Within-subject errors

What one observes is a measured value that differs from the model-predicted value by some amount called a residual error (also called intrasubject error or within-subject error). There are many reasons why the actual observation may not correspond to the predicted value. The structural model may only be approximate/ or the plasma concentrations may have been measured with error. It is too difficult to model all the sources of error separately/ so the simplifying assumption is made that each difference between an observation and its prediction is random. When the data are from an individual/ and the error model is the additive error model/ the error is denoted by s. 

Figure 6.1 illustrates this relationship graphically for a 1-compartment open model plotted on a semi-log scale for two different subjects. Equation (6.15) has fixed effects (3 and random effects U . Note that if z = 0, then Eq. (6.15) simplifies to a general linear model. If there are no fixed effects in the model and all model parameters are allowed to vary across subjects, then Eq. (6.16) is referred to as a random coefficients model. It is assumed that U is normally distributed with mean 0 and variance G (which assesses between-subject variability), s is normally distributed with mean 0 and variance R (which assesses residual variability), and that the random effects and residuals are independent. Sometimes R is referred to as within-subject or intrasubject variability but this is not technically correct because within-subject variability is but one component of residual variability. There may be other sources of variability in R, sometimes many others, like model misspecification or measurement variability. However, in this book within-subject variability and residual variability will be used interchangeably. Notice that the model assumes that each subject follows a linear regression model where some parameters are population-specific and others are subject-specific. Also note that the residual errors are within-subject errors. 

In the model presented above, the R matrix elements, or matrix of within-subject errors, are uncorrelated and of constant variance across all subjects. When this is the case, Eq. (6.15) is sometimes called the conditional independence model since it assumes that responses for the ith subject are independent of and conditional on the U s and (3. At times, however, this may be an unrealistic assumption since it seems more likely that observations within a subject are correlated. For example, if the model were misspecified, then parts of the data over time would be more correlated than other parts (Karls-son, Beal, and Sheiner, 1995). Hence, it is more realistic to allow the within-subject errors to be correlated. 

Only the within-subject variation is included in the mean square error (MSE) term. 

The maximum entropy principle To produce an image or map which is maximally non-committal or minimally biased with respect to missing data, maximize the entropy of the map subject to the constraint that this map must reproduce the data which generated it within experimental error. 

Population pharmacokinetic parameters quantify population mean kineticS/ between-subject variability (intersubject variability)/ and residual variability. Residual variability includes within-subject variability/ model misspecification/ and measurement error. This information is necessary to design a dosage regimen for a drug. If all patients were identical/ the same dose would be appropriate for all. However/ since... 

The results of this sampler study are summarized in Table I, where the value for the mercury concentration in each case is the mean mercury concentration found by triplicate analysis. The mercury concentrations found in the surface seawater collected in Teflon agree within experimental error with those for this same water after 1 hr in the PVC sampler subjected to the hydrocast procedure and sample transfer usually used. The mercury concentrations found at other depths (Table II) show little variation from the surface values. This suggests that the open samplers are not contaminated or affected signiflcantly by the hydrocast procedure. Additional details regarding this sampling study can be found in Fitzgerald and Lyons (27). 

As an alternative to the normal-mode method, Monte Carlo and molecular dynamics calculations have been performed on small clusters. Monte Carlo and molecular dynamics methods have the virtue of being exact, within calculable error bars, subject to the constraint of the approximate intermo-lecular interactions that are used. Prior to about six years ago both methods were restricted to systems projjerly described by classical mechanics. This restriction implied that systems for which tunneling or low-temjjerature vibrations were important at best could be treated approximately. 

Inherent in any reported laboratory test results on patients are influences of (1) biological variation, (2) inherent analytical error, (3) preanalytical and postanalytical sources of variation, and (4) possible pathophysiological alterations. When repeated measurements are made over time in one individual, even under standardized conditions, there is a considerable variability in the test results. The variability within the individual is attributable to both analytical and intraindividual factors but the intraindividual (within-subject) variability is typically less than the variability among a group of individuals. This means that when ana-... 

Direct experimental verification of the Gibbs adsorption equation in aqueous solutions is difficult, because physical separation of the monomolecular layer at the water surface is required to compare the concentration differences between the surface layer and the bulk solution. Several attempts have been made on this subject from 1910 to the present day, and although an exact fit has never been obtained, the results show a good agreement with the theory. McBain and co-workers used a suitable microtome to cut off a thin layer of approximately 50-100 4m from the surface of phenol, p-toluidine etc. solutions and verified the Gibbs equation within experimental error in 1932. Later, isotopically labeled solute molecules were employed for this purpose. Beta-emitter molecules, such as 3H, 14C and 35S have also been used and the radioactivity close to the surface measured. Since electrons only travel a short distance, the recorded radioactivity comes from the interface or very near the interface. 

To obtain initial estimates, an Emax model was fit to the data set in a na ive-pooled manner, which does not take into account the within-subject correlations and assumes each observation comes from a unique individual. The final estimates from this nonlinear model, 84% maximal inhibition and 0.6 ng/mL as the IC50, were used as the initial values in the nonlinear mixed effects model. The additive variance component and between-subject variability (BSV) on Emax was modeled using an additive error models with initial values equal to 10%. BSV in IC50 was modeled using an exponential error model with an initial estimate of 10%. The model minimized successfully with R-matrix singularity and an objective function value (OFV) of 648.217. The standard deviation (square root of the variance component) associated with IC50 was 6.66E-5 ng/mL and was the likely source of the... 

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