光滑样条非参数回归方法及医学应用

目的 改进回归分析的经典最小二乘估计方法和探讨光滑样条非参数回归分析方法。方法 利用三次函数和粗糙度惩罚方法的有机结合，构造惩罚平方和，通过广义交互有效得分函数和模式搜索法自动选择光滑参数值。结果 用ＳＡＳ程序实现了光滑样条非参数回归分析，得到了回归函数的最小惩罚二乘估计，实例表明，该方法优于传统方法和非参数Ｍｏｎｏｔｏｎｉｃ回归。结论 非参数回归分析方法能够最佳地兼顾拟合优度和光滑度，改进了经典     

多维样条回归模型及医学应用

目的放宽经典线性模型中的多个解释变量的线性假定和探讨多维样条回归分析模型.方法利用最小惩罚二乘原理构造惩罚残差平方和,通过广义交互有效得分函数自动选择光滑参数值,对有关矩阵进行QR分解、Cholesky分解以及奇异值分解.结果用SAS程序实现了多维样条回归分析,得到了模型系数向量和多维样条函数的最小惩罚二乘估计,实例分析表明,多维样条回归模型较一般线性模型有更强的适应性.结论多维样条回归模型是一般线性模型的全面扩展,为探索医学指标间的关系以及进行预测提供了可靠的线索和有效的途径.     

半参数回归模型及模拟实例分析

目的 放宽经典线性模型中的解释变量的线性假定和探讨半参数回归分析模型。方法 利用最小惩罚二乘原理构造加权惩罚平方和,通过广义交互有效得分函数自动选择光滑参数值,用直接法求解方程组。结果 用SAS程序实现了半参数回归分析。得到了回归系数向量和样条函数的最小惩罚二乘估计,模拟实例表明,半参数回归模型较传统的线性模型有较强的适应性。结论 半参数回归模型是经典线性模型和非参数回归模型的一个混合体。可作为回归分析的一种新技术得到广泛应用。     

Controlling for seasonal patterns and time varying confounders in time‐series epidemiological models: a simulation study

An important topic when estimating the effect of air pollutants on human health is choosing the best method to control for seasonal patterns and time varying confounders, such as temperature and humidity. Semi‐parametric Poisson time‐series models include smooth functions of calendar time and weather effects to control for potential confounders. Case‐crossover (CC) approaches are considered efficient alternatives that control seasonal confounding by design and allow inclusion of smooth functions of weather confounders through their equivalent Poisson representations. We evaluate both methodological designs with respect to seasonal control and compare spline‐based approaches, using natural splines and penalized splines, and two time‐stratified CC approaches. For the spline‐based methods, we consider fixed degrees of freedom, minimization of the partial autocorrelation function, and general cross‐validation as smoothing criteria. Issues of model misspecification with respect to weather confounding are investigated under simulation scenarios, which allow quantifying omitted, misspecified, and irrelevant‐variable bias. The simulations are based on fully parametric mechanisms designed to replicate two datasets with different mortality and atmospheric patterns. Overall, minimum partial autocorrelation function approaches provide more stable results for high mortality counts and strong seasonal trends, whereas natural splines with fixed degrees of freedom perform better for low mortality counts and weak seasonal trends followed by the time‐season‐stratified CC model, which performs equally well in terms of bias but yields higher standard errors. Copyright © 2014 John Wiley & Sons, Ltd.     

Derivative estimation for longitudinal data analysis: Examining features of blood pressure measured repeatedly during pregnancy

Estimating velocity and acceleration trajectories allows novel inferences in the field of longitudinal data analysis, such as estimating change regions rather than change points, and testing group effects on nonlinear change in an outcome (ie, a nonlinear interaction). In this article, we develop derivative estimation for 2 standard approaches—polynomial mixed models and spline mixed models. We compare their performance with an established method—principal component analysis through conditional expectation through a simulation study. We then apply the methods to repeated blood pressure (BP) measurements in a UK cohort of pregnant women, where the goals of analysis are to (i) identify and estimate regions of BP change for each individual and (ii) investigate the association between parity and BP change at the population level. The penalized spline mixed model had the lowest bias in our simulation study, and we identified evidence for BP change regions in over 75% of pregnant women. Using mean velocity difference revealed differences in BP change between women in their first pregnancy compared with those who had at least 1 previous pregnancy. We recommend the use of penalized spline mixed models for derivative estimation in longitudinal data analysis.     

Semi‐parametric latent process model for longitudinal ordinal data: Application to cognitive decline

Ordinal and quantitative discrete data are frequent in biomedical and neuropsychological studies. We propose a semi‐parametric model for the analysis of the change over time of such data in longitudinal studies. A threshold model is defined where the outcome value depends on the current value of an underlying Gaussian latent process. The latent process model is a Gaussian linear mixed model with a non‐parametric function of time, f(t), to model the expected change over time. This model includes random‐effects and a stochastic error process to flexibly handle correlation between repeated measures. The function f(t) and all the model parameters are estimated by penalized likelihood using a cubic‐spline approximation for f(t). The smoothing parameter is estimated by an approximate cross‐validation criterion. Confidence bands may be computed for the estimated curves for the latent process and, using a Monte Carlo approach, for the outcome in its natural scale. The method is applied to the Paquid cohort data to compare the time‐course over 14 years of two cognitive scores in a sample of 350 future Alzheimer patients and in a matched sample of healthy subjects. Copyright © 2010 John Wiley & Sons, Ltd.     

Flexible assessment of trends in age-specific HIV incidence using two-dimensional penalized likelihood

We present a method for estimating age- and time-specific HIV incidence using back-calculation of AIDS incidence data. Two-dimensional penalized likelihood is employed, using a flexible bivariate step function model of HIV incidence, together with a quadratic roughness penalty which leads to thin-plate spline smoothing. This allows incidence estimates to vary flexibly and smoothly in both age and time. We propose generalized cross-validation as a guide for choice of an appropriate level of smoothing and describe an EM algorithm for computing the estimates. We propose the method primarily for qualitative assessment of trends in age-specific incidence over time and apply it to a small Italian data set on men who have sex with men. The analysis suggests a trend over time of increasing relative incidence among younger individuals, consistent with incidence patterns observed in other countries. © 1998 John Wiley & Sons, Ltd.     

Advanced methods in meta-analysis: multivariate approach and meta-regression

This tutorial on advanced statistical methods for meta-analysis can be seen as a sequel to the recent Tutorial in Biostatistics on meta-analysis by Normand, which focused on elementary methods. Within the framework of the general linear mixed model using approximate likelihood, we discuss methods to analyse univariate as well as bivariate treatment effects in meta-analyses as well as meta-regression methods. Several extensions of the models are discussed, like exact likelihood, non-normal mixtures and multiple endpoints. We end with a discussion about the use of Bayesian methods in meta-analysis. All methods are illustrated by a meta-analysis concerning the efficacy of BCG vaccine against tuberculosis. All analyses that use approximate likelihood can be carried out by standard software. We demonstrate how the models can be fitted using SAS Proc Mixed.     

Flexible estimation of covariance function by penalized spline with application to longitudinal family data

Longitudinal data are routinely collected in biomedical research studies. A natural model describing longitudinal data decomposes an individual's outcome as the sum of a population mean function and random subject-specific deviations. When parametric assumptions are too restrictive, methods modeling the population mean function and the random subject-specific functions nonparametrically are in demand. In some applications, it is desirable to estimate a covariance function of random subject-specific deviations. In this work, flexible yet computationally efficient methods are developed for a general class of semiparametric mixed effects models, where the functional forms of the population mean and the subject-specific curves are unspecified. We estimate nonparametric components of the model by penalized spline (P-spline, Biometrics 2001; 57:253-259), and reparameterize the random curve covariance function by a modified Cholesky decomposition (Biometrics 2002; 58:121-128) which allows for unconstrained estimation of a positive-semidefinite matrix. To provide smooth estimates, we penalize roughness of fitted curves and derive closed-form solutions in the maximization step of an EM algorithm. In addition, we present models and methods for longitudinal family data where subjects in a family are correlated and we decompose the covariance function into a subject-level source and observation-level source. We apply these methods to the multi-level Framingham Heart Study data to estimate age-specific heritability of systolic blood pressure nonparametrically.     

HISTORICAL HIV INCIDENCE MODELLING IN REGIONAL SUBGROUPS: USE OF FLEXIBLE DISCRETE MODELS WITH PENALIZED SPLINES BASED ON PRIOR CURVES

This paper presents an approach to back-projection (back-calculation) of human immunodeficiency virus (HIV) person-year infection rates in regional subgroups based on combining a log-linear model for subgroup differences with a penalized spline model for trends. The penalized spline approach allows flexible trend estimation but requires far fewer parameters than fully non-parametric smoothers, thus saving parameters that can be used in estimating subgroup effects. Use of a reasonable prior curve to construct the penalty function minimizes the degree of smoothing needed beyond model specification. The approach is illustrated in application to acquired immunodeficiency syndrome (AIDS) surveillance data from Los Angeles County.     

Flexible regression models with cubic splines

We describe the use of cubic splines in regression models to represent the relationship between the response variable and a vector of covariates. This simple method can help prevent the problems that result from inappropriate linearity assumptions. We compare restricted cubic spline regression to non-parametric procedures for characterizing the relationship between age and survival in the Stanford Heart Transplant data. We also provide an illustrative example in cancer therapeutics.     

Modelling bivariate ordinal responses smoothly with examples from ophthalmology and genetics.

A non-parametric implementation of the bivariate Dale model (BDM) is presented as an extension of the generalized additive model (GAM) of Hastie and Tibshirani. The original BDM is an example of a bivariate generalized linear model. In this paper smoothing is introduced on the marginal as well as on the association level. Our non-parametric procedure can be used as a diagnostic tool for identifying parametric transformations of the covariates in the linear BDM, hence it also provides a kind of goodness-of-fit test for a bivariate generalized linear model. Cubic smoothing spline functions for the covariates are estimated by maximizing a penalized version of the log-likelihood. The method is applied to two studies. The first study is the classical Wisconsin Epidemiologic Study of Diabetic Retinopathy. The second study is a twin study, where the association between the elements of twin pairs is of primary interest. The results show that smoothing on the association level can give a significant improvement to the model fit.     

A semiparametric recurrent events model with time‐varying coefficients

We consider a recurrent events model with time‐varying coefficients motivated by two clinical applications. We use a random effects (Gaussian frailty) model to describe the intensity of recurrent events. The model can accommodate both time‐varying and time‐constant coefficients. We use the penalized spline method to estimate the time‐varying coefficients. We use Laplace approximation to evaluate the penalized likelihood without a closed form. We estimate the smoothing parameters in a similar way to variance components. We conduct simulations to evaluate the performance of the estimates for both time‐varying and time‐independent coefficients. We apply this method to analyze two data sets: a stroke study and a child wheeze study. Copyright © 2012 John Wiley & Sons, Ltd.     

Quantifying epidemiologic risk factors using non-parametric regression: model selection remains the greatest challenge

Logistic regression is widely used to estimate relative risks (odds ratios) from case-control studies, but when the study exposure is continuous, standard parametric models may not accurately characterize the exposure-response curve. Semi-parametric generalized linear models provide a useful extension. In these models, the exposure of interest is modelled flexibly using a regression spline or a smoothing spline, while other variables are modelled using conventional methods. When coupled with a model-selection procedure based on minimizing a cross-validation score, this approach provides a non-parametric, objective, and reproducible method to characterize the exposure-response curve by one or several models with a favourable bias-variance trade-off. We applied this approach to case-control data to estimate the dose-response relationship between alcohol consumption and risk of oral cancer among African Americans. We did not find a uniquely 'best' model, but results using linear, cubic, and smoothing splines were consistent: there does not appear to be a risk-free threshold for alcohol consumption vis-à-vis the development of oral cancer. This finding was not apparent using a standard step-function model. In our analysis, the cross-validation curve had a global minimum and also a local minimum. In general, the phenomenon of multiple local minima makes it more difficult to interpret the results, and may present a computational roadblock to non-parametric generalized additive models of multiple continuous exposures. Nonetheless, the semi-parametric approach appears to be a practical advance.     

Comparing smoothing techniques in Cox models for exposure-response relationships

To allow for non-linear exposure-response relationships, we applied flexible non-parametric smoothing techniques to models of time to lung cancer mortality in two occupational cohorts with skewed exposure distributions. We focused on three different smoothing techniques in Cox models: penalized splines, restricted cubic splines, and fractional polynomials. We compared standard software implementations of these three methods based on their visual representation and criterion for model selection. We propose a measure of the difference between a pair of curves based on the area between them, standardized by the average of the areas under the pair of curves. To capture the variation in the difference over the range of exposure, the area between curves was also calculated at percentiles of exposure and expressed as a percentage of the total difference. The dose-response curves from the three methods were similar in both studies over the denser portion of the exposure range, with the difference between curves up to the 50th percentile less than 1 per cent of the total difference. A comparison of inverse variance weighted areas applied to the data set with a more skewed exposure distribution allowed us to estimate area differences with more precision by reducing the proportion attributed to the upper 1 per cent tail region. Overall, the penalized spline and the restricted cubic spline were closer to each other than either was to the fractional polynomial.     

Penalized likelihood approach to estimate a smooth mean curve on longitudinal data

This paper aims to propose a penalized likelihood approach to estimate a smooth mean curve for the evolution with time of a Gaussian variable taking into account the correlation structure of longitudinal data. The model is an extension of the mixed effects linear model including an unspecified function of time f(t). The estimator (circumflex)f(t) is defined as the solution of the maximization of the penalized likelihood and is approximated on a basis of cubic M-spline with a reduced number of knots. We present modifications of four criteria (cross-validation, generalized cross-validation, T of Rice, Akaike's criterion) to estimate the smoothing parameter when data are correlated; these four criteria gave very similar results in the simulation study. The simulation study showed also the superiority of the Bayesian confidence bands of the mean curve over the frequentist ones. We develop empirical Bayes estimates of subject-specific deviations. This approach was applied to study the progression of CD4+ lymphocyte counts in a cohort of HIV patients treated with protease inhibitors.     

Generalized survival models for correlated time‐to‐event data

Our aim is to develop a rich and coherent framework for modeling correlated time‐to‐event data, including (1) survival regression models with different links and (2) flexible modeling for time‐dependent and nonlinear effects with rich postestimation. We extend the class of generalized survival models, which expresses a transformed survival in terms of a linear predictor, by incorporating a shared frailty or random effects for correlated survival data. The proposed approach can include parametric or penalized smooth functions for time, time‐dependent effects, nonlinear effects, and their interactions. The maximum (penalized) marginal likelihood method is used to estimate the regression coefficients and the variance for the frailty or random effects. The optimal smoothing parameters for the penalized marginal likelihood estimation can be automatically selected by a likelihood‐based cross‐validation criterion. For models with normal random effects, Gauss‐Hermite quadrature can be used to obtain the cluster‐level marginal likelihoods. The Akaike Information Criterion can be used to compare models and select the link function. We have implemented these methods in the R package rstpm2. Simulating for both small and larger clusters, we find that this approach performs well. Through 2 applications, we demonstrate (1) a comparison of proportional hazards and proportional odds models with random effects for clustered survival data and (2) the estimation of time‐varying effects on the log‐time scale, age‐varying effects for a specific treatment, and two‐dimensional splines for time and age.     

Penalized likelihood estimation for semiparametric mixed models,with application to alcohol treatment research

In this article, we implement a practical computational method for various semiparametric mixed effects models, estimating nonlinear functions by penalized splines. We approximate the integration of the penalized likelihood with respect to random effects with the use of adaptive Gaussian quadrature, which we can conveniently implement in SAS procedure NLMIXED. We carry out the selection of smoothing parameters through approximated generalized cross‐validation scores. Our method has two advantages: (1) the estimation is more accurate than the current available quasi‐likelihood method for sparse data, for example, binary data; and (2) it can be used in fitting more sophisticated models. We show the performance of our approach in simulation studies with longitudinal outcomes from three settings: binary, normal data after Box–Cox transformation, and count data with log‐Gamma random effects. We also develop an estimation method for a longitudinal two‐part nonparametric random effects model and apply it to analyze repeated measures of semicontinuous daily drinking records in a randomized controlled trial of topiramate. Copyright © 2012 John Wiley & Sons, Ltd.     

Variational methods for fitting complex Bayesian mixed effects models to health data

We consider approximate inference methods for Bayesian inference to longitudinal and multilevel data within the context of health science studies. The complexity of these grouped data often necessitates the use of sophisticated statistical models. However, the large size of these data can pose significant challenges for model fitting in terms of computational speed and memory storage. Our methodology is motivated by a study that examines trends in cesarean section rates in the largest state of Australia, New South Wales, between 1994 and 2010. We propose a group‐specific curve model that encapsulates the complex nonlinear features of the overall and hospital‐specific trends in cesarean section rates while taking into account hospital variability over time. We use penalized spline‐based smooth functions that represent trends and implement a fully mean field variational Bayes approach to model fitting. Our mean field variational Bayes algorithms allow a fast (up to the order of thousands) and streamlined analytical approximate inference for complex mixed effects models, with minor degradation in accuracy compared with the standard Markov chain Monte Carlo methods. Copyright © 2015 John Wiley & Sons, Ltd.     

Improving the mapping of condition-specific health-related quality of life onto SF-6D score

Background

This study sought to improve the predicative performance and goodness-of-fit of mapping models, as part of indirect valuation, by introducing cubic spline smoothing to map a group of health-related quality of life (HRQOL) measures onto a preference-based measure.

Methods

This study was a secondary analysis of a cross-sectional health survey data assessing the HRQOL for patients with colorectal neoplasms. Mapping functions of condition-specific functional assessment of cancer therapy—colorectal (FACT-C) onto preference-based SF-6D measure were developed using a dataset of 553 Chinese subjects with different stages of colorectal neoplasm. The missing values of FACT-C were imputed using multiple imputation. Then three widely applicable models (ordinary least square (OLS), Tobit and two-part models) were employed for the mapping function after applying the cubic spline smoothing on the data. For the evaluation of the effectiveness of cubic spline smoothing and multiple imputation, the goodness-of-fit and prediction performance of each model were compared.

Results

Analyses showed that the models fitted with transformed data from cubic spline smoothing offered better performance in goodness-of-fit and prediction than the models fitted with the original data. The values of \(R^2\) were improved by over 10 %, and the root mean square error and the mean absolute error were both reduced. The best goodness-of-fit and performance were achieved by OLS model using transformed data from cubic spline smoothing.

Conclusions

Cubic spline smoothing and multiple imputation were recommended for the mapping of HRQOL measures onto the preference-based measure. Among the three mapping models, the simple-to-use OLS model had the best performance.