Variance estimation for the instrumental variables approach to measurement error in generalized linear models

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1 he Stata Journal (2003) 3, Number 4, pp Varance estmaton for the nstrumental varables approach to measurement error n generalzed lnear models James W. Hardn Arnold School of Publc Health Unversty of South Carolna Columba, SC Raymond J. Carroll Department of Statstcs MS-3143 exas A&M Unversty College Staton, Abstract. hs paper derves and gves explct formulas for a derved sandwch varance estmate. hs varance estmate s approprate for generalzed lnear addtve measurement error models ftted usng nstrumental varables. We also generalze the known results for lnear regresson. As such, ths artcle explans the theoretcal justfcaton for the sandwch estmate of varance utlzed n the software for measurement error developed under the Small Busness Innovaton Research Grant (SBIR) by StataCorp. he results admt estmaton of varance matrces for measurement error models where there s an nstrument for the unknown covarate. Keywords: st0048, sandwch estmate of varance, measurement error, Whte s estmator, robust varance, generalzed lnear models, nstrumental varables 1 Introducton hs s the second of fve papers descrbng software for fttng measurement error models. Software producton by StataCorp was funded by a Natonal Insttutes of Health (NIH) Small Busness Innovaton Research Grant (SBIR). he goal of the work descrbed n the grant s the producton of software to analyze statstcal models where one or more covarates are measured wth error. he software development ncludes two major features. he frst development feature s the development of Stata programs to support communcaton to dynamcally lnked user-wrtten computer code. StataCorp was responsble for ths development and support for user-wrtten code n the C/C++ programmng languages was added to Stata verson 8. Stata refers to compled user-wrtten code as plugns and mantans documentaton on ther web ste at In ths paper, we nvestgate the dervaton of the sandwch estmate of varance for a generalzed lnear addtve measurement error model ftted usng nstrumental varables (IV). he general dea n ths context has been proposed, for example, n Carroll, Ruppert, and Stefansk (1995), and explctly descrbed for two-stage models he project descrbed was supported by Grant Number R44 RR12435 from the Natonal Insttutes of Health, Natonal Center for Research Resources. he contents of ths artcle are solely the responsblty of the authors and do not necessarly represent the offcal vews of the Natonal Center for Research Resources. c 2003 StataCorp LP st0048

2 J. W. Hardn and R. J. Carroll 343 n Hardn (2002); though ths latter reference does not explctly menton measurement error as motvaton. We also nclude a restatement of the fndngs for the smple case of lnear regresson wth nstrumental varables and dscuss the mplcatons. In secton 2, we summarze the methods for fttng GLMs. In secton 3, we state the nstrumental varables technque and provde explct formulas for the sandwch estmate of varance for GLMs wth nstrumental varables followng the arguments of Hardn (2002) and Murphy and opel (1985). In secton 4, we summarze the smplfcaton of our general formula for the case of lnear regresson. Secton 5 presents an example applcaton, and we present a summary n secton 6. We nvestgate the case of fttng a generalzed lnear model (GLM) where one or more of the covarates are measured wth error. We have nstruments avalable that are uncorrelated wth the error term of the model but correlated wth the covarates measured wth error. Usng the nstrumental varables approach, we have an estmatng equaton that ncludes the nstrumental varables regressons n the estmaton of the GLM. he usual varance estmate of the coeffcent vector from the GLM does not take nto account the estmaton of the nstrumental varables regressons. We must derve a varance estmate that takes nto account these regressons as well as the GLM estmaton. As we show, the sandwch estmate of varance for estmatng equatons s a vald estmator. hs varance estmate defnes estmatng equatons that nclude all of the parameters the parameters from the nstrumental varables regressons and the parameters from the GLM. Hardn and Carroll (2003) present an overvew of both generalzed lnear models and the notatonal conventons that we employ n the present dscusson. A second source of nformaton n the case of unvarate measurement error s the gllamm command; see Rabe-Hesketh, Skrondal, and Pckles (2003). 2 Estmaton Hardn and Carroll (2003) reference sources for estmaton algorthms for the classcal GLM. In those references, t s assumed that the lst of covarates s measured wthout error. Recall that we use z for covarates measured wthout error, w for covarates measured wth error, s for the nstruments of w,and r for the augmented matrx of exogenous varables [ z s ]. We begn wth an n p matrx of covarates measured wthout error gven by the augmented matrx =( z u ), where u s unobserved, and w = u plus measurement error. z s an n p z matrx of covarates measured wthout error (possbly ncludng a constant), and w s an n p x (p z + p x = p) matrx of covarates wth classcal measurement error that estmates u. We wsh to employ an n p s (where p s p x ) matrx of nstruments s for w.

3 344 Robust varance estmate for GLM-IV Greene (2003) dscusses nstrumental varables and provdes a clear presentaton to supplement the followng concse descrpton. he method of nstrumental varables assumes that some subset w of the ndependent varables s correlated wth the error term n the model. In addton, we have a matrx s of ndependent varables that are correlated wth w.gven, Y and w are uncorrelated. Usng these relatonshps, we can construct an approxmately consstent estmator that may be succnctly descrbed. One performs a regresson for each of the ndependent varables (each column) of w on the nstruments and the ndependent varables not correlated wth the error term ( z s ). Predcted values are obtaned from each regresson and substtuted for the assocated column of w n the analyss of the GLM of nterest. hs constructon provdes an approxmately consstent estmator of the coeffcents n the GLM (t s consstent n the lnear case). Secton 3 addresses formng a vald varance estmator for the coeffcents n the GLM. If we have access to the complete matrx of covarates measured wthout error (f we know u nstead of usng nstruments s ), we denote the lnear predctor η = p j=1 [ z u ] j β j, and the assocated dervatve as / β j =[ z u ] j. he estmatng equaton for β s then n =1 (y µ )/V(µ )( µ/) [ z u ] j. However, snce we do not observe u,weuse r =( z s ) to denote the augmented matrx of exogenous varables, whch combnes the covarates measured wthout error and the nstruments. We regress each of the p x components (each of the j columns) of w on r to obtan an estmated (p z +p s ) 1 coeffcent vector γ j for j =1,...,p x. he complete coeffcent vector γ =(γ 1 γ 2 γ p x ) for these IV regressons s descrbed by the estmatng equaton r ( w 1 r γ 1 ) r ( w 2 r γ 2 ) Ψ 2 = (1). r ( wpx r γ px ) We may then form an n p x matrx u =[ r γ 1 r γ 2 r γ px ] of predcted values from the nstrumental varables regressons to estmate u. Combnng the (predcted value) regressors wth the ndependent varables measured wthout error, we may wrte the estmatng equaton of the GLM as where Ψ 1 = n =1 y µ V(µ ) ( ) µ [ z r γ] j (2) { (z ) [ z r γ] j = j f 1 j p z ( r γ j pz ) f p z <j p z + p x Operatonally, we obtan a two-stage estmate β by frst replacng each unknown covarate w for =1,...,p x wth the ftted values of the regresson of w on ( z s ). We call the resultng n p x matrx of ftted values u. We then perform

4 J. W. Hardn and R. J. Carroll 345 the (second stage) usual GLM ft of Y on ( z u ). hs GLM ft provdes an estmate of β. Our goal s to construct a vald varance estmate of β. 3 he sandwch varance estmate he varance matrx estmate from the IRLS algorthm used to compute the (second stage) GLM ft assumes that u = u. hs s clearly unacceptable. An alternatve approach s accounted for n Murphy and opel (1985); ths varance estmator s ncluded n the software developed as part of ths small busness nnovaton research project. We derve an approprate sandwch estmate of varance that takes nto account the estmaton of u. Excellent revews of the sandwch varance estmator and ts propertes are gven n Carroll and Kauermann (2001)and Carroll et al. (1998), whle the classc references are Ecker (1963), Ecker (1967), Huber (1967), and Whte (1980). Applcaton of the sandwch estmate of varance for panel data s dscussed n e, Smpson, and Carroll (2000). Our applcaton s a specal case of Hardn (2002), whch descrbes the general dervaton of asymptotc and sandwch varance estmators for two-stage models. In fact, the nstrumental varables approach to measurement error s a specal case of two-stage estmaton. he two-stage dervaton resultng n an estmate for β nvolves estmatng the combned parameter vector gven by Θ = (β γ ). hese results are from the estmatng equatons gven n equatons 1 and 2. Whle we are ultmately nterested n β,wemust consder all of the parameters n formng the assocated varance matrx. Our goal s the constructon of the sandwch estmate of varance gven by V S = A 1 BA. We form the varance matrx, A, for Θ by obtanng the necessary dervatves. he varance matrx A (nformaton matrx) may be calculated numercally, but the analytc dervatves are not dffcult and are gven by A = Ψ 1 β (pz+p x) (p z+p x) Ψ 2 β {px(p s+p z)} (p z+p x) Ψ 1 γ (pz+p x) {p x(p z+p s)} Ψ 2 γ {px(p z+p s)} {p x(p z+p s)} 1 where Ψ 1 β k = [ n 1 =1 V(µ ) { (µ y ) ( ) 2 µ 1 V(µ ) 2 ( ) 2 µ V(µ ) 1 µ V(µ ) ( 2 ) }] µ 2 [ Z R γ] j [ Z R γ] k j =1,...,p z + p x ; k =1,...,p z + p x yelds a matrx of sze (p z + p x ) (p z + p x ) (3)

5 346 Robust varance estmate for GLM-IV Ψ 1 γ lk = [ n 1 =1 V(µ ) { (µ y ) ( ) 2 µ 1 V(µ ) 2 [ Z R γ] j R k β l+pz ( ) 2 µ V(µ ) 1 µ V(µ ) ( 2 ) }] µ 2 j =1,...,p z + p x ; k =1,...,p z + p s ; l =1,...,p x yelds a matrx of sze (p z + p x ) {p x (p z + p s )} (4) Ψ 2 β k = 0 k =1,...,p z + p x yelds a matrx of sze {p x (p z + p s )} (p z + p x ) (5) Ψ 2 n = R j R k γ lk =1 j =1,...,p z + p s ; k =1,...,p z + p s ; l =1,...,p x yelds a block dagonal matrx of sze {p x (p z + p s )} {p x (p z + p s )} where each block matrx s of sze (p z + p s ) (p z + p s ) (6) he elements of the varance matrx are formed from the defntons above. Mappng these equatons s accomplshed by defnng the matrx A usng [ Z R γ] j = n whch we apply the notaton { Z (Rj f 1 j p z γ (j pz)) f p z <j p z + p x Z j = th observaton of the jth column of Z R j = th observaton of the jth column of R γ j pz = IV coeffcent vector from regressng W(j pz) on R ( β l+pz ) = (l + p z )th coeffcent of β R γ (j pz) = th observaton of (the predcted values from) R γ j pz γ lk = kth coeffcent of the lth IV coeffcent vector Equaton 3 defnes the (j,k) elementsofa for j,k =1,...,p z + p x. Equaton 4 defnes the (j,k) elementsofa for j =1,...,p z + p x and k = p z + p x + l, where l = 1,...,p x (p z +p s ). hs notaton addresses the cross dervatves for all of the (p z +p s ) 1 coeffcent vectors γ m for m =1,...,p x. Equaton 5 calculates the (j,k)elementsofa for j = p z + p x + l, where l =1,...,p x (p z + p s )andk =1,...,p z + p x. Equaton 6 defnes the (j,k)elementsofa for j,k = p z +p x +l, where l =1,...,p x (p z +p s ). hese are the covarances of all of the (p z + p s ) 1 coeffcent vectors γ m for m =1,...,p x.

6 J. W. Hardn and R. J. Carroll 347 Notng that Ψ =[Ψ 1 Ψ 2 ], the mddle of the sandwch estmate of varance s then gven by B = n =1 Ψ Ψ. A sutable estmate may be formed usng Ψ 1 = { y µ V(µ ) ( ) } j=1,...,(pz+p µ x) [ Z R γ] j (p z+p x) 1 [( W1 R γ 1j ) R j ] j=1,...,(pz+p s) Ψ 2 = (p z+p s) 1 [( W2 R γ 2j ) R j ] j=1,...,(pz+p s) (p z+p s) 1. [ ] j=1,...,(pz+p s) ( W px R γ pxj) R j (p z+p s) 1 {p x(p z+p s)} 1 he sandwch estmate of varance for β s then the upper (p z + p x ) (p z + p x ) matrx of V S obtaned from the derved estmates of A and B. he precedng descrpton s mplemented n the qvf command provded as part of the newly developed software. 4 he lnear-regresson case In the lnear-regresson case (dentty lnk and Gaussan varance), the dervaton of the sandwch estmate of varance greatly smplfes. hs smplfcaton s shown n detal n Whte (1982). Here, we present the results of the smplfcaton and summarze the mplcatons. he nave (model-based) covarance matrx utlzes the fact that V (Y W β)= V{Y W β} I = σ 2 I where σ 2 s the mean square of Y W β. hus, V( β) σ 2 ( P P) 1, where P = S ( S S ) 1 S W s a matrx of the predcted values from usng the nstruments S. herefore, the correct asymptotc varance can be obtaned smply by performng a standard lnear regresson of Y on P. he sandwch estmate of varance s then clearly gven by ( P P ) 1 P (Y W β)(y W β) P ( P P ) 1 such that the usual sandwch estmate from the lnear regresson of Y on P s correct. hus, for the standard lnear regresson case, we may obtan both a model-based and a sandwch estmate of varance by consderng only the second stage regresson. hese smplfcatons are not true n the general case of a GLM and may be dscerned from the two-stage regresson formula gven n StataCorp (2003).

7 348 Robust varance estmate for GLM-IV 5 Example applcaton Carroll, Ruppert, and Stefansk (1995) (hereafter, CRS) present several examples usng data from the Framngham Heart Study (see chapters 4 and 5). hs dataset conssts of three measurements taken two years apart on 1,615 men aged he outcome varable s chd, ndcatng the presence of coronary heart dsease wthn an eght-year perod followng the thrd set of collected measurements. he predctors nclude age, the patent s age n years; smoke, an ndcator of whether the patent smokes; sbp, the systolc blood pressure; and chol, a categorcal (three categores) of cholesterol. In the examples, CRS uses the transformed predctor lbsp gven by log(sbp 50). For llustraton, we use the ndcator varable smoke as an nstrument for the transformed systolc blood pressure lbsp where the systolc blood pressure s the mean of two measurements for the patent by dfferent techncans. Our use of the smoke varable as an nstrument s for llustratve purposes, as ths s not a good nstrument. he purpose n choosng smoke as the nstrument s to magnfy the comparatve results of the nave (GLM varance estmate gnorng the estmaton from the IV regressons) and sandwch estmate of varance. able 1 lsts the coeffcent estmates and standard errors. able 1: Instrumental varables logstc regresson results. Nave standard errors are the result of gnorng the estmaton from the nstruments. Sandwch standard errors are the result from the sandwch estmate of varance presented n the prevous secton. Nave Sandwch Varable Coeff Std. Error Std. Error lbsp age chol constant hs example demonstrates the dfference between the two approaches. he nave standard errors are calculated from the varance matrx resultng from the GLM ft usng the ftted values for the lbsp varable. hs nave varance estmate does not take nto account the estmaton of lbsp and s nvald. he assocated software developed for measurement error analyss ncludes support for varous models and varance estmates. hese estmates nclude the sandwch varance estmate descrbed here, as well as bootstrap, asymptotc (model-based), and Murphy opel (for nstrumental varables approach to measurement error). 6 Summary hs paper presents a sandwch estmate of varance for generalzed lnear models wth nstrumental varables. he presentaton ncludes detaled formulas for the calculaton of the varance estmate. hese formulas admt the calculaton of a vald varance

8 J. W. Hardn and R. J. Carroll 349 estmate for the coeffcent vector for any regresson model wthn the generalzed lnear models framework. hese results may be extended to quas-lkelhood cases as well. As Bnder (1992) ponts out, the bread of the sandwch estmate of varance s not, n general, symmetrc. We have asymmetry for the case of GLMs wth nstrumental varables due to the augmented matrces of cross dervatves. We have also shown that n the specal case of lnear regresson, our dervaton smplfes such that we may use the results of the second stage regresson wthout modfcaton. 7 References Bnder, D. A Fttng Cox s proportonal hazards models from survey data. Bometrka 79(1): Carroll, R. J. and G. Kauermann A note on the effcency of sandwch covarance matrx estmaton. Journal of the Amercan Statstcal Assocaton 96: Carroll, R. J., D. Ruppert, and L. A. Stefansk Measurement Error n Nonlnear Models. London: Chapman & Hall. Carroll, R. J., S. Wang, D. G. Smpson, A. J. Stromberg, and D. Ruppert he sandwch (robust covarance matrx) estmator. Unpublshed echncal Report. Ecker, F Asymptotc normalty and consstency of the least squares estmators for famles of lnear regressons. he Annals of Mathematcal Statstcs 34(2): Lmt theorems for regressons wth unequal and dependent errors. In Proceedngs of the Ffth Berkeley Symposum on Mathematcal Statstcs and Probablty, vol. 1, Berkeley, CA: Unversty of Calforna Press. Greene, W. H Econometrc Analyss. 5th ed. Upper Saddle Rver, NJ: Prentce Hall. Hardn, J. W he robust varance estmator for two-stage models. Stata Journal 2(3): Hardn, J. W. and R. J. Carroll Measurement error, GLMs, and notatonal conventons. Stata Journal 3(4): Huber, P. J he behavor of maxmum lkelhood estmates under nonstandard condtons. In Proceedngs of the Ffth Berkeley Symposum on Mathematcal Statstcs and Probablty, vol. 1, Berkeley, CA: Unversty of Calforna Press. Murphy, K. M. and R. H. opel Estmaton and nference n two-step econometrc models. Journal of Busness and Economc Statstcs 3(4):

9 350 Robust varance estmate for GLM-IV Rabe-Hesketh, S., A. Skrondal, and A. Pckles Maxmum lkelhood estmaton of generalzed lnear models wth covarate measurement error. Stata Journal 3(4): StataCorp he Stata Reference Manual. Verson 8 ed. College Staton, : Stata Press. Whte, H A heteroskedastcty-consstent covarance matrx estmator and a drect test for heteroskedastcty. Econometrca 48(4): Instrumental varables regresson wth ndependent observatons. Econometrca 50(2): e, M., D. G. Smpson, and R. J. Carroll Random effects n censored ordnal regresson: Latent structure and Bayesan approach. Bometrcs 56: About the Authors James W. Hardn (jhardn@gwm.sc.edu), s an Assocate Research Professor, Department of Epdemology and Bostatstcs, and a Research Scentst, Center for Health Servces and Polcy Research, Arnold School of Publc Health, Carolna Plaza Sute 1120, Unversty of South Carolna, Columba, SC 29208, USA. Raymond J. Carroll (carroll@stat.tamu.edu) s a Dstngushed Professor, Department of Statstcs, MS 3143, exas A&M Unversty, College Staton, , USA. Research by StataCorp was supported by the Natonal Insttutes of Health (NIH) Small Busness Innovaton Research Grant (SBIR) (2R44RR ).