ordinary differential equation pk/pd models using the sas macro nlinmix

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Ordinary Differential Equation PKPD Models Usingthe SAS Macro NLINMIXAndrzej T Galecki a e Russell D Wolfinger b Oscar A Linares c d Marla J Smith c ampJeffrey B Halter a ca Institute of Gerontology University of Michigan Ann Arbor Michigan USAb SAS Institute Inc SAS Campus Drive Cary North Carolina USAc Geriatrics Center University of Michigan Ann Arbor Michigan USAd Michigan Horace H Rackham School of Graduate Studies University of Michigan AnnArbor Michigan USAe Institute of Gerontology University of Michigan Ann Arbor MI 48109-2007 USAPublished online 02 Feb 2007

To cite this article Andrzej T Galecki Russell D Wolfinger Oscar A Linares Marla J Smith amp Jeffrey B Halter (2004)Ordinary Differential Equation PKPD Models Using the SAS Macro NLINMIX Journal of Biopharmaceutical Statistics 142483-503 DOI 101081BIP-120037194

To link to this article httpdxdoiorg101081BIP-120037194

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Ordinary Differential Equation PK=PD ModelsUsing the SAS Macro NLINMIX

Andrzej T Galecki1 Russell D Wolfinger2 Oscar A Linares34

Marla J Smith3 and Jeffrey B Halter13

1Institute of Gerontology University of Michigan Ann Arbor Michigan USA2SAS Institute Inc SAS Campus Drive Cary North Carolina USA

3Geriatrics Center University of Michigan Ann Arbor Michigan USA4Michigan Horace H Rackham School of Graduate Studies University of

Michigan Ann Arbor Michigan USA

ABSTRACT

We describe some theory and recent enhancements for the SAS macro NLINMIX(Wolfinger R D (1993) Laplacersquos approximation for nonlinear mixed effectsmodels Biometrika 80791ndash795) that enable model calculation to take placewithin the interaction matrix language SAS=IML (SAS Institute Inc (1999a)SAS=IML Userrsquos Guide Version 7 Cary NC SAS Institute Inc) They providegreater flexibility and scope for the specification and analysis of complexnonlinear mixed models For example using data from a frequently sampledintravenous glucose test we fit a two-compartment kinetics model that has noclosed-form representation It is derived as the solution of a system of ordinarydifferential equations and specified as such in SAS=IML Additional details andexample NLINMIX code are available in Appendix A

Correspondence Andrzej T Galecki Institute of Gerontology University of MichiganAnn Arbor MI 48109-2007 USA E-mail agaleckiumichedu

JOURNAL OF BIOPHARMACEUTICAL STATISTICS

Vol 14 No 2 pp 483ndash503 2004

483

DOI 101081BIP-120037194 1054-3406 (Print) 1520-5711 (Online)

Copyright 2004 by Marcel Dekker Inc wwwdekkercom

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Key Words Nonlinear mixed effects model Longitudinal data analysisPopulation pharmacokinetics and pharmacodynamics models Linearizationmethod SAS=IML macro Glucose disappearance model

Mathematics Subject Classification 62-07 62H12 62P10 65Y15 65L99

INTRODUCTION

Population pharmacokinetics and pharmacodynamics (PK=PD) data consist ofrepeated measurements taken on each of a number of individuals In PK studiesblood samples are drawn several times from each subject to investigate levels of a testagent in the blood and=or in the body over time In PD studies we measure aphysiological effect of a drug administered using different doses

Both population PK and PD data can be analyzed in the framework of nonlinearmixed effects models (NonLinMEM) (Davidian and Giltinan 1995) Inference aboutNonLinMEM parameters is one of the most challenging tasks in statistical modelingas it is necessary to take into account several aspects of the study design includingthe nonlinear shape of the response with respect to a set of parameters of interestand the fact that measurements taken from the same individual are likely to becorrelated In addition overall variation between measurements has to be properlypartitioned into between-and within-subject variation Population PK=PD data areespecially interesting examples of NonLinMEM in which a model for each subject isoften represented as a solution of a system of ordinary differential equations (ODE)

There are two common groups of methods to make inference on parametricNonLinMEM models those based on individual estimates and those based onapproximation of a marginal likelihood function The first group of methods is oftenreferred as two-stage approaches where parameters for each subject are estimated inthe first stage and these estimates are used as data in the second stage of analysis toestimate population parameters (Davidian and Giltinan 1995) These techniques areappealing in their computational simplicity and they naturally reflect hierarchies in aNonLinMEM model (eg within-and between-subject models) Two-stage methodsperform fairly well when the number of observations per individual is sufficient toreliably estimate individual parameters

The second group of estimation methods is based on approximation of amarginal likelihood function This group of methods is usually computationallymore intensive than the first one and are the methods of choice when data are sparseor when time-dependent covariates are present The main difficulty is that the mar-ginal likelihood function involves integration over the distribution of the randomeffects In the context of NonLinMEM this integral usually does not have a closedform solution so approximation of this integral is necessary Possible integralapproximations of the marginal likelihood function include those based on quadra-ture methods and on Taylor-series linearization (Davidian and Giltinan 1995Pinheiro and Bates 1995 Vonesh and Chinchilli 1997)

In this paper we focus our attention on linearization approximations in particu-lar those based on iteratively fitting a set of corresponding estimating equations(Wolfinger and Lin 1997) This kind of approach is often less accurate than other

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more complex approximations but is usually more computationally robust It ismost effective in the common case in which the within- and between-subject errorsare approximately normally distributed and is similar but not identical to the first-order method popularized by the NONMEM package of Beal and Sheiner (1992)and also available in the NLMIXED procedure (SAS Institute Inc 1999b) It isimplemented in the nlme function in Splus and R (Pinheiro and Bates 2000) andin the SAS macros MIXNLIN (Vonesh and Chinchilli 1997) and NLINMIX

The main purpose of this paper is to present recently implemented new featuresto NLINMIX enabling a wide range of population PK=PD models One of thecritical developments is that a formerly independent NLMEM macro (Galecki1998) has been integrated into NLINMIX so that models specified as a solutionof a system of ODE can now be addressed The next sections describe the modeland estimation algorithms and these are then illustrated with a new data set onglucose elimination Other improvements and implemented features of NLINMIXare presented in detail in the Appendix A

STATISTICAL MODEL

Using the notation presented in Table 1 the hierarchical nonlinear modelconsidered here is specified as follows

Stage 1 Within-subject (WS) variation

yij frac14 f ethxij d iTHORN thorn eij

for j frac14 1 mi or in vector notation

yi frac14 f iethd iTHORN thorn ei eth1THORNei jd iNeth0Riethd i hRTHORNTHORN

Stage 2 Between-subjects (BS) variation

d i frac14 dethai b biTHORNbi Neth0G iethhGTHORNTHORN

The model is similar to that specified in Davidian and Giltinan (1995) The onlydifference is how the intra-individual covariance structure Ri is factored In thisfactorization the diagonal weight matrix W i and covariance matrix Ri are usedconsistent with the syntax of the NLINMIX macro In Davidian and Giltinan(1995) the specification of the variance function is used instead of a weight functionIn contrast to the specification of Lindstrom and Bates (1990) a nonlinear model isallowed at the second stage of the hierarchy for between-subject variation

The specification of the model is general and can be further extended to addressother aspects of longitudinal data analysis In particular multiple outcomes canbe accommodated by allowing the model function f to be a vector This extensionleads to stacking response vectors for different dependent variables from thesame individual in the model formula (1) In addition to time-dependent covariates

Ordinary Differential Equation PKPD Using SAS Macro NLINMIX 485

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being addressed in stage 1 they also can be addressed at stage 2 of the hierarchyby allowing the individual regression parameters d i to depend on time-varyingindividual-specific information Another possible extension is to introduce morethan two stages of hierarchy so multilevel hierarchical models can be addressedAll these extensions of the two-stage model can be addressed using the NLINMIXmacro presented in this paper

ESTIMATION

This section describes the linear approximation of the hierarchical nonlinearmodel This approximation is a basis of the estimation algorithm presented laterin this section

Linearization of the Model

Given b frac14 b and bi frac14 bi the two stages of hierarchical model presented in theprevious section can be approximated using a Taylor-series linearization in the

Table 1 Notation for nonlinear mixed effects model specification and related derivativefunctions

i Subject indexmi Number of measurements per subject ij Measurement index for subject i j frac14 1 2 mi

yi A mi 1 vector of responses yij for subject ixij A vector of time-dependent covariates for a subject i

describing condition(s) at which jth measurement was takend i A p 1 vector of subject-specific regression parametersf ethxij d iTHORN An expected conditional mean of the jth response yij for subject if iethd iTHORN A nonlinear vector function of d i whose jth component is equal to f ethxij d iTHORNDiethd iTHORN frac14 f ethd iTHORN=d i is a mi p matrix function of derivatives with respect to (wrt) d i

ei A mi 1 vector of residuals eij j frac14 1 2 mi for subject ihR Vector of covariance parameters for within-subject variationRiethhR d iTHORN A mi mi covariance matrix function for within-subject

heterogeneity of varianceW iethd iTHORN A mi mi diagonal weight matrix function depending on d i

Riethd i hRTHORN frac14W1=2i ethd iTHORNRiethd i hRTHORNW1=2

i ethd iTHORNai A a 1 vector of time-independent covariates for subject ib A r 1 vector regression parameters for the fixed effectsbi A q 1 vector of random effects associated with subject idethai b biTHORN A nonlinear vector function of ai b and bi differentiable wrt b and biAethai b biTHORN frac14 dethai b biTHORN=b a p r matrix function of partial derivatives wrt bBethai b biTHORN frac14 dethai b biTHORN=bi a p q matrix function of partial derivatives wrt bihG Vector of covariance parameters for between-subject variationG iethhGTHORN A q q covariance matrix function for the between-subjects heterogeneity

of variance depending on a vector of covariance parameters hG

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neighborhood of b and bi as follows

Stage 1 Approximation

yi frac14 f iethdi THORN thornDi ethd i di THORN thorn ei eth2THORNei jd i Neth0Ri ethhRTHORNTHORNStage 2 Approximation

d i frac14 di thorn Ai eth b bTHORN thorn Bi ethbi bi THORN eth3THORNbi Neth0G iethhGTHORNTHORN

where di frac14 dethai b bi THORN Di frac14 Diethdi THORN Ai frac14 Aethai b bi THORN Bi frac14 Bethai b bi THORN Wi frac14

W iethdi THORN Ri ethYRTHORN frac14 Riethdi YRTHORN and Ri ethYRTHORN frac14Weth1=2THORNi Ri ethYRTHORNWeth1=2THORN

i

The two stages of the hierarchy (Eqs 2 and 3) can be combined into thefollowing model

yi frac14 f iethdi THORN thorn Xi ethb bTHORN thorn Zi ethbi bi THORN thorn ei eth4THORNwhere Xi frac14 DiA

i and Zi frac14 Di B

i

After rearranging terms in Eq (4) the model can be expressed as a linear mixedeffects model in terms of b and the birsquos

vi frac14 Xi b thorn Zi bi thorn ei eth5THORNwhere

vi frac14 yi f iethdi THORN thorn Xi b thorn Zi b

i

Note that in this approximation the matrix of derivatives Bi and consequentlyexpression Zi b

i are evaluated at bi frac14 bi This approximation is referred to as a

second-order linearization (Lindstrom and Bates 1990) or a conditional first-orderlinearization (Davidian and Giltinan 1995) The second-order approximation maybe simplified by omitting term Zi bi from expression (5) which is equivalent tosetting bi frac14 0 This simplified approximation is referred as a first-order linearizationAn intermediate case in which selected elements of vector bi are set to zero in Zi b

i is

referred to as a hybrid linearization The linear approximation Eq (5) is used in thelinear mixed effects (LME) step of the iterative algorithm presented in the nextsection Additional details can be found in Vonesh and Chinchilli (1997)

Computational Algorithm

Several iterative algorithms have been proposed to implement a linearizationmethod The algorithm used in NLINMIX are presented in Wolfinger and Lin(1997) Details of the algorithm are illustrated in Fig 1

The algorithm optionally uses an initial call to the nonlinear regression (NLINstep) to refine initial values for b estimates Predictors of bi are set to zero and are notupdated at the initial NLIN step By default during subsequent iterations the


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algorithm alternates between the following two steps (1) the pseudo-data (P-D)generation step and (2) the linear mixed effects (LME) step

In the P-D generation step derivative matrices Ai Bi D

i X

i Z

i and weighting

matrix Wi are calculated at current values of b frac14 b and bi frac14 0 (first-order lineariza-

tion) In case we want to apply second-order linearization we set bi frac14 bi Similarilyin case of hybrid linearization selected elements of bi are set to zero Matrices Xiand Zi are subsequently used to calculate a working dependent vector vi so theLME step can be performed using Eq (5)

In the LME step ML or REML estimates of covariance parameters hR and hGfrom a previous iteration are refined iteratively Given hR and hG vectors b

and biare updated by solving mixed model equations (MME) which completes LME step

New b and bi values are then used in subsequent P-D step New YR and YG

estimates are used in a subsequent LME step to initiate ML or REML subiterationsThe iterative process is continued until the difference between estimates of b and thecovariance parameter estimates between the current and previous iteration satisfiesconvergence criteria

EXAMPLE

The use of NLINMIX for models specified using SAS=IML is illustrated usingoriginal data from a population pharmacokinetics study of the glucose disappear-ance in old subjects with a normal glucose tolerance Each subject underwentfrequently sampled intravenous glucose test (FSIVGTT) An initial bolus dose ofglucose was administered intravenously to each of 10 subjects

For a given subject glucose and insulin concentration plasma levels are mea-sured m frac14 30 times between 0 minutes and 180 minutes after glucose injectionFor notational simplicity and to be consistent with IML modules syntax the subjectsubscript i frac14 1 10 is omitted in considerations below

The data consist of 30 glucose and insulin plasma concentrationsGeth1THORN Geth30THORN from each subject measured at prespecified time points t1 t30

Figure 1 Computational algorithm

488 Galecki et al

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from 0 to 180 minutes Patientrsquos age is used as a covariate in this example Profiles ofthe 10 patients are displayed in Fig 2

Glucose disappearance is assumed to conform a nonlinear two-compartmentmodel with intravenous bolus of glucose administration Following Wastney et al(1999) the model is graphically presented in Fig 3 This model is also referred asa minimal model proposed by Bergman and Bowden (1981)

The model implies that a predicted glucose concentration gethtTHORN at time t in aprimary compartment and insulin level xethtTHORN in a so-called remote compartment

Figure 2 Individual profiles for 10 subjects


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conform to the following system of two nonlinear ODE after a glucose bolus

_ggethtTHORN frac14 ethp1 thorn xethtTHORNTHORNgethtTHORN thorn p1Gb

_xxethtTHORN frac14 p2xethtTHORN thorn p3ethIethtTHORN IbTHORN

eth6THORN

with initial values at t frac14 0 geth0THORN frac14 g0 and xeth0THORN frac14 0 The system does not have aclosed form solution Gb and Ib called basal glucose and insulin respectively aresubject characteristics derived empirically We denote by d frac14 frac12p1 p2 p3 g00 a vectorof parameters for this model The parameter p1 is interpreted as glucose sensitivityand g0 is an estimated glucose level at time t1 frac14 0 The ratio of parameters sI frac14 p2=p3is interpreted as insulin sensitivity

Note that so far in this section we have assumed that the parameter vector d isgiven and consequently gdethtTHORN frac14 gethtTHORN and xdethtTHORN frac14 xethtTHORN are functions of time only Byallowing d to vary the model functions can be defined as follows

getht dTHORN frac14 gdethtTHORNxetht dTHORN frac14 xdethtTHORN

(

Then the population model for IVGTT is specified in the following two stages

Stage 1 Within-subject (WS) model

y frac14 gtethdTHORN thorn e

e j d Neth0 s2THORN

e j d N 0 s2g2etht1 dTHORN 0

0 g2ethtm dTHORN

2664

3775

0BB

1CCA eth7THORN

where y frac14 frac12y1 ym0 and gtethdTHORN frac14 frac12getht1 dTHORN gethtm dTHORN0

Figure 3 Model of glucose disappearance

490 Galecki et al

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Stage 2 Nonlinear model for between-subjects (BS) variation

p1 frac14 b1 thorn u1

log10ethp2THORN frac14 b2 thorn u2

log10ethsI THORN frac14 b3 thorn ethage 70THORNb5 thorn u3

log10ethp3THORN frac14 log10ethp2THORN thorn log10ethsITHORN

u frac14u1u2u3

24

35 N 0

s211 s212 s213

s212 s222 s223

s213 s223 s233

2664

3775

0BB

1CCA

Note that the covariance matrix in Eq (7) implies conditional independence of thewithin-subject residual errors with a constant coefficient of variation

Calculations are performed using the NLINMIX macro in SAS=IML executionmode refer to Appendix A for details The remaining part of this sectiondescribes how to use the macro to fit this model to the IVGTT data The data arestored in a SAS dataset called IVGTT containing 300 records and five variablesID TIME AGE G T I T The code in Appendix A includes both the user-providedIML functions and the NLINMIX macro invocation for the two-compartmentmodel

The key part is the user-defined IML functions invoked from the useriml

macro The function GLCD defines the integrand function Eq (6) for an ODE Itis invoked from the function GLC SOLV which solves the ODE using theSAS=IML ODE function and returns the vector PREDV of predicted glucose valuesin compartment 1 Integration for each subject is performed numerically over timeintervals determined by the vector TIME with 30 elements Note that the functionGLCD contains a definition of six additional differential equations the so-calledsensitivity equations The sensitivity equations are integrated in the GLC SOLVfunction to calculate the partial derivatives getht dTHORN=d and xetht dTHORN=d Thefunction BSM defines the between-subject model and its derivatives with respectto bi

The SAS=IML functions defined in the useriml macro are then used in the model

argument to define a mapping from b bi ( B and U row vectors respectively in SASnotation) to f iethdethai b biTHORNTHORN (PREDV column vector in SAS notation) Partial deriva-tives of the d function with respect to the fixed effects and random effects arecomputed analytically by BSM SAS=IML function and returned by the DBSM func-tion Partial derivatives for superimposed functions f i and d are calculated using thechain rule and are stored in the matrix DPREDV Note that DPREDV is listed inthe globvar argument so that it can be passed from a SAS=IML function definedby model to a function defined by the derivs argument In this way the derivativematrices DB and DU can be extracted from DPREDV in derivs argument asneeded To address rounding errors during numerical integration the convergencecriterion is set to a more liberal value 1E-3 increased from the default value of 1E-8


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Calculations were performed on a Sun Blade 1000 workstation (Solaris 8) usingSAS Version 82 The algorithm converged successfully in five iterations (27 secCPU) and produces the following parameter estimates

Iteration History Extracted from SAS Log

The NLINMIX Macro

Data Set fsivgtt

Response g_t

Fixed-Effect Parameters beta1 beta2 beta3 beta4 beta5

Random-Effect Parameters u1 u2 u3

Expansion Point zero

Iteratively calling PROC MIXED

PROC MIXED call 0

iteration = 0

convergence criterion =

beta1=002 beta2=-2 beta3=-3 beta4=250 beta5=0

COVP1=0000117471

COVP2=-000244529 COVP3= 0128204017

COVP4= 000282178 COVP5=-0084482893 COVP6=00917336449

COVP7=00070584159

(iterations 1 through 3 omitted)

iteration = 4

beta1=0023464368 beta2=-1647843655 beta3=-3402938866

beta4=2788343188 beta5=-0213441218

COVP1= 000012790

COVP2=-000209429 COVP3 =0110625648

COVP4=0001486708 COVP5=-0031203057 COVP6=00535009216

COVP7=00064359279

NLINMIX convergence criteria met

Parameter Estimates Extracted from SAS Output

Standard 95 CI

Effect Estimate Error Lower Upper

D_BETA1 00235 00040 00156 00313 b1D_BETA2 -16476 01346 -19129 -13823 b2D_BETA3 -34031 00795 -35598 -32468 b3D_BETA4 27884 221 27448 28319 b4D_BETA5 -02135 08619 -19116 14846 b5

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Covariance Parameter Estimates

Cov Parm Subject Estimate

UN(11) id 0000128 s211UN(21) id -0210 s212UN(22) id 01108 s222UN(31) id 01487 s213UN(32) id -00313 s223UN(33) id 00536 s233Residual 00064 s2

Population estimate for glucose sensitivity p1 is 00235E(SEfrac14 0004) Similarlythe estimate for a population insulin sensitivity sI expressed on an original scale is0003954 (0002763 000567 95 CI)

DISCUSSION

The SAS=IML execution mode of the NLINMIX macro presented in this paperis a substantial new feature which allows the fitting of ODE and other models thatare not tractable in the non-SAS=IML execution mode This new execution modemay be considered as an interface to IML for the NLINMIX macro The programretains all the advantages of the non-SAS=IML execution mode and adds the flexibleand powerful IML environment In particular the program can fit models in whichfunctions f i and d are defined implicitly Important examples are when f i isexpressed as a numerical solution to a set of ODEs for pharmacokinetics andpharmacodynamics modeling

SAS=IML also substantially simplifies the specification of the model Whenusing a naming convention for selected variables (eg Y for the dependent variableTIME for experiment time or DOSE for instantaneous bolus doses into dosecompartment) a library of SAS=IML functions for selected within-subject modelscan be created and user-defined IML code can be further simplified The use ofthe SAS environment also allows users to use several powerful features to processoutput datasets from NLINMIX especially in combination with the accumulationof intermediate results using the append argument The macro has been extensivelytested using several population PK=PD data In one group of tests parameterestimates and their standard errors were compared to those obtained using othersoftware Other tests involved comparing results for ODE models which had bothnumerical and analytical solution All tests led to very similar results As with mostiterative numerical procedures stability and performance can degrade when theselected model (including starting values) and data do not match well Examplesof this include cases when the data are too noisy to extract pharmacokinetic trendsor when an overly complex model is fitted to a small data set However for standardPK=PD data sets like the one considered in the example the method convergesrobustly and parameter estimates are stable To conclude the NLINMIX macroexecuted in SAS=IML mode constitutes an attractive option for the analysis ofpopulation pharmacokinetics and pharmacodynamics studies


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APPENDIX A

THE NLINMIX MACRO

NLINMIX is a SAS macro that estimates nonlinear mixed models using alinearization approximation It has been under development for about 10 yearsand the most current version is compatible with SAS Version 8 Source code fornon-SAS=IML including documentation and examples is available at http==ftpsascom=techsup=download=stat=nlmm800sas

Additional code that allows NLINMIX to run in SAS=IML mode is available inthe NLMEM macro (Galecki 1998) Starting with SAS Version 8 NLMEM hasbeen redesigned so that it can be called directly from NLINMIX Relevant NLMEMcode can be downloaded at http==www-personalumichedu=agalecki Examplesand documentation are also available

SYNTAX

The macro can be executed in two different modes depending on the value of thenlmem macro argument By default nlmem is blank and NLINMIX is executed instandard (non-SAS=IML) mode otherwise NLINMIX loads a source file with aset of NLMEM macros from the file indicated by the nlmem argument and NLIN-MIX is executed in SAS=IML mode Since the syntax of several of the macroarguments is different in these two modes we describe the syntax separately fornon-SAS=IML and SAS=IML execution modes

Standard (Non-SAS=IML) Execution Mode

Non-SAS=IML execution mode has been available in all previous versions of theNLINMIX macro Similar to previous versions the execution in a default mode iscontrolled by several macroarguments listed in Table 2

Arguments are arranged in five groups depending on their role in the macro-execution and default values for selected arguments are provided Detailed syntaxfor these macroarguments can be found in the NLINMIX documentation andexamples Note that all arguments listed in groups 1ndash4 are also valid in SAS=IMLexecution mode

The key macroargument introduced in NLINMIX Version 8 and controllingexecution of LME step is called stmts It contains SAS code executed as a part ofPROC MIXED Another new argument called append allows accumulation of awide range of results from all iterations This feature may serve as a debugging toolor to provide insight into execution of the macro Other recently added featuresspecified by arguments (specified in parantheses) include linear nonequality con-straints (neq constraintsfrac14) extensive user-defined debugging features (user debugfrac14)and choosing parameters not to be adjusted during NLINMIX iterations(paramfixedfrac14)

The syntax of the derivs and expand arguments is now more flexible In previousNLINMIX releases the derivs argument required specification of analytical

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derivatives ethf ifrac12dethai b biTHORNTHORN=b and ethf ifrac12dethai b biTHORNTHORN=bi with respect to all fixedand random effects If none of the derivatives were specified numerical derivativeswere computed instead In the new release selected derivatives can be specifiedanalytically and the remaining ones are calculated numerically The new syntaxfor the expand argument enables a hybrid linearization approach in the estimationprocess

SAS=IML Execution Mode

If the macroargument nlmem is not blank and points to a file containingsource code with a set of NLMEM macros NLINMIX is executed in SAS=IML mode

Table 2 NLINMIX macro arguments and their role in a default (non-SAS=IML) modelspecification

Group 1 User Input Informationdata Input datasetparmsa Specifies initial values and variables containing b estimatesparamfixed Specifies b parameters not adjusted during NLINMIX iterations

Group 2 Pseudo Data (P-D) Stepmodela Systematic part of the model f iethdethai b biTHORNTHORNmodinita Auxiliary to model argument useful in recursively defined modelsderivsa Derivatives ethf iethdethai b biTHORNTHORNTHORN=b and eth f iethdethai b biTHORNTHORNTHORN=bitolfrac14 1endash5 Specifies tolerance if finite differences derivatives calculatedexpandfrac14 zero Expansion method first-order second-order or hybrid linearizationweighta Weight matrix function

Group 3 LME Stepstmts Specifies LME step (Eq 5)procopt Auxiliary to stmts argumentneq constraints Points to a macro containing linear nonequality constraints

Group 4 Iteration and Output Controlconvergefrac14 1endash8 Convergence criterionmaxitfrac14 30 Maximum number of iterationsswitchfrac14 0 Specifies iteration number at which expansion method indicated

by expand argument switches to first-order linearizationappend Specifies datasets to append during subsequent iterationsuser debug Points to a dataset containing list of user defined

debugging macrosoutdatafrac14 nlinmix Working datasetoptions Options skipnlin and others

Group 5a NLIN Step (non-IML Specification Only)nlinopt PROC NLIN optionsnlinstmt Additional PROC NLIN statements

aMacro arguments that have different syntax in non-SAS=IML and SAS=IML specification


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All arguments listed in Table 2 (except group 5a) can be used in SAS=IMLexecution mode The user should be aware that some of these arguments havedifferent syntax in non-SAS=IML and SAS=IML modes More specifically in modelmodinit derivs and weight arguments SAS=IML as opposed to PROC NLINprogramming statements are used In the parms argument one initial value per fixedeffect in SAS=IML mode is allowed as opposed to multiple values in non-SAS=IML mode All remaining arguments from groups 1 through 4 in Table 2 playthe same role and have exactly the same syntax in both non-SAS=IML andSAS=IML execution modes

In addition to arguments from groups 1ndash4 listed in Table 2 all arguments listedin Table 3 can also be used in SAS=IML execution mode The macroarguments inGroup 1 from Table 3 enhance the way the input dataset can be read in duringNLIN and P-D steps In non-SAS=IML mode the only option available is thatNLINMIX reads in one record at a time from an input dataset and consequentlyone record is processed at a time In SAS=IML mode one or more records can beread at a time and subsequently multiple records can be processed at once If nextis a number a fixed number of records are read at a time By default this numberis 1 which means that one record is read at a time If next is a variable which iden-tifies a subject in a dataset multiple records are read at a time Note that the next

argument affects the way NLIN and P-D steps are executed and it does not directlyaffect the LME step execution

The arguments tvar and xvar indicate disjoint lists of time-dependent variablesand time-independent variables respectively By default tvar and xvar are blankand all variables are read in from a dataset and considered to be time-dependentexcept for the subject-identifying variable

Table 3 Additional NLINMIX macro arguments used for a model specified using SAS=IMLsyntax (SAS=IML execution model)

Group 1 User Input Informationnlmem File with a source code for a set of NLMEM macrosnextfrac14 1 Indicates how many records are read at a timexvar List of time-independent covariatestvar List of time-dependent covariates

Group 2 Pseudo Data (P-D) Stepimlfun Macro containing user-defined SAS=IML functionsglobvar List of user-defined global vectors or matricesdtmplt Indicates which derivatives returned by derivsfrac14 argumant are not

recomputed using finite difference derivativesretain For recursively specified models

Group 4 Output Controlsavevar List of output vectors to save in outdatafrac14 datasetdebug Points to a user-defined SAS=IML function for debugging model specification

Group 5b NLIN Step (SAS=IML Specification Only)imlnlin NLIN step using SAS=IML statements stored in default nlin default objf and

default initobjf macros

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If more than one record is read at a time then for each time-dependent variablea SAS=IML column vector with the same name is extracted from the input datasetThe number of elements in a vector is determined by the number of records read at atime For time-independent variables an IML constant with the same name repre-sents an entire vector All constants and vectors created this way are put in a listof global SAS=IML objects and can be potentially used in specification of NLINor P-D step By using constants instead of vectors for time-independent variablesthe specification of the model and calculations in the NLIN and P-D steps can besimplified

The arguments model modinit derivs and weight play a key role in specifyingNLIN and P-D step Auxiliary argument globvar provides the user with a mechanismof sharing SAS=IML data objects eg matrices vectors and scalars betweenSAS=IML function arguments defining the NLIN and P-D step

EXECUTION

During the initial step common to both non-SAS=IML and SAS=IMLexecution NLINMIX extracts the names of fixed and random effects from theparms and RANDOM statements specified in the stmt macro argument Otherinformation related to controlling the P-D and LME execution is also extractedat this stage

Initial Values

Initial values for fixed effects are usually specified through the parms argumentBy default random effects are assumed to be zero If variables corresponding to fixedand=or random effects are already included in an input dataset it is not necessary tospecify initial values in the parms argument If present initial values for the fixedand=or random effects will be taken from the input dataset and those from parms willbe ignored

Psuedo-Data and NLIN Steps

In a non-SAS=IML execution mode programming statements from modelmodinit derivs and weight are built into PROC NLIN to optionally execute anNLIN step A refined set of fixed parameter estimates and other auxiliary variablesare stored in a working dataset specified by the outdata macroargument Similarilythe same statements are included into an SAS DATA step so that a P-D step can beexecuted The P-D step is completed by storing resulting derivatives weight vectorand working dependent variables in a working dataset

In SAS=IML execution mode an optional NLIN step is executed after program-ming statements from model modinit derivs and weight are incorporated intoSAS=IML functions Note that the generated SAS=IML function returns a vectorof weighted differences and their corresponding derivatives By default the imlnlin


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argument implies that a least squares optimization is performed using theLevenbergndashMarquardt algorithm By specifying a different set of SAS=IMLstatements in macros specified by the imlnlin argument the user can implementdifferent optimization methods for the NLIN step

The P-D step is executed as previously except that contents of the modelmodinit derivs and weight arguments are incorporated into analogous SAS=IMLfunctions The resulting functions are executed in a separate PROC IML call andthe resulting matrix derivatives weight vector and other auxiliary variables arestored in a working dataset to complete the P-D step

LME Step

In both SAS=IML and non-SAS=IML runs the LME step is executed usingPROC MIXED applied to a working dependent variable created in a workingdataset Derivatives with respect to fixed effects are used to construct a design matrixspecified as effects in the MODEL statement Derivatives with respect to randomeffects are used to specify effects in the RANDOM statement By default the YR

and YG estimates from the previous iteration are introduced through a PARMSstatement to initiate ML or REML subiterations

EXAMPLE CODE

This section details the code needed to execute the NLINMIX macro inSAS=IML mode for the glucose disappearance data The within-subject two-compartment model is specified as an ODE solution

filename nlmem rsquonlmemsasrsquo NLMEM source

filename nlinmix rsquonlmm800sasrsquo NLINMIX source

include nlinmix nosource

data fsivgtt1

----------------------------------------------------------------

Data from 10 old subjects with normal glucose tolerance

collected by Dr Halter and his collaborators

at the University of Michigan

----------------------------------------------------------------

input id time age g_t i_t

cards

2 0 82 99333 8333

28 2 82 363000 112900

more data

run

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macro useriml

----------------------------------------------------------------

-- --

-- MINMOD Glucose disappearance model --

-- Coded using SASIML A Galecki June 2002 --

-- --

-- glcd - integrand function for MINMOD model --

-- and sensitivity eqs --

-- glc_solv - solves glucose disappearance model --

-- wghtc - weight for observations --

-- interpolate - interpolation Auxiliary function --

-- --

----------------------------------------------------------------

start glcd(ty) global(ddI_bG_bI_ttime)

Integrand for glucose disappearance model

time is a vector

t is a scalar

function interpolate interpolates insulin level at time t

d=dd

p1=d[1] S_G Glucose sensitivity

p2=d[2] p_2

p3=d[3] p_3

G0=d[4] G_0 Predicted glucose at time 0

g_t =y[1]

x_t =y[2]

idiff= interpolate(ttimeI_t) - I_b

Prevents idiff to be less than zero

idiff= choose(idiffgt0idiff0)

dg_dt= -(p1+x_t)g_t + p1G_b

dx_dt= -p2x_t + p3idiff

--- Integrand function for sensitivity equations ---

g_1=y[3]

g_2=y[4] x_2=y[7]

g_3=y[5] x_3=y[8]

g_4=y[6]

c=p1+x_t

dg_1=-cg_1 - (g_t - G_b)

dg_2=-cg_2 - g_tx_2

dg_3=-cg_3 - g_tx_3

dg_4=-cg_4


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dx_2=-p2x_2-x_t

dx_3=-p2x_3+idiff

return(dg_dtdx_dt

dg_1dg_2dg_3dg_4

dx_2dx_3)

finish glcd

start glc_solv(d) global(ddI_tg_ttimeid)

--- Function returns solution vector for G_t and X_t ---

--- and solves sensitivity eqs Invokes GLCD function ---

dd=d

hmin hmax hinit step size

h1=10(-12)

h2=1

h3=10(-5)

h=h1||h2||h3

c= d[4] 0000100 Initial condition y(0)=c

eps=10(-4)

call ode(resglcdctimeh) eps=eps

glc_solv=t(c)t(res)

return(glc_solv)

finish glc_solv

start interpolate(x0xy)

-- This is an auxiliary function to perform linear interpolation --

statements omitted

finish interpolate

------ Between-subject model -----------

start bsm(bu) global(der_buage)

Derivs for p1p2p3G0

wrt b and u stored in der_bu

p1 = b[1] + u[1]

logp2 = b[2] + u[2]

logsi = b[3] + (age-70)b[5]100 + u[3] Linear model for log(si)

G0 = b[4]

db = i(4) || j(410)

db[35] = (age-70)100

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du = 1 0 0

0 1 0

0 0 1

0 0 0

logp3 = logp2 + logsi

dlogp3b = 1 1 db[23]

dlogp3u = 1 1 du[23]

p2 = exp(logp2)

db[22] = p2 db[22]

du[22] = p2 du[22]

p3 = exp(logp3)

db[3] = p3 dlogp3b

du[3] = p3 dlogp3u

der_bu = db || du

return(p1p2p3G0)

finish bsm

start dbsm(bu) global(der_bu)

return(der_bu)

finish

start wghtc(dummy) global(g_ttime)

Vector of weights

g_weight= choose(timegt=810) Zero-weighted measures for

timelt8 mins

g_weight[6]=50 Overweighted measure for time=8

w= g_weight(g_tg_t)

return(w)

finish

mend useriml

nlinmix(data = fsivgtt1

next = id

xvar = age

globvar = predv dpredv dd g_b i_b x_solv ins_action

stmts = str(

class id

model pseudo_g_t = d_beta1 d_beta2 d_beta3 d_beta4 d_beta5

noint notest solution cl

random d_u1 d_u2 d_u3 subject=id solution cl type=un

weight _weight_

)

model = str(

g_b=g_t[1] Baseline glucose

i_b=i_t[1] Baseline insulin


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Between subject model

_d=bsm(_b_u)

der_bu=dbsm(_b_u)

Within-subject model

d=_d

g_solv= glc_solv(d) Solving ODE

predv = g_solv[1] Predicted glucose

x_solv= g_solv[2] Predicted insulin in a remote

compartment

jacs = g_solv[36] Solution for sensitivity eqs

dpredv= jacs der_bu

ins_action = 100x_solv(d[1]+x_solv) Insulin action

)

derivs = str(

_db=dpredv[15] Derivatives wrt _b

_du=dpredv[68] Derivatives wrt _u

)

savevar = x_solv ins_action

weight = str( _weight_= wghtc())

parms = str(beta1=2 beta2=0 beta3=0 beta4=25 beta5=0)

expand = zero

imlfun = useriml Points to a user defined SAS macro

options = skipnlin

converge = 1e-3

nlmem = nlmem nosource Points to NLMEM macro

)

ACKNOWLEDGMENTS

The authors thank Jose Pinheiro and Richard Watanabe for their helpfulcomments and suggestions and Shu Chen for technical assistance Support for thisresearch was provided in part by NIA grant No P30 AG08808

REFERENCES

Beal S L Sheiner L B (1992) NONMEM Userrsquos Guides San FranciscoNONMEM Project Group University of California

Bergman R N Bowden C R (1981) The minimal model approach to quantifica-tion of factors controlling glucose disposal in man In Cobelli C BergmanR N eds Carbohydrate Metabolism Wiley pp 269ndash296

Davidian M Giltinan D M (1995) Nonlinear Models for Repeated MeasurementData London Chapman amp Hall

502 Galecki et al

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ORDER REPRINTS

Galecki A T (1998) NLMEM A new SAS=IML macro for hierarchical modelsComput Methods Progra Biomed 55207ndash216

Lindstrom M J Bates D M (1990) Nonlinear mixed effects models for repeatedmeasures data Biometrics 46673ndash687

Pinheiro J C Bates D M (1995) Approximations to the log-likelihood functionin the nonlinear mixed-effects model J Computat Graph Statist 412ndash35

Pinheiro J C Bates D M (2000) Mixed-Effects Models in S and S-PLUSNew York Springer-Verlag

SAS Institute Inc (1999a) SAS=IML Userrsquos Guide Version 7 Cary NC SASInstitute Inc

SAS Institute Inc (1999b) SAS=STAT Userrsquos Guide Version 8 Cary NC SASInstitute Inc

Vonesh E F Chinchilli V M (1997) Linear and Nonlinear Models for the Analysisof Repeated Measurements New York Marcel Decker Inc

Wastney M E Patterson B H Linares O A Greif P C Boston R C (1999)Investigating Biological Systems Using Modelling San Diego Academic Press

Wolfinger R D (1993) Laplacersquos approximation for nonlinear mixed effectsmodels Biometrika 80791ndash795

Wolfinger R D Lin X (1997) Two Taylor-series approximation methods fornonlinear mixed models Computat Statist Data Analy 25465ndash490

Received February 2003Accepted July 2003


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Request PermissionOrder Reprints

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The Materials are for your personal use only and cannot be reformatted reposted resold or distributed by electronic means or otherwise without permission from Marcel Dekker Inc Marcel Dekker Inc grants you the limited right to display the Materials only on your personal computer or personal wireless device and to copy and download single copies of such Materials provided that any copyright trademark or other notice appearing on such Materials is also retained by displayed copied or downloaded as part of the Materials and is not removed or obscured and provided you do not edit modify alter or enhance the Materials Please refer to our Website User Agreement for more details

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Ordinary Differential Equation PK=PD ModelsUsing the SAS Macro NLINMIX

Andrzej T Galecki1 Russell D Wolfinger2 Oscar A Linares34

Marla J Smith3 and Jeffrey B Halter13

1Institute of Gerontology University of Michigan Ann Arbor Michigan USA2SAS Institute Inc SAS Campus Drive Cary North Carolina USA

3Geriatrics Center University of Michigan Ann Arbor Michigan USA4Michigan Horace H Rackham School of Graduate Studies University of

Michigan Ann Arbor Michigan USA

ABSTRACT

We describe some theory and recent enhancements for the SAS macro NLINMIX(Wolfinger R D (1993) Laplacersquos approximation for nonlinear mixed effectsmodels Biometrika 80791ndash795) that enable model calculation to take placewithin the interaction matrix language SAS=IML (SAS Institute Inc (1999a)SAS=IML Userrsquos Guide Version 7 Cary NC SAS Institute Inc) They providegreater flexibility and scope for the specification and analysis of complexnonlinear mixed models For example using data from a frequently sampledintravenous glucose test we fit a two-compartment kinetics model that has noclosed-form representation It is derived as the solution of a system of ordinarydifferential equations and specified as such in SAS=IML Additional details andexample NLINMIX code are available in Appendix A

Correspondence Andrzej T Galecki Institute of Gerontology University of MichiganAnn Arbor MI 48109-2007 USA E-mail agaleckiumichedu

JOURNAL OF BIOPHARMACEUTICAL STATISTICS

Vol 14 No 2 pp 483ndash503 2004

483

DOI 101081BIP-120037194 1054-3406 (Print) 1520-5711 (Online)

Copyright 2004 by Marcel Dekker Inc wwwdekkercom

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INTRODUCTION






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STATISTICAL MODEL











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ESTIMATION













486 Galecki et al

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i




i




i




i is






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i X

i Z

i and weighting







EXAMPLE





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eth6THORN




(




e j d Neth0 s2THORN


0 g2ethtm dTHORN

2664

3775

0BB

1CCA eth7THORN



490 Galecki et al

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u frac14u1u2u3

24

35 N 0

s211 s212 s213

s212 s222 s223

s213 s223 s233

2664

3775

0BB

1CCA








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The NLINMIX Macro

Data Set fsivgtt

Response g_t





PROC MIXED call 0

iteration = 0



COVP1=0000117471

COVP2=-000244529 COVP3= 0128204017

COVP4= 000282178 COVP5=-0084482893 COVP6=00917336449

COVP7=00070584159


iteration = 4

beta1=0023464368 beta2=-1647843655 beta3=-3402938866

beta4=2788343188 beta5=-0213441218

COVP1= 000012790

COVP2=-000209429 COVP3 =0110625648

COVP4=0001486708 COVP5=-0031203057 COVP6=00535009216

COVP7=00064359279



Standard 95 CI



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DISCUSSION




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APPENDIX A

THE NLINMIX MACRO



SYNTAX







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EXECUTION


Initial Values






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LME Step



EXAMPLE CODE





data fsivgtt1

----------------------------------------------------------------




----------------------------------------------------------------


cards

2 0 82 99333 8333

28 2 82 363000 112900

more data

run

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macro useriml

----------------------------------------------------------------

-- --



-- --






-- --

----------------------------------------------------------------



time is a vector

t is a scalar


d=dd


p2=d[2] p_2

p3=d[3] p_3


g_t =y[1]

x_t =y[2]







g_1=y[3]

g_2=y[4] x_2=y[7]

g_3=y[5] x_3=y[8]

g_4=y[6]

c=p1+x_t

dg_1=-cg_1 - (g_t - G_b)

dg_2=-cg_2 - g_tx_2

dg_3=-cg_3 - g_tx_3

dg_4=-cg_4


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dx_2=-p2x_2-x_t

dx_3=-p2x_3+idiff

return(dg_dtdx_dt

dg_1dg_2dg_3dg_4

dx_2dx_3)

finish glcd




dd=d


h1=10(-12)

h2=1

h3=10(-5)

h=h1||h2||h3


eps=10(-4)


glc_solv=t(c)t(res)

return(glc_solv)

finish glc_solv



statements omitted

finish interpolate



Derivs for p1p2p3G0


p1 = b[1] + u[1]

logp2 = b[2] + u[2]


G0 = b[4]

db = i(4) || j(410)

db[35] = (age-70)100

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du = 1 0 0

0 1 0

0 0 1

0 0 0




p2 = exp(logp2)

db[22] = p2 db[22]

du[22] = p2 du[22]

p3 = exp(logp3)

db[3] = p3 dlogp3b

du[3] = p3 dlogp3u

der_bu = db || du

return(p1p2p3G0)

finish bsm


return(der_bu)

finish


Vector of weights


timelt8 mins


w= g_weight(g_tg_t)

return(w)

finish

mend useriml


next = id

xvar = age


stmts = str(

class id




weight _weight_

)

model = str(




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_d=bsm(_b_u)

der_bu=dbsm(_b_u)


d=_d




compartment


dpredv= jacs der_bu


)

derivs = str(



)




expand = zero


options = skipnlin

converge = 1e-3


)

ACKNOWLEDGMENTS


REFERENCES




502 Galecki et al

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ORDER REPRINTS













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14









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ORDER REPRINTS



INTRODUCTION






484 Galecki et al

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ORDER REPRINTS



STATISTICAL MODEL











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ORDER REPRINTS


ESTIMATION













486 Galecki et al

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ORDER REPRINTS







i




i




i




i is






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ORDER REPRINTS



i X

i Z

i and weighting







EXAMPLE





488 Galecki et al

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ORDER REPRINTS




eth6THORN




(




e j d Neth0 s2THORN


0 g2ethtm dTHORN

2664

3775

0BB

1CCA eth7THORN



490 Galecki et al

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ORDER REPRINTS






u frac14u1u2u3

24

35 N 0

s211 s212 s213

s212 s222 s223

s213 s223 s233

2664

3775

0BB

1CCA








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ORDER REPRINTS



The NLINMIX Macro

Data Set fsivgtt

Response g_t





PROC MIXED call 0

iteration = 0



COVP1=0000117471

COVP2=-000244529 COVP3= 0128204017

COVP4= 000282178 COVP5=-0084482893 COVP6=00917336449

COVP7=00070584159


iteration = 4

beta1=0023464368 beta2=-1647843655 beta3=-3402938866

beta4=2788343188 beta5=-0213441218

COVP1= 000012790

COVP2=-000209429 COVP3 =0110625648

COVP4=0001486708 COVP5=-0031203057 COVP6=00535009216

COVP7=00064359279



Standard 95 CI



492 Galecki et al

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ORDER REPRINTS





DISCUSSION




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ORDER REPRINTS

APPENDIX A

THE NLINMIX MACRO



SYNTAX







494 Galecki et al

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ORDER REPRINTS














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ORDER REPRINTS












496 Galecki et al

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ORDER REPRINTS



EXECUTION


Initial Values






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ORDER REPRINTS



LME Step



EXAMPLE CODE





data fsivgtt1

----------------------------------------------------------------




----------------------------------------------------------------


cards

2 0 82 99333 8333

28 2 82 363000 112900

more data

run

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ORDER REPRINTS

macro useriml

----------------------------------------------------------------

-- --



-- --






-- --

----------------------------------------------------------------



time is a vector

t is a scalar


d=dd


p2=d[2] p_2

p3=d[3] p_3


g_t =y[1]

x_t =y[2]







g_1=y[3]

g_2=y[4] x_2=y[7]

g_3=y[5] x_3=y[8]

g_4=y[6]

c=p1+x_t

dg_1=-cg_1 - (g_t - G_b)

dg_2=-cg_2 - g_tx_2

dg_3=-cg_3 - g_tx_3

dg_4=-cg_4


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ORDER REPRINTS

dx_2=-p2x_2-x_t

dx_3=-p2x_3+idiff

return(dg_dtdx_dt

dg_1dg_2dg_3dg_4

dx_2dx_3)

finish glcd




dd=d


h1=10(-12)

h2=1

h3=10(-5)

h=h1||h2||h3


eps=10(-4)


glc_solv=t(c)t(res)

return(glc_solv)

finish glc_solv



statements omitted

finish interpolate



Derivs for p1p2p3G0


p1 = b[1] + u[1]

logp2 = b[2] + u[2]


G0 = b[4]

db = i(4) || j(410)

db[35] = (age-70)100

500 Galecki et al

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ORDER REPRINTS

du = 1 0 0

0 1 0

0 0 1

0 0 0




p2 = exp(logp2)

db[22] = p2 db[22]

du[22] = p2 du[22]

p3 = exp(logp3)

db[3] = p3 dlogp3b

du[3] = p3 dlogp3u

der_bu = db || du

return(p1p2p3G0)

finish bsm


return(der_bu)

finish


Vector of weights


timelt8 mins


w= g_weight(g_tg_t)

return(w)

finish

mend useriml


next = id

xvar = age


stmts = str(

class id




weight _weight_

)

model = str(




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ORDER REPRINTS


_d=bsm(_b_u)

der_bu=dbsm(_b_u)


d=_d




compartment


dpredv= jacs der_bu


)

derivs = str(



)




expand = zero


options = skipnlin

converge = 1e-3


)

ACKNOWLEDGMENTS


REFERENCES




502 Galecki et al

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ORDER REPRINTS













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14









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ORDER REPRINTS



STATISTICAL MODEL











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ORDER REPRINTS


ESTIMATION













486 Galecki et al

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ORDER REPRINTS







i




i




i




i is






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at 2

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embe

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14

ORDER REPRINTS



i X

i Z

i and weighting







EXAMPLE





488 Galecki et al

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ORDER REPRINTS






Dow

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at 2

004

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embe

r 20

14

ORDER REPRINTS




eth6THORN




(




e j d Neth0 s2THORN


0 g2ethtm dTHORN

2664

3775

0BB

1CCA eth7THORN



490 Galecki et al

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ded

by [

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at 2

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02

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embe

r 20

14

ORDER REPRINTS






u frac14u1u2u3

24

35 N 0

s211 s212 s213

s212 s222 s223

s213 s223 s233

2664

3775

0BB

1CCA








Dow

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by [

Nor

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at 2

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embe

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ORDER REPRINTS



The NLINMIX Macro

Data Set fsivgtt

Response g_t





PROC MIXED call 0

iteration = 0



COVP1=0000117471

COVP2=-000244529 COVP3= 0128204017

COVP4= 000282178 COVP5=-0084482893 COVP6=00917336449

COVP7=00070584159


iteration = 4

beta1=0023464368 beta2=-1647843655 beta3=-3402938866

beta4=2788343188 beta5=-0213441218

COVP1= 000012790

COVP2=-000209429 COVP3 =0110625648

COVP4=0001486708 COVP5=-0031203057 COVP6=00535009216

COVP7=00064359279



Standard 95 CI



492 Galecki et al

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ORDER REPRINTS





DISCUSSION




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ORDER REPRINTS

APPENDIX A

THE NLINMIX MACRO



SYNTAX







494 Galecki et al

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ORDER REPRINTS














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ORDER REPRINTS












496 Galecki et al

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ORDER REPRINTS



EXECUTION


Initial Values






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embe

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ORDER REPRINTS



LME Step



EXAMPLE CODE





data fsivgtt1

----------------------------------------------------------------




----------------------------------------------------------------


cards

2 0 82 99333 8333

28 2 82 363000 112900

more data

run

498 Galecki et al

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embe

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ORDER REPRINTS

macro useriml

----------------------------------------------------------------

-- --



-- --






-- --

----------------------------------------------------------------



time is a vector

t is a scalar


d=dd


p2=d[2] p_2

p3=d[3] p_3


g_t =y[1]

x_t =y[2]







g_1=y[3]

g_2=y[4] x_2=y[7]

g_3=y[5] x_3=y[8]

g_4=y[6]

c=p1+x_t

dg_1=-cg_1 - (g_t - G_b)

dg_2=-cg_2 - g_tx_2

dg_3=-cg_3 - g_tx_3

dg_4=-cg_4


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ORDER REPRINTS

dx_2=-p2x_2-x_t

dx_3=-p2x_3+idiff

return(dg_dtdx_dt

dg_1dg_2dg_3dg_4

dx_2dx_3)

finish glcd




dd=d


h1=10(-12)

h2=1

h3=10(-5)

h=h1||h2||h3


eps=10(-4)


glc_solv=t(c)t(res)

return(glc_solv)

finish glc_solv



statements omitted

finish interpolate



Derivs for p1p2p3G0


p1 = b[1] + u[1]

logp2 = b[2] + u[2]


G0 = b[4]

db = i(4) || j(410)

db[35] = (age-70)100

500 Galecki et al

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ORDER REPRINTS

du = 1 0 0

0 1 0

0 0 1

0 0 0




p2 = exp(logp2)

db[22] = p2 db[22]

du[22] = p2 du[22]

p3 = exp(logp3)

db[3] = p3 dlogp3b

du[3] = p3 dlogp3u

der_bu = db || du

return(p1p2p3G0)

finish bsm


return(der_bu)

finish


Vector of weights


timelt8 mins


w= g_weight(g_tg_t)

return(w)

finish

mend useriml


next = id

xvar = age


stmts = str(

class id




weight _weight_

)

model = str(




Dow

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embe

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14

ORDER REPRINTS


_d=bsm(_b_u)

der_bu=dbsm(_b_u)


d=_d




compartment


dpredv= jacs der_bu


)

derivs = str(



)




expand = zero


options = skipnlin

converge = 1e-3


)

ACKNOWLEDGMENTS


REFERENCES




502 Galecki et al

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by [

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vers

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at 2

004

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ORDER REPRINTS













Dow

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by [

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vers

ity]

at 2

004

02

Dec

embe

r 20

14









Dow

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by [

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at 2

004

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Dec

embe

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ORDER REPRINTS


ESTIMATION













486 Galecki et al

Dow

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at 2

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ORDER REPRINTS







i




i




i




i is






Dow

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by [

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vers

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at 2

004

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Dec

embe

r 20

14

ORDER REPRINTS



i X

i Z

i and weighting







EXAMPLE





488 Galecki et al

Dow

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by [

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at 2

004

02

Dec

embe

r 20

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ORDER REPRINTS






Dow

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by [

Nor

th D

akot

a St

ate

Uni

vers

ity]

at 2

004

02

Dec

embe

r 20

14

ORDER REPRINTS




eth6THORN




(




e j d Neth0 s2THORN


0 g2ethtm dTHORN

2664

3775

0BB

1CCA eth7THORN



490 Galecki et al

Dow

nloa

ded

by [

Nor

th D

akot

a St

ate

Uni

vers

ity]

at 2

004

02

Dec

embe

r 20

14

ORDER REPRINTS






u frac14u1u2u3

24

35 N 0

s211 s212 s213

s212 s222 s223

s213 s223 s233

2664

3775

0BB

1CCA








Dow

nloa

ded

by [

Nor

th D

akot

a St

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Uni

vers

ity]

at 2

004

02

Dec

embe

r 20

14

ORDER REPRINTS



The NLINMIX Macro

Data Set fsivgtt

Response g_t





PROC MIXED call 0

iteration = 0



COVP1=0000117471

COVP2=-000244529 COVP3= 0128204017

COVP4= 000282178 COVP5=-0084482893 COVP6=00917336449

COVP7=00070584159


iteration = 4

beta1=0023464368 beta2=-1647843655 beta3=-3402938866

beta4=2788343188 beta5=-0213441218

COVP1= 000012790

COVP2=-000209429 COVP3 =0110625648

COVP4=0001486708 COVP5=-0031203057 COVP6=00535009216

COVP7=00064359279



Standard 95 CI



492 Galecki et al

Dow

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by [

Nor

th D

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at 2

004

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ORDER REPRINTS





DISCUSSION




Dow

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at 2

004

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embe

r 20

14

ORDER REPRINTS

APPENDIX A

THE NLINMIX MACRO



SYNTAX







494 Galecki et al

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by [

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at 2

004

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Dec

embe

r 20

14

ORDER REPRINTS














Dow

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by [

Nor

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Uni

vers

ity]

at 2

004

02

Dec

embe

r 20

14

ORDER REPRINTS












496 Galecki et al

Dow

nloa

ded

by [

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Uni

vers

ity]

at 2

004

02

Dec

embe

r 20

14

ORDER REPRINTS



EXECUTION


Initial Values






Dow

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vers

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at 2

004

02

Dec

embe

r 20

14

ORDER REPRINTS



LME Step



EXAMPLE CODE





data fsivgtt1

----------------------------------------------------------------




----------------------------------------------------------------


cards

2 0 82 99333 8333

28 2 82 363000 112900

more data

run

498 Galecki et al

Dow

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ded

by [

Nor

th D

akot

a St

ate

Uni

vers

ity]

at 2

004

02

Dec

embe

r 20

14

ORDER REPRINTS

macro useriml

----------------------------------------------------------------

-- --



-- --






-- --

----------------------------------------------------------------



time is a vector

t is a scalar


d=dd


p2=d[2] p_2

p3=d[3] p_3


g_t =y[1]

x_t =y[2]







g_1=y[3]

g_2=y[4] x_2=y[7]

g_3=y[5] x_3=y[8]

g_4=y[6]

c=p1+x_t

dg_1=-cg_1 - (g_t - G_b)

dg_2=-cg_2 - g_tx_2

dg_3=-cg_3 - g_tx_3

dg_4=-cg_4


Dow

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at 2

004

02

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embe

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14

ORDER REPRINTS

dx_2=-p2x_2-x_t

dx_3=-p2x_3+idiff

return(dg_dtdx_dt

dg_1dg_2dg_3dg_4

dx_2dx_3)

finish glcd




dd=d


h1=10(-12)

h2=1

h3=10(-5)

h=h1||h2||h3


eps=10(-4)


glc_solv=t(c)t(res)

return(glc_solv)

finish glc_solv



statements omitted

finish interpolate



Derivs for p1p2p3G0


p1 = b[1] + u[1]

logp2 = b[2] + u[2]


G0 = b[4]

db = i(4) || j(410)

db[35] = (age-70)100

500 Galecki et al

Dow

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ded

by [

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at 2

004

02

Dec

embe

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14

ORDER REPRINTS

du = 1 0 0

0 1 0

0 0 1

0 0 0




p2 = exp(logp2)

db[22] = p2 db[22]

du[22] = p2 du[22]

p3 = exp(logp3)

db[3] = p3 dlogp3b

du[3] = p3 dlogp3u

der_bu = db || du

return(p1p2p3G0)

finish bsm


return(der_bu)

finish


Vector of weights


timelt8 mins


w= g_weight(g_tg_t)

return(w)

finish

mend useriml


next = id

xvar = age


stmts = str(

class id




weight _weight_

)

model = str(




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ORDER REPRINTS


_d=bsm(_b_u)

der_bu=dbsm(_b_u)


d=_d




compartment


dpredv= jacs der_bu


)

derivs = str(



)




expand = zero


options = skipnlin

converge = 1e-3


)

ACKNOWLEDGMENTS


REFERENCES




502 Galecki et al

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ORDER REPRINTS













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ORDER REPRINTS







i




i




i




i is






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ORDER REPRINTS



i X

i Z

i and weighting







EXAMPLE





488 Galecki et al

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ORDER REPRINTS




eth6THORN




(




e j d Neth0 s2THORN


0 g2ethtm dTHORN

2664

3775

0BB

1CCA eth7THORN



490 Galecki et al

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ORDER REPRINTS






u frac14u1u2u3

24

35 N 0

s211 s212 s213

s212 s222 s223

s213 s223 s233

2664

3775

0BB

1CCA








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ORDER REPRINTS



The NLINMIX Macro

Data Set fsivgtt

Response g_t





PROC MIXED call 0

iteration = 0



COVP1=0000117471

COVP2=-000244529 COVP3= 0128204017

COVP4= 000282178 COVP5=-0084482893 COVP6=00917336449

COVP7=00070584159


iteration = 4

beta1=0023464368 beta2=-1647843655 beta3=-3402938866

beta4=2788343188 beta5=-0213441218

COVP1= 000012790

COVP2=-000209429 COVP3 =0110625648

COVP4=0001486708 COVP5=-0031203057 COVP6=00535009216

COVP7=00064359279



Standard 95 CI



492 Galecki et al

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ORDER REPRINTS





DISCUSSION




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ORDER REPRINTS

APPENDIX A

THE NLINMIX MACRO



SYNTAX







494 Galecki et al

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ORDER REPRINTS














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ORDER REPRINTS












496 Galecki et al

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ORDER REPRINTS



EXECUTION


Initial Values






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ORDER REPRINTS



LME Step



EXAMPLE CODE





data fsivgtt1

----------------------------------------------------------------




----------------------------------------------------------------


cards

2 0 82 99333 8333

28 2 82 363000 112900

more data

run

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ORDER REPRINTS

macro useriml

----------------------------------------------------------------

-- --



-- --






-- --

----------------------------------------------------------------



time is a vector

t is a scalar


d=dd


p2=d[2] p_2

p3=d[3] p_3


g_t =y[1]

x_t =y[2]







g_1=y[3]

g_2=y[4] x_2=y[7]

g_3=y[5] x_3=y[8]

g_4=y[6]

c=p1+x_t

dg_1=-cg_1 - (g_t - G_b)

dg_2=-cg_2 - g_tx_2

dg_3=-cg_3 - g_tx_3

dg_4=-cg_4


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ORDER REPRINTS

dx_2=-p2x_2-x_t

dx_3=-p2x_3+idiff

return(dg_dtdx_dt

dg_1dg_2dg_3dg_4

dx_2dx_3)

finish glcd




dd=d


h1=10(-12)

h2=1

h3=10(-5)

h=h1||h2||h3


eps=10(-4)


glc_solv=t(c)t(res)

return(glc_solv)

finish glc_solv



statements omitted

finish interpolate



Derivs for p1p2p3G0


p1 = b[1] + u[1]

logp2 = b[2] + u[2]


G0 = b[4]

db = i(4) || j(410)

db[35] = (age-70)100

500 Galecki et al

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ORDER REPRINTS

du = 1 0 0

0 1 0

0 0 1

0 0 0




p2 = exp(logp2)

db[22] = p2 db[22]

du[22] = p2 du[22]

p3 = exp(logp3)

db[3] = p3 dlogp3b

du[3] = p3 dlogp3u

der_bu = db || du

return(p1p2p3G0)

finish bsm


return(der_bu)

finish


Vector of weights


timelt8 mins


w= g_weight(g_tg_t)

return(w)

finish

mend useriml


next = id

xvar = age


stmts = str(

class id




weight _weight_

)

model = str(




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ORDER REPRINTS


_d=bsm(_b_u)

der_bu=dbsm(_b_u)


d=_d




compartment


dpredv= jacs der_bu


)

derivs = str(



)




expand = zero


options = skipnlin

converge = 1e-3


)

ACKNOWLEDGMENTS


REFERENCES




502 Galecki et al

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ORDER REPRINTS













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ORDER REPRINTS



i X

i Z

i and weighting







EXAMPLE





488 Galecki et al

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ORDER REPRINTS






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embe

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ORDER REPRINTS




eth6THORN




(




e j d Neth0 s2THORN


0 g2ethtm dTHORN

2664

3775

0BB

1CCA eth7THORN



490 Galecki et al

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ORDER REPRINTS






u frac14u1u2u3

24

35 N 0

s211 s212 s213

s212 s222 s223

s213 s223 s233

2664

3775

0BB

1CCA








Dow

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ORDER REPRINTS



The NLINMIX Macro

Data Set fsivgtt

Response g_t





PROC MIXED call 0

iteration = 0



COVP1=0000117471

COVP2=-000244529 COVP3= 0128204017

COVP4= 000282178 COVP5=-0084482893 COVP6=00917336449

COVP7=00070584159


iteration = 4

beta1=0023464368 beta2=-1647843655 beta3=-3402938866

beta4=2788343188 beta5=-0213441218

COVP1= 000012790

COVP2=-000209429 COVP3 =0110625648

COVP4=0001486708 COVP5=-0031203057 COVP6=00535009216

COVP7=00064359279



Standard 95 CI



492 Galecki et al

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ORDER REPRINTS





DISCUSSION




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ORDER REPRINTS

APPENDIX A

THE NLINMIX MACRO



SYNTAX







494 Galecki et al

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ORDER REPRINTS














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ORDER REPRINTS












496 Galecki et al

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ORDER REPRINTS



EXECUTION


Initial Values






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ORDER REPRINTS



LME Step



EXAMPLE CODE





data fsivgtt1

----------------------------------------------------------------




----------------------------------------------------------------


cards

2 0 82 99333 8333

28 2 82 363000 112900

more data

run

498 Galecki et al

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ORDER REPRINTS

macro useriml

----------------------------------------------------------------

-- --



-- --






-- --

----------------------------------------------------------------



time is a vector

t is a scalar


d=dd


p2=d[2] p_2

p3=d[3] p_3


g_t =y[1]

x_t =y[2]







g_1=y[3]

g_2=y[4] x_2=y[7]

g_3=y[5] x_3=y[8]

g_4=y[6]

c=p1+x_t

dg_1=-cg_1 - (g_t - G_b)

dg_2=-cg_2 - g_tx_2

dg_3=-cg_3 - g_tx_3

dg_4=-cg_4


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ORDER REPRINTS

dx_2=-p2x_2-x_t

dx_3=-p2x_3+idiff

return(dg_dtdx_dt

dg_1dg_2dg_3dg_4

dx_2dx_3)

finish glcd




dd=d


h1=10(-12)

h2=1

h3=10(-5)

h=h1||h2||h3


eps=10(-4)


glc_solv=t(c)t(res)

return(glc_solv)

finish glc_solv



statements omitted

finish interpolate



Derivs for p1p2p3G0


p1 = b[1] + u[1]

logp2 = b[2] + u[2]


G0 = b[4]

db = i(4) || j(410)

db[35] = (age-70)100

500 Galecki et al

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ORDER REPRINTS

du = 1 0 0

0 1 0

0 0 1

0 0 0




p2 = exp(logp2)

db[22] = p2 db[22]

du[22] = p2 du[22]

p3 = exp(logp3)

db[3] = p3 dlogp3b

du[3] = p3 dlogp3u

der_bu = db || du

return(p1p2p3G0)

finish bsm


return(der_bu)

finish


Vector of weights


timelt8 mins


w= g_weight(g_tg_t)

return(w)

finish

mend useriml


next = id

xvar = age


stmts = str(

class id




weight _weight_

)

model = str(




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ORDER REPRINTS


_d=bsm(_b_u)

der_bu=dbsm(_b_u)


d=_d




compartment


dpredv= jacs der_bu


)

derivs = str(



)




expand = zero


options = skipnlin

converge = 1e-3


)

ACKNOWLEDGMENTS


REFERENCES




502 Galecki et al

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ORDER REPRINTS













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ORDER REPRINTS






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004

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embe

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ORDER REPRINTS




eth6THORN




(




e j d Neth0 s2THORN


0 g2ethtm dTHORN

2664

3775

0BB

1CCA eth7THORN



490 Galecki et al

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at 2

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embe

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ORDER REPRINTS






u frac14u1u2u3

24

35 N 0

s211 s212 s213

s212 s222 s223

s213 s223 s233

2664

3775

0BB

1CCA








Dow

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14

ORDER REPRINTS



The NLINMIX Macro

Data Set fsivgtt

Response g_t





PROC MIXED call 0

iteration = 0



COVP1=0000117471

COVP2=-000244529 COVP3= 0128204017

COVP4= 000282178 COVP5=-0084482893 COVP6=00917336449

COVP7=00070584159


iteration = 4

beta1=0023464368 beta2=-1647843655 beta3=-3402938866

beta4=2788343188 beta5=-0213441218

COVP1= 000012790

COVP2=-000209429 COVP3 =0110625648

COVP4=0001486708 COVP5=-0031203057 COVP6=00535009216

COVP7=00064359279



Standard 95 CI



492 Galecki et al

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ORDER REPRINTS





DISCUSSION




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ORDER REPRINTS

APPENDIX A

THE NLINMIX MACRO



SYNTAX







494 Galecki et al

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ORDER REPRINTS














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ORDER REPRINTS












496 Galecki et al

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ORDER REPRINTS



EXECUTION


Initial Values






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ORDER REPRINTS



LME Step



EXAMPLE CODE





data fsivgtt1

----------------------------------------------------------------




----------------------------------------------------------------


cards

2 0 82 99333 8333

28 2 82 363000 112900

more data

run

498 Galecki et al

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ORDER REPRINTS

macro useriml

----------------------------------------------------------------

-- --



-- --






-- --

----------------------------------------------------------------



time is a vector

t is a scalar


d=dd


p2=d[2] p_2

p3=d[3] p_3


g_t =y[1]

x_t =y[2]







g_1=y[3]

g_2=y[4] x_2=y[7]

g_3=y[5] x_3=y[8]

g_4=y[6]

c=p1+x_t

dg_1=-cg_1 - (g_t - G_b)

dg_2=-cg_2 - g_tx_2

dg_3=-cg_3 - g_tx_3

dg_4=-cg_4


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ORDER REPRINTS

dx_2=-p2x_2-x_t

dx_3=-p2x_3+idiff

return(dg_dtdx_dt

dg_1dg_2dg_3dg_4

dx_2dx_3)

finish glcd




dd=d


h1=10(-12)

h2=1

h3=10(-5)

h=h1||h2||h3


eps=10(-4)


glc_solv=t(c)t(res)

return(glc_solv)

finish glc_solv



statements omitted

finish interpolate



Derivs for p1p2p3G0


p1 = b[1] + u[1]

logp2 = b[2] + u[2]


G0 = b[4]

db = i(4) || j(410)

db[35] = (age-70)100

500 Galecki et al

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ORDER REPRINTS

du = 1 0 0

0 1 0

0 0 1

0 0 0




p2 = exp(logp2)

db[22] = p2 db[22]

du[22] = p2 du[22]

p3 = exp(logp3)

db[3] = p3 dlogp3b

du[3] = p3 dlogp3u

der_bu = db || du

return(p1p2p3G0)

finish bsm


return(der_bu)

finish


Vector of weights


timelt8 mins


w= g_weight(g_tg_t)

return(w)

finish

mend useriml


next = id

xvar = age


stmts = str(

class id




weight _weight_

)

model = str(




Dow

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embe

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14

ORDER REPRINTS


_d=bsm(_b_u)

der_bu=dbsm(_b_u)


d=_d




compartment


dpredv= jacs der_bu


)

derivs = str(



)




expand = zero


options = skipnlin

converge = 1e-3


)

ACKNOWLEDGMENTS


REFERENCES




502 Galecki et al

Dow

nloa

ded

by [

Nor

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a St

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vers

ity]

at 2

004

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Dec

embe

r 20

14

ORDER REPRINTS













Dow

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by [

Nor

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at 2

004

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Dec

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14









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ORDER REPRINTS




eth6THORN




(




e j d Neth0 s2THORN


0 g2ethtm dTHORN

2664

3775

0BB

1CCA eth7THORN



490 Galecki et al

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ORDER REPRINTS






u frac14u1u2u3

24

35 N 0

s211 s212 s213

s212 s222 s223

s213 s223 s233

2664

3775

0BB

1CCA








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ORDER REPRINTS



The NLINMIX Macro

Data Set fsivgtt

Response g_t





PROC MIXED call 0

iteration = 0



COVP1=0000117471

COVP2=-000244529 COVP3= 0128204017

COVP4= 000282178 COVP5=-0084482893 COVP6=00917336449

COVP7=00070584159


iteration = 4

beta1=0023464368 beta2=-1647843655 beta3=-3402938866

beta4=2788343188 beta5=-0213441218

COVP1= 000012790

COVP2=-000209429 COVP3 =0110625648

COVP4=0001486708 COVP5=-0031203057 COVP6=00535009216

COVP7=00064359279



Standard 95 CI



492 Galecki et al

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ORDER REPRINTS





DISCUSSION




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ORDER REPRINTS

APPENDIX A

THE NLINMIX MACRO



SYNTAX







494 Galecki et al

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ORDER REPRINTS














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ORDER REPRINTS












496 Galecki et al

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ORDER REPRINTS



EXECUTION


Initial Values






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ORDER REPRINTS



LME Step



EXAMPLE CODE





data fsivgtt1

----------------------------------------------------------------




----------------------------------------------------------------


cards

2 0 82 99333 8333

28 2 82 363000 112900

more data

run

498 Galecki et al

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ORDER REPRINTS

macro useriml

----------------------------------------------------------------

-- --



-- --






-- --

----------------------------------------------------------------



time is a vector

t is a scalar


d=dd


p2=d[2] p_2

p3=d[3] p_3


g_t =y[1]

x_t =y[2]







g_1=y[3]

g_2=y[4] x_2=y[7]

g_3=y[5] x_3=y[8]

g_4=y[6]

c=p1+x_t

dg_1=-cg_1 - (g_t - G_b)

dg_2=-cg_2 - g_tx_2

dg_3=-cg_3 - g_tx_3

dg_4=-cg_4


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ORDER REPRINTS

dx_2=-p2x_2-x_t

dx_3=-p2x_3+idiff

return(dg_dtdx_dt

dg_1dg_2dg_3dg_4

dx_2dx_3)

finish glcd




dd=d


h1=10(-12)

h2=1

h3=10(-5)

h=h1||h2||h3


eps=10(-4)


glc_solv=t(c)t(res)

return(glc_solv)

finish glc_solv



statements omitted

finish interpolate



Derivs for p1p2p3G0


p1 = b[1] + u[1]

logp2 = b[2] + u[2]


G0 = b[4]

db = i(4) || j(410)

db[35] = (age-70)100

500 Galecki et al

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ORDER REPRINTS

du = 1 0 0

0 1 0

0 0 1

0 0 0




p2 = exp(logp2)

db[22] = p2 db[22]

du[22] = p2 du[22]

p3 = exp(logp3)

db[3] = p3 dlogp3b

du[3] = p3 dlogp3u

der_bu = db || du

return(p1p2p3G0)

finish bsm


return(der_bu)

finish


Vector of weights


timelt8 mins


w= g_weight(g_tg_t)

return(w)

finish

mend useriml


next = id

xvar = age


stmts = str(

class id




weight _weight_

)

model = str(




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ORDER REPRINTS


_d=bsm(_b_u)

der_bu=dbsm(_b_u)


d=_d




compartment


dpredv= jacs der_bu


)

derivs = str(



)




expand = zero


options = skipnlin

converge = 1e-3


)

ACKNOWLEDGMENTS


REFERENCES




502 Galecki et al

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ORDER REPRINTS













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14









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ORDER REPRINTS






u frac14u1u2u3

24

35 N 0

s211 s212 s213

s212 s222 s223

s213 s223 s233

2664

3775

0BB

1CCA








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ORDER REPRINTS



The NLINMIX Macro

Data Set fsivgtt

Response g_t





PROC MIXED call 0

iteration = 0



COVP1=0000117471

COVP2=-000244529 COVP3= 0128204017

COVP4= 000282178 COVP5=-0084482893 COVP6=00917336449

COVP7=00070584159


iteration = 4

beta1=0023464368 beta2=-1647843655 beta3=-3402938866

beta4=2788343188 beta5=-0213441218

COVP1= 000012790

COVP2=-000209429 COVP3 =0110625648

COVP4=0001486708 COVP5=-0031203057 COVP6=00535009216

COVP7=00064359279



Standard 95 CI



492 Galecki et al

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ORDER REPRINTS





DISCUSSION




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ORDER REPRINTS

APPENDIX A

THE NLINMIX MACRO



SYNTAX







494 Galecki et al

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ORDER REPRINTS














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ORDER REPRINTS












496 Galecki et al

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ORDER REPRINTS



EXECUTION


Initial Values






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ORDER REPRINTS



LME Step



EXAMPLE CODE





data fsivgtt1

----------------------------------------------------------------




----------------------------------------------------------------


cards

2 0 82 99333 8333

28 2 82 363000 112900

more data

run

498 Galecki et al

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ORDER REPRINTS

macro useriml

----------------------------------------------------------------

-- --



-- --






-- --

----------------------------------------------------------------



time is a vector

t is a scalar


d=dd


p2=d[2] p_2

p3=d[3] p_3


g_t =y[1]

x_t =y[2]







g_1=y[3]

g_2=y[4] x_2=y[7]

g_3=y[5] x_3=y[8]

g_4=y[6]

c=p1+x_t

dg_1=-cg_1 - (g_t - G_b)

dg_2=-cg_2 - g_tx_2

dg_3=-cg_3 - g_tx_3

dg_4=-cg_4


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ORDER REPRINTS

dx_2=-p2x_2-x_t

dx_3=-p2x_3+idiff

return(dg_dtdx_dt

dg_1dg_2dg_3dg_4

dx_2dx_3)

finish glcd




dd=d


h1=10(-12)

h2=1

h3=10(-5)

h=h1||h2||h3


eps=10(-4)


glc_solv=t(c)t(res)

return(glc_solv)

finish glc_solv



statements omitted

finish interpolate



Derivs for p1p2p3G0


p1 = b[1] + u[1]

logp2 = b[2] + u[2]


G0 = b[4]

db = i(4) || j(410)

db[35] = (age-70)100

500 Galecki et al

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ORDER REPRINTS

du = 1 0 0

0 1 0

0 0 1

0 0 0




p2 = exp(logp2)

db[22] = p2 db[22]

du[22] = p2 du[22]

p3 = exp(logp3)

db[3] = p3 dlogp3b

du[3] = p3 dlogp3u

der_bu = db || du

return(p1p2p3G0)

finish bsm


return(der_bu)

finish


Vector of weights


timelt8 mins


w= g_weight(g_tg_t)

return(w)

finish

mend useriml


next = id

xvar = age


stmts = str(

class id




weight _weight_

)

model = str(




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ORDER REPRINTS


_d=bsm(_b_u)

der_bu=dbsm(_b_u)


d=_d




compartment


dpredv= jacs der_bu


)

derivs = str(



)




expand = zero


options = skipnlin

converge = 1e-3


)

ACKNOWLEDGMENTS


REFERENCES




502 Galecki et al

Dow

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ORDER REPRINTS













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14









Dow

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004

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Dec

embe

r 20

14

ORDER REPRINTS



The NLINMIX Macro

Data Set fsivgtt

Response g_t





PROC MIXED call 0

iteration = 0



COVP1=0000117471

COVP2=-000244529 COVP3= 0128204017

COVP4= 000282178 COVP5=-0084482893 COVP6=00917336449

COVP7=00070584159


iteration = 4

beta1=0023464368 beta2=-1647843655 beta3=-3402938866

beta4=2788343188 beta5=-0213441218

COVP1= 000012790

COVP2=-000209429 COVP3 =0110625648

COVP4=0001486708 COVP5=-0031203057 COVP6=00535009216

COVP7=00064359279



Standard 95 CI



492 Galecki et al

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by [

Nor

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ORDER REPRINTS





DISCUSSION




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ORDER REPRINTS

APPENDIX A

THE NLINMIX MACRO



SYNTAX







494 Galecki et al

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ORDER REPRINTS














Dow

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ORDER REPRINTS












496 Galecki et al

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ORDER REPRINTS



EXECUTION


Initial Values






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ORDER REPRINTS



LME Step



EXAMPLE CODE





data fsivgtt1

----------------------------------------------------------------




----------------------------------------------------------------


cards

2 0 82 99333 8333

28 2 82 363000 112900

more data

run

498 Galecki et al

Dow

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at 2

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embe

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ORDER REPRINTS

macro useriml

----------------------------------------------------------------

-- --



-- --






-- --

----------------------------------------------------------------



time is a vector

t is a scalar


d=dd


p2=d[2] p_2

p3=d[3] p_3


g_t =y[1]

x_t =y[2]







g_1=y[3]

g_2=y[4] x_2=y[7]

g_3=y[5] x_3=y[8]

g_4=y[6]

c=p1+x_t

dg_1=-cg_1 - (g_t - G_b)

dg_2=-cg_2 - g_tx_2

dg_3=-cg_3 - g_tx_3

dg_4=-cg_4


Dow

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ORDER REPRINTS

dx_2=-p2x_2-x_t

dx_3=-p2x_3+idiff

return(dg_dtdx_dt

dg_1dg_2dg_3dg_4

dx_2dx_3)

finish glcd




dd=d


h1=10(-12)

h2=1

h3=10(-5)

h=h1||h2||h3


eps=10(-4)


glc_solv=t(c)t(res)

return(glc_solv)

finish glc_solv



statements omitted

finish interpolate



Derivs for p1p2p3G0


p1 = b[1] + u[1]

logp2 = b[2] + u[2]


G0 = b[4]

db = i(4) || j(410)

db[35] = (age-70)100

500 Galecki et al

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ORDER REPRINTS

du = 1 0 0

0 1 0

0 0 1

0 0 0




p2 = exp(logp2)

db[22] = p2 db[22]

du[22] = p2 du[22]

p3 = exp(logp3)

db[3] = p3 dlogp3b

du[3] = p3 dlogp3u

der_bu = db || du

return(p1p2p3G0)

finish bsm


return(der_bu)

finish


Vector of weights


timelt8 mins


w= g_weight(g_tg_t)

return(w)

finish

mend useriml


next = id

xvar = age


stmts = str(

class id




weight _weight_

)

model = str(




Dow

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embe

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14

ORDER REPRINTS


_d=bsm(_b_u)

der_bu=dbsm(_b_u)


d=_d




compartment


dpredv= jacs der_bu


)

derivs = str(



)




expand = zero


options = skipnlin

converge = 1e-3


)

ACKNOWLEDGMENTS


REFERENCES




502 Galecki et al

Dow

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by [

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at 2

004

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embe

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14

ORDER REPRINTS













Dow

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vers

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at 2

004

02

Dec

embe

r 20

14









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004

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embe

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14

ORDER REPRINTS





DISCUSSION




Dow

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vers

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at 2

004

02

Dec

embe

r 20

14

ORDER REPRINTS

APPENDIX A

THE NLINMIX MACRO



SYNTAX







494 Galecki et al

Dow

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ded

by [

Nor

th D

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a St

ate

Uni

vers

ity]

at 2

004

02

Dec

embe

r 20

14

ORDER REPRINTS














Dow

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ded

by [

Nor

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akot

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Uni

vers

ity]

at 2

004

02

Dec

embe

r 20

14

ORDER REPRINTS












496 Galecki et al

Dow

nloa

ded

by [

Nor

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Uni

vers

ity]

at 2

004

02

Dec

embe

r 20

14

ORDER REPRINTS



EXECUTION


Initial Values






Dow

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Uni

vers

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at 2

004

02

Dec

embe

r 20

14

ORDER REPRINTS



LME Step



EXAMPLE CODE





data fsivgtt1

----------------------------------------------------------------




----------------------------------------------------------------


cards

2 0 82 99333 8333

28 2 82 363000 112900

more data

run

498 Galecki et al

Dow

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at 2

004

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embe

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14

ORDER REPRINTS

macro useriml

----------------------------------------------------------------

-- --



-- --






-- --

----------------------------------------------------------------



time is a vector

t is a scalar


d=dd


p2=d[2] p_2

p3=d[3] p_3


g_t =y[1]

x_t =y[2]







g_1=y[3]

g_2=y[4] x_2=y[7]

g_3=y[5] x_3=y[8]

g_4=y[6]

c=p1+x_t

dg_1=-cg_1 - (g_t - G_b)

dg_2=-cg_2 - g_tx_2

dg_3=-cg_3 - g_tx_3

dg_4=-cg_4


Dow

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embe

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ORDER REPRINTS

dx_2=-p2x_2-x_t

dx_3=-p2x_3+idiff

return(dg_dtdx_dt

dg_1dg_2dg_3dg_4

dx_2dx_3)

finish glcd




dd=d


h1=10(-12)

h2=1

h3=10(-5)

h=h1||h2||h3


eps=10(-4)


glc_solv=t(c)t(res)

return(glc_solv)

finish glc_solv



statements omitted

finish interpolate



Derivs for p1p2p3G0


p1 = b[1] + u[1]

logp2 = b[2] + u[2]


G0 = b[4]

db = i(4) || j(410)

db[35] = (age-70)100

500 Galecki et al

Dow

nloa

ded

by [

Nor

th D

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Uni

vers

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at 2

004

02

Dec

embe

r 20

14

ORDER REPRINTS

du = 1 0 0

0 1 0

0 0 1

0 0 0




p2 = exp(logp2)

db[22] = p2 db[22]

du[22] = p2 du[22]

p3 = exp(logp3)

db[3] = p3 dlogp3b

du[3] = p3 dlogp3u

der_bu = db || du

return(p1p2p3G0)

finish bsm


return(der_bu)

finish


Vector of weights


timelt8 mins


w= g_weight(g_tg_t)

return(w)

finish

mend useriml


next = id

xvar = age


stmts = str(

class id




weight _weight_

)

model = str(




Dow

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ORDER REPRINTS


_d=bsm(_b_u)

der_bu=dbsm(_b_u)


d=_d




compartment


dpredv= jacs der_bu


)

derivs = str(



)




expand = zero


options = skipnlin

converge = 1e-3


)

ACKNOWLEDGMENTS


REFERENCES




502 Galecki et al

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ORDER REPRINTS













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ORDER REPRINTS

APPENDIX A

THE NLINMIX MACRO



SYNTAX







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ORDER REPRINTS














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ORDER REPRINTS












496 Galecki et al

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ORDER REPRINTS



EXECUTION


Initial Values






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ORDER REPRINTS



LME Step



EXAMPLE CODE





data fsivgtt1

----------------------------------------------------------------




----------------------------------------------------------------


cards

2 0 82 99333 8333

28 2 82 363000 112900

more data

run

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ORDER REPRINTS

macro useriml

----------------------------------------------------------------

-- --



-- --






-- --

----------------------------------------------------------------



time is a vector

t is a scalar


d=dd


p2=d[2] p_2

p3=d[3] p_3


g_t =y[1]

x_t =y[2]







g_1=y[3]

g_2=y[4] x_2=y[7]

g_3=y[5] x_3=y[8]

g_4=y[6]

c=p1+x_t

dg_1=-cg_1 - (g_t - G_b)

dg_2=-cg_2 - g_tx_2

dg_3=-cg_3 - g_tx_3

dg_4=-cg_4


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ORDER REPRINTS

dx_2=-p2x_2-x_t

dx_3=-p2x_3+idiff

return(dg_dtdx_dt

dg_1dg_2dg_3dg_4

dx_2dx_3)

finish glcd




dd=d


h1=10(-12)

h2=1

h3=10(-5)

h=h1||h2||h3


eps=10(-4)


glc_solv=t(c)t(res)

return(glc_solv)

finish glc_solv



statements omitted

finish interpolate



Derivs for p1p2p3G0


p1 = b[1] + u[1]

logp2 = b[2] + u[2]


G0 = b[4]

db = i(4) || j(410)

db[35] = (age-70)100

500 Galecki et al

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ORDER REPRINTS

du = 1 0 0

0 1 0

0 0 1

0 0 0




p2 = exp(logp2)

db[22] = p2 db[22]

du[22] = p2 du[22]

p3 = exp(logp3)

db[3] = p3 dlogp3b

du[3] = p3 dlogp3u

der_bu = db || du

return(p1p2p3G0)

finish bsm


return(der_bu)

finish


Vector of weights


timelt8 mins


w= g_weight(g_tg_t)

return(w)

finish

mend useriml


next = id

xvar = age


stmts = str(

class id




weight _weight_

)

model = str(




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ORDER REPRINTS


_d=bsm(_b_u)

der_bu=dbsm(_b_u)


d=_d




compartment


dpredv= jacs der_bu


)

derivs = str(



)




expand = zero


options = skipnlin

converge = 1e-3


)

ACKNOWLEDGMENTS


REFERENCES




502 Galecki et al

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ORDER REPRINTS













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ORDER REPRINTS














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ORDER REPRINTS












496 Galecki et al

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ORDER REPRINTS



EXECUTION


Initial Values






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ORDER REPRINTS



LME Step



EXAMPLE CODE





data fsivgtt1

----------------------------------------------------------------




----------------------------------------------------------------


cards

2 0 82 99333 8333

28 2 82 363000 112900

more data

run

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ORDER REPRINTS

macro useriml

----------------------------------------------------------------

-- --



-- --






-- --

----------------------------------------------------------------



time is a vector

t is a scalar


d=dd


p2=d[2] p_2

p3=d[3] p_3


g_t =y[1]

x_t =y[2]







g_1=y[3]

g_2=y[4] x_2=y[7]

g_3=y[5] x_3=y[8]

g_4=y[6]

c=p1+x_t

dg_1=-cg_1 - (g_t - G_b)

dg_2=-cg_2 - g_tx_2

dg_3=-cg_3 - g_tx_3

dg_4=-cg_4


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ORDER REPRINTS

dx_2=-p2x_2-x_t

dx_3=-p2x_3+idiff

return(dg_dtdx_dt

dg_1dg_2dg_3dg_4

dx_2dx_3)

finish glcd




dd=d


h1=10(-12)

h2=1

h3=10(-5)

h=h1||h2||h3


eps=10(-4)


glc_solv=t(c)t(res)

return(glc_solv)

finish glc_solv



statements omitted

finish interpolate



Derivs for p1p2p3G0


p1 = b[1] + u[1]

logp2 = b[2] + u[2]


G0 = b[4]

db = i(4) || j(410)

db[35] = (age-70)100

500 Galecki et al

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ORDER REPRINTS

du = 1 0 0

0 1 0

0 0 1

0 0 0




p2 = exp(logp2)

db[22] = p2 db[22]

du[22] = p2 du[22]

p3 = exp(logp3)

db[3] = p3 dlogp3b

du[3] = p3 dlogp3u

der_bu = db || du

return(p1p2p3G0)

finish bsm


return(der_bu)

finish


Vector of weights


timelt8 mins


w= g_weight(g_tg_t)

return(w)

finish

mend useriml


next = id

xvar = age


stmts = str(

class id




weight _weight_

)

model = str(




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ORDER REPRINTS


_d=bsm(_b_u)

der_bu=dbsm(_b_u)


d=_d




compartment


dpredv= jacs der_bu


)

derivs = str(



)




expand = zero


options = skipnlin

converge = 1e-3


)

ACKNOWLEDGMENTS


REFERENCES




502 Galecki et al

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ORDER REPRINTS













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ORDER REPRINTS












496 Galecki et al

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ORDER REPRINTS



EXECUTION


Initial Values






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ORDER REPRINTS



LME Step



EXAMPLE CODE





data fsivgtt1

----------------------------------------------------------------




----------------------------------------------------------------


cards

2 0 82 99333 8333

28 2 82 363000 112900

more data

run

498 Galecki et al

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ORDER REPRINTS

macro useriml

----------------------------------------------------------------

-- --



-- --






-- --

----------------------------------------------------------------



time is a vector

t is a scalar


d=dd


p2=d[2] p_2

p3=d[3] p_3


g_t =y[1]

x_t =y[2]







g_1=y[3]

g_2=y[4] x_2=y[7]

g_3=y[5] x_3=y[8]

g_4=y[6]

c=p1+x_t

dg_1=-cg_1 - (g_t - G_b)

dg_2=-cg_2 - g_tx_2

dg_3=-cg_3 - g_tx_3

dg_4=-cg_4


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ORDER REPRINTS

dx_2=-p2x_2-x_t

dx_3=-p2x_3+idiff

return(dg_dtdx_dt

dg_1dg_2dg_3dg_4

dx_2dx_3)

finish glcd




dd=d


h1=10(-12)

h2=1

h3=10(-5)

h=h1||h2||h3


eps=10(-4)


glc_solv=t(c)t(res)

return(glc_solv)

finish glc_solv



statements omitted

finish interpolate



Derivs for p1p2p3G0


p1 = b[1] + u[1]

logp2 = b[2] + u[2]


G0 = b[4]

db = i(4) || j(410)

db[35] = (age-70)100

500 Galecki et al

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ORDER REPRINTS

du = 1 0 0

0 1 0

0 0 1

0 0 0




p2 = exp(logp2)

db[22] = p2 db[22]

du[22] = p2 du[22]

p3 = exp(logp3)

db[3] = p3 dlogp3b

du[3] = p3 dlogp3u

der_bu = db || du

return(p1p2p3G0)

finish bsm


return(der_bu)

finish


Vector of weights


timelt8 mins


w= g_weight(g_tg_t)

return(w)

finish

mend useriml


next = id

xvar = age


stmts = str(

class id




weight _weight_

)

model = str(




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ORDER REPRINTS


_d=bsm(_b_u)

der_bu=dbsm(_b_u)


d=_d




compartment


dpredv= jacs der_bu


)

derivs = str(



)




expand = zero


options = skipnlin

converge = 1e-3


)

ACKNOWLEDGMENTS


REFERENCES




502 Galecki et al

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ORDER REPRINTS













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ORDER REPRINTS



EXECUTION


Initial Values






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ORDER REPRINTS



LME Step



EXAMPLE CODE





data fsivgtt1

----------------------------------------------------------------




----------------------------------------------------------------


cards

2 0 82 99333 8333

28 2 82 363000 112900

more data

run

498 Galecki et al

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ORDER REPRINTS

macro useriml

----------------------------------------------------------------

-- --



-- --






-- --

----------------------------------------------------------------



time is a vector

t is a scalar


d=dd


p2=d[2] p_2

p3=d[3] p_3


g_t =y[1]

x_t =y[2]







g_1=y[3]

g_2=y[4] x_2=y[7]

g_3=y[5] x_3=y[8]

g_4=y[6]

c=p1+x_t

dg_1=-cg_1 - (g_t - G_b)

dg_2=-cg_2 - g_tx_2

dg_3=-cg_3 - g_tx_3

dg_4=-cg_4


Dow

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ORDER REPRINTS

dx_2=-p2x_2-x_t

dx_3=-p2x_3+idiff

return(dg_dtdx_dt

dg_1dg_2dg_3dg_4

dx_2dx_3)

finish glcd




dd=d


h1=10(-12)

h2=1

h3=10(-5)

h=h1||h2||h3


eps=10(-4)


glc_solv=t(c)t(res)

return(glc_solv)

finish glc_solv



statements omitted

finish interpolate



Derivs for p1p2p3G0


p1 = b[1] + u[1]

logp2 = b[2] + u[2]


G0 = b[4]

db = i(4) || j(410)

db[35] = (age-70)100

500 Galecki et al

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ORDER REPRINTS

du = 1 0 0

0 1 0

0 0 1

0 0 0




p2 = exp(logp2)

db[22] = p2 db[22]

du[22] = p2 du[22]

p3 = exp(logp3)

db[3] = p3 dlogp3b

du[3] = p3 dlogp3u

der_bu = db || du

return(p1p2p3G0)

finish bsm


return(der_bu)

finish


Vector of weights


timelt8 mins


w= g_weight(g_tg_t)

return(w)

finish

mend useriml


next = id

xvar = age


stmts = str(

class id




weight _weight_

)

model = str(




Dow

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004

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embe

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14

ORDER REPRINTS


_d=bsm(_b_u)

der_bu=dbsm(_b_u)


d=_d




compartment


dpredv= jacs der_bu


)

derivs = str(



)




expand = zero


options = skipnlin

converge = 1e-3


)

ACKNOWLEDGMENTS


REFERENCES




502 Galecki et al

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ORDER REPRINTS













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ORDER REPRINTS



LME Step



EXAMPLE CODE





data fsivgtt1

----------------------------------------------------------------




----------------------------------------------------------------


cards

2 0 82 99333 8333

28 2 82 363000 112900

more data

run

498 Galecki et al

Dow

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embe

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ORDER REPRINTS

macro useriml

----------------------------------------------------------------

-- --



-- --






-- --

----------------------------------------------------------------



time is a vector

t is a scalar


d=dd


p2=d[2] p_2

p3=d[3] p_3


g_t =y[1]

x_t =y[2]







g_1=y[3]

g_2=y[4] x_2=y[7]

g_3=y[5] x_3=y[8]

g_4=y[6]

c=p1+x_t

dg_1=-cg_1 - (g_t - G_b)

dg_2=-cg_2 - g_tx_2

dg_3=-cg_3 - g_tx_3

dg_4=-cg_4


Dow

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ORDER REPRINTS

dx_2=-p2x_2-x_t

dx_3=-p2x_3+idiff

return(dg_dtdx_dt

dg_1dg_2dg_3dg_4

dx_2dx_3)

finish glcd




dd=d


h1=10(-12)

h2=1

h3=10(-5)

h=h1||h2||h3


eps=10(-4)


glc_solv=t(c)t(res)

return(glc_solv)

finish glc_solv



statements omitted

finish interpolate



Derivs for p1p2p3G0


p1 = b[1] + u[1]

logp2 = b[2] + u[2]


G0 = b[4]

db = i(4) || j(410)

db[35] = (age-70)100

500 Galecki et al

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ORDER REPRINTS

du = 1 0 0

0 1 0

0 0 1

0 0 0




p2 = exp(logp2)

db[22] = p2 db[22]

du[22] = p2 du[22]

p3 = exp(logp3)

db[3] = p3 dlogp3b

du[3] = p3 dlogp3u

der_bu = db || du

return(p1p2p3G0)

finish bsm


return(der_bu)

finish


Vector of weights


timelt8 mins


w= g_weight(g_tg_t)

return(w)

finish

mend useriml


next = id

xvar = age


stmts = str(

class id




weight _weight_

)

model = str(




Dow

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004

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embe

r 20

14

ORDER REPRINTS


_d=bsm(_b_u)

der_bu=dbsm(_b_u)


d=_d




compartment


dpredv= jacs der_bu


)

derivs = str(



)




expand = zero


options = skipnlin

converge = 1e-3


)

ACKNOWLEDGMENTS


REFERENCES




502 Galecki et al

Dow

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by [

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Uni

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at 2

004

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embe

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ORDER REPRINTS













Dow

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ORDER REPRINTS

macro useriml

----------------------------------------------------------------

-- --



-- --






-- --

----------------------------------------------------------------



time is a vector

t is a scalar


d=dd


p2=d[2] p_2

p3=d[3] p_3


g_t =y[1]

x_t =y[2]







g_1=y[3]

g_2=y[4] x_2=y[7]

g_3=y[5] x_3=y[8]

g_4=y[6]

c=p1+x_t

dg_1=-cg_1 - (g_t - G_b)

dg_2=-cg_2 - g_tx_2

dg_3=-cg_3 - g_tx_3

dg_4=-cg_4


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ORDER REPRINTS

dx_2=-p2x_2-x_t

dx_3=-p2x_3+idiff

return(dg_dtdx_dt

dg_1dg_2dg_3dg_4

dx_2dx_3)

finish glcd




dd=d


h1=10(-12)

h2=1

h3=10(-5)

h=h1||h2||h3


eps=10(-4)


glc_solv=t(c)t(res)

return(glc_solv)

finish glc_solv



statements omitted

finish interpolate



Derivs for p1p2p3G0


p1 = b[1] + u[1]

logp2 = b[2] + u[2]


G0 = b[4]

db = i(4) || j(410)

db[35] = (age-70)100

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ORDER REPRINTS

du = 1 0 0

0 1 0

0 0 1

0 0 0




p2 = exp(logp2)

db[22] = p2 db[22]

du[22] = p2 du[22]

p3 = exp(logp3)

db[3] = p3 dlogp3b

du[3] = p3 dlogp3u

der_bu = db || du

return(p1p2p3G0)

finish bsm


return(der_bu)

finish


Vector of weights


timelt8 mins


w= g_weight(g_tg_t)

return(w)

finish

mend useriml


next = id

xvar = age


stmts = str(

class id




weight _weight_

)

model = str(




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ORDER REPRINTS


_d=bsm(_b_u)

der_bu=dbsm(_b_u)


d=_d




compartment


dpredv= jacs der_bu


)

derivs = str(



)




expand = zero


options = skipnlin

converge = 1e-3


)

ACKNOWLEDGMENTS


REFERENCES




502 Galecki et al

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ORDER REPRINTS













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ORDER REPRINTS

dx_2=-p2x_2-x_t

dx_3=-p2x_3+idiff

return(dg_dtdx_dt

dg_1dg_2dg_3dg_4

dx_2dx_3)

finish glcd




dd=d


h1=10(-12)

h2=1

h3=10(-5)

h=h1||h2||h3


eps=10(-4)


glc_solv=t(c)t(res)

return(glc_solv)

finish glc_solv



statements omitted

finish interpolate



Derivs for p1p2p3G0


p1 = b[1] + u[1]

logp2 = b[2] + u[2]


G0 = b[4]

db = i(4) || j(410)

db[35] = (age-70)100

500 Galecki et al

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ORDER REPRINTS

du = 1 0 0

0 1 0

0 0 1

0 0 0




p2 = exp(logp2)

db[22] = p2 db[22]

du[22] = p2 du[22]

p3 = exp(logp3)

db[3] = p3 dlogp3b

du[3] = p3 dlogp3u

der_bu = db || du

return(p1p2p3G0)

finish bsm


return(der_bu)

finish


Vector of weights


timelt8 mins


w= g_weight(g_tg_t)

return(w)

finish

mend useriml


next = id

xvar = age


stmts = str(

class id




weight _weight_

)

model = str(




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ORDER REPRINTS


_d=bsm(_b_u)

der_bu=dbsm(_b_u)


d=_d




compartment


dpredv= jacs der_bu


)

derivs = str(



)




expand = zero


options = skipnlin

converge = 1e-3


)

ACKNOWLEDGMENTS


REFERENCES




502 Galecki et al

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ORDER REPRINTS













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ORDER REPRINTS

du = 1 0 0

0 1 0

0 0 1

0 0 0




p2 = exp(logp2)

db[22] = p2 db[22]

du[22] = p2 du[22]

p3 = exp(logp3)

db[3] = p3 dlogp3b

du[3] = p3 dlogp3u

der_bu = db || du

return(p1p2p3G0)

finish bsm


return(der_bu)

finish


Vector of weights


timelt8 mins


w= g_weight(g_tg_t)

return(w)

finish

mend useriml


next = id

xvar = age


stmts = str(

class id




weight _weight_

)

model = str(




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ORDER REPRINTS


_d=bsm(_b_u)

der_bu=dbsm(_b_u)


d=_d




compartment


dpredv= jacs der_bu


)

derivs = str(



)




expand = zero


options = skipnlin

converge = 1e-3


)

ACKNOWLEDGMENTS


REFERENCES




502 Galecki et al

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ORDER REPRINTS













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at 2

004

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embe

r 20

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ORDER REPRINTS


_d=bsm(_b_u)

der_bu=dbsm(_b_u)


d=_d




compartment


dpredv= jacs der_bu


)

derivs = str(



)




expand = zero


options = skipnlin

converge = 1e-3


)

ACKNOWLEDGMENTS


REFERENCES




502 Galecki et al

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ORDER REPRINTS













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ORDER REPRINTS













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ordinary differential equation pk/pd models using the sas macro nlinmix

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