Residual error models

The assumption of homoscedasticity means that the residual variability should be constant over all available data dimensions (predictions, covariates, time, etc). If we observe heteroscedasticity, then we need to change the residual error model to account for this. In practice, this means that we should weight the data differently by using a different model for the residual variability. 

The above code specifies a constant CV residual error model and theta(y) will be the standard deviation of the residual error (note the sigma i fix). The del variable protects from division by zero in the line with iwres. ipred and iwres can be output in a table file. 

To implement a slope-intercept residual error model is as simple as replacing the w= line above with... 

A residual error model should, by necessity, be part of the basic PM model. It is useful to start with a combination of additive and proportional error models. If the data does not support either of the error models, the estimate of one of the errors would tend toward zero. As a note of caution, if the base model has not been optimized, especially the structural model component, an initial estimate of an infi-nitely small value for the additive component of the residual error model may lead to an erroneous elimination of that component of the error model. This should be avoided. It is important to let the nature of the data determine the type of error model to be used. For instance, radioactive decay may be better characterized with a power error model. 

The population pharmacokinetics of enoxaparin was described by a one-compartment IV bolus model, the parameters of which are presented in Table 12.1. The interindividual variability parameters specify variances in the log-scale of the lognormaUy distributed PK parameters, and the residual error parameter specihes the variance of the proportional error. This model was implemented in ADAPT II using the Fortran code, provided in Appendix 12.1, which is identical to the ICOMPCL.FOR code provided as part of the software distribution, except for the residual error model. 

C Residual error model changed to Proportional error. . . ... 

The covariates were coded such that tmt = 0 if without RTV, tmt = 1 if with RTV, POP = 0 if healthy, pop = 1 if HIV-infected, and tji, 7/2, tjs, and were independently normally distributed. The residual error model took the form... 

Means. First, take geometric means of all measurements at each time of measurement. We thus have mro, mri, mr2, mto, mti, mt2. Then use Eq. (16.2) to jointly estimate all parameters under a chosen residual error model. 

This section defines the residual error models for the pharmacokinetics and pharmacodynamics. Note that the PD observations have a simple additive error function, which is different from the PK residual error function. Residual error models must be selected separately for the PK and PD models. 

A proportional error, a constant additive error, and a combination of both error models were evaluated for the residual error model. Between-subject random effects were explored on the clearance of parent drug and metabolite, the volume of distribution of the parent drug, and the absorption rate constant. An exponential model was preferred. Interoccasion random effects were explored on the clearance of the parent drug and of the metabolite, the volume of distribution of the parent drug, and the absorption rate constant. An exponential model was preferred. The joint distribution of the between-subject random effect, the interoccasion random effects, and the residual error were assumed normal with mean 0 and variance-covariance matrices O for the between-subject and interoccasion random effects, and I, for the residual error to be estimated. The FO method was used for the estimation of the parameters. 

The selection of the structural PK model and residual error models was based on the goodness-of-fit plots and on the difference in NONMEM objective function (approximately -2 x log likelihood) between hierarchical models (i.e., the likelihood ratio test). This difference is asymptomatically distributed with a degree of freedom equal to the number of additional parameters of the full compared to the reduced model. A p-value of 0.05 was chosen for one additional parameter, corresponding to a difference in the objective function of 3.84. Potential covariates were selected by univariate analysis, testing the addition of each covariate on each of the relevant PK parameters. When a set of covariates, identified by the... 

In the third step, the residual covariance structure (the R matrix) is chosen, conditional on the set of random effects chosen for the model. In helping to choose a covariance model, one guideline provided by Verbeke and Molenberghs (2000) is that for highly unbalanced data with many measurements made on each subject, it can be assumed that the random effects predominate and can account for most of the variability in the data. In this case, one usually chooses a simple residual error model, usually the simple covariance matrix. Indeed, the first model examined should be one where the residual... 

Under all these models, the generic residuals are assumed to be independent, have zero mean, and constant variance residual variance components, a2, is referred to as the residual variance matrix (X), the elements of which are not necessarily independent, i.e., the residual variance components can be correlated, which is referred to as autocorrelation. It is generally assumed that and r are independent, but this condition may be relaxed for residual error models with proportional terms (referred to as an -q — s interaction, which will be discussed later). 

Another example of distinct residual error models is when data from two different populations are studied. For example, data from healthy volunteers and subjects may be analyzed together. For whatever reason, possibly due to better control over dosing and sampling in healthy volunteers or because of inherent homogeneity in healthy volunteers, data from healthy volunteers have smaller residual variability than subjects. Hence, a residual error of the type... 

Figure 7.22 Scatter plots and box and whiskers plots of true clearance against the EBE for clearance using FO-approximation. A total of 250 or 75 subjects were simulated with each subject having either 2, 4, or 8 samples collected. In this example, all pharmacokinetic parameters had 40% CV whereas residual error had a 10% CV. A 2-compartment model was fit to the data with all pharmacokinetic parameters treated as log-normal random effects and residual error modeled using an additive and proportional error model where the additive component was fixed to 0.00001 (ng / mL)2 to avoid infinite objective functions. Solid line is the line of unity.

Type I error rates with FOCE-I were consistently near nominal values and were unaffected by number of subjects or number of observations per subject. With large residual variability (42%) and two observations per subject, Type I error rates for FOCE-I were higher than nominal, about 0.075 instead of 0.05. But when the number of observations was increased to four, the Type I error rate decreased to the nominal value and remained there as further increases in the number of observations were examined. Also, when the residual variance was modeled using a proportional residual error model, instead of an exponential residual variance model, the Type I error rate decreased. The major conclusion of this analysis was that FOCE-I should be preferred as an estimation method over FO-approximation and FOCE. 

In their third simulation, they examined the Type I error rate for inclusion of a false covariance term between CL and V in a 1-compartment model. FOCE-I Type I error rates were dependent on the number of samples collected per subject (more samples tended to decrease the Type I error rate), degree of residual variability (as residual variability increased so did the Type I error rate), and whether the residual error was defined using an exponential or proportional model (exponential residual error models always produced larger Type I errors than proportional error models). With 100 subjects, two observations per subject (at 1.75 and 7 h after... 

This indicates that the residual sum of squares for the individual is near zero. Check the initial value of the residual error and see if it is small. If so, increase its value. Alternatively, try fitting the logarithms of the concentration with an additive residual error model. 

Karlsson, M.O., Beal, S.L., and Sheiner, L.B. Three new residual error models for population PK/PD analyses. Journal of Pharmacokinetics and Biopharmaceutics 1995 23 651-672. 

Examination of residual error model shape of distribution about residual = 0 line can suggest error model representation (i.e., additive for uniform distribution proportional for cone shape)... 

Examination of residual error model weighted residuals have unit variance and are uncorrelated so that the shape of distribution about residual = 0 line should be uniform if error model is adequate WRES scale can be viewed as an approximate SD scale (i.e., outside the 3 unit range should be further explored as a potential outlier, see Figure 15.3) Goodness-of-fit and examination of time dependencies can be used to discern tolerance or induction phenomenon Goodness-of-fit and examination of time dependencies residuals should be randomly dispersed about the residual = 0 line... 

---