Journal of Statistical Software

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1 JSS Journal of Statstcal Software July 2010, Volume 35, Issue 2. %QLS SAS Macro: A SAS Macro for Analyss of Correlated Data Usng Quas-Least Squares Hanjoo Km Forest Research Insttute, Inc. Justne Shults Unversty of Pennsylvana Abstract Quas-least squares (QLS) s an alternatve computatonal approach for estmaton of the correlaton parameter n the framework of generalzed estmatng equatons (GEE). QLS overcomes some lmtatons of GEE that were dscussed n Crowder (1995). In addton, t allows for easer mplementaton of some correlaton structures that are not avalable for GEE. We descrbe a user wrtten SAS macro called %QLS, and demonstrate applcaton of our macro usng a clncal tral example for the comparson of two treatments for a common toenal nfecton. %QLS also computes the lower and upper boundares of the correlaton parameter for analyss of longtudnal bnary data that were descrbed by Prentce (1988). Furthermore, t dsplays a warnng message f the Prentce constrants are volated. Ths warnng s not provded n exstng GEE software packages and other packages that were recently developed for applcaton of QLS (n Stata, MATLAB, and R). %QLS allows for analyss of contnuous, bnary, or count data wth one of the followng workng correlaton structures: the frst-order autoregressve, equcorrelated, Markov, or tr-dagonal structures. Keywords: longtudnal data, generalzed estmatng equatons, quas-least squares, SAS. 1. Introducton Quas-least squares (QLS, Chaganty 1997; Shults and Chaganty 1998; Chaganty and Shults 1999) s a two-stage approach for fttng longtudnal data that are correlated over tme usng an assumed workng correlaton structure n the framework of generalzed estmatng equatons (GEE, Lang and Zeger 1986). Both QLS and GEE estmate the regresson parameter β va soluton of the same estmatng equaton whch contans an addtonal unknown correlaton parameter α. These methods alternate between updatng the estmate of β, and updatng the estmate of α teratvely, but they dffer wth respect to estmaton of α. GEE uses a moment estmate of α based on the Pearson resdual, whle QLS solves estmatng equatons for α that

2 2 %QLS: Analyss of Correlated Data Usng Quas-Least Squares n SAS are orthogonal to the GEE estmatng equaton of β. There are prmarly two mportant reasons to consder the mplementaton of QLS over GEE. Crowder (1995) ponted out that there may be no feasble estmate of α under msspecfcaton of the workng correlaton structure, whch could result n a falure to converge for the teratve GEE estmaton procedure. On the other hand, the QLS estmates of α are guaranteed to be feasble for several structures, ncludng the frst-order autoregressve (AR(1)), equcorrelated, Markov, and tr-dagonal structures. The other man advantage of the applcaton of QLS s that t allows for relatvely straghtforward mplementaton of more complex and/or bologcally plausble correlaton structures than are currently avalable n GEE. For example, Shults and Chaganty (1998) and Chaganty and Shults (1999) descrbed estmatng equatons for α for the Markov correlaton structure that s a generalzaton of the AR(1) structure for measurements that are unequally spaced n tme. Furthermore, QLS has been successfully mplemented n mult-outcome longtudnal studes whch contan multple sources of correlatons, e.g., correlatons between multple outcomes as well as wthn subject for each outcome over tme, usng Kronecker product-based correlaton structures (Shults and Morrow 2002; Chaganty and Nak 2002; Shults et al. 2004; Shults and Ratclffe 2009). In addton, Shults et al. (2006) mplemented a banded Toepltz structure that had prevously not been mplemented for GEE. There has been a consderable effort to dssemnate and promote the use of QLS va the development of user wrtten QLS programs n some of the major statstcal software packages. For example, Shults et al. (2007) developed xtqls usng Stata s propretary hgher programmng language (StataCorp. 2003). Xe and Shults (2009) provde a user wrtten QLS program called qlspack usng R. These programs estmate the regresson parameter β by callng exstng GEE procedures, e.g., xtgee n Stata and the user wrtten package geepack (Halekoh et al. 2006) n R, and then solvng the stage one and two estmatng equatons for α teratvely. Lastly, Ratclffe and Shults (2008) developed GEEQBOX software n MATLAB for the mplementaton of both GEE and QLS. Pror to ntroducton of GEEQBOX, there was no software avalable for the mplementaton of GEE n MATLAB. All these QLS programs can be downloaded from whle GEEQBOX s also avalable for download on the webste of the Journal of Statstcal Software provded n Ratclffe and Shults (2008). Ths manuscrpt reflects our ongong effort to develop QLS software n SAS (SAS Insttute Inc. 2003) whch s one of the most wdely used software packages, due to ts versatlty ncludng vared and powerful procedures for data management and statstcal analyss. In ths manuscrpt, we present our user wrtten SAS macro called %QLS for the mplementaton of QLS developed under SAS verson 9.1 usng SAS/IML, an nteractve matrx language n SAS. We demonstrate %QLS usng a randomzed clncal tral reported by Backer et al. (1996) for the comparson of two treatments for a common toenal nfecton. (See Secton 4 for more descrpton of the study). %QLS can be used for analyss of contnuous, bnary, or count data, wth an AR(1), equcorrelated, Markov, or tr-dagonal correlaton structure to descrbe the pattern of assocaton among the repeated measurements on each subject, or cluster. A unque feature of %QLS, whch s not avalable n current GEE packages and other userwrtten QLS software, s that t computes the so called Prentce boundary (Prentce 1988) of the estmate of α for analyss of bnary data (see Secton 2.5 for more detals). Our outlne for ths manuscrpt s as follows: Secton 2 descrbes each workng correlaton

3 Journal of Statstcal Software 3 structure mplemented n %QLS. Then t brefly descrbes QLS and the Prentce constrants for α. Secton 3 descrbes a complete lst of the parameters n %QLS. Fnally, Secton 4 demonstrates %QLS usng the clncal tral data (toenal data) for testng the tme-averaged dfference between the two treatment groups. Ths ncludes a demonstraton of an analyss that results n a volaton of the Prentce constrants, n addton to an applcaton of the Markov structure that currently s unavalable for GEE. 2. Descrpton of quas-least squares In ths secton, we provde a bref descrpton of QLS smlar to that provded n Shults et al. (2007); Ratclffe and Shults (2008) and Xe and Shults (2009). For a more through treatment of the development of QLS and ts related asymptotc dstrbutonal propertes, see Chaganty (1997) for a descrpton of stage one of QLS for balanced and equally spaced data, Shults and Chaganty (1998) for stage one of QLS for unbalanced and unequally spaced data, and Chaganty and Shults (1999) for the second stage of QLS. Pror to Chaganty and Shults (1999), the QLS estmate of α was n general not consstent, even f the correlaton structure was correctly specfed. As noted earler, QLS s a method n the framework of GEE. For an excellent text on GEE, see Hardn and Hlbe (2002). We consder outcome measurements y = (y 1,..., y n ) wth assocated covarates x j = (x j1,..., x jk ) collected at measurement occasons j = 1,..., n, for each subject = 1,..., m. For the model specfcaton, we assume that the mean and varance of the outcome varable satsfy E(y j ) = g 1 (x j β) = µ j and Var(y j ) = φh(µ j ), respectvely, where φ s known as the dsperson parameter. Unlke the generalzed lnear model (GLM), both GEE and QLS assume that the covarance matrx Cov(y ) = φa 1/2 R (α)a 1/2 where A = dag(h(µ 1 ),..., h(µ n )), and R (α) s known as the workng correlaton matrx that descrbes the pattern of assocaton among the repeated measurements on each subject Workng correlaton structures mplemented n %QLS %QLS currently allows for applcaton of the AR(1), equcorrelated, Markov, and tr-dagonal structures that are descrbed as follows: 1. The frst-order autoregressve (AR(1)) structure: Ths structure assumes that Corr(y j, y k ) = α j k, wth feasble regon ( 1, 1). The feasble regon for α s defned as the nterval on whch α yelds a postve defnte correlaton matrx. 2. The equcorrelated structure: Ths structure assumes that Corr(y j, y k ) = α. The feasble regon for ths structure s ( 1/(n m 1), 1), where n m s the maxmum value of n over = 1, 2,..., m. Note that for large n m, ths structure may be unrealstc unless all parwse correlatons are roughly dentcal across all tme ponts. 3. The Markov correlaton structure: Ths structure assumes that Corr(y j, y k ) = α t j t k, wth feasble regon ( 1, 1). Ths structure s a generalzaton of the AR(1) structure and useful n modelng data that are unequally spaced n tme. In addton, ths structure has not been mplemented n any of the currently avalable packages that mplement GEE.

4 4 %QLS: Analyss of Correlated Data Usng Quas-Least Squares n SAS 4. The tr-dagonal correlaton structure: Ths structure assumes that Corr(y j, y k ) = α for j k = 1 and s zero otherwse. The feasble regon for ths structure s ( 1/c m, 1/c m ), where ( ) π[nm 1] c m = 2 sn 2[n m + 1] and n m s the maxmum value of n over = 1, 2,..., m. Ths nterval s approxmately ( 1/2, 1/2) for large n and contans ( 1/2, 1/2) for all n. %QLS does not allow for applcaton of the ndependent structure (dentty matrx) because QLS s dentcal to GEE for ths structure. In addton, applcaton of the unstructured matrx s complex for QLS. In SAS, we therefore suggest applcaton of PROC GENMOD wth the repeated statement and the opton corr=nd for applcaton of the ndependent correlaton structure, or corr=un for applcaton of the unstructured correlaton matrx for GEE Estmatng equatons and the algorthm for quas-least squares QLS s a two-stage procedure for estmaton of the regresson parameter β and the correlaton parameter α. In stage one t alternates between updatng the GEE estmatng equaton for β and the estmatng equaton for α. After convergence n stage one, an updated estmate of α s obtaned by solvng the stage two estmatng equaton for α. A fnal estmate of β s then obtaned by solvng the GEE estmatng equaton for β. The estmatng equatons are as follows: ˆ The GEE estmatng equaton for β: n =1 ˆ The stage one estmatng equaton for α: ( ) µ A 1/2 R 1 (α)a 1/2 [y µ (β)] = 0 (1) β { n Z α =1 (β) R 1 (α) Z (β) where Z (β) = (z 1,..., z n ) and z j = (y j µ j )/h(µ j ). } = 0, (2) ˆ The stage two estmatng equaton for α (evaluated at the stage one estmate ˆα): n { R 1 (δ) trace R (α)} = 0. (3) δ δ=ˆα =1 Note that the stage one and two estmatng equatons (2) and (3) nvolve the frst dervatve of the nverse of R (α), whch may not be easly obtaned for some correlaton structures, e.g., the tr-dagonal structure. However, t can be easly shown that R 1 (α) α = R 1 (α) R (α) α R 1 (α) (4)

5 Journal of Statstcal Software 5 where R (δ) δ s computed by takng the dervatve of each element n R (α). For example, for the tr-dagonal structure, R (δ) δ s an n n matrx wth ones on the off-dagonal and zeros elsewhere,.e., the (j, k)th element of R (α) α s 1 f j k = 1 and s 0 otherwse. %QLS nvolves applcaton of the followng algorthm to obtan the estmates of β and α: 1. Ft a generalzed lnear model wth an approprate lnk and varance functon usng PROC GENMOD n SAS to obtan an ntal estmate for β. 2. In stage one of QLS, repeat the followng steps untl a pre-specfed convergence crteron s met for estmatng β and α. () Compute the Pearson resduals at the current estmate of β, where the jth Pearson resdual on subject s gven by z j = (y j û j )/h(û j ). () Compute the estmate of α by solvng the QLS stage one estmatng equaton (2) for α usng a pre-specfed workng correlaton structure. () Update the estmate of β by solvng the GEE estmatng equaton (1) for β at the current estmate of α. 3. After convergence n the estmates of β and α n stage one, obtan an updated estmate of α by solvng the stage two estmatng equaton (3) for α. Then obtan a fnal estmate of β by solvng the GEE estmatng for β that s evaluated at the stage two estmate of α Algebrac expressons of the estmatng equatons for β and α At each teraton, %QLS uses the Gauss-Newton method to compute the current estmate β usng the prevous estmates β and α as follows: { n ( ) ( ) } 1 β µ = β + A 1/2 R 1 (α)a 1/2 µ β β =1 { n ( ) } µ A 1/2 R 1 (α)a 1/2 [y µ (β)] β =1 Although (4) can be used to obtan solutons to the stage one and two estmatng equatons (2) and (3), explct expressons for α are preferred snce ther applcaton s computatonally more effcent. Except for the tr-dagonal structure, %QLS mplements explct expressons for α that were obtaned by solvng the stage one and two estmatng equatons (2) and (3) for α, for partcular workng structures. For the AR(1) structure wth unbalanced data, Shults and Chaganty (1998) proved that the feasble stage one estmate ˆα A-ONE can be expressed as: m n m (z j + z j 1 ) 2 n (z j + z j 1 ) 2 m n (z j z j 1 ) 2 ˆα A-ONE = =1 j=2 =1 j=2 2 m n =1 j=2 =1 j=2 z j z j 1,

6 6 %QLS: Analyss of Correlated Data Usng Quas-Least Squares n SAS whle Chaganty and Shults (1999) showed that the stage two estmate ˆα A-TWO s gven by ˆα A-TWO = 2ˆα A-ONE 1 + ˆα A-ONE 2. For the equcorrelated structure wth unbalanced data, Shults (1996) showed that the stage one estmatng equaton of α has a unque feasble soluton ˆα A-ONE gven by :n >1 Z Z :n >1 1 + α 2 (n 1) (1 + α(n 1)) 2 (Z (β) e ) 2 = 0 where I n s the dentty matrx and e s a n 1 column vector of ones. Shults and Morrow (2002) obtaned the stage two estmate ˆα E-TWO gven by n (n 1) ˆα E-ONE (ˆα E-ONE (n 2) + 2) (1 + ˆα(n 1)) 2 / n (n 1) ( 1 + ˆα E-ONE 2 (n 1) ) (1 + ˆα E-ONE (n 1)) 2. :n >1 m n e j α e j =1 j=2 :n >1 For the Markov structure wth unbalanced data, Shults (1996) obtaned the QLS stage one estmatng equaton for α as follows: [ α 2e j z j z,j 1 α e j ( ) ] zj 2 + z2,j 1 + z j z,j 1 (1 α 2e j) 2 = 0 where e j = t j t,j 1. The stage two estmatng equaton of the Markov structure (Chaganty and Shults 1999) s then gven by m n 2e j δ 2ej 1 α ej [ e j δ e j 1 + δ 3e j 1 ] (1 δ 2e j) 2 = 0. δ=ˆαm-one =1 j= Estmates of the covarance matrx %QLS provdes two types of estmated covarance matrces of the estmated regresson parameter, the model-based and robust sandwch-based estmates. The robust sandwch-based estmate of the covarance matrx s the default matrx. It s often preferred when there s any uncertanty n the choce of workng correlaton structure. However, the standard errors are not necessarly smaller for the sandwch covarance matrx. Therefore, applcaton of both the model based and sandwch based covarance matrces mght be consdered n an analyss. %QLS computes the model-based covarance matrx as follows: Ĉov M ( ˆβ) = ˆφ n =1 X A 1/2 R 1 (ˆα)A 1/2 X. where ˆφ s the estmate of the dsperson parameter wth or wthout the bas correcton gven by ˆφ BC = 1 n Z ( N p ˆβ) Z ( ˆβ), or ˆφ B = 1 n Z ( N ˆβ) Z ( ˆβ) =1 =1

7 Journal of Statstcal Software 7 respectvely, where p s the dmenson of the regresson parameter β and N = m =1 n. By default, %QLS provdes the bas corrected estmate of the dsperson parameter ˆφ BC. For the robust sandwch-based estmate of the covarance matrx, %QLS computes } Ĉov R ( ˆβ) = W 1 n { n =1 X A 1/2 R 1 (ˆα)Z ( ˆβ)Z 1 ( ˆβ)R (ˆα)A 1/2 X W 1 n. where W n = n =1 X A 1/2 R 1 (ˆα)A 1/2 X. %QLS also computes the (1 α)100% confdence nterval, and a p value for testng each ndvdual regresson parameter β j = 0, based on ether the model-based or the robust sandwchbased estmate of the covarance matrx of β Prentce boundary of the estmate of α for analyzng bnary data Consder longtudnal bnary measurements y 1,..., y n wth expected values E(y j ) = Pr(y j = 1) = p j, wth q j = Pr(y j = 0) = 1 p j, and the correlaton between measurements y j and y k denoted by Corr(y j, y k ). An mportant feature of correlated bnary data s that the p j, q j and Corr(y j, y k ) completely determne the bvarate dstrbuton of y j and y k because the par-wse probabltes pr(y j, y k ) can be expressed as Pr(y j, y k ) = p y j j q1 y j j p y k k q1 y k k { 1 + Corr(y j, y k ) (y } j p j )(y k p k ) (p j p k q j q k ) 1/2. (5) Prentce (1988) ponted out that the probabltes n (5) wll be non-negatve,.e., Pr(y j, y k ) 0, only f the correlatons satsfy the followng constrants that depend on the margnal means: for all = 1,..., n. { } { } pj p k qj q k pj q k qj p k max, Corr(y j, y k ) mn, j k q j q k p j p k j k q j p k p j q k A man problem wth volaton of the Prentce constrants s that ths leads to a lack of theoretcal justfcaton for the exstence of a vald jont probablty mass functon, gven the estmated parameter values of β and α. However, as Molenberghs and Verbeke (2005) ponted out, the estmate of the workng correlaton s merely a devce to provde consstent and asymptotcally normal pont estmates for the margnal regresson parameters whch should not the made a part of formal nference. Rochon (1998) also noted that volaton of the Prentce constrants appears to cause no dffculty n practce, although the potental for volaton should be taken nto account n the desgn phase of a study. For example, when conductng sample sze calculatons, values of α that satsfy the Prentce constrants should be consdered. Shults et al. (2009) also suggest that a severe volaton of bounds mght be used to remove a partcular workng structure from a lst of canddate workng structures. As noted n Secton 1, the exstng GEE and user-wrtten QLS software (n Stata, MATLAB, and R) do not provde the estmated Prentce constrants (6) for analyss of bnary data. %QLS, on the other hand, computes the estmated Prentce constrants for each correlaton structure, (6)

8 8 %QLS: Analyss of Correlated Data Usng Quas-Least Squares n SAS and alerts users to a potental problem by provdng a warnng message f the estmate of α does not fall wthn the nterval. 3. Lst of parameters n the macro A complete lst of the parameters n %QLS s as follows: %QLS(data, y, x, d, tme, lnk, corr, robust, dsperson, alpha, ntalout, stage1out, stage2out, cmatrx, reference, converge, maxter) where ˆ data s the name of the data set n the usual longtudnal data format to be read n PROC GENMOD. The data set must not contan any mssng values. ˆ y s the outcome varable. ˆ x are the predctors (covarates) n the regresson model. ˆ d s the ID varable. ˆ tme s the tme varable. ˆ lnk equals 1 for the dentty lnk, 2 for the logt lnk, and 3 for the log lnk (default s 1). ˆ corr equals 1 for the AR(1), 2 for the equcorrelated, 3 for the Markov, 4 for the tr-dagonal (default s 1). ˆ robust equals 1 for robust sandwch-based standard errors, 2 for model-based standard errors (default s 1). ˆ dsperson equals 1 for bas not corrected, 0 for bas-corrected (default s 1). ˆ alpha s the sgnfcance level to be used n testng each regresson coeffcent (default s 0.05). ˆ ntalout equals 1 creates a SAS permanent data set n the current work space for the ntal output, 0 otherwse (default s 0). ˆ stage1out equals 1 creates a SAS permanent data set n the current work space for the stage 1 output, 0 otherwse (default s 0). ˆ stage2out equals 1 creates a SAS permanent data set n the current work space for the stage 2 output, 0 otherwse (default s 0). ˆ cmatrx equals 1 creates a SAS permanent data set n the current work space for the stage 2 correlaton matrx, 0 otherwse (default s 0). ˆ reference equals 1 prnts out the reference nformaton, 0 otherwse (default s 0). ˆ converge s the convergence crteron for estmaton of β and of α (default s ).

9 Journal of Statstcal Software 9 ˆ maxter s the maxmum number of allowable teratons for estmaton of β and α (default s 100). Note that many of the parameters have default values, so that they do not have to be specfed. %QLS assumes the usual longtudnal data format to be read n PROC GENMOD wthout any mssng observaton contaned n the data. If there are mssng observatons n the data that are coded as mssng, these must be deleted pror to mplementaton of %QLS. Ths s equvalent to assumng that the observatons are mssng completely at random (MCAR), as n the usual GEE analyss mplemented by PROC GENMOD wth the repeated statement. 4. Clncal tral example Backer et al. (1996) reported a 12-week, randomzed, double-blnd, mult-center tral for the comparson between the standard oral drug (terbnafne 250mg daly) and the expermental oral drug (thertraconazole 200mg daly) n the treatment of a common toenal nfecton called dermatophyte toe onychomycoss (DTO) whch affects more than 2% of the Brtsh populaton (Roberts 1992). The data was also descrbed n Molenberghs and Verbeke (2005), s avalable along wth ths manuscrpt, and can also be downloaded from ~sratclf/qlsproject.html. A total of 189 patents were randomzed to each treatment group and followed over 12 weeks, wth measurements taken at baselne, and at months 1, 2, 3, 6, 9, and 12. The prmary outcome measure was the severty of the toe nal nfecton, that was defned as 1 f the nfecton was severe, and 0 otherwse. For the purpose of demonstraton, we frst consder a smple logstc regresson model for comparson of the tme-averaged treatment dfference between the standard treatment group and the expermental treatment group. The toenal data, toenal.txt, contans 4 varables: tme, treatment, y, and d, where tme s the tme varable, treatment s the treatment ndcator (1 for the standard arm, and 0 otherwse), y s the outcome varable (1 f the nfecton s severe, and 0 otherwse), and d s the ID number assgned to each patent. Let y j follow the Bernoull dstrbuton wth pr(y j = 1) = p j such that ( ) pj log = β 0 + β 1 treatment (7) 1 p j where treatment s the treatment ndcator that equals 1 f the th subject s assgned to the standard drug and 0 otherwse. One advantage of mplementng the model n (7) s that the upper lmt of the Prentce constrant for α s always equal to 1. In general, any model that nvolves covarates that do not vary wthn clusters wll have an upper Prentce boundary (for α) of 1, due the fact that the estmated probabltes p j wll not vary wthn subjects (clusters) when only cluster level covarates are consdered n the model. Before we demonstrate %QLS to ft the model (7), frst we assume that the data, s read nto the current SAS workspace, e.g., data toenal; nfle "D:\toenal.txt"; nput tme treatment y d; run; toenal.txt,

10 10 %QLS: Analyss of Correlated Data Usng Quas-Least Squares n SAS where we assume that the toenal.txt s stored n the D:\ drectory Applcaton of the AR(1) correlaton structure The followng codes can be used to analyze the toenal data usng the QLS regresson model (7) wth the AR1 structure: %QLS(data = toenal, y = y, x = treatment, d = d, tme = tme, lnk = 2, corr = 1); The estmated standard errors are the robust sandwch-based estmates that are set by default. The outputs from the code are as follows: Quas-Least Squares SAS Macro Verson 1.0 Regresson Analyss usng Quas-Least Squares (QLS) QLS Model Informaton Varance Functon Lnk Functon Dependent Varable Correlaton Structure : Bnomal : Logt : Y : AR(1) Number of Observaton Read : 1907 Number of Clusters : 294 Maxmum Cluster Sze : 7 Mnmum Cluster Sze : 1 Correlaton Matrx Dmenson : 7 Number of Dstnct Tme Ponts : 7 TIME Number of Events : 408 Number of Trals : 1907 Analyss of Intal Parameter Estmates Parameter Estmate Stand Err Z Pr> Z [95% Con. Interval] INTERCEPT TREATMENT %QLS s modelng the probablty that Y= page 1/

11 Journal of Statstcal Software 11 Correlaton converged after 1 teratons ( tolerance = 0 ) Reg. coeff. converged after 2 teratons ( tolerance = ) Analyss of Stage 1 QLS Parameter Estmates Parameter Estmate Stand Err Z Pr> Z [95% Con. Interval] INTERCEPT TREATMENT Stage 1 Correlaton Parameter Estmate Dsperson Parameter Estmate at Stage 1 1 Stage 1 Workng Correlaton Matrx page 2/ Correlaton converged after 1 teratons ( tolerance = 0 ) Reg. coeff. converged after 3 teratons ( tolerance = E-9 ) Analyss of Stage 2 QLS Parameter Estmates Parameter Estmate Stand Err Z Pr> Z [95% Con. Interval] INTERCEPT TREATMENT Prentce Boundary Stage 2 Correlaton Parameter Estmate

12 12 %QLS: Analyss of Correlated Data Usng Quas-Least Squares n SAS Dsperson Parameter Estmate at Stage 2 1 Stage 2 Workng Correlaton Matrx page 3/ The output of %QLS contans the model nformaton followed by the estmates of the stage one and two estmates of β and of α. As noted earler, the upper lmt of the Prentce nterval s equal to 1 n the above output. From the stage two output, the p value correspondng to the tme-averaged treatment effect s equal to 0.38, whch suggests that there s no sgnfcant tme-averaged treatment dfference between the standard drug versus the expermental drug Applcaton of the Markov correlaton structure Here we demonstrate applcaton of the Markov correlaton structure, whch s currently unavalable for GEE. Ths s mportant because the toenal data s unequally spaced n tme, e.g., the varable tme n ths data set ndcates the month of measurement (post baselne) and takes value n {0, 1, 2, 3, 6, 9, 12} for each subject. Therefore, the Markov correlaton structure s preferable for the analyss of ths tral. The followng code can be used to ft the model (7) wth the Markov correlaton structure: %QLS(data = toenal, y = y, x = treatment, d = d, tme = tme, lnk = 2, corr = 3); Here we omt the stage one output, and present the ntal and stage two outputs. Quas-Least Squares SAS Macro Verson 1.0 Regresson Analyss usng Quas-Least Squares (QLS) QLS Model Informaton Varance Functon Lnk Functon Dependent Varable : Bnomal : Logt : Y

13 Journal of Statstcal Software 13 Correlaton Structure : Markov Number of Observaton Read : 1907 Number of Clusters : 294 Maxmum Cluster Sze : 7 Mnmum Cluster Sze : 1 Correlaton Matrx Dmenson : 7 Number of Dstnct Tme Ponts : 7 TIME Number of Events : 408 Number of Trals : 1907 Analyss of Intal Parameter Estmates Parameter Estmate Stand Err Z Pr> Z [95% Con. Interval] INTERCEPT TREATMENT %QLS s modelng the probablty that Y= page 1/ Correlaton converged after 27 teratons ( tolerance = ) Reg. coeff. converged after 3 teratons ( tolerance = E-7 ) Analyss of Stage 2 QLS Parameter Estmates Parameter Estmate Stand Err Z Pr> Z [95% Con. Interval] INTERCEPT TREATMENT Prentce Boundary Stage 2 Correlaton Parameter Estmate Dsperson Parameter Estmate at Stage 2 1

14 14 %QLS: Analyss of Correlated Data Usng Quas-Least Squares n SAS Stage 2 Workng Correlaton Matrx page 3/ The results are smlar to those for the AR(1) structure, wth an estmated α n stage two of ˆα = 0.79 versus ˆα = 0.74 for the AR(1) structure. Further, the p value wth respect to the tme-averaged treatment effect s Hence, the same concluson follows as wth the AR(1) structure Applcaton of the equcorrelated and tr-dagonal structures Although the equcorrelated and tr-dagonal structures may not be the best canddate correlaton structures for the toenal data, we nclude the mplementaton of theses structures for the purpose of demonstraton. Here we only present the codes for fttng the model (7) wth the equcorrelated and tr-dagonal correlaton structures, but omt ther outputs. For the equcorrelated correlaton structure, we have %QLS(data = toenal, y = y, x = treatment, d = d, tme = tme, lnk = 2, corr = 2); For the tr-dagonal correlaton structure, we have %QLS(data = toenal, y = y, x = treatment, d = d, tme = tme, lnk = 2, corr = 4); 4.4. Volaton of the Prentce boundary Here we brefly demonstrate volaton of the Prentce constrants usng the toenal data. Consder the followng model for testng the treatment effect over tme: ( ) πj log = β 0 + β 1 (treatment ) + β 2 (tme j ) + β 3 (treatment tme j ) (8) 1 π j where treatment s the treatment ndcator that equals 1 f the th subject s assgned to the standard drug and 0 otherwse, tme j represents the tme of the measurement collected on subject at the jth measurement occason, and treatment tme j s the treatment by tme nteracton. To ft the model n (8) usng %QLS, a new varable correspondng to the nteracton term must be created frst, e.g.,

15 Journal of Statstcal Software 15 data toenal; nfle "D:\toenal.txt"; nput tme treatment y d; nteracton=treatment*tme; run; To ft the model (8) wth the AR(1) structure, we use %QLS(data = toenal, y = y, x = treatment tme nteracton, d = d, tme = tme, lnk = 2, corr = 1); Here we provde the ntal and stage two outputs for the AR(1) structure. Quas-Least Squares SAS Macro Verson 1.0 Regresson Analyss usng Quas-Least Squares (QLS) QLS Model Informaton Varance Functon Lnk Functon Dependent Varable Correlaton Structure : Bnomal : Logt : Y : AR(1) Number of Observaton Read : 1907 Number of Clusters : 294 Maxmum Cluster Sze : 7 Mnmum Cluster Sze : 1 Correlaton Matrx Dmenson : 7 Number of Dstnct Tme Ponts : 7 TIME Number of Events : 408 Number of Trals : 1907 Analyss of Intal Parameter Estmates Parameter Estmate Stand Err Z Pr> Z [95% Con. Interval] INTERCEPT TREATMENT TIME INTERACTION

16 16 %QLS: Analyss of Correlated Data Usng Quas-Least Squares n SAS %QLS s modelng the probablty that Y= page 1/ Correlaton converged after 1 teratons ( tolerance = 0 ) Reg. coeff. converged after 5 teratons ( tolerance = ) Analyss of Stage 2 QLS Parameter Estmates Parameter Estmate Stand Err Z Pr> Z [95% Con. Interval] INTERCEPT TREATMENT TIME INTERACTION Prentce Boundary Stage 2 Correlaton Parameter Estmate Dsperson Parameter Estmate at Stage 2 1 Stage 2 Workng Correlaton Matrx Warnng! Correlaton parameter estmate s not wthn the boundary. The exstence of a multvarate bnary dstrbuton s questonable page 3/ From the stage two output, the estmated stage two α s 0.71, whch exceeds the upper lmt (0.31) of the Prentce constrants. It s also mportant to note that although the results are not shown here, applcaton of GEE for the AR(1) structure would also result n a volaton of the Prentce constrants.

17 Journal of Statstcal Software 17 The above results suggest somethng dfferent than the tme-averaged model, whch s that there s a dfference n the lkelhood of hgh severty between the two treatment condtons. However, graphcal dsplays (not shown) suggest that the assumpton of lnearty n the logt s not approprate for these data. For an extensve dscusson of approaches for assessment of the lnearty n the logt assumpton, see Hlbe (2009). For demonstraton purposes, we now present a model that dd not volate the lnearty n the logt assumpton, and that also dd not result n a volaton of the Prentce bounds for α. Ths model contans ndcator varables for the second (1 month), thrd (2 month), ffth (6 month), and seventh (12 month) measurements on each subject, an ndcator varable for the standard treatment, and a vst seven (12 month) by treatment ndcator varable. (All other treatment by vst ndcator varables dd not dffer sgnfcantly from zero). The correspondng data set, toenal2.txt, can be also downloaded from html. Here we present the code, and the ntal and stage two outputs. Frst, we read n the data: data toenal2; nfle "D:\toenal2.txt" delmter="," ; nput y tme2 tme3 tme5 tme7 treatment tme7_trt d tme; run; Next, we ft the QLS model. %QLS(data = toenal2, y = y, x = tme2 tme3 tme5 tme7 treatment tme7_trt, d = d, tme = tme, lnk = 2, corr = 1); Quas-Least Squares SAS Macro Verson 1.0 Regresson Analyss usng Quas-Least Squares (QLS) QLS Model Informaton Varance Functon Lnk Functon Dependent Varable Correlaton Structure : Bnomal : Logt : Y : AR(1) Number of Observaton Read : 1907 Number of Clusters : 294 Maxmum Cluster Sze : 7 Mnmum Cluster Sze : 1 Correlaton Matrx Dmenson : 7 Number of Dstnct Tme Ponts : 7 TIME Number of Events : 408

18 18 %QLS: Analyss of Correlated Data Usng Quas-Least Squares n SAS Number of Trals : 1907 Analyss of Intal Parameter Estmates Parameter Estmate Stand Err Z Pr> Z [95% Con. Interval] INTERCEPT TIME TIME TIME TIME TREATMENT TIME7_TRT %QLS s modelng the probablty that Y= page 1/ Correlaton converged after 1 teratons ( tolerance = 0 ) Reg. coeff. converged after 8 teratons ( tolerance = ) Analyss of Stage 2 QLS Parameter Estmates Parameter Estmate Stand Err Z Pr> Z [95% Con. Interval] INTERCEPT TIME TIME TIME TIME TREATMENT TIME7_TRT Prentce Boundary Stage 2 Correlaton Parameter Estmate Dsperson Parameter Estmate at Stage 2 1

19 Journal of Statstcal Software 19 Stage 2 Workng Correlaton Matrx page 3/ For the above model, the Prentce constrants are not volated. In addton, the results seem more n agreement wth the tme-averaged model, whch also dd not dentfy a sgnfcant dfference between the two treatment condtons wth respect to severty of toenal nfecton. 5. Dscusson %QLS can ft a model to longtudnal data usng the method of quas-least squares, and can consder data whch follows the normal, Bernoull, or Posson dstrbuton wth the AR(1), Markov, equcorrelated, and tr-dagonal structures. The syntax and the output of %QLS are smlar to the exstng GEE procedures n SAS,.e., PROC GENMOD wth the repeated statement, that would be famlar to SAS users. %QLS assumes that there are no mssng observatons n the dataset. Hence, any observatons that are coded as mssng should be deleted pror to the mplementaton of the macro. As noted earler, ths s equvalent to assumng that the data s mssng completely at random (MCAR), whch s a typcal assumpton n a GEE analyss. Further updates of %QLS wll be made to allow for mplementaton of other structures that are not currently avalable for GEE, ncludng the famlal structure, and Kronecker product-based structures whch account for multple sources of correlaton, e.g., mult-outcome longtudnal data for whch the sources of correlatons come from wthn subjects over tme, and between multple outcomes. It wll also be of nterest to develop and compare exstng methods for selecton of a workng correlaton structure and for assessment of goodness of ft of QLS (and GEE) models. Acknowledgments Work on ths manuscrpt was supported by the NIH grant R01CA Longtudnal Analyss for Dverse Populatons. We are also extremely grateful to the assocate edtor and referee who revewed our manuscrpt and software and provded very helpful suggestons for mprovement. We also thank edtor Achm Zeles for hs techncal suggestons.

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21 Journal of Statstcal Software 21 Shults J (1996). The Analyss of Unbalanced and Unequally Spaced Longtudnal Data Usng Quas-Least Squares. Ph.D. thess, Department of Mathematcs and Statstcs, Old Domnon Unversty, Norfolk, Vrgna. Shults J, Chaganty NR (1998). Analyss of Serally Correlated Data Usng Quas-Least Squares. Bometrcs, 54(4), Shults J, Mazurck C, Lands R (2006). Analyss of Repeated Bouts of Measurements n the Framework of Generalzed Estmatng Equatons. Statstcs n Medcne, 25(23), Shults J, Morrow A (2002). Use of Quas-Least Squares to Adjust for Two Levels of Correlaton. Bometrcs, 58, Shults J, Ratclffe S (2009). Analyss of Mult-Level Correlated Data n the Framework of Generalzed Estmatng Equatons va xtmultcorr Procedures n Stata and qls Functons n MATLAB. Statstcs and Its Interface, 2, Shults J, Ratclffe S, Leonard M (2007). Improved Generalzed Estmatng Equaton Analyss va xtqls for Implementaton of Quas-Least Squares n Stata. Stata Journal, 7(2), Shults J, Sun W, Tu X, Km H, Amsterdam J, Hlbe JM, Ten-Have T (2009). A Comparson of Several Approaches for Choosng Between Workng Correlaton Structures n Generalzed Estmatng Equaton Analyss of Longtudnal Bnary Data. Statstcs n Medcne, 28(18), Shults J, Whtt M, Kumanyka S (2004). Analyss of Data wth Multple Sources of Correlaton n the Framework of Generalzed Estmatng Equatons. Statstcs n Medcne, 23(20), StataCorp (2003). Stata Statstcal Software: Release 8. StataCorp LP, College Staton, TX. URL Xe J, Shults J (2009). Implementaton of Quas-Least Squares wth the R Package qlspack. Workng Paper 32, UPenn Bostatstcs Workng Papers. URL com/upennbostat/papers/art32. Afflaton: Han-Joo Km Department of Bostatstcs Forest Research Insttute, Inc. Fnancal Harborsde Center Plaza V Jersey Cty, NJ , Unted States of Amerca Telephone: +1/201/427/8356 E-mal: hanjookm@gmal.com

22 22 %QLS: Analyss of Correlated Data Usng Quas-Least Squares n SAS Justne Shults Department of Bostatstcs and Epdemology Center for Clncal Epdemology and Bostatstcs Unversty of Pennsylvana School of Medcne 423 Guardan Drve, 610 Blockley Hall Phladelpha, PA , Unted States of Amerca Telephone: +1/215/573/6526 E-mal: jshults@mal.med.upenn.edu URL: Journal of Statstcal Software publshed by the Amercan Statstcal Assocaton Volume 35, Issue 2 Submtted: July 2010 Accepted: