Statistical Analysis With Missing Data 3rd Edition Roderick J. A. Little, Donald B. Rubin - Solutions

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1.3. What assumptions about the missing-data mechanism are implied by the statistical analyses used in Problem 1.2? Do these assumptions appear realistic?
1.2. List methods for handling missing values in an area of statistical application of interest to you, based on experience or relevant literature.
1.1. Find the monotone pattern for the data of Table 1.1 that involves minimal deletion of observed values. Can you think of better criteria for deleting values than this one?
define the following
15.17 Repeat the calculations in Problem 15.16 for the MNAR model in Table 15.12.
15.16 Reproduce the Bayes’ estimates for the ignorable model in Table 15.12 by data augmentation, and provide a histogram of draws. Use 10 draws of the missing data under DA to create multiple imputations of the missing data, and apply the general combining rules to draw an inference for the
15.15 Reproduce the EM estimates for the ignorable model in Table 15.12.Use the bootstrap to estimate the standard error of the estimated proportion voting yes, and compare itwith the large sample standard error.
15.13 Verify the five sets of estimated cell probabilities in Table 15.10.15.14 Redo Table 15.10 for the data in Table 15.9 with m1 and m2 multiplied by a factor of 10.
15.12 For suitable parameterizations of the models, write down factored likelihoods for the models {Y1Y2, Y1M}, {Y1Y2, Y2M}, {Y1M, Y1M},{Y1, Y2M} in Example 15.17. State for each model which parameters (if any) are inestimable, in the sense that they do not enter the likelihood.
12⋅2. Deduce situations where the available case estimate is the preferred estimate (see Little 1994).
12 ); the IML estimate when ????=0; and the available case estimate ????̂1 − ȳ2 when ???? = −????(0)
15.11 For the pattern-mixture model (15.15) where missingness of Y2 depends on Y1 + ????Y2, show that the ML estimate of c1????1 + c2????2 is c1 ̄y1 + c2 ̄y2 + (c1 + c2b(̂ ????)21⋅1)(????̂1 − ȳ1).Hence, show that theML estimate of ????1 − ????2 is the complete case estimate, ̄y1 −
12⋅2, substituting the ML estimate of ????(0)12⋅2 in Eqs.(15.27)–(15.29) yields complete-case estimates. That is, if ???? = −????(0)12⋅2 is thought to be more plausible than ????=0, then the complete case estimate of ????2 is better than the ML estimate assuming ignorable nonresponse.
15.10 Show that if ???? = −????(0)
15.9 Fill in the details leading to the ML estimates (15.27)–(15.29) for the pattern-mixture model (15.15) under restriction (15.26).
15.8 Fill in the details leading to the ML estimates (15.22)–(15.24) for the pattern-mixture model (15.15) under restrictions (15.20).404 15 Missing Not at RandomModels
15.7 In Example 15.13, it is shown that, for the pattern-mixture model(15.15) with MAR restrictions (15.18), the ML estimate of ????2 is the same as for the ignorable selection model in Section 7.2.1. Show explicitly that this statement also applies for the mean of Y1 and the covariance matrix of
15.6 Derive the expressions for the posterior mean and variance of the finite population mean estimand ̄y in Example 15.10. What is the posterior mean and variance of variable 32D when ????1 = ????2 = 0.5?
15.5 Suppose that for the model of Example 15.7, a random subsample of nonrespondents to Y1 is followed up and values of Y obtained. Write the likelihood for the resulting data and describe the E and M steps of the EM algorithm.
15.4 Review the two-step fitting method for the model of Example 15.7 of Heckman (1976). Contrast the assumptions made by that method and by the ML fitting procedure in Problem 15.3 (see e.g., Little 1985a).
15.7.
15.3 Derive the E and M steps of the EM algorithm for the model in Example
15.2 Derive the expressions for the E step in Example 15.4. Also display the M step for this example explicitly.
15.1 Carry out the integrations needed to derive the E step in Example 15.3.
14.12 Derive Eq. (14.18) from Eqs. (14.1) and (14.2), and hence express the parameters {????d0, ????dj} as functions of the general location model parametersθ = (Π, Γ, Ω). Consider the impact of imposing constraints on θand on the parameters of the logistic model (14.18).
14.11 Derive Eq. (14.17) from Eqs. (14.1) and (14.2), and hence express the parameters {????c0, ????j, ????2} as functions of the general location model parameters θ = (Π, Γ, Ω). Consider the impact of imposing constraints on θ, the parameters of the linear model (14.17).
14.10 Describe the Bayesian analog of the simplified ML algorithm in Section 14.3.5.Problems 349
14.9 Derive themaximized loglikelihood of the data in Problem 14.7 for the models of Problems 14.7 and 14.8, and hence derive the likelihood ratio statistic for testing independence of race and sex. Note that the sample size is too small for this statistic to be considered distributed as chisquared
14.8 Repeat (b) of Problem 14.7, with the restriction that the variables race and sex are independent.
(b) Develop explicit formulas for the E and M steps (14.5)–(14.8) for these data, and carry out three steps of the EM algorithm, starting from estimates found in (a).
(a) Compute ML estimates for the general location model applied to these data, based on complete units only.
14.7 A survey of 20 graduates of a university class five years after graduation yielded the following results for the variables sex (1 = male, 2 = female), race (1 = white, 2 = other), and annual income,measured on a log scale(? denotes missing):Unit 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
14.6 Derive the expressions in Section 14.2.3 for the conditional expectations of wimxij and xijxik given x(0),i, Si and θ(t), from properties of the general location model.
14.5 Using Bayes’ theorem, show that Eq. (14.9) follows from the definition of the general location model, Eqs. (14.1) and (14.2).
14.4 Compare the properties of discriminant analysis and logistic regression for classifying units into groups on the basis of known covariates (see, for example Press andWilson 1978; Krzanowski 1980, 1982).
14.3 Suppose that in Problem 14.2, X is fully observed and Y has somemissing values. Show that ML estimates for the general location model cannot be found by factoring the likelihood, because the parameters of the appropriate factorization are not distinct. Suggest an alternative model for which
14.2 Using the factored likelihood methods of Chapter 7, derive ML estimates of the general location model for the special case of one fully observed categorical variable Y and one continuous variable X with some missing values.
14.1 Show that Eq. (14.4) provides ML estimates of the parameters for the complete-data loglikelihood Eq. (14.3).
(b) Show that (13.7) is asymptotically equal to Var(????jk − ̂????jk) ≈̂????jk(1 − ̂????jk)r[1 −̂????k⋅j − ̂????jk 1 − ̂????jk n − r n].Hence, state the proportionate reduction in asymptotic sampling variance of ̂????jk over the complete-case estimate, and describe situations
(a) Show that cj in (13.7) is of smaller order than other terms in the expression.
13.17 Consider bivariate monotone data as in Section 13.2, and suppose the data are MCAR.
13.16 Why can starting values including zero probabilities disrupt proper performance of EM? (Hint: Consider the loglikelihood.)
13.15 Compute ML estimates for the model {SP, SC} for the full data in Table 13.8, with the counts in the supplemental Table 13.8(b) increased by a factor of 10.
13.14 Using results from Problem 13.13, derive the estimates in Table 13.9 for the models {SPC}, {SC, PC}, and {SP, SC}.
13.13 Display explicitML estimates for all themodels in Table 13.7 except for{12, 23, 31}.
13.11 Implement the EM algorithm for the data in Table 13.5 with values superscripteda, b andc, d in the supplemental margins interchanged.Compare theMLestimateof the odds ratio????11????22????−1 12 ????−1 21 with the estimate from complete units. Are they identical?328 13 Models for Partially
13.10 Redo Example 13.4 assuming that the coarsely classified data in Table 13.4 were summarized as “Improvement” or “No Improvement”(stationary or worse).
13.9 Replicate the calculations of Example 13.4 for estimates of ????12.
13.8 Fill in the details in the derivation of Eq. (13.7).
13.7 State inwords the assumption about the missingness mechanismunder which the estimates in Table 13.4(c) are ML for Example 13.4.
13.6 Suppose that in Example 13.3 there are no units with patternd. Which parameters are inestimable, in the sense that they do not appear in the likelihood? Estimate the cell probabilities, assuming specific values for the inestimable parameters.
13.5 Calculate the expected cell frequencies in the first column of data in Table 13.3• (b), and compare the answers with those obtained from complete units.
13.4 Compute the fraction of missing information in Example 13.2, using the methods of Section 9.1.
13.3 Verify the results of the LR test for the MCAR assumption in Example 13.2.
13.2 Derive ML estimates and associated asymptotic sampling variances for the likelihood (13.1). (Hint: Remember the constraint that the cell probabilities sum to 1.)
13.1 Show that for complete data, the Poisson and multinomial models for multiway count data yield the same likelihood-based inferences for the cell probabilities. Show that the result continues to hold when data are MAR.
12.7 Derive the E step equations in Section 12.2.3.
12.6 Derive the weighting functions (12.11) and (12.13) for the models of Section 12.2.2.
12.5 Extend the t model with known degrees of freedom and the contaminated normal to the case of simple linear regression of X on a fixed observed covariate Z. Derive the EMalgorithm for this model. Do units with Z observed, but X missing, contribute information? (Hint: Review Section 11.4.1.)
12.1 with (a) ???? known and ???? unknown and estimated by ML, and (b)???? unknown and estimated byMLand ???? known. (Does the case with both???? and ???? unknown involve toomuch missing information to be practical?)
12.4 Explore ML estimation for the contaminated normal model of Example
12.3 Outline a program to compute ML estimates for the contaminated normal model of Example 12.4. Simulate data from the contaminated normal model and explore sensitivity of inferences to different choices of true and assumed values of ???? and ????.
12.2 Describe the data augmentation algorithm for Example 12.1.
12.1 Derive the weighting function (12.5) for the model of Example 12.1.
11.13 Develop a Gibbs’ sampler for simulating the posterior distributions of the parameters and predictions of the {zi} for Example 11.9. Compare the posterior distributions for the predictions for years 1949 and 1981 with the EM predictions in the last two columns of Table 11.6.
11.12 For Example 11.8, extend the results of Problem 11.11 to compute the means, variances, and covariance of yj+1 and yj+2 given yj, yj+3 and θ, for a sequence where yj and yj+3 are observed, and yj+1 and yj+2 are missing.
11.11 Fill in the details leading to the expressions for the mean and variance of yj+1 given yj, yj+2, and θ in Example 11.8. Comment on the form of the expected values of yj+1 as ???? ↑1 and ???? ↓ 0.
11.10 Examine Beale and Little’s (1975) approximate method for estimating the covariancematrix of estimated slopes in Section 11.4.2, for a single predictor X, and data with (a) Y completely observed and X subject to missing values, and (b) X completely observed and Y subject tomissing values.
11.9 Derive the EMalgorithm for the model of Example 11.4 extended with the specification that ???? ∼ N(0, ????2), where ???? is treated as missing data.Then consider the case where ????2→∞, yielding a flat prior on ????.
11.8 Review the discussion in Rubin and Thayer (1978, 1982) and Bentler and Tanaka (1983) on EM for factor analysis.
11.7 Prove the statement before Eq. (11.9) that complete-data ML estimates of Σ are obtained from C by simple averaging. (Hint: Consider the covariance matrix of the four variables U1 = Y1 +Y2 +Y3 +Y4, U2 = Y1 −Y2 +Y3 −Y4, U3 = Y1 −Y3, and U4 = Y2 −Y4.)
(b) a bivariate sample of size r, and effectively infinite supplemental samples from the marginal distributions of both variables. Note the rather surprising fact that (a) and (b) yield different answers.
11.6 For bivariate data, find the ML estimate of the correlation ???? for (a)a bivariate sample of size r, with known means (????1, ????2) and known variances (????2 1 , ????2 2 ), and
11.2.2, for the special case of bivariate data.
11.5 Derive the expression for the expected information matrix in Section
11.4 Describe the EM algorithm for bivariate normal data with means(????1, ????2), correlation ????, and common variance ????2, and an arbitrary pattern of missing values. If you did Problem 11.2, modify the program you wrote to handle this model. (Hint: For the M step, transform to U1 = Y1 +Y2, U2
11.2 Write a computer program for the EM algorithm for bivariate normal data with an arbitrary pattern of missing values.11.3 Write a computer program for generating draws from the posterior distribution of the parameters, for bivariate normal data with an arbitrary pattern of missing values, and a
11.1 Show that the available-case estimates of the means and variances of an incomplete multivariate sample, discussed in Section 3.4, are ML when the data are specified as multivariate normal with unrestricted means and variances, and zero correlations, with ignorable nonresponse.(This result
10.5 Modify the multiple-imputation approach of Problem 10.4 to give the correct answer for large r and N/r. (Hint: For example, add a random residual sRr−1/2zd to the imputed value for unit i.)
(e) Assume r and N/r are large, and tabulate true coverages and significance levels of themultiple-imputation inference. Compare with the results in Problem 10.3, part (d).
(c) Tabulate values of the relative efficiency of y∗ to yR for different values of D, assuming large r and large N/r.(d) Show that the sampling variance of y∗ (conditional on n, r, and the population Y-values) is greater than the expectation of T∗ by approximately s2 R(1 − r∕n)2∕r.
(b) Show that the variance of y∗ (conditional on n, r, and the population Y-values) is D−1 Var( y∗) + (1 − D−1) Var(yR), and conclude that y∗is more efficient than the single-imputation estimate y∗.
(a) Show that, conditional on the data, the expected value of B∗ equals the variance of the y(d)∗ .
10.4 Suppose D multiple imputations are created using the method of Problem 10.3, and let y(d)∗ and U(d)∗ be the values of y∗ and U∗ for the dth completed dataset. Let y∗ =ΣDd=1 y(d)∗ ∕D, and T∗ be themultiple-imputation estimate of the sampling variance of the y∗. That is T∗ =
(d) Assume r and N/r are large, and show that interval estimates of Y based on using U∗ as the estimated sampling variance of y∗ are too short by a factor (1+nr−1 −rn−1)1/2. Note that there are two reasons:n>r, and y∗ is not as efficient as yR. Tabulate true coverages and true
(c) Show that conditional on the sample sizes n and r (and the population Y-values), the sampling variance of y∗ is the variance of yR times (1+(r−1)n−1(1−r/n)(1−r/N)−1), and show that this is greater than the expectation of U∗ = s2∗(n−1 − N−1).
(b) Show that conditional on the observed data, the sampling variance of y∗ is ms2 R(1 − r−1)∕n2, and that the expectation of s2∗ is s2 R(1 − r−1)(1 + rn−1(n − 1)−1).
(a) Show that y∗ is unbiased for the population mean Y.
10.3 Suppose in Problem 10.2 that imputations are randomly drawn with replacement from the r respondents’ values. Assume the missing data are MACAR.
10.2 Consider a simple random sample of size n sampled froma finite population of size N, with r respondents andm= n−r nonrespondents, and let yR and s2 R be the samplemean and variance of the respondents’ data, and yNR and s2 NR the sample mean and variance of the imputed data. Show that the
10.1 Reproduce the posterior distribution in Figure 10.1, and compare the posterior mean and standard deviation with that given in Table 10.1.Recalculate the posterior distribution of θ using the improper prior distribution with ????1 = ????2 = 0. Is the resulting posterior distribution proper?
9.8 Using the reasons given at the end of Section 9.2.1, explain why SEM is more computationally stable than simply numerically differentiating????(θ ∣ Y(0)) twice.
9.7 Suppose the model is misspecified, but the ML estimate found by EM is a consistent estimate of the parameter θ. Which method of estimating the large-sample covariance matrix of (???? − ̂????) is preferable? Explain your reasoning.
9.6 Suppose PX-EM is used to find the ML estimate of θ in Example 8.10.Further suppose that SEM is applied to the sequence of PX-EM iterates, assuming the algorithm was EM. Would the resulting estimated of standard errors more likely be over or underestimates of the corresponding asymptotic
9.5 The SEM algorithm can be extended to the SECM algorithm when ECM is used rather than EM. Details are provided in Van Dyk et al. (1995), but it ismore complicated than SEM. Describe howthe bootstrap can be used to estimate the variance of (???? − ̂????), where ̂????is theML estimate of θ
9.4 Compute standard errors for the data in Table 7.1 using the bootstrap, and compare the results with the standard errors in Table 9.1.
9.3 Suppose ECMis used to find the ML estimate of θ in Example 8.6. Further suppose that SEM is applied to the sequence of ECM iterates, assuming they were EMiterates. Are the iterates likely to converge more quickly or more slowly than EM?Would the resulting asymptotic covariancematrix more
9.2 Apply SEM to Example 9.2, but without the normalizing transformations on ????22 and ????. Compare the intervals for ????22 and ???? based on the results of Example 9.2 and the results in this problem. In theory, which are preferable?
9.1 In Example 9.1, show how the EM and SEM answers were obtained for logit(̂????). Compare the interval estimates for θ using EM/SEM on the raw and logit scales.
8.16 Suppose that (a) X is Bernoulliwith Pr(X = 1)= 1−Pr(X = 0) = ????, and(b)Y given X = j is normal with mean ????j, variance ????2, a simple formof the discriminant analysis model. Consider now the monotone missing-data pattern with Y completely observed but n−r values of X missing, and an

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Statistical Analysis With Missing Data 3rd Edition Roderick J. A. Little, Donald B. Rubin - Solutions

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