Statistical Analysis Using Combined Data Sources Ray Chambers Centre for Statistical and Survey Methodology University of Wollongong

Transcription

1 Statitical Analyi Uing Combined Data Source Ray Chamber Centre for Statitical and Survey Methodology Univerity of Wollongong JPSM Preentation, Univerity of Maryland, April 7,

2 What Are The Data? The claical perpective - a (random) rectangular window on the population of interet A Meier Real World Multiple ource of data with varying level of aggregation Typically a ditorted window on the target population + window on related population 2

3 Example 1 Sample value Y and X for two correlated variable at t 1 Regiter value of X at time t k 1 available at time t k Focu on etimation of population total of Y at t 1 Example 2 Value of Y from regiter A Value of X from regiter B Sample of unit from regiter A linked to regiter B Interet in modelling Y - X relationhip at regiter level Example 3 Value of Y from regiter A Value of X from regiter B Regiter probabilitically linked Interet in modelling Y - X relationhip at regiter level 3

4 Example 4 Value of Y and auxiliarie X and C from urvey A Value of Y and auxiliarie Z and C from urvey B Marginal etimate for Y baed on combined ample required Example 5 Value of 'accurate' zero-one variable Y from a mall urvey A Value of 'rough' zero-one approximation X from a much larger urvey B Small area etimate of Y required Example 6 Value of Y and Z from a large national urvey Value of X and Z from another, ditinct, large national urvey Value of Y, X and Z from a mall, non-repreentative, third urvey National model relating Y, X and Z required 4

5 How do we tackle inference uing thee complex data ource? 5

6 General Approach 1: Calibrated Sample Weighting Sample weighting i a tandard method of urvey etimation Many urvey etimation ytem ue calibrated ample weight - i.e. they are capable of exactly reproducing known population quantitie - Typically, thee are either total or mean of auxiliary variable - Natural way of integrating external information into urvey weight 6

7 Model-Aited Approach Cloet Calibrated Weighting Deville and Särndal (1992) - Ue calibrated ample weight w i that are cloet to the expanion weight π i 1 (i.e. the weight that define the Horvitz-Thompon etimator for the urvey variable) A metric for cloene Q = (w π 1 ) Ω(w π 1 ) Minimiing Q ubject to calibration lead to GREG weight ( ) 1 ( - w = π 1 + Ω 1 Z Z Ω 1 Z Z U 1 N Z π 1 ) - Z U = matrix of population value of auxiliary variable - Z = correponding matrix of ample value - Ω= poitive definite matrix (typically diag( π i v i )) 7

8 Model-Baed Calibration y U = Z U β + e U Unbiaed etimation under thi model i a good thing... E( ˆt y t y Z ) U = 0 Calibration with repect to the auxiliary variable in Z U i equivalent to unbiaedne with repect to the linear model E(y U Z U ) = Z U β ( ) = E E ˆt y t y Z U ( w y 1 N y U Z ) U = ( w Z 1 N Z U )β = 0 So calibration i a good thing provided the linear model aumption i valid! 8

9 Efficient Model-Baed Calibration Royall (1976) - Bet linear unbiaed predictor (BLUP) of t y i defined by weight of the form opt w = 1 n + H ( Z U 1 N Z 1 n ) + ( I n H Z )V 1 V r 1 N n.. I n = identity matrix of order n.. 1 k = k-vector of one ( ) 1.. H = Z V 1 Z.. V U Var( y U ) = Z V 1 V V r V r V rr Model unbiaedne thee weight are calibrated on Z U opt Z w = Z U 1 N 9

10 Example 1: Calibrating to Out of Date Contraint The population marginal information ued to define a calibration contraint i often not exact... We want on calibrate on X, but do not know the population total of thi variable. Intead, we know the population total of Z, a variable that i approximately the ame a X - Should we try to predict the population total of X or jut calibrate to Z? Y = WAGES (wage bill for a buine) X = EMP (# employee of the buine) Z = Regiter EMP (# employee of the buine at lat update of the buine regiter) 10

11 Sector K1 (N = 1005) WAGES log(wages) EMP 0 log(emp) WAGES log(wages) Regiter EMP 0 log(regiter EMP) 11

12 Preferred Model (A) E(Y X) = α X + β X X Var(Y X) = σ 2 X X 2 - Calculation of Preferred BLUP require value of t X (unknown) - Ue Predicted value of t X? - Or Subtitute t for t Z X? Alternative Model (B) E(Y Z) = α Z + β Z Z Var(Y Z) = σ Z 2 Z 2 - Calculation of thi Alternative BLUP require value of t Z (known) σ X 2 < σ Z 2, o BLUP under (A) expected to be more efficient than BLUP under (B) Value of Y, X and Z available on the ample 12

13 Deign-Baed Simulation %RelBia %RelRMSE Pref Alt Sub Pred Pref Alt Sub Pred STR1: tratified on Z / equal allocation / acro tratum etimation GREG BLUP STR2: tratified on Z / equal allocation / within tratum etimation Both PPZ: no tratification / probability proportional to Z ampling GREG BLUP

14 Reality? Overall accuracy i all well and good, but a more practical requirement i that our etimate hould deviate a little a poible from it preferred value... - preferred value i etimate defined by calibrating on X - we want to minimie reviion when t X doe become available... 14

15 Ditribution of %RelDiff for STR1 GREG BLUP 15

16 General Approach 2: Likelihood-Baed Information Pooling The data we have A (confuing) mix of individual Y-value, value of other, related, variable, ummary tatitic, metadata (e.g. data definition), paradata (e.g. information about how the data were obtained, ample weight, auxiliary data for the target population), related data from other urvey and other population, etc, etc... The data we d like to have (for likelihood inference) A clean rectangular databae with unit record data for the target population and related population 16

17 The Miing Information Principle Likelihood-baed inference uing a mey oberved dataet d can be achieved by carrying out likelihood-baed inference uing a larger clean dataet d U but with the ufficient tatitic defined by d U replaced by their expected value given d Note 1. It doen t matter what d U i. The only requirement i that d (the data we have) i a ubet of d U (the data we would like to have) 2. Firt developed (Orchard and Woodbury, 1972) for inference with miing data, and form bai for EM algorithm (Dempter, Laird and Rubin, 1977) ued widely with miing data 3. Application to analyi of urvey data by Breckling et al (1994) 4. Bai for data augmentation algorithm ued to generate poterior ditribution (Tanner and Wong, 1987) 17

18 MIP Identitie Provided the ideal data d U include the available data d, the available data core c for the parameter Θ of the ditribution of d U i the conditional expectation, given thee data, of the ideal data core c U for Θ, i.e. c = E{ Θ log f (d U ) d } = E ( c U ) Furthermore, the available data information info for Θ i the conditional expectation, given thee data, of the ideal data information info U for Θ minu the correponding conditional variance of the ideal data core c U, i.e. { } Var{ Θ log f (d U ) d } info = E ΘΘ log f (d U ) d ( ) Var ( c U ) = E info U 18

19 Example 2: Combining Survey Data and Marginal Population Information Comparing the MIP With Calibrated Weighting Motivating Scenario Population U i uch that value y i and x i of two calar variable, Y and X are tored on eparate regiter, each of ize N. A ample of n unit from one regiter i linked to the other via a unique common identifier, thu defining n matched (y i,x i ) pair Aim To ue thee linked ample data to etimate the parameter α, β and σ 2 that characterie the population regreion model where ε i ~ iid N(0,1) y i = α + βx i + σε i 19

20 Extra Information Regiter ummary data are available. In particular, we know the population mean y U and x U of Y and X... - OLS etimate are no longer the full information MLE - Can ue the MIP to combine thi population marginal information with the urvey data to obtain full information MLE - Ue E and Var to denote conditioning on ample value of Y and X + population mean of thee variable 20

21 MIP component of the available data core function are c 1 = 1 σ 2 { E (y i ) α βx } U i { } c 2 = 1 x σ 2 i E (y i ) α βx U i c 3 = N 2σ σ 4 { } 2 E (y i ) α βx U i + U Var (y i ) For non-ample i y i x U, y U N y U + β(x i x U ),σ N n 21

22 Lead to an available data core with component c 1 = 1 (y σ 2 i α βx i ) + (N n) ( y U α βx U ) { } ( ) { } c 2 = 1 x σ 2 i (y i α βx i ) + (N n)x U y U α βx U c 3 = (n + 1) 2σ { + (N n) ( y U α βx U ) 2 } 2σ 4 (y i α βx i ) 2 Full information MLE defined by etting thee core component to zero and olving for α, β and σ 2 22

23 Full Information MLE ˆβ fimle = (x i x )(y i y ) + nx (y y U ) + (N n)x U (y U y U ) (x i x ) 2 + nx (x x U ) + (N n)x U (x U x U ) and ˆα fimle = y U ˆβ fimle x U 2 ˆσ fimle = 1 n + 1 ( y i ˆα fimle ˆβ fimle x ) 2 i + (N n) ( y U ˆα fimle ˆβ fimle x ) 2 U 23

24 Peudo-Likelihood Inference Kih and Frankel (1974), Binder (1983), Godambe and Thompon (1986) f (y U ;θ) = probability denity of population Y value If y U were oberved, θ would be etimated by olution of c U = θ log f (y U ;θ) = 0 For any pecified value of θ, c U define a finite population parameter ( cenu core ), which we can etimate from the ample data c w = ample-weighted etimator of c U maximum peudo-likelihood etimator (MPLE) of θ i olution to c w = 0 24

25 Calibration + Peudo-Likelihood Aume SRSWOR. There are three calibration contraint - the population ize N - population mean of X - population mean of Y [ ] w cal = N n 1 n + N 1 n y x 1 n 1 n 1 n y 1 n x y 1 n y y y x x 1 n x y x x 1 0 y U y x U x ˆβ cal = { w cal i x i (x i x w )} 1 w cal i x i (y i y w ) ˆα cal = y w ˆβ cal x w 2 ˆσ cal = N 1 w cal i (y i ˆα cal ˆβ cal x i ) 2 25

26 Model-Baed Simulation % relative efficiencie with repect to 5% trimmed RMSE of MPLE (π 1 -weighted) σ 2 Parameter Sampling N = 500, n = 20 N = 1000, n = 50 N = 5000, n = 200 α Scheme CAL MIP CAL MIP CAL MIP SRS PPX PPY β SRS PPX PPY SRS PPX PPY

27 Example 3: Modelling Probability-Linked Data Fellegi and Sunter (1969) Record Linkage i a olution to the problem of recognizing thoe record in two file which repreent identical peron, object, or event... Y-regiter contain value of calar random variable Y X-regiter contain value of vector random variable X Modelling of (Y, X) relationhip traightforward given a random ample of (Y, X) value. But, we do not have uch a ample intead probabilitic record linkage ued to link record from the two regiter 27

28 Toy Aumption Both regiter contain N record, with no duplication Complete linkage: All record on both regiter can be linked Categorical 'blocking' variable Z recorded on both regiter meaured without error on both take Q ditinct value q = 1,2,,Q M q record in each regiter with Z = q (o N = ) M q q Linkage error can only occur within block' We index the record on the linked data et in exactly the ame way a we index the X-regiter 28

29 A Model for Linkage Error y q = A q y q A q i an unknown random permutation matrix of order M q, i.e. entrie of A q are either zero or one, with a value of one occurring jut once in each row and column - E X (A q ) = E q (aumed known for the moment) Non-Informative Linkage A q y q X q E X (y q ) = E q E X (y q ) 29

30 Exchangeable Linkage Error Since linkage proce maximie the probability that a declared link i a true link, correct linkage hould be more likely than incorrect linkage... Pr(correct linkage in block q) = λ q Pr(wrong linkage in block q) = γ q E q = E X ( A ) q = λ q γ q γ q γ q λ q γ q γ q γ q λ q 30

31 Etimating Function with Linked Data Unbiaed etimating function given correctly-linked data H(θ) = N { } G i (θ) y i f i (x i ;θ) = G q (θ) y q f q (X q ;θ) i=1 q { } When ued with probability-linked data, thi become H (θ) = q G q (θ){ y q f q (X q ;θ)} 31

32 Etimating function i no longer unbiaed... { } = G q (θ 0 ) E q I q E X H (θ 0 ) {( )f q (X q ;θ 0 )} 0 q A bia-corrected etimating function H adj (θ) = H (θ) G q (θ) ( E q I q )f q (X q ;θ) q { } = G q (θ){ y q E q f q (X q ;θ)} q Variance etimation: Standard Taylor erie approximation, leading to a plug-in andwich etimator 32

33 Logitic Regreion { } G q (β) y q E q f q (X q ;β) = 0 q M (define MLE when data are correctly linked): G q (β) = X q A (lead to unbiaed etimator in linear model): G q (β) = X q E q C (econd-order optimal) where G q (β) = β E ( X y ) q { }{ Var ( X y )} 1 q = X q D q (β) E q Σ 1 q (β) D q (β) = diag f i (β){ 1 f i (β)};i q Σ q (β) = Var( y q X ) q 33

34 Some Simulation Reult Three block, M 1 = 1500, M 2 = 300 and M 3 = 200, with independent exchangeable linkage error in each block Two cenario o λ q correctly pecified (λ 1 = 1.0, λ 2 = 0.95, λ 3 = 0.75) o λ q etimated by ˆλ q = min{ m 1 ( q m q 0.5),max( M 1 q,l ) q }, with l q equal to the number of correct link in a random ample of m q = 20 linked record in each of block 2 and 3 logit { E( y i x )} i = 1 5x i, with x i Uniform[0,1] 34

35 Focu on Etimation of Slope Parameter Etimator Relative Bia Relative RMSE Coverage Scenario 1: Linkage Probabilitie Correctly Specified Perfect Linkage Naive M A C Scenario 2: Linkage Probabilitie Etimated Perfect Linkage Naive M A C

36 Related Reult MIP - Chamber (2009), Chamber et al. (2009) Sample Regiter linkage, with non-linkage + linkage error o Kim and Chamber (2009) Allowing for linkage error (Regiter Regiter) in linear mixed modelling o Samart and Chamber (2010) Longitudinal linkage with error (Sample multiple Regiter) o Kim and Chamber (2009) 36

37 The Other Example Example 4: Merkouri (JASA, 2004) "Combining independent regreion etimator from multiple urvey" GREG( X,C Sugget compoite GREG etimator: φˆt ) GREG(Z,C ) Ay + (1 φ)ˆt By Alternative BLUP baed on Or perhap y A y B = X A C A 0 0 C B Z B β + e A e B? y A y B = X A C A Ẑ A ˆX B C B Z B β + e A e B? 37

38 Example 5: Elliot and Davi (Applied Statitic, 2005) "Obtaining cancer rik factor prevalence etimate in mall area: combining data from two urvey" Bae SAE on urvey B data, but ue urvey A data to modify urvey B weight to get rid of the Y-X bia Implemented via propenity weight adjutment that enure marginal probability that Y = 1 in area g i the ame for both urvey A and urvey B Ue of propenity-baed bia correction increae variance relative to urvey B etimator that do not ue thi correction. However, reduction in bia lead to maller MSE If it i poible to identify value of X for urvey A ample, then thi i item non-repone for Y in urvey B. MIP can be applied directly. 38

39 Example 6: Strau, Carroll, Bortnick, Menkedick & Schultz (Biometric, 2001): "Combining data et to predict the effect of regulation of environmental lead expoure in houing tock" Y = log(blood lead concentration) - NHANES, Rocheter Study H = log(environmental expoure) a meaured by HUD G = log(environmental expoure) a meaured by Rocheter Study C = expoure variable common to HUD and Rocheter Study Target Model Y = α + βh + γ C + e Gauian meaurement error model + Rocheter/HUD data ued to get etimate of β and γ. Marginal data on Y from NHANES then ued to etimate α Complex, but MIP hould be applicable... 39

40 THANK YOU! 40

41 Reference Binder, D.A. (1983). On the variance of aymptotically normal etimator from complex urvey. International Statitical Review, 51, Breckling, J.U., Chamber, R.L., Dorfman, A.H., Tam, S.M. and Welh, A.H. (1994). Maximum likelihood inference from urvey data. International Statitical Review, 62, Chamber, R. (2009). Regreion analyi of probability-linked data. Official Statitic Reearch Serie, Vol 4, No. 2, Statitic New Zealand, Wellington. ( Chamber, R., Chipperfield, J., Davi, W. and Kovacevic, M. (2009). Inference baed on etimating equation and probability-linked data. Working Paper 18-09, Centre for Statitical and Survey Methodology. Dempter, A.P., Laird, N.M. and Rubin, D.B. (1977). Maximum likelihood from incomplete data via the EM algorithm (with dicuion). Journal of the Royal Statitical Society Serie B, 39, Deville, J.C. and Särndal, C.E. (1992). Calibration etimator in urvey ampling. Journal of the American Statitical Aociation, 87, Felligi, I.P. and Sunter, A.B. (1969). A theory for record linkage. Journal of the American Statitical Aociation, 64,

42 Godambe, V.P. and Thompon, M.E. (1986). Parameter of uper population and urvey population: their relationhip and etimation. International Statitical Review, 54, Kim, G. and Chamber, R. (2009). Regreion analyi under incomplete linkage. Working Paper 17-09, Centre for Statitical and Survey Methodology. Kim, G. and Chamber, R. (2010). Regreion analyi for longitudinally linked data. Working Paper 22-10, Centre for Statitical and Survey Methodology. Kih, L. and Frankel, M.R. (1974). Inference from complex ample (with dicuion). Journal of the Royal Statitical Society, Serie B, 36, Orchard, T. and Woodbury, M.A. (1972). A miing information principle: theory and application. Proc. 6th Berkeley Symp. Math. Statit., 1, Royall, R.M. (1976). The linear leat quare prediction approach to two-tage ampling. Journal of the American Statitical Aociation}, 71, Samart, K. and Chamber, R. (2010). Fitting linear mixed model uing linked data. Working Paper 18-10, Centre for Statitical and Survey Methodology. Tanner, M.A. and Wong, W.H. (1987). The calculation of poterior ditribution by data augmentation (with dicuion). Journal of the American Statitical Aociation, 82,