The predict method can also be used for plotting smooth fitted curves by calculating fitted values at closely spaced concentrations. Figure 7.2.5 presents the individual fitted curves for all twelve plants using a total of 200 concentra-tions between 50 and 1000µL/L.
Ambient CO2 (uL/L)
CO2 uptake rate (umol/(m2 sec))
200 400 600 800 1000
-10 0 10 20 30 40 50
Mississippi Control Chilled Quebec Control Chilled
Figure 7.2.5: Individual fitted curves for the twelve plants in the CO2 uptake data based on theco2.fit2 object.
all the powerful analytical and graphical machinery present in S is simultane-ously available. The analyses of the dental data and CO2 uptake data illustrate some of the available features, but many other features are available.
The code presented here was developed to handle primarily repeated mea-sures data, i.e. data generated by observing a number of clusters repeatedly under varying experimental conditions. More general mixed effects models (e.g.
with different levels of nesting) can be analyzed using the functions described here, but the code will not be computationally efficient for that purpose.
There are several directions in which the software can be expanded to handle more general mixed effects models and/or incorporate other estimation tech-niques. These include, but are not limited to,
• Mixed effects models with autocorrelated cluster errors (Chi and Reinsel, 1989). The current version of the code only handles the i.i.d. case;
• More accurate approximations to the loglikelihood in the nonlinear mixed effects model (cf. chapter 5). These include Laplacian and Gaussian quadrature approximations to the integral that defines the likelihood of the data in the nonlinear mixed effects model. The current version uses an alternating algorithm suggested by Lindstrom and Bates (1990);
• Profiling methods (Bates and Watts, 1988) for deriving confidence regions on the parameters in the model and assessing the normality of the param-eter estimates. These methods are computationally intensive, especially for the nonlinear mixed effects model, and efficient programming is needed to make them feasible to use;
• Update methods for refitting the model when only small changes in the
original calling sequence are necessary. These methods are particular use-ful for model building, when several similar models are fitted sequentially;
• Methods for deriving confidence and prediction intervals for predicted val-ues.
We plan to incorporate all these features in future releases of the software to be contributed to the S collection at StatLib. The autocorrelation structure for the cluster errors has already been incorporated in an experimental version currently undergoing tests. Ccode to calculate Laplacian and Gaussian quadra-ture approximations to the integral in the nonlinear mixed effects has already been developed, but needs to be incorporated into theScode. For the profiling methods we plan to use a linear mixed effects approximation to the marginal density in the nonlinear mixed effects, suggested in Lindstrom and Bates (1990), to speed up the calculations.
Model Building in Mixed Effects Models
Model building in mixed effects models involves questions that do not have a parallel in (fixed effects) linear and nonlinear models. Some of these questions are:
• determining which effects should have an associated random component and which should be purely fixed;
• using covariates to explain cluster-to-cluster parameter variability;
• using structured random effects variance-covariance matrices (e.g. diago-nal matrices) to reduce the number of parameters in the model.
In this chapter we consider strategies for addressing these questions in the con-text of nonlinear mixed effects models, though most of the techniques described are also applicable to linear mixed effects models.
Any model building strategy is by nature iterative: a tentative model is initially fitted and modified to generate possibly better models (according to some goodness-of-fit criterion) and the process is repeated until no further im-provements are possible. In comparing alternative models one must also analyze the residuals from the fit, checking for departures from the assumptions in the model. It is also highly recommended that any model building analysis be done in conjunction with experts in the field of application of the model, to ensure the practical usefulness of the chosen model.
The use of the model building techniques described in this chapter is illus-trated through the analysis of four real data examples. These data sets are described in section 8.1. In section 8.2 we describe techniques that can be used to model the variance-covariance matrix of the random effects and to choose which random effects should be incorporated in the model. The use of covari-ates to model cluster-to-cluster parameter variability is considered in section 8.3.
Our conclusions are included in section 8.4.
8.1 Examples
We make extensive use of real data examples to illustrate the model building techniques presented in this chapter. We now introduce the data sets that will be used throughout this chapter.
8.1.1 Pine Trees
The pine trees growth data are described in Kung (1986). A total of 14 sources (seeds) of Loblolly pine were planted in the southern United States and the tree heights (in ft.) were measured at 3, 5, 10, 15, 20, and 25 years of age.
Figure 8.1.1 shows a plot of these data.
Age (years)
Height (feet)
5 10 15 20 25
10 20 30 40 50 60
a a
a
a
a
a
b b
b
b
b
b
c c
c
c
c
c
d d
d
d
d
d
e e
e
e
e
e
f f
f
f
f
f
g g
g
g
g
g
h h
h
h
h
h
i i
i
i
i
i
j j
j
j
j
j
k k
k
k
k
k
l l
l
l
l
l
m m
m
m
m
m
n n
n
n
n
n
Figure 8.1.1: Loblolly pine heights at different ages.
Kung (1986) used a logistic curve to model the trees’ growth, but an asymptotic regression model seems to explain the observed growth pattern better. We also tried the logistic model, the Gompertz model, the Morgan, Mercer, and Flodin model, and the Weibull type model (Ratkowsky, 1990), but the asymptotic regression gave the best overall fit. This model can be expressed as
f(t,φ) =φ1−φ2exp (−φ3t) (8.1.1) where t denotes the tree’s age, φ1 the asymptotic height, φ2 the difference be-tween φ1 and the height at age zero, andφ3 the growth rate.
8.1.2 Theophylline
The Theophylline data were described in section 5.1. We reproduce the plot of the data in Figure 8.1.2.
Time (hrs)
Concentration (mg/L)
0 5 10 15 20 25
0 2 4 6 8 10
a a
a a
a
a a
a a
a
a
b b
b b b
b b
b b
b
b c
c c
c
c c
c c
c
c
c
d d
d d d
d d
d d
d
d
e e
e e
e e
e e
e
e
e
f f
f f f
f f
f f
f
f g
g g
g g
g g
g g
g
g
h hh
h h h
h
h h
h
h
i i
i
i i
i i
i i
i
i j
j j
j j
j
j
j j
j
j
k k
k k
k k
k k
k k
k l
l l
l
l l
l
l l
l
l
Figure 8.1.2: Theophylline concentrations (in mg/L) of twelve patients over time.
We recall from section 5.1 that a first order compartment model with absorption in a peripheral compartment is used to represent the variation in the drug concentration with time. The model equation is reproduced next
Ct= DKka
Cl(ka−K)[exp (−Kt)−exp (−kat)] (8.1.2) where Ct is the observed concentration at time t (mg/L), t is the time (hr), D is the dose (mg/kg), Cl is the clearance (L/kg), K is the elimination rate constant (1/hr), and ka is the absorption rate constant (1/hr). In order to
ensure positivity of the rate constants and the clearance, the logarithms of these quantities can be used in (8.1.2), giving the reparametrized model
Ct= Dexp (lka+lK−lCl)
exp (lka)−exp (lK) (8.1.3)
× {exp [−exp (lK)t]−exp [−exp (lka)t]}
where lCl= log(Cl), lka= log(ka), and lK = log(K).
8.1.3 Quinidine
The third data set comes from a pharmacokinetics clinical study of the antiar-rhytimic drug Quinidine. A total of 361 Quinidine concentration measurements were made on 136 hospitalized patients under varying dosage regimens. Addi-tional data were collected on a set of nine covariates: age, height, weight, race, smoking status, ethanol abuse, congestive heart failure, creatinine clearance, and α-1-acid glycoprotein concentration. Some of these covariates varied for the same patient during the course of the study, while others remained con-stant. One of the main objectives of the study was to investigate relationships between the individual pharmacokinetics parameters and the covariates. A full description of the data can be found in Verme, Ludden, Clementi and Harris (1992). Statistical analyses of these data using alternative modeling approaches are given in Davidian and Gallant (1993) and Wakefield (1993).
The model that has been suggested for the Quinidine data is the one-compartment open model with first-order absorption. This model can be defined in a recursive way as follows.
Suppose that, at time t, the patient receives at dose dt and prior to that time the last dose was given at time t. The expected concentration, Ct, and
the apparent concentration in the absorption compartment, Cat are given by Ct = Ctexp [−K(t−t)] + Catka
ka−K {exp [−K(t−t)]−exp [−ka(t−t)]} Cat = Catexp [−ka(t−t)] +dt
V (8.1.4)
where V represents the apparent volume in distribution and ka and K are respectively the absorption and the elimination rate constants.
When a patient receives the same dose d at regular time intervals ∆, the model (8.1.4) converges to the so-called steady state model, where the expected concentrations are given by
Ct = dka V (ka−K)
$ 1
1−exp (−K∆) − 1
1−exp (−ka∆)
%
Cat = d
V [1−exp (−ka∆)] (8.1.5)
Patients considered to be in steady state conditions have concentrations modeled as above.
Finally, for a between-dosages time t, the model for the expected concentra-tion Ct, given that the last dose was received at time t, is identical to (8.1.4).
Using the fact that the elimination rate constant K is equal to the ra-tio between the clearance (Cl) and the volume of distribura-tion (V), we can reparametrize models (8.1.4) and (8.1.5) in terms of V, ka, andCl.
In order to ensure that the estimates of V, ka, and Cl are positive, we can rewrite models (8.1.4) and (8.1.5) in terms of lV = log(V), lka = log(ka), and lCl= log(Cl).
The initial conditions for the recursive model are C0 = 0 and Ca0 = d0/V, with d0 denoting the first dose received by the patient. It has been assumed
throughout the model definition that the bioavailability of the drug, i.e. the per-centage of the administered dose that reaches the measurement compartment, is equal to one.
8.1.4 CO
2Uptake
The last data set considered here is the CO2uptake data described in section 7.2.
The data, presented in Figure 8.1.3, consist of measurements of CO2 uptake (in µmol/m2s) for six Echinochloa crus-galli plants from Qu´ebec and six plants from Mississippi at seven different concentrations of ambient CO2. Half the plants from each type were chilled before the measurements were taken, while the other half stayed at room temperature.
Ambient CO2 (uL/L)
CO2 uptake rate (umol/(m2 sec))
200 400 600 800 1000
-10 0 10 20 30 40 50
Mississippi Control Chilled Quebec Control Chilled
Figure 8.1.3: CO2uptake rates (inµmol/m2s) for Qu´ebec and Mississippi plants of Echinochloa crus-galli, control and chilled at different ambient CO2 concen-trations.
The nonlinear mixed effects model used to describe the CO2 uptake as a function of the ambient CO2 concentration is
Uij =φ1i{1−exp [−φ2i(Cj−φ3i)]}+ij (8.1.6) whereUij denotes the CO2 uptake rate of theith plant at thejth CO2 ambient concentration;φ1i, φ2i, andφ3i denote respectively the asymptotic uptake rate, the uptake growth rate, and the maximum ambient CO2 concentration at which no uptake is verified for the ith plant; Cj denotes the jth ambient CO2 level;
and theij are i.i.d.error terms with distribution N(0, σ2). The main purpose of the study was to estimate the effect of the plant type (P) and the chilling treatment (T) on the parameters φ1, φ2, and φ3.