Testing the Need for a Random Effects Modelin in a Two Compartment Model

Thammasat Int. J. Sc. Tech., Vol. 9, No. 2, April-June 2004

Testing the Need for a RandomEffects Modelin in a Two

Compartment ModelKamon Budsaba

Department of Mathematics and Statistics, Thammasat University,Pathum Thani 12121

Charles E. SmithDepartment of Statistics, North Carolina State University,

Raleigh, NC 27695-8273

AbstractThe objective of this paper is to find a simple way to test whether random effects are needed in a

nonlinear mixed effects model. We proposed a test statistic, approximately an F random variable, fromthe fixed parameter approach which compares the residual sum of squares from the full model and thereduced model. From the difference of exponentials model simulations, the empirical size of the test isslightly higher than the nominal level a. The test offers very good power for detection. Theachieved power depends on the error variance, the population coefficient of variation (CV) of therandom effects, and the number of random effects in the model. For a fixed error variance, powerincreases as the population CV increases and/or the number of random effects increases. From oursensitivity analysis the performance of these test statistics is similar when the modei has fwo rateconstants that are almost equal, or when the model is close to a one compartment model.

Keywords: Compartment model, Difference of exponentials model, Fixed parameter approach,Nonlinear mixed effects mode

l.IntroductionA nonlinear mixed effects model is often

used to model repeated-measurement responsedata. In these types of studies, one is usuallyinterested in estimating the underlyingpopulation response curye. Individuals arerandomly sampled from the population as awhole, so the parameters could be considered asrandom effects. Variance-covariance parametersare also specified for these random effects.Treating the parameters as fixed and equal forall individuals implies the variances of theseparameters are zero. Therefore testing whetherrandom effects are needed is a test of whetherthe parameter's variance is non-zero. Theobjective of this research is to find a simpleapproximate way to test for non-zero variance-covariance parameters.

Under the null hypothesis, the true value ofa parameter is on the boundary of the parameterspace defined by the alternative hypothesis. Inthe case of the linear mixed effects model [],

where random effects are associated withparameters in the mean response only, Stramand Lee [2], applying results of Self and Liang[3], showed that the likelihood ratio test statisticfor non-zero variance components under theassumption of normality does not follow a 12distribution, but instead has the distribution ofamixture of ;X2 random variables. Morgan et al.[4] developed Stram and Lee's test by using adifferent approach namely reparameterizing thecovariance matrix of random effects. Theycompared the maximum likelihood approach toa fixed parameter approach. They found that theF test from the fixed parameter approach is anexact test given the normality assumption. It issimple to implement and invariant to thebehavior of the parameters not being tested.Power can be approximated by usingSatterthwaite's approximation. However, fromtheir simulations, the fixed parameter approachF test had slightly lower power than thelikelihood ratio test.

The same idea applies in principle to thecase of a nonlinear mixed effects model.However, the non-linearity of the model causesseveral complications, even under normalityassumptions; as discussed by Davidian andGiltinan [5]. Approximate methods depend onthe particular model used and may becomputationally intensive. Zeng and Davidian

[6] adapted a simple method for testing variancecomponents associated with random regressionparameters in the linear random coefficientmodels suggested by Swamy [7] and Morgan etal. [a]. They concluded that the approximatedtest based on 12 random variable achieves thecorrect cr, level and exhibits fairly good powerfor detection of moderate to large departuresfrom the homogeneity assumption.

2. A General NonJinear MixedEffects Model

Several different nonlinear mixed effectsmodels have been proposed in recent years(Lindstrom and Bates, [8]; Vonesh and Carter,

[9]; Davidian and Gallant, [0]; Wakefield,Smith, Racine-Poon and Gelfand, I l]; Pinheiroand Bates, [2];Davidian and Giltinan, [5]). Weslightly modified the model proposed inPinheiro and Bates [2]. This model is ahierarchical model. At the first stage the jth

observation on the ith individual is modeled as:

/ r = . f ( f ' ,x , j )+ s i j , i = 1, . . ' ,m; i = l , . . n i

where / is a nonlinear function of an individual

parameter vector /J. and the covariate vector y'',

eris a normally distributed noise term, rz is the

total number of individuals, and ni is the

number of observation in the ith individual. Forthe second stage, the individual parameter vectoris modeled as:

f i1 = ArP + Brb,, b, * (0, D)

where p is a p-dimensional vector of fixed

population parameters, b, is a q-dimensional

random effects vector associated with the ith

individual, I-' and B, are design matrices for

the fixed and random effects respectively, andD is a variance-covariance matrix. It is alsoassumed that observations made on different


individuals are independent and that the arrare

independent of b,.

The vector bl can be partitioned as

b, = (b,.,,br,, )' where br,,contains the r elements

that are of interest to test as fixed *d bt,t

contains the other (q-r) elements in the model.The matrix D canbe partitioned accordingly:

where D,, =Var(b,,), Dr, =Yar(b2.,), and

D,, =Cov(b,.,,br,).To test whether the variance

associated with the parameters in br., are zero,

the hypothesis can be written as:

H . : L D L ' = 0 ( l )

where L=(o,r_,r,r_,,,1^,). under the null

hypothesis, D, equal zero implies D* is zero.

3. Fixed Parameter ApproachThe fixed parameter approach treats the

parameters as fixed but different for everyindividual. Swamy [7] suggested this approachas a preliminary test as a diagnostic for whetherthe assumptions of the random coefficientregression model are reasonable. A model is fitusing,ordinary least squares (OLS) in which allparameters are estimated as different acrossindividuals. Then, a model in which theparameters are equal across individuals is fit byOLS. The residual sum of squares between thesefwo models is then compared and forms a test.We will follow the fixed parameter approach inthe non-linear model and investigate thisapproximate test with regard to its correct olevel and power.

Let SSEr denote the residual sum ofsquares when the model in which all parametersare different across individuals is fit, and has

ln,-mrdegrees of freedom. SSEn is the

residual sum of squares when the model inwhich all parameters are equal acrossindividuals is fit and has )n,

-r degrees of

freedom. Let Q : SSER - SSEp, the difference inthe residual sum of squares between these two

ln , , D, .1D = l " " I

lD,' D,,)

models. For testing the hypothesis in (l), thisapproach yields the test statistic of the form:

T c _ Q / r ( m - l )-S S E o / ) , n , - m r

This test is approximated as an F randomvariable with r(m-l) and ln,

-rzr degrees of

freedom under the null hypothesis.

4. Simulation StudyWe conducted simulations to evaluate the

performance of the test statistic from the fixedparameter approach by consider its empiricalsize and power under certain conditions. Thesimulation is based on the model:

/ i1 = A,{exp( -b1 , )

- exp( -d ,t u )} + eu ,i= 1, . . . ,8 ; j = 1, . . . ,23

where I, is normal with mean 1.5. b and d are

normal with mean 0.0065 and 0.044respectively. Ai,bi and di are considered

random effects with Pearson's correlationcoefficient (p) between any relevant pair ofrandom parameters is -.90, 0, or .90. Therandom effects are independent in thispreliminary study. The independent norma-random variables eo have mean zero and four

choices of variance, i.e. Ve : 5.50287 x 10-6, l0x Vs, 100 x Vs, and 1000 x V6. These errorterms are also independent of the randomeffects. The model and its parameters includingthe approximate value of the error termsvariance were generated based on a porcine skinflaps experiment. With these scenarios andseveral choices of the coefficients of variation(CV) of the random effects across individuals,1000 Monte Carlo replications were realized att ime (11. ; ) : {0,5, 10, 15, 30,45,60,75,90, 105,120, 150, lg0, 210, 240, 270,300, 330, 360,390, 420, 450, 480). When only one randomeffect is considered, the results are shown inTable L

The model we propose for the flux rateprofile of the porcine flaps experiment is the


difference of exponentials model [3]. Thismodel is a compartment model. Compartmentmodels are commonly used in pharmacokinetics,where the exchange of materials in biologicalsystems is studied. A system is divided intocompartments, and it is assumed that the rates offlow of drugs between compartments followfirst order kinetics. so that the rate oftransfer toa receiving compartment is proportional to theconcentration in the supplying compartment.The transfer coefficients, which are assumedconstant with respect to time, are called rateconstants.

The reciprocal of a rate constant is called atime constant. Our model has two constant rates(b and d). We also assume that d > b. Since d isgreater than b, this model can be considered as atwo compartment model with a faster absorptionconstant rates than elimination constant rates. Ais mathematically explained as a function of b,d, and an initial unobservable quantify of thesupplying compartment. This model allows theresponse to be zero at time zero.

The empirical size of the test slightlyincreases when the variance of the error termincreases. It is greater than level .05 but it is stillwithin three standard errors of the nominal levelWhen variance of the error terms equal V6 =

5.50287 x 10-6, the achieved power is very highand close to one in all random effects when CVis only .01. If the error variance is l0 x Vs, theachieved power is close to one when CV is .05.If the error variance is 100 x Vs the achievedpower is close to one when CV is .10. If theeiror variance is 1000 x Ve, the achieved poweris close to one when CV is .10 for only when Aor b is random. The information from 8individuals each at 23 time points is sufficient toallow detection ofslight departures from the nullhypothesis. The test offers very good power fordetection. Power increases as the population CVincreases as expected. The random effect d is theslowest effect to achieve a higher power with anincrease in the population CV, in contrast withthe random effect A that is the fastest effect toachieve the desired power.


Table 1. Simulation results : the proportion of times the null hypothesis is rejected at level 0.05(F.ss.zr,roo : 1.622) when a single random effect is considered at different levels of the population CVand variance of error terms.

Yar(er) Random Effect C V : O C V : , O I CV: .05 CV = . 1 0

Vo: 5.50287 x l0-" A(Median)b(Median)d(Median)

.068(1 .0r 8 )

.068(1 .018)

.068( l .018)

1 . 0(37.'733)

1 . 0(20.643)

.992(7.433)

1 . 0(922.s20)

1 . 0(498.480)

1 . 0(r62.31'�7'l

1 . 0(3709.266)

1 . 0(2023.392)

1 . 0(6s5.314)

l 0 x V s A(Median)b(Median)d(Median)

( l

( l

I

.0690l 8).0690r 8).069018)

.978(4.730)

.906(3.0s2)

.552( l .655)

1 . 0(92.742)

1 . 0(s0.875)

1 . 0(r 7. l 03)

1 . 0(36e. l 58)

1 . 0(203.448)

1 .0(66.047)

100 x Ve A(Median)b(Median)d(Median)

.070( 1 . 0 2 1 )

.070( l . 0 2 1 )

.070( l .021)

.345( 1 .418)

.215(1.226)

. l 1 50.082)

.999 1.0(10.27s) (37.768)

.993 1.0(s.es3) (2r.2r7)

.868 .991(2.656) (7.s51)

1000 x Ve A(Median)b(Median)d(Median)

.092(1.047)

.092(1.047)

.092(r.047)

. 1 0 8(1 .087)

.094(1.074)

.095( l .052)

.720(2.040)

.473(1 .s88)

. 1 9 8(1.220)

.980(4.e34)

.916(3. l 63)

.577(r.7 44)

Tables 2 and 3 show the preliminary studyof 1000 Monte Carlo replications when two ormore random effects are in the model and theyare negatively correlated, independent, orpositively conelated ( p = -.90, 0, .90

respectively) at the error variance 100 x Vs, and1000 x V6 , respectively. Not surprisingly, thepower increases when more random effectsterms are in the model. In Table 2, withp:O

and CV: .01, the power increases by.l57 (.502-.345) when A and b are both random, comparedto when only A is random (Table l). Similarly,the power increases by .056 (.401-.345) when Aand d are both random effects, compared towhen only effect A is random (Table 1). When bis the only random effect in the model, adding das an additional random effect slightly increasesthe power form .21 5 to .259 . Adding both b andd as random effects in the model that has only Aas a random effect increases the power about the

same amount as adding only b effect. Similarpattern when CV : .01 and .05 is presented inTable 3.

When random effects are highly negativelyor highly positively correlated, we observe thatpositive correlation between each random pairof effects causes the power to be lower than theindependent random effects case when (A,b),(b,d), and (A,b,d) are random but not for (A,d).Negative correlation between (A,b) and (b,d)causes the power to be higher than when theyare independent random but not for (A,d). Forexample, when CV=.01 the power when both Aand b are random is .165, .502, and .698,respectively, when p = .90, 0, and -.90. If both

A and d are random when CV: .01, the power is.547, .401, and .225, respectively, when p --.90,

0. and -.90. Similar results were obtained inTable 3 when the error variance is 1000 x Vo.

Table 2. Simulation results: the proportion of times the null .hypothesis is(F.ss,zr,roo :1.622) When more than one random effect are considered atpopulation CV and variance oferror term equal 100 x Ve.

rejected at level 0.05different levels of the

Random Effects CV: .01 CV: .05 CV: .10p : . 9 0 A,b

(Median)A,d(Median)b,d(Median)A,b,d(Median)

. 165(1 .201)

.547(r.707)

.193(r.228)

.301f l .335)

.994( 5 . 1 6 1 )

1 . 0( l 7.085)

.997(s.762)

1 . 000.041)

1 . 0(r7.826)

1 . 0(6s.04s)

1 . 0(20.s71)

1 . 0(37. l 86)

p : 0 A,b(Median)A,d(Median)b,d(Median)A,b,d(Median)

.502(r.623)

.401( 1.486)

.259(r.2e8)

.546(r.677)

1 . 0( l s . l 6 6 )

1 . 0(r2.t37)

1 . 0(7.674)

1 . 0(17.090)

1 .0(57.e10)

.999(45. l e l )

1 . 0(28.0se)

1 . 0(66.1 05)

p : - .90 A,b(Median)A,d(Median)b,d(Median)

.698(2.032)

.22s(r.267)

.303(1 .355)

1 . 0(25.631)

.994(6.72r)

1 . 0(9. l 89)

1 . 0(100.295)

1 . 0(23.288)

1 . 0(34.781)

5. Sensitivity AnalysisThe performance of the approximate F-

test was also evaluated at other different valuesof the absorption rate parameter d. First, whend* : d/2 : 0.022 and secondly when d* : 4 x d :0.176. Since b : .0065 for both cases. then thefirst model can be considered as a model wheretwo constant rates are almost equal (a Gammamodel when two constant rates are equal), andthe second model can be considered as a modelthat comes close to the one compartment model(a model when d approaches to infinity).Simulations were conducted at the errorvariance 10xVe for both cases. When only one

random effect is considered, the empirical sizeand power are shown in Table 4.

The empirical size of the test for bothmodels is about .07. The achieved power is veryhigh and close to one in all random effects whenCV is .05. Power increases as the CV increases.The random effect d is the slowest effect toachieve a higher power with an increase in CVfor both models. The random effect A is thefastest effect to achieve the desired power forthe model with d* : 0.176. Both A and b arethe quickest to achieve the desired power for themodel with d* :0.022.


Table 3. Simulation results: the proportion of times the null hypothesis is rejected at level 0.05

(F.ss,zr.roo = 1.622) when more than one random effect are considered at different levels of the

population CV and variance of error terms equal 1000 x Vs.

Random Effects CV: .01 CV: .05 CV = .10

p : ,90 A,b(Median)A,d(Median)b,d(Median)A,b,d(Median)

. 103(1.073)

.12' l(1.126)

.100(r.014)

. 1 0 8(1.072)

.410( 1 .s07)

.864(2.767)

.456(1.54e)

.689( 1.945)

.903(2.786)

.993(7.826)

. 9 1 8(3.048)

.988(4.793)


.126114). l 1 6103). 1 0 1078). l l 9r22)

.846(2.s6r)

.765(2.20e)

.576( r .7s2)

.896(2.748)

.998(7.030)

.993(5.6e3)

.980(3.863)

1 .0(66. l 05)

( l

( l

( r

p: - .90 A,b(Median)A,d(Median)b,d(Median)

.143(1 . l 57)

. 1 0 1(1.076)

. 1 0 3(1.0e6)

.945(3.63e)

.530(1 .688)

.661(r.e62)

1 . 0(tr.473)

.939(3.466)

.984(4.551)

Table 4. Simulation results: the proportion of times the null hypothesis is rejected at level 0.05

(F.ss,zr,roo :1.622) when a single random effect is considered at different levels of the population CV

and variance of error terms equal I 0 x Ve for d* : 0.022, and d* : 0. I 76

d'l' Random Effects C V = 0.067

( 1 . 0 1 8 ).067

( 1 . 0 1 8 ).06't

C V : . O I CV = .05 C V =

(198.4

:1II

l 0.07).0

0.022 A(Median)b(Median)d

.904(3.038)

.91 5(3.026)

.715

1 . 0(50.682)

1 . 0(s0.7s2)

1 . 0(202.s34)

1.0Median 1 . 0 1 8 046 . 1 8 5 102.

0 .176 A(Median)b(Median)d

.068(1.03s)

.068(1.03s)

.068

.992(7.48s)

.912(3.032)

. ) z+

1.0(159.e61)

1 .0(s0.8se)

.998

1.0(640.0e5)

1.0(202.788)

.998Median 1.035 1.385 .770 36.470


Table 5. Simulation results: the proportion of times the null hypothesis is rejected at level 0.05(F.ss.ur,'eo : 1.622) when more than one random effect are considered at different levels of thepopulation CV and variance of error terms equal l0 x V6 for d* : 0.022.

Random Effects CV: .01 CV: .05 CV = .10p : . 9 0 A,b


.602( l .786)

.990(6 .411)

. 8 1 0(2.404)

.946(3.376\

1 . 0( le.78e)

1 . 0(r32.s28)

1 . 0(34.s78)

1 . 0(60.273)

1 . 0(76. r 30)

1 . 0(s26.178)

1 . 0(r3s.e4e)

1 . 0(238.023\


.989(5 .014)

.964(4.106)

.981(4.076)

.996(6 . r l6 )

1 . 0(e8.288)

1 . 0(77.407)

1 . 0(76.4t6)

1 . 0(r25.743)

1 .0(3es.e6l)

1 . 0(310.201)

1 . 0(30s.277)

1 .0G99.374\


.992(8.2e5)

.593( r .774)

.992(s.s7 4)

1 . 0(181.673)

1 . 0(19 .415)

1 . 00 1 6 . 2 0 1 )

1 . 0(726.r13)

1 .0(7s.66r)

1 . [email protected]\

Tables 5 and 6 show the power when twoor more random effects are in the model andthey are negatively, independent, or positivelycorrelated (p: -.90,0,.90 respectively) at theerror variance l0xV6 for d* = 0.022 and d* :0.176 respectively. The power increases whenmore random effect terms are in the models.When random effects are highly negatively orhighly positively correlated, we observe thatpositive correlation between each random pairsof effects causes the power to be lower than the

independent random effects case when (A,b),(b,d), and (A,b,d) are random but not for (A,d)for both models. Negative correlation between(A,b) and (b,d) causes the power to be higherthan when they are independent random but notfor (A,d) for both models. For example, whenCV = .01 and d* :0.022, the power when bothA and b are random is .602, .989, and .992,respectively, when p= -.90, 0, .90. Similarresults were obtained in Table 6 when d* =0 .176 .


Table 6. Simulation results: the proportion of times the null hypothesis is rejected at level 0.05(F.ss,zr.roo = 1.622) when more than one random effect are considered at different levels of thepopulation CV and variance of error terms equal l0 x Vs for d* : 0. 176.

Random Effects CV = .01 CV: .05 CV = .10p : . 9 0 A,b


.987(4.es7)

.998(e.304)

.929(3.232)

.997(6.597\

1 . 0(ee.270)

1 . 0(205.1 l4)

1 . 0(56.e03)

1 . 0(141.r52)

1 .0(3e4.322)

1 . 0(815.640)

1 . 0(222.9r0)

1 . 0(560.222\

p : 0 A,b(Median)A,d(Median)b,d(Median)A'b,d(Median)

1 . 0(e.s6t4)

.998(7.er3)

.956(3.346)

.99900. l l4 )

1 . 0(213.113)

1 . 0( l6e .8e l )

1 . 0(60.7s3)

1 . 0(222.472\

1 . 0(8s4.044)

1 . 0(687.277)

1 . 0(241.080)

1 . 0(901.780)


.999(14.075)

.991(6.3e6)

.9488.466)

1 . 0(329.347)

1 . 0(r32.766)

1 . 0(61 .865)

1 .0(1330.202)

1 . 0(s25.s88)

1 . 0(249.087)

6. ExampleWe applied the method we proposed to the

methyl salicylate data (MS). 400 1tg / cm2 ofto C - MS in ethanol were topically applied to8 isolated perfused porcine skin flaps andexperiments terminated at 8 hrs. Perfusate wascol lected over t ime (5, 10, 20, 30,45,60,75,90,105, 120 minutes and then every 30 minutesuntil termination of the experiment). Perfusateflux profiles were fitted to an exponentialdifference model,

! i1 = A,(exp( -b,t,i ) - exp( -d / u D + €,i .

We performed the test statistic from 5 flaps forthe final analysis since three flaps are outliers.Prior to analysis, time was converted to hoursand percent of dose was multiplied by 100.

The individual estimates are shown inTable 7 and the profile fitting is shown inFigure 1.

Table 7. Parameter estimates for each flap of 8hr. MS data.

Flap A b aI2J

45

1 . 0 5 1 61.62301.73461.76421.7109

0.30070.33970.44140.30760.2978

3.60953.222010.14355.69089.4859

The approximate F test statistic is18.419 with p-value close to 0since F,.or.,r.,rr, :1.850. The result suggests

that a random effects model is needed forthese data under model assumptions.


Figure 1. Individual profile fitting for 5 replications for 8 hr MS data (original scale)

7. Conclusion and DiscussionTo model and to estimate an underlying

population response curye from repeated-measurement response data, a nonlinear mixedeffects model is often used. Its parametersmight be considered as random effects sinceindividuals are randomly sampled from thepopulation. A simple way to test whetherrandom effects are needed is therefore of interestto investigate. We proposed a test statistic,approximately an F random variable, from thefixed parameter approach which compares theresidual sum of squares from the full model (themodel in which all parameters are differentacross individuals) and the reduced model (themodel in which all parameters are equal acrossindividuals).

From the difference of exponentials modelsimulations, the empirical size of the test isslightly higher than the nominal level a. Thetest offers very good power for detection. Theachieved power depends on the error variance,the population coefficient of variation (CV) ofthe random effects. and the number of random

effects in the model. For a fixed error variance,power increases as the population CV increaseand/or the number of random effects increases.The random effect d is the slowest and therandom effect A is the fastest parameter toachieve a higher power with an increase in CVwhen only one random effect is considered.When more than one independent randomeffects are considered, adding parameter d as anadditional random effect slightly increases thepower. High positive or negative correlationbetween each pair of effects causes the power tobe lower or higher respectively, than theindependent random effects case except for thepair (A,d). From our sensitivity analysis theperformance of these test statistics is similarwhen the model has two rate constants that arealmost equal, or when the model is close to aone compartment model. However, now A andb are the quickest to achieve a higher power forthe approximate gamma model.

Individual Profile Fitting for 8 hr MS Data

0.02

0 . 0 1 5

oot o.o1s

0.005

200 300 400

Time (Mins)

8. References

[1] Laird, N. and Ware, J.H., Random-EffectsModels for Longitudinal Data, Biometrics,Vol. 38, pp. 963-97 4, 1982.

l2l Stram, D. O. and Lee, J.W., VarianceComponents Testing in the LongitudinalMixed Effects Model, Biometrics, Vol. 50,p p . l l T l - 1 1 1 7 , 1 9 9 4 .

t3l Sell S. G. and Liang, K.-Y., AsymptoticProperties of Maximum LikelihoodEstimators and Likelihood Ratio TestsUnder Non-Standard Conditions, Journal ofthe American Statistical Association, Vol.82, pp.605-610, 1987.

[4] Morgan, B., Pantula, S. and Gumpertz,M.,Testing whether Coefficients are Randomin a Linear Random Coefhcient RegressionModel, Proceedings of the BiometricsSection of the American StatisticalAssociation, pp.1 5-24, 1995.

[5] Davidian, M. and Giltinan, D. M., Non-linear Models for Repeated MeasurementData, Chapman and Hall, London,1995.

[6] Zeng, Q., and Dvidian, M., TestingHomogeneity of Intra-Run VarianceParameters in Immunoassay, Statistics inMedicine, Vol. 16, pp. 17 65'177 6, 1997.

Ul Swamy, P. A. V. B., Statistical Inference inRandom Coefficient Regression Models,Springer-Verlag, Berlin, 197 l.


[8] Lindstrom, M. J., and Bates, D. M., Non-linear Mixed Effects Models for RepeatedMeasures Data, Biometrics, Vol. 46, pp.613-687.1990.

[9] Vonesh, E. F., and Carter, R. L., MixedEffects Nonlinear Regression forUnbalanced Repeated Measures,Biometrics, Vol.48, pp. l-18, 1992.

[0] Davidian, M., and Gallant, A. R., SmoothNonparametric Maximum LikelihoodEstimation for. PopulationPharmacokinetics, with Application to

Quinidine, Journal of PharmacokineticsandBiopharmaceutics, Vol. 20, pp. 529-556,1992.

I l ]Wakef ie ld, J . C. , Smith, A. F. M. , Racine-Poon, A., and Gelfand, A. 8., BayesianAnalysis of Linear and NonlinearPopulation Models Using the GibbsSampler, Applied Statistics, Vol. 43, pp.201-221,1994.

[2] Pinheiro, J.C., and Bates, D.M., ModelBuilding in Nonlinear Mixed EffectsModels, Proceedings of theBiopharmaceutical Section of the AmericanStatistical Association, pp. 1-8, 1994.

[3] Bates, D. M. and Watts, D. G., NonlinearRegression Analysis and Its Applications,Wiley, New York, 1988.

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Testing the Need for a Random Effects Modelin in a Two Compartment Model

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