JSM Survey Research Methods Section. Is it MAR or NMAR? Michail Sverchkov

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1 JSM Survey Researh Methods Seton Is t MAR or NMAR? Mhal Sverhkov Bureau of Labor Statsts 2 Massahusetts Avenue, NE, Sute 1950, Washngton, DC , Sverhkov.Mhael@bls.gov Abstrat Most methods that deal wth the estmaton of response probabltes assume ether expltly or mpltly that the mssng data are mssng at random (MAR). However, n many pratal stuatons ths assumpton s not vald, sne the probablty of respondng often depends on the outome value or on latent varables related to the outome. The ase where the mssng data are not MAR (NMAR) an be treated by postulatng a parametr model for the dstrbuton of the outomes under full response and a model for the response probabltes. The two models defne a parametr model for the jont dstrbuton of the outome and the response ndator, and therefore the parameters of ths model an be estmated by maxmzaton of the lkelhood orrespondng to ths dstrbuton. Modelng the dstrbuton of the outomes under full response, however, an be problemat sne no data are avalable from ths dstrbuton. Sverhkov (2008) proposed a new approah that permts estmatng the parameters of the model for the response probabltes wthout modellng the dstrbuton of the outomes under full response. The approah utlzes relatonshps between the populaton, the sample and the sample-omplement dstrbuton derved n Pfeffermann and Sverhkov (1999) and Sverhkov and Pfeffermann (2004). The present paper nvestgates how ths approah an be used for testng whether response s MAR or NMAR. Key words: sample dstrbuton, omplement-sample dstrbuton, predton under nformatve samplng and non-response, estmatng equatons, mssng nformaton prnple, non-parametr estmaton 1. Introduton There s almost no survey wthout nonresponse, but n prate most methods that deal wth ths problem assume ether expltly or mpltly that the mssng data are mssng at random (MAR, Rubn, 1976; Lttle, 1982). However, n many pratal stuatons ths assumpton s not vald, sne the probablty of respondng often depends dretly on the outome value. In ths ase, the use of methods that assume that the nonresponse s MAR an lead to large bases of populaton parameter estmators and large mputaton bas. The ase where the mssng data are not MAR (NMAR) an be treated by postulatng a parametr model for the dstrbuton of the outomes before non-response and a model for the response mehansm. These two models defne a parametr model for the jont dstrbuton of the outomes and response ndators, and therefore the parameters of these models an be estmated by maxmzaton of the lkelhood based on ths jont dstrbuton. See, Greenlees et al. (1982), Rubn (1987), Lttle (1993), Beaumont (2000), Lttle and Rubn (2002) and Qn et al. (2002). Modelng the dstrbuton of the outomes before non-response an be problemat sne t refers to the partly unobserved data. Qn et al. (2002) suggests usng a non-parametr model for ths dstrbuton (empral lkelhood approah). Sverhkov (2008) suggests an alternatve approah that allows one to estmate the parameters of the response model by ndependent parametr or non-parametr estmaton of the outomes dstrbuton after non-response (whh an be done by 2306

2 JSM Survey Researh Methods Seton use of lass statstal nferenes sne the latter refers to the observed data) and then by solvng estmatng equatons obtaned from the ensus lkelhood funton of the response ndators. The dervaton of these estmatng equatons utlzes the relatonshps between the populaton, the sample and the sample-omplement dstrbutons, as n Pfeffermann and Sverhkov (1999, 2003), Sverhkov and Pfeffermann (2004). Even under ths approah one needs to assume a model for the response mehansm whh annot be heked from the observed data n ase of NMAR. Therefore t s mportant to know whether the response s MAR or NMAR. The present paper nvestgates how the Sverhkov (2008) approah an be used for testng whether response s MAR or NMAR. 2. Notaton Let Y denote the value of an outome varable Y assoated wth unt belongng to a sample S {1,..., n}, drawn from a fnte populaton U {1,..., N}. Let X denote the orrespondng values of ovarates X ( X1,..., X K ). In what follows we assume that the populaton outome values are ndependent realzatons from dstrbutons wth unknown probablty densty funtons (pdf), f ( Y X ). We use the abbrevaton pdf for the probablty densty funton when Y s ontnuous and the probablty funton when Y s dsrete. Let R {1,..., n r } defne the sample of respondents (the sample wth observed outome values), and R { nr 1,..., n} defne the sample of nonrespondents. The response proess s assumed to our stohastally, ndependently between unts. The observed sample of respondents an be vewed therefore as the result of a two-phase samplng proess where n the frst phase the sample S s seleted from U wth known nluson probabltes Pr( S) and n the seond phase the sample R s self seleted wth unknown response probabltes (Särndal and Swensson, 1987). Denote by p( Y, X) Pr( R Y, X, S) and let u and v be any random vetors suh that ( u, v ) and response ndators, R ( R 1 f R and 0 otherwse), are ndependent gven ( Y, X, S). For example, u and v are funtons of ( Y, X ), or the responses are ompletely defned by ( Y, X ). In what follows we use the followng relatonshps between populaton and sample dstrbuton (Pfeffermann and Sverhkov 1999, 2003 and Sverhkov and Pfeffermann 2004) whh an be wrtten n terms of response probabltes as, E( p ( Y, X ) u v, R) E( u v, S), (2.1) E( p ( Y, X ) v, R) E( u v, R ) E{[ p ( Y, X ) ] u v, R) E{[ p ( Y, X ) ] v, R). (2.2) Note that (2.1) mples E[ p ( Y, X ) R] / E[ p( Y, X ) S]. (2.3) Remark 2.1 In the followng setons we onentrate on estmaton of the response probabltes p( Y, X ). Note that f the response probabltes or ther estmates are known then the sample respondents an be onsdered as a sample from the fnte populaton wth known, p( Y, X), or estmated seleton probabltes, ˆ pˆ( Y, X). Then populaton model parameters (or fnte populaton parameters) an be estmated as f there were no non-response wth these new nluson probabltes, see Särndal and Swensson (1987). One an use these probabltes for mputaton also usng the relatonshp between the sample and sampleomplement dstrbutons derved n Sverhkov and Pfeffermann (2004), 2307

3 JSM Survey Researh Methods Seton f ( u v, R ) [ p ( Y, X ) ] f ( u v, R) E{[ p ( Y, X ) ] v, R}. (2.4) 3. Estmaton of the Response Probabltes when Non-Response s NMAR Let p( Y, X ; ) Pr( R Y, X, S; ) be a parametr set of pdf s and suppose that p( Y, X ; ) s dfferentable wth respet to (vetor) parameter. For smplty we onsder the followng senaro: The ovarates are observed for all nonrespondents,.e. Observed Data={ Y, R, X, k S}. k Under ths senaro, f the mssng data were later observed, ould be estmated by solvng the lkelhood equatons, log p( Y, X ; ) log[1 p( Y, X ; )] 0 R R. (3.1) Smlarly to the Mssng Informaton Prnple (Cplln et al, 1955, Orhard and Woodbury 1972), sne the outome values are mssng for j R, we propose to solve nstead, log p( Y, X ; ) log[1 p( Y, X ; )] 0 E[ Observed Data],.e., R R log p( Y, X ; ) log[1 p( Y, X ; )] 0 E[ { Y, R, X k, k S}] R R log p( Y, X ; ) log[1 p( Y, X ; )] E[ R,{ X k, k S}] R R R log p( Y, X ; ) p( Y, X ; ) p R log[1 p( Y, X ; )] E{[ p ( Y, X ; ) ] X, R} (3.2a) R E{[ p ( Y, X ; ) ] X, R} ( Y, X ; ) p( Y, X ; ) p ( Y, X ; ) f ( Y X, R) dy p ( Y, X ; ) f ( Y X, R) dy. (3.2b) R The thrd equaton follows from (2.2) where we assume for smplty that p( Y, X; ) and ( X k, k S) are ndependent gven X. Note that the seond sum n (3.2a) and (3.2b) predts the unobserved seond sum n (3.1). Note also that f p( Y, X; ) s a funton of X and only (mssng data are MAR) then (3.2b) redues to a ommon system of log-lkelhood equatons, log p( X ; ) log[1 p( X ; )] 0. (3.3) R R Estmatng funtons (3.2b) suggest the followng two-step estmaton proedure: Step 1. Ft the model fr( Y X) f ( Y X, R). Note that ths pdf refers to the respondents sample and therefore an be dentfed and estmated from the observed data usng lass statstal nferene. Step 2. Approxmate (3.2b) by replang fr( Y X ) by ts estmate, f ˆ r( Y X ), and solve (3.2b) for. 2308

4 JSM Survey Researh Methods Seton Note that nstead of estmaton of f r n (3.2b) one an estmate the expetatons n (3.2a) nonparametrally, and after substtutng the estmates n (3.2a) solve them for. For example, for dsrete X -s and an arbtrary funton g, E[ g( Y, X, ) X x, j R] an be estmated by the j j j respetve mean, 1 g( Y j, X j, ). For ontnuous X -s let mx (, ) be an jr: X jx jr: X jx estmator of E( g( Y, X, ) X x, j R), for example the Nadaraya-Watson estmator, mx (, ) jr j j j K[( x X j ) / h] g( Y j, X j, ), where h and K are a sale-fator and a kernel. K[( x X ) / h] jr j Estmatng the respetve ondtonal expetatons n the seond sum of (3.2a) by mx (, ) one obtans the followng estmatng equatons, p( Yk, X k; ) p( Y, X ; ) K[( X k X j ) / h] p ( Yk, X k; ) kr p ( Y, X ; ) 0, (3.4) 1 K[( X X ) / h][ p ( Y, X ; ) ] R whh defnes an estmator of. jr k j k k kr Estmatng equatons (3.4) do not requre any knowledge of the model for the respondents. On the other hand one an expet that the estmates obtaned by solvng (3.4) wll be less stable than the estmates obtaned from (3.2b) by the above two step estmaton proedure when the model for the respondents an be ftted well. See Sverhkov (2008) for the detals. 4. Is t MAR or NMAR? The proposed approah requres knowledge of the parametr form of the response model whh refers to the unobserved data n the ase of NMAR. On the other hand, f response s MAR, the propensty sore, px ( ; ) Pr( R X, S; ), an be estmated from the observed data for example by solvng a ommon system of log-lkelhood equatons (3.3). Note that the latter estmator muh more stable than the estmators assumng NMAR. Therefore t s very mportant to know whether response s MAR or NMAR. We suggest usng the followng proedure for testng the latter: Step 1. Ft the model for propensty sore, px ( ; ) Pr( R X, S; ), and estmate the parameter from the observed data assumng MAR. Step 2. Defne a lass of models for p( Y, X; ) Pr( R Y, X, S; ),, n suh way that for some, p( Y, X ; ) px ( ; ). It reommended to use models that nlude the Y- omponent n a smple form, for example, f logt[ p( X; )] g( X; ) then one an onsder logt[ p( Y, X; )] g(x ; ) Y, (, ), so n ths ase for (,0), p( Y, X ; ) px ( ; ). Step 3. Obtan estmatng equatons (3.2a) based on the lass of models defned n Step 2. Step 4.1. Solve them and hek whether Y-omponent s sgnfant (n whh ase the response s for sure NMAR ) or not (the response s MAR or not very nformatve ). 2309

5 JSM Survey Researh Methods Seton The latter an be done by a bootstrap proedure: one an take B smple random samples wth replaement from the orgnal sample and repeat steps 1 4 above n order to get a varane estmate for the Y-omponent. Remark 4.1. Sne the parametr famly defned n Step 2 does not neessarly nlude the true response probablty Pr( R Y, X, S), even f the Y-omponent s nsgnfant we annot onlude for sure that response s MAR. Nevertheless, we reommend to use propensty sore assumng MAR n ths ase. If response s very nformatve then one an expet that the Y- omponent wll be sgnfant even n a smplfed model. Instead of Step 4.1 one an do Step 4.2. Substtute from Step 2 nto (3.2a-b) obtaned n Steps 1-3 and hek whether the result s sgnfantly non-zero (response s NMAR) or not (response seems to be MAR sne orresponds to the propensty sore). The latter an also be done by use of a bootstrap. Aknowledgements The opnons expressed n ths paper are those of the author and do not neessarly represent the poles of the Bureau of Labor Statsts. Referenes Beaumont, J.F. (2000). An estmaton method for nongnorable nonresponse, Survey Methodology, 26, Ceplln, R., Snsalo, M., and Smth, C.A.B. (1955). The estmaton of gene frequenes n a random matng populaton, Annals of Human Genets, 20, Greenlees, J.S. Reee, W.S. and Zeshang, K.D. (1982). Imputaton of mssng values when the probablty of response depends on the varable beng mputed, Journal of the Ameran Statstal Assoaton, 77, Lttle, R.J.A. (1982). Models for nonresponse n sample surveys, Journal of the Ameran Statstal Assoaton 77, Lttle, R.J.A. (1993) Pattern-mxture models for multvarate nomplete data, Journal of the Ameran Statstal Assoaton, 88, Lttle, R.J.A. and Rubn, D.B. (2002). Statstal analyss wth mssng data, New York: Wley. Rubn, D.B. (1976). Inferene and mssng data, Bometrka 63, Rubn, D.B. (1987). Multple mputaton for nonresponse n surveys, New York: Wley Qn, J., Leung, D. and Shao, J. (2002). Estmaton wth Survey data under nongnorable nonresponse or nformatve samplng, Journal of the Ameran Statstal Assoaton 97, Orhard, T., and Woodbury, M.A. (1972). A mssng nformaton prnple: theory and applaton, Proeedngs of the 6 th Berkeley Symposum on Mathematal Statsts and Probablty, 1, Pfeffermann, D., and Sverhkov, M. (1999). Parametr and sem-parametr estmaton of regresson models ftted to survey data, Sankhya 61, Pfeffermann, D., and Sverhkov, M. (2003). Fttng generalzed lnear models under nformatve probablty samplng, In: Analyss of Survey Data, eds. C. J. Sknner and R. L. Chambers. New York: John Wley & Sons. pp

6 JSM Survey Researh Methods Seton Sarndal C.E., and Swensson B. (1987). A general vew of estmaton for two fases of seleton wth applatons to two-phase samplng and nonresponse, Internatonal Statstal Revew 55, Sverhkov, M., and Pfeffermann, D. (2004). Predton of fnte populaton totals based on the sample dstrbuton, Survey Methodology 30,