An Introduction To Categorical Data Analysis 2nd Edition Alan Agresti - Solutions

9.17 What is wrong with this statement: “For a first-order Markov chain, Yt is independent of Yt−2”?
9.18 True, or false? With repeated measures data having multiple observations per subject, one can treat the observations as independent and still get valid estimates, but the standard errors based on the independence assumption may be badly biased.
10.1 Refer back to Table 8.10 from a recent General Social Survey that asked subjects whether they believe in heaven and whether they believe in hell.a. Fit model (10.3). If your software uses numerical integration, report ˆ β, ˆσ , and their standard errors for 2, 10, 100, 400, and 1000
10.2 You plan to apply the matched-pairs model (10.3) to a data set for which yi1 is whether the subject agrees that abortion should be legal if the woman cannot afford the child (1 = yes, 0 = no), and yi2 is whether the subject opposes abortion if a woman wants it because she is unmarried (1 =
10.3 A dataset on pregnancy rates among girls in 13 north central Florida counties has information on the total in 2005 for each county i on Ti = number of births and yi = number of those for which mother’s age was under 18. Let πi be the probability that a pregnancy in county i is to a mother
10.4 Table 10.9 shows the free-throw shooting, by game, of Shaq O’Neal of the Los Angeles Lakers during the 2000 NBA (basketball) playoffs. In game i, let yi = number made out of Ti attempts.a. Fit the model, logit(πi ) = ui + α, where {ui } are independent N(0, σ), and given {ui }, {yi } are
10.5 For 10 coins, let πi denote the probability of a head for coin i. You flip each coin five times. The sample numbers of heads are {2, 4, 1, 3, 3, 5, 4, 2, 3, 1}.a. Report the sample proportion estimates of πi . Formulate a model for which these are the ML estimates.b. Formulate a random
10.6 For Table 7.3 from the survey of high school students, let yit = 1 when subject i used substance t (t = 1, cigarettes; t = 2, alcohol; t = 3, marijuana).Table 10.10 shows output for the logistic-normal model, logit[P(Yit = 1)] =ui + βt .a. Interpret { ˆ βt }. Illustrate odds ratio
10.7 Refer to the previous exercise.a. Compare { ˆ βt } to the estimates for the marginal model in Problem 9.1.Why are they so different?b. How is the focus different for the model in the previous exercise than for the loglinear model (AC,AM,CM) that Section 7.1.6 used?c. If ˆσ = 0 in the GLMM
10.8 For the student survey summarized by Table 7.13, (a) analyze using GLMMs,(b) compare results and interpretations to those with marginal models in Problem 9.2.
10.9 For the crossover study summarized by Table 9.9 (Problem 9.6), fit the model logit[P(Yi(k)t = 1)] = ui(k) + αk + βt where {ui(k)} are independent N(0, σ). Interpret { ˆ βt } and ˆσ .
10.10 For the previous exercise, compare estimates of βB − βA and βC − βA and their SE values to those using the corresponding marginal model of Problem 9.6.
10.11 Refer to Table 5.5 on admissions decisions for Florida graduate school applicants.For a subject in department i of gender g (1 = females, 0 = males), let yig = 1 denote being admitted.a. For the fixed effects model, logit[P(Yig = 1)] = α + βg + βD i , the estimated gender effect is ˆ β =
10.12 Consider Table 8.14 on premarital and extramarital sex. Table 10.11 shows the results of fitting a cumulative logit model with a random intercept.a. Interpret ˆ β.b. What does the relatively large ˆσ value suggest?
10.13 Refer to the previous exercise. Analyze these data with a corresponding cumulative logit marginal model.a. Interpret ˆ β.b. Compare ˆ β to its value in the GLMM. Why are they so different?
10.14 Refer to Problem 9.11 for Table 7.25 on government spending.a. Analyze these data using a cumulative logit model with random effects.Interpret.b. Compare the results with those with a marginal model (Problem 9.11).
10.15 Refer to Table 4.16 and Problem 4.20, about an eight-center clinical trial comparing a drug with placebo for curing an infection. Model the data in a way that allows the odds ratio to vary by center. Summarize your analyses and interpretations in a one-page report.
10.16 See http://bmj.com/cgi/content/full/317/7153/235 for a meta analysis of studies about whether administering albumin to critically ill patients increases or decreases mortality. Analyze the data for the 13 studies with hypovolemia patients using logistic models with (i) fixed effects, (ii)
10.17 Refer to the insomnia example in Section 10.3.1.a. From results in Table 10.7 for the GLMM, explain how to get the interpretation quoted in the text that “The response distributions are similar initially for the two treatment groups, but the interaction suggests that at the follow-up
10.18 Analyze Table 9.8 with age and maternal smoking as predictors using a (a)logistic-normal model, (b) marginal model, (c) transitional model. Summarize your analyses in a two-page report, explaining how the interpretation of the maternal smoking effect differs for the three approaches.
10.19 Refer to the toxicity study with data summarized in Table 10.12. Collapsing the ordinal response to binary in terms of whether with data summarized in the outcome is normal, consider logistic models as a linear function of the dose level.a. Does the ordinary binomial GLM show evidence of
10.20 Refer to the previous exercise. Analyze the data with a marginal model and with a GLMM, both of cumulative logit form, for the ordinal response.Summarize analyses in a two-page report.
10.21 Table 10.13 reports 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 by day 10. The three treatments were (1) carbon dioxide and oxygen removed, (2) carbon dioxide only
10.22 Problem 3.15 analyzed data from a General Social Survey on responses of 1308 subjects to the question, “Within the past 12 months, how many people have you known personally that were victims of homicide?” It used Poisson and negative binomial GLMs for count data. Here is a possible GLMM:
10.23 A crossover study compares two drugs on a binary response variable. The study classifies subjects by age as under 60 or over 60. In a GLMM, these two age groups have the same conditional effect comparing the drugs, but the older group has a much larger variance component for its random
In the following examples, identify the response variable and the explanatory variables.a. Attitude toward gun control (favor, oppose), Gender (female, male), Mother’s education (high school, college).b. Heart disease (yes, no), Blood pressure, Cholesterol level.c. Race (white, nonwhite),
Which scale of measurement is most appropriate for the following variables –nominal, or ordinal?a. Political party affiliation (Democrat, Republican, unaffiliated).b. Highest degree obtained (none, high school, bachelor’s, master’s, doctorate).c. Patient condition (good, fair, serious,
Each of 100 multiple-choice questions on an exam has four possible answers but one correct response. For each question, a student randomly selects one response as the answer.a. Specify the distribution of the student’s number of correct answers on the exam.b. Based on the mean and standard
A coin is flipped twice. Let Y = number of heads obtained, when the probability of a head for a flip equals π.a. Assuming π = 0.50, specify the probabilities for the possible values for Y , and find the distribution’s mean and standard deviation.b. Find the binomial probabilities for Y when π
Refer to the previous exercise. Suppose y = 0 in 2 flips. Find the ML estimate of π. Does this estimate seem “reasonable”? Why? [The Bayesian estimator is an alternative one that combines the sample data with your prior beliefs about the parameter value. It provides a nonzero estimate of π,
Genotypes AA, Aa, and aa occur with probabilities (π1, π2, π3). For n = 3 independent observations, the observed frequencies are (n1, n2, n3).a. Explain how you can determine n3 from knowing n1 and n2. Thus, the multinomial distribution of (n1, n2, n3) is actually two-dimensional.b. Show the set
In his autobiography A Sort of Life, British author Graham Greene described a period of severe mental depression during which he played Russian Roulette.This “game” consists of putting a bullet in one of the six chambers of a pistol, spinning the chambers to select one at random, and then
When the 2000 General Social Survey asked subjects whether they would be willing to accept cuts in their standard of living to protect the environment, 344 of 1170 subjects said “yes.”a. Estimate the population proportion who would say “yes.”b. Conduct a significance test to determine
A sample of women suffering from excessive menstrual bleeding have been taking an analgesic designed to diminish the effects.Anewanalgesic is claimed to provide greater relief. After trying the new analgesic, 40 women reported greater relief with the standard analgesic, and 60 reported greater
Refer to the previous exercise. The researchers wanted a sufficiently large sample to be able to estimate the probability of preferring the new analgesic to within 0.08, with confidence 0.95. If the true probability is 0.75, how large a sample is needed to achieve this accuracy? (Hint: For how
When a recent General Social Survey asked 1158 American adults, “Do you believe in Heaven?”, the proportion who answered yes was 0.86. Treating this as a random sample, conduct statistical inference about the true proportion of American adults believing in heaven. Summarize your analysis and
To collect data in an introductory statistics course, recently I gave the students a questionnaire. One question asked whether the student was a vegetarian. Of 25 students, 0 answered “yes.” They were not a random sample, but let us use these data to illustrate inference for a proportion. (You
Refer to the previous exercise, with y = 0 in n = 25 trials.a. Showthat 0, the maximized likelihood underH0, equals (1 − π0)25, which is (0.50)25 for H0: π = 0.50.b. Show that 1, the maximum of the likelihood function over all possible πvalues, equals 1.0. (Hint: This is the value at the ML
Sections 1.4.4 and 1.4.5 found binomial P-values for a clinical trial with y = 9 successes in 10 trials. Suppose instead y = 8. Using the binomial distribution shown in Table 1.2:a. Find the P-value for (i) Ha:π > 0.50, (ii) Ha:π < 0.50.b. Find the mid P-value for (i) Ha:π > 0.50, (ii) Ha:π <
If Y is a variate and c is a positive constant, then the standard deviation of the distribution of cY equals cσ(Y ). Suppose Y is a binomial variate, and let p = Y/n.a. √Based on the binomial standard deviation for Y , show that σ(p) =[π(1 − π)/n].b. Explain why it is easier to estimate π
Using calculus, it is easier to derive the maximum of the log of the likelihood function, L = log , than the likelihood function itself. Both functions have maximum at the same value, so it is sufficient to do either.a. Calculate the log likelihood function L(π) for the binomial distribution
Suppose a researcher routinely conducts significance tests by rejecting H0 if the P-value satisfies P ≤ 0.05. Suppose a test using a test statistic T and righttail probability for the P-value has null distribution P(T = 0) = 0.30, P(T =3) = 0.62, and P(T = 9) = 0.08.a. Show that with the usual
For a given sample proportion p, showthat a value π0 for which the test statistic z = (p − π0)/√[π0(1 − π0)/n] takes some fixed value z0 (such as 1.96) is a solution to the equation (1 + z2 0/n)π2 0+ (−2p − z2 0/n)π0 + p2 = 0. Hence, using the formula x = [−b ±√(b2 − 4ac)]/2a
An article in the NewYork Times (February 17, 1999) about the PSA blood test for detecting prostate cancer stated that, of men who had this disease, the test fails to detect prostate cancer in 1 in 4 (so called false-negative results), and of men who did not have it, as many as two-thirds receive
For diagnostic testing, let X = true status (1 = disease, 2 = no disease)and Y = diagnosis (1 = positive, 2 = negative). Let πi = P(Y = 1|X = i), i = 1, 2.a. Explain why sensitivity = π1 and specificity = 1 − π2.b. Let γ denote the probability that a subject has the disease. Given that the
According to recent UN figures, the annual gun homicide rate is 62.4 per one million residents in the United States and 1.3 per one million residents in the UK.a. Compare the proportion of residents killed annually by guns using the(i) difference of proportions, (ii) relative risk.b. When both
Consider the following two studies reported in the New York Times:a. A British study reported (December 3, 1998) that, of smokers who get lung cancer, “women were 1.7 times more vulnerable than men to get small-cell lung cancer.” Is 1.7 an odds ratio, or a relative risk?b. A National Cancer
In the United States, the estimated annual probability that awoman over the age of 35 dies of lung cancer equals 0.001304 for current smokers and 0.000121 for nonsmokers [M. Pagano and K. Gauvreau, Principles of Biostatistics, Belmont, CA: Duxbury Press (1993), p. 134].a. Calculate and interpret
For adults who sailed on the Titanic on its fateful voyage, the odds ratio between gender (female, male) and survival (yes, no) was 11.4. (For data, see R. Dawson, J. Statist. Educ. 3, no. 3, 1995.)a. What is wrong with the interpretation, “The probability of survival for females was 11.4 times
A research study estimated that under a certain condition, the probability a subject would be referred for heart catheterization was 0.906 for whites and 0.847 for blacks.a. A press release about the study stated that the odds of referral for cardiac catheterization for blacks are 60% of the odds
An estimated odds ratio for adult females between the presence of squamous cell carcinoma (yes, no) and smoking behavior (smoker, nonsmoker) equals 11.7 when the smoker category consists of subjects whose smoking level s is 0 < s < 20 cigarettes per day; it is 26.1 for smokers with s ≥ 20
Data posted at the FBI website (www.fbi.gov) stated that of all blacks slain in 2005, 91% were slain by blacks, and of all whites slain in 2005, 83% were slain by whites. Let Y denote race of victim and X denote race of murderer.a. Which conditional distribution do these statistics refer to, Y
A20-year study of British male physicians (R. Doll and R. Peto, British Med. J., 2: 1525–1536, 1976) noted that the proportion who died from lung cancer was 0.00140 per year for cigarette smokers and 0.00010 per year for nonsmokers.The proportion who died from heart disease was 0.00669 for
A statistical analysis that combines information from several studies is called a meta analysis. A meta analysis compared aspirin with placebo on incidence of heart attack and of stroke, separately for men and for women(J. Am. Med. Assoc., 295: 306–313, 2006). For the Women’s Health Study,
Refer to Table 2.1 about belief in an afterlife.a. Construct a 90% confidence interval for the difference of proportions, and interpret.b. Construct a 90% confidence interval for the odds ratio, and interpret.c. Conduct a test of statistical independence. Report the P-value and interpret.
A poll by Louis Harris and Associates of 1249 adult Americans indicated that 36% believe in ghosts and 37% believe in astrology. Can you compare the proportions using inferential methods for independent binomial samples? If yes, do so. If not, explain why not.
A large-sample confidence interval for the log of the relative risk isAntilogs of the endpoints yield an interval for the true relative risk. Verify the 95% confidence interval of (1.43, 2.30) reported for the relative risk in Section 2.2.3 for the aspirin and heart attack study. 1- P1 1-P2
Table 2.12 comes from one of the first studies of the link between lung cancer and smoking, by Richard Doll and A. Bradford Hill. In 20 hospitals in London, UK, patients admitted with lung cancer in the previous year were queried about their smoking behavior. For each patient admitted, researchers
Refer to Table 2.3. Find the P-value for testing that the incidence of heart attacks is independent of aspirin intake using (a) X2, (b) G2. Interpret results.
Table 2.13 shows data from the 2002 General Social Survey cross classifying a person’s perceived happiness with their family income. The table displays the observed and expected cell counts and the standardized residuals for testing independence.a. Show how to obtain the estimated expected cell
Table 2.14 was taken from the 2002 General Social Survey.a. Test the null hypothesis of independence between party identification and race. Interpretb. Use standardized residuals to describe the evidence.c. Partition the chi-squared into two components, and use the components to describe the
In an investigation of the relationship between stage of breast cancer at diagnosis(local or advanced) and a woman’s living arrangement (D. J. Moritz and W. A. Satariano, J. Clin. Epidemiol., 46: 443–454, 1993), of 144 women living alone, 41.0% had an advanced case; of 209 living with spouse,
Each subject in a sample of 100 men and 100 women is asked to indicate which of the following factors (one or more) are responsible for increases in teenage crime: A, the increasing gap in income between the rich and poor;B, the increase in the percentage of single-parent families; C, insufficient
Table 2.15 classifies a sample of psychiatric patients by their diagnosis and by whether their treatment prescribed drugs.a. Conduct a test of independence, and interpret the P-value.b. Obtain standardized residuals, and interpret.c. Partition chi-squared into three components to describe
Table 2.16, from a recent General Social Survey, cross-classifies the degree of fundamentalism of subjects’ religious beliefs by their highest degree of education. The table also shows standardized residuals. For these data, X2 =69.2. Write a report of about 200words, summarizing description and
Formula (2.8) has alternative formula X2 = n(pij − pi+p+j )2/pi+p+j .Hence, for given {pij }, X2 is large when n is sufficiently large, regardless of whether the association is practically important. Explain why chi-squared tests, like other tests, merely indicate the degree of evidence against
For tests of H0: independence, { ˆ μij = ni+n+j/n}.a. Show that { ˆ μij } have the same row and column totals as {nij }.b. For 2 × 2 tables, show that ˆμ11ˆμ22/ˆμ12 ˆμ21 = 1.0. Hence, { ˆ μij } satisfy H0.
A chi-squared variate with degrees of freedom equal to df has representation Z2 1+· · ·+Z2 df , where Z1, . . . , Zdf are independent standard normal variates.a. If Z has a standard normal distribution, what distribution does Z2 have?b. Show that, if Y1 and Y2 are independent chi-squared
A study on educational aspirations of high school students (S. Crysdale, Int. J.Comp. Sociol., 16: 19–36, 1975) measured aspirations using the scale (some high school, high school graduate, some college, college graduate). For students whose family income was low, the counts in these categories
By trial and error, find a 3 × 3 table of counts for which the P-value is greater than 0.05 for the X2 test but less than 0.05 for the M2 ordinal test. Explain why this happens.
A study (B. Kristensen et al., J. Intern. Med., 232: 237–245, 1992) considered the effect of prednisolone on severe hypercalcaemia in women with metastatic breast cancer. Of 30 patients, 15 were randomly selected to receive prednisolone, and the other 15 formed a control group. Normalization in
Table 2.17 contains results of a study comparing radiation therapy with surgery in treating cancer of the larynx. Use Fisher’s exact test to testH0: θ = 1 against Ha: θ > 1. Interpret results.
Refer to the previous exercise.a. Obtain and interpret a two-sided exact P-value.b. Obtain and interpret the one-sided mid P-value. Give advantages of this type of P-value, compared with the ordinary one.
Of the six candidates for three managerial positions, three are female and three are male. Denote the females by F1, F2, F3 and the males by M1, M2, M3.The result of choosing the managers is (F2, M1, M3).a. Identify the 20 possible samples that could have been selected, and construct the
In murder trials in 20 Florida counties during 1976 and 1977, the death penalty was given in 19 out of 151 cases in which a white killed a white, in 0 out of 9 cases in which a white killed a black, in 11 out of 63 cases in which a black killed a white, and in 6 out of 103 cases in which a black
Smith and Jones are baseball players. Smith had a higher batting average than Jones in 2005 and 2006. Is it possible that, for the combined data for these two years, Jones had the higher batting average? Explain, and illustrate using data.
At each age level, the death rate is higher in South Carolina than in Maine, but overall the death rate is higher in Maine. Explain how this could be possible.(For data, see H.Wainer, Chance, 12: 44, 1999.)
Give a “real world” example of three variables X, Y , and Z, for which you expect X and Y to be marginally associated but conditionally independent, controlling for Z.
Based on murder rates in the United States, the Associated Press reported that the probability a newborn child has of eventually being a murder victim is 0.0263 for nonwhite males, 0.0049 for white males, 0.0072 for nonwhite females, and 0.0023 for white females.a. Find the conditional odds ratios
For three-way contingency tables:a. When any pair of variables is conditionally independent, explain why there is homogeneous association.b. When there is not homogeneous association, explain why no pair of variables can be conditionally independent.
True, or false?a. In 2 × 2 tables, statistical independence is equivalent to a population odds ratio value of θ = 1.0.b. We found that a 95% confidence interval for the odds ratio relating having a heart attack (yes, no) to drug (placebo, aspirin) is (1.44, 2.33). If we had formed the table with
Describe the purpose of the link function of a GLM. Define the identity link, and explain why it is not often used with the binomial parameter.
In the 2000 US Presidential election, Palm Beach County in Florida was the focus of unusual voting patterns apparently caused by a confusing “butterfly ballot.” Many voters claimed they voted mistakenly for the Reform party candidate, Pat Buchanan, when they intended to vote for Al Gore. Figure
Refer to Table 2.7 on x = mother’s alcohol consumption and Y = whether a baby has sex organ malformation. WIth scores (0, 0.5, 1.5, 4.0, 7.0) for alcohol consumption, ML fitting of the linear probability model has the output:a. State the prediction equation, and interpret the intercept and
Refer to the previous exercise and the solution to (b).a. The sample proportion of malformations is much higher in the highest alcohol category than the others because, although it has only one malformation, its sample size is only 38. Is the result sensitive to this single malformation
For Table 3.1 on snoring and heart disease, re-fit the linear probability model or the logistic regression model using the scores (i) (0, 2, 4, 6), (ii) (0, 1, 2, 3),(iii) (1, 2, 3, 4). Compare the model parameter estimates under the three choices. Compare the fitted values. What can you conclude
In Section 3.2.2 on the snoring and heart disease data, refer to the linear probability model. Would the least squares fit differ from the ML fit for the 2484 binary observations? (Hint: The least squares fit is the same as the ML fit of the GLM assuming normal rather than binomial random
Access the horseshoe crab data in Table 3.2 at www.stat.ufl.edu/∼aa/introcda/appendix.html. Let Y = 1 if a crab has at least one satellite, and let Y = 0 otherwise. Using weight as the predictor, fit the linear probability model.a. Use ordinary least squares. Interpret the parameter estimates.
Refer to the previous exercise for the horseshoe crab data.a. Report the fit for the probit model, with weight predictor.b. Find ˆπ at the highest observed weight, 5.20 kg.c. Describe the weight effect by finding the difference between the ˆπ values at the upper and lower quartiles of weight,
Table 3.6 refers to a sample of subjects randomly selected for an Italian study on the relation between income and whether one possesses a travel credit card(such as American Express or Diners Club). At each level of annual income in millions of lira, the table indicates the number of subjects
Refer to Problem 4.1 on cancer remission. Table 3.7 shows output for fitting a probit model. Interpret the parameter estimates (a) finding the remission value at which the estimated probability of remission equals 0.50, (b) finding the difference between the estimated probabilities of remission at
An experiment analyzes imperfection rates for two processes used to fabricate silicon wafers for computer chips. For treatment A applied to 10 wafers, the numbers of imperfections are 8, 7, 6, 6, 3, 4, 7, 2, 3, 4. Treatment B applied to 10 other wafers has 9, 9, 8, 14, 8, 13, 11, 5, 7, 6
Refer to Problem 3.11. The wafers are also classified by thickness of silicon coating (z = 0, low; z = 1, high). The first five imperfection counts reported for each treatment refer to z = 0 and the last five refer to z = 1. Analyze these data, making inferences about the effects of treatment type
Access the horseshoe crab data of Table 3.2 at www.stat.ufl.edu/∼aa/introcda/appendix.html.a. Using x = weight and Y = number of satellites, fit a Poisson loglinear model. Report the prediction equation.b. Estimate the mean of Y for female crabs of average weight 2.44 kg.c. Use ˆ β to describe
Refer to the previous exercise. Allow overdispersion by fitting the negative binomial loglinear model.a. Report the prediction equation and the estimate of the dispersion parameter and its SE. Is there evidence that this model gives a better fit than the Poisson model?b. Construct a 95% confidence
A recent General Social Survey asked subjects, “Within the past 12 months, how many people have you known personally that were victims of homicide”? The sample mean for the 159 blacks was 0.522, with a variance of 1.150. The sample mean for the 1149 whites was 0.092, with a variance of 0.155.a.
One question in a recent General Social Survey asked subjects howmany times they had had sexual intercourse in the previous month.a. The sample means were 5.9 for males and 4.3 for females; the sample variances were 54.8 and 34.4. Does an ordinary Poisson GLM seem appropriate? Explain.b. The GLM
A study dealing with motor vehicle accident rates for elderly drivers (W. Ray et al., Am. J. Epidemiol., 132: 873–884, 1992) indicated that the entire cohort of elderly drivers had 495 injurious accidents in 38.7 thousand years of driving.Using a Poisson GLM, find a 95% confidence interval for
Table 3.8 lists total attendance (in thousands) and the total number of arrests in a season for soccer teams in the Second Division of the British football league.a. Let Y denote the number of arrests for a team with total attendance t . Explain why the model E(Y) = μt might be plausible. Show
Table 3.4 showed data on accidents involving trains.a. Is it plausible that the collision counts are independent Poisson variates with constant rate over the 29 years? Respond by comparing a Poisson GLM for collision rates that contains only an intercept term to a Poisson GLM that contains also a

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An Introduction To Categorical Data Analysis 2nd Edition Alan Agresti - Solutions

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