Maximum Likelihood Logistic Regression With Auxiliary Information

Transcription

1 niveity of Wollongong Reeach Online Cente fo Statitical Suvey Methodology Woking Pape Seie Faculty of Engineeing and Infomation Science 2008 Maximum Likelihood Logitic Regeion With Auxiliay Infomation R Chambe niveity of Wollongong, ay@uoweduau S Wang Texa AM niveity Recommended Citation Chambe, R and Wang, S, Maximum Likelihood Logitic Regeion With Auxiliay Infomation, Cente fo Statitical and Suvey Methodology, niveity of Wollongong, Woking Pape 12-08, 2008, 22p Reeach Online i the open acce intitutional epoitoy fo the niveity of Wollongong Fo futhe infomation contact the OW Libay: eeach-pub@uoweduau

2 Cente fo Statitical and Suvey Methodology The niveity of Wollongong Woking Pape Maximum Likelihood Logitic Regeion With Auxiliay Infomation Ray Chambe and Suojin Wang Copyight 2008 by the Cente fo Statitical Suvey Methodology, OW Wok in poge, no pat of thi pape may be epoduced without pemiion fom the Cente Cente fo Statitical Suvey Methodology, niveity of Wollongong, Wollongong NSW 2522 Phone , Fax anica@uoweduau

3 Maximum Likelihood Logitic Regeion With Auxiliay Infomation Ray Chambe Cente fo Statitical and Suvey Methodology, niveity of Wollongong Wollongong NSW 2522, Autalia and Suojin Wang Depatment of Statitic, Texa AM niveity College Station, Texa 77843, SA Summay In thi pape we ue the geneal appoach to maximum likelihood etimation fo complex uvey data decibed in Beckling et al 1994 to develop method fo efficiently incopoating extenal population infomation into linea logitic egeion model fitted via ample uvey data In paticula, we ue innovative addlepoint and meaing method to deive highly accuate appoximation to the coe and infomation function defined by the model paamete unde andom ampling and unde cae-contol ampling when auxiliay data on population moment ae available Simulation-baed eult illutating the eulting gain in efficiency ae povided Some key wod: Sample uvey; Cae-contol; Saddlepoint appoximation; Smeaing etimate; Scoe function; Infomation function 1 Intoduction Analyi of uvey data doe not happen in a vacuum A model fo the pobability that a woman ue a paticula contaceptive method will typically depend on a numbe of facto, eg he age, he education level, he labou foce tatu, he houehold income, he ethnic backgound and he acce to family planning infomation, pehap meaued by peence o

4 abence of a family planning clinic within a pecified ditance of he home All of thee vaiable ae meaued fo women taking pat in the uvey, and the claical appoach i to conide them in iolation in the modelling poce, implicitly auming that the model fitted to thee ample data i alo appopiate fo the population fom which the ample i dawn Sometime, if thi i felt to be too big an aumption, and uvey weight ae available, thee ae included in the model fitting poce, auming that they coect the paamete etimation poce fo potential ample election bia Howeve, we typically know a lot moe about the taget population than jut the data obeved in the uvey In paticula we may know the total numbe of women in the population, the popotion of women in the population that ue the contaceptive method of inteet, thei aveage age, thei labou foce paticipation ate and thei ethnic ditibution in the population By know hee we mean eithe the actual population value o at leat an accuate etimate The quetion hee i how to efficiently integate thi auxiliay population infomation into the model fitting poce decibed above In ome cae, thi infomation i incopoated in the uvey weight, though the poce of calibation Deville and Sändal, 1992; Chambe, 1996 That i, thee weight ae contucted o that weighted aveage fo elected vaiable meaued in the uvey equal coeponding known o highly accuate etimate of population value One appoach to uing thi auxiliay infomation would theefoe be to ue uch calibated weight in etimation Howeve, thi ha two majo poblem Fit, uch weight typically lead to inceaed tandad eo compaed to unweighted analyi Second, uvey weight ae uually calibated to a fixed and elatively mall et of vaiable eg age by ex population ditibution, egional population ditibution, while population data ae often known fo many moe vaiable 2

5 Altenative, moe model-baed, way of incopoating auxiliay population infomation when modelling uvey data have been exploed in the econometic liteatue, mainly in the context of analyi of linked data et An ealy example i Imben and Lancate 1994, who ugget a genealied method of moment appoach to the poblem of incopoating knowledge of the population expected value of the epone vaiable Y into a ample-baed linea egeion of Y on an explanatoy vaiable X Moe ecently, Qin 2000 ha conideed the ame poblem uing a combination of empiical and paametic likelihood See Handcock, Rendall and Cheadle 2005 fo a compehenive eview of ecent development in thi aea Thi pape focue on developing method fo efficiently uing auxiliay population infomation when uvey data ae ued to fit a linea logitic egeion model to a taget population In paticula, we look at how maximum likelihood method fo fitting uch model can be modified to incopoate thi infomation The appoach we take i baed on the geneal appoach to maximum likelihood etimation fo complex uvey decibed in Beckling et al 1994, heeafte efeed to a BCDTW In the following ection we develop the geneal theoy fo maximum likelihood etimation of a linea logitic model when the ample data ae obtained via imple andom ampling In Section 3 we develop coeponding eult when cae-contol ampling i ued In both ection we peent imulation eult that illutate the efficiency gain fom incopoating auxiliay population infomation Section 4 conclude the pape with a dicuion of elated iue and extenion of the method 2 MLE unde andom ampling Conide the following imple ituation A uvey meaue the value y i and of a zeo-one vaiable, Y, and a cala vaiable, X, epectively, fo a ample of n unit fom a population of N unit The vaiable X i a population covaiate, ie we know the value of X 3

6 fo evey unit in the population and the ampling method i imple andom ampling without eplacement Ou aim i to ue thee ample data to fit a linea logitic model to the population value of Y and X That i, we want to ue thee data to etimate the paamete! =! 0,! 1 T that chaacteie the linea logitic population egeion model! = P y i = 1 = exp" 0 + " 1 { 1+ exp" 0 + " 1 } #1 Auming ditinct population unit ae independently ditibuted, the maximum likelihood etimate MLE fo! i then the olution to the etimating equation c 1n! = $ { y i " # } = 0, c 2n! = $ y i " # = 0 whee the ummation ae ove the ampled unit We efe to thi etimate a the ample MLE, o SMLE, in what follow to indicate that it i the MLE that only ue the infomation in the n ample value of Y and X Suppoe that we alo know the population total t y =! y i of Y Thi can happen, fo example, if the vaiable Y i alo meaued in a cenu, and cenu tabulation ae publihed Suppoe a well that the individual population value of X ae known and identifiable a would be the cae if they wee held on a egite and the ample elected fom that egite Now the SMLE i no longe the MLE fo! In ode to obtain the full infomation MLE that include thi additional infomation, we fit obeve that the population level coe function fo thi paamete i c! = c 1!, c 2! T, whee c 1! = $ { y i " # } $ c 2! = y i " # and the ummation ae ove the N unit defining the population In what follow we let E and Va denote the expectation and vaiance opeato epectively that condition on the 4

7 available data fo ue in analyi In thi cae thee data coepond to the ample value of Y and X, the non-ample value of X and the population mean of Y We efe to the coe function fo! given thee data a the full infomation coe function fo thi paamete BCDTW how that the full infomation coe function i the conditional expectation of the coeponding population level coe function given the available data That i, the full infomation coe function c! = c 1!, c 2! T, whee c 1! = E c 1! c 2! = E c 2! = y i " # " $ 1a = { y i " # } $ $ + E y i " # $ 1b whee denote the et of non-ampled population unit Let t y =! = t y "! y i denote the known total of Y fo thee non-ample unit Alo, fo abitay non-ample population unit i, let i denote the emaining N n 1 non-ampled population unit Without lo of geneality we aume that t y > 0, and obeve that the conditional expectation in 1b can then be witten E! y i x i! y i,x =! E y i! y j,x! P y i = 1,! y i j " 1 x = P! y j x =! # R 1i, whee R 1i = P {! y j x } "1 P y i j function component defined by 1 ae theefoe y i! " 1 x i The full infomation coe c 1! = $ { y i " # } 2a c 2! = $ y i " # " $ # 1" R 1i 2b A addlepoint appoximation to the econd tem on the ight hand ide of 2b i developed in the Appendix Thi i 5

8 c 2! " % { y i # $ } # % $ 1# [1+ { 1# $ }{bt y # 1}] #1 2c with bt y = exp $ #!x j 1"!x j * % ' "1 + {#!x j " t y }, - We denote by FIMLE the coeponding appoximation to the full infomation MLE of! obtained by etting 2a and 2c to zeo and olving fo! It i unlikely in pactice that the actual non-ample X value will be known Since the full infomation coe function 1 depend diectly on thee value, we need to evie thi function when non-ample X value ae unavailable In geneal, the coe function fo! i then defined by c 1! = " y i # " $ # E $ 3a " c 2! = $ y i " # + E y i $ " E { # } 3b whee E now denote expectation afte conditioning on the actual auxiliay infomation that we have we continue to aume that t y i known Suppoe that we know the non-ample $ mean x of X We can then appoximate the conditional expectation E {"! } and E {"! } in 3 uing a meaing appoach Duan, 1983 Thi i baed on the aumption that, fo an abitay function f of x that depend on ome paamete!, we can wite 1 # f," = N! n 1 # f x +! x," $ 1 # f x! x +," N! n n Put! = x " x The meaing appoximation to E {"! } i then # N $ n E "! n "!% + We theefoe eplace the coe component 3a by 6

9 c mea1! " # y i $ # % $ N $ n # % + 4a n A coeponding meaing appoximation to 3b that include a addlepoint appoximation i given by A7 in the Appendix Thi allow u to eplace thi component coe by % c mea 2! = $ y i " # " N " n ' n * + + % + N " n ' n * $ + + #+ +, 1+ 1" #+ + - $ #+ + b mea t y " 1 / "1 4b whee / b mea t y = exp 1 %$!" + { 1#!" + }' 0 #1 * + $!" + # n N # n t, 2 y -4 3 We efe to the olution to etting 4 to zeo and olving fo! a the SMEAR etimato fo thi paamete In pactice population benchmak like t y and x may in fact be etimated Thi can aie, fo example, if cenu coveage i incomplete, and o cenu output ae adjuted fo coveage eo It can alo be the cae that we have acce to etimate deived fom anothe lage uvey athe than cenu value fo thee benchmak A long a the eo o impeciion of uch etimation i mall, the etimato defined by 4 i till valid In paticula, if benchmak etimate!t y,!x fo t y,x that atify!t y = t y + o p n 1/ 2 and!x = x + o p n 1/2 / N ae ued in 4, then, apat fom a negligible eo of o p n!1/2, the eulting etimate of! i aymptotically equivalent to the SMEAR etimato Thi comment alo applie to the FIMLE when the benchmak t y i aumed to be ubject to eo Finally, thee i the cae whee even x i unknown In thi cae we can till ue 4, but eplace x by an appopiate ample-baed etimate Thi will depend on the chaacteitic of the ample deign and the natue of the auxiliay population infomation available to u Fo the cae of imple andom ampling and no auxiliay infomation it i 7

10 natual to etimate x by x, ie ue expanion etimation Thi i equivalent to etting! = 0 in 4 We efe to the etimato of! obtained by eplacing x by x and etting 4 to zeo a the expanion MLE fo thi paamete, and denote it by EXP Reult fom a imulation tudy of the pefomance of the etimato decibed above ae et out in Table 1 A total of 1,000 independent imulation wee caied out, with N = 5,000 population value fo X geneated fom the tandad lognomal ditibution and coeponding value fo Y geneated unde the linea logitic model A ample of n = 200 wa then taken fom each population uing imple andom ampling without eplacement SRSWOR The impact of benchmak eo on thee elative efficiencie wa aeed by conideing thee level of impeciion in the benchmak value ued a auxiliay infomation no eo in the benchmak, benchmak ubject to cenu-level eo benchmak ued equal to tue value plu a andom eo with zeo mean and tandad deviation equal to the actual maginal tandad deviation divided by N!1/2 and benchmak ubject to with lage uvey eo benchmak ued equal to tue value plu a andom eo with zeo mean and with tandad deviation equal to the actual maginal tandad deviation divided by N / 5!1/2 The value hown in Table 1 ae elative efficiencie, defined a the atio of the 5% timmed oot mean quaed eo 5%RMSE of a efeence etimato to the coeponding 5%RMSE of an altenative etimato, expeed a a pecentage Value ove 100 theefoe indicate upeio elative efficiency fo the altenative etimato The 5%RMSE i the quae oot of the 5% timmed mean of the quaed eo geneated by an etimato, ie afte timming the top 5% and bottom 5% of thee quaed eo A timmed RMSE wa ued to meaue efficiency in ode to avoid ditotion caued by a mall numbe of outlying eo value geneated in the imulation The efeence etimation method in Table 1 i SMLE ie the uual etimato of! given ampling i SRSWOR, computed uing the glm function in R 8

11 with it default option The EXP, SMEAR and FIMLE etimato wee calculated by uing the nlm function in R to olve thei epective etimating equation, with tating value! 0 = logy " log1" y and! 1 = 0 Table 1 about hee Fom Table 1 we clealy ee that, even when the benchmak data contain eo, all thee etimation method that ue thi infomation ae upeio to tandad logitic modelling in tem of efficiency Futhemoe, thee i little to chooe between any of the thee etimation method, with the FIMLE method maginally upeio at leat fo etimation of! 0 3 MLE unde cae-contol ampling In the peviou ection we aumed imple andom ampling fom the population of inteet Howeve, in many impotant application of logitic modelling, paticulaly in medicine, the ample data ae obtained via ome fom of cae-contol ampling In thi ituation the aumption undepinning the addlepoint and meaing appoximation ued in the development in the peviou ection ae no longe valid Howeve, the baic tategy of uing the appoach of BCDTW to incopoate auxiliay population infomation into infeence can till be ued, povided the fact that the ample data ae obtained via an infomative ampling method cae-contol ampling i allowed fo when taking conditional expectation Moe pecifically, we adopt the etup decibed in Scott and Wild 1997, and aume the exitence of two ampling fame, one fo the N 1 population unit with value Y = 1 and one fo the N 0 unit with Y = 0 Independent imple andom ample of ize n 1 and n 0 epectively ae then taken fom thee fame Value of X ae obeved on the ample, and the aim again i to fit a linea logitic model to thee data By definition, we know N 1 and hence t y = N 1! n 1 9

12 Again, we conide the ame thee ituation coeponding to diffeent level of knowledge of X The fit i whee we know the non-ample value of thi vaiable In the tandad cae-contol ituation thi i highly unlikely Howeve, it could coepond to a ituation whee a epaate adminitative egite contain thee value, and the cae-contol tudy i being ued to foge a link between the Y egite and the X egite The econd i whee no X egite exit, but the value of x o an accuate etimate of thi quantity i known The thid i the conventional cae-contol ituation, whee no X knowledge i available outide the ample In all thee cae, the ML etimating equation fo the paamete! of the aumed population level linea logitic model ae theoetically defined a the conditional expectation of the population level ML etimating equation given the ample data and the known population infomation Howeve, in thi cae the andom vaiable undepinning thee conditional expectation no longe follow the ame logitic model a in the population, o the appoximation to the ML coe function deived in the peviou ection need modification To tat, conide the fit ituation decibed above, whee individual X value fo non-ample population unit ae known, but the coeponding value of Y ae not We continue to ue the notation intoduced in the peviou ection Fom 1, we ee that the key unknown quantity in the coe function i E y i!, whee now, becaue of the caecontol ampling, the y i value in the ummation no longe follow the aumed population level logitic model Following Scott and Wild 1997, we ue Baye Theoem to appoximate the ditibution of thee value a N n independent Benoulli ealiation with! = P y i = 1 i ", = N 1 #1 N 1 # n 1! N 1 #1 N 1 # n 1! + N 0 #1 N 0 # n 0 1#! 10

13 With thi et up, we can ue the ame addlepoint agument a in the peviou ection to appoximate E, eplacing! in that development by! above Thi lead to y i! a full infomation coe function with component 2a a befoe, but with 2c eplaced by b t y " 1 "1 c 2! = $ y i " # + x $ i # % 1+ 1" # ' " x $ i #, 5 "1 $# {#! " t y } whee b t y = exp %! 1"! ' In the peviou ub-ection, we ued meaing to appoximate the coe function in the cae whee the individual non-ample X value ae unknown, but thei mean x i known Thi appoach need modification unde cae-contol, becaue ample and non-ample aveage no longe have the ame expected value In paticula, fo the cae-contol deign aumed hee, we need to apply meaing appoximation epaately fo cae and contol That i, fo an abitay function f of x chaacteied by a paamete!, we ue the appoximation $1 $1 " f,! # N 1 $ n 1 n 1 " f % 1 +,! + N 0 $ n 0 n 0 " f % 0 +,! 1 0 Hee d denote the ample unit with Y = d and! d denote ou bet etimate of the diffeence between the non-ample and ample mean of X fo thoe unit with Y = d Since we know the oveall non-ample mean x of X, we calculate! d uing a egeion type etimate, ie! d = " d n #1 2 d xd " 2 1 n # " 2 n #1 2 x1 #1 x 0 0 x0 # " 1 x 1 # " 0 x 0 2 whee! d = N d " n d / N " n and x d, xd denote the mean and vaiance of X fo the ample unit with Y = d The cae-contol veion of the meaing appoximation 4a i then 1 N c mea1! = " y i # " $ # d # n d " " $% d + n d 6a d while the coeponding cae-contol veion of 4b i d =0 11

14 c mea 2! = $ y i " # " + 1 $ d =0 % ' N d " n d n d 1 $ d =0 % ' N d " n d n d * + d + $ d #+ d + * $ + d + # + d +, 1+ 1" # + d + - cc b mea { t y " 1} / "1 6b whee cc b mea 2 1 % N t y = exp d! n d '" " # d =0 n d $ d + { 1! # $ d + }* 3 4 d! N d! n d / 5 -," " # n d $ d +! t y 0 - d =0 d When x i alo unknown, we eplace x d by x d above Thi i equivalent to etting! d = 0 in 20 and coepond to uing tatified expanion etimato fo the expected value of the unknown non-ample component of the coe function In what follow we ue the ame notation a in the peviou ection, denoting etimate obtained by etting 2a and 5 to zeo by FIMLE, and efeing to them a full infomation MLE Etimate obtained by etting 6 to zeo and olving ae efeed to a meaing MLE and ae denoted by SMEAR Finally, thoe obtained by olving 6 with! d = 0 ae efeed to a expanion MLE and ae denoted by EXP Table 2 et out imulation eult fo the above etimato a well a fo the tandad ample-baed MLE of! 1 denoted SMLE Pentice and Pyke 1979 howed that the SMLE of! 1 povide a good appoximation to the actual MLE of thi paamete unde cae-contol ampling We do not povide eult fo the SMLE of! 0 ince, a i well known, thi etimato i eiouly biaed unde cae-contol ampling The entie in Table 2 ae elative 5%RMSE, whee the efeence etimation method i maximum peudo-likelihood, defined by olving weighted veion of the SMLE etimating equation, with weight given by w i = N 0 n!1 0 Iy i = 0 + N 1 n!1 1 Iy i = 1, and denoted by WTD We alo computed the maximum peudo-model likelihood etimate popoed by Scott and Wild 1997 fo cae-contol 12

15 ampling, but do not how eult fo them ince thee wee almot identical to thoe fo the SMLE fo! 1 and tended to be untable fo! 0 The imulation methodology ued to obtain the eult in Table 2 i identical to that ued in Table 1, with the exception that ampling hee i caied out uing the tatified caecontol deign decibed at the tat of thi ection The SMLE and WTD etimate wee computed uing the glm function in R without and with weight epectively with default etting The FIMLE, SMEAR and EXP etimato wee all computed by uing the nlm function in R to olve the elevant etimating equation Table 2 about hee The eult et out in Table 2 confim once again that incluion of population level auxiliay infomation can bing ubtantial gain in maximum likelihood-baed infeence Thi i paticulaly the cae whee thi infomation i tong, a in the FIMLE Howeve, thee ae till ubtantial gain when the auxiliay infomation ued i much weake, a in SMEAR and EXP 4 Dicuion The two mot impotant concluion that we daw fom the eult et out in thi pape i that it pay to include population level auxiliay infomation when modelling ample uvey data, and that the BCDTW likelihood famewok offe a viable appoach to achieving thi aim Obviouly, the moe auxiliay infomation one ha available, the moe ignificant the impovement in one infeence Howeve, even maginal infomation eg knowledge of population mean fo the model vaiable can be vey ueful when integated with the ample data within thi famewok An impotant apect of auxiliay infomation i it accuacy In thi pape we ue imulation to ae the enitivity of likelihood method to eo in population benchmak 13

16 Thi i becaue thee i uually little o no knowledge of the mechanim undelying benchmak eo, and o one ha to accept benchmak on tut In thi context we note that ou likelihood method appea geneally quite obut to benchmak eo Howeve, if we do have infomation about the poce that gave ie to thee eo, then thi infomation can to be included in the conditioning poce undepinning the BCDTW famewok Fo example, if the obeved population mean of Y and X ae thei actual population value plu nomally ditibuted eo with zeo mean and known vaiance, then we can modify the development leading to the FIMLE and SMEAR etimato to condition on thee obeved mean, athe than the tue one Thi lead to diffeent et of etimating equation fo! that depend on thee known vaiance A much moe difficult poblem i whee the benchmak eo ae tue population value, but ae biaed, eg becaue of ubtle diffeence in the way Y and X ae meaued in the uvey and in the ouce of the benchmak Futhe eeach i neceay to ee whethe thi type of auxiliay population infomation i ueful in infeence In geneal, ue of the BCDTW famewok equie one to evaluate conditional expectation that depend both on the aumed population model a well a on the method ued to elect the ample Fo the impotant cae of a logitic population model, the addlepoint and meaing appoximation to thee conditional expectation that we decibe in thi pape eem to wok well and hould be ueful in extending ou eult in pactice While the eult in thi pape have been peented in the famewok whee X i a cala fo implicity, it i taightfowad to put them in the multiple egeion etting Extenion of ou appoach to moe complex ampling deign eg unequal pobability ampling of contol in cae-contol ituation hould be poible, but will equie appopiate adjutment to the addlepoint and meaing appoximation that we ue Thi pape doe not include eult on inteval etimation when auxiliay population data ae integated into likelihood infeence The BCDTW famewok alo cove thi 14

17 ituation, and in the Appendix we how how the infomation function can be extended to allow fo the auxiliay infomation in the cae of a logitic model, including appopiate addlepoint appoximation An impotant ue of thi function i in evaluating the exta infomation fo paametic infeence povided by the auxiliay infomation, eg along the line et out in Steel et al 2004 Finally, we point out that ou ue of the BCDTW famewok fo integation of auxiliay population infomation into maximum likelihood infeence i quite geneal We have focued on the logitic model ituation in thi pape becaue of it pactical application Howeve, othe type of modelling, eg linea modelling, can benefit jut a much fom ue of thi auxiliay infomation In fact, we have obtained paallel eult fo the linea model cae that uppot thi agument Detail ae available fom the autho on equet Acknowledgement Pat of thi eeach wa caied out when Wang wa at the niveity of Wollongong a a Viiting Pofeoial Fellow In addition, thi eeach wa uppoted in pat by the TAM Cente fo Envionmental and Rual Health via a gant fom the National Intitute of Envionmental Health Science P30 ES09106 Appendix A Saddlepoint Appoximation We fit conide an appoximation of R 1i Let y v be the mean of Y ove the et v, with N v the coeponding numbe of obevation Futhe, let g v d = Py v = d x v and! i =! Then, fo t y > 0 15

18 R 1i = g i " i g i t y! 1 / N i {t y! 1 / N i } + 1! " i g i t y / N i = 1+ 1! " g t / N i y i i % + % g i {t y! 1 / N i }! 1 It follow that the majo poblem i to appoximate " # g i t y! 1 / N i * # $, ' -!1 A1!1 $ % gi t y / N i accuately The cumulant geneating function of " i K v u = $ log{! j e u + 1"! j } y j! # Fo any d!0,1 the addlepoint appoximation to g v d i then h v d = N v {2! K v ""u d } exp{k u # N u d}, 1/2 v d v d whee u d i called the addlepoint, and i defined a the olution of K v!u / N v = d A2 Standad agument can be ued to how that h v d = g v d{1+ O 1 N v } unde geneal egulaity condition That i, the addlepoint appoximation ha a elative eo of ode N v!1 Subtituting d = d 1 / N i o d = d 2 = t y! 1 / N i in h i d, we then have g i g i t y / N i {t y! 1 / N i } = h it y / N i h i t y! 1 / N i whee the lat equation i due to the identity {1+ O 1 N } = exp{!u d 1 }{1 + O 1 N }, A3 K i u d1! N i u d1 d 1! { K i u d2! N i u d2 d 2 }= N i u d1 d 2! d 1 + O 1 =!u N d 1 + O 1 N "1/2 Fom the cental limit theoem N! $ y j " # j % N0, 2 a N v v! ", whee! 2 #1 = lim N " % $ j 1 # $ j It follow that we can focu on the nomal deviation value of v t y : t y! # " j = O N Fo uch value of t y, u d1 = O N!1/2 i In fact, fom A2, u d1 = t! " y i# j " i # j 1! # j + O $ 1 ' % N = t! " y # j " # j 1! # j + O $ 1 ' % N A4 By A1, A3 and A4, an appoximation to R 1i i then 16

19 R 1i = # $ 1+ 1! " i bt y! 1 %!1 - ' 1 + O 1 * 0 N +, 1 / 2 A5 "1 #! j " t y with bt y = exp $ #! j 1"! j % appoximated by ' It immediately follow that 2b can be c 2! " % y i # $ i # x % i $ i 1# [1+ 1# $ i {bt y # 1}] #1 A6 When non-ample value of X ae unavailable, but thei mean x i known, we can combine the addlepoint appoximation developed above with a meaing appoximation to again appoximate the logitic coe function In paticula, thi pocedue can be ued togethe with A6 to appoximate the econd pat of 3b We continue to ue 4a to appoximate 3a By A6, * c 2! " % y i # $ i # x % { + # x }$ i 1# 1+ 1# $ i bt y # 1 +, ' #1 - / * " % y i # $ i # N # n - * +, n / % x # x + $ i,adj 1# 1+ 1# $ i,adj bt y # 1 +, ' #1 * " % y i # $ i # N # n - * +, n / % x # x + $ i,adj 1# 1+ 1# $ i,adj b adj t y # 1 +, ' - / #1 - / A7 whee / 1 + exp {" 1 x # x + " 0 + " 1 }! i,adj = exp " 1 x # x + " 0 + " 1 $% ' and { } "1 $! i,adj * b adj t y = exp, #! i,adj 1"! i,adj + # " % n N " n t ' - y / Note that the lat two appoximation tep in A7 ued meaing appoximation epeatedly B The Infomation Function Within the BCDTW famewok the infomation function fo paametic likelihood infeence i the conditional expectation of the population level infomation function, 17

20 info!, minu the conditional vaiance of the coeponding population level coe function A alway, thi conditioning i with epect to the obeved uvey data a well a the auxiliay infomation Fo the imple logitic model conideed in thi pape the component of the population infomation function ae defined by the decompoition " info! = info 11! info 12!% $ # info 12! info 22! ' We ue a ubcipt of to denote the coeponding component of the infomation function defined by ou available data Thee ae info 11! = E info 11! " Va { c 1!} $ " Va %$ { y i " # } = E # 1" # = $ # 1" #, info 12! = E info 12! " Cov { c 1!,c 2!} $ " Cov % { y i " # } = E # 1" # = # { 1" # }, $ info 22! = E info 22! " Va { c 2!} %$ ' $, $ y i " # ' = $ x 2 i # 1" # " Va y i " # ' = $ x 2 i # 1" # " Va y i, $ whee with Va y i = Va! y i! y i,x!! i"! j"! y i,x # E! y i x i! y i,x = E y i y j x j 2 E " " j! y i y j x i! j " y i,x = " x 2 i E y i " y k,x + x j E y i y j " y j,x " i! " j #i! = " x 2 i $ R 1i + " i! " j #i! x j $ $x j R 2ij 18

21 2 $! { E! y i x i! y i,x } 2 " P! y i j # 1 x i = % P! y k x ' * = x 2 i " 2 2! R 1i +! i,! j +i, x j " "x j R 1i R 1 j and It follow R 2ij = P {! y k x } "1 P y ij k! " 2 x ij Va y i =! x 2 i " R 1i { 1# " R 1i } +! x j " "x j R 2ij # R 1i R 1 j! i%! j$i% A addlepoint appoximation to R 2ij imila to that developed above fo R 1i can be witten down Thi i baed on the fact that the denominato of R 2ij can be expeed a P leading to = " i " j P y k # 2 x ij y k x!! ij! ij + {" i 1 # " j + 1# " i " j }P y k # 1 x ij + 1# " i 1# " j P! ij y k x ij $! i! j +! i +! j " 2! i! j P # y = t " 1 x ij k y ij P # ij y k " 2 x ij R 2ij = % P # ij y k x ij +1 "! i 1 "! j ' P # ij y k " 2 x ij ing the ame addlepoint appoximation technique a that ued fo R 1i, the two atio in thi expeion can be appoximated by bt y! 1 and b 2 t y! 1 epectively That i, R 2ij = {! i! j +! i +! j " 2! i! j bt y " 1 +1 "! i 1"! j b 2 t y " 1} { 1+ O 1 N } * "1 19

22 Refeence Beckling, J, Chambe, RL, Dofman, AH, Tam, SM and Welh, AH 1994 Maximum likelihood infeence fom uvey data Intenational Statitical Review, 62, Chambe, RL 1996 Robut cae-weighting fo multipupoe etablihment uvey Jounal of Official Statitic, 12, 3-32 Deville, JC and Sändal, CE 1992 Calibation etimato in uvey ampling Jounal of the Ameican Statitical Aociation, 87, Duan, N 1983 Smeaing etimate: A nonpaametic etanfomation etimate Jounal of the Ameican Statitical Aociation, 78, Handcock, M, Rendall, M and Cheadle, J 2005 Impoved egeion etimation of a multivaiate elationhip with population data on the bivaiate elationhip Sociological Methodology, 35, Imben, GW and Lancate, T 1994 Combining mico and maco data in micoeconometic model Review of Economic Studie, 61, Pentice, RL and Pyke, R 1979 Logitic dieae incidence model and cae-contol tudie Biometika, 66, Qin, J 2000 Combining paametic and empiical likelihood Biometika, 87, Scott, AJ and Wild, CJ 1997 Fitting egeion model to cae-contol data by maximum likelihood Biometika, 84, Steel, DG, Beh, E J and Chambe, RL 2004 The infomation in aggegate data In Ecological Infeence: New Methodological Stategie ed G King, O Roen and M Tanne Cambidge niveity Pe: Cambidge 20

23 Table 1 Linea logitic population model unde SRSWOR and population benchmak of vaying quality Value in table ae pecent elative efficiencie with epect to 5% timmed oot mean quaed eo of SMLE Value of X dawn fom the tandad lognomal ditibution In all cae N = 5000 and n = 200 Value of X dawn fom the tandad lognomal ditibution Tue! 0,! 1 3, 1 5, 2 5, 1 8, 2 Population Benchmak Known Peciely! 0 EXP SMEAR FIMLE ! 1 EXP SMEAR FIMLE Population Benchmak Subject to Cenu-level Eo! 0 EXP SMEAR FIMLE ! 1 EXP SMEAR FIMLE Population Benchmak Subject to Lage Suvey Eo! 0 EXP SMEAR FIMLE ! 1 EXP SMEAR FIMLE

24 Table 2 Linea logitic model unde cae-contol ampling with population benchmak known peciely Value in table ae pecent elative efficiencie with epect to 5% timmed oot mean quaed eo of WTD In all cae N = 5000 and n 1 = n 0 = 100 Value of X dawn fom the tandad lognomal ditibution Tue! 0,! 1 3, 1 5, 2 5, 1 8, 2! 0 EXP SMEAR FIMLE ! 1 SMLE EXP SMEAR FIMLE