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In PK/PD data analysis, statistical models involving random variables are developed. At times an analysis can be carried out by conditioning on certain random events involving these variables. This paper attempts to clarify issues regarding conditioning. In particular, conditioning is examined as it relates to a number of disparate practical matters: missing covariate values, dose titration, BQL data, "no change from baseline" data, and the use of a truncated intraindividual probability distribution for PK observations.

A model can be fit to joint PK/PD data (concentration and effect) either simultaneously or sequentially. The results of a companion paper suggested that when the data-analytic and true models agree, a particular sequential approach is computationally faster than the simultaneous one, yet produces hardly less precise PD parameter estimates, and for suitable designs, about as accurate PD standard error estimates. In this paper, we compare the performance of various methods for the case that the data-analytic model is misspecified. We illustrate these methods by applying them to a set of real data. Using NONMEM, population PK/PD observations were simulated under various study designs according to a one- or two-compartment PK model and direct Emax or sigmoid Emax model. A one-compartment PK model and Emax PD model were fit to the simulated observations by simultaneous and sequential methods. Predictive performance (interpolation and extrapolation) of PD and the type-I error rate of a likelihood ratio test are compared. The real data set consists of PK and (more frequent) PD observations after administration of the muscle relaxant vecuronium. When only the PK data-analytic model is misspecified, the simultaneous method has greater precision than the sequential methods. However a sequential method that uses a non-parametric PK model performs better than both other methods when PK model misspecification is severe. When the PD data-analytic model is misspecified, sequential and simultaneous methods perform similarly. The analysis of the real data shows that the PK fitted with the simultaneous method can be quite sensitive to PD model misspecification, yielding a possible diagnostic for this type of misspecification.

Dose [-concentration]-effect relationships can be obtained by fitting a predictive pharmacokinetic (PK)-pharmacodynamic (PD) model to both concentration and effect observations. Either a model can be fit simultaneously to all the data (“simultaneous” method), or first a model can be fit to the PK data and then a model can be fit to the PD data, conditioning in some way on the PK data or on the estimates of the PK parameters (“sequential” method). Using simulated data, we compare the performance of the simultaneous method with that of three sequential method variants with respect to computation time, estimation precision, and inference. Using NONMEM, under various study designs, observations of one type of PK and one type of PD response from different numbers of individuals were simulated according to a one-compartment PK model and direct Emax PD model, with parameters drawn from an appropriate population distribution. The same PK and PD models were fit to these observations using simultaneous and sequential methods. Performance measures include computation time, fraction of cases for which estimates are successfully obtained, precision of PD parameter estimates, precision of PD parameter standard error estimates, and type-I error rates of a likelihood ratio test. With the sequential method, computation time is less, and estimates are more likely to be obtained. Using the First Order Conditional Estimation (FOCE) method, a sequential approach that conditions on both population PK parameter estimates and PK data, estimates PD parameters and their standard errors about as well as the “gold standard” simultaneous method, and saves about 40% computation time. Type-I error rates of likelihood ratio test for both simultaneous and sequential approaches are close to the nominal rates.

When modeling new data with a complex population pharmacokinetic/pharmacodynamic model, there may not be sufficient information to obtain estimates of all parameters. In this case information from previous studies can also be used to help stabilize estimation. Using simulated data, we explored three different ways to do this. (i) Some parameter values were fixed to estimates obtained from earlier data. (ii) The earlier data were combined with the current data. (iii) The objective function based on the current data was augmented by a penalty function expressing summary information obtained from the earlier data. This last method is similar to the use of a Bayesian prior. It may be particularly useful when either the combined data set of method (ii) is very large and leads to large computation times or when the early data are not readily available. With this method, two different types of penalty functions were used. With our examples, the three methods all resulted in stabilized estimation. Methods (ii) and (iii) gave similar results for parameter and standard error estimation, especially with respect to fixed effects parameters. For hypothesis testing, results obtained with method (i) are very problematic. There are also problems with the results obtained with method (iii), but they are much less severe, and when the design for the earlier data is known, they can be corrected by using a computer-intensive simulation test procedure.

The posterior predictive check (PPC) is a model evaluation tool. It assigns a value (p PPC ) to the probability that the value of a given statistic computed from data arising under an analysis model is as or more extreme than the value computed from the real data themselves. If this probability is too small, the analysis model is regarded as invalid for the given statistic. Properties of the PPC for pharmacokinetic (PK) and pharmacodynamic (PD) model evaluation are examined herein for a particularly simple simulation setting: extensive sampling of a single individual's data arising from simple PK/PD and error models. To test the performance characteristics of the PPC, repeatedly, “real” data are simulated and for a variety of statistics, the PPC is applied to an analysis model, which may (null hypothesis) or may not (alternative hypothesis) be identical to the simulation model. Five models are used here: (PK1) mono-exponential with proportional error, (PK2) biexponential with proportional error, (PK2e) biexponential with additive error, (PD1) E max model with additive error under the logit transform, and (PD2) sigmoid E max model with additive error under the logit transform. Six simulation/analysis settings are studied. The first three, (PK1/PK1), (PK2/PK2), and (PD1/PD1) evaluate whether the PPC has appropriate type-I error level, whereas the second three (PK2/PK1), (PK2e/PK2), and (PD2/PD1) evaluate whether the PPC has adequate power. For a set of 100 data sets simulated/analyzed under each model pair according to a stipulated extensive sampling design, the p PPC is computed for a number of statistics in three different ways (each way uses a different approximation to the posterior distribution on the model parameters). We find that in general; (i) The PPC is conservative under the null in the sense that for many statistics, prob(p PPC ≤α)

Over the past 25 years sophisticated data analytic techniques have been developed which can lead to improved analyses, but at additional computational cost. In particular, this applies to the approach where interindividual random effects are included in a data analytic model for population pharmacokinetic data, which can often lead to substantially improved estimates of fixed-effect parameters. However, there are also commonly occurring situations, notably with some types of pharmacodynamic data, where such improvement is not realized. This study simulates some simple population dichotomous data, and secondarily, some related continuous data. These data are analyzed using both mixed-effect (ME) models that include interindividual random effects and naive (NA) models that do not include interindividual random effects, and it is seen that use of an ME model does not inevitably lead to gains over use of an NA model. In fact, using maximum likelihood estimation with both types of models, the root mean square estimation errors for fixed effect parameters can actually be larger with an ME model than with the corresponding NA model. Using a form of restricted maximum likelihood estimation with the ME model, the two types of models yield root mean square errors which are comparable, but which still do not suggest that there is always marked advantage in using the ME model.

Pharmacokinetic data consist of drug concentration measurements, as well as reports of some measured concentrations being below the quantification limit of the assay (BQL). A pharmacokinetic model may befit to these data, and for this purpose, the BQL observations must be either discarded or handled in a special way. In this paper, seven methods for dealing with BQL observations are evaluated. Both single-subject and population data are simulated from a one-compartment model. A moderate amount of data is simulated for each individual. The actual cv of concentration measurements at the quantification limit is assumed to be no greater than 20%, in accord with the FDA Guidance. The results of this paper should be interpreted in this context. The methods include handling BQL observations as fixed-point censored observations, i.e., by using the likelihoods that these observations are in fact BQL. This method is shown to have some overall statistical advantage. However, the gain in using this method over that of simply discarding the BQL observations is not always much, and this is especially so when the frequency of BQL observations is small. Some simple methods entailing (i) replacing one or more BQL observations with the value 0, or (ii) replacing them with the value QL/2, where QL is the quantification limit, are also included. The first of these two approaches should not be used With population data, use of the second approach can result in some noticeably improved estimation of the typical value of a parameter, but then there is also marked degradation in the estimation of the population variance of the parameter.

This paper presents and illustrates methodology for specifying, estimating, and evaluating a predictive model for repeated measures time-to-event responses. The illustrative example specifies a model of the antiemetic effect vs. concentration relationship for the 5-HT3 antagonist ondansetron in the human ipecac model for emesis. A key part of this model is a time-dependent log hazard function for emesis that is increased by ipecac administration and decreased by ondansetron concentration. The model is fit using an approximate maximum likelihood method. The data consist of the time free of emeses and, for those individuals with emetic episodes, the time(s) of the episode(s). Model evaluation is accomplished using residual plots adapted to time-to-event data and a "posterior predictive check" wherein observed data statistics are compared to those obtained from data simulated from the fitted model. The ondansetron concentration required to obtain a 50% reduction in the hazard of emesis is estimated to be 1.4 +/- 0.2 ng/ml, and the rate constant for elimination of ipecac-induced hazard is 1.5 +/- 0.2 hr-1.

A clinical trial of an analgesic agent compares pain relief scores over time among groups of patients. All subjects experience the same painful procedure, but different subjects are given different randomly assigned doses of active agent or placebo when they first request it. The data are short individual time series of ordered categorical pain relief scores subsequent to dosing. Nonrandom right censoring may be present because patients can elect to remedicate with an active agent if their pain relief is insufficient. The trial is meant to address two questions: (a) Is there proof that the drug relieves pain? If so, (b) What dosage patterns should be investigated further, or recommended for use by a typical patient? Marginal models of human pharmacology are basically empirical models, and although an analysis of a study based on such a model can adequately address the first question, such is not the case for the second question, because this question requires extrapolation to untested dosing patt...

In pharmacokinetic data analysis, it is frequently necessary to select the number of exponential terms in a polyexponential expression used to describe the concentration-time relationship. The performance characteristics of several selection criteria, the Akaike Information Criterion (AIC), and the Schwarz Criterion (SC), and the F test (alpha = 0.05), were examined using Monte Carlo simulations. In particular, the ability of these criteria to select the correct model, to select a model allowing estimation of pharmacokinetic parameters with small bias and good precision, and to select a model allowing precise predictions of concentration was evaluated. To some extent interrelationships among these procedures is explainable. Results indicate that the F test tends to choose the simpler model more often than does either the AIC or SC, even when the more complex model is correct. Also, the F test is more sensitive to deficient sampling designs. Clearance estimates are generally very robust to the choice of the wrong model. Other pharmacokinetic parameters are more sensitive to model choice, particularly the apparent elimination rate constant. Prediction of concentrations is generally more precise when the correct model is chosen. The tendency for the F test (alpha = 0.05) to choose the simpler model must be considered relative to the objectives of the study.

This note gives expressions for recirculation mean time parameters of the disposition kinetics of particles in a semihomogeneous stationary linear system. In such a system each compartment may have an arbitrary single-pass disposition function, rather than a known parametric (usually monoexponential) one. Such systems provide a generalization of physiological flow models. Given observations of arterial blood concentrations and tissue amounts, and making the additional assumptions that (i) the fraction of total blood flow exiting each tissue that goes to each other tissue is constant and known, and (ii) the fraction of drug entering each tissue that is eliminated to the outside is constant and known, the input to each tissue can be known, and therefore both its total blood flow and its single-pass disposition function can be estimated. Recirculation mean time parameters can be computed from these estimates. Application to real thiopental data is presented as an example.

Objective: To characterize change from baseline weight over time for pregabalin and placebo administration. Methods: Asymptotic fraction of baseline weight was modeled with a nonmixture model and a mixture model as a function of baseline weight, exposure, time, covariate effects, and subject-specific random effects. Model fit was assessed using standard diagnostic plots. Predictive performance was assessed using both data similar to the original data, and open-label data. Results: The nonmixture model indicated that a typical patient (baseline weight 82 kg) receiving placebo or 300 mg/day pregabalin approached an asymptotic fractional change from baseline weight of [mean (95% prediction interval for typical individual)] 0.7% (−5.5% to 7.4%) or 2.5% (−3.8% to 9.1%), respectively, with a half-life of 17 days. Substantial between-subject variability is observed, with some drug-treated subjects remaining weight neutral or losing weight, at all levels of exposure. Structural fixed effects parameters for the two submodels (mixture model) were in close agreement with each other and with those for the nonmixture model. The mixture model described two subpopulations differing in interindividual variability. No significant interindividual-varying covariates influencing the mixture probabilities were identified other than exposure. Both models had adequate fit; both models performed well during external validation. Predictive performance (nonmixture model) was adequate to ~900 days. Conclusions: The weight of a typical 82-kg patient receiving placebo or pregabalin (300 mg/day) approaches an asymptotic fractional change from baseline weight of 0.7%, or 2.5%, respectively, with a half-life of 17 days. Substantial between-subject variability remains unexplained.

Objective Our objective was to develop a population 1-compartment pharmacokinetic (PK) method of analysis to deal with suspect or missing prior dosage history. Methods Population PK data from a 1-compartment model with first-order elimination and absorption, described by PK parameters clearance, volume of distribution, and absorption rate constant, are simulated. A PK sample is drawn just before a test dose (Dt), followed by a (varying) number of additional samples over 1 interdose interval (τ). For 60% of the subjects, the true history of the scheduled dose (Ds) preceding Dt differs from that prescribed, whereas doses taken before Ds do not. Two settings are evaluated: considerable accumulation of drug in the body (typical drug half-life t½ approximately equal to τ) and very little such accumulation (t½ approximately equal to τ/5). Precision and bias of several PK analysis methods—Missing Dose Method (MDM), Missing Dose Mixture Method (MDMM) and Extrapolation-Subtraction Method (ESM), all of which essentially do not use prior dose history—are compared with those of the Prescribed Dose Method (PDM), which assumes nominal dosage, and an Ideal Method (IDM), which uses true (but unknown) pre-test dose history. Results At t½ approximately equal to τ, MDM and MDMM are the most precise methods. The accuracy of ESM and PDM is poor. At t½ approximately equal to τ/5, no significant differences, in terms of precision or bias, are observed between methods. Misspecification of the structural or statistical model seems not to influence these results. The results of analysis of a real (caffeine) data set are compatible with the findings from the simulations. Conclusion When a test dose is given and a predose baseline observation is taken as part of an “intensive” PK study during outpatient therapy of a 1-compartment drug, an analysis that assumes that the nominal dose history is correct is not robust to past dosage history misspecification, whereas methods that do not do this are robust and reliable. Clinical Pharmacology & Therapeutics (2004) 76, 441–451; doi: 10.1016/j.clpt.2004.07.010

When modeling new data with a complex population pharmacokinetic/pharmacodynamic model, there may not be sufficient information to obtain estimates of all parameters. In this case information from previous studies can also be used to help stabilize estimation. Using simulated data, we explored three different ways to do this. (i) Some parameter values were fixed to estimates obtained from earlier data. (ii) The earlier data were combined with the current data. (iii) The objective function based on the current data was augmented by a penalty function expressing summary information obtained from the earlier data. This last method is similar to the use of a Bayesian prior. It may be particularly useful when either the combined data set of method (ii) is very large and leads to large computation times or when the early data are not readily available. With this method, two different types of penalty functions were used. With our examples, the three methods all resulted in stabilized estimation. Methods (ii) and (iii) gave similar results for parameter and standard error estimation, especially with respect to fixed effects parameters. For hypothesis testing, results obtained with method (i) are very problematic. There are also problems with the results obtained with method (iii), but they are much less severe, and when the design for the earlier data is known, they can be corrected by using a computer-intensive simulation test procedure.

SUMMARY This bibliography lists all published methodological works (statistical methodology, implementation methodology, review papers, descriptions of software) in the area of population pharmacokinetics/ pharmacodynamics up to 1993 and all published works describing applications of population pharmacokinetics/pharmacodynamics up to 1992.

The influence of the choice of pharmacokinetic model on subsequent determination of covariate relationships in population pharmacokinetic analysis was studied using both simulated and real data sets. Simulations and data analysis were both performed with the program NONMEM. Data were simulated using a two-compartment model, but at late sample times, so that preferential selection of the two-compartment model should have been impossible. A simple categorical covariate acting on clearance was included. Initially, on the basis of a difference in the objective function values, the two-compartment model was selected over the one-compartment model. Only when the complexity of the one-compartment model was increased in terms of the covariate and statistical models was the difference in objective function values of the two structural models negligible. For two real data sets, with which the two-compartment model was not selected preferentially, more complex covariate relationships were supported with the one-compartment model than with the two-compartment model. Thus, the choice of structural model can be affected as much by the covariate model as can the choice of covariate model be affected by the structural model; the two choices are interestingly intertwined. A suggestion on how to proceed when building population pharmacokinetic models is given.

Only with knowledge of the (prior) distribution of dose-response parameters in a population, can one determine both the initial dose of a drug for chronic administration to an individual (such as the dose producing a fixed degree of response in a fixed proportion of the population) and an appropriate subsequent (adjusted) dose (such as the dose yielding a desirable response according to the posterior parameter distribution, given an observed response to an initial dose). The currently FDA-sanctioned design for a dose-ranging study, the parallel-dose design, assigns just one of several doses to each patient. It does not provide good information on the distribution of individual dose-response parameters. A cross-over design assigns several dose levels to each patient. It therefore can provide better information, but does not resemble clinical practice. Consequently, study participants must be restricted to patients who can tolerate such non-therapeutic drug exposure, posing problems in extrapolation of study results to other types of patients. A titration or dose-escalation design begins all patients on placebo and, except for those patients assigned to a placebo-only group, escalates the dose for a patient at preset intervals only when clinical response at lower doses is inadequate. It both exposes patients to several dose levels and resembles good clinical practice, allowing study of a representative patient sample. We report here the simulation results of parameter estimation for the three designs when the data arise from complex and realistic dose-response models and/or with certain complications in study execution. The dose-escalation design clearly performs better overall than the parallel-dose design for the models considered here, and generally, just a little worse than the cross-over design. These results support the conclusion that for dose ranging, depending on the demands of the clinical situation, one should use either the cross-over or the dose-escalation design.

In a recent paper, Beal (1989, Biometrics 45, 969-977) considers the problem of determining the appropriate sample size when inference about a parameter theta is to be made on the basis of a confidence interval (CI). He suggests that the sample size should be chosen so that the probability that the length of the CI is less than a given value, conditional on the interval including the true theta, is greater than a specified level. In this note, in which we concentrate on two-sided intervals, this suggestion is examined, as is the effect of uncertainty in our knowledge of the population variance sigma 2 on estimates of sample size.

SUMMARY Use of the maximum likelihood method for parameter estimation in a general mixed effects model requires initial estimates for all the model parameters. When the model is complicated, it may not be easy to obtain such estimates, especially estimates of the variances and covariances. A computational strategy is given for obtaining initial estimates. It is based on a general formulation of the method of moments for mixed effects models which allows the variances and covariances to be estimated for fixed values of the regression parameters. A number of numerical algorithms has been proposed for obtaining maximum, or restricted maximum, likelihood estimates of the parameters of a linear mixed effects model (Hemmerle & Hartley, 1973; Jennrich & Sampson, 1976; Corbeil & Searle, 1976; Laird & Ware, 1982). The maximum likelihood method can also be used with a model in which some fixed effects enter nonlinearly, while all random effects enter linearly; see ? 2. With any of the algorithms it is important to have good initial estimates for all the model parameters. One computational strategy sometimes used for finding initial estimates is to first maximize the full likelihood or log likelihood function, referred to here as the objective function, over a mesh, or grid, of points covering the parameter space and then to use the mesh point corresponding to the maximum as the initial estimate. However, with a complicated mixed effects model the dimension of the parameter space can be large enough to make this strategy impractical. Also, this strategy implies that practical bounds can be placed on all variances and covariances of the model, and though this may be possible, it may not be easy to do, especially when the model is complicated. This paper presents a modification of this strategy. It makes use of a general formulation of the method of moments for mixed effects models which is applicable to obtaining estimates of the variances and covariances of such models for fixed values of the regression parameters. Thus initial estimates of the variances-covariances are obtained without recourse to a grid search over at least those dimensions of the parameter space corresponding to the variances and covariances. The method of moments formulation permits estimates to be obtained under various constraints which might be imposed on the variances and covariances. It is also applicable to multivariate mixed effects models, and

Simultaneous dual immunofluorescence and flow cytometry was used to study sixteen lymphocyte phenotypes in 209 men including: healthy homosexuals, lymphadenopathy patients (LAN), and AIDS patients. Significant differences between the distribution of lymphocytes in healthy homosexuals and healthy heterosexuals were decreased percentages of helper/inducer T cells (Leu 3), increased cytotoxic/suppressor T cells (Leu 2), and consequently a decreased Leu 3/Leu 2 ratio. The increased Leu 2 cells were identified as functionally cytotoxic subset Leu 2+ 15- phenotype rather than suppressor cells which are Leu 15+. Leu 2 and Leu 3 bearing cells exhibited an excess of membrane-bound immunoglobulins which were easily elutable at 37 degrees C. An increased percentage of an HLA-DR framework determinant bearing T cells were also detected. Within the NK cell family, Leu 7 cells were moderately increased and the functionally unidentified Leu 2+ 7+ population was strikingly elevated. LAN or AIDS patients were compared to healthy homosexual controls. Lower percentages of Leu 3 cells and higher percentages of Leu 2 cells were evident in LAN patients. These subsets were similar in LAN and AIDS patients. The increase in Leu 2+ cells was due to the Leu 2+ 15- cytotoxic subset. Fewer T cells had immunoglobulin in LAN and AIDS. A definite increase in Leu 2+ DR+ cells but not Leu 3+ DR+ cells occurred in AIDS compared to LAN or healthy controls. NK cell changes already present in healthy homosexuals persisted in LAN and AIDS patients. No differences in the distribution of B cells was detected in any intergroup comparisons. Changes in monocytes or pan-T cells were relatively insensitive measures of immunologic alterations among any of the groups. These results indicate many of the changes in lymphocyte subsets seen in AIDS and LAN subjects are already present in a carefully screened population of healthy homosexuals in San Francisco. Many of the changes in Leu 2 and NK family of cells suggest a possible adaptive response to viral or neoplastic challenge. Whether these interesting phenotypic alterations relate to functional changes in response to such challenge of the identified subsets waits further investigation.

Individual pharmacokinetic parameters quantify the pharmacokinetics of an individual, while population pharmacokinetic parameters quantify population mean kinetics, interindividual kinetic variability, and residual variability, including intraindividual variability and measurement error. Individual pharmacokinetics are estimated by fitting a pharmacokinetic model to individual data. Population pharmacokinetic parameters have traditionally been estimated by doing this separately for each individual, and then combining the individual parameter estimates, the Standard Two Stage (STS) approach. Another approach, NONMEM, appropriately pools data across individuals and is therefore less dependent on individual parameter estimates. This study provides further evidence of NONMEM's validity and usefulness by comparing both approaches on simulated routine-type pharmacokinetic data arising from a monoexponential model. The estimates of population parameters (notably those describing interindividual variability) provided by the STS method are poorer than those provided by NONMEM, especially when there is considerable residual error. Further, NONMEM's estimates of population parameters do not require that the data be restricted to special types of routine data such as those obtained only at steady state, or only at peak or trough, nor do the estimates improve with such data. NONMEM's estimates do improve, however, when a data set is enhanced by the addition of single-observation-per-individual type data. Thus, population parameters can be estimated efficiently from data that simulate real clinical pharmacokinetic conditions.

The stochastic formulation of linear kinetic models is elaborated in order to introduce some new concepts and help clarify the meaning and role of residence time moments. Certain conditional moments are introduced. Multicompartment and steady-state dosing within the stochastic context are considered. A general model-independent formula for steady state volume of distribution and a new concept of steady-state moments are presented. A technique for constructing a model of a given topology from its moments is also given.

One may attempt to individualize drug dosage by estimating an individual's pharmacokinetic parameters. Information useful for this purpose consists of certain population pharmacokinetic parameters (notably those describing the typical relationship between dosage and drug concentrations) and also measured drug concentrations from the individual of concern. Both types of information should be used. A (Bayesian) method that does so has been described in the pharmacokinetic literature. In this report an implementation of the Bayesian method that is readily adapted to a microcomputer is presented. Using simulated data it is compared with two other methods proposed by others, for estimating individual theophylline clearances. Both previously suggested methods are shown to be less precise than the Bayesian method: their typical error magnitudes are 20-70% larger.

Consider a finite population, called here the target population, and consider two conditions, each of which we may imagine applying to all the units of this population. Suppose that under each condition, each population unit will exhibit some binary (0-1) response. The problem is to "compare" the set of responses of all the units under one condition with the set of responses of all the units under the second condition. We call these two sets the two population responses. In order to make the comparison, an experiment, in which some responses are actually obtained, must be performed. The population may be large enough to make it impractical to obtain an entire population response, and consequently, part of the experiment may entail determining a sample of units whose response under one or both conditions will be obtained. Also, in a real experiment it may be impossible in general to actually apply both conditions to a unit. (The actual application of a condition to a particular unit may result in altering the unit so that the other condition cannot be applied to it later). Consequently part of the experiment may entail choosing the conditions which shall be applied to particular experimental units. A common experimental design in these situations involves 1) obtaining a random sample from the target population, 2) randomly allocating the sample units to two experimental groups, and 3) applying one condition to the units of one group, while applying the other condition to the units of the other group. Even when both conditions may actually be applied to all units, this design may still be helpful since both groups may undergo experimentation during the same time period, and therefore, total experiment time may be reduced. Fisher's well known hypergeometric test (2,3,8) or a test which is approximate to Fisher's test, such as the chi-squared test with Yates correction (1,2,3,8), is usually used to analyze the responses obtained from an experiment with the above design. When the sample size, N, is sufficiently small compared to the size, M, of the target population, one may regard the responses of the first (second) group as being approximately a sequence of Bernoulli 0-1 outcomes, with probability m1!M (m2!M) for outcome 1, where mi = the number of I's in the ith population response. Thus Fisher's test may be regarded to be approximately testing the null hypothesis m1!M = m2!M (or ml = M2). In this article we explain that the appropriateness of using Fisher's test to test m1 = m2 may be examined without recourse to the Bernoulli approximation. Such examination is especially important when NIM is so large that it is not really meaningful to talk about Bernoulli outcomes. An important requirement of the inferential procedure is that the sample be random. The usual assumption that the sample is indeed random is often rather shaky in application. This is often the case in medical experiments where an accessible group of patients is used for a sample, rather than a group of patients chosen from either a well-defined target population or sampling frame, and where the inference is about some population of interest which is characterized after the sample is determined and which should be identified here with the target population. For a proper interpretation of the experimental result, one may need to redefine the target population in such a way that the randomness assumption becomes more tenable (or, as it is said, in such a way that the sample becomes more representative of the population). In so doing, one may well add restrictions to the original definition of the target population, thereby decreasing the population size. In fact, it may be necessary to define the target population to be, in effect, the set of units comprising the sample; the sample is then undoubtedly random. Of course, when the population is made less encompassing, it may lose some interest value with respect to the conclusion to be drawn about it. However, this loss is often necessary for a valid inference to result. While constricting the population, the ratio of sample size to population size may increase to the point where the inferential argument should not depend on the use of the Bernoulli approximation. These considerations motivated this author's inquiry into the role of Fisher's test in the inference problem to compare the population responses.

It has previously been shown that the extended least squares (ELS) method for fitting pharmacokinetic models behaves better than other methods when there is possible heteroscedasticity (unequal error variance) in the data. Confidence intervals for pharmacokinetic parameters, at the target confidence level of 95%, computed in simulations with several pharmacokinetic and error variance models, using a theoretically reasonable approximation to the asymptotic covariance matrix of the ELS parameter estimator, are found to include the true parameter values considerably less than 95% of the time. Intervals with the ordinary least squares method perform better. Two adjustments to the ELS confidence intervals, taken together, result in better performance. These are: (i) apply a bias correction to the ELS estimate of variance, which results in wider confidence intervals, and (ii) use confidence intervals with a target level of 99% to obtain confidence intervals with actual level closer to 95%. Kineticists wishing to use the ELS method may wish to use these adjustments.

A statistical model is set forth that can be taken as a very general description of most data sets that arise from population pharmacokinetic studies. A (nontraditional) method for estimating the parameters of the model—called the NONMEM method in previously published papers—is described. This method involves a linearization of the model. An investigation of the effect of this linearization is reported. This investigation is an empirical one, based on the simulation of data from special cases of the model and the application of a few different estimation methods to these data. From the limited evidence in this investigation it appears that the linearization per se does not significantly adversely affect the estimates.

It is often difficult to specify weights for weighted least squares nonlinear regression analysis of pharmacokinetic data. Improper choice of weights may lead to inaccurate and/or imprecise estimates of pharmacokinetic parameters. Extended least squares nonlinear regression provides a possible solution to this problem by allowing the incorporation of a general parametric variance model. Weighted least squares and extended least squares analyses of data from a simulated pharmacokinetic experiment were compared. Weighted least squares analysis of the simulated data, using commonly used weighting schemes, yielded estimates of pharmacokinetic parameters that were significantly biased, whereas extended least squares estimates were unbiased. Extended least squares estimates were often significantly more precise than were weighted least squares estimates. It is suggested that extended least squares regression should be further investigated for individual pharmacokinetic data analysis.

An explicit solution to the Michaelis-Menten differential equation with bolus and zero-order input is presented. This solution involves certain simple functions whose values are not readily available. Efficient algorithms for computing values of these functions to any desired degree of accuracy and FORTRAN codes for implementing them are also given.

The precision of pharmacokinetic parameter estimates from several least squares parameter estimation methods are compared. The methods can be thought of as differing with respect to the way they weight data. Three standard methods, Ordinary Least Squares (OLS-equal weighting), Weighted Least Squares with reciprocal squared observation weighting [WLS(y-2)], and log transform OLS (OLS(ln))--the log of the pharmacokinetic model is fit to the log of the observations--are compared along with two newer methods, Iteratively Reweighted Least Squares with reciprocal squared prediction weighting (IRLS,(f-2)), and Extended Least Squares with power function "weighting" (ELS(f-xi)--here xi is regarded as an unknown parameter). The values of the weights are more influenced by the data with the ELS(f-xi) method than they are with the other methods. The methods are compared using simulated data from several pharmacokinetic models (monoexponential, Bateman, Michaelis-Menten) and several models for the observation error magnitude. For all methods, the true structural model form is assumed known. Each of the standard methods performs best when the actual observation error magnitude conforms to the assumption of the method, but OLS is generally least perturbed by wrong error models. In contrast, WLS(y-2) is the worst of all methods for all error models violating its assumption (and even for the one that does not, it is out performed by OLS(ln)). Regarding the newer methods, IRLS(f-2) improves on OLS(ln), but is still often inferior to OLS. ELS(f-xi), however, is nearly as good as OLS (OLS is only 1-2% better) when the OLS assumption obtains, and in all other cases ELS(f-xi) does better than OLS. Thus, ELS(f-xi) provides a flexible and robust method for estimating pharmacokinetic parameters.

SummaryC3H/HeJ mice show a high incidence of spontaneously occurring mammary tumors. Mice made experimentally obese were observed to have an earlier appearance and a greater incidence of tumors than were the normal-weight mice. A fatenriched diet increased the observed tumor incidence in normal-weight mice to about that of obese mice given a regular diet, while a fat-enriched diet further enhanced the already augmented observed yield of mammary tumors in the obese mice given a regular diet. Statistical analysis supported the dependence of tumor incidence and appearance time on the type of diet. The calories consumed by normal mice on a fat-enriched diet was considerably below that of obese mice on a regular diet, yet produced about the same tumor incidence. A specific effect of the fat-enriched diet is suggested.High caloric diets, particularly those modified to include a high portion of fat, have been reported to promote or enhance the occurrence of several types of tumors in various species of animals (...

Often drug dosage may be chosen rationally by use of plasma concentration (CP) as the "therapeutic" end point. The ability to accurately forecast the CP resulting from a dosage regimen is central to choosing that regimen. Tradionally forecasting has been attempted only by accounting for known influences on pharmacokinetics, such as sex, age, and renal disease. One must also adjust for previously observed CPs. Herein, we discuss and explain an approach to both of these tasks, mainly focusing on the latter. The approach balances observed outcomes against prior expectations taking account of observation CP error. For digoxin, use of 1 measured CP, as opposed to none, improves forecast precision for future CPs by 40% (decrement in variance of forecast error), and 2 CPs improve it by 67%. There is also an increase in forecast accuracy (decrement in mean of forecast error) as the number of CPs used increases. After only 2, forecast accuracy and precision are as good as theoretically possible. Moreover, information from CPs is far more valuable for forecasting than that from observable patient features-sex, age, and the like; use of all the latter information does not improve accuracy and precision as much as only 1 CP.

Premarketing dose-ranging studies of a drug are done to establish a reasonable initial dose. According to the current procedure sanctioned by the Food and Drug Administration, each patient is given one of several possible doses, including placebo, after an initial placebo run-in period. Data analysis is based on a model in which the mean response at each dose is independent of the magnitude of the dose. The initial dose is the lowest dose tested that has a response that is statistically significantly greater than the response after placebo administration. We suggest that the present conceptual approach to, and standard study design and analysis for, dose-ranging studies be changed. We believe one must begin with a parametric model for patient-specific dose-response curves. Knowledge of the distribution of these curves in a population provides a basis for choice of an initial dose (e.g., the dose that achieves a given response in a given fraction of patients) and, after observation of response to an initial dose, for choice of an incremental dose for a specific patient (by use of Bayes rule). The current parallel-dose design can provide only poor information about the distribution of dose-response curves, biased estimates of the typical curve, and little information on interpatient variability. Crossover studies provide better information. In studies in which a parametric patient-specific dose-response model is used, a dose-escalation design provides no less information than a crossover design, and it has ethical advantages that allow a more representative patient group and clinical setting to be studied.

To accurately forecast, or predict the pharmacokinetics of a drug in an individual patient implies that one can accurately forecast the plasma drug concentration (PDC) time course that one will see in response to an arbitrary dosage regimen. Why might one wish to do this? The reasons concern clinical management of patients as well as research.

An “explicit” solution to the Michaelis-Menten differential equation is presented. Instead of involving the exponential function, the solution involves another simple function. A table for this function is presented.

A semiparametric model for analysis of pharmacokinetic (PK) and pharmacodynamic (PD) data arising from non-steady-state experiments is presented. The model describes time lag between drug concentration in a sampling compartment, e.g., venous blood (Cv), and drug effect (E). If drug concentration at the effect site (Ce) equilibrates with arterial blood concentration (Ca) slower than with Cv, a non-steady-state experiment yields E vs. Cv data describing a counterclockwise hysteresis loop. If Ce equilibrates with Ca faster than with Cv, clockwise hysteresis is observed. To model hysteresis, a parametric model is proposed linking (unobserved) Ca to Cv with elimination rate constant kappa ov and also linking Ca to Ce with elimination rate constant kappa oe. When kappa oe is greater than (or less than) kappa ov clockwise (or counterclockwise) hysteresis occurs. Given kappa oe and kappa ov, numerical (constrained) deconvolution is used to obtain the disposition function of the arterial compartment (Ha), and convolution is used to calculate Ce given Ha. The values of kappa oe and kappa ov are chosen to collapse the hysteresis loops to single curves representing the Ce-E (steady-state) concentration-response curve. Simulations, and an application to real data, are reported.

The performance of a prediction or measurement method is often evaluated by computing the correlation coefficient and/or the regression of predictions on true (reference) values. These provide, however, only a poor description of predictive performance. The mean squared prediction error (precision) and the mean prediction error (bias) provide better descriptions of predictive performance. These quantities are easily computed, and can be used to compare prediction methods to absolute standards or to one another. The measures, however, are unreliable when the reference method is imprecise. The use of these measures is discussed and illustrated.

Individual pharmacokinetic par parameters quantify the pharmacokinetics of an individual, while population pharmacokinetic parameters quantify population mean kinetics, interindividual variability, and residual intraindividual variability plus measurement error. Individual pharmacokinetics are estimated by fitting individual data to a pharmacokinetic model. Population pharmacokinetic parameters are estimated either by fitting all individual's data together as though there was no individual kinetic differences (the naive pooled data approach), or by fitting each individual's data separately, and then combining the individual parameter estimates (the two-stage approach). A third approach, NONMEM, takes a middle course between these, and avoids shortcomings of each of them. A data set consisting of 124 steady-state phenytoin concentration-dosage pairs from 49 patients, obtained in the routine course of their therapy, was analyzed by each method. The resulting population parameter estimates differ considerably (population mean Km, for example, is estimated as 1.57, 5.36, and 4.44 micrograms/ml by the naive pooled data, two-stage, and NONMEN approaches, respectively). Simulations of the data were analyzed to investigate these differences. The simulations indicate that the pooled data approach fails to estimate variabilities and produces imprecise estimates of mean kinetics. The two-stage approach produces good estimates of mean kinetics, but biased and imprecise estimates of interindividual variability. NONMEN produces accurate and precise estimates of all parameters, and also reasonable confidence intervals for them. This performance is exactly what is expected from theoretical considerations and provides empirical support for the use of NONMEM when estimating population pharmacokinetics from routine type patient data.