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     

Estimated estimating equations: semiparametric inference for clustered and longitudinal data

Summary.  We introduce a flexible marginal modelling approach for statistical inference for clustered and longitudinal data under minimal assumptions. This estimated estimating equations approach is semiparametric and the proposed models are fitted by quasi-likelihood regression, where the unknown marginal means are a function of the fixed effects linear predictor with unknown smooth link, and variance–covariance is an unknown smooth function of the marginal means. We propose to estimate the nonparametric link and variance–covariance functions via smoothing methods, whereas the regression parameters are obtained via the estimated estimating equations. These are score equations that contain nonparametric function estimates. The proposed estimated estimating equations approach is motivated by its flexibility and easy implementation. Moreover, if data follow a generalized linear mixed model, with either a specified or an unspecified distribution of random effects and link function, the model proposed emerges as the corresponding marginal (population-average) version and can be used to obtain inference for the fixed effects in the underlying generalized linear mixed model, without the need to specify any other components of this generalized linear mixed model. Among marginal models, the estimated estimating equations approach provides a flexible alternative to modelling with generalized estimating equations. Applications of estimated estimating equations include diagnostics and link selection. The asymptotic distribution of the proposed estimators for the model parameters is derived, enabling statistical inference. Practical illustrations include Poisson modelling of repeated epileptic seizure counts and simulations for clustered binomial responses.     

Penalized Spline Estimation for Varying-Coefficient Models

Varying-coefficient models are useful extensions of classical linear models. They arise from multivariate nonparametric regression, nonlinear time series modeling and forecasting, longitudinal data analysis, and others. This article proposes the penalized spline estimation for the varying-coefficient models. Assuming a fixed but potentially large number of knots, the penalized spline estimators are shown to be strong consistency and asymptotic normality. A systematic optimization algorithm for the selection of multiple smoothing parameters is developed. One of the advantages of the penalized spline estimation is that it can accommodate varying degrees of smoothness among coefficient functions due to multiple smoothing parameters being used. Some simulation studies are presented to illustrate the proposed methods.     

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.     

Inference in generalized additive mixed modelsby using smoothing splines

Generalized additive mixed models are proposed for overdispersed and correlated data, which arise frequently in studies involving clustered, hierarchical and spatial designs. This class of models allows flexible functional dependence of an outcome variable on covariates by using nonparametric regression, while accounting for correlation between observations by using random effects. We estimate nonparametric functions by using smoothing splines and jointly estimate smoothing parameters and variance components by using marginal quasi-likelihood. Because numerical integration is often required by maximizing the objective functions, double penalized quasi-likelihood is proposed to make approximate inference. Frequentist and Bayesian inferences are compared. A key feature of the method proposed is that it allows us to make systematic inference on all model components within a unified parametric mixed model framework and can be easily implemented by fitting a working generalized linear mixed model by using existing statistical software. A bias correction procedure is also proposed to improve the performance of double penalized quasi-likelihood for sparse data. We illustrate the method with an application to infectious disease data and we evaluate its performance through simulation.     

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.     

Nonparametric modeling of the gap time in recurrent event data

Recurrent event data arise in many biomedical and engineering studies when failure events can occur repeatedly over time for each study subject. In this article, we are interested in nonparametric estimation of the hazard function for gap time. A penalized likelihood model is proposed to estimate the hazard as a function of both gap time and covariate. Method for smoothing parameter selection is developed from subject-wise cross-validation. Confidence intervals for the hazard function are derived using the Bayes model of the penalized likelihood. An eigenvalue analysis establishes the asymptotic convergence rates of the relevant estimates. Empirical studies are performed to evaluate various aspects of the method. The proposed technique is demonstrated through an application to the well-known bladder tumor cancer data.     

Penalized pseudo-likelihood hazard estimation: A fast alternative to penalized likelihood

Penalized likelihood method has been developed previously for hazard function estimation using standard left-truncated, right-censored lifetime data with covariates, and the functional ANOVA structures built into the log hazard allows for versatile nonparametric modeling in the setting. The computation of the method can be time-consuming in the presence of continuous covariates; however, due to the repeated numerical integrations involved. Adapting a device developed by Jeon and Lin [An effective method for high dimensional log-density ANOVA estimation, with application to nonparametric graphical model building. Statist. Sinica 16, 353–374] for penalized likelihood density estimation, we explore an alternative approach to hazard estimation where the log likelihood is replaced by some computationally less demanding pseudo-likelihood. An assortment of issues are addressed concerning the practical implementations of the approach including the selection of smoothing parameters, and extensive simulations are presented to assess the inferential efficiency of the “pseudo” method as compared to the “real” one. Also noted is an asymptotic theory concerning the convergence rates of the estimates parallel to that for the original penalized likelihood estimation.     

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.     

Mixed inverse problems arising in the estimation of calibration factors in positron emission tomography

A penalized likelihood approach to the estimation of calibration factors in positron emission tomography (PET) is considered, in particular the problem of estimating the efficiency of PET detectors. Varying efficiencies among the detectors create a non-uniform performance and failure to account for the non-uniformities would lead to streaks in the image, so efficient estimation of the non-uniformities is desirable to reduce the propagation of noise to the final image. The relevant data set is provided by a blank scan, where a model may be derived that depends only on the sources affecting non-uniformities: inherent variation among the detector crystals and geometric effects. Physical considerations suggest a novel mixed inverse model with random crystal effects and smooth geometric effects. Using appropriate penalty terms, the penalized maximum likelihood estimates are derived and an efficient computational algorithm utilizing the fast Fourier transform is developed. Data-driven shrinkage and smoothing parameters are chosen to minimize an estimate of the predictive loss function. Various examples indicate that the approach proposed works well computationally and compares well with the standard method.     

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.     

随机效应半参数logit模型的惩罚似然估计研究

 在颇具争议的收入差距和健康关系研究中,为了降低可能存在的模型设定和遗漏变量偏误,本文提出了随机效应半参数logit模型,其中非参数的设定还可用于数据的初探性分析。随后本文提出了模型非参数和参数部分的估计方法。这里涉及的难点是随机效应的存在导致似然函数中的积分没有解析式,而非参数的存在更加大了估计难度。本文基于惩罚样条非参数估计方法和四阶Laplace近似方法建立了惩罚对数似然函数,其最大化采用了Newton_Raphson近似方法。文章还建立了惩罚样条中重要光滑参数的选取准则。模型在收入差距和健康实例中的估计结果表明数据支持收入差距弱假说,且非参数估计结果表明其具有U型形式,与实例估计结果的比较指出本文提出的估计方法是较准确的。     

Influence diagnostics and outlier tests for semiparametric mixed models

Summary. Semiparametric mixed models are useful in biometric and econometric applications, especially for longitudinal data. Maximum penalized likelihood estimators (MPLEs) have been shown to work well by Zhang and co-workers for both linear coefficients and nonparametric functions. This paper considers the role of influence diagnostics in the MPLE by extending the case deletion and subject deletion analysis of linear models to accommodate the inclusion of a nonparametric component. We focus on influence measures for the fixed effects and provide formulae that are analogous to those for simpler models and readily computable with the MPLE algorithm. We also establish an equivalence between the case or subject deletion model and a mean shift outlier model from which we derive tests for outliers. The influence diagnostics proposed are illustrated through a longitudinal hormone study on progesterone and a simulated example.     

ON CONFIDENCE INTERVALS FOR GENERALIZED ADDITIVE MODELS BASED ON PENALIZED REGRESSION SPLINES

Generalized additive models represented using low rank penalized regression splines, estimated by penalized likelihood maximisation and with smoothness selected by generalized cross validation or similar criteria, provide a computationally efficient general framework for practical smooth modelling. Various authors have proposed approximate Bayesian interval estimates for such models, based on extensions of the work of Wahba, G. (1983) [Bayesian confidence intervals for the cross validated smoothing spline. J. R. Statist. Soc. B 45 , 133–150] and Silverman, B.W. (1985) [Some aspects of the spline smoothing approach to nonparametric regression curve fitting. J. R. Statist. Soc. B 47 , 1–52] on smoothing spline models of Gaussian data, but testing of such intervals has been rather limited and there is little supporting theory for the approximations used in the generalized case. This paper aims to improve this situation by providing simulation tests and obtaining asymptotic results supporting the approximations employed for the generalized case. The simulation results suggest that while across‐the‐model performance is good, component‐wise coverage probabilities are not as reliable. Since this is likely to result from the neglect of smoothing parameter variability, a simple and efficient simulation method is proposed to account for smoothing parameter uncertainty: this is demonstrated to substantially improve the performance of component‐wise intervals.     

Improved estimation procedures for multilevel models with binary response: a case-study

During recent years, analysts have been relying on approximate methods of inference to estimate multilevel models for binary or count data. In an earlier study of random-intercept models for binary outcomes we used simulated data to demonstrate that one such approximation, known as marginal quasi-likelihood, leads to a substantial attenuation bias in the estimates of both fixed and random effects whenever the random effects are non-trivial. In this paper, we fit three-level random-intercept models to actual data for two binary outcomes, to assess whether refined approximation procedures, namely penalized quasi-likelihood and second-order improvements to marginal and penalized quasi-likelihood, also underestimate the underlying parameters. The extent of the bias is assessed by two standards of comparison: exact maximum likelihood estimates, based on a Gauss–Hermite numerical quadrature procedure, and a set of Bayesian estimates, obtained from Gibbs sampling with diffuse priors. We also examine the effectiveness of a parametric bootstrap procedure for reducing the bias. The results indicate that second-order penalized quasi-likelihood estimates provide a considerable improvement over the other approximations, but all the methods of approximate inference result in a substantial underestimation of the fixed and random effects when the random effects are sizable. We also find that the parametric bootstrap method can eliminate the bias but is computationally very intensive.     

Sparse Functional Principal Component Analysis via Regularized Basis Expansions and Its Application

This article introduces principal component analysis for multidimensional sparse functional data, utilizing Gaussian basis functions. Our multidimensional model is estimated by maximizing a penalized log-likelihood function, while previous mixed-type models were estimated by maximum likelihood methods for one-dimensional data. The penalized estimation performs well for our multidimensional model, while maximum likelihood methods yield unstable parameter estimates and some of the parameter estimates are infinite. Numerical experiments are conducted to investigate the effectiveness of our method for some types of missing data. The proposed method is applied to handwriting data, which consist of the XY coordinates values in handwritings.     

A Mixed Model Approach for Geoadditive Hazard Regression

Abstract.  Mixed model based approaches for semiparametric regression have gained much interest in recent years, both in theory and application. They provide a unified and modular framework for penalized likelihood and closely related empirical Bayes inference. In this article, we develop mixed model methodology for a broad class of Cox-type hazard regression models where the usual linear predictor is generalized to a geoadditive predictor incorporating non-parametric terms for the (log-)baseline hazard rate, time-varying coefficients and non-linear effects of continuous covariates, a spatial component, and additional cluster-specific frailties. Non-linear and time-varying effects are modelled through penalized splines, while spatial components are treated as correlated random effects following either a Markov random field or a stationary Gaussian random field prior. Generalizing existing mixed model methodology, inference is derived using penalized likelihood for regression coefficients and (approximate) marginal likelihood for smoothing parameters. In a simulation we study the performance of the proposed method, in particular comparing it with its fully Bayesian counterpart using Markov chain Monte Carlo methodology, and complement the results by some asymptotic considerations. As an application, we analyse leukaemia survival data from northwest England.     

A test for the linearity of the nonparametric part of a semiparametric logistic regression model

A semiparametric logistic regression model is proposed in which its nonparametric component is approximated with fixed-knot cubic B-splines. To assess the linearity of the nonparametric component, we construct a penalized likelihood ratio test statistic. When the number of knots is fixed, the null distribution of the test statistic is shown to be asymptotically the distribution of a linear combination of independent chi-squared random variables, each with one degree of freedom. We set the asymptotic null expectation of this test statistic equal to a value to determine the smoothing parameter value. Monte Carlo experiments are conducted to investigate the performance of the proposed test. Its practical use is illustrated with a real-life example.     

Ordinal ridge regression with categorical predictors

In multi-category response models, categories are often ordered. In the case of ordinal response models, the usual likelihood approach becomes unstable with ill-conditioned predictor space or when the number of parameters to be estimated is large relative to the sample size. The likelihood estimates do not exist when the number of observations is less than the number of parameters. The same problem arises if constraint on the order of intercept values is not met during the iterative procedure. Proportional odds models (POMs) are most commonly used for ordinal responses. In this paper, penalized likelihood with quadratic penalty is used to address these issues with a special focus on POMs. To avoid large differences between two parameter values corresponding to the consecutive categories of an ordinal predictor, the differences between the parameters of two adjacent categories should be penalized. The considered penalized-likelihood function penalizes the parameter estimates or differences between the parameter estimates according to the type of predictors. Mean-squared error for parameter estimates, deviance of fitted probabilities and prediction error for ridge regression are compared with usual likelihood estimates in a simulation study and an application.     

Maximum penalized likelihood estimation of mixed proportional hazard models

Unobservable individual effects in models of duration will cause estimation bias that include the structural parameters as well as the duration dependence. The maximum penalized likelihood estimator is examined as an estimator for the survivor model with heterogeneity. Proofs of the existence and uniqueness of the maximum penalized likelihood estimator in duration model with general forms of unobserved heterogeneity are provided. Some small sample evidence on the behavior of the maximum penalized likelihood estimator is given. The maximum penalized likelihood estimator is shown to be computationally feasible and to provide reasonable estimates in most cases.