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PubMed1988

A more direct approach to compartmental modelling.

By first using some simple linear curve fitting method, (such as spline fitting), on data following a compartmental model, direct application of linear regression can then be made to the system of differential equations describing this data. This allows information about the parameters governing the system to be obtained from tracer and tracee measurements. It simplifies both the process of determining which parameters are estimable from various measurements, as well as the estimation process itself. Since it does not rely on knowledge of a simple closed form of the solution it has the potential to make full use of data measured over very long time periods under nonequilibrium conditions. Essentially it allows introduction of sequential estimation methods of statistics. These results can then be used to predict future substrate concentration from known substrate production, or determine substrate production from concentration measurements. A byproduct of these methods is the ability to estimate parameters in data assumed to be a finite linear combination of exponentials or sinusoids of unknown exponents or frequencies, without use of complicated nonlinear regression methods.

Animals↗

PubMed1997

Random effects probit and logistic regression models for three-level data.

In analysis of binary data from clustered and longitudinal studies, random effect models have been recently developed to accommodate two-level problems such as subjects nested within clusters or repeated classifications within subjects. Unfortunately, these models cannot be applied to three-level problems that occur frequently in practice. For example, multicenter longitudinal clinical trials involve repeated assessments within individuals and individuals are nested within study centers. This combination of clustered and longitudinal data represents the classic three-level problem in biometry. Similarly, in prevention studies, various educational programs designed to minimize risk taking behavior (e.g., smoking prevention and cessation) may be compared where randomization to various design conditions is at the level of the school and the intervention is performed at the level of the classroom. Previous statistical approaches to the three-level problem for binary response data have either ignored one level of nesting, treated it as a fixed effect, or used first- and second-order Taylor series expansions of the logarithm of the conditional likelihood to linearize these models and estimate model parameters using more conventional procedures for measurement data. Recent studies indicate that these approximate solutions exhibit considerable bias and provide little advantage over use of traditional logistic regression analysis ignoring the hierarchical structure. In this paper, we generalize earlier results for two-level random effects probit and logistic regression models to the three-level case. Parameter estimation is based on full-information maximum marginal likelihood estimation (MMLE) using numerical quadrature to approximate the multiple random effects. The model is illustrated using data from 135 classrooms from 28 schools on the effects of two smoking cessation interventions.

Clinical Trials as Topic↗

PubMed1985

Evaluation of nonlinear regression with extended least squares: simulation study.

A new approach to nonlinear least-squares regression analysis using extended least squares (ELS) was compared with three conventional methods: ordinary least squares (OLS); weighted least squares 1/C (WLS-1) and weighted least squares 1/C2 (WLS-2). With Monte Carlo simulation techniques, 3 X 200 data sets were constructed with constant proportional error (5, 10, and 15% error) and 3 X 200 with constant additive error (0.05, 0.10, and 0.15 g/mL) from an initial (perfect) data set based on known parameters. Two sampling strategies were employed: one with 17 time points and one with 10 time points. All data sets were fitted by each of the four methods, and parameter estimation bias was assessed by comparing the mean parameter estimate with the known value. The relative precision of each method was investigated by examination of the absolute deviations of each individual parameter estimate from the known value. ELS performed as well as the appropriate weighting scheme (WLS-2 for constant proportional error sets and OLS for constant additive error sets) and was superior with regard to both bias and precision to less appropriate methods.

Kinetics↗

PubMed2004

Simulation of post-dialysis urea rebound using regional flow model.

BACKGROUND: A regional flow model (RFM) can establish the missing link between hemodynamics and solute removal. We tried to simulate post-dialysis urea rebound using a RFM for the purpose of evaluating the validity of this model. METHODS: Eight patients on maintenance hemodialysis with negligible renal function were investigated. The parameters of the RFM were estimated so as to fit the calculated values of urea nitrogen to the measured values during a dialysis session. The estimated parameters were total urea distribution volume (TUV), systemic blood flow (Qsys), flow fraction (fQH) and volume fraction (fVH) of the high-flow system. Thirteen types of parameter sets were used for the estimation. The urea rebound at 60 min after a dialysis session (Creb) and the rebound ratio (RR) were calculated using these estimated parameters. The accuracy of the calculated Creb and RR was assessed. RESULTS: The accuracy of Creb and RR determined using estimated TUV, by taking Qsys as systemic blood flow calculated from ultrasonic echo cardiogram (Qucg), fQH as 0.8, and fVH as 0.2, was insufficient (method 1a). The accuracy of these values was significantly increased by taking fQH as 0.85 (method 1b). The estimation of Qsys with TUV did not improve the accuracy of Creb and RR (methods 2a and 2b). The estimation of fQH, fVH, and TUV (method 8) increased the accuracy of Creb and RR significantly compared with method 1a, but not compared with method 1b. Even with method 1b or method 8, the percentage RR was less than 90% in two patients. CONCLUSIONS: By taking fQH as 0.85, an acceptably accurate simulation of urea rebound can be accomplished with the necessity to estimate only TUV. The simulation was not significantly improved by the estimation of Qsys, fQH, and fVH. The RFM is useful in practice, although it has some limitations.

Blood Urea Nitrogen↗

PubMed2005

Sampling plans for fitting the psychometric function.

Research on estimation of a psychometric function psi has usually focused on comparing alternative algorithms to apply to the data, rarely addressing how best to gather the data themselves (i.e., what sampling plan best deploys the affordable number of trials). Simulation methods were used here to assess the performance of several sampling plans in yes-no and forced-choice tasks, including the QUEST method and several variants of up-down staircases and of the method of constant stimuli (MOCS). We also assessed the efficacy of four parameter estimation methods. Performance comparisons were based on analyses of usability (i.e., the percentage of times that a plan yields usable data for the estimation of all the parameters of psi) and of the resultant distributions of parameter estimates. Maximum likelihood turned out to be the best parameter estimation method. As for sampling plans, QUEST never exceeded 80% usability even when 1000 trials were administered and rendered accurate estimates of threshold but misestimated the remaining parameters. MOCS and up-down staircases yielded similar and acceptable usability (above 95% with 400-500 trials) and, although neither type of plan allowed estimating all parameters with optimal precision, each type appeared well suited to estimating a distinct subset of parameters. An analysis of the causes of this differential suitability allowed designing alternative sampling plans (all based on up-down staircases) for yes-no and forced-choice tasks. These alternative plans rendered near optimal distributions of estimates for all parameters. The results just described apply when the fitted psi has the same mathematical form as the actual psi generating the data; in case of form mismatch, all parameters except threshold were generally misestimated but the relative performance of all the sampling plans remained identical. Detailed practical recommendations are given.

Algorithms↗

PubMed2006

Symbolic-numeric estimation of parameters in biochemical models by quantifier elimination.

The sequencing of complete genomes allows analyses of the interactions between various biological molecules on a genomic scale, which prompted us to simulate the global behaviors of biological phenomena on the molecular level. One of the basic mathematical problems in the simulation is the parameter optimization in the kinetic model for complex dynamics, and many estimation methods have been designed. We introduce a new approach to estimate the parameters in biological kinetic models by quantifier elimination (QE), in combination with numerical simulation methods. The estimation method was applied to a model for the inhibition kinetics of HIV proteinase with ten parameters and nine variables, and attained the goodness of fit to 300 points of observed data with the same magnitude as that obtained by the previous estimation methods, remarkably by using only one or two points of data. Furthermore, the utilization of QE demonstrated the feasibility of the present method for elucidating the behavior of the parameters and the variables in the analyzed model. Therefore, the present symbolic-numeric method is a powerful approach to reveal the fundamental mechanisms of kinetic models, in addition to being a computational engine.

Algorithms↗

PubMed2006

A combined FEM/genetic algorithm for vascular soft tissue elasticity estimation.

Tissue elasticity reconstruction is a parameter estimation effort combining imaging, elastography, and computational modeling to build maps of soft tissue mechanical properties. One application is in the characterization of atherosclerotic plaques in diseased arteries, wherein the distribution of elastic properties is required for stress analysis and plaque stability assessment. In this paper, a computational scheme is proposed for elasticity reconstruction in soft tissues, combining finite element modeling (FEM) for mechanical analysis of soft tissues and a genetic algorithm (GA) for parameter estimation. With a model reduction of the discrete elasticity values into lumped material regions, namely the plaque constituents, a robust, adaptive strategy can be used to solve inverse elasticity problems involving complex and inhomogeneous solution spaces. An advantage of utilizing a GA is its insistence on global convergence. The algorithm is easily implemented and adaptable to more complex material models and geometries. It is meant to provide either accurate initial guesses of low-resolution elasticity values in a multi-resolution scheme or as a replacement for failing traditional elasticity estimation efforts.

Algorithms↗

PubMed2002

Comparison of two population pharmacokinetic programs, NONMEM and P-PHARM, for tacrolimus.

OBJECTIVES: To compare the population modelling programs NONMEM and P-PHARM during investigation of the pharmacokinetics of tacrolimus in paediatric liver-transplant recipients. METHODS: Population pharmacokinetic analysis was performed using NONMEM and P-PHARM on retrospective data from 35 paediatric liver-transplant patients receiving tacrolimus therapy. The same data were presented to both programs. Maximum likelihood estimates were sought for apparent clearance (CL/F) and apparent volume of distribution (V/F). Covariates screened for influence on these parameters were weight, age, gender, post-operative day, days of tacrolimus therapy, transplant type, biliary reconstructive procedure, liver function tests, creatinine clearance, haematocrit, corticosteroid dose, and potential interacting drugs. RESULTS: A satisfactory model was developed in both programs with a single categorical covariate--transplant type--providing stable parameter estimates and small, normally distributed (weighted) residuals. In NONMEM, the continuous covariates--age and liver function tests--improved modelling further. Mean parameter estimates were CL/F (whole liver) = 16.3 l/h, CL/F (cut-down liver) = 8.5 l/h and V/F = 565 l in NONMEM, and CL/F = 8.3 l/h and V/F = 155 l in P-PHARM. Individual Bayesian parameter estimates were CL/F (whole liver) = 17.9 +/- 8.8 l/h, CL/F (cut-down liver) = 11.6 +/- 8.8 l/h and V/F = 712 +/- 792 l in NONMEM, and CL/F (whole liver) = 12.8 +/- 3.5 l/h, CL/F (cut-down liver) = 8.2 +/- 3.4 l/h and V/F = 221 +/- 164 l in P-PHARM. Marked interindividual kinetic variability (38-108%) and residual random error (approximately 3 ng/ml) were observed. P-PHARM was more user friendly and readily provided informative graphical presentation of results. NONMEM allowed a wider choice of errors for statistical modelling and coped better with complex covariate data sets. CONCLUSION: Results from parametric modelling programs can vary due to different algorithms employed to estimate parameters, alternative methods of covariate analysis and variations and limitations in the software itself.

Adolescent↗

PubMed2002

Estimation of parameters and unobserved components for nonlinear systems from noisy time series.

We study the problem of simultaneous estimation of parameters and unobserved states from noisy data of nonlinear time-continuous systems, including the case of additive stochastic forcing. We propose a solution by adapting the recently developed statistical method of unscented Kalman filtering to this problem. Due to its recursive and derivative-free structure, this method minimizes the cost function in a computationally efficient and robust way. It is found that parameters as well as unobserved components can be estimated with high accuracy, including confidence bands, from heavily noise-corrupted data.

Journal Article↗

PubMed2000

Estimating the parameters of aerobic function during exercise using an exponentially increasing work rate protocol.

A new exercise protocol has been proposed, with respect to cardiopulmonary exercise testing, which starts at a low work rate (WR) and increases exponentially by a standard percentage of the previous work rate every minute: the test is termed STEEP (standardised exponential exercise protocol). The potential advantage of this protocol is that it can accommodate a wide range of subjects, since it allows a maximum to be attained with a relatively narrow variation of tolerance time, regardless of subjects exercise capacity. To date, only the VO2max has been compared with that from the current standard ramp protocol. The ramp, however, also allows other important parameters of aerobic function to be estimated: the anaerobic threshold (AT); the response time constant; and delta VO2/delta WR. The aim of this study was, therefore, to clarify whether these aerobic parameters can be readily discerned from the responses to the STEEP protocol both from a theoretical and practical viewpoint. As a result of theoretical considerations, we demonstrated that the VO2 time constant and delta VO2/delta WR may not both be estimated uniquely. As a practical expedient, a procedure was proposed for estimating the parameter analogues. The preliminary results for six subjects between the STEEP and ramp protocols showed consistent positive correlation for VO2max (r = 0.997) and AT-VO2 (r = 0.980), whereas the correlation for the VO2 time constant and delta VO2/delta WR were not significant. Further study is needed to clarify the reason(s) for the discrepancies both from a theoretical and practical viewpoint.

Clinical Protocols↗

PubMed1975

Simultaneous estimation of parameters in different linear models and applications to biometric problems.

Empirical Bayes procedure is employed in simultaneous estimation of vector parameters from a number of Gauss-Markoff linear models. It is shown that with respect to quadratic loss function, empirical Bayes estimators are better than least squares estimators. While estimating the parameter for a particular linear model, a suggestion has been made for distinguishing between the loss due to decision maker and the loss due to individual. A method has been proposed but not fully studied to achieve balance between the two losses. Finally the problem of predicting future observations in a linear model has been considered.

Bayes Theorem↗

PubMedDate not supplied

Estimation of parameters of the inflated geometric distribution for rural out-migration.

"Several attempts have been made in the past at studying trends in rural-urban migration through the use of probability models.... In this paper, an alternative method (based on maximum likelihood) of estimating the parameters of the inflated geometric distribution is proposed.... Section 2 describes the model. The method of estimating the parameters and the associated variance-covariance matrix is contained in section 3. Section 4 gives an illustrative example. Section 5 contains the conclusion." Data are from a 1978 survey conducted in three types of villages in India.

Asia↗

PubMed1992

Methods for estimating the parameters of a linear model for ordered categorical data.

In many empirical analyses, the response of interest is categorical with an ordinal scale attached. Many investigators prefer to formulate a linear model, assigning scores to each category of the ordinal response and treating it as continuous. When the covariates are categorical, Haber (1985, Computational Statistics and Data Analysis 3, 1-10) has developed a method to obtain maximum likelihood (ML) estimates of the parameters of the linear model using Lagrange multipliers. However, when the covariates are continuous, the only method we found in the literature is ordinary least squares (OLS), performed under the assumption of homogeneous variance. The OLS estimates are unbiased and consistent but, since variance homogeneity is violated, the OLS estimates of variance can be biased and may not be consistent. We discuss a variance estimate (White, 1980, Econometrica 48, 817-838) that is consistent for the true variance of the OLS parameter estimates. The possible bias encountered by using the naive OLS variance estimate is discussed. An estimated generalized least squares (EGLS) estimator is proposed and its efficiency relative to OLS is discussed. Finally, an empirical comparison of OLS, EGLS, and ML estimators is made.

Abnormalities, Drug-Induced↗

PubMed2000

Estimation of kinetic parameters by progress curve analysis for the synthesis of (R)-mandelonitrile by Prunus amygdalus hydroxynitrile lyase.

Consistent sets of kinetic parameters were estimated for the synthesis of (R)-mandelonitrile, catalyzed by Prunus amygdalus hydroxynitrile lyase, at 5 and 25 degrees C and pH 5.5 by progress curve analysis. The rate constants and equilibrium constants of the nonenzymatic reaction were determined separately to reduce the number of parameters to be estimated simultaneously. At a lower temperature the equilibrium is much more favorable and the formation of rac-mandelonitrile by the nonenzymatic reaction is suppressed. The estimated kinetic parameters were used to identify that the rate determining step in the catalytic cycle is the release of (R)-mandelonitrile from the ternary complex.

Journal Article↗

PubMed1992

Analysis of algebraic weighted least-squares estimators for enzyme parameters.

An algorithm for the least-squares estimation of enzyme parameters Km and Vmax. is proposed and its performance analysed. The problem is non-linear, but the algorithm is algebraic and does not require initial parameter estimates. On a spreadsheet program such as MINITAB, it may be coded in as few as ten instructions. The algorithm derives an intermediate estimate of Km and Vmax. appropriate to data with a constant coefficient of variation and then applies a single reweighting. Its performance using simulated data with a variety of error structures is compared with that of the classical reciprocal transforms and to both appropriately and inappropriately weighted direct least-squares estimators. Three approaches to estimating the standard errors of the parameter estimates are discussed, and one suitable for spreadsheet implementation is illustrated.

Algorithms↗

PubMed2001

Bias in estimating association parameters for longitudinal binary responses with drop-outs.

This paper considers the impact of bias in the estimation of the association parameters for longitudinal binary responses when there are drop-outs. A number of different estimating equation approaches are considered for the case where drop-out cannot be assumed to be a completely random process. In particular, standard generalized estimating equations (GEE), GEE based on conditional residuals, GEE based on multivariate normal estimating equations for the covariance matrix, and second-order estimating equations (GEE2) are examined. These different GEE estimators are compared in terms of finite sample and asymptotic bias under a variety of drop-out processes. Finally, the relationship between bias in the estimation of the association parameters and bias in the estimation of the mean parameters is explored.

Algorithms↗

PubMed2003

An approach to improve the offshore platform coordinates accuracy by using multichannel Kalman filtering.

In this paper, multichannel Kalman filters for estimation of offshore platform (OP) coordinates are designed. The complete OP motion is assumed to be composed of the low-frequency motion caused by the wind and undercurrent, and the high-frequency motion caused by the sea. The mathematical model of the low-frequency OP motion is given by the normal differential equation system, and the high-frequency OP motion is represented by a moving-average multivariable autoregression model. The parameter estimation problem for the model of the low-frequency OP motion, on which the in-service control is performed, is solved through two jointly operating Kalman filters: the first one is for the estimation of the parameters of the low-frequency motion model, and the second one is for the parameter estimation for the high-frequency model. The parameters of the first filter are automatically adapted to variations of the second filter, i.e., they are adapted to disturbances from the sea. Two algorithms for the OP motion parameter estimation (parallel and with preliminary data compression). employed for several measuring channels data estimation, are developed, and simulated on a computer. Some recommendations on their use are given.

Algorithms↗

PubMed1984

Experience with NONMEM: analysis of routine phenytoin clinical pharmacokinetic data.

NONMEM, a program package the produces the extended least squares estimates of population parameters for a nonlinear mixed-effect model, has been applied to two data sets from patients routinely receiving phenytoin. A general model for the data is proposed. The models used in previous, standard-method analyses of each data set are compared to the general model using NONMEM. The comparison involves two questions: The first asks whether the parameters estimated previously agree with NONMEM estimates when the original model is used. We find that for fixed-effect parameters they generally do, while for interindividual random-effect parameters the previous methods' estimates appear upward biased relative to NONMEM. Second, the original model per se is compared to the general model by comparing the best fit to each. The general model is clearly superior. NONMEM's ability to distinguish among models, and to precisely estimate their parameters from sparse individual data, is illustrated and verified.

Computers↗