A refined parameter estimating approach for HIV dynamic model

HIV dynamic models, a set of ordinary differential equations (ODEs), have provided new understanding of the pathogenesis of HIV infection and the treatment effects of antiviral therapies. However, to estimate parameters for ODEs is very challenging due to the complexity of this nonlinear system. In this article, we propose a comprehensive procedure to deal with this issue. In the proposed procedure, a series of cutting-edge statistical methods and techniques are employed, including nonparametric mixed-effects smoothing-based methods for ODE models and stochastic approximation expectation–maximization (EM) approach for mixed-effects ODE models. A simulation study is performed to validate the proposed approach. An application example from a real HIV clinical trial study is used to illustrate the usefulness of the proposed method.     

Parameter estimation for semiparametric ordinary differential equation models

Abstract

We propose a new class of two-stage parameter estimation methods for semiparametric ordinary differential equation (ODE) models. In the first stage, state variables are estimated using a penalized spline approach; In the second stage, form of numerical discretization algorithms for an ODE solver is used to formulate estimating equations. Estimated state variables from the first stage are used to obtain more data points for the second stage. Asymptotic properties for the proposed estimators are established. Simulation studies show that the method performs well, especially for small sample. Real life use of the method is illustrated using Influenza specific cell-trafficking study.     

Smoothing Spline Estimation for Partially Linear Single-index Models

In this article, the partially linear single-index models are discussed based on smoothing spline and average derivative estimation method. This proposed technique consists of two stages: one is to estimate the vector parameter in the linear part using the smoothing cubic spline method, simultaneously, obtaining the estimator of unknown single-index function; the other is to estimate the single-index coefficients in the single-index part by the using average derivative estimator procedure. Some simulated and real examples are presented to illustrate the performance of this method.     

A note on asymptotics for quantile smoothing splines

When cubic smoothing splines are used to estimate the conditional quantile function, thereby balancing fidelity to the data with a smoothness requirement, the resulting curve is the solution to a quadratic program. Using this quadratic characterization and through comparison with the sample conditional quan-tiles, we show strong consistency and asymptotic normality for the quantile smoothing spline.     

Variance Reduction in Smoothing Splines

Abstract.  We develop a variance reduction method for smoothing splines. For a given point of estimation, we define a variance-reduced spline estimate as a linear combination of classical spline estimates at three nearby points. We first develop a variance reduction method for spline estimators in univariate regression models. We then develop an analogous variance reduction method for spline estimators in clustered/longitudinal models. Simulation studies are performed which demonstrate the efficacy of our variance reduction methods in finite sample settings. Finally, a real data analysis with the motorcycle data set is performed. Here we consider variance estimation and generate 95% pointwise confidence intervals for the unknown regression function.     

A Two-Stage Estimation Method for Random Coefficient Differential Equation Models with Application to Longitudinal HIV Dynamic Data

We propose a two-stage estimation method for random coefficient ordinary differential equation (ODE) models. A maximum pseudo-likelihood estimator (MPLE) is derived based on a mixed-effects modeling approach and its asymptotic properties for population parameters are established. The proposed method does not require repeatedly solving ODEs, and is computationally efficient although it does pay a price with the loss of some estimation efficiency. However, the method does offer an alternative approach when the exact likelihood approach fails due to model complexity and high-dimensional parameter space, and it can also serve as a method to obtain the starting estimates for more accurate estimation methods. In addition, the proposed method does not need to specify the initial values of state variables and preserves all the advantages of the mixed-effects modeling approach. The finite sample properties of the proposed estimator are studied via Monte Carlo simulations and the methodology is also illustrated with application to an AIDS clinical data set.     

Estimation of parameters in incomplete data models defined by dynamical systems

Parametric incomplete data models defined by ordinary differential equations (ODEs) are widely used in biostatistics to describe biological processes accurately. Their parameters are estimated on approximate models, whose regression functions are evaluated by a numerical integration method. Accurate and efficient estimations of these parameters are critical issues. This paper proposes parameter estimation methods involving either a stochastic approximation EM algorithm (SAEM) in the maximum likelihood estimation, or a Gibbs sampler in the Bayesian approach. Both algorithms involve the simulation of non-observed data with conditional distributions using Hastings–Metropolis (H–M) algorithms. A modified H–M algorithm, including an original local linearization scheme to solve the ODEs, is proposed to reduce the computational time significantly. The convergence on the approximate model of all these algorithms is proved. The errors induced by the numerical solving method on the conditional distribution, the likelihood and the posterior distribution are bounded. The Bayesian and maximum likelihood estimation methods are illustrated on a simulated pharmacokinetic nonlinear mixed-effects model defined by an ODE. Simulation results illustrate the ability of these algorithms to provide accurate estimates.     

A model-averaging approach for smoothing spline regression

ABSTRACT

This article considers nonparametric regression problems and develops a model-averaging procedure for smoothing spline regression problems. Unlike most smoothing parameter selection studies determining an optimum smoothing parameter, our focus here is on the prediction accuracy for the true conditional mean of Y given a predictor X. Our method consists of two steps. The first step is to construct a class of smoothing spline regression models based on nonparametric bootstrap samples, each with an appropriate smoothing parameter. The second step is to average bootstrap smoothing spline estimates of different smoothness to form a final improved estimate. To minimize the prediction error, we estimate the model weights using a delete-one-out cross-validation procedure. A simulation study has been performed by using a program written in R. The simulation study provides a comparison of the most well known cross-validation (CV), generalized cross-validation (GCV), and the proposed method. This new method is straightforward to implement, and gives reliable performances in simulations.     

A semiparametric stochastic mixed effects model for bivariate cyclic longitudinal data

We propose a flexible semiparametric stochastic mixed effects model for bivariate cyclic longitudinal data. The model can handle either single cycle or, more generally, multiple consecutive cycle data. The approach models the mean of responses by parametric fixed effects and a smooth nonparametric function for the underlying time effects, and the relationship across the bivariate responses by a bivariate Gaussian random field and a joint distribution of random effects. The proposed model not only can model complicated individual profiles, but also allows for more flexible within-subject and between-response correlations. The fixed effects regression coefficients and the nonparametric time functions are estimated using maximum penalized likelihood, where the resulting estimator for the nonparametric time function is a cubic smoothing spline. The smoothing parameters and variance components are estimated simultaneously using restricted maximum likelihood. Simulation results show that the parameter estimates are close to the true values. The fit of the proposed model on a real bivariate longitudinal dataset of pre-menopausal women also performs well, both for a single cycle analysis and for a multiple consecutive cycle analysis. The Canadian Journal of Statistics 48: 471–498; 2020 © 2020 Statistical Society of Canada     

Estimation of Median in Two-Phase Sampling Using Two Auxiliary Variables

In recent years, zero-inflated count data models, such as zero-inflated Poisson (ZIP) models, are widely used as the count data with extra zeros are very common in many practical problems. In order to model the correlated count data which are either clustered or repeated and to assess the effects of continuous covariates or of time scales in a flexible way, a class of semiparametric mixed-effects models for zero-inflated count data is considered. In this article, we propose a fully Bayesian inference for such models based on a data augmentation scheme that reflects both random effects of covariates and mixture of zero-inflated distribution. A computational efficient MCMC method which combines the Gibbs sampler and M-H algorithm is implemented to obtain the estimate of the model parameters. Finally, a simulation study and a real example are used to illustrate the proposed methodologies.     

Spatially-adaptive Penalties for Spline Fitting

The paper studies spline fitting with a roughness penalty that adapts to spatial heterogeneity in the regression function. The estimates are p th degree piecewise polynomials with p − 1 continuous derivatives. A large and fixed number of knots is used and smoothing is achieved by putting a quadratic penalty on the jumps of the p th derivative at the knots. To be spatially adaptive, the logarithm of the penalty is itself a linear spline but with relatively few knots and with values at the knots chosen to minimize the generalized cross validation (GCV) criterion. This locally-adaptive spline estimator is compared with other spline estimators in the literature such as cubic smoothing splines and knot-selection techniques for least squares regression. Our estimator can be interpreted as an empirical Bayes estimate for a prior allowing spatial heterogeneity. In cases of spatially heterogeneous regression functions, empirical Bayes confidence intervals using this prior achieve better pointwise coverage probabilities than confidence intervals based on a global-penalty parameter. The method is developed first for univariate models and then extended to additive models.     

Penalized models to estimate customer survival

In this paper we propose a novel procedure, for the estimation of semiparametric survival functions. The proposed technique adapts penalized likelihood survival models to the context of lifetime value modeling. The method extends classical Cox model by introducing a smoothing parameter that can be estimated by means of penalized maximum likelihood procedures. Markov Chain Monte Carlo methods are employed to effectively estimate such smoothing parameter, using an algorithm which combines Metropolis–Hastings and Gibbs sampling. Our proposal is contextualized and compared with conventional models, with reference to a marketing application that involves the prediction of customer’s lifetime value estimation.     

A note on mse coverage intervals in a partial spline model

A partial spline model is used to estimate an unknown function which is smooth except for some break points. Assuming the break points are known, a Generalized Cross-Validated smoothing spline estimation method is proposed. Some interval estimation methods for the magnitude of the discontinuities based on the mean square error are introduced and investigated.     

Estimating curves and derivatives with parametric penalized spline smoothing

Accurate estimation of an underlying function and its derivatives is one of the central problems in statistics. Parametric forms are often proposed based on the expert opinion or prior knowledge of the underlying function. However, these strict parametric assumptions may result in biased estimates when they are not completely accurate. Meanwhile, nonparametric smoothing methods, which do not impose any parametric form, are quite flexible. We propose a parametric penalized spline smoothing method, which has the same flexibility as the nonparametric smoothing methods. It also uses the prior knowledge of the underlying function by defining an additional penalty term using the distance of the fitted function to the assumed parametric function. Our simulation studies show that the parametric penalized spline smoothing method can obtain more accurate estimates of the function and its derivatives than the penalized spline smoothing method. The parametric penalized spline smoothing method is also demonstrated by estimating the human height function and its derivatives from the real data.     

Semiparametric inference for open populations using the Jolly–Seber model: a penalized spline approach

When there are frequent capture occasions, both semiparametric and nonparametric estimators for the size of an open population have been proposed using kernel smoothing methods. While kernel smoothing methods are mathematically tractable, fitting them to data is computationally intensive. Here, we use smoothing splines in the form of P-splines to provide an alternate less computationally intensive method of fitting these models to capture–recapture data from open populations with frequent capture occasions. We fit the model to capture data collected over 64 occasions and model the population size as a function of time, seasonal effects and an environmental covariate. A small simulation study is also conducted to examine the performance of the estimators and their standard errors.     

Semiparametric Estimation with Profile Algorithm for Longitudinal Binary Data

This article considers analyzing longitudinal binary data semiparametrically and proposing GEE-Smoothing spline in the estimation of parametric and nonparametric components. The method is an extension of the parametric generalized estimating equation to semiparametric. The nonparametric component is estimated by smoothing spline approach, i.e., natural cubic spline. We use profile algorithm in the estimation of both parametric and nonparametric components. Properties of the estimators are evaluated by simulation.     

Estimation of HIV seroconversion and effects of age in the San Francisco homosexual population

SUMMARY Using San Francisco city clinic cohort data, we estimate the HIV seroconversion distribution by both non-parametric and parametric methods, and illustrate the effects of age on this distribution. The non-parametric methods include the Turnbull method, the Bacchetti method, the expectation, maximization and smoothing (EMS) method and the penalized spline method. The seroconversion density curves estimated by these nonparametric methods are of bimodal nature with obvious effects of age. As a result of the bimodal nature of the seroconversion curves, the parametric models considered are mixtures of two distributions taken from the generalized log-logistic distribution with three parameters, the Weibull distribution and the log-normal distribution. In terms of the logarithm of the likelihood values, it appears that the non-parametric methods with smoothing as well as without smoothing (i.e. the Turnbull method) provided much better fits than did the parametric models. Among the non-parametric methods, the EMS and the spline estimates are more appealing, because the unsmoothed Turnbull estimates are very unstable and because the Bacchetti estimates have a longer tail. Among the parametric models, the mixture of a generalized log-logistic distribution with three parameters and a Weibull distribution or a log-normal distribution provided better fits than did other mixtures of parametric models.     

Mixed effects smoothing spline analysis of variance

We propose a general family of nonparametric mixed effects models. Smoothing splines are used to model the fixed effects and are estimated by maximizing the penalized likelihood function. The random effects are generic and are modelled parametrically by assuming that the covariance function depends on a parsimonious set of parameters. These parameters and the smoothing parameter are estimated simultaneously by the generalized maximum likelihood method. We derive a connection between a nonparametric mixed effects model and a linear mixed effects model. This connection suggests a way of fitting a nonparametric mixed effects model by using existing programs. The classical two-way mixed models and growth curve models are used as examples to demonstrate how to use smoothing spline analysis-of-variance decompositions to build nonparametric mixed effects models. Similarly to the classical analysis of variance, components of these nonparametric mixed effects models can be interpreted as main effects and interactions. The penalized likelihood estimates of the fixed effects in a two-way mixed model are extensions of James–Stein shrinkage estimates to correlated observations. In an example three nested nonparametric mixed effects models are fitted to a longitudinal data set.     

A semiparametric estimator of the distribution function of a variable measured with error

The estimation of the distribution functon of a random variable X measured with error is studied. Let the i-th observation on X be denoted by Y_iX_i+ε_i where ε_i is the measuremen error. Let {Y_i} (i=1,2,…,n) be a sample of independent observations. It is assumed that {X_i} and {∈_i} are mutually independent and each is identically distributed. As is standard in the literature for this problem, the distribution of e is assumed known in the development of the methodology. In practice, the measurement error distribution is estimated from replicate observations.

The proposed semiparametric estimator is derived by estimating the quantises of X on a set of n transformed V-values and smoothing the estimated quantiles using a spline function. The number of parameters of the spline function is determined by the data with a simple criterion, such as AIC. In a simulation study, the semiparametric estimator dominates an optimal kernel estimator and a normal mixture estimator for a wide class of densities.

The proposed estimator is applied to estimate the distribution function of the mean pH value in a field plot. The density function of the measurement error is estimated from repeated measurements of the pH values in a plot, and is treated as known for the estimation of the distribution function of the mean pH value.     

Period analysis of variable stars by robust smoothing

Summary.  The objective is to estimate the period and the light curve (or periodic function) of a variable star. Previously, several methods have been proposed to estimate the period of a variable star, but they are inaccurate especially when a data set contains outliers. We use a smoothing spline regression to estimate the light curve given a period and then find the period which minimizes the generalized cross-validation (GCV). The GCV method works well, matching an intensive visual examination of a few hundred stars, but the GCV score is still sensitive to outliers. Handling outliers in an automatic way is important when this method is applied in a 'data mining' context to a vary large star survey. Therefore, we suggest a robust method which minimizes a robust cross-validation criterion induced by a robust smoothing spline regression. Once the period has been determined, a nonparametric method is used to estimate the light curve. A real example and a simulation study suggest that the robust cross-validation and GCV methods are superior to existing methods.