Weighted Estimating Equations with Response Propensities in Terms of Covariates Observed only for Responders
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1 Weghted Estmatng Equatons wth Response Propenstes n Terms of Covarates Observed only for Responders Erc V. Slud, U.S. Census Bureau, CSRM Unv. of Maryland, Mathematcs Dept. NISS Mssng Data Workshop, November
2 Dsclamer Ths report s released to nform nterested partes of ongong research and to encourage dscusson of work n progress. The vews expressed are the author s and not necessarly the Census Bureau s. 2
3 Outlne 1. Standard Household Survey Data Structure Propensty Covarates observed at Intervew MAR & Condtonal Independence of Y, R gven X 2. Modfed Estmatng Equatons Alternatve Forms 3. Consequences for Nonresponse Adjustment n Complex Surveys 3
4 Survey- or Based- Samplng Motvaton Data {X (1, R, R (X (2, Y : S} S U probablty sample from frame U Incluson prob s π, R response ndcator (lkely depend on both X (1, and X (2 X (1, X (2 predctve (unt-level covarates Y attrbute of nterest wth desred populaton mean µ Y 4
5 Predctve covarates X (1 ( X (11 X (12, X (2 ( X (21 X (22 Known totals µ X (11, µ X (21 of X (11 and X (21 X (11 ncludes 1 (ntercept R response ndcator condtonally ndep. of Y gven X (1 X (1 components, e.g., from paradata on modes of nterm refusal n multple contact attempts, wthout known means. (2 Regresson on X (a = (X (11, X (21 may leave resduals dependent on propensty predctors X. (3 Cond. ndep. R, Y may hold gven X but not gven X (1. (therefore nformatve 5
6 Problem Settng Workng lnear outcome model n terms of X (a = (X (11, X (21 E(Y X (a = β X (a E(X (a = µ a known Estmate mean of Y as ˆβ µ a Nonresponse adjustment va Inverse Probablty Weghtng wth respect to propensty p 0 (X, γ = P (R = 1 X Survey analysts do not use estmatng equatons wth such propenstes; nstead, do post-stratfed rato adjustment for nonresponse, followed by regresson estmaton. 6
7 Amercan Communty Survey Varables Covarates: X (1 = (Mult-unt, Base-Wt, URBAN, CTY, Nghbd*, X (11 = (Geography down to block-group, Mult-Unt X (2 = (BLD-type, OWNER, AGE, SEX, HISP, RACE X (21 = (AGE, SEX, HISP, RACE * summary n plannng data base (PDB at block-gp level Housng-type covarates not avalable n ACS before ntervew Indvdual ACS covarates may be mssng and mputed unt-level covarates dsplace PDB covarates Imputatons do not much affect block-group ACS covarates 7
8 Notes from Semparametrc Theory, I I.I.D. Data (R, X (1, R (X (2, Y observable X = (X (1, X (2, X (j = (X (j1, X (j2, X (a = (X (11, X (21 Ignore survey (based-samplng aspect and restrct (X, Y, R only by jont denstes satsfyng ( Y, R condtonally ndependent gven X ( µ a = E(X (a known ( E(Y X (a = β X (a (v p 2 (x 2 x 1 P (X (2 = x 2 X (1 = x 1 known 8
9 Semparametrc theory (Tsats 2006 Regular Asympt. Lnear Estmators of β satsfy estmatng equaton n =1 R p 0 (X g(x(a (Y β X (a = 0 (1 In regresson-type estmator ˆµ Y = ˆβ µ a = n 1 n =1 R Y /p 0 (X + ˆβ (µ a ˆµ IPW X (a s double-robust by def n because model-asssted desgn-based. Optmal Estmatng Equaton of form (1 has / ( g(x (a = X (a (Y X (a β 2 E p 0 (X wth n { / a.var( (ˆβ β = E [X (a 2 E ( (Y X (a β 2 p 0 (X X(a } 1 X(a] 9
10 Semparametrc Theory, II Idea of Pfeffermann and Sverchkov (1999, 2009: to estmate µ Y by ˆβ µ a wth ˆβ coeffcents estmated from S w R Ê RS (w X (a X (a (Y β X (a = 0 where w / Ê RS (w X (a s a smoothed weght, wth cond. exp. gven sample-ncluson and response. w may depend on (Y, X ; denomnator uses (msspecfed nsample parametrc model, e.g. WLS regresson of w on X (a. If denom. converges n prob. to nonrandom functon of X (a, at 1/ n rate, then β estmator s consstent n superpopulaton f lnear outcome model E(Y X (a = β X (a holds. 10
11 Alternatve Estmatng Equatons for γ For β use (1. Forms for γ nclude the followng: (I (wth thanks to Z. Tan When enough totals are known dm(h(x (1 + dm(x (21 = dm(γ one general form s based on external calbraton totals: n =1 h(x (1 ( R p 0 (X, γ 1 = 0 (2 n =1 ( X (21 R p 0 (X, γ µ X (21 = 0 (3 11
12 (II If p 2 (x 2 x 1 completely known, p 0 (, γ = P (R=1,x 2 x 1, γ. p 2 (x 2 x 1 For suffcent set of q s, n =1 q(x (1 ( R I (2 [X =x 2 ] P (R = 1, x 2 X (1, γ = 0 More often, not all jont cell-values are known ( rakng. (III Treat external calbraton data as over-determnng a model p 2 (x 2 x 1, α. Compatblty condtons between external (α and nternal (γ survey models: (for suffcently large set of q, B n =1 q(x (1 ( R I (2 [X B] p 0(X, γ p 2 (X (2 B X (1, α = 0 12
13 External versus Current Data Model α n p 2 (x 2 x 1, α based on hgh-qualty external data; varablty not always quantfed estmaton may also use current-survey data γ must use nternal survey data relatng X to R Control nformaton may be very hghly detaled from sources such as US Pop. Estmates down to county-level demographcally cross-classfed (13 Age-Gp by 6 Race/Hsp by Sex, but many cells are too small to be 100% relable, so can work wth model p(x, α suppressng hghest-order nteractons. THEN eq ns n (II, (III can be used, to solve exactly or to mnmze weghted sum of squares to estmate survey propensty parameters γ. 13
14 Survey Forms of Estmatng Equatons 1st step n transton: Posson samplng, effcency results 2nd step: hgh-entropy samplng (Hajek 1964, Tan 2014 (ncludes SRS and other PPS rejectve samplng stll mantan effcency results General complex surveys: no lkelhood-based optmalty results, but apply same nverse-propensty-weghted estmatng equatons wth survey weghts. w : (possbly adjusted, not yet calbrated weghts p 0 (x, γ d-dm logstc regresson, dm(x (a d dm(x 14
15 Estmatng equatons S w h(x (1 ( R p 0 (X, γ 1 = 0 S w ( X (21 R p 0 (X, γ µ X (21 = 0 S w X (1 ( R I (2 [X B k ] p 0(X, γ p 2 (X (2 B k X (1, α = 0 and S w R p 0 (X, γ g(x(a (Y β X (a = 0 15
16 Dscusson on Search for Covarates Kreuter, Olson, Wagner et al. (2010, Usng Proxy Measures and other Correlates of Survey Outcomes to Adjust for Non-response, JRSSA hghly cted argue va correlatons that varables hghly dependent both on Survey Outcomes and Response ndcator are hard to fnd. same asserton dffcult to justfy n large surveys f sngle varables can be replaced by blocks of nteractng varables. X (a outcome varables could smultaneously nteract wth a subset of X varables that strongly nteract wth block of key varables n propensty p 0 (X = P (R = 1 X Stronger possblty f propensty nvolves outcome varables. 16
17 Summary (1 Propenstes may nvolve covarates observed at ntervew; survey world does ths only through poststratfed regresson. (2 In IID/Posson-samplng settngs, weghted regresson estmates from S w R h(x (a p 0 (X,ˆγ (Y β X (a are effcent. (3 Weght-smoothng strateges may help but do not mprove on (2 n nonnformatve-samplng settngs. (4 External control data can usually not supply fully crossclassfed totals or stable calbrated survey weghts. Must be ncorporated through (α models forced to be compatble wth propensty (γ parameter estmates. Ths s a drecton of further research. 17
18 References ACS Desgn & Methodology (2014, Ch. 11, Weghtng Kreuter, F. et al. (2010, JRSSA Pfeffermann, D. and Sverchkov (1999, Sankhya B (2009 chapter n: Handbook of Statstcs: Survey Samplng chapter, North-Holland Tan, Z. (2014, calbraton... hgh-entropy samplng, Bometrka Tsats, A. (2006 Semparametrc Theory and Mssng Data, Sprnger 18
19 Thank you! 19
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