Biometrics Unit Technical Reports: Number BU-1560-M: Self-Modeling Regression with Random Effects (original) (raw)

Self-Modeling Regression with Random Effects Using Penalized Splines

2000

In many longitudinal studies, the response can be modeled as a (discretely sampled) curve over time for each subject. Often these curves have a common shape function and individual subjects differ from the common shape by a transformation ofthe time and response scales. Lindstrom (1995) represented the common shape by a free-knot regression spline, and used a parametric random effects model to represent the differences between curves. We extend Lindstrom's work by representing the common shape by a penalized regression spline, and use a parametric random effects model to represent the differences between curves. The use of penalized regression splines allows for a generalization in the modeling, estimation, and testing of parameters and is easily implemented. An iterative two-step algorithm is proposed for fitting the model.

Self-Modeling Regression with Random Effects

2001

The Analysis of Designed Experiments and Longitudinal Data by Using Smoothing Splines

Journal of the Royal Statistical Society: Series C (Applied Statistics), 1999

Smoothing splines and other non-parametric smoothing methods are well accepted for exploratory data analysis. These methods have been used in regression, in repeated measures or longitudinal data analysis, and in generalized linear models. However, a major drawback is the lack of a formal inferential framework. An exception which has not been fully exploited is the cubic smoothing spline. The cubic smoothing spline admits a mixed model formulation, which places this non-parametric smoother firmly in a parametric setting. The formulation presented in this paper provides the mechanism for including cubic smoothing splines in models for the analysis of designed experiments and longitudinal data. Thus nonlinear curves can be included with random effects and random coefficients, and this leads to very flexible and informative modelling within the linear mixed model framework. Variance heterogeneity can also be accommodated. The advantage of using the cubic smoothing spline in the case of longitudinal data is particularly pronounced, because covariance modelling is achieved implicitly as for random coefficient models. Several examples are considered to illustrate the ideas. Verbyla, Cullis, Kenward and Welham mechanism driving the process being observed. A major difficulty is that both estimation and inference are approximate, because the nonlinearity precludes exact likelihoods from being found. In addition, the assumptions made for the random effects may be questionable. Methods which avoid these problems are therefore very attractive.

Tutorial in biostatistics: spline smoothing with linear mixed models

Statistics in Medicine, 2005

The semi-parametric regression achieved via penalized spline smoothing can be expressed in a linear mixed models framework. This allows such models to be ÿtted using standard mixed models software routines with which many biostatisticians are familiar. Moreover, the analysis of complex correlated data structures that are a hallmark of biostatistics, and which are typically analysed using mixed models, can now incorporate directly smoothing of the relationship between an outcome and covariates. In this paper we provide an introduction to both linear mixed models and penalized spline smoothing, and describe the connection between the two. This is illustrated with three examples, the ÿrst using birth data from the U.K., the second relating mammographic density to age in a study of female twin-pairs and the third modelling the relationship between age and bronchial hyperresponsiveness in families. The models are ÿtted in R (a clone of S-plus) and using Markov chain Monte Carlo (MCMC) implemented in the package WinBUGS.

A framework for longitudinal data analysis via shape regression

SPIE Proceedings, 2012

Traditional longitudinal analysis begins by extracting desired clinical measurements, such as volume or head circumference, from discrete imaging data. Typically, the continuous evolution of a scalar measurement is estimated by choosing a 1D regression model, such as kernel regression or fitting a polynomial of fixed degree. This type of analysis not only leads to separate models for each measurement, but there is no clear anatomical or biological interpretation to aid in the selection of the appropriate paradigm. In this paper, we propose a consistent framework for the analysis of longitudinal data by estimating the continuous evolution of shape over time as twice differentiable flows of deformations. In contrast to 1D regression models, one model is chosen to realistically capture the growth of anatomical structures. From the continuous evolution of shape, we can simply extract any clinical measurements of interest. We demonstrate on real anatomical surfaces that volume extracted from a continuous shape evolution is consistent with a 1D regression performed on the discrete measurements. We further show how the visualization of shape progression can aid in the search for significant measurements. Finally, we present an example on a shape complex of the brain (left hemisphere, right hemisphere, cerebellum) that demonstrates a potential clinical application for our framework.

Analyzing Longitudinal Data using Gee-Smoothing Spline

… . Proceedings. Mathematics and Computers in Science …, 2009

This paper considers nonparametric regression to analyze longitudinal data. Some developments of nonparametric regression have been achieved for longitudinal or clustered categorical data. For exponential family distribution, Lin & Carroll [6] considered nonparametric regression for longitudinal data using GEE-Local Polynomial Kernel (LPK). They showed that in order to obtain an efficient estimator, one must ignore within subject correlation. This means within subject observations should be assumed independent, hence the working correlation matrix must be an identity matrix. With Lin & Carroll , to obtain efficient estimates we should ignore correlation that exist in longitudinal data, even if correlation is the interest of the study. In this paper we propose GEE-Smoothing spline to analyze longitudinal data and study the property of the estimator such as the bias, consistency and efficiency. We use natural cubic spline and combine with GEE of Liang & Zeger [5] in estimation. We want to explore numerically, whether the properties of GEE-Smoothing spline are better than of GEE-Local Polynomial Kernel that proposed by Lin & Carrol [6]. Using simulation we show that GEE-Smoothing Spline is better than GEE-local polynomial. The bias of pointwise estimator is decreasing with increasing sample size. The pointwise estimator is also consistent even with incorrect correlation structure, and the most efficient estimate is obtained if the true correlation structure is used.

The estimation of branching curves in the presence of subject-specific random effects

Statistics in Medicine, 2014

Branching curves are a technique for modeling curves that change trajectory at a change (branching) point. Currently, the estimation framework is limited to independent data, and smoothing splines are used for estimation. This article aims to extend the branching curve framework to the longitudinal data setting where the branching point varies by subject. If the branching point is modeled as a random effect, then the longitudinal branching curve framework is a Semiparametric Nonlinear Mixed Effects Model. Given existing issues with using random effects within a smoothing spline, we express the model as a B-spline Based Semiparametric Nonlinear Mixed Effects Model. Simple, clever smoothness constraints are enforced on the Bsplines at the change point. The method is applied to Women's Health data where we model the shape of the labor curve (cervical dilation measured longitudinally) before and after treatment with oxytocin (a labor stimulant).

Marginal Longitudinal Curves Estimated via Bayesian Penalized Splines

The Six Cites air pollution data is used to estimate and investigate the marginal curve of a function describing lung growth for set of children in a longitudinal study. This article proposes penalized regression spline technique based on a semiparametric mixed models (MM) framework for an additive model. This smoothing approach fits marginal models for longitudinal unbalanced measurements by using a Bayesian inference approach, implemented using a Markov chain Monte Carlo (MCMC) approach with the Gibbs sampler. The unbalanced case in which missing or different number of measurements for a set of subjects is more practical and common in real life studies. This methodology makes it possible to establish a straightforward approach to estimating similar models using R programming, when it is not possible to do so using existing codes like lme().

On the Estimation of Nonlinear Mixed-Effects Models and Latent Curve Models for Longitudinal Data

Nonlinear models are effective tools for the analysis of longitudinal data. These models provide a flexible means for describing data that follow complex forms of change. Exponential and logistic functions that include a parameter to represent an asymptote, for instance, are useful for describing responses that tend to level off with time. There are forms of nonlinear latent curve models and nonlinear mixed-effects model that are equivalent, and so given the same set of data, growth function, distributional assumptions, and method of estimation, the two models yield equivalent results. There are also forms that are strikingly different and can yield different interpretations for a given set of data. This paper discusses cases in which nonlinear mixed-effects models and nonlinear latent curve models are equivalent and those in which they are different and clarifies the estimation needs of the different models. Examples based on empirical data help to illustrate these points.

Biometrics Unit Technical Reports: Number BU-1560-M: Self-Modeling Regression with Random Effects (original) (raw)

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