The SAEM algorithm is a stochastic algorithm which is used to estimate the ML estimates (Lavielle, 2014). This algorithm is implemented in a number of softwares including Monolix, NONMEM, MATLAB and R. This is an algo- rithm based on iterations and requires an initial guess of the PK parameters. The iteration k of SAEM comprises of three stages:
1. the simulation stage: for i =1,2,...,N, draw ψk
i from the conditional distribution P (ψi|yi; θk−1);.
2. stochastic approximation:
Update Qk−1(θ) according to Qk(θ) = Qk−1(θ) + γk(logp(y, ψk; θ) −
Qk−1(θ)), where γk represents a sequence of decreasing positive numbers
with γ1 = 1; and
3. maximisation step:
Update θk−1 as stated by θk = argmax Qk(θ).
For the SAEM algorithm to converge, the requirement is that P∞
k=1γk = ∞
and P∞
k=1γ 2
k < ∞ (Delyon et al., 1999). This condition is satisfied if, for
instance, γk decreases as 1/k. In order for SAEM to converge, the choice of
step-size (γk) is very important. Choosing smaller step-sizes ensure there is
almost a guarantee of convergence of the algorithm to the ML estimates.
3.11
Model Evaluation
It is very important to evaluate the performance of any model that is developed. In this, the model should be able to explain a phenomenon and the data used in the modelling process. Model evaluation, therefore, is concerned with whether the model best describes the observed data satisfactorily, whether the model is simple enough for extrapolation and whether the model could be used for
the reason it has been developed (Comets et al., 2010). It is essential to check whether the data are in agreement with the model and also if the model explains the data well (Lavielle, 2014). To be able to do this effectively, model diagnostics are used to choose the model that best describes the data and eliminate the models that are not able to reproduce the data (Comets et al., 2010). These diagnostic plots are able to explain whether the model addresses every relevant aspect of the data or if the model needs any further attention. It is also important to be able to conclude if the data best describe a one- or two-compartment model. A model selection process should be developed to select the best model. Whenever several models are valid, it is often desirable to select the model with the simplest assumptions. This selection process is often done using model diagnostics. Another process could be using selection tools to compute a criterion for comparing the models with one another. Some of the criteria which would be used include Akaike information criteria (AIC) and the Bayesian information criteria (BIC).
3.12
Model Diagnostics
The study examines several diagnostic plots and applies them to the data used in this dissertation. These model diagnostics plots would be used to select the best model among the different models which are used. The diagnostic plots which are used in this dissertation are discussed below.
3.12.1
Spaghetti Plot
This is a plot of the concentration time curve for all subjects plotted on the same panel. It is a plot that shows the effects of drugs on subjects after the administration of the drug. This plot can be used to track results of drugs amongst subjects. Spaghetti plot shows variability between individual concentration data at a given time. It gives a better picture of the variability between the individuals in the study. It is easy to select individuals who deviate from the central tendency with respect to half-life, absorption, distribution,
clearance,etc. using this plot.
3.12.2
Individual Fits
For the model defined in Equation 3.40, estimating the individual (ψi) and
population parameters ψpop enables the computation for each individual, the
predicted profile which is given by the estimated population model (f (t, ˆψpop)),
and the predicted profile which is given by the estimated individual model, where ˆψi is an estimate of ψi (Lavielle, 2014).
3.12.3
Observation vs Prediction
The population and individual models enable the calculation of the predic- tions f (tij; ˆψpop) for the population and f (tij; ˆψ) for each individual at the
observation times tij.
3.12.4
Residuals
There are a number of residuals but two of them are used in this disserta- tion. The individual weighted residuals (IWRES) and normalised prediction distribution errors (NPDE).
• IWRES are the estimates of the standardised residual (εij) which is based
on individual predictions:
IW RESij =
yij − f (tij; ˆψ)
g(tij; ˆψ)
.
Whenever the residuals are assumed to be correlated, it can be decorre- lated by multiplying each individual vector IW RESi = (IW RESij, 1 ≤
j ≤ ni) by
ˆ
R−1/2i , where ˆRi represents the estimated correlation matrix
of the vector of residuals (Lavielle, 2014).
• Population weighted residuals (PWRES) are defined as the normalised difference between observations and their mean. Let yi = (yij, 1 ≤
mean of yi is the vector E(yi) = (E(f (tij; ψi), 1 ≤ j ≤ ni). Let Vi
be the ni × ni variance-covariance matrix of yi. Then the ith vector of
population weighted residuals P W RESi = (P W RESij, 1 ≤ j ≤ ni)
is therefore defined as:
P W RESi = V −1/2
i (yi− E(yi)).
E(yi) and Vi are unknown but can be estimated by a Monte Carlo sim-
ulation.
• NPDE are a non parametric category of PWRES which depends on rank statistics (Lavielle, 2014). For any (i, j), let Fij = FP W RESij(P W RESij)
where FP W RESijis the cumulative distribution function (cdf) of PWRESij.
The NPDE are defined as the empirical estimates of Φ−1(Fij) where the
Fij are obtained using Monte Carlo simulation: a large number of repli-
cates y1, y2, ..., yk, of the original data yobs are drawn under the model and Fij estimated by:
ˆ Fij = 1 K K X k=1 1yk ij≤yobsij .
The NPDE are then defined as the empirical estimation of Φ−1(Fij), that is,
NPDEij=Φ−1( ˆFij).