h for each h. Interpret.
For an ordinal multinomial response with c categories, let????ij = P(yi = j ∣ yi ≥ j) = ????ij????ij + ⋯ + ????ic, j = 1,…, c − 1.The continuation-ratio logit model is logit(????ij) = ????j + xi????j. j = 1,…, c − 1.a. Interpret (i) ????j, (ii) ???? for the simpler model with
Consider the null multinomial model, having the same probabilities {????j}for every observation. Let ???? = ∑j bj????j, and suppose that ????j = fj(????) > 0, j =1,…,c. For sample proportions {pj = nj∕N}, let S = ∑j bjpj. Let T = ∑j bj ̂????j, where ̂????j = fj(????̂), for the ML
A response scale has the categories (strongly agree, mildly agree, mildly disagree, strongly disagree, do not know). A two-part model uses a logistic regression model for the probability of a don’t know response and a separate ordinal model for the ordered categories conditional on response in
The file Alligators2.dat at the text website is an expanded version of Table 6.1 that also includes the alligator’s gender. Using all the explanatory variables, use model-building methods to select a model for predicting primary food choice. Conduct inference and interpret effects in that model.
For 63 alligators caught in Lake George, Florida, the file Alligators3.dat at the text website classifies primary food choice as (fish, invertebrate, other)and shows alligator length in meters. Analyze these data.
The following R output shows output from fitting a cumulative logit model to data from the US 2008 General Social Survey. For subject i let yi = belief in existence of heaven (1 = yes, 2 = unsure, 3 = no), xi1 = gender (1 = female, 0 = male) and xi2 = race (1 = black, 0 = white). State the model
Refer to the previous exercise. Consider the model log(????ij∕????i3) = ????j + ????G j xi1 + ????R j xi2, j = 1, 2.a. Fit the model and report prediction equations for log(????i1∕????i3), log(????i2∕????i3), and log(????i1∕????i2).b. Using the “yes”and “no” response categories,
Refer to Exercise 5.33. The color of the female crab is a surrogate for age, with older crabs being darker. Analyze whether any characteristics or combinations of characteristics of the attached male crab can help to predict a female crab’s color. Prepare a short report that summarizes your
A 1976 article by M. Madsen (Scand. J. Stat. 3: 97–106) showed a 4 × 2 × 3 × 3 contingency table (the file Satisfaction.dat at the text website) that cross classifies a sample of residents of Copenhagen on the type of housing, degree of contact with other residents, feeling of influence on
At the website sda.berkeley.edu/GSS for the General Social Survey, download a contingency table relating the variable GRNTAXES (about paying higher taxes to help the environment) to two other variables, using the survey results from 2010 by specifying year(2010) in the “Selection Filter.” Model
Suppose {yi} are independent Poisson observations from a single group. Find the likelihood equation for estimating ???? = E(yi). Show that ̂???? = ȳ regardless of the link function.
Suppose {yi} are independent Poisson variates, with ???? = E(yi), i = 1,…, n.For testing H0: ???? = ????0, show that the likelihood-ratio statistic simplifies to−2(L0 − L1) = 2[n(????0 − ȳ) + nȳ log(ȳ∕????0)].Explain how to use this to obtain a large-sample confidence interval for
Refer to the previous exercise. Explain why, alternatively, for large samples you can test H0 using the standard normal test statistic z = √n(ȳ − ????0)∕√????0.Explain how to invert this test to obtain a confidence interval. (These are the score test and score-test based confidence
When y1 and y2 are independent Poisson with means ????1 and ????2, find the likelihood-ratio statistic for testing H0: ????1 = ????2. Specify its asymptotic null distribution, and describe the condition under which the asymptotics apply.
For the one-way layout for Poisson counts (Section 7.1.5), using the identity link function, show how to obtain a large-samples confidence interval for????h − ????i. If there is overdispersion, explain why it is better to use a formula(ȳh − ȳi) ± z????∕2√(s2 h∕nh) + (s2 i ∕ni)
For the one-way layout for Poisson counts, derive the likelihood-ratio statistic for testing H0: ????1 = ⋯ = ????c.
For the one-way layout for Poisson counts, derive a test of H0: ????1 = ⋯ = ????c by applying a Pearson chi-squared goodness-of-fit test (with df = c − 1) for a multinomial distribution that compares sample proportions in c categories against H0 values of multinomial probabilities, (a) when n1
In a balanced two-way layout for a count response, let yijk be observation k at level i of factor A and level j of factor B, k = 1,…, n. Formulate a Poisson loglinear main-effects model for {????ijk = E(yijk)}. Find the likelihood equations, and show that {????ij+ = ∑k E(yijk)} have fitted
Refer to Note 1.5. For a Poisson loglinear model containing an intercept, show that the average estimated rate of change in the mean as a function of explanatory variable j satisfies 1 n∑i(???? ̂????i∕????xij) = ????̂jȳ.
A method for negative exponential modeling of survival times relates to the Poisson loglinear model for rates (Aitkin and Clayton 1980). Let T denote the time to some event, with pdf f and cdf F. For subject i, let wi = 1 for death and 0 for censoring, and let t = ∑i ti and w = ∑i wi.a. Explain
Consider the loglinear model of conditional independence between A and B, given C, in a r × c × ???? contingency table. Derive the likelihood equations, and interpret. Give the solution of fitted values that satisfies the model and the equations. (From Birch (1963), it follows that only one
Two balanced coins are flipped, independently. Let A = whether the first flip resulted in a head (yes, no), B = whether the second flip resulted in a head, and C = whether both flips had the same result. Using this example, show that marginal independence for each pair of three variables does not
For three categorical variables A, B, and C:a. When C is jointly independent of A and B, show that A and C are conditionally independent, given B.b. Prove that mutual independence of A, B, and C implies that A and B are (i)marginally independent and (ii) conditionally independent, given C.c.
Express the loglinear model of mutual independence for a 2 × 2 × 2 table in the formlog ???? = X????. Show that the likelihood equations equate {yijk} and { ̂????ijk} in the one-dimensional margins, and their solution is { ̂????ijk = yi++y+j+y++k∕n2}.
For a 2 × c × ???? table, consider the loglinear model by which A is jointly independent of B and C. Treat A as a response variable and B and C as explanatory, conditioning on {n+jk}. Construct the logit for the conditional distribution of A, and identify the corresponding logistic model.
For the homogeneous association loglinear model (7.7) for a r × c × ???? contingency table, treating A as a response variable, find the equivalent baselinecategory logit model.
For a four-way contingency table, consider the loglinear model having AB, BC, and CD two-factor terms and no three-factor interaction terms. Explain why A and D are independent given B alone or given C alone or given both B and C. When are A and C conditionally independent?
Suppose the loglinear model (7.7) of homogeneous association holds for a three-way contingency table. Find log ????ij+ and explain why marginal associations need not equal conditional associations for this model.
Consider the loglinear model for a four-way table having AB, AC, and AD two-factor terms and no three-factor interaction term. What is the impact of collapsing over B on the other associations? Contrast that with what the collapsibility condition in Section 7.2.7 suggests, treating group S3 = {B},
A county’s highway department keeps records of the number of automobile accidents reported each working day on a superhighway that runs through the county. Describe factors that are likely to cause the distribution of this count over time to show overdispersion relative to the Poisson
Show that a gamma mixture of Poisson distributions yields the negative binomial distribution.
Given u, y is Poisson with E(y ∣ u) = u????, where u is a positive random variable with E(u) = 1 and var(u) = ????. Show that E(y) = ???? and var(y) = ???? + ????????2.Explain how you can formulate the negative binomial distribution and a negative binomial GLM using this construction.
For discrete distributions, Jørgensen (1987) showed that it is natural to define the exponential dispersion family as f(yi; ????i, ????) = exp[yi????i − b(????i)∕a(????) + c(yi, ????)].a. For fixed k, show that the negative binomial distribution (7.8) has this form with ????i =
For a sequence of independent Bernoulli trials, let y = the number of successes before the kth failure. Show that y has the negative binomial distribution, f(y; ????, k) = Γ(y + k)Γ(k)Γ(y + 1)????y(1 − ????)k, y = 0, 1, 2,… .(The geometric distribution is the special case k = 1.) Relate ????
With independent negative binomial observations from a single group, find the likelihood equation and show that ̂???? = ȳ. (ML estimation for ???? requires iterative methods, as R. A. Fisher showed in an appendix to Bliss (1953). See also Anscombe (1950).) How does this generalize to the one-way
For the ZIP null model (i.e., without explanatory variables), show from the likelihood equations that the ML-fitted 0 count equals the observed 0 count.
The text website contains an expanded version (file Drugs3.dat) of the student substance use data of Table 7.3 that also has each subject’s G = gender(1 = female, 2 = male) and R = race (1 = white, 2 = other). It is sensible to treat G and R as explanatory variables. Explain why any loglinear
Other than a formal goodness-of-fit test, one analysis that provides a sense of whether a particular GLM is plausible is the following: Suppose the ML fitted equation were the true equation. At the observed x values for the n observations, randomly generate n variates with distributions specified
Another model (Dobbie and Welsh 2001) for zero-inflated count data uses the Neyman type A distribution, which is a compound Poisson–Poisson mixture. For observation i, let zi denote a Poisson variate with expected value????i. Conditional on zi, let wij (j = 1,…,zi) denote independent
A headline in The Gainesville Sun (February 17, 2014) proclaimed a worrisome spike in shark attacks in the previous 2 years. The reported total number of shark attacks in Florida per year from 2001 to 2013 were 33, 29, 29, 12, 17, 21, 31, 28, 19, 14, 11, 26, 23. Are these counts consistent with a
Table 7.5, also available at www.stat.ufl.edu/~aa/glm/data, summarizes responses of 1308 subjects to the question: within the past 12 months, how many people have you known personally that were victims of homicide?The table shows responses by race, for those who identified their race as white or as
For the horseshoe crab data, the negative binomial modeling shown in the R output first treats color as nominal-scale and then in a quantitative manner, with the category numbers as scores. Interpret the result of the likelihoodratio test comparing the two models. For the simpler model, interpret
For the horseshoe crab data, the following output shows a zero-inflated negative binomial model using quantitative color for the zero component. Interpret results, and compare with the NB2 model fitted in the previous exercise with quantitative color. Can you conduct a likelihood-ratio test
Refer to Section 7.5.2. Redo the zero-inflated NB2 model building, deleting the outlier crab weighing 5.2 kg. Compare results against analyses that used this observation and summarize conclusions.
A question in a GSS asked subjects how many times they had sexual intercourse in the preceding month. The sample means were 5.9 for males and 4.3 for females; the sample variances were 54.8 and 34.4. The mode for each gender was 0. Specify a GLM that would be inappropriate for these data,
Table 7.6 is based on a study involving British doctors.Table 7.6 Data for Exercise 7.36 on Coronary Death Rates Person-Years Coronary Deaths Age Nonsmokers Smokers Nonsmokers Smokers 35–44 18,793 52,407 2 32 45–54 10,673 43,248 12 104 55–64 5710 28,612 28 206 65–74 2585 12,663 28 186
Does the inflated-variance QL approach make sense as a way to generalize the ordinary normal linear model with v(????i) = ????2? Why or why not?
Using E(y) = E[E(y|x)] and var(y) = E[var(y|x)] + var[E(y|x)], derive the mean and variance of the beta-binomial distribution.
Let y1 and y2 be independent negative binomial variates with common dispersion parameter ????.a. Show that y1 + y2 is negative binomial with dispersion parameter ????∕2.b. Conditional on y1 + y2, show that y1 has a beta-binomial distribution.c. State the multicategory extension of (b) that yields
Altham (1978) introduced the discrete distribution f(x; ????, ????) = c(????, ????)( n x)????x(1 − ????)n−x????x(n−x), x = 0, 1,…, n, where c(????, ????) is a normalizing constant. Show that this is in the two-parameter exponential family and that the binomial occurs when ???? = 1. (Altham
Sometimes sample proportions are continuous rather than of the binomial form (number of successes)/(number of trials). Each observation is any real number between 0 and 1, such as the proportion of a tooth surface that is covered with plaque. For independent responses {yi}, Bartlett (1937) modeled
Motivation for the quasi-score equations (8.2): suppose we replace v(????i) by known variance vi. Show that the equations result from the weighted least squares approach of minimizing ∑i[(yi − ????i)2∕vi].
Before R. A. Fisher introduced the method of maximum likelihood in 1922, Karl Pearson had proposed the method of moments as a general-purpose method for statistical estimation2. Explain how this method can be formulated as having estimating equations with an unbiased estimating function.
Ordinary linear models assume that v(????i) = ????2 is constant. Suppose instead that actually var(yi) = ????i. Using the QL approach for the null model ????i = ????, i = 1,…, n, show that u(????) = (1∕????2)∑i(yi − ????), so ????̂ = ȳ and V = ????2∕n. Find the model-based estimate of
Suppose we assume v(????i) = ????i but actually var(yi) = ????2. For the null model????i = ????, find the model-based var(????̂), the actual var(????̂), and the robust estimate of that variance.
Suppose we assume v(????i) = ????i but actually var(yi) = v(????i) for some unspecified function v. For the null model ????i = ????, find the model-based var(????̂), the actual var(????̂), and the robust estimate of that variance.
Consider the null model ????i = ???? when the observations are independent counts.Of the Poisson-model-based and robust estimators of the variance of ????̂ = ȳpresented in Section 8.3.3, which would you expect to be better (a) if the Poisson model truly holds, (b) if there is severe
Let yij denote the response to a question about belief in life after death (1 =yes, 0 = no) for person j in household i, j = 1,…, ni, i = 1,…, n. In modeling P(yij = 1) with explanatory variables, describe a scenario in which you would expect binomial overdispersion. Specify your preferred
Use QL methods to construct a model for the horseshoe crab satellite counts, using weight, color, and spine condition as explanatory variables. Compare results with those obtained with zero-inflated GLMs in Section 7.5.
Use QL methods to analyze Table 7.5 on counts of homicide victims. Interpret, and compare results with Poisson and negative binomial GLMs.
Refer to Exercise 7.35 on the frequency of sexual intercourse. Use QL methods to obtain a confidence interval for the (a) difference, (b) ratio of means for males and females.
For the teratology study analyzed in Section 8.2.4, analyze the data using only the group indicators as explanatory variables (i.e., ignoring hemoglobin).Interpret results. Is it sufficient to use the simpler model having only the placebo indicator for the explanatory variable?

Showing 1000 - 1100 of 1264

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

Step by Step Answers