saemix, an R version of the SAEM algorithm for parameter estimation in nonlinear mixed effect models

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saemix, an R version of the SAEM algorithm for

parameter estimation in nonlinear mixed effect models

Audrey Lavenu, Emmanuelle Comets, Marc Lavielle

To cite this version:

Audrey Lavenu, Emmanuelle Comets, Marc Lavielle. saemix, an R version of the SAEM algorithm

for parameter estimation in nonlinear mixed effect models. 1ères Rencontres R, Jul 2012, Bordeaux,

France. �hal-00717539�

for parameter estimation in nonlinear mixed ee t models A. Lavenu

a

, E. Comets

b

and M. Lavielle

c

a

University Rennes-I &INSERM CIC 0203 Rennes,Fran e

audrey.lavenuuniv-rennes1.fr

b

INSERM, UMR 738 &Univ Paris Diderot, Sorbonne ParisCité Paris, Fran e emmanuelle. ometsinserm.fr

c

INRIA Sa lay, Fran e Mar .Lavielleinria.fr

Mots lefs : Nonlinear mixed ee t models, parameter estimation, SAEM algorithm, R, R pa kage, pharma okineti s, pharma odynami s,longitudinal data.

Introdu tion: The use of modelling and simulation in lini al drug development is now well established. Regardless of whether a single out ome is onsidered at the end of the study, lini altrialsoften olle tlongitudinaldata,withea hsubje t providingseveral measurements throughoutthe study. Longitudinal dataisastapleinparti ular ofpharma okineti (PK) and pharma odynami (PD)studies,whi harearequired partofa newdrugappli ationle. Non-linearmixed ee t models an help to hara teriseand tounderstandmany omplex nonlinear biologi al pro esses, su h as biomarkers or surrogate endpoints, and are ru ial in des ribing and quantifying the me hanismsof drug a tionand the dierent sour es of variation,e.g., the interindividualvariability. Overthe pastde ade, new andpowerfulestimationalgorithmshave been proposed to estimate the parameters of these models. The Sto hasti Approximation Expe tation Maximization(SAEM) algorithmhas proven very e ient, qui kly onverging to the maximum likelihood estimators [1℄ and performing better than linearisation-based algo-rithms[2℄. It hasbeenimplemented intheMonolixsoftware [3℄whi hhas enjoyed in reasingly widespread use over the lastfewyears, morere entlyin theStatisti stoolbox ofMatlab (nlm-etsa.m), andisalsoavailableinNONMEM version7[4℄. The obje tiveof thepresentpa kage was toimplement SAEMin the R software [5℄.

Methods: Detailed and omplete presentations of the nonlinear mixed ee ts model an be found in several referen e textbooks, for instan e [6℄. We onsider the following general non-linear mixed ee ts model for ontinuous outputs:

y

ij

= f (x

ij

, ψ

i

) + g(x

ij

, ψ

i

, ξ)ε

ij

,

1 ≤ i ≤ N , 1 ≤ j ≤ n

i

(1)

where

y

ij

is the

j

th observation of subje t

i

,

N

is the number of subje ts,

n

i

is the number of observations of subje t

i

,

x

ij

are known regression variables, and

ψ

i

is the ve tor of indi-vidual parameters. The SAEM algorithm is used to obtain maximum likelihood estimates of the parameters of nonlinear mixed ee ts models withoutany linearisationof the model. The log-likelihoodfornonlinearmixedee t models isanalyti allyintra tablesin eitrequires

inte-usual E-step with a sto hasti approximation step [8℄. The missing parameters are simulated atea hiterationviaaMCMC pro edure, whi h anbeusedafterthe algorithmhas onverged toobtainthe onditionalmodes,the onditionalmeansandthe onditionalstandarddeviations of the individual parameters.

Results: Thelibrary usesthe S4 lasssystem ofR toprovideauser-friendlyinput andoutput system,with methodslikesummaryorplotforttedobje ts. Thepa kageprovidessummaries of the results, individual parameter estimates, standard errors (obtained using a linearised omputation of the Fisher information matrix) Wald tests for xed ee ts, and a number of diagnosti plots, in luding VPC plots and npde [9℄. The log-likelihood an be omputed by three methods: alinearisation ofthe model, animportan esamplingpro edure, oraGaussian quadrature. Thediagnosti graphs anbetailoredtotheuser'sindividualpreferen esbysetting a number of options,and are easily exported toa le.

Weillustratetheuseofthelibrarywith thewellknown PKdatasetoftheophylline. Thesedata in ludesthe on entration versus time data olle ted in12 subje ts given a singleoral dose of theophylline,and forwhom11bloodsampleswere olle tedoveraperiodof24h. Wemodelled this data using aone- ompartment model with rst-order absorption, parameterised ask

a

, V, CL. The IIV was modelled using an exponential model with diagonal varian e- ovarian e ma-trix,whiletheresidualvariabilitywasmodelledwitha ombinederrormodel. Many diagnosti plots are available to evaluate onvergen e or model adequa y, su h as individualplots, using a plotfun tion through whi h user-spe i options an be set.

Con lusion: The saemix pa kage provides the SAEM algorithm for R users. The urrent version handlesmodels in analyti alform, with ontinuous or binary ovariates.

Référen es

[1℄ Delyon B., Lavielle M., Moulines E. (1999). Convergen e of a sto hasti approximation version of the EM algorithm.Annals of Statisti s 27, 94128

[2℄ GirardP.,MentréF.(2005).A omparisonofestimationmethodsinnonlinearmixedee ts modelsusingablind analysis(oralpresentation).Meetingof thePopulationApproa hGroup in Europe (PAGE), Pamplona

[3℄ Lavielle M. (2010). MONOLIX (MOdèles NOn LInéaires á eets miXtes) User Guide. MONOLIX group, Orsay, Fran e. URL: http://software.monolix.org/

[4℄ Beal S., SheinerL.B.,Boe kmann A.,BauerR.J.(2009).NONMEM User's Guides. (1989-2009), I on DevelopmentSolutions, Elli ott City, MD, USA

[5℄ R Development Core Team (2006). R: A Language and Environment for Statisti al Com-puting. R Foundation for Statisti alComputing,Vienna, Austria

[6℄ DavidianM.,GiltinanD(1995).Nonlinearmodelsforrepeatedmeasurementdata.Chapman & Hall,London

[7℄ Dempster A. P., Laird N. M., Rubin D.B. (1977). Maximum likelihood from in omplete data viathe EM algorithm.Journal of the Royal Statisti al So iety Series B 39, 138 [8℄ Kuhn E., Lavielle M (2005). Maximum likelihood estimation in nonlinear mixed ee ts

models. ComputationalStatisti s and Data Analysis 49, 102038

[9℄ Brendel K., CometsE.,LaontC., LaveilleC.,Mentré M..Metri sfor externalmodel eval-uationwith anappli ationtothepopulationpharma okineti sof gli lazide(2006).