A note on multiple imputation for method of moments estimation

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1 Statstcs Publcatons Statstcs A note on multple mputaton for method of moments estmaton Shu Yang Harvard Unversty Jae Kwang Km Iowa State Unversty, jkm@astate.edu Follow ths and addtonal works at: Part of the Desgn of Experments and Sample Surveys Commons, Statstcal Methodology Commons, and the Statstcal Models Commons The complete bblographc nformaton for ths tem can be found at stat_las_pubs/120. For nformaton on how to cte ths tem, please vst howtocte.html. Ths Artcle s brought to you for free and open access by the Statstcs at Iowa State Unversty Dgtal Repostory. It has been accepted for ncluson n Statstcs Publcatons by an authorzed admnstrator of Iowa State Unversty Dgtal Repostory. For more nformaton, please contact dgrep@astate.edu.

2 Bometrka (20xx), xx, x, pp. 1 8 C 2012 Bometrka Trust Prnted n Great Brtan A note on multple mputaton for method of moments estmaton arxv: v1 [stat.me] 27 Aug 2015 BY S. YANG Department of Bostatstcs, Harvard T. H. Chan School of Publc Health, Boston, Massachusetts 02115, U.S.A. shuyang@hsph.harvard.edu J. K. KIM Department of Statstcs, Iowa State Unversty, Ames, Iowa 50010, U.S.A. jkm@astate.edu SUMMARY Multple mputaton s a popular mputaton method for general purpose estmaton. Rubn (1987) provded an easly applcable formula for the varance estmaton of multple mputaton. However, the valdty of the multple mputaton nference requres the congenalty condton of Meng (1994), whch s not necessarly satsfed for method of moments estmaton. Ths paper presents the asymptotc bas of Rubn s varance estmator when the method of moments estmator s used as a complete-sample estmator n the multple mputaton procedure. A new varance estmator based on over-mputaton s proposed to provde asymptotcally vald nference for method of moments estmaton. Some key words: Bayesan method; Congenalty; Mssng at random; Proper mputaton; Survey samplng. 1. INTRODUCTION Imputaton s often used to handle mssng data. For nference, f mputed values are treated as f they were observed, varance estmates wll generally be underestmates (Ford, 1983). To account for the uncertanty due to mputaton, Rubn (1987, 1996) proposed multple mputaton whch creates multply completed datasets to allow assessment of mputaton varablty. Multple mputaton s motvated n a Bayesan framework; however, ts frequentst valdty s controversal. Rubn (1987) clamed that multple mputaton can provde vald frequentst nference n varous applcatons (for example, Clogg et al., 1991). On the other hand, as dscussed by Fay (1992), Kott (1995), Fay (1996), Bnder & Sun (1996), Wang & Robns (1998), Robns & Wang (2000), Nelsen (2003), and Km et al. (2006), the multple mputaton varance estmator s not always consstent. For multple mputaton nference to be vald, mputatons must be proper (Rubn, 1987). A suffcent condton s gven by Meng (1994), the so-called congenalty condton, mposed on both the mputaton model and the form of subsequent complete-sample analyses, whch s qute restrctve for general purpose estmaton. Rubn s varance estmator s otherwse nconsstent. Km (2011) ponted out that multple mputaton that s congenal for mean estmaton s not necessarly congenal for proporton estmaton. Therefore, some common statstcal procedures,

3 2 S. YANG AND J. K. KIM such as the method of moments estmators, can be ncompatble wth the multple mputaton framework. In ths paper, we characterze the asymptotc bas of Rubn s varance estmator when the method of moments estmator s used n the complete-sample analyss. We also dscuss an alternatve varance estmator that can provde asymptotcally vald nference for method of moments estmaton. The new varance estmator s compared wth Rubn s varance estmator through two lmted smulaton studes n BASIC SETUP Suppose that the sample conssts of n observatons (x 1,y 1 ),...,(x n,y n ), whch s an ndependent realzaton of a random vector (X, Y). For smplcty of presentaton, assume that Y s a scalar outcome varable and X s a p-dmensonal covarate. Suppose that x s fully observed and y s not fully observed for all unts n the sample. Wthout loss of generalty, assume the frst r unts of y are observed and the remanng n r unts of y are mssng. Let δ be the response ndcator of y, that s, δ = 1 f y s observed and δ = 0 otherwse. Denote y obs = (y 1,...,y r ) T and X n = (x 1,...,x n ). We further assume that the mssng mechansm s mssng at random n the sense of Rubn (1976). The parameter of nterest s η = E{g(Y)}, whereg( ) s a known functon. For example, fg(y) = y, thenη = E(Y) s the populaton mean ofy,and fg(y) = I(y < 1), thenη = pr(y < 1) s the populaton proporton ofy less than1. Assume that the condtonal densty f(y x) belongs to a parametrc class of models ndexed by θ such that f(y x) = f(y x;θ) for some θ Ω and the margnal dstrbuton of x s completely unspecfed. To generate mputed values for mssng outcomes from f(y x; θ), we need to estmate the unknown parameter θ, ether by lkelhood-based methods or by Bayesan methods. The multple mputaton procedure employs a Bayesan approach to deal wth the unknown parameter θ, whch unfolds n three steps: Step 1. (Imputaton) Create M complete datasets by fllng n mssng values wth mputed values generated from the posteror predctve dstrbuton. Specfcally, to create thejth mputed dataset, frst generateθ (j) from the posteror dstrbutonp(θ X n,y obs ), and then generatey (j) from the mputaton model f(y x ;θ (j) ) for each mssng y. Step 2. (Analyss) Apply the user s complete-sample estmaton procedure to each mputed dataset. Let ˆη (j) be the complete-sample estmator of η = E{g(Y)} appled to the jth mputed dataset and ˆV (j) be the complete-sample varance estmator of ˆη (j). Step 3. (Summarze) Use Rubn s combnng rule to summarze the results from the multply mputed datasets. The multple mputaton estmator ofη s ˆη MI = M 1 M j=1 ˆη(j), and Rubn s varance estmator s ˆV MI (ˆη MI ) = W M + ( 1+M 1) B M, (1) where W M = M 1 M ˆV j=1 (j) and B M = (M 1) 1 M j=1 (ˆη(j) ˆη MI ) 2. If the method of moments estmator ofη = E{g(Y)} s used n step 2, the multple mputaton estmator of η becomes M r n M ˆη MI = M 1 ˆη (j) = n 1 g(y )+ M 1 g(y (j) ), (2) j=1 =1 =r+1 where ˆη (j) = n 1 { r =1 g(y )+ n =r+1 g(y (j) )}. To derve the frequentst property of ˆη MI, we rely on the Bernsten-von Mses theorem (van der Vaart, 2000; Chapter 10), whch clams j=1

4 Multple Imputaton 3 that under regularty condtons and condtonal on the observed data, the posteror dstrbuton p(θ X n,y obs ) converges to a normal dstrbuton wth mean ˆθ and varance I 1 obs, where ˆθ s the maxmum lkelhood estmator of θ from the observed data and I 1 obs s the nverse of the observed Fsher nformaton matrx wth I obs = r =1 2 logf(y x ; ˆθ)/ θ θ T. As a result, assume that E{g(Y) x ;θ} s suffcently smooth n θ, condtonal on the observed data, we have plm M M 1 M j=1 g(y (j) ) = E[E{g(Y) x ;θ } X n,y obs ] = E{g(Y) x ; ˆθ}, where A n = Bn means A n = B n +o p (1). Therefore, for M, ˆη MI converges to ˆη MI, = n 1 { r =1 y + n =r+1 m(x ; ˆθ)}, where m(x;θ) = E{g(Y) x;θ}. The varance estmaton of ˆη MI, needs to approprately account for the uncertanty assocated wth the estmate of θ, whch s usually done usng lnearzaton methods f the mputaton models are known (Robns & Wang, 2000; Km & Rao, 2009). In the multple mputaton procedure, ths s characterzed n the varablty between the multply mputed datasets wthout referrng to the mputaton models. However, Rubn s varance estmator (1) requres restrctve condtons for vald nference, whch we dscuss n the next secton. 3. MAIN RESULT Rubn s varance estmator s based on the followng decomposton, var(ˆη MI ) = var(ˆη n )+var(ˆη MI ˆη n )+2cov(ˆη MI ˆη n,ˆη n ), (3) where ˆη n s the complete-sample estmator ofη. Bascally, n Rubn s varance estmator (1),W M estmates the frst term of (3) and(1+m 1 )B M estmates the second term of (3). In partcular, Km et al. (2006) proved that E{(1+M 1 )B M } = var(ˆη MI ˆη n ) for a farly general class of estmators. Thus, f the complete-sample varance estmator satsfes the condton E(ˆV (j) ) = var(ˆη n ) for j = 1,...,M, the bas of Rubn s varance estmator s bas(ˆv MI ) = 2cov(ˆη MI ˆη n,ˆη n ). (4) Rubn s varance estmator s asymptotcally unbased f cov(ˆη MI ˆη n,ˆη n ) = 0, whch s called the congenalty condton by Meng (1994). However, the congenalty condton does not hold for some common estmators such as the method of moments estmators. Theorem 1 gves ths asymptotc bas of Rubn s varance estmator for M, wth the proof outlned n the onlne supplementary materal. THEOREM 1. Let ˆη n = n 1 n =1 g(y ) be the method of moments estmator ofη = E{g(Y)} under complete response. Assume that E(ˆV (j) ) = var(ˆη n ) holds for j = 1,...,M. Then for M, the bas of Rubn s varance estmator s bas(ˆv MI ) = 2n 1 (1 p) ( E[var{g(Y) X} δ = 0] ṁ T θ,0 I 1 θ ṁ θ,1 ), (5) where p = r/n, I θ = E{ 2 logf(y X;θ)/ θ θ T }, m(x;θ) = E{g(Y) x;θ}, ṁ θ (x) = m(x;θ)/ θ, ṁ θ,0 = E{ṁ θ (X) δ = 0}, and ṁ θ,1 = E{ṁ θ (X) δ = 1}. Remark 1. Under mssng completely at random, the bas n (5) smplfes to bas(ˆv MI ) = 2p(1 p){var(ˆη r,mme ) var(ˆη r,mle )}, (6) where ˆη r,mme = r 1 r =1 g(y ) and ˆη r,mle = r 1 r =1 E{g(Y) x ; ˆθ}, because var(ˆη r,mme ) = r 1 var{g(y)} = r 1 var[e{g(y) X}]+r 1 E[var{g(Y) X}],

5 4 S. YANG AND J. K. KIM and var(ˆη r,mle ) = r 1 var[e{g(y) X}]+r 1 ṁ T θ I 1 θ ṁ θ, whereṁ θ = E{ṁ θ (X)}. Result (6) explctly shows that Rubn s varance estmator s unbased f and only f the method of moments estmator s as effcent as the maxmum lkelhood estmator, that s, var(ˆη r,mme ) = var(ˆη r,mle ). Otherwse, Rubn s varance estmator s postvely based. Remark 2. Under mssng at random, the bas of Rubn s varance estmator can be zero, postve or negatve. Consder a smple lnear regresson model Y = X T β +ǫ, whereǫ N(0,σ 2 ). For g(y) = Y, f X contans 1, then the method of moments estmator n 1 n =1 y s dentcal to the maxmum lkelhood estmator n 1 n =1 xt ˆβ wth ˆβ beng the maxmum lkelhood estmator of β under complete response. By Theorem 1, let E 0 ( ) = E( δ = 0) and E 1 ( ) = E( δ = 1), the bas of Rubn s varance estmator n (5) s bas(ˆv MI ) = 2n 1 (1 p)σ 2 {1 E 0 (X) T E 1 (XX T ) 1 E 1 (X)} = 0, by drect calculaton consderng that X contans 1. Ths s consstent wth the theory n Wang & Robns (1998) and Nelsen (2003). Now consder a smple lnear regresson model whch contans one covarate X and no ntercept, then the method of moments estmator s strctly less effcent than the maxmum lkelhood estmator (Matloff, 1981). The bas of Rubn s varance estmator s bas(ˆv MI ) = 2n 1 (1 p)σ 2 E 1 (X 2 ) 1 {E 1 (X 2 ) E 0 (X) T E 1 (X)}, (7) whch can be zero, postve or negatve dependng on the nformaton of X n the respondent and non-respondent groups. See the frst smulaton study n ALTERNATIVE VARIANCE ESTIMATION In ths secton, we consder an alternatve varance estmaton method that leads to an unbased varance estmator for multple mputaton regardless of whether the method of moments estmator or the maxmum lkelhood estmator s used as the complete-sample estmator n the multple mputaton procedure. We frst decompose the multple mputaton estmator as, ˆη MI = ˆη MI, +(ˆη MI ˆη MI, ). The two terms are uncorrelated usng the law of total covarance and the fact that ˆη MI, s the condtonal expectaton of ˆη MI, condtonal on the observed data. Therefore, we have var(ˆη MI ) = var(ˆη MI, )+var(ˆη MI ˆη MI, ). (8) Note thatvar(ˆη MI ˆη MI, ) can be estmated bym 1 B M (Km et al., 2006; Lemma 2). We now focus on estmatngvar(ˆη MI, ) n (8). For smplcty of presentaton, all detals of dervaton are to be found n supplementary materal. We show that the varance of ˆη MI, s a sum of two terms, var(ˆη MI, ) = n 1 V 1 +r 1 V 2, (9) where V 1 = var{g(y)} (1 p)e[var{g(y) X} δ = 0], and V 2 = ṁ T θ I 1 θ ṁ θ p 2 ṁ T θ,1 I 1 θ ṁ θ,1. The frst term, n 1 V 1, s the varance of the sample mean of g(y ) (1 δ ){g(y ) m(x ;θ)}. To estmate ths term, consder W M = M 1 M ˆV j=1 (j) as n (1), and { 2 1 M n C M = n 2 g(y (k) ) 1 M g(y (k) )}. (10) (M 1) M k=1=r+1 k=1

6 Multple Imputaton 5 We have E{W M } = n 1 var{g(y)} and E(C M ) = n 1 (1 p)e[var{g(y) X} δ = 0]. Therefore, the frst term n 1 V 1 can be estmated by W M = W M C M. By the strong law of large numbers, pr( W M 0) 1 asn. The second term, r 1 V 2, reflects the varablty assocated wth the estmated value of θ nstead of the true value θ n the mputed values. To estmate ths term, we use overmputaton n the sense that the mputaton s carred out not only for the unts wth mssng outcomes, but also for the unts wth observed outcomes. Over-mputaton has been used n model dagnostcs for multple mputaton (Honaker et al., 2010; Blackwell et al., 2015). Let d (k) = g(y (k) ) M 1 M l=1 g(y (l) ) for = 1,...,n and k = 1,...,M. De- ) 2 (M 1) 1 M k=1 n 2 n =1 (d (k) ) 2, and fne D M,n = (M 1) 1 M k=1 (n 1 n =1 d (k) D M,r = (M 1) 1 M k=1 (n 1 r =1 d (k) =1 (d (k) ) 2. The key nsght s based on the followng observatons: E(D M,n ) = r 1 ṁ T θ I 1 θ ṁ θ and E(D M,r ) = r 1 p 2 ṁ T θ,1 I 1 θ ṁ θ,1 ; therefore, the second term of (9) can be estmated byd M = D M,n D M,r. Combnng the estmators of the two terms n (9), we have the new multple mputaton varance estmator, gven n the followng theorem. ) 2 (M 1) 1 M k=1 n 2 r THEOREM 2. Under the assumptons of Theorem 1, the new multple mputaton varance estmator s ˆV MI = W M +D M +M 1 B M, (11) where W M = W M C M, wthc M defned n (10) andb M beng the usual between-mputaton varance n (1). ˆV MI s asymptotcally unbased for estmatng the varance of the multple mputaton estmator n (2) as n. Remark 3. To account for the uncertanty n the varance estmator wth a small to moderate mputaton sze, a 100(1 α)% nterval estmate for η s ˆη MI ±t df,1 α/2 ˆVMI, where df s an approxmate number of degrees of freedom based on Satterthwate s method (1946) gven n supplementary materal. From smulaton studes, we fnd that usng df = M 1 gves smlar satsfactory results as usng the formula we provded. As a practcal matter, df = M 1 s preferred. Remark 4. The proposed varance estmator n (11) s also asymptotcally unbased when ˆη n s the maxmum lkelhood estmator of η = E{g(Y)} (see supplementary materal for proof). Therefore, the proposed varance estmator s applcable regardless of whether the maxmum lkelhood estmator or the method of moments estmator s used for the complete-sample estmator. The prce we pay for the better performance of our varance estmator s an ncrease n computatonal complexty and data storage space, whch requres M + 1 datasets, wth M of them ncludng the over-mputatons and the last one contanng the orgnal observed data. However, when one s concern s wth vald nference of multple mputaton, as n ths paper, our proposed varance estmator based on over-mputaton s preferred over that of Rubn s. In addton, gven over-mputatons, the subsequent nference does not requre the knowledge of the mputaton models. Ths s mportant because data analysts typcally do not have access to all the nformaton that the mputers used for mputaton. Our study would promote the use of over-mputaton at the tme of mputaton, whch not only allows the mputers to assess the adequacy of the mputaton models, but also enables the analysts to carry out vald nference wthout knowledge of the mputaton models.

7 6 S. YANG AND J. K. KIM 5. SIMULATION STUDY To test our theory, we conduct two lmted smulaton studes. In the frst smulaton, 5, 000 Monte Carlo samples of szen = 2,000 are ndependently generated fromy = βx +e, where β = 0.1, X exp(1) and e N(0,σe 2) wth σ2 e = 0.5. In the sample, we assume that X s fully observed, but Y s not. Let δ be the response ndcator of y and δ Bernoull(p ), where p = 1/{1+exp( φ 0 φ 1 x )}. We consder two scenaros: () (φ 0,φ 1 ) = ( 1.5,2) and () (φ 0,φ 1 ) = (3, 3), wth the average response rate about 0.6. The parameters of nterest are η 1 = E(Y) and η 2 = pr(y < 0.15). For multple mputaton, M = 500 mputed values are ndependently generated from the lnear regresson model usng the Bayesan regresson mputaton procedure dscussed n Schenker & Welsh (1998), where β and σe 2 are treated as ndependent wth pror densty proportonal to σe 2. In each mputed dataset, we adopt the followng complete-sample pont estmators and varance estmators: ˆη 1,n = n 1 n =1 y, ˆη 2,n = n 1 n =1 I(y < 0.15), ˆV(ˆη 1,n ) = n 1 (n 1) 1 n =1 (y ˆη 1,n ) 2, and ˆV(ˆη 2,n ) = (n 1) 1ˆη 2,n (1 ˆη 2,n ). The relatve bas of the varance estmator s calculated as {E(ˆV MI ) var(ˆη MI )}/var(ˆη MI ) 100%. The100(1 α)% confdence ntervals are calculated as (ˆη MI t ν,1 α/2 ˆVMI,ˆη MI +t ν,1 α/2 ˆVMI ), where t ν,1 α/2 s the 100(1 α/2)% quantle of the t dstrbuton wth ν degrees of freedom. For Rubn s method, ν = ν 1 ν 2 /(ν 1 +ν 2 ) wth ν 1 = (M 1)λ 2, ν 2 = (ν com +1)(ν com +3) 1 ν com (1 λ), ν com = n 3, and λ = (1+M 1 )B M /{W M +(1+M 1 )B M } (Barnard & Rubn, 1999). In our new method, ν = M 1. The coverage s calculated as the percentage of Monte Carlo samples where the estmate falls wthn the confdence nterval. From Table 1, for η 1 = E(Y), under scenaro (), the relatve bas of Rubn s varance estmator s 96.8%, whch s consstent wth our result n (7) wth E 1 (X 2 ) E 0 (X) T E 1 (X) > 0, where E 1 (X 2 ) = 3.38, E 1 (X) = 1.45, and E 0 (X) = Under scenaro (), the relatve bas of Rubn s varance estmator s 19.8%, whch s consstent wth our result n (7) wth E 1 (X 2 ) E 0 (X) T E 1 (X) < 0, where E 1 (X 2 ) = 0.37, E 1 (X) = 0.47, and E 0 (X) = The emprcal coverage for Rubn s method can be over or below the nomnal coverage due to varance overestmaton or underestmaton. On the other hand, the new varance estmator s essentally unbased for these scenaros. In the second smulaton, 5,000 Monte Carlo samples of sze n = 200 are ndependently generated from Y = β 0 +β 1 X +e, where β = (β 0,β 1 ) = (3, 1), X N(2,1) and e N(0,σe) 2 wthσe 2 = 1. The parameters of nterest areη 1 = E(Y)andη 2 = pr(y < 1). We consder two dfferent factors for smulaton. One s the response mechansm: mssng completely at random and mssng at random. For mssng completely at random, δ Bernoull(0.6). For mssng at random, δ Bernoull(p ), where p = 1/{1+exp( φ 0 φ 1 x )} and (φ 0,φ 1 ) = (0.28,0.1) wth the average response rate about 0.6. The other factor s the sze of multple mputaton, wth two levels M = 10 and M = 30. From Table 2, regardng the relatve bas, Rubn s varance estmator s unbased for η 1 = E(Y), wth absolute relatve bas of less than1%, and our new varance estmator s comparable wth Rubn s varance estmator wth absolute relatve bas of less than 1.68%. Rubn s varance estmator s based upward for η 2 = pr(y < 1), wth absolute relatve bas as hgh as 24%; whereas our new varance estmator reduces absolute relatve bas to less than1.74%. Regardng confdence nterval estmates, for η 1 = E(Y), the confdence nterval calculated from our new method s slghtly wder than that from Rubn s method, because our new method uses a smaller number of degrees of freedom n thetdstrbuton. However, forη 2 = pr(y < 1), the confdence nterval calculated from our new method s narrower than that from Rubn s method even wth a smaller number of degrees of freedom n the t dstrbuton, due to the overestmaton n Rubn s

8 Multple Imputaton 7 Table 1. Relatve bases of two varance estmators and mean wdth and coverages of two nterval estmates under two scenaros n smulaton one Relatve bas Mean Wdth Mean Wdth Coverage Coverage (%) for 90% C.I. for 95% C.I. for 90% C.I. for 95% C.I. Scenaro Rubn New Rubn New Rubn New Rubn New Rubn New 1 η η η η C.I., confdence nterval; η 1 = E(Y);η 2 = pr(y < 0.15); Rubn/New, Rubn s/new varance estmator. Table 2. Relatve bases of two varance estmators and mean wdth and coverages of two nterval estmates under two scenaros of mssngness n smulaton two Relatve Bas Mean Wdth Mean Wdth Coverage Coverage (%) for 90% C.I. for 95% C.I. for 90% C.I. for95% C.I. M Rubn New Rubn New Rubn New Rubn New Rubn New Mssng completely at random η η Mssng at random η η C.I., confdence nterval; η 1 = E(Y);η 2 = pr(y < 1); Rubn/New, Rubn s/new varance estmator. method. Rubn s method provdes good emprcal coverage for η 1 = E(Y) n the sense that the emprcal coverage s close to the nomnal coverage; however, the emprcal coverage for η 2 = pr(y < 1) reaches to 95% for 90% confdence ntervals, and 98% for 95% confdence ntervals, due to varance overestmaton. In contrast, our new method provdes more accurate coverage of confdence nterval for bothη 1 = E(Y) andη 2 = pr(y < 1) at90% and95% levels. 6. DISCUSSION Our method can be extended to a more general class of parameters obtaned from estmatng equatons. Let η be defned as a soluton to the estmatng equaton n =1 U(η;x,y ) = 0. Examples of η nclude mean of y, proporton of y less than q, pth quantle, regresson coeffcents, and doman means. A smlar approach can be used to characterze the bas of Rubn s varance estmator and to develop a bas-corrected varance estmator. Another extenson would be developng unbased varance estmaton for the vector case of η wth q > 1 components. As n the scalar case, we can construct the multvarate analogues of the multple mputaton estmator and the varance estmator; however, fndng an adequate reference dstrbuton for the statstc (ˆη MI η) T 1 ˆV MI (ˆη MI η)/q s more subtle n the vector case than n the scalar case. One potental soluton s to make a smplfyng assumpton that the fracton of mssng nformaton s equal for all the components of η, as dscussed n Xe & Meng (2014) and L et al. (1994).

9 8 S. YANG AND J. K. KIM ACKNOWLEDGMENT We are grateful to Xanchao Xe and Xaol Meng for many helpful conversatons and to the Bometrka edtors and four referees for ther valuable comments that helped to mprove ths paper. The research of the second author was partally supported by a grant from US Natonal Scence Foundaton and also by a Cooperatve Agreement between the U.S. Department of Agrculture Natural Resources Conservaton Servce and Iowa State Unversty. SUPPLEMENTARY MATERIAL The supplementary materal avalable at Bometrka onlne ncludes the proof of Theorem 1, the proof of Theorem 2, verfcaton of the new varance estmator beng unbased when ˆη n s the maxmum lkelhood estmator of η = E{g(Y)}, and an approxmate number of degrees of freedom. REFERENCES BARNARD, J. & RUBIN, D. B. (1999). Small-sample degrees of freedom wth multple mputaton. Bometrka 86, BINDER, D. A. & SUN, W. (1996). Frequency vald multple mputaton for surveys wth a complex desgn. Proc. Surv. Res. Meth. Sect. Am. Statst. Ass., BLACKWELL, M., HONAKER, J. & KING, G. (2015). A Unfed Approach to Measurement Error and Mssng Data: Detals and Extensons. Soc. Meth. Res., In press. CLOGG, C. C., RUBIN, D. B., SCHENKER, N., SCHULTZ, B. & WEIDMAN, L. (1991). Multple mputaton of ndustry and occupaton codes n census publc-use samples usng Bayesan logstc regresson. J. Am. Statst. Assoc. 86, FAY, R. E.(1992). When are nferences from multple mputaton vald?. Proc. Surv. Res. Meth. Sect. Am. Statst. Ass., FAY, R. E.(1993). Vald nference from mputed survey data. Proc. Surv. Res. Meth. Sect. Am. Statst. Ass., FAY, R. E.(1996). Alternatve paradgms for the analyss of mputed survey data. J. Am. Statst. Assoc. 91, FORD, B. L.(1983). An overvew of hot-deck procedures. Incomplete data n sample surveys 2, HONAKER, J. AND KING, G. & BLACKWELL, M. M.(2010). Package Amela. KIM, J. K.(2011). Parametrc fractonal mputaton for mssng data analyss. Bometrka 98, KIM, J. K., BRICK, J., FULLER, W. A., & KALTON, G.(2006). On the bas of the multple-mputaton varance estmator n survey samplng. J. R. Statst. Soc. B 68, KIM, J. K. & RAO, J. N. K.(2009). A unfed approach to lnearzaton varance estmaton from survey data after mputaton for tem nonresponse. Bometrka 96, KOTT, P. S.(1995). A paradox of multple mputaton. Proc. Surv. Res. Meth. Sect. Am. Statst. Ass., LI,K. H., RAGHUNATHAN, T. E. & RUBIN, D. B.(1994). Large-sample sgnfcance levels from multply mputed data usng moment-based statstcs and an F reference dstrbuton. J. Am. Statst. Assoc. 86, MATLOFF, N. S.(1981). Use of regresson functons for mproved estmaton of means. Bometrka 68, MENG, X.(1994). Multple-mputaton nferences wth uncongenal sources of nput. Statst. Sc., 9, NIELSEN, S. F.(2003). Proper and mproper multple mputaton. Int. Statst. Rev. 71, ROBINS, J. M. & WANG, N.(2000). Inference for mputaton estmators. Bometrka 87, RUBIN, D. B.(1976). Inference and mssng data. Bometrka 63, RUBIN, D. B. (1987). Multple Imputaton for Nonresponse n Surveys. John Wley & Sons. RUBIN, D. B.(1996). Multple mputaton after 18+ years. J. Am. Statst. Assoc. 91, RUBIN, D. B. & SCHENKER, N.(1986). Multple mputaton for nterval estmaton from smple random samples wth gnorable nonresponse. J. Am. Statst. Assoc. 81, SATTERTHWAITE, F. E.(1946). An approxmate dstrbuton of estmates of varance components. Bometrcs bulletn 2, SCHENKER, N. & WELSH, A. H.(1998). Asymptotc results for multple mputaton. Ann. Statst. 16, VAN DER VAART, A. W.(2000). Asymptotc Statstcs. Cambrdge Unversty Press. WANG, N. & ROBINS, J. M.(1998). Large-sample theory for parametrc multple mputaton procedures. Bometrka 85, XIE, X. & MENG, X. L.(2014). Dssectng multple mputaton from a mult-phase nference perspectve: what happens when God s, mputer s and analyst s models are uncongenal?. Statstca Snca, to appear.

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