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PubMed2004

The back-step method--method for obtaining unbiased population parameter estimates for ordered categorical data.

A significant bias in parameters, estimated with the proportional odds model using the software NONMEM, has been reported. Typically, this bias occurs with ordered categorical data, when most of the observations are found at one extreme of the possible outcomes. The aim of this study was to assess, through simulations, the performance of the Back-Step Method (BSM), a novel approach for obtaining unbiased estimates when the standard approach provides biased estimates. BSM is an iterative method involving sequential simulation-estimation steps. BSM was compared with the standard approach in the analysis of a 4-category ordered variable using the Laplacian method in NONMEM. The bias in parameter estimates and the accuracy of model predictions were determined for the 2 methods on 3 conditions: (1) a nonskewed distribution of the response with low interindividual variability (IIV), (2) a skewed distribution with low IIV, and (3) a skewed distribution with high IIV. An increase in bias with increasing skewness and IIV was shown in parameters estimated using the standard approach in NONMEM. BSM performed without appreciable bias in the estimates under the 3 conditions, and the model predictions were in good agreement with the original data. Each BSM estimation represents a random sample of the population; hence, repeating the BSM estimation reduces the imprecision of the parameter estimates. The BSM is an accurate estimation method when the standard modeling approach in NONMEM gives biased estimates.

Bias↗

PubMed1992

Bayesian parameter estimation and population pharmacokinetics.

The widespread application of Bayesian parameter estimation in the area of therapeutic drug monitoring (TDM) has prompted the need for well conducted population studies to obtain relevant prior pharmacokinetic parameter estimates. In many cases the population has consisted of a relatively small number of subjects. This may be unavoidable for drugs used in cancer chemotherapy or in small, specific populations of patients. In contrast, information about drugs which are used extensively, such as the aminoglycosides, can be obtained by population studies which involve a large number of individuals. Indeed, this technique has proved particularly useful for determining parameter estimates which can be employed in neonatal TDM. Bayesian parameter estimation has been most frequently used for drugs with narrow therapeutic ranges such as the aminoglycosides, cyclosporin, digoxin, anticonvulsants (especially phenytoin), lithium and theophylline. However, the technique has now been extended to cytotoxic drugs, Factor VIII and warfarin. Bayesian methods have also been used to limit the number of samples required in more conventional pharmacokinetic studies with new drugs. Further advances in the use of these methods are likely to include measures of drug response and toxicity requiring population studies which also include relevant pharmacodynamic information.

Bayes Theorem↗

PubMed1985

Simulation studies of errors of parameter estimates in pharmacokinetics.

Pharmacokinetic parameters are indispensable for the development and evaluation of drugs. Therefore, they are required by international drug agencies and may influence relevant decisions. Their determination from experimental results is only possible with statistical errors. For the estimation of these errors only approximative procedures are available. A widespread approximation (information inequality) is based on specific assumptions which are invalid in practice. Erroneous estimations of unknown amount may consequently result. Alternative methods more adaptable to reality are simulation techniques with random numbers. This technique was used to calculate the statistical variability of parameter estimates for the usual compartment models. The results were compared with the variation of the parameter estimates from a descriptive model function recently developed. Adequate curves were predefined by appropriate parameter sets. Around these curves, a random generator simulated 500 data sequences. Fitting with the appertaining model function resulted in 500 sets of estimated parameters. The statistical behavior of the parameter estimates was investigated. The parameters computed by the fitting procedure--in the compartment models the rate constants--showed a large variation so that an estimate from a single data sequence does barely reproduce the "true" value of the parameter. In contrast, the parameters derived from the geometric properties of the fitted curves (e.g., area under curve, time and height of maximum) showed considerable less variation. As far as these investigations allow a conclusion, the latter parameters seem to be sufficiently reliable for practical purposes.

Half-Life↗

PubMed1989

A comparison of results from parameter estimations of impulse responses of the transient visual system.

Parameter estimations of a fourth-order linear model are applied to data from subthreshold measurements of impulse responses of the transient visual system. These impulse responses were obtained experimentally by several subjects, at two different background luminance levels and for different field sizes. The parameter estimations show consistent results over different subjects. For both different background levels and field sizes there are consequent changes in the estimated parameters. On the basis of these changes a proposal is made for a spatiotemporal model of the transient visual system.

Humans↗

PubMed1997

Computer simulation of clonal growth cancer models. I. Parameter estimation using an iterative absolute bisection algorithm.

Quantitative models of the relationship between exposure to chemical carcinogens and carcinogenic response are useful for hypothesis evaluation and risk assessment. The degree to which such models accurately depict the underlying biology is often a function of the need for mathematical tractability. When closed-form expressions are used, the need for tractability may significantly limit their complexity. This problem can be minimized by using numerical computer simulation methods to solve the model, thereby allowing more complex and realistic descriptions of the biology to be used. Unfortunately, formal methods of parameter estimation for numerical models are not as well developed as they are for analytical models. In this report, we develop a formal parameter estimation routine and apply it to a numerical clonal growth simulation (CGS) model of the growth of preneoplastic lesions consisting of initiated cells. An iterative bisection algorithm was used to estimate parameters from time-course data on the number of initiated cells and the number of clones of these cells. The algorithm successfully estimated parameter values to give a best fit to the observed dataset and was robust vis-à-vis starting values of the parameters. Furthermore, the number of data points to which the model was fit, the number of stochastic repetitions and other variables were examined with respect to their effects on the parameter estimates. This algorithm facilitates the application of CGS models for hypothesis evaluation and risk assessment by ensuring uniformity and reproducibility of parameter estimates.

Algorithms↗

PubMed2004

Statistical methods of parameter estimation for deterministically chaotic time series.

We discuss the possibility of applying some standard statistical methods (the least-square method, the maximum likelihood method, and the method of statistical moments for estimation of parameters) to deterministically chaotic low-dimensional dynamic system (the logistic map) containing an observational noise. A "segmentation fitting" maximum likelihood (ML) method is suggested to estimate the structural parameter of the logistic map along with the initial value x(1) considered as an additional unknown parameter. The segmentation fitting method, called "piece-wise" ML, is similar in spirit but simpler and has smaller bias than the "multiple shooting" previously proposed. Comparisons with different previously proposed techniques on simulated numerical examples give favorable results (at least, for the investigated combinations of sample size N and noise level). Besides, unlike some suggested techniques, our method does not require the a priori knowledge of the noise variance. We also clarify the nature of the inherent difficulties in the statistical analysis of deterministically chaotic time series and the status of previously proposed Bayesian approaches. We note the trade off between the need of using a large number of data points in the ML analysis to decrease the bias (to guarantee consistency of the estimation) and the unstable nature of dynamical trajectories with exponentially fast loss of memory of the initial condition. The method of statistical moments for the estimation of the parameter of the logistic map is discussed. This method seems to be the unique method whose consistency for deterministically chaotic time series is proved so far theoretically (not only numerically).

Journal Article↗

PubMed2006

On parameter estimation in population models.

We describe methods for estimating the parameters of Markovian population processes in continuous time, thus increasing their utility in modelling real biological systems. A general approach, applicable to any finite-state continuous-time Markovian model, is presented, and this is specialised to a computationally more efficient method applicable to a class of models called density-dependent Markov population processes. We illustrate the versatility of both approaches by estimating the parameters of the stochastic SIS logistic model from simulated data. This model is also fitted to data from a population of Bay checkerspot butterfly (Euphydryas editha bayensis), allowing us to assess the viability of this population.

Animals↗

PubMed2002

Genetic parameter estimates of yearling live animal ultrasonic measurements in Brangus cattle.

The objective of this study was to estimate genetic parameters for real-time ultrasound measurements of longissimus muscle area (LMA), 12th rib backfat thickness (FT), percent intramuscular fat (IMF), and yearling weight (YW) for 1,299 yearling Brangus bulls and heifers. A single ultrasound technician performed all measurements. The number of observations was 1,298, 1,298, 1,215, and 1,170 for LMA, FT, IMF, and YW, respectively. Genetic parameters were estimated for each trait using single- and multiple-trait derivative-free restricted maximal likelihood. Fixed effects were contemporary group (defined as same sex, same age within six months, and same environment), and days of age as a covariate. Correlations were estimated from two-trait models. Heritabilities for LMA, FT, IMF, and YW were 0.31, 0.26, 0.16, and 0.53, respectively. Genetic correlations between LMA and FT, LMA and IMF, LMA and YW, FT and IMF, FT and YW, and IMF and YW were 0.09, 0.25, 0.44, 0.36, 0.42, and 0.31, respectively. Yearling live animal ultrasonic measurements can be used as a selection tool in breeding cattle for the improvement of carcass traits.

Adipose Tissue↗

PubMed2006

Crossbreeding parameter estimation for functional longevity in rabbits using survival analysis methodology.

A complete diallel cross involving 3 maternal lines of rabbit was performed to estimate cross-breeding parameters for functional longevity. This trait was defined as the ability to delay involuntary culling. The lines considered, A, V, and Prat, had all been selected by litter size at weaning for a long period. Data were related to a total of 653 does belonging to the 9 genetic types from the diallel cross; does were reared and bred on the same commercial farm. Survival analysis was performed using a Cox proportional hazard model. The model incorporated time-dependent factors, such as year-season, litter size, and the interaction between cycle and physiological status of the female; time-independent factors, such as the genetic type of the doe; and sire and dam random factors. Crossbreeding parameters were estimated from the solutions obtained for the type of doe and its estimated variance-covariance matrix, using a generalized least squares procedure. The estimated parameters were the differences between lines in direct genetic effects and maternal genetic effects and individual heterosis. Relevant differences were observed in direct genetic effects between lines A and Prat but not in any maternal effects. Heterosis was found to be significant and favorable between lines A and Prat, and between the lines V and Prat. The magnitude of this effect was variable but important, especially in the first cross. Results stress the importance of using crosses between specialized lines to produce does for intensive meat rabbit production.

Animals↗

PubMed1993

A model of human microvascular exchange: parameter estimation based on normals and nephrotics.

A mathematical model is formulated and used to describe the distribution and transport of fluid and albumin in the human circulation, interstitium and lymphatics. Two transcapillary mass exchange mechanisms are investigated: a homoporous 'coupled Starling model', in which transcapillary albumin diffusion and convection occur within the same pathway, and a heteroporous 'plasma leak model', in which variations in structure and pressure are permitted along the length of the capillary. Parameters used in the transport models are determined based on statistical fitting of simulation predictions to experimental data from normal humans and nephrotic patients. The data consists of interstitial fluid volumes and interstitial colloid osmotic pressures as functions of plasma colloid osmotic pressure. Model validation is carried out based on comparison of (i) simulation predictions with experimental data used in parameter estimation, (ii) estimated transport parameters with experimentally determined values, and (iii) simulation predictions with a set of dynamic data from an albumin infusion study. While both models with their best-fit parameter estimates provide a good representation of experimental data, the drawbacks of the plasma leak model are three-fold: it requires more estimated parameters than the coupled Starling model, little experimental information exists with which to compare these parameters and, with the best fit values obtained, the plasma leak mechanism becomes insignificant. The model that employs a Starling-type exchange mechanism will therefore be favoured in future applications.

Arteries↗

PubMed1999

Individual based modeling and parameter estimation for a Lotka-Volterra system.

Stochastic component, inevitable in biological systems, makes problematic the estimation of the model parameters from a single sequence of measurements, despite the complete knowledge of the system. We studied the problem of parameter estimation using individual-based computer simulations of a 'Lotka-Volterra world'. Two kinds (species) of particles--X (preys) and Y (predators)--moved on a sphere according to deterministic rules and at the collision (interaction) of X and Y the particle X was changed to a new particle Y. Birth of preys and death of predators were simulated by addition of X and removal of Y, respectively, according to exponential probability distributions. With this arrangement of the system, the numbers of particles of each kind might be described by the Lotka-Volterra equations. The simulations of the system with low (200-400 particles on average) number of individuals showed unstable oscillations of the population size. In some simulation runs one of the species became extinct. Nevertheless, the oscillations had some generic properties (e.g. mean, in one simulation run, oscillation period, mean ratio of the amplitudes of the consecutive maxima of X and Y numbers, etc.) characteristic for the solutions of the Lotka-Volterra equations. This observation made it possible to estimate the four parameters of the Lotka-Volterra model with high accuracy and good precision. The estimation was performed using the integral form of the Lotka-Volterra equations and two parameter linear regression for each oscillation cycle separately. We conclude that in spite of the irregular time course of the number of individuals in each population due to stochastic intraspecies component, the generic features of the simulated system evolution can provide enough information for quantitative estimation of the system parameters.

Animals↗

PubMed2001

Parameter estimation in spatially extended systems: the Karhunen-Lóeve and Galerkin multiple shooting approach.

Parameter estimation for spatiotemporal dynamics for coupled map lattices and continuous time domain systems is shown using a combination of multiple shooting, Karhunen-Loéve decomposition and Galerkin's projection methodologies. The resulting advantages in estimating parameters have been studied and discussed for chaotic and turbulent dynamics using small amounts of data from subsystems, availability of only scalar and noisy time series data, effects of space-time parameter variations, and in the presence of multiple time scales.

Journal Article↗

PubMed1980

Analysis of errors in parameter estimation with application to physiological systems.

The accuracy of parameter estimation applied to physiological systems is analyzed. The method of analysis is applicable to procedures utilizing minimization of squared output error and a nonlinear dynamic system model. Three major sources of estimation error are described: 1) measurement error, 2) modeling error, and 3) optimization error. Measurement errors affect values used for the system output, the model input, and nonestimated parameters of the model. Modeling errors are due to failure to adequately describe the structure of the system and to numerical errors that occur in the digital computer solution of the model equations. Linearization by use of Taylor series expansions in the region of the nominal solution is used to obtain an expression for the covariance matrix of the parameter estimates in terms of the covariance matrix of each error source. The analysis is applied to the example of cardiac output estimation from respiratory measurements. The results demonstrate that an analysis of system identifiability is not sufficient to ensure usable estimates and that systematic error analysis is essential for assessing the usefulness of parameter estimation techniques.

Cardiac Output↗

PubMed2000

Parameter estimates for genetic effects on carcass traits of Korean native cattle.

Data (n = 1,746) collected from 1985 through 1995 on Korean Native Cattle by the National Livestock Research Institute of Korea were used to estimate genetic parameters for marbling score, dressing percentage, and longissimus muscle area, with backfat thickness, slaughter age, or slaughter weight as covariates. Estimates were obtained with REML. Model 1 included animal genetic and residual random effects. Model 2 was extended to include an uncorrelated random effect of the dam. Model 3 was based on Model 1 but also included sire x region x year-season interaction effects. Model 4 combined Models 2 and 3. All models included fixed effects for region x year-season and age of dam x sex combinations. From single-trait analyses, estimates of heritability with covariates to adjust for backfat thickness, slaughter age, and slaughter weight from Model 4 were, respectively, .10, .08, and .01 for marbling score; .09, .12, and .16 for dressing percentage; and .18, .17, and .24 for longissimus muscle area. From three-trait analyses, estimates of genetic correlations between marbling score and dressing percentage, marbling score and longissimus muscle area, and dressing percentage and longissimus muscle area were, respectively, -.99, .20, and -.11 with backfat thickness as covariate; -.88, .47, and .01 with slaughter age as covariate; and -.03, .39, and .91 with slaughter weight as covariate. Results of this study suggest that choice of covariate (backfat thickness, slaughter age, or slaughter weight) for the model seems to be important for carcass traits for Korean Native Cattle. Including sire x region x year-season interaction effects in the model for marbling score and dressing percentage may be important because whether sire x region x year-season interaction effects were in the model affected estimates of other variance components for the three carcass traits. Whether the maternal effect was in the model had little effect on estimates of other parameters. With backfat thickness and slaughter age end points, selection for increasing marbling score would be expected to result in decreasing dressing percentage for Korean Native Cattle. With slaughter weight as a covariate for end point, increased longissimus muscle area would be associated with increased dressing percentage, and increased marbling score would be related to increased longissimus muscle area. The differences in estimates associated with choice of end point, however, need further study.

Animals↗

PubMed1983

Parameter estimation of the threshold time function in the neural system.

An algorithm for parameter estimation is presented for the neural system model. Because of its firing mechanism analogous to that of the model based on the first time crossing problem, this problem is solved numerically for our model according to the results of Kostyukov et al. (1981). We propose the algorithm that estimates the parameters of the model considering the equivalence between the probability density function of the 1st crossing time and that of the interspike interval, which is derived from the interspike interval histogram by making use of the spline function technique. The ability of the algorithm is ensured by the application to the simulated interspike interval data. The parameter estimation is carried out also for the practical neural data recorded in the cat's optic tract fibers in both the spontaneous and the stimulated cases. These applications will show the effectiveness of the algorithm in practical cases.

Animals↗

PubMed2000

Parameter estimates for direct and maternal genetic effects on yearling, eighteen-month, and slaughter weights of Korean native cattle.

Data collected by the National Livestock Research Institute of the Rural Development Administration of Korea were used to estimate genetic parameters for yearling (YWT, n = 5,848), 18-mo (W18, n = 4,585), and slaughter (SWT, n = 2,279) weights for Korean Native cattle. Nine animal models were used to obtain REML estimates of genetic parameters: DP-2 included genetic, uncorrelated dam, and residual random effects; DQ-2 included genetic, sire x region x year-season interaction, and residual random effects; DPQ-2 was based on DQ-2 but included both interaction and dam effects; DMP-2 was based on DP-2 but with dam effect partitioned to include maternal genetic and permanent environmental effects; and DMPQ-2 was based on DMP-2 but also included sire interaction effects. Those five models included two fixed factors: region x year-season and age of dam x sex effects. Models DP-3, DQ-3, DPQ-3, and DMPQ-3 were based on DP-2, DQ-2, DPQ-2, and DMPQ-2 but included as a third fixed factor whether or not identification of the sire was known. Estimates of heritability with DMPQ-3 for YWT, with DPQ-3 for W18 and SWT when analyzed with single-trait analyses were .14, .11, and .17, respectively, and were nearly the same with bivariate analyses. Estimate of maternal heritability for YWT from single-trait analysis was .04, with estimates for other traits near zero. For bivariate analyses, the estimate for YWT was .01. With single trait analysis, estimate of the direct-maternal genetic correlation for YWT was negative (-.81). Estimates of direct genetic correlations between YWT and W18, YWT and SWT, and W18 and SWT were .99, 1.00, and .97, respectively. Estimates of environmental correlations varied from .60 to .81; the largest was between W18 and SWT. Including a fixed factor for whether sire identification was missing or not missing reduced the estimate of heritability for slaughter weight. The results suggest that the sire x region x year-season interaction is important for yearling weight and may be needed in a model for slaughter weight. Maternal effects may be of slight importance for yearling weight but of no importance for W18 and SWT. Models for national cattle evaluations for Korean Native cattle for YWT should be considered that include maternal genetic and permanent environmental as well as sire x region x year-season interaction effects, but those effects seem not to be needed for models for W18 and SWT. Not much reranking of sires occurred when ranked was based on the different models for W18 and SWT.

Animals↗

PubMed1997

Parameter estimation by reduced-order linear associative memory (ROLAM).

In a series of papers we have shown that nonlinear parameter estimation by linear association provides accurate estimates of the parameters in complex systems described by nonlinear differential equations even in the presence of additive white noise of considerable power. The technique is based on linearly associating the system's output with a set of parameter values spanning the region of interest. When an actual output is measured, the system's unknown parameters could be estimated by a matrix inversion. The size of the inverted matrix, being equal to the length of the output vector, poses a limiting factor upon the generalization of the technique. In this paper we propose a modification which requires the inversion of a matrix whose dimension equals the number of model parameters. The modified version is called reduced-order associative memory (ROLAM). The technique is applied to two complex lumped-parameter nonlinear models: the Van der Pol relaxation oscillator and the passive neuron model of the granule cells. Results validate ROLAM as a parameter-estimation tool which is especially suited in cases where the number of parameters is large, the number of samples in the observation signal is high, or when on-line parameter estimation is required. It is also shown that ROLAM provides an optimal parameter estimate in the special case of single-parameter nonlinear models.

Algorithms↗

PubMed2000

Improved kinetic parameter estimation in pH-profile data treatment.

Statistical problems in temperature stability parameter estimation have been the subject of many papers whereas statistics in, pH-profile parameter estimation have focused little attention. However, the conventional two step method used in data treatment in both cases leads to identical statistical problems. The aim of this study is then to introduce a method that improves statistics in pH-profile parameter estimation. A one step non-linear method that takes into account the errors in drug content determination is proposed. A mathematical relationship between drug content C, pH and time t is tested. The proposed method allows the estimation of the specific kinetic constants and the dissociation constant (pK(a)) in a single run. The most likely experimental initial drug contents C(0j),. where j is the index of a given experiment, are also determined. This approach that takes into account all relevant experimental information for the estimation of kinetic parameters is more rigorous from a statistical viewpoint than the classical two step methods. Kinetic data from acetylsalicylic acid (ASA) hydrolysis was used for the tests.

Algorithms↗