Statistical Analysis with Missing Data

We propose a general, nonlinear mixed effects model for repeated measures data and define estimators for its parameters. The proposed estimators are a natural combination of least squares estimators for nonlinear fixed effects models and maximum likelihood (or restricted maximum likelihood) estimators for linear mixed effects models. We implement Newton-Raphson estimation using previously developed computational methods for nonlinear fixed effects models and for linear mixed effects models. Two examples are presented and the connections between this work and recent work on generalized linear mixed effects models are discussed.

https://www.jstor.org/stable/pdfplus/2532087.pdf

Nonlinear Mixed Effects Models for Repeated Measures Data

PKSolver: An add-in program for pharmacokinetic and pharmacodynamic data analysis in Microsoft Excel

Many types of data are best analyzed by fitting a curve using nonlinear regression, and computer programs that perform these calculations are readily available. Like every scientific technique, however, a nonlinear regression program can produce misleading results when used inappropriately. This article reviews the use of nonlinear regression in a practical and nonmathematical manner to answer the following questions: Why is nonlinear regression superior to linear regression of transformed data? How does nonlinear regression differ from polynomial regression and cubic spline? How do nonlinear regression programs work? What choices must an investigator make before performing nonlinear regression? What do the final results mean? How can two sets of data or two fits to one set of data be compared? What problems can cause the results to be wrong? This review is designed to demystify nonlinear regression so that both its power and its limitations will be appreciated.

Fitting curves to data using nonlinear regression: a practical and nonmathematical review.

Mathematical fundamentals on pharmacokinetics introduction to biopharmaceutics and pharmacokinetics one compartment model physiological factors relating to drug absorption biopharmaceutic considerations in drug product design modified release drug products delivery of biotechnical products and targetted drug delivery systems pharmacokinetics of drug absorption bioavailability and bioequivalence multicompartmental models drug distribution and protein binding drug clearance drug metabolism and hepatic clearance pharmacokinetics after intravenous infusion pharmacokinetics after multiple dosing non-linear pharmacokinetics introduction to clinical pharmacokinetics application of pharmacokinetics to clinical situations relationship between pharmacokinetic parameters and pharmacologic response physiologic pharmacokinetic model and statistical moment computer applications in pharmacokinetics biopharmaceutics and pharmacokinetic studies in drug development appendix.

Applied biopharmaceutics and pharmacokinetics

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.

Some suggestions for measuring predictive performance

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.

Ways to fit a PK model with some data below the quantification limit.

Individual pharmacokinetic 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 were 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 μg/ml by the naive pooled data, two-stage, and NONMEM 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 appproach produces good estimates of mean kinetics, but biased and imprecise estimates of interindividual variability. NONMEM 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.

Evaluation of methods for estimating population pharmacokinetic parameters. I. Michaelis-menten model: Routine clinical pharmacokinetic data

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.

Forecasting individual pharmacokinetics

This paper will review the methods that have been advanced for the estimation of parameters of models quantifying the population characteristics of the kinetic behavior of endogenous and exogenous substances in individuals of the population. Such methods are used frequently, for example, in pharmacokinetic studies. In certain populations, especially biological ones, considerable kinetic variability between population members is present. The models with which we are concerned describe this variability. Some interindividual kinetic variability may be explainable on the basis of measureable concomitant variables, but much of it may remain unexplainable on such a basis. We give this matter particular attention. The assumptions underlying the models are critically discussed.

Stuart L. Beal

Papers

Some suggestions for measuring predictive performance

Ways to fit a PK model with some data below the quantification limit.

Evaluation of methods for estimating population pharmacokinetic parameters. I. Michaelis-menten model: Routine clinical pharmacokinetic data

Forecasting individual pharmacokinetics

Estimating population kinetics.