Fixed Effect Parameter Estimates

For inferences regarding the fixed effect parameters in the model, one common method to assess the significance of the estimates is to use a Z-test in which the parameter estimate (p) is divided by its asymptotic standard error [SE(p)] and compared to a Z-distribution 

Z-tests are valid for large sample sizes, but are typically biased upwards, resulting in inflated Z-scores and Type I error rate, because of their underestimation in the variability of 3. Note that the covariance matrix C [Eq. (6.37)] uses an estimate of G and R in its calculation and that no correction in Z is made for the uncertainty in G and R. Rather than adjust for the inflation, most software packages account for the bias by comparing Z to a t-distribution, which has wider tails than a Z-distribution, and adjust the degrees of freedom accordingly. 

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FIGURE 12.1 Comparison of bias in fixed effect parameter estimates for the PFIM-4, IBR-4A, and IBR-4B designs (see text for description of the designs). 

Recalling that one cannot use zero as a starting estimate for the transformation parameter (A=theta(4>), the new model (see c6.txt r6.txt) has an objective function value that is 16 points lower than the original model that we started with (ci. TXT Ri. TXT), similar fixed effects parameter estimates, but substantially reduced skewness in the density of the random effect on K (see Figure 28.3). 

Table 7.5 Fixed effect parameter estimates and variance components from a 4-compartment model fit using data in Table 7.4.

The PPK approach estimates the joint distribution of population specific pharmacokinetic model parameters for a given drug. Fixed effect parameters quantify the relationship e.g. of clearance to individual physiology like function of liver, kidney, or heart. The volume of distribution is typically related to body size. Random effect parameters quantify the inter-subject variability which remains after the fixed effects have been taken into account. Then the observed concentrations will still be randomly distributed around the concentration time course predicted by the model for an individual subject. This last error term called residual variability... 

Therefore if one has an estimate of clearance and volume of distribution the plasma concentration can be predicted at different times after administration of any selected dose. The quantities that are known because they are either measured or controlled such as dose and time/ are called fixed effects/" in contrast to effects that are not known and are regarded as random. The parameters CLe and are called fixed-effect parameters because they quantify the influence of the fixed effects on the dependent variable/ Cp. 

NONMEM is a one-stage analysis that simultaneously estimates mean parameters, fixed-effect parameters, interindividual variability, and residual random effects. The fitting routine makes use of the EES method. A global measure of goodness of fit is provided by the objective function value based on the final parameter estimates, which, in the case of NONMEM, is minus twice the log likelihood of the data (1). Any improvement in the model would be reflected by a decrease in the objective function. The purpose of adding independent variables to the model, such as CLqr in Equation 10.7, is usually to explain kinetic differences between individuals. This means that such differences were not explained by the model prior to adding the variable and were part of random interindividual variability. Therefore, inclusion of additional variables in the model is warranted only if it is accompanied by a decrease in the estimates of the intersubject variance and, under certain circumstances, the intrasubject variance. 

If the number of concentration measurements per subject were smaller than twice the number of fixed effects parameters and the standard errors of the parameter estimates were not provided, then some downweighting would be necessary. It is possible to compute the expected standard errors for any given study (e.g., see Retout et al. (23), but this is beyond the scope of this chapter). 

The application of the empirical approaches yielded interesting and informative findings. For instance, Al-Banna et al. (3) found that the accuracy and precision of random effect parameter estimates from PPK studies improved dramatically when the number of sampling time points for each subject was increased by a single observation beyond the minimum number of 2 required to estimate the individual parameters in the open one-compartment intravenous (IV) bolus model they examined. They examined several three-sample point designs in which the first and the second time points were fixed, while the third time point was varied. They found that the exact location of the third time point was not critical to parameter estimation. 

Uncertainty for random effects was defined as high for all evaluations. The high level of uncertainty for random effects represented a worst case scenario, but also was representative of the greater uncertainty in these parameters at this stage in development relative to the fixed effects parameters. In an actual trial simulation, prior estimates of these uncertainties could be used, and possibly inflated to accommodate the additional parameter uncertainty associated with extrapolation to a new trial scenario or population. To illustrate a local SA, the parameter describing the maximum drug effect on the hazard parameter (ZDVSL) was fixed at values ranging from 0.25 to 1.0. 

In most cases, the fixed effect parameters are the parameters of interest. However, adequate modeling of the variance-covariance structure is critical for assessment of the fixed effects and is useful in explaining the variability of the data. Indeed, sometimes the fixed effects are of little interest and the variance components are of primary importance. Covariance structures that are overparameterized may lead to poor estimation of the standard errors for estimates of the fixed effects (Altham, 1984). However, covariance matrices that are too restrictive may lead to invalid inferences about the fixed effects because the assumed covariance structure does not exist and is not valid. For this reason, methods need to be available for testing the significance of the variance components in a model. 

Three parameters thus need to be estimated, namely the scalar factor a, the compression factor c, and the shift d. Parameter b was dropped for two reasons (1) the effect of this exponent is to be explored, so it must remain fixed during a parameter-fitting calculation, and (2) the parameter estimation decreases in efficiency for every additional parameter. Therefore the model takes on the form... 

The process rnust be iterated until convergence and the final estimates are denoted with Plb, bi,LB, and colb- The individual regression parameter can be therefore estimated by replacing the final fixed effects and random effects estimates in the function g so that ... 

Occasionally it is convenient to refer to the p function in (11.21), but generally the form (11.22) is used in robust M-estimation. The use of the t(r form is due to Hampel s concept of the influence function (Hampel et al., 1986). According to the IF concept, the value of it represents the effect of the residuals on the parameter estimation. If iff is unbounded, it means that an outlier has an infinite effect on the estimation. Thus, the most important requirement for robustness is that iff must be bounded and should have a small value when the residual is large. In fact, the value of the iff function corresponds to the gross error sensitivity (Hampel etal., 1986), which measures the worst (approximate) influence that a small amount of contamination of fixed size can have on the value of the estimator. 

The art of experimental design is made richer by a knowledge of how the placement of experiments in factor space affects the quality of information in the fitted model. The basic concepts underlying this interaction between experimental design and information quality were introduced in Chapters 7 and 8. Several examples showed the effect of the location of one experiment (in an otherwise fixed design) on the variance and co-variance of parameter estimates in simple single-factor models. 

In the panel data models estimated in Example 21.5.1, neither the logit nor the probit model provides a framework for applying a Hausman test to determine whether fixed or random effects is preferred. Explain. (Hint Unlike our application in the linear model, the incidental parameters problem persists here.) Look at the two cases. Neither case has an estimator which is consistent in both cases. In both cases, the unconditional fixed effects effects estimator is inconsistent, so the rest of the analysis falls apart. This is the incidental parameters problem at work. Note that the fixed effects estimator is inconsistent because in both models, the estimator of the constant terms is a function of 1/T. Certainly in both cases, if the fixed effects model is appropriate, then the random effects estimator is inconsistent, whereas if the random effects model is appropriate, the maximum likelihood random effects estimator is both consistent and efficient. Thus, in this instance, the random effects satisfies the requirements of the test. In fact, there does exist a consistent estimator for the logit model with fixed effects - see the text. However, this estimator must be based on a restricted sample observations with the sum of the ys equal to zero or T muust be discarded, so the mechanics of the Hausman test are problematic. This does not fall into the template of computations for the Hausman test. 

In most models developed for pharmacokinetic and pharmacodynamic data it is not possible to obtain a closed form solution of E(yi) and var(y ). The simplest algorithm available in NONMEM, the first-order estimation method (FO), overcomes this by providing an approximate solution through a first-order Taylor series expansion with respect to the random variables r i,Kiq, and Sij, where it is assumed that these random effect parameters are independently multivariately normally distributed with mean zero. During an iterative process the best estimates for the fixed and random effects are estimated. The individual parameters (conditional estimates) are calculated a posteriori based on the fixed effects, the random effects, and the individual observations using the maximum a posteriori Bayesian estimation method implemented as the post hoc option in NONMEM [10]. 

The advantages of this method are its simplicity, familiarity, and the fact that it can be used with sparse data and differing numbers of data points per individual. The disadvantages are that it is not possible to determine the fixed-effect sources of inter individual variability, such as creatinine clearance (CLcr). It also cannot distinguish between variability within and between individuals, and an imbalance between individuals results in biased parameter estimates. 

Population pharmacokinetic analysis provides not only an opportunity to estimate variability, but also to explain it. Variability is usually characterized in terms of fixed and random effects. The fixed effects are the population average values of pharmacokinetic parameters, which may in turn be a function of patient characteristics discussed above. The random effects... 

The first attempt at estimating interindividual pharmacokinetic variability without neglecting the difficulties (data imbalance, sparse data, subject-specific dosing history, etc.) associated with data from patients undergoing drug therapy was made by Sheiner et al. " using the Non-linear Mixed-effects Model Approach. The vector 9 of population characteristics is composed of all quantities of the first two moments of the distribution of the parameters the mean values (fixed effects), and the elements of the variance-covariance matrix that characterize random effects.f " " ... 

Selecting a mixed effects model means identifying a structural or mean model, the components of variance, and the covariance matrix for the residuals. The basic rationale for model selection will be parsimony in parameters, i.e., to obtain the most efficient estimation of fixed effects, one selects the covariance model that has the most parsimonious structure that fits the data (Wol-finger, 1996). Estimation of the fixed effects is dependent on the covariance matrix and statistical significance may change if a different covariance structure is used. The general strategy to be used follows the ideas presented in... 

The linear mixed effect model assumes that the random effects are normally distributed and that the residuals are normally distributed. Butler and Louis (1992) showed that estimation of the fixed effects and covariance parameters, as well as residual variance terms, were very robust to deviations from normality. However, the standard errors of the estimates can be affected by deviations from normality, as much as five times too large or three times too small (Verbeke and Lesaffre, 1997). In contrast to the estimation of the mean model, the estimation of the random effects (and hence, variance components) are very sensitive to the normality assumption. Verbeke and Lesaffre (1996) studied the effects of deviation from normality on the empirical Bayes estimates of the random effects. Using computer simulation they simulated 1000 subjects with five measurements per subject, where each subject had a random intercept coming from a 50 50 mixture of normal distributions, which may arise if two subpopulations were examined each having equal variability and size. By assuming a unimodal normal distribution of the random effects, a histogram of the empirical Bayes estimates revealed a unimodal distribution, not a bimodal distribution as would be expected. They showed that the correct distributional shape of the random effects may not be observed if the error variability is large compared to the between-subject variability. 

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