Distribution of the Random Effects

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. 

At this point no assumptions have been made regarding the distribution of the random effects other than their scale. 

The first step towards inclusion of a mixture model in a PopPK analysis is often graphical examination of the histograms of the EBEs of the model parameters that are treated as random effects assuming no mixture is present in the distribution of the random effects (i.e., the model does not include a mixture distribution), with clear multimodality used as strong evidence that a mixture distribution should be included in the model. This approach is not ideal since in order to visually detect a bimodal distribution of two normally distributed random variates, the subpopulation means must be separated by at least two standard deviations (Schilling,... 

The assumptions of the structural model regarding distribution of the random effects should be examined. Most population parameters are modeled assuming the random effects are log-normal. The random effects are assumed to be independent and have a normal distribution with mean 0 and variance co2. These assumptions should be tested. Other assumptions that should be tested include testing the residuals for homoscedasticity, normality, and lack of systematic deviations in the residuals over time. More about assumption testing will be presenting later in the chapter. If the assumptions are violated, remedial measures should be taken. 

At the heart of any analysis lies the question of whether the structural model was adequate. Notice that it was not said that the model was correct. No model is correct. The question is whether the model adequately characterizes the data (a descriptive model) and is useful for predictive purposes (a predictive model). Adequacy of the structural model for descriptive purposes is typically made through goodness of fit plots, particularly observed versus predicted plots, residual plots, histograms of the distribution of the random effects, and histograms of the distribution of the residuals. Adequacy of the model for predictive purposes is done using simulation and predictive checks. 

It is assumed that any correlation between the random effects is accounted for in the final model. Adding a covariance term to a random effect is usually made on the basis of examination of the distribution of EBEs of the random effects, e.g., clearance versus volume of distribution, or the distribution of the random effects, e.g., r cl versus t v. If the scatter plot indicates a trend or if a correlation analysis indicates a significant correl-... 

In addition it is now time to think about the two assumption models, or types of analysis of variance. ANOVA type 1 assumes that all levels of the factors are included in the analysis and are fixed (fixed effect model). Then the analysis is essentially interested in comparing mean values, i.e. to test the significance of an effect. ANOVA type 2 assumes that the included levels of the factors are selected at random from the distribution of levels (random effect model). Here the final aim is to estimate the variance components, i.e. the variance fractions with respect to total variance caused by the samples taken or the measurements made. In that case one is well advised to ensure balanced designs, i.e. equally occupied cells in the above scheme, because only then is the estimation process straightforward. 

It is difficult to calculate the likelihood of the data for most pharmacokinetic models because of the nonlinear dependence of the observations on the random parameters rj,- and, possibly, Sy. To deal with these problems, several approximate methods have been proposed. These methods, apart from the approximation, differ widely in their representation of the probability distribution of interindividual random effects. 

Fattinger, K.E. Sheiner, L.B. Verotta, D. A new method to explore the distribution of interindividual random effects in non-linear mixed effects models. Biometrics 1995, 51, 1236-1251. 

The use of mixture models is not limited to identification of important subpopulations. A common assumption in modeling pharmacokinetic parameters is that the distribution of a random effect is log-normal, or approximately normal on a log-scale. Sometimes, the distribution of a random effect is heavy tailed and when examined on a log-scale, is skewed and not exactly normal. A mixture distribution can be used to account for the large skewness in the distribution. However, the mixture used in this way does not in any way imply the distribution consists of two populations, but acts solely to account for heavy tails in the distribution of the parameter. 

At the second step, the model assumptions must be examined and confirmed. The reader is referred to the Section on Testing the Model Assumptions for details. Briefly, informative graphics are essential (Ette and Lud-den, 1995). Scatter plots of individual versus predicted concentrations, weighted residuals versus predicted concentrations, and weighted residuals versus time provide evidence of the goodness of fit of the model. Histograms and possibly QQ plots of the distribution of the residuals, the r s (deviations from the mean), and the EBEs of the random effects are used to examine the assumptions of normality. Further, sensitivity analysis can be done to assess the stability of the model. 

Another consideration when using the approach is the assumption that stress and strength are statistically independent however, in practical applications it is to be expected that this is usually the case (Disney et al., 1968). The random variables in the design are assumed to be independent, linear and near-Normal to be used effectively in the variance equation. A high correlation of the random variables in some way, or the use of non-Normal distributions in the stress governing function are often sources of non-linearity and transformations methods should be considered. 

The fact that there are no characteristic length scales immediately implies a similar lack of any characteristic time scales for the fluctuations. Consider the effect of a single perturbation of a random site of a system in the critical state. The perturbation will spread to the neighbors of the site, to the next nearest neighbors, and so on, until, after a time r and a total of / sand slides, the effects will die out. The distribution of the life-times of the avalanches, D t), obeys the power law... 

Recently the effect of intrinsic traps on hopping transport in random organic systems was studied both in simulation and experiment [72]. In the computation it has been assumed that the eneigy distribution of the traps features the same Gaussian profile as that of bulk states. 

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