New Semester Started Get 50% OFF Study Help! --h --m --s Claim Now

Question Answers

Textbooks

Find textbooks, questions and answers

Oops, something went wrong!

Change your search query and then try again

Result Not Found

Books FREE
Study Help
Tutors
AI Study Planner NEW
Sell Books

Search
Sign In
Register

study help
business
statistical techniques in business

Statistical Analysis With Missing Data 2nd Edition Roderick J. A. Little, Donald B. Rubin - Solutions

7.18. Outline extensions of Problem 7.16 to multivariate monotone patterns where the factored likelihood method works.
7.17. For the model of Problem 7.16, consider now the reverse monotone missingdata pattern, with Y completely observed but n r values of X missing, and an ignorable mechanism. Does the factored likelihood method provide closedform expressions for ML estimates for this pattern? (Hint: find the
7.16. (i) Consider the following simple form of the discriminant analysis model for bivariate data with binary X and continuous Y: (a) X is Bernoulli with PrðX ¼ 1Þ ¼ 1 Prðx ¼ 0Þ ¼ p, and (b) Y given X ¼ j is normal with mean mj , variance s2 ( j ¼ 1; 0). Derive ML estimates of ðp, m0,
7.15. If data are MAR and the data analyst discards values to yield a data set with all complete-data factors, then are the resultant missing data necessarily MAR? Provide an example to illustrate important points.
7.14. Create a factorization table (see Rubin, 1974) for the data in Example 7.11. State why the estimates produced in Example 7.11 are ML.
7.13. Estimate the parameters of the distribution of X1, X2, X3, and X5 in Example 7.8, pretending X4 is never observed. Would the calculations be more or less work if X3 rather than X4 was never observed?
7.12. Show how to compute partial correlations and multiple correlations using SWP.
7.11. Show that RSW is the inverse operation to SWP.
7.10. Prove that SWP is commutative and conclude that the order in which a set of sweeps is taken is irrelevant algebraically. (However, it can be shown that the order can matter for computational ease and accuracy.)
7.9. Using the computer or otherwise, generate a bivariate normal sample of 20 cases with parameters m1 ¼ m2 ¼ 0, s11 ¼ s12 ¼ 1, s22 ¼ 2, and delete values of Y2 so that Prðy2 missing j y1, y2) equals 0.2 if y1 < 0 and 0.8 if y1 0. (a) Construct a test for whether the data are MCAR and
7.8. Show that the factorization of Example 7.1 does not yield distinct parameters ffjg for a bivariate normal sample with means ðm1, m2), correlation r and common variance s2, with missing values on Y2.
7.7. Show that for the set-up of Problem 7.6, the estimate of b122 obtained by maximizing the complete-data loglikelihood over parameters and missing data is ~ b122 ¼ ^ b122s^ 22=s^*22, where (in the notation of Section 7.2), s^*22 ¼ ^ b2112s^ 11 þ n1 Pr i¼1 ½yi2 y2 ^ b211ðyi1
7.6. Compute the large sample variance of ^ b122 in Problem 7.5, and compare with the variance of the complete-case estimate, assuming MCAR.
7.5. For the bivariate normal distribution, express the regression coefficient b122 of Y1 on Y2 in terms of the parameters f in Section 7.2, and hence derive its ML estimate for the data in Example 7.2.
7.4. Prove the six results on Bayes inference for monotone bivariate normal data after Eq. (7.17) in Section 7.3 (For help, see Chapter 2 of Box and Tiao (1973), and the material in Section 6.1.4.)
7.3. Compare the asymptotic variance of m^ 2 m2 given by (7.13) and (7.14) with the small-sample variance computed in Problem 7.2.
7.2. Assume the data in Example 7.1 are MCAR. By first conditioning on ðy11; ... ; yn1Þ, find the exact small sample variance of m^ 2. (Hint: If u is chisquared on d degrees of freedom, then Eð1=uÞ ¼ 1=ðd 2Þ.) (See Morrison, 1971.) Hence show that m^ 2 has a smaller sampling variance than
7.1. Assume the data in Example 7.1 are MAR. Show that given ðy11; ... ; yn1Þ, ^ b201 and ^ b211 are unbiased for b201 and b211. Hence show that m^ 2 is unbiased for m2.
6.20. The definition of MAR can depend on how the complete data are defined. Suppose that X ¼ ðxi ; ... ; xnÞ is an iid random sample, Z ¼ ðz1; ... ;znÞ are completely unobserved latent variables, and ðxi , ziÞ are bivariate normal with means ðmx, 0), variances ðs2 x ; 1Þ and correlation
6.19. Describe ML estimates of the parameters in Example 6.24 under (a) the fixed effects model of Eq. (6.56) with fixed parameters ðfmi g, s2Þ and (b) the random effects model of Eqs. (6.53)–(6.55) with parameters y ¼ ðm, s2, t2Þ, assuming ignorable nonresponse. Show that for the mechanism
6.18. Suppose that given sets of (possibly overlapping) covariates X1 and X2, yi1 and yi2 are bivariate normal with means xi1b1 and xi2b2, variances s2 1 and s2 2 ¼ 1, and correlation r. The data consist of a random sample of observations i with xi1 and xi2 always present, yi2 always missing and
6.17. For a bivariate normal sample on ðY1; Y2Þ with parameters y ¼ ðm1, m2, s11, s12, s22) and values of Y2 missing, state for the following missing-data mechanisms whether (i) the data are MAR, and (ii) the missing-data mechanism is ignorable for likelihood-based inference. (a) Prðy2 missing
6.16. Find large-sample variance estimates for the two ML estimates in Example 6.22.
6.15. Suppose the following data are a random sample of n ¼ 7 from the Cauchy distribution with median y: Y ¼ ð4:2, 3:2, 2:0, 0.5, 1.5, 1.5, 3.5). Compute and compare 90% intervals for y using (i) the asymptotic distribution based on the observed information, (b) the asymptotic distribution
intercept (b0 ¼ 0), a single covariate X, and weight for observation i wi ¼ xi, the ratio estimator y=x is (a) the ML estimate of b1, and (b) the posterior mean of b1 when the prior distribution is pðb1, log s2Þ ¼ const.
6.14. Derive the modifications of the posterior distributions in Eqs. (6.30–6.32) for weighted linear regression, discussed at the end of Example 6.16. Show that for special case of weighted linear regression with no
6.13. Derive the posterior distributions in Eqs. (6.30–6.32) for Example 6.16.
6.11. In Example 6.14, show that for large n, LR ¼ t 2. 6.12. Derive the posterior distributions in Eqs. (6.22–6.24) for Example 6.15.
6.10. Show that, for random samples from ‘‘regular’’ distributions (differentials can be passed through the integral), the expected squared score function equals the expected information.
6.9. For the distributions of Problem 6.1, calculate the observed information and the expected information.
6.8. Summarize the theoretical and practical differences between the frequentist and Bayesian interpretation of Approximation 6.1. Which is closer to the direct likelihood interpretation?
6.7. Show, by similar arguments to those in Problem 6.6, that for the model of Eq. (6.9), Varðyijdi; fÞ ¼ fb00ðdiÞ, where di ¼ dðxi; bÞ, and double prime denotes differentiation twice with respect to the function argument.
6.6. Show that for the GLIM model of Eq. (6.9), Eðyij xi; bÞ ¼ b0 ½dðxi; bÞ, where prime denotes differentiation with respect to the function argument. Conclude that the canonical link Eq. (6.12) is obtained by setting g1ð Þ ¼ b0 ð Þ. (Hint: consider the density of yi in Eq. (6.9) as a
6.5. Suppose the data are a random sample of size n from the uniform distribution between 0 and y; y > 0. Show that the ML estimate of y is the largest data value. (Hint: differentiation of the score function does not work for this problem!) Find the posterior mean of y, assuming a uniform prior on
6.4. (a) Relate ML and least squares estimates for the model of Example 6.10. (b) Show that if the data are iid with the Laplace (double exponential) distribution, f ðyi jyÞ ¼ 0:5 expðj yi mðxi ÞjÞ; where mðxiÞ ¼ b0 þ b1xi1 þ þ bpxip, then ML estimates of b0; ... ; bp are obtained
6.3. For a univariate normal sample, find the ML estimate of the coefficient of variation, s=m.
6.2. Find the score function for the distributions in Problem 6.1. Which have closed-form ML estimates? Find the ML estimates for those distributions that have closed-form estimates.
6.1. Write the likelihood function for an iid sample from the (a) beta distribution; (b) Poisson distribution; (c) Cauchy distribution.
5.14. Is MI better than single imputation of a draw from the predictive distribution of the missing values (SI) because (i) It yields more efficient estimates from the filled-in data? (ii) Unlike SI, it yields consistent estimates of quantities that are not linear in the data? (iii) It allows valid
5.13. Is multiple imputation (MI) better than imputation of a mean from the conditional distribution of the missing value because (i) It yields more efficient estimates from the filled-in data? (ii) It yields consistent estimates of quantities that are not linear in the data? (iii) It allows valid
5.12. (a) Modify the multiple imputation approach of Problem 5.11 to give the correct inferences for large r and N=r. Hint: For example, add sRr1=2zd to the imputed values for the dth multiply-imputed dataset, where zd is a standard normal deviate. (b) Justify the adjustment in (a) based on the
5.11. Suppose multiple imputations are created using the method of Problem 5.10 D times, and let y ðdÞ and UðdÞ be the values of y and U for the dth imputed data set. Let y ¼ PðDÞ d¼1 y ðdÞ =D, and T be the multiple imputation estimate of variance of y. That is, T ¼
5.10. Suppose in Problem 5.9, imputations are randomly drawn with replacement from the r respondents’ values. (a) Show that y is unbiased for the population mean Y . (b) Show that conditional on the observed data, the variance of y is ms2 Rð1 r1Þ=n2, and that the expectation of s2 is
5.9. Consider a simple random sample of size n with r respondents and m ¼ n r nonrespondents, and let yR and s2 R be the sample mean and variance of the respondents’ data, and yNR and s2 NR the sample mean and variance of the imputed data. Show that the mean and variance y and s2 of all
5.6. Repeat Problem 5.4 or 5.5 for D ¼ 2, 5, 10, 20, and 50 multiple imputes, and compare answers. For what value of D does the inference stabilize? PROBLEMS 91 5.7. Repeat Problem 5.4 or 5.5 using the more refined degrees of freedom formula (5.24) for the multiple imputation inference, and
5.5. As discussed in Section 5.4, the imputation method in Problem 5.4 is improper, since it does not propagate the uncertainty in the regression parameter estimates. One way of making it proper is to compute the regression parameters for each set of imputations using a bootstrap sample of the
5.4. For the data in Problem 5.3, create 10 multiply-imputed data sets with different sets of conditional draws of the parameters, using the method of Problem 5.3. Compute 90% confidence intervals for the mean and coefficient of variation of Y2 using Eqs. (5.17)–(5.23), and compare with the
5.3. Repeat Problem 5.2, with the same observed data, but with missing values imputed using conditional draws rather than conditional means. That is, add a random normal deviate with mean zero and variance given by the estimated residual variance to the conditional mean imputations.
5.2. Create missing values of Y2 for the data in Problem 5.1 by generating a latent variable U with values ui ¼ 2*ðyi1 1Þ þ zi3, where zi3 is a standard normal deviate, and setting yi2 as missing when ui < 0. Since U depends on Y1 but not Y2 this mechanism is MAR, and since U has mean zero,
5.1. As in Problem 1.6, generate 100 bivariate normal observations fðyi1; yi2Þ, i ¼ 1; ... ; 100g on ðY1; Y2Þ as follows: yi1 ¼ 1 þ zi1 yi2 ¼ 5 þ 2*zi1 þ zi2; 90 ESTIMATION OF IMPUTATION UNCERTAINTY where fðzi1;zi 2 Þ, i ¼ 1; ... ; 100g are independent standard normal (mean 0, variance
4.15. For the artificial data sets generated for Problem 1.6, compute and compare estimates of the mean and variance of Y, from the following methods: (a) Complete-case analysis; (b) Buck's method, imputing the conditional mean of Y given Y from the linear regression based on complete cases; (c)
4.14. Outline a situation where the "Last Observation Carried Forward" method of Example 4.11 gives poor estimates. (See, for example, Little and Yau, 1996).
4.13. Which of the metrics in Example 4.9 give the best imputations for a particular outcome Y? Propose an extension of the predictive mean matching metric to handle a set of missing outcomes Y..... Yk. (See Little, 1986, for details).
4.12. Another method for generating imputations is the sequential hot deck, where responding and nonresponding units are treated in a sequence, and a missing value of Y is replaced by the nearest responding value preceding it in the sequence. For example, if n = 6, r = 3, y, y4, and ys are present
4.11. Consider a hot deck like that of Example 4.7, except that imputations are by random sampling of donors without replacement. To define the procedure when there are fewer donors than recipients, write n r ¼ kr þ t, where k is a non-negative integer and 0 < t < r. The hot deck without
4.10. Derive the expressions (4.6)-(4.8) for the simple hot deck where imputations are by simple random sampling with replacement. Show that the proportionate variance increase of YHDI over y is at most 0.25, and this maximum is attained when r/n = 0.5. How do these numbers change when the hot deck
4.9. Derive the expressions for large-sample variance of the Cmean and Cdraw estimates of , in the discussion of Example 4.5.
4.8. Derive the expressions for large-sample bias in Table 4.1.
4.7. Buck's method (Example 4.3) might be applied to data with both continuous and categorical variables, by replacing the categorical variables by a set of dummy variables, numbering one less than the number of categories. Consider properties of this method when (a) the categorical variables are
4.6. Show that Buck's (1960) method yields consistent estimates of the means when the data are MCAR and the distribution of the variables has finite fourth moments.
4.5. Suppose data are an incomplete random sample on Y1 and Y2, where Y1 given y ¼ ðm1, s11, b2013, b2113, b2313, s2213) is Nðm1, s11) and Y2 given Y1 and y is Nðb2013 þ b2113Y1 þ b23:13Y2 1 , s2213). The data are MCAR, the first r cases are complete, the next r1 cases record Y1 only,
4.4. Derive the expressions for the biases of Buck’s (1960) estimators of sjj and sjk , stated in Example 4.3.
4.3, and compare the answers. 4.3. Describe the circumstances where Buck’s (1960) method clearly dominates both complete-case and available-case analysis.
4.2. Assuming MCAR, determine the percentage bias of estimates of the following quantities computed from the filled-in data: (a) the variance of Y1 ðs11Þ; (b) the covariance of Y1 and Y2 ðs12Þ; (c) the slope of the regression of Y2 on Y1 ðs12=s11Þ. You can ignore bias terms of order 1=n. 4.2.
4.1. Consider a bivariate sample with n ¼ 45, r ¼ 20 complete cases, 15 cases with only Y1 recorded, and 10 cases with only Y2 recorded. The data are filled in using unconditional means, as in Section
3.18. Review the results of Haitovsky (1968), Kim and Curry (1977), and Azen and Van Guilder (1981). Describe situations where complete-cases analysis is more sensible than available-case analysis, and vice versa.
3.17. Consider the relative merits of complete-case analysis and available-case analysis for estimating (a) means, (b) correlations, and (c) regression coefficients when the data are not MCAR.
3.15. Construct a data set where the estimated correlation (3.18) lies outside the range (1, 1). 3.16. (a) Why does the estimated correlation (3.19) always lie in the range (1, 1)? (b) Suppose the means y ð jkÞ j and y ð jkÞ k in (3.17) are replaced by estimates from all available cases. Is
3.13, compute the odds ratio of response rates discussed in Problem 3.12. Repeat the computation with the respondent counts 5 and 8 in the second row of (b) in Problem 3.13 interchanged. By comparing these odds ratios, predict which set of the raked respondent counts will be closer to the raked
3.14. For the data in Problem
3.13. Compute raked estimates of the class counts from the sample counts and respondent counts in (a) and (b) below, using population marginal counts in (c):(a) sample fnjlg (b) respondent frjlg (c) population fNjlg 8 10 18 5 9 14 ? ? 300 15 17 32 5 8 13 ? ? 700 23 27 50 10 17 27 500 500 1000
3.12. Show that raking the class sample sizes and raking the class respondent sample sizes (as in the previous example) yields the same answer if and only if pijpkl=ðpilpjk Þ ¼ 1; for all i; j; k and l; where pij is the response rate in class (i; j) of the table.
3.11. Oh and Scheuren (1983) propose an alternative to the raked estimate yrake in Section 3.3.3, where the estimated counts N*jl are found by raking the respondent sample sizes frjlg instead of fnjlg. Show that (i) unlike yrl, this estimator exists when rjl ¼ 0, njl 6¼ 0 for some j, l, and
3.10. Generalize the response propensity method in Example 3.7 to a monotone pattern of missing data. (See Little, 1986; Robins, Rotnitsky and Zhao 1995).
3.9. The following table shows respondent means of an incomplete variable Y (say, income in $1000), and response rates (respondent sample size=sample size), classified by three fully observed covariates: Age (30), marital status (single, married), and gender (male, female). Note that weighting
3.8. Apply the Cassell, Sarndal and Wretman (1983) estimator discussed in Example 3.7 to the data of Problem 3.7. Comment on the resulting weights as compared with those of the weighting class estimator.
3.7. Calculate Horvitz–Thompson and weighting class estimators in the following artificial example of a stratified random sample, where the xi and yi values displayed are observed, the selection probabilities pi are known, and the response probabilities fi are known for the Horvitz–Thompson
3.6. Suppose census data yield the following age distribution for the county of interest in problems 3.4 and 3.5: 20–30: 20%; 30–40: 40%; 40–50: 30%; 50– 60: 10%. Calculate the post-stratified estimate of mean cholesterol, its associated standard error, and a 95% confidence interval for the
3.5 Compute the weighting class estimate (3.6) of the mean cholesterol level in the population and its estimated mean squared error (3.7). Hence construct an approximate 95% confidence interval for the population mean and compare it with the result of Problem 3.4. What assumptions are made about
3.4. Compute the mean cholesterol for the respondent sample and its standard error. Assuming normality, compute a 95% confidence interval for the mean cholesterol for respondents in the county. Can this interval be applied to all individuals in the county?
3.3. Show that for dichotomous Y1 and Y2, the odds ratio based on complete cases is a consistent estimate of the population odds ratio if the log probability of response logarithm of the probability of response is an additive function of Y1 and Y2. Data for Problems 3.4–3.6: A simple random
3.2. Show that if missingness (of Y1 or Y2) depends only on Y2, and Y1 has a linear regression on Y2, then the sample regression of Y1 on Y2 based on complete cases yields unbiased estimates of the regression parameters.
3.1. List some standard multivariate statistical analyses that are based on the sample means, variances, and correlations.
Squarea Column Row1 2 3 4 5 1 B: — E: 230 A: 279 C: 287 D: 202 2 D: 245 A: 283 E: 245 B: 280 C: 260 3 E: 182 B: — C: 280 D: 246 A: 250 4 A: — C: 204 D: 227 E: 193 B: 259 5 C: 231 D: 271 B: 266 A: 334 E: 338 a Spacings (in.): A, 2; B, 4; C, 6; D, 8; E, 10.
2.13. Carry out a standard ANOVA for the following data, where three values have been deleted from a (5 5) Latin square (Snedecor and Cochran, 1967, p. 313). Yields (Grams) of Plots of Millet Arranged in a Latin
2.12. Carry out the computations leading to the results of Example 2.3.
2.11. Carry out the computations leading to the results of Example 2.2.
2.10. Show Eq. (2.22) and then Eq. (2.23) and (2.24).
2.9. Justify Eqs. (2.17)–(2.20).
2.8. Carry out the computations leading to the results of Example 2.1.
2.7. Using the notation and results of Section 2.5.4, justify Eq. (2.16) and the method for calculating B and r that follows it.
2.6. Provide intermediate steps leading to Eqs. (2.13), (2.14), and (2.15).
2.5. Prove that Eq. (2.12) follows from the definition of U1.
2.4. Summarize the argument that Bartlett’s ANCOVA method leads to correct least squares estimates of missing values.
2.3. Outline the distributional results leading to Eq. (2.6) being distributed as F.
2.2. Prove that b^ in Eq. (2.2) is (a) the least squares estimate ofb, (b) the minimum variance unbiased estimate, and (c) the maximum likelihood estimate under normality. Which of these properties does s^ 2 possess, and why?
2.1. Review the literature on missing values in ANOVA from Allan and Wishart (1930) through Dodge (1985).
1.6. One way to understand missing data mechanisms is to generate hypothetical complete data and then create missing values by specific mechanisms. This is common in simulation studies of missing-data methods, since the deleted values can be retained and used to compare methods; missing-data
1.5. Let Y ¼ ðyijÞ be a data matrix and let M ¼ ðmijÞ be the corresponding missingdata indicator matrix, where mij ¼ 1 indicates missing and mij ¼ 0 indicates present. (a) Propose situations where two values of mij are not sufficient. (Hint: see Heitjan and Rubin, 1991). (b) Nearly always
1.4. What impact does the occurrence of missing values have on (a) estimates and (b) tests and confidence intervals for the analyses in Problem 1.2? For example, are estimates consistent for underlying population quantities, and do tests have the stated significance levels?

Showing 900 - 1000 of 5757

First
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
Last

Step by Step Answers

Services

Sitemap
Fun
Definitions
Become Tutor
Used Textbooks
Study Help Categories
Recent Questions
Expert Questions
Campus Wear
Sell Your Books

Company Info

Security
Copyrights
Privacy Policy
Terms & Conditions
SolutionInn Fee
Scholarship
Online Quiz
Give Feedback, Get Rewards

Get In Touch

About Us
Contact Us
Career
Jobs
FAQ
Student Discount
Campus Ambassador

Secure Payment

Download Our App

© 2026 SolutionInn. All Rights Reserved