Parameter estimation for incomplete bivariate longitudinal data in clinical trials

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1 Parameter estmaton for ncomplete bvarate longtudnal data n clncal trals Naum M. Khutoryansky Novo Nordsk Pharmaceutcals, Inc., Prnceton, NJ ABSTRACT Bvarate models are useful when analyzng longtudnal data of two correlated endponts. Longtudnal data n clncal trals are often ncomplete whch can create bas n estmaton of ther means and varances. The ncompleteness s partally caused by wthdrawal from trals. In some clncal trals wth two endponts, only one of them s used as a marker for wthdrawal. The drop-out mechansm plays an mportant role n correct estmaton of statstcal parameters of bvarate longtudnal data. In ths paper, we compare several methods of parameter estmaton for bvarate data ncludng lnear mxed models (proc MIED), multple mputaton method (proc MI) and ncremental methods. The comparson s done on smulated longtudnal data that are created to be the bvarate Markov processes. The dropout mechansm depends on ndvdual prevously observed values of one of the endponts. INTRODUCTION In much of clncal research, repeated observatons of response varables and a set of covarates are taken from each patent over a certan tme. Such studes are commonly referred to as longtudnal studes n whch the prmary objectve s to descrbe the dependence of response varables on tme, treatment effects and other possble covarates. The response varables are often correlated. Many stuatons arse n whch two response varables are observed smultaneously on each patent at each occason. For example, n dabetes longtudnal studes two glycemc varables are of most mportance: Glycosylated Haemoglobn (HbA1c) and Fastng Plasma Glucose (FPG). These varables are correlated at each vst. In some trals, only the FPG level s used for wthdrawal crteron: f the FPG response of a patent at a vst exceeds a pre-specfed threshold, the patent s wthdrawn from the tral after ths vst. Bvarate response (HbA1c, FPG) can often be consdered as a Markov process whch means that the dependence of the current value of ths response on ts hstory of durng the tral can be reduced to the dependence on ts most recent prevous value. The drop-out mechansm descrbed above can also be presented n the framework of Markov processes. Ths paper s an attempt to compare three methods of means and varances estmaton (two of whch are presented n SAS procedures) for smulated longtudnal bvarate ncomplete data satsfyng the Markov process assumptons wth the dropout mechansm descrbed above. The methods under consderaton are the lnear mxed models method (represented by PROC MIED [1]), multple mputaton method (represented by PROCs MI and MIANALYZE [2]) and ncremental method [3]. The comparson of these methods for smulated longtudnal unvarate data were presented n [4]. Consderaton of bvarate correlated responses wth the drop-out mechansm nvolvng only one varable much closer resembles the practce of the dabetes clncal trals than the unvarate approach.

2 BIVARIATE LONGITUDINAL DATA AND DROP-OUT MECHANISM Consder tme ponts t 1, t 2,..., t n. Now suppose that two response varables and Y are represented by random varables and Y at tme ponts t (=1,, n). Denote par {, Y } by Z.. In ths paper, we assume that Z s a nhomogeneous Markov process whch means that P( = P( < x, Y < x, Y < y Z 1 < y Z,..., Z 1 ) 1 ) (1) Random varables and Y are assumed to be correlated wth correlaton coeffcent?. The drop-out mechansm mplemented s based on a threshold value T for random varable. Let j and Y j be the values of and Y for the jth subject. If kj > T then the jth subject s a drop-out after tme pont t k.. Therefore, the followng propertes hold for a drop-out at tme t k : j T, = 1,2,..., k 1, kj > T j and Y j are mssng for >k. (2) The goal of the paper s to consder and compare estmaton of the means, varances and correlaton coeffcents of random varables and Y satsfyng (1) f the date ncludes drop-outs of type (2), when three methods of estmaton are used: the lnear mxed models method (represented by PROC MIED), multple mputaton method (represented by PROCs MI and MIANALYZE) and a generalzaton of the ncremental methods. DATA SET SIMULATION We consder the smulaton based on longtudnal data sets resemblng that n dabetes clncal trals. Two endponts comprsng the bvarate response are FPG (Fastng Plasma Glucose) and HbA1c (Glycosylated Haemoglobn). The drop-out mechansm of type (2) s based on values of FPG wth threshold T. As an example we consder such a tral wth sx tme ponts of measurements startng from baselne. The frst three tme ncrements are equal to four weeks, the next two are sx weeks. We assume that the baselne dstrbutons of FPG and HbA1c can be approxmated by normal dstrbutons and specfy tme behavor of these varables resemblng ther dependence on tme n real dabetes trals for dfferent treatments. The followng SAS data step bulds a bvarate longtudnal data set wth 6 tme ponts t. Two longtudnal varables a and b are correlated at each tme pont t wth correlaton coeffcent? =?. Coeffcent? s presented n the program by macro varable &rho. Unvarate normal varables are generated by SAS functon rannor(seed) and used to smulate standard bvarate normal varable (x,y) wth correlaton coeffcent ρ. Ths varable s used to smulate two correlated normal varables wth specfed means and standard devatons at each tme pont (see the program). The ntal value of the seed for the random generator s gven by macro varable &seed. The baselne and ncremental values of means and standard devatons are specfed n the data step below to resemble behavor of HbA1c and FPG for dabetes patents treated by some drugs used n clncal practce. data ntal; drop seed j ma1 mb1 c rho x y sa1 sb1 sda sdb da db mda1 mda2 mda3 mda4 mda5

3 mdb1mdb2 mdb3 mdb4 mdb5; retan seed &seed; array a{6} a1 a2 a3 a4 a5 a6; array mda{5} mda1 mda2 mda3 mda4 mda5; array b{6} b1 b2 b3 b4 b5 b6; array mdb{5} mdb1 mdb2 mdb3 mdb4 mdb5; do d=1 to &tot; ma1=9; mda1=0.25; mda2=-0.25; mda3=-0.5; mda4=-0.5; mda5=0; mb1=220; mdb1=-5; mdb2=-10; mdb3=-10; mdb4=-5; mdb5=0; rho=ρ c=sqrt(1-rho**2); sa1=1.4; sb1=60; sda=0.5; sdb=30; x = rannor(seed); y = rho*x+c*rannor(seed); a1 = ma1 + sa1*x; b1 = mb1 + sb1*y; do j = 1 to 5; x = rannor(seed); y = rho*x+c*rannor(seed); da = mda{j} + sda*x; db = mdb{j} + sdb*y; a{j+1}=a{j}+da; b{j+1}=b{j}+db; end; output; end; run; It s mportant to emphasze that the data generated by ths program represent a bvarate Markov process. The next task s to smulate an ncomplete data set usng the wthdrawal crteron based on a threshold value for one of the varable (for example, for b). The followng SAS data step executes ths task. run; set ntal; drop j; array a{6} a1 a2 a3 a4 a5 a6; array b{6} b1 b2 b3 b4 b5 b6; drop=0; do j = 2 to 6; f b{j-1} > &thres then drop=1; f drop=1 then a{j}=.; f drop=1 then b{j}=.; end; The threshold value for varable b here s specfed by macro varable &thres. The data sets (complete and ncomplete) smulated by these programs can be used to compare dfferent methods of parameter estmaton for the complete data set usng only the ncomplete data set. The results obtaned by each method under consderaton could be assessed by ther comparson wth the correspondng results for the complete data set. LINEAR MIED MODELS ESTIMATION SAS PROC MIED based on the lnear mxed models theory provdes tools for estmatng the mean and standard devaton of an ncomplete longtudnal data set usng drect maxmzaton of the observed lkelhood [1]. In general, the results depend on the covarance structure used for estmaton. MULTIPLE IMPUTATION TECHNIQUES Some multple mputaton technques are mplemented now n expermental SAS PROC MI [2]. data ncomplete;

4 INCREMENTAL METHODS Let D j be a bvarate ncrement of Z=(,Y) for subject from tme pont j to tme pont j+1. Next, let Z * j and Z j are observed and mssng values of Z, respectvely. Smlarly, let D * j and D j be observed and mssng bvarate ncrements, respectvely. Denote by D *.j the bvarate sample mean of observed ncrements D * j at tme pont j. Suppose that values Z 1 are all observed. The ncremental mean method [3] creates two matrces: a basc mputed matrx and an uncertanty matrx. To buld the basc mputed matrx B the unknown bvarate ncrements D j are prescrbed to be equal to D *.j and, next, the mssng values Z,j+1 are replaced by Z^,j+1= Z j + D *.j for j = 1,,k - 1 (3) Where bvarate values Z j are observed or calculated on the prevous step. The elements Bj that concde ether wth Z * j or wth Z^j (calculated step by step as specfed above) create the basc mputed matrx B. The approxmate bvarate mean values of Z at each tme pont j wll be calculated as the bvarate sample mean B.j usng the bvarate jth column of matrx B. The columns of matrx B can be used to calculate the varances of Z at each tme pont. However, these calculated values V^j underestmate the real varances due to the lack of uncertanty n (3) and wll be named the partal varances. To nduce an addtonal uncertanty t s proposed to buld an uncertanty matrx V. Let U * j be the sample varance of the set {D * j} for fxed j. Let R j be equal to 1 f Z j s observed and 0 otherwse. Then the elements V j of matrx V are calculated as follows: V j = 0; V j = (V I,j-1 + U * j) (1 R j ) (4) for j = 2,,k. Therefore, f Z j s observed then V j = 0 (there s no uncertanty). Otherwse, an addtonal varance (as an uncertanty measure) s accumulated over tme untl tme pont j for each subject. The pooled addtonal varance V j at tme pont j s defned as the average of V j over all the subjects. The total varance V j at tme pont j s defned as V j = V^j + V j. (5) RESULTS OF SIMULATION Comparson of the three methods mensoned above were done for dfferent sample szes, correlaton coeffcents and threshold values. The range of sample szes was from 30 to 120 subjects. The correlaton coeffcent was chosen to be 0.8. The threshold for the dropout process was 250. To make the comparsons, we consdered the bas and mean square error (MSE) of the mean and standard devaton for each estmaton method nvolved n comparson. The results of the comparson showed the followng: 1. For the data smulated as above PROC MIED gave unstable results dependng heavly on the covarance structure ndcated n the REPEATED statement opton type=covarance structure. There was often no convergence f type=un. 2. The ncremental mean method rendered the most precse results (n average). 3. The multple mputaton method gave sutable results, however less precse than the ncremental mean method for standard devaton estmaton. CONCLUSION The paper s concerned wth comparson of several mputaton and estmaton technques appled to ncomplete longtudnal data sets wth two correlated varables. The data sets

5 are smulated to resemble tme behavor of HbA1c and FPG n dabetes clncal trals. The mssngness mechansms employed resemble the process of wthdrawal from trals based on a threshold for only one varable. The mputaton technques beng compared nclude the mxed effects model repeated measures method, the multple mputaton method and the ncremental method. The results of smulaton presented n the paper show that all the methods for dfferent numbers of patents (30 to 120) and a relatvely large percentage of mssng values the ncremental mean method gve, n average, more precse estmatons of the means and standard devatons (as measured by MSE) than the other estmaton methods under comparson. CONTACT INFORMATION Your comments and questons are valued and encouraged. Contact the author at: Naum Khutoryansky Novo Nordsk Pharmaceutcals, Inc. 100 College Road West Prnceton NJ Work Phone: Emal: nakh@nnp.com Web: REFERENCES [1] Verbeke, G., Molenberghs, G. Lnear mxed models n practce. A SAS-orented approach, Sprnger, New York, [2] Yuan, Y.C. Multple Imputaton for Mssng Data: Concepts and New Development. SUGI Proceedngs 2000; P [3] Khutoryansky, N.M., and Huang, W.C. Imputaton Technques Usng SAS Software For Incomplete Data In Dabetes Clncal Trals, Pharmaceutcal Industry SAS Users Group, 2001; [4] Khutoryansky, N. M., and Chernck, M.R. Incremental methods of mputaton n longtudnal clncal trals, JSM 2002, Conference Proceedngs.

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