贝叶斯非线性混合效应模型及其应用研究

由于常用的线性混合效应模型对具有非线性关系的纵向数据建模具有一定的局限性,因此对线性混合效应模型进行扩展,根据变量间的非线性关系建立不同的非线性混合效应模型,并根据因变量的分布特征建立混合分布模型。基于一组实际的保险损失数据,建立多项式混合效应模型、截断多项式混合效应模型和B样条混合效应模型。研究结果表明,非线性混合效应模型能够显著改进对保险损失数据的建模效果,对非寿险费率厘定具有重要参考价值。     

A class of mixed models for recurrent event data

In this article, we propose a class of mixed models for recurrent event data. The new models include the proportional rates model and Box–Cox transformation rates models as special cases, and allow the effects of covariates on the rate functions of counting processes to be proportional or convergent. For inference on the model parameters, estimating equation approaches are developed. The asymptotic properties of the resulting estimators are established and the finite sample performance of the proposed procedure is evaluated through simulation studies. A real example with data taken from a clinic study on chronic granulomatous disease (CGD) is also illustrated for the use of the proposed methodology. The Canadian Journal of Statistics 39: 578–590; 2011. © 2011 Statistical Society of Canada     

A note on transformed likelihood approach in linear dynamic panel models

The transformed likelihood approach to estimation of fixed effects dynamic panel data models is shown to present very good inferential properties but it is not directly implemented in the most diffused statistical software. The present paper aims at showing how a simple model reformulation can be adopted to describe the problem in terms of classical linear mixed models. The transformed likelihood approach is based on the first differences data transformation, the following results derive from a convenient reformulation in terms of deviations from the first observations. Given the invariance to data transformation, the likelihood functions defined in the two cases coincide. Resulting in a classical random effect linear model form, the proposed approach significantly improves the number of available estimation procedures and provides a straightforward interpretation for the parameters. Moreover, the proposed model specification allows to consider all the estimation improvements typical of the random effects model literature. Simulation studies are conducted in order to study the robustness of the estimation method to mean stationarity violation.     

Nonlinear regression modeling via regularized radial basis function networks

The problem of constructing nonlinear regression models is investigated to analyze data with complex structure. We introduce radial basis functions with hyperparameter that adjusts the amount of overlapping basis functions and adopts the information of the input and response variables. By using the radial basis functions, we construct nonlinear regression models with help of the technique of regularization. Crucial issues in the model building process are the choices of a hyperparameter, the number of basis functions and a smoothing parameter. We present information-theoretic criteria for evaluating statistical models under model misspecification both for distributional and structural assumptions. We use real data examples and Monte Carlo simulations to investigate the properties of the proposed nonlinear regression modeling techniques. The simulation results show that our nonlinear modeling performs well in various situations, and clear improvements are obtained for the use of the hyperparameter in the basis functions.     

Influence diagnostics in nonlinear reproductive dispersion mixed models

The class of nonlinear reproductive dispersion mixed models (NRDMMs) is an extension of nonlinear reproductive dispersion models and generalized linear mixed models. This paper discusses the influence analysis of the model based on Laplace approximation. The equivalence of case-deletion models and mean-shift outlier models in NRDMMs is investigated, and some diagnostic measures are proposed via the case-deletion method. We also investigate the assessment of local influence of various perturbation schemes. The proposed method is illustrated with an example.     

Boosting techniques for nonlinear time series models

Many of the popular nonlinear time series models require a priori the choice of parametric functions which are assumed to be appropriate in specific applications. This approach is mainly used in financial applications, when sufficient knowledge is available about the nonlinear structure between the covariates and the response. One principal strategy to investigate a broader class on nonlinear time series is the Nonlinear Additive AutoRegressive (NAAR) model. The NAAR model estimates the lags of a time series as flexible functions in order to detect non-monotone relationships between current and past observations. We consider linear and additive models for identifying nonlinear relationships. A componentwise boosting algorithm is applied for simultaneous model fitting, variable selection, and model choice. Thus, with the application of boosting for fitting potentially nonlinear models we address the major issues in time series modelling: lag selection and nonlinearity. By means of simulation we compare boosting to alternative nonparametric methods. Boosting shows a strong overall performance in terms of precise estimations of highly nonlinear lag functions. The forecasting potential of boosting is examined on the German industrial production (IP); to improve the model’s forecasting quality we include additional exogenous variables. Thus we address the second major aspect in this paper which concerns the issue of high dimensionality in models. Allowing additional inputs in the model extends the NAAR model to a broader class of models, namely the NAARX model. We show that boosting can cope with large models which have many covariates compared to the number of observations.     

Self‐modelling regression for longitudinal data with time‐invariant covariates

The authors propose the use of self‐modelling regression to analyze longitudinal data with time invariant covariates. They model the population time curve with a penalized regression spline and use a linear mixed model for transformation of the time and response scales to fit the individual curves. Fitting is done by an iterative algorithm using off‐the‐shelf linear and nonlinear mixed model software. Their method is demonstrated in a simulation study and in the analysis of tree swallow nestling growth from an experiment that includes an experimentally controlled treatment, an observational covariate and multi‐level sampling.     

Nonlinear mixed‐effects models with misspecified random‐effects distribution

Nonlinear mixed‐effects models are being widely used for the analysis of longitudinal data, especially from pharmaceutical research. They use random effects which are latent and unobservable variables so the random‐effects distribution is subject to misspecification in practice. In this paper, we first study the consequences of misspecifying the random‐effects distribution in nonlinear mixed‐effects models. Our study is focused on Gauss‐Hermite quadrature, which is now the routine method for calculation of the marginal likelihood in mixed models. We then present a formal diagnostic test to check the appropriateness of the assumed random‐effects distribution in nonlinear mixed‐effects models, which is very useful for real data analysis. Our findings show that the estimates of fixed‐effects parameters in nonlinear mixed‐effects models are generally robust to deviations from normality of the random‐effects distribution, but the estimates of variance components are very sensitive to the distributional assumption of random effects. Furthermore, a misspecified random‐effects distribution will either overestimate or underestimate the predictions of random effects. We illustrate the results using a real data application from an intensive pharmacokinetic study.     

Influence Diagnostics of Semiparametric Nonlinear Reproductive Dispersion Models

This article proposes a semiparametric nonlinear reproductive dispersion model (SNRDM) which is an extension of nonlinear reproductive dispersion model and semiparametric regression model. Maximum penalized likelihood estimators (MPLEs) of unknown parameters and nonparametric functions in SNRDMs are presented. Some novel diagnostic statistics such as Cook distance and difference deviance for parametric and nonparametric parts are developed to identify influence observations in SNRDMs on the basis of case-deletion method, and some formulae readily computed with the MPLEs algorithm for diagnostic measures are given. The equivalency of case-deletion models and mean-shift outlier models in SNRDM is investigated. A simulation study and a real example are used to illustrate the proposed diagnostic measures.     

Bayesian quantile regression for parametric nonlinear mixed effects models

We propose quantile regression (QR) in the Bayesian framework for a class of nonlinear mixed effects models with a known, parametric model form for longitudinal data. Estimation of the regression quantiles is based on a likelihood-based approach using the asymmetric Laplace density. Posterior computations are carried out via Gibbs sampling and the adaptive rejection Metropolis algorithm. To assess the performance of the Bayesian QR estimator, we compare it with the mean regression estimator using real and simulated data. Results show that the Bayesian QR estimator provides a fuller examination of the shape of the conditional distribution of the response variable. Our approach is proposed for parametric nonlinear mixed effects models, and therefore may not be generalized to models without a given model form.     

Exploratory model analysis using cdf knotting with applications to distinguishability,limiting forms,and moment ratios to the three parameter weibull family

An exploratory model analysis device we call CDF knotting is introduced. It is a technique we have found useful for exploring relationships between points in the parameter space of a model and global properties of associated distribution functions. It can be used to alert the model builder to a condition we call lack of distinguishability which is to nonlinear models what multicollinearity is to linear models. While there are simple remedial actions to deal with multicollinearity in linear models, techniques such as deleting redundant variables in those models do not have obvious parallels for nonlinear models. In some of these nonlinear situations, however, CDF knotting may lead to alternative models with fewer parameters whose distribution functions are very similar to those of the original overparameterized model. We also show how CDF knotting can be exploited as a mathematical tool for deriving limiting distributions and illustrate the technique for the 3-parameterWeibull family obtaining limiting forms and moment ratios which correct and extend previously published results. Finally, geometric insights obtained by CDF knotting are verified relative to data fitting and estimation.     

MODEL SELECTION FOR PENALIZED SPLINE SMOOTHING USING AKAIKE INFORMATION CRITERIA

Two different forms of Akaike's information criterion (AIC) are compared for selecting the smooth terms in penalized spline additive mixed models. The conditional AIC (cAIC) has been used traditionally as a criterion for both estimating penalty parameters and selecting covariates in smoothing, and is based on the conditional likelihood given the smooth mean and on the effective degrees of freedom for a model fit. By comparison, the marginal AIC (mAIC) is based on the marginal likelihood from the mixed‐model formulation of penalized splines which has recently become popular for estimating smoothing parameters. To the best of the authors' knowledge, the use of mAIC for selecting covariates for smoothing in additive models is new. In the competing models considered for selection, covariates may have a nonlinear effect on the response, with the possibility of group‐specific curves. Simulations are used to compare the performance of cAIC and mAIC in model selection settings that have correlated and hierarchical smooth terms. In moderately large samples, both formulations of AIC perform extremely well at detecting the function that generated the data. The mAIC does better for simple functions, whereas the cAIC is more sensitive to detecting a true model that has complex and hierarchical terms.     

Local influence of nonlinear mixed effects model based on M-estimation

Abstract

In this work we mainly study the local influence in nonlinear mixed effects model with M-estimation. A robust method to obtain maximum likelihood estimates for parameters is presented, and the local influence of nonlinear mixed models based on robust estimation (M-estimation) by use of the curvature method is systematically discussed. The counting formulas of curvature for case weights perturbation, response variable perturbation and random error covariance perturbation are derived. Simulation studies are carried to access performance of the methods we proposed. We illustrate the diagnostics by an example presented in Davidian and Giltinan, which was analyzed under the non-robust situation.     

Estimation for semiparametric varying coefficient models with different smoothing variables under random right censoring

In this paper we study a semiparametric varying coefficient model when the response is subject to random right censoring. The model gives an easy interpretation due to its direct connectivity to the classical linear model and is very flexible since nonparametric functions which accommodates various nonlinear interaction effects between covariates are admitted in the model. We propose estimators for this model using mean-preserving transformation and establish their asymptotic properties. The estimation procedure is based on the profiling and the smooth backfitting techniques. A simulation study is presented to show the reliability of the proposed estimators and an automatic bandwidth selector is given in a data-driven way.     

Estimating consumer acceptance limits

A methodology is developed for estimating consumer acceptance limits on a sensory attribute of a manufactured product. In concept these limits are analogous to engineering tolerances. The method is based on a generalization of Stevens' Power Law. This generalized law is expressed as a nonlinear statistical model. Instead of restricting the analysis to this particular case, a strategy is discussed for evaluating nonlinear models in general since scientific models are frequently of nonlinear form. The strategy focuses on understanding the geometrical contrasts between linear and nonlinear model estimation and assessing the bias in estimation and the departures from a Gaussian sampling distribution. Computer simulation is employed to examine the behavior of nonlinear least squares estimation. In addition to the usual Gaussian assumption, a bootstrap sample reuse procedure and a general triangular distribution are introduced for evaluating the effects of a non-Gaussian or asymmetrical error structure. Recommendations are given for further model analysis based on the simulation results. In the case of a model for which estimation bias is not a serious issue, estimating functions of the model are considered. Application of these functions to the generalization of Stevens’ Power Law leads to a means for defining and estimating consumer acceptance limits, The statistical form of the law and the model evaluation strategy are applied to consumer research data. Estimation of consumer acceptance limits is illustrated and discussed.     

Applying functional data analysis to assess tele-interpersonal psychotherapy's efficacy to reduce depression

The use of parametric linear mixed models and generalized linear mixed models to analyze longitudinal data collected during randomized control trials (RCT) is conventional. The application of these methods, however, is restricted due to various assumptions required by these models. When the number of observations per subject is sufficiently large, and individual trajectories are noisy, functional data analysis (FDA) methods serve as an alternative to parametric longitudinal data analysis techniques. However, the use of FDA in RCTs is rare. In this paper, the effectiveness of FDA and linear mixed models (LMMs) was compared by analyzing data from rural persons living with HIV and comorbid depression enrolled in a depression treatment randomized clinical trial. Interactive voice response systems were used for weekly administrations of the 10-item Self-Administered Depression Scale (SADS) over 41 weeks. Functional principal component analysis and functional regression analysis methods detected a statistically significant difference in SADS between telphone-administered interpersonal psychotherapy (tele-IPT) and controls but linear mixed effects model results did not. Additional simulation studies were conducted to compare FDA and LMMs under a different nonlinear trajectory assumption. In this clinical trial with sufficient per subject measured outcomes and individual trajectories that are noisy and nonlinear, we found FDA methods to be a better alternative to LMMs.     

Multi-step forecasts from threshold ARMA models using asymmetric loss functions

The forecasts generation from nonlinear time series models is investigated under general loss functions. After presenting the main results and some relevant features of these functions, the Linex loss has been used to generate multi-step forecasts from threshold autoregressive moving average models showing their main properties and some results connected to a proper transformation of the forecast errors. A simulation exercise highlights interesting properties of the proposed predictors, both in terms of their bias and their distribution, further clarifying how the Linex predictor can be helpful in empirical applications.     

On the relative efficiency of a monotone parameter curve estimator in a functional nonlinear model

Functional regression models that relate functional covariates to a scalar response are becoming more common due to the availability of functional data and computational advances. We introduce a functional nonlinear model with a scalar response where the true parameter curve is monotone. Using the Newton-Raphson method within a backfitting procedure, we discuss a penalized least squares criterion for fitting the functional nonlinear model with the smoothing parameter selected using generalized cross validation. Connections between a nonlinear mixed effects model and our functional nonlinear model are discussed, thereby providing an additional model fitting procedure using restricted maximum likelihood for smoothing parameter selection. Simulated relative efficiency gains provided by a monotone parameter curve estimator relative to an unconstrained parameter curve estimator are presented. In addition, we provide an application of our model with data from ozonesonde measurements of stratospheric ozone in which the measurements are biased as a function of altitude.     

A multifactor transformed diffusion model with applications to VIX and VIX futures

Abstract

Transformed diffusions (TDs) have become increasingly popular in financial modeling for their model flexibility and tractability. While existing TD models are predominately one-factor models, empirical evidence often prefers models with multiple factors. We propose a novel distribution-driven nonlinear multifactor TD model with latent components. Our model is a transformation of a underlying multivariate Ornstein–Uhlenbeck (MVOU) process, where the transformation function is endogenously specified by a flexible parametric stationary distribution of the observed variable. Computationally efficient exact likelihood inference can be implemented for our model using a modified Kalman filter algorithm and the transformed affine structure also allows us to price derivatives in semi-closed form. We compare the proposed multifactor model with existing TD models for modeling VIX and pricing VIX futures. Our results show that the proposed model outperforms all existing TD models both in the sample and out of the sample consistently across all categories and scenarios of our comparison.     

A unified approach to rank tests for mixed models

The nonparametric version of the classical mixed model is considered and the common hypotheses of (parametric) main effects and interactions are reformulated in a nonparametric setup. To test these nonparametric hypotheses, the asymptotic distributions of quadratic forms of rank statistics are derived in a general framework which enables the derivation of the statistics for the nonparametric hypotheses of the fixed treatment effects and interactions in an arbitrary mixed model. The procedures given here are not restricted to semiparametric models or models with additive effects. Moreover, they are robust to outliers since only the ranks of the observations are needed. They are also applicable to pure ordinal data and since no continuity of the distribution functions is assumed, they can also be applied to data with ties. Some approximations for small sample sizes are suggested and analyzed in a simulation study. The application of the statistics and the interpretation of the results is demonstrated in several worked-out examples where some data sets given in the literature are re-analyzed.