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Statistical methods such as regression analysis are often used to estimate cell kinetic parameters, such as cell birth rate or tumour growth rate. The standard errors which the statistical procedures provide may be considerable underestimates of the true uncertainty surrounding the parameter of interest.
A recent series of papers has raised issues regarding estimation of the key epidemiological parameters of variant Creutzfeldt-Jakob disease (vCJD) from fitting survival models to case data. In particular, it was stated that the scale of the epidemic cannot be estimated and must be fixed in any analysis. We show that this problem is an artefact of the approximate likelihood used in these papers to facilitate model-fitting, and is not a concern if estimation is based on the full likelihood. We also address comments in these papers regarding the choice of incubation period distribution.
By formalizing the relation among components and 'borrowing information' among them, Bayes and empirical Bayes methods can produce more valid, efficient and informative statistical evaluations than those based on traditional methods. In addition, Bayesian structuring of complicated models and goals guides development of appropriate statistical approaches and generates summaries which properly account for sampling and modelling uncertainty. Computing innovations enable implementation of complex and relevant models, thereby substantially increasing the role of Bayes/empirical Bayes methods in important statistical assessments. Policy-relevant statistical assessments involve synthesis of information from a set of related components such as medical clinics, geographic regions or research studies. Typical assessments include inference for individual parameters, synthesis over the collection of components (for example, the parameter histogram) and comparisons among parameters (for example, ranks). The relative importance of these goals depends on the context. Bayesian structuring provides a guide to valid inference. For example, while posterior means are the 'obvious' and optimal estimates for individual components under squared error loss, their empirical distribution function (EDF) is underdispersed and never valid for estimating the EDF of the true, underlying parameters. Effective histogram estimates result from optimizing a loss function based in a distance between the histogram and its estimate. Similarly, ranking observed data usually produces poor estimates and ranking posterior means can be inappropriate. Effective estimates should be based on a loss function that caters directly to ranks. Using examples of 'borrowing information', shrinkage and the variance/bias trade-off we motivate Bayes and empirical Bayes analysis. Then, we outline the formal approach and discuss 'triple-goal' estimates with values that when ranked produce optimal ranks, for which the EDF is an optimal estimate of the parameter EDF and such that the values themselves are effective estimates of co-ordinate-specific parameters. We use basic models and data analysis examples to highlight the conceptual and structural issues.
The observations in an experiment define a set of observational parameters that are functions of the basic kinetic parameters of the model of the system. The problem of identifiability is concerned with whether the observational parameters uniquely specify the basic kinetic parameters. As such, it depends only on the functional relation between the two levels of parameters and not on errors of observation and the estimation procedure. It should be checked before doing the experiment. Given initial estimates of the basic kinetic parameters, identifiability can be checked, in a local sense, from data generated by simulating the experiment on the model.
A method is developed to evaluate system stability for the CO2 control of breathing in individuals by using data from the dynamics of CO2 rebreathing and elimination. The theoretical basis of the method is a physiological model of the CO2 respiratory control system and an explicit stability index (SI). The SI is algebraically related to the model parameters: system volume (Vs), cardiac output (Q), circulatory transit time (ts), and controller gain (G). A sequential optimization scheme is shown to yield estimates of the model parameters by comparing the alveolar ventilation and PCO2 of the model output with corresponding experimental data. Model simulation of CO2 rebreathing and elimination with different parameter values demonstrate that all parameters except ts have significant effect on the outputs. Least-squares estimation of the parameters using model-generated data with added noise showed good precision for all parameters (except ts). This analysis is performed with parameter values chosen to produce overdamped and underdamped responses that would occur in normal and abnormal respiratory control systems, respectively. It is anticipated that SI values of the (overdamped) normal and (underdamped) abnormal systems differ by much more than the variation produced by imprecision of the parameter estimates. For this circumstance, the method is expected to be sensitive enough to distinguish normal from abnormal CO2 respiratory control of individual subjects.
Three algorithms for scatter compensation in Tc-99m brain single-photon emission computed tomography (SPECT) were optimized and compared on the basis of the accuracy and precision with which lesion and background activity could be simultaneously estimated. These performance metrics are directly related to the clinically important tasks of activity quantitation and lesion detection, in contrast to measures based solely on the fidelity of image pixel values. The scatter compensation algorithms were (a) the Compton-window (CW) method with a 20% photopeak window, a 92-126 keV scatter window, and an optimized "k-factor," (b) the triple-energy window (TEW) method, with optimized widths of the photopeak window and the abutting scatter window, and (c) a general spectral (GS) method using seventeen 4 keV windows with optimized energy weights. Each method was optimized by minimizing the sum of the mean-squared errors (MSE) of the estimates of lesion and background activity concentrations. The accuracy and precision of activity estimates were then determined for lesions of different size, location, and contrast, as well as for a more complex Bayesian estimation task in which lesion size was also estimated. For the TEW and GS methods, parameters optimized for the estimation task differed significantly from those optimized for global normalized pixel MSE. For optimal estimation, the CW bias of activity estimates was larger and varied more (-2% to 22%) with lesion location and size than that of the other methods. The magnitude of the TEW bias was less than 7% across most conditions, although its precision was worse than that of CW estimates. The GS method performed best, with bias generally less than 4% and the lowest variance; its root-mean square (rms) estimation error was within a few percent of that achievable from primary photons alone. For brain SPECT, estimation performance with an optimized, energy-based, subtractive correction may approach that of an ideal scatter-rejection procedure.
Accurate estimates of human body segment parameters (BSPs) are required for kinetic analyses of motion. The purpose of this study was to develop a geometric model of the human lower leg based on the mass distribution properties of the segment. Forty subjects were recruited from 4 human populations. Each population was randomly divided equally into model development (MD) and model validation (MV) groups. Participants underwent frontal and sagittal plane dual energy X-ray absorptiometry (DEXA) scans and anthropometric measurements. Leg BSPs were calculated from the scan information and mass distribution properties in the two planes were determined. Further, a geometric model was developed based on the ensemble averages of the mass distribution information from the MD groups. The model was applied to the MV groups and mean absolute errors were calculated for each BSP and each population. Finally, BSP estimates from literature sources were also determined and compared against DEXA. The model developed produced the lowest errors overall. Additionally, the results showed that the model developed estimated BSPs for all four populations with consistent accuracy whereas the other 4 models tested provided different levels of accuracy depending on the age and gender categories of the group tested. The results of this study present a model that accurately estimates BSPs of the lower leg for individuals varying in age, gender, race, and morphology. This study also presents a modelling technique that may successfully provide similar results for other body segments.
Genetic parameters were estimated using REML with animal models for number of lambs born and 18-mo body weight in Rambouillet sheep. Number of lambs born was modeled either as repeated measurements on the same trait or as different traits at different ages. The original data for number of lambs born were separated according to age of the ewe into two classes: 2 and 3 yr, and older than 3 yr. Numbers of ewes with lambing records for the age classes were 653 and 466 with 1,106 and 1,118 records, respectively. For the data set that included all ages, the number of ewes was 684 with 2,224 records, and for 18-mo body weight the number of ewes measured was 557. For number of lambs born, the animal model included random genetic, permanent environmental, and residual environmental effects and fixed effects for age of ewe, year of lambing, and month of year of lambing. Lambing day within season was used as a covariate. For 18-mo body weight, year of birth of ewe was used as a fixed effect. Actual age in days when the ewe was weighed was used as a covariate. Estimates of heritability for number of lambs born by age group were .04, for 2- and 3-yr old ewes, and .06, for ewes greater than 3 yr old, from the two-trait (two age of ewe classes) analyses and .06 when all ages were included. Estimates of heritability for number of lambs born from the single-trait analyses were somewhat less than estimates from two-trait analyses. Estimate of genetic correlation between number of lambs born for the 2 and 3 yr and the >3 yr classes was near unity (1.00), which suggests that a repeated measures model for number of lambs born is adequate for making selection decisions. Estimate of genetic correlation between number of lambs born and 18-mo body weight was .35 with a heritability estimate of .48 for 18-mo body weight. The estimate of genetic correlation suggests that selection for increased number of lambs born would result in increased 18-mo body weight.
The actual values of internal metabolites and fluxes can be measured by a number of experimental techniques and they provide important information for evaluating the properties of a metabolic pathway in situ. In this paper we propose a strategy to properly exploit this information. The suggested approach permits estimation of a set of parameters on the whole system so that a useful model can be constructed and used to describe its components and systemic properties and to predict its behavior under new conditions. A simulated reference pathway is provided to validate this method and to show its utility in metabolic studies.
Accurate estimations of body segment inertial parameters (BSPs) are required to calculate the kinetics of motion. The purpose of this study was to develop a geometric model of the human thigh segment based on mass distribution properties determined from dual energy x ray absorptiometry (DEXA). One hundred subjects from four populations underwent a DEXA scan and anthropometric measurements were taken. The mass distribution properties of the thigh segment were determined for 20 subjects, a geometric model was developed, and the model was applied to the remaining 80 subjects. The model was validated by comparing to benchmark DEXA measurements. Four other popular models in the literature were also evaluated in the same manner No one set of predictors performed best for a particular group or BSP, however modeling the mass distribution properties of the segment allows the assumption of constant density while still accurately representing the inertial properties of the segment and provides promise for future development of BSP models.
An optimization method for the sensitivity of diffraction efficiency measurements is presented. I define the sensitivity as the estimation precision of the grating parameters. The optimization method called sensitivity analysis for fitting scans all the possible measurement configurations and selects the configuration that yields the best sensitivity. The scan is made over the domain of the experimental parameters of the arrangement, such as the azimuth angle of the grating and the orientation angles of the analyzer and the polarizer. These parameters can be freely varied, and among the multitude of possible combinations there is one configuration that provides optimum sensitivity. Comparison with experimental results reveals a qualitative agreement between theory and practice.
Selection of dairy sires is based on the production records of their female ancestors, half-sibs and daughters. No trait expressed by the sire is used. Concentrations of growth hormone (GH) and prolactin (PRL), hormones produced in both males and females that are fundamental in lactation, may be correlated with production. A study was conducted to determine whether measures of these hormones in the sire would be useful predictors of lactational ability of daughters. Blood samples were collected at 15-min intervals for 8 h from 26 Holstein bulls (5.5 yr of age) that had one progeny summary available. Plasma concentrations of GH and PRL were quantified and the mean and baseline concentrations and the frequency and mean amplitude of the secretory peaks were determined for each bull. Concentrations among these values and bulls' predicted differences (PD) were determined. Significant negative correlations were detected for frequency of GH peaks and PD for yield of milk, fat and protein; correlations were positive for PRL baseline concentrations and PD for fat and protein (P less than .10), and correlations were negative for frequency of PRL peaks and PD for milk, fat and protein (P less than .10). Addition of estimates of bull hormone secretory parameters to breeding values based on performance of relatives considerably improved the accuracy (R2) for predicting progeny performance from sire information. Certain characteristics of the patterns of GH and PRL secretion may be heritable and aid in identification of superior dairy animals.
We present here a direct least-squares estimation (DLSE) method for the determination of renal kinetic parameters from sequences of very fast acquisitions performed with a three-headed single photon emission computed tomography (SPECT) system. A simple linear model for the behavior of the radiopharmaceutical, as well as a spatial model for its spatial distribution are defined. The model enables one to estimate the kinetic parameters directly from the projections, once the plasma concentration function is known. A new technique for the accurate reconstruction of time-radioactivity curves based on the direct reconstruction of the region-of-interest contents from a series of data from three-projections is presented. The technique is used to determine the plasma concentration function with a sub-second time resolution. The spatially-variant geometrical response is also included in the model to compensate for the spatial resolution of the SPECT system. Results obtained from simulations are presented. Basic spatial and time features of the simulations are derived from a patient study. Noise and segmentation errors are also simulated. The DLSE method is compared with the conventional one of deriving kinetic parameters from the time series of reconstructed images. The standard deviation of results given by DLSE is less than 2%, whereas with the conventional method it is between 5% and 6%. Within the limit of statistical fluctuations, DLSE results are unbiased whereas those of the conventional method are overestimated by 24%.
The amplitude and frequency contents of a received ultrasound pulse depends on the spatial pressure amplitude distribution of the sound field, as produced by the transducer in the medium, and on the position and orientation of the reflecting surface in this field. Often, the geometry of the reflecting surface and the acoustical properties of the medium are known or can be estimated. Then it is practical to determine the transducer parameters in order to calculate the distortion of a reflected ultrasound pulse. Apart from geometrical parameters such as size and mechanical focusing, the surface velocity amplitude distribution (SVAD) of a transducer is of major importance. For some transducer configurations this SVAD may be described with a limited number of parameters. This paper presents a method, based on a Kalman filter algorithm, to assess the transducer parameters by measurement of the spatial sound field pressure amplitude distribution in various planes for different emission frequencies. Comparison of the measured pressure amplitudes with those calculated using the estimated values of the parameters shows that this method yields reliable results.
A novel method is described for solving systems of differential equations pertaining to organism development, where this development is assumed to be directly influenced by fluctuation in measurable environmental variables. The system parameters are written as functions of these variables and, because these functions involve the accumulation of "environment time" (e.g., "day-degrees"), the system is therefore regulated by the prevailing environmental conditions. This method contrasts with the more usual descriptions of development along a time-line. The parameters of the differential equations involved are estimated by modelling data, which show evidence of changes in the dependent variable(s), i.e. the components of the system. They are expressed in terms of their response to continuous fluctuations in one or more independent, environmental variables. Accumulated thermal time (including day-degrees) or more complex units may be derived by using either linear or nonlinear functions. Critical environmental parameters such as the basal thresholds of a given developmental process or parameters describing a nonlinear relationship with the environment may then be estimated. This paper develops the methodology of this environmentally driven approach to describing organism development in general terms, and gives a specific example of its application with reference to the cellular development within the secondary vascular tissues in the stems of young hybrid aspen trees.
Estimates of direct and maternal genetic parameters in beef cattle were obtained with a random regression model with a linear spline function (SFM) and were compared with those obtained by a multitrait model (MTM). Weight data of 18,900 Gelbvieh calves were used, of which 100, 75, and 17% had birth (BWT), weaning (WWT), and yearling (YWT) weights, respectively. The MTM analysis was conducted with a three-trait maternal animal model. The MTM included an overall linear partial fixed regression on age at recording for WWT and YWT, and direct-maternal genetic and maternal permanent environmental effects. The SFM included the same effects as MTM, plus a direct permanent environmental effect and heterogeneous residual variance. Three knots, or breakpoints, were set to 1, 205, and 365 d. (Co)variance components in both models were estimated with a Bayesian implementation via Gibbs sampling using flat priors. Because BWT had no variability of age at recording, there was good agreement between corresponding components of variance estimated from both models. For WWT and YWT, with the exception of the sum of direct permanent environmental and residual variances, there was a general tendency for SFM estimates of variances to be lower than MTM estimates. Direct and maternal heritability estimates with SFM tended to be lower than those estimated with MTM. For example, the direct heritability for YWT was 0.59 with MTM, and 0.48 with SFM. Estimated genetic correlations for direct and maternal effects with SFM were less negative than those with MTM. For example, the direct-maternal correlation for WWT was -0.43 with MTM and -0.33 with SFM. Estimates with SFM may be superior to MTM due to better modeling of age in both fixed and random effects.
In experiments on biological systems one often cannot measure all state variables (compartments). Given a particular experiment of that type, a basic kinetic parameter may have no effect on the observations; such a parameter is an insensible parameter for that experiment. A parameter may influence the observations and not be uniquely determinable; such a parameter is nonidentifiable for that experiment. Only identifiable parameters can be estimated uniquely, by that experiment. I review the basic theory to check identifiability for a nominal value of a parameter (local identifiability), and present some examples of problems that may arise in estimation.
The system that regulates plasma calcium in the bird has been formalized into a model based on a series of differential equations and solved by computer simulation. Bone, kidney, and intestine have been considered as the control subsystems, with parathyroid hormone and 1,25-dihydroxycholecalciferol as the regulating hormones. The parameters used in the simulation model have been computed either from published results or by specifically designed experiments described here. For the estimation of parameters, an iterative procedure has been developed that was designed to minimize the sum of square errors between observed and system-simulated values. Parameters of 1,25-dihydroxycholecalciferol metabolism were experimentally obtained from the kinetic behavior of the 3H-labeled hormone in rachitic birds after a single dose. Model parameters have been adjusted using the results of in vivo calcium loading and validated by an EDTA infusion experiment. The simulation model has been used to study the hierarchy of the activities of the three control subsystems and of the regulating hormones, at different calcium intakes. Positive or negative errors in plasma calcium resulted in an asymmetry in the activities of the controlling systems, bone and kidney, whereas the intestine is characterized by its relatively long response time.