Nonlinear Mixed Effects Models, a Tool for …

1 Nonlinear Mixed Effects Models, a tool for analyzing repeated MeasurementsA Brief Tutorial Using SAS Software Peter Gaccione, Harvard School of Public Health, Boston, Blanchard, Frontier Science & Technology Research Foundation, Inc. Chestnut Hill, MAAbstractThe exponential increase in computing powerhas led analysts to consider statistical techniques which until recently were not evenconceived of. One area of exciting work is theuse of Nonlinear models to explain biologicallyderived data, particularly with the inclusion ofrandom, individual Effects . Programs foranalyzing Nonlinear Mixed Effects models arenow becoming more prevalent and available tothe general user, and for more general uses. This paper will outline this methodology, anddescribe an application using SAS Institute slatest contribution to the field, the NLINMIX macro.

2 Introduction repeated measurements are data whereindividuals have multiple measurements overtime or space. analyzing these data requiresrecognizing and estimating variability bothbetween and within individuals. Further, it isnot uncommon for the relationship between anexplanatory variable ( , time) and a responsevariable ( , drug concentration or growth) tobe Nonlinear in the parameters. Nonlinearmixed Effects models provide a tool foranalyzing repeated measurements data by takinginto consideration these two types of variabilityas well as the Nonlinear relationship between theexplanatory variable and the response Model Let y be the j = 1 to n responses for individualiji, and let there be i = 1 to m individuals. Thenonlinear function f (x , ) models theijirelationship between the response variable yij and the explanatory variable x.

3 The variabilityijbetween individuals is included in the model byallowing the p x 1 parameter vector to varyibetween individuals. Thus the mean responsefor the individual i at time j, given the parametervector is i E(y | ) = f (x , ) (1) ijiiji As described by Davidian and Giltinan(1993a; 1993b; 1995), the variability in y mayijbe a function of f (x , ). For example, theijiwithin patient variability may be proportional tothe power of the mean and modeled as Var(y | ) = f (x , ), (2) ijiiji22 with scale parameter and power parameter ,or more generally as Var(y | ) = g (f (x , ), ). (3)ijiiji22 Thus, the jth measurement on individual i canbe written as y = f (x , ) + g(f (x , ), ), (4) ijijiijiijwhere , is a random error term with mean zeroijand variance between individual variation is modeledbased on the assumption of individual specificsets of regression parameters.

4 This variability inthe parameters can be the result of systematicdependence on individual attributes (Davidianand Giltinan, 1993a; 1993b), , weight or may also vary due to unexplainedrandom variation, ( , individual variation orrun to run variation in an assay). To account forthis we can write the following model = A + B b , (5) ii iiStatistics, Data Analysis, and ModelingStatistics, Data Analysis, and Modelingwhere is the vector of typical values for thepopulation parameters, A is the design matrixicorresponding to the systematic portion of themodels for , b is a random vector with meanii0 and covariance matrix D, and B is the designimatrix associated with b defining the noiseiportion of the model for (Davidian andiGiltinan, 1995; Lindstrom and Bates, 1990).

5 Note the model for does not have to be When using Nonlinear Mixed Effects models, two types of inference may be distinguished,population and individual inference. Withpopulation inference, the interest focuses onestimation of typical values for the population, and D. Individual inference focuses on theparameters associated with a particularindividual, in which case would be theistatistic of interest. The Mixed Effects modelsmethodology is implemented here to improvethe inference by utilizing common informationacross subjects. For example, when consideringclearance as a possible parameter in describing adrug concentration time curve, the populationinference would be the estimation of the meanand variance of clearance for the populationbeing studied, while the individual inferencewould center on predicting the value ofclearance for a particular of Estimation Two types of estimation methods are quicklynoted here: individual estimation methods, andmethods based on linearization using a Taylorseries expansion.

6 Individual estimation methodsdepend on the ability to estimate from theidata from each subject; the estimated are thenithe data used to estimate , D, , and . Twomain linearization methods are popular: they useTaylor series expansions in the random effect b .iOne is a first order method proposed by Beal(1984) and Sheiner, Rosenberg and Melmon(1972), in which a Taylor series is taken aboutb set to 0 (the expected value). The second is aifirst order conditional method presented byLindstrom and Bates (1990), in which a Taylorseries is taken about b set to the conditionaliestimate of the between individual randomeffects (Roe, 1997). These methods areappealing in that they are applicable to sparsedata problems and the estimation methods usedparallel those of linear hierarchical models(Davidian and Giltinan, 1995).

7 NLINMIX macro NLINMIX was originally written to implementthe algorithm of Lindstrom and Bates (Wolfinger, 1997). Their estimation schemeiterates to convergence between a nonlinearleast squares step and linear Mixed model step,corresponding to steps in PROC NLIN and inPROC Mixed . The NLINMIX macro,however, only uses PROC NLIN to computestarting values, since solving the Mixed modelsequations corresponds to taking one Gauss-Newton step in a Nonlinear least squaresoptimization (Wolfinger, 1997). TheNLINMIX macro also includes two othermethods of estimation, the first order method,and a GEE method (Littell, Milliken, Stroup,and Wolfinger, 1996; Wolfinger, 1993). It islisted and documented in Littell, Milliken,Stroup and Wolfinger, 1996, and is available byanonymous ftp ( ), on the WorldWide Web, ( ) in the SASS upplemental Library, or from the SAST echnical Support date most of the applications of nonlinearmixed Effects modeling have been in the area ofpharmacokinetics, where the relationshipbetween drug concentration and time dependsupon an overall biological mechanism as well asindividual subject characteristics.

8 This createsan ideal setting for the consideration of bothfixed (population attributable) and random(subject specific) parameters. As mentionedabove, software developed to analyze such datahave taken several approaches. The applicationStatistics, Data Analysis, and ModelingStatistics, Data Analysis, and Modelingfor this paper is an analysis of data collected onstatement. Note that subject 2 has only twoAIDS patients receiving triple combinationobservations. If there are many subjects withantiviral treatment. Their viral load (V(t)) - isfew observations estimation and convergencequantified on days 2, 7, 10, 14, 21, 48, 84, 168,problems could result during an 336 after initiation of , Data Setup and ImplementationThe problem involves modeling viral load overthe fixed and random parameters.

9 One call istime as a function of four fixed and randommade to PROC NLIN to perform standardeffects. In particular, the model isnonlinear least squares estimation, in effectlog V(t) =exp(p )exp(-d t) + exp(p )exp(-d t) .10 11 22 p and p are viral loads in two compartments, d121and d are their rates of decline, and t is the time2in days after initiation of therapy. The programexpects data to be set up as repeated measures. For example, data for the first four patientswould look as follows:patid lgcopy day 1 2 1 7 1 8 1 16 1 22 1 29 1 57 1 91 2 2 2 7 3 2 3 7 3 14 4 7 4

10 9 4 14 4 21 4 28 4 56 When the SAS program for the analysis is runthe response variable (lgcopy) will bereferenced in the NLINMIX macro by thestatement response=lgcopy, and t in the modelwill be the day variable. The model functionitself is given to the macro in the pred=The NLINMIX macro implements a two stepprocess for the estimation of the variancecomponents of the model and the estimation ofgetting default starting values for the regressionparameters. (This call is optional and is skippedin this paper s example. The parms= statement allows the user to supply startingvalues.)