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

8.15 Prove the PX-E and PX-M steps in Example 8.10.
8.14 Write the complete-data loglikelihood in Example 8.6, and verify the two CM steps, Eqs. (8.34) and (8.35), in that example.
Find the observed information for ???? by computing the unconditional and conditional variance of s(Y ) and subtracting, as in (8.32). Hence, find the proportion of missing information from the censoring, and the asymptotic sampling variances of ̂????− ???? and ̂????− ????.
8.13 For the censored exponential sample in the second part of Example 6.22, suppose y1,…, yr are observed and yr+1,…, yn are censored atc. Show that the complete-data sufficient statistic Σ for this problem is s(Y) = ni=1 yi, and the natural parameter is ???? = 1/θ, the reciprocal of themean.
8.12 Write the asymptotic sampling variance of theMLestimate of θ in Example 8.2, and compare it with the sampling variance of the ML estimate when the first and third counts (namely, 38 and 125) are combined, yielding counts (163, 34) from a binomial distribution with probabilities(1−θ/4,
8.11 Verify the E and M steps in Example 8.4.
8.10 Write the loglikelihood of θ for the observed data in Example 8.2. Show directly by differentiating this function that I(θ ∣ Y(0)) = 435.3, as found in Example 8.5.
8.9 By hand calculation, carry out the multivariate normal EM algorithm for the data set in Table 7.1, with initial estimates based on the complete observations. Hence verify that, for this pattern of data and choice of starting values, the algorithm converges after one iteration (i.e., subsequent
8.8 Suppose values yi in Problem 8.7 aremissing if and only if yi >c, for some known censoring pointc. Explore the E step of the EM algorithm for estimating (a) ????1,…, ????J when k is known; (b) ????1,…, ????J and k, when k is unknown.
8.7 Suppose Y = ( y1,…, yn)T are independent gamma random variableswith density f ( yi ∣ k, ????i) = xk−1exp(−yi∕????i)∕(????k i Γ(k)), with unknown shape k, scale θi, andmean kθi = g(Σj????jxij), where g is a known function, ???? = (????1,…, ????J) are unknown regression
8.6 Show that (8.20) and (8.21) are the E and M steps for the regular exponential family (8.19).
8.5 Review results concerning the convergence of EM.
8.4 Show how Corollaries 8.1 and 8.2 follow from Theorem 8.1.
8.3 Prove that the loglikelihood in Example 8.3 is linear in the statistics in Eq. (8.12).
8.2 Describe in words the purpose of the E andMsteps of the EMalgorithm.
8.1 Show that for a scalar parameter, the Newton–Raphson algorithm converges in one step if the loglikelihood is quadratic.
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 missing data pattern, with Y completely observed but n−r values of X missing, and an ignorable mechanism. Does the factored likelihood method provide closed-form expressions for ML estimates for this pattern?(Hint: Find the
(iii) Describe how to generate draws from the posterior distribution of the parameters (????, ????0, ????1, ????2), when the prior distribution takes the form p(????, ????0, ????1, log ????2)∝????1/2(1−????)1/2.
(ii) Suppose now that X is completely observed, but n−r values of Y are missing, with an ignorable mechanism. Use the methods of Chapter 7 to derive the ML estimates of the marginal mean and variance of Y for this monotone pattern.
(b) Y given X = j is normal with mean ????j and variance ????2 ( j = 1, 0).Derive ML estimates of (????, ????0, ????1, ????2) and the marginal mean and variance of Y for a complete random sample (xi, yi), i = 1,…, n on X and Y.
(a) X is Bernoulli with Pr(X = 1 ∣ ????) = 1−Pr(X = 0 ∣ ????) = ???? and
7.16 (i) Consider the following simple form of the discriminant analysis model for bivariate data with binary X and continuous Y:
7.15 If data are MAR and the data analyst discards values to yield a data set with all complete-data factors, are the resultant missing data necessarily MAR? Provide an example to illustrate important points.
7.11. State why the estimates produced in Example 7.11 are ML.184 7 Factored Likelihood MethodsWhen theMissingness Mechanism Is Ignorable
7.14 Create a factorization table (see Rubin 1974) for the data in Example
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 were never observed?
7.12 Show how to compute partial correlations and multiple correlations using SWP.
7.11 Show that RSWis 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.
(b) Compute 95% confidence intervals for ????2 using (i) the data before values were deleted; (ii) the complete units; and (iii) the t-approximation in (2) of Table 7.2. Summarize the properties of these intervals for this missingness mechanism.
(a) Construct a test for whether the data are MACAR and implement the test on your data set.
7.9 By simulation, generate a bivariate normal sample of 20 units with parameters ????1 = ????2 = 0, ????11 = ????12 = 1, and ????22 = 2, and delete values of Y2 so that Pr(mi2 = 1 ∣ yi1, yi2) equals 0.2 if yi1
7.8 Show that the factored likelihood of Example 7.1 does not yield distinct parameters {????j} for a bivariate normal sample with means (????1, ????2), correlation????, and common variance ????2, with missing values on Y2.
7.7 Show that for the setup of Problem 7.6, the estimate of ????12⋅2 obtained by maximizing the complete data loglikelihood over parameters and missing data is ????̃12⋅2 = ????̂12⋅2????̂22∕????̂ ∗22, where (in the notation of Section 7.2),̂ ????∗22 = ????̂2 21⋅1 ̂ ????11 +
7.6 Compute the large-sample variance of ????̂12⋅2 in Problem7.5 and compare with the variance of the complete-case estimate, assuming MACAR.
7.5 For the bivariate normal distribution, express the regression coefficient????12⋅2 of Y1 on Y2 in terms of the parameters ???? in Section 7.2 and hence, derive its ML estimate for the data in Example 7.2.Problems 183
7.4 Prove the six results on Bayes’ inference for monotone bivariate normal data in Section 7.3 (For help, see chapter 2 of Box and Tiao (1973) or chapter 18 of Gelman et al. (2013); also see the material in Section 6.1.4.)
7.3 Compare the asymptotic variance of ????̂2 − ????2 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 missing always completely at random(MACAR). By first conditioning on ( y11,…, yn1), find the exact small-sample variance of ????̂2. (Hint: If u is chi-squared on d degrees of freedom, then E(1/u) = 1/(d−2)) (seeMorrison 1971). Hence show that????̂2 has a
7.1 Assume the data in Example 7.1 are missing always at random (MAAR).Show that given ( y11,…, yn1), ????̂20⋅1 and ????̂21⋅1 are unbiased for ????20⋅1 and????20⋅1. Hence, show that ????̂2 is unbiased for ????2, assuming appropriate random sampling.
Step 3: Find theML estimates ????̂30⋅12, ????̂31⋅12, ????̂32⋅12, and Σ̂33⋅12 of the intercepts, regression coefficients, and residual covariance matrix for the regression of Y3 on Y1 and Y2. These can be found by sweeping the variables Y1 and Y2 out of the augmented covariance matrix
Step 2: Find the ML estimates ????̂20⋅1, ????̂21⋅1, and Σ̂22⋅1 of the intercepts, regression coefficients, and residual covariance matrix for the regression of Y2 on Y1. These can be found by sweeping the variables Y1 out of the augmented covariance matrix of Y1 and Y2 based on the units
Step 1: Find theML estimates ????̂1 and Σ̂11 of themean ????1 and covariancematrixΣ11 of the first block of variables, which are completely observed.These are simply the sample mean and sample covariance matrix of Y1 based on all the units.
4. Calculate themultivariate linear regression of block 4 variables on block 1–3 variables from units with all variables recorded
3. Calculate the multivariate linear regression of block 3 variables on block 1 and 2 variables, from units with block 1–3 variables recorded.
2. Calculate the multivariate linear regression of the next most observed variables, block 2 variables on block 1 variables, from units with both block 1 and block 2 variables recorded.
1. Calculate the mean vector and covariance matrix for the fully observed block 1 variables, from all the units.
6.22 In Example 6.29, derive the estimates maximizing over parameters and missing data.
6.21 For Example 6.28, derive the estimates maximizing over parameters and missing data, as described in the example.
6.20 For Example 6.27, derive the estimates maximizing over parameters and missing data in Eq. (6.71).
6.19 The definition of MAR can depend on how the complete data are defined. Suppose that X = (x1,…, xn), Z = (z1,…, zn) (xi, zi) are a random sample from a bivariate normal distribution with means (????x, 0), variances (????2 x, 1), and correlation 0 (so xi and zi are independent). Suppose that
6.18 Suppose that, given sets of covariates X1 and X2 (possibly overlapping, that is, not disjoint), yi1 and yi2 are bivariate normal with means xi1????1 and xi2????2, variances ????2 1 and ????2 2 = 1, and correlation ????. The data consist of a random sample of units i with xi1 and xi2 always
6.17 For a bivariatenormal sample ( yi1, yi2), i=1,…, n on (Y1, Y2)with parametersθ = (????1, ????2, ????11, ????12, ????22), missing values of Y2, andmi2 themissingness indicator for yi2, state for the following missingnessmechanisms (i)whether the data areMAR and (ii) whether the missingness
6.16 Find large-sample sampling variance estimates for the two ML estimates in Example 6.23.
6.15 Suppose the following data are a random sample of n= 7 from the Cauchy distribution with median θ : Y= (−4.2, −3.2, −2.0, 0.5, 1.5, 1.5, 3.5). Compute and compare 90% intervals for θ using (a) the asymptotic distribution based on the observed information, (b) the asymptotic
(b) the posterior mean of ????1 when the prior distribution for (????1, ????2) is p(????1, log ????2) = const.
6.17. Show that for the special case of weighted linear regression with no intercept (????0 = 0), a single covariate X, and weight for observation i wi = xi, the ratio estimator y∕x is (a) the ML estimate of ????1 and
6.14 Derive the modifications of the posterior distributions in Eqs. (6.34)–(6.36) for weighted linear regression, discussed at the end of Example
6.13 Derive the posterior distributions in Eqs. (6.34)–(6.36) for Example 6.17.
6.12 Derive the posterior distributions in Eqs. (6.26)–(6.28) for Example 6.16.
6.11 In Example 6.15, show that for large n, LR = t2.Problems 149
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), Var( yi ∣ ????i, ????) = ????b″(????i), where ????i = ????(xi, ????), and double prime denotes differentiation twice with respect to the function’s argument.
6.6 Show that for the GLIM model of Eq. (6.9), E( yi ∣ xi, ????) = b′(????(xi, ????)), where prime denotes differentiation with respect to the function’s argument.Conclude that the canonical link Eq. (6.12) is obtained by setting g−1(⋅) = b′(⋅). (Hint: Consider the density of yi in
6.5 Suppose the data are a random sample of size n from the uniform distribution between 0 and θ, θ>0. Show that the ML estimate of θ is the largest data value. (Hint: Differentiation of the score function does not work for this problem!) Find the posterior mean of θ, assuming a uniform prior
(b) Show that if the data are iid with the Laplace (double exponential)distribution, f ( yi ∣ ????) = 0.5exp(−||yi − ????(xi)||), where ????(xi) = ????0 +????1xi1 +⋯+????pxip, thenML estimates of ????0,…, ????p are obtained by minimizing the sum of absolute deviations of the y values from
6.10.
6.4 (a) Compare ML and least squares estimates for the model of Example
6.3 For a univariate normal sample, find the ML estimate of the coefficient of variation, ????/????.148 6 Theory of Inference Based on the Likelihood Function
6.2 Find the score function for the distributions in Problem 6.1. Find the ML estimates for those distributions that have closed-form estimates.
6.1 Write the likelihood function for an iid sample fromthe (a) beta distribution;(b) Poisson distribution; and (c) Cauchy distribution with locationθ and scale 1.
(iv) it yields inferences that are robust against model misspecification?
(iii) it allows valid inferences from the filled-in data, if the imputation model is correct?
(ii) unlike SI, it yields consistent estimates of quantities that are not linear in the data?
(i) it yields more efficient estimates from the filled-in data?
5.14 Is MI better than single imputation of a draw from the predictive distribution of the missing values (SI) because
(iv) it yields inferences that are robust against model misspecification?Problems 105
(iii) it allows valid inferences from the filled-in data, if the imputation model is correct?
(ii) it yields consistent estimates of quantities that are not linear in the data?
(i) it yields more efficient estimates from the filled-in data?
5.13 Is multiple imputation (MI) better than imputation of a mean from the conditional distribution of the missing value because
(b) Justify the adjustment in (a) based on the sampling variability of( yR − Y).
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 sRr−1/2zd to the imputed values for the dth multiply-imputed data set, where zd is a standard normal deviate.)
(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).
(d) Show that the 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.
(c) Tabulate values of the relative efficiency of y∗ to yR for different values of D, assuming large r and N/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 y∗.
5.10 D times, and let y(d)∗ and U(d)∗ be the values of y∗ and U∗ for the dth imputed data set. Let y∗ =ΣDd=1 y(d)∗ ∕D, and T∗ be the multiple imputation estimate of variance of y∗. That is, T∗ = U∗ + (1 + D−1)B∗, where U∗ =ΣDd=1 U(d)∗ ∕D, B∗ =ΣDd=1 ( y(d)∗ −
5.11 Suppose multiple imputations are created using the method of Problem
(d) Assume r and N/r are large, and show that interval estimates of Y based on U∗ as the estimated 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 104 5 Accounting for Uncertainty fromMissing Data not as efficient as yR.
(b) Show that conditional on the observed data, the 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).(c) Show that conditional on the sample sizes n and r (and the population Y-values), the variance of y∗ is the variance of
(a) Show that y∗ is unbiased for the population mean Y.
5.10 Suppose in Problem 5.9, imputations are randomly drawn with replacement from the r respondents’ values.
5.9 Consider a simple random sample of size n with r respondents and m = n−r nonrespondents, and let yR and s2 R be the sample mean and variance of the respondents’ data, and yNR and s2 NR be the sample mean and variance of the imputed data. Show that the mean and variance y∗and s2∗ of all
5.8 Apply the methods in Problems 5.1–5.5 to 500 replicate data sets generated as in Problem 5.2 and assess the bias of the estimates and the coverage of intervals. Interpret the results given your understanding of the properties of the various methods.
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 compare the resulting 90% nominal intervals with the simpler intervals based on(5.23).

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