A data assimilation technique applied to a predator-prey model

A new approach for data assimilation, which is based on the adjoint method, but allows the computer code for the adjoint to be constructed directly from the model computer code, is described. This technique is straightforward and reduces the chance of introducing errors in the construction of the adjoint code. Implementation of the technique is illustrated by applying it to a simple predator-prey model in a model fitting mode. A series of identical twin numerical experiments are used to show that this data assimilation approach can successfully recover model parameters as well as initial conditions. However, the ease with which these values are recovered is dependent on the form of the model equations as well as on the type and amount of data that are available. Additional numerical experiments show that sufficient coefficient and parameter recoveries are possible even when the assimilated data contain significant random noise. Thus, for biological systems that can be described by ecosystem models, the adjoint method represents a powerful approach for estimating values for little-known biological parameters, such as initial conditions, growth rates, and mortality rates.     

Modeling kappa for measuring dependent categorical agreement data

A method for analysing dependent agreement data with categorical responses is proposed. A generalized estimating equation approach is developed with two sets of equations. The first set models the marginal distribution of categorical ratings, and the second set models the pairwise association of ratings with the kappa coefficient (kappa) as a metric. Covariates can be incorporated into both sets of equations. This approach is compared with a latent variable model that assumes an underlying multivariate normal distribution in which the intraclass correlation coefficient is used as a measure of association. Examples are from a cervical ectopy study and the National Heart, Lung, and Blood Institute Veteran Twin Study.     

Fitting nonlinear and constrained generalized estimating equations with optimization software

In this article, we present an estimation approach for solving nonlinear constrained generalized estimating equations that can be implemented using object-oriented software for nonlinear programming, such as nlminb in Splus or fmincon and lsqnonlin in Matlab. We show how standard estimating equation theory includes this method as a special case so that our estimates, when unconstrained, will remain consistent and asymptotically normal. To illustrate this method, we fit a nonlinear dose-response model with nonnegative mixed bound constraints to clustered binary data from a developmental toxicity study. Satisfactory confidence intervals are found using a nonparametric bootstrap method when a common correlation coefficient is assumed for all the dose groups and for some of the dose-specific groups.     

Censored linear regression in the presence or absence of auxiliary survival information

There has been a rising interest in better exploiting auxiliary summary information from large databases in the analysis of smaller-scale studies that collect more comprehensive patient-level information. The purpose of this paper is twofold: first, we propose a novel approach to synthesize information from both the aggregate summary statistics and the individual-level data in censored linear regression. We show that the auxiliary information amounts to a system of nonsmooth estimating equations and thus can be combined with the conventional weighted log-rank estimating equations by using the generalized method of moments (GMM) approach. The proposed methodology can be further extended to account for the potential inconsistency in information from different sources. Second, in the absence of auxiliary information, we propose to improve estimation efficiency by combining the overidentified weighted log-rank estimating equations with different weight functions via the GMM framework. To deal with the nonsmooth GMM-type objective functions, we develop an asymptotics-guided algorithm for parameter and variance estimation. We establish the asymptotic normality of the proposed GMM-type estimators. Simulation studies show that the proposed estimators can yield substantial efficiency gain over the conventional weighted log-rank estimators. The proposed methods are applied to a pancreatic cancer study for illustration.     

Determination of the unknown age at first capture of western rock lobsters (Panulirus cygnus) by random effects model

We propose a method for fitting growth curves to multiple recapture data of lobsters when the age at first capture is unknown. The von Bertalanffy growth curve is used to model the growth. To account for individual variability, the unknown age in logarithmic scale of a lobster at first capture, the individual asymptotic size, and the individual growth coefficient of its carapace length are modeled as random effects with a trivariate normal distribution. Unlike previously suggested models, the present model permits correlation between the growth coefficient and the age at first capture and can be fitted readily using existing software. The error structures between consecutive recaptures of a lobster are assumed to be a first-order autoregressive process with unequally spaced time points. A comparison between this model and the Fabens growth equation is given. The proposed method is a flexible method and can be applied to fit different growth equations when the age at first capture is unknown.     

Estimating the effect of continual disturbances on discrete-time population models

In this paper, methods for estimating the effect of continual (but bounded) disturbances on ecosystems modelled by difference equations are discussed. The approach adopted is to estimate the region of state space (called a reachable set) which can be reached by the disturbed system from an initial healthy state in a given time period. Liapunov stability methods for estimating these reachable sets are presented and applied to two specific population models.     

三参数增长模型拟合:以季风常绿阔叶林中两个优势乔木种群为例

Logistic、Mitscherlich、Gompertz方程是一类三参数饱和增长曲线模型,广泛地应用于许多学科领域．本文基于logistic方程饱和值K估计的三点法、四点法,推导出Mitscherlich、Gompertz方程K值的三点法、四点法估计公式,并以南亚热带季风常绿阔叶林中两种优势乔木厚壳桂、黄果厚壳桂种群为例,先用三点法或四点法估计出K值,再通过线性回归与非线性回归相结合的方法,可获得三个增长模型中三个参数的最优无偏估计．实例研究表明,两个优势种群增长数据均符合三个增长模型,但更符合增长曲线呈S形的logistic、Gompertz方程,且以logistic方程最适合于观察;黄果厚壳桂种群增长快于厚壳桂种群．     

Growth rates of young-of-year shovelnose sturgeon in the Upper Missouri River

Information on growth during the larval and young‐of‐year life stages in natural river environments is generally lacking for most sturgeon species. In this study, methods for estimating ages and quantifying growth were developed for field‐sampled larval and young‐of‐year shovelnose sturgeon Scaphirhynchus platorynchus in the upper Missouri River. First, growth was assessed by partitioning samples of young‐of‐year shovelnose sturgeon into cohorts, and regressing weekly increases in cohort mean length on sampling date. This method quantified relative growth because ages of the cohorts were unknown. Cohort increases in mean length among sampling dates were positively related (P < 0.05, r² > 0.59 for all cohorts) to sampling date, and yielded growth rate estimates of 0.80–2.95 mm day⁻¹ (2003) and 0.44–2.28 mm day⁻¹ (2004). Highest growth rates occurred in the largest (and earliest spawned) cohorts. Second, a method was developed to estimate cohort hatch dates, thus age on date of sampling could be determined. This method included quantification of post‐hatch length increases as a function of water temperature (growth capacity; mm per thermal unit, mm TU⁻¹), and summation of mean daily water temperatures to achieve the required number of thermal units that corresponded to post‐hatch lengths of shovelnose sturgeon on sampling dates. For six of seven cohorts of shovelnose sturgeon analyzed, linear growth models (r² ≥ 0.65, P < 0.0001) or Gompertz growth models (r² ≥ 0.83, P < 0.0001) quantified length‐at‐age from hatch through 55 days post‐hatch (98–100 mm). Comparisons of length‐at‐age derived from the growth models indicated that length‐at‐age was greater for the earlier‐hatched cohorts than later‐hatched cohorts. Estimated hatch dates for different cohorts were corroborated based on the dates that newly‐hatched larval shovelnose sturgeon were sampled in the drift. These results provide the first quantification of growth dynamics for field‐sampled age‐0 shovelnose sturgeon in a natural river environment, and provide an accurate method for estimating age of wild‐caught individuals. Methods of age determination used in this study have applications to sturgeons in other regions, but require additional testing and validation.     

Local estimation of age-dependent variance components from longitudinal twin data

In the study of longitudinal twin and family data, interest is often in the covariance structure of the data and the decomposition of this covariance structure into genetic and environmental components rather than in estimating the mean function. Various parametric models for covariance structures have been proposed but, e.g., in studies of children where growth spurts occur at various ages, it is difficult to a priori determine an appropriate parametric model for the covariance structure. In particular, there is a general lack of the visualization procedures, such as lowess, that are invaluable in the initial stages of constructing a parametric model for a mean function. Here we use kernel smoothing to modify a cross-sectional approach based on the sample covariance matrices to obtain smoothed estimates of the genetic and environmental variances and correlations for longitudinal twin data. The methods are proposed to be exploratory as an aid to parametric modeling rather than inferential, although approximate asymptotic standard errors are derived in the Appendix.     

Testing for bias in weighted estimating equations

It is very common in regression analysis to encounter incompletely observed covariate information. A recent approach to analyse such data is weighted estimating equations (Robins, J. M., Rotnitzky, A. and Zhao, L. P. (1994), JASA, 89, 846-866, and Zhao, L. P., Lipsitz, S. R. and Lew, D. (1996), Biometrics, 52, 1165-1182). With weighted estimating equations, the contribution to the estimating equation from a complete observation is weighted by the inverse of the probability of being observed. We propose a test statistic to assess if the weighted estimating equations produce biased estimates. Our test statistic is similar to the test statistic proposed by DuMouchel and Duncan (1983) for weighted least squares estimates for sample survey data. The method is illustrated using data from a randomized clinical trial on chemotherapy for multiple myeloma.     

Statistical Inference in a Two-Compartment Model for Hematopoiesis

We present a method for parameter estimation in a two-compartment hidden Markov model of the first two stages of hematopoiesis. Hematopoiesis is the specialization of stem cells into mature blood cells. As stem cells are not distinguishable in bone marrow, little is known about their behavior, although it is known that they have the ability to self-renew or to differentiate to more specialized (progenitor) cells. We observe progenitor cells in samples of bone marrow taken from hybrid cats whose cells contain a natural binary marker. With data consisting of the changing proportions of this binary marker over time from several cats, estimates for stem cell self-renewal and differentiation parameters are obtained using an estimating equations approach.     

Modelling and analysis of time-lags in some basic patterns of cell proliferation

 In this paper, we present a systematic approach for obtaining qualitatively and quantitatively correct mathematical models of some biological phenomena with time-lags. Features of our approach are the development of a hierarchy of related models and the estimation of parameter values, along with their non-linear biases and standard deviations, for sets of experimental data. We demonstrate our method of solving parameter estimation problems for neutral delay differential equations by analyzing some models of cell growth that incorporate a time-lag in the cell division phase. We show that these models are more consistent with certain reported data than the classic exponential growth model. Although the exponential growth model provides estimates of some of the growth characteristics, such as the population-doubling time, the time-lag growth models can additionally provide estimates of: (i) the fraction of cells that are dividing, (ii) the rate of commitment of cells to cell division, (iii) the initial distribution of cells in the cell cycle, and (iv) the degree of synchronization of cells in the (initial) cell population. Received: 15 September 1997/Revised version: 1 April 1998     

Estimating Cytomegalovirus Growth Rates by Using Only a Single Point

Calculation of pathogen growth rates is important in understanding the natural history of infection and effects of therapy. However, it is often difficult to estimate pathogen growth because patients are treated immediately upon the detection of infection, leaving only one nonzero untreated reading. Previous approaches have relied on the flawed assumption that pathogen loads just prior to detection are at the assay detection threshold. We have developed a novel method for estimating the pathogen growth rate from a single reading and investigated the initial growth of cytomegalovirus (CMV) in allogeneic hematopoietic stem cell transplant (HSCT) patients. We applied this approach to CMV viral loads measured at least weekly in 122 patients in the 3 months posttransplant. Viral growth rates were estimated by using a modeling approach that accounts for the viral load and the time since the last negative reading. Viral growth rates decreased rapidly within the first week, from 0.72/day (doubling time, 0.96 day) at the point of reactivation to 0.22/day (doubling time, 3.1 days) at 1 week. Results from this method correlated closely with a two-point regression analysis of a subset of 58 patients with detectable subthreshold viral loads immediately prior to overt reactivation. Patients with lymphocyte counts of ≥0.5 × 10⁹/liter had significantly slower viral growth than patients with low lymphocyte counts (0.612/day versus 0.325/day, P < 0.0001). Thus, our novel method of estimating pathogen growth rates reveals a rapid slowing of CMV growth during reactivation in HSCT patients and a significant impact of the lymphocyte count on CMV growth.     

Genome-wide linkage analysis of systolic blood pressure: a comparison of two approaches to phenotype definition

Problem 1 of the Genetic Analysis Workshop 13(GAW13) contains longitudinal data of cardiovascular measurements from 330 pedigrees. The longitudinal data complicates the phenotype definition because multiple measurements are taken on each individual. To address this complication, we propose an approach that uses generalized estimating equations to obtain residuals for each time point for each person. The mean residual is then taken as the new phenotype with which to use in a variance components linkage analysis. We compare our phenotype definition approach to an approach that first reduces the multiple measurements to a single measurement and then models these summary statistics as regression terms in a variance components analysis. For each approach, multipoint linkage analysis was performed using the residuals and the SOLAR computer program. Our results show little difference between the methods based on the LOD scores.     

A Note on the Use of Unbiased Estimating Equations to Estimate Correlation in Analysis of Longitudinal Trials

Longitudinal trials can yield outcomes that are continuous, binary (yes/no), or are realizations of counts. In this setting we compare three approaches that have been proposed for estimation of the correlation in the framework of generalized estimating equations (GEE): quasi‐least squares (QLS), pseudo‐likelihood (PL), and an approach we refer to as Wang–Carey (WC). We prove that WC and QLS are identical for the first‐order autoregressive AR(1) correlation structure. Using simulations, we then develop guidelines for selection of an appropriate method for analysis of data from a longitudinal trial. In particular, we argue that no method is uniformly superior for analysis of unbalanced and unequally spaced data with a Markov correlation structure. Choice of the best approach will depend on the degree of imbalance and variability in the temporal spacing of measurements, value of the correlation, and type of outcome, e.g. binary or continuous. Finally, we contrast the methods in analysis of a longitudinal study of obesity following renal transplantation in children (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Modeling concordance correlation via GEE to evaluate reproducibility

Clinical studies are often concerned with assessing whether different raters/methods produce similar values for measuring a quantitative variable. Use of the concordance correlation coefficient as a measure of reproducibility has gained popularity in practice since its introduction by Lin (1989, Biometrics 45, 255-268). Lin's method is applicable for studies evaluating two raters/two methods without replications. Chinchilli et al. (1996, Biometrics 52, 341-353) extended Lin's approach to repeated measures designs by using a weighted concordance correlation coefficient. However, the existing methods cannot easily accommodate covariate adjustment, especially when one needs to model agreement. In this article, we propose a generalized estimating equations (GEE) approach to model the concordance correlation coefficient via three sets of estimating equations. The proposed approach is flexible in that (1) it can accommodate more than two correlated readings and test for the equality of dependent concordant correlation estimates; (2) it can incorporate covariates predictive of the marginal distribution; (3) it can be used to identify covariates predictive of concordance correlation; and (4) it requires minimal distribution assumptions. A simulation study is conducted to evaluate the asymptotic properties of the proposed approach. The method is illustrated with data from two biomedical studies.     

A simple imputation method for longitudinal studies with non-ignorable non-responses

Missing data are a common problem in longitudinal studies in the health sciences. Motivated by data from the Muscatine Coronary Risk Factor (MCRF) study, a longitudinal study of obesity, we propose a simple imputation method for handling non-ignorable non-responses (i.e., when non-response is related to the specific values that should have been obtained) in longitudinal studies with either discrete or continuous outcomes. In the proposed approach, two regression models are specified; one for the marginal mean of the response, the other for the conditional mean of the response given non-response patterns. Statistical inference for the model parameters is based on the generalized estimating equations (GEE) approach. An appealing feature of the proposed method is that it can be readily implemented using existing, widely-available statistical software. The method is illustrated using longitudinal data on obesity from the MCRF study.     

应用Wyatt-White方法计算寄生物种群γ_m的研究(英文)

本文是关于应用Wyatt-White方法计算寄生物种群内禀增长率的研究。研究表明:r_m的精确值与ln(M_d)/d或ln(M_(d/ 2))/d之间存在着线性关系,这种关系可表达如下:(1)r_m=0.845ln(M_d)/d;(2)r_m=0.880ln(M_(d/2)/d。这里d为生殖前期;M_d为生殖起d天内每个原始雌虫产下的平均雌性后代数;M_(d/2)为生殖起初d/2天内每个原始雌虫产下的平均雌性后代数。运用23组生殖力表资料,研究表明公式1—2可以给出r_m的精确估计值,公式2的估计效果更好。这种方法不要求组建生殖力表。该方法是否适用于其它寄生物种群或其它生物类群有待进一步研究。     

Hierarchical Bayesian methods for estimation of parameters in a longitudinal HIV dynamic system

HIV dynamics studies have significantly contributed to the understanding of HIV infection and antiviral treatment strategies. But most studies are limited to short-term viral dynamics due to the difficulty of establishing a relationship of antiviral response with multiple treatment factors such as drug exposure and drug susceptibility during long-term treatment. In this article, a mechanism-based dynamic model is proposed for characterizing long-term viral dynamics with antiretroviral therapy, described by a set of nonlinear differential equations without closed-form solutions. In this model we directly incorporate drug concentration, adherence, and drug susceptibility into a function of treatment efficacy, defined as an inhibition rate of virus replication. We investigate a Bayesian approach under the framework of hierarchical Bayesian (mixed-effects) models for estimating unknown dynamic parameters. In particular, interest focuses on estimating individual dynamic parameters. The proposed methods not only help to alleviate the difficulty in parameter identifiability, but also flexibly deal with sparse and unbalanced longitudinal data from individual subjects. For illustration purposes, we present one simulation example to implement the proposed approach and apply the methodology to a data set from an AIDS clinical trial. The basic concept of the longitudinal HIV dynamic systems and the proposed methodologies are generally applicable to any other biomedical dynamic systems.