1,218 research outputs found
Functional Regression
Functional data analysis (FDA) involves the analysis of data whose ideal
units of observation are functions defined on some continuous domain, and the
observed data consist of a sample of functions taken from some population,
sampled on a discrete grid. Ramsay and Silverman's 1997 textbook sparked the
development of this field, which has accelerated in the past 10 years to become
one of the fastest growing areas of statistics, fueled by the growing number of
applications yielding this type of data. One unique characteristic of FDA is
the need to combine information both across and within functions, which Ramsay
and Silverman called replication and regularization, respectively. This article
will focus on functional regression, the area of FDA that has received the most
attention in applications and methodological development. First will be an
introduction to basis functions, key building blocks for regularization in
functional regression methods, followed by an overview of functional regression
methods, split into three types: [1] functional predictor regression
(scalar-on-function), [2] functional response regression (function-on-scalar)
and [3] function-on-function regression. For each, the role of replication and
regularization will be discussed and the methodological development described
in a roughly chronological manner, at times deviating from the historical
timeline to group together similar methods. The primary focus is on modeling
and methodology, highlighting the modeling structures that have been developed
and the various regularization approaches employed. At the end is a brief
discussion describing potential areas of future development in this field
A semiparametric regression model for paired longitudinal outcomes with application in childhood blood pressure development
This research examines the simultaneous influences of height and weight on
longitudinally measured systolic and diastolic blood pressure in children.
Previous studies have shown that both height and weight are positively
associated with blood pressure. In children, however, the concurrent increases
of height and weight have made it all but impossible to discern the effect of
height from that of weight. To better understand these influences, we propose
to examine the joint effect of height and weight on blood pressure. Bivariate
thin plate spline surfaces are used to accommodate the potentially nonlinear
effects as well as the interaction between height and weight. Moreover, we
consider a joint model for paired blood pressure measures, that is, systolic
and diastolic blood pressure, to account for the underlying correlation between
the two measures within the same individual. The bivariate spline surfaces are
allowed to vary across different groups of interest. We have developed related
model fitting and inference procedures. The proposed method is used to analyze
data from a real clinical investigation.Comment: Published in at http://dx.doi.org/10.1214/12-AOAS567 the Annals of
Applied Statistics (http://www.imstat.org/aoas/) by the Institute of
Mathematical Statistics (http://www.imstat.org
A Flexible Semi-Parametric Approach to Estimating a Dose-Response Relationship: the Treatment of Childhood Amblyopia.
In a study of a dose-response relationship, flexibility in modelling is essential to capturing the treatment effect when the mean effect of other covariates is not fully understood, so that observed treatment effect is not due to the imposition of a rigid model for the relationship between response, treatment, and other variables. A semiparametric additive linear mixed (SPALM) model (Ruppert et al. 2003) provides a tractable and flexible approach to modelling the influence of potentially confounding variables. In this paper, we present pure likelihood and Bayesian versions of the SPALM model. Both methods of inference are readily implementable, but the Bayesian approach allows coherent propagation of uncertainty in the model, and, more importantly, allows prediction of future experimental results for as yet untreated individuals, thus allowing an assessment of the merits of different dosing strategies. We motivate the use of the methodology with the Monitored Occlusion Treatment of Amblyopia Study (MOTAS), which investigated the relationship between duration of occlusion and improvement in visual acuity
Joint Dispersion Model with a Flexible Link
The objective is to model longitudinal and survival data jointly taking into
account the dependence between the two responses in a real HIV/AIDS dataset
using a shared parameter approach inside a Bayesian framework. We propose a
linear mixed effects dispersion model to adjust the CD4 longitudinal biomarker
data with a between-individual heterogeneity in the mean and variance. In doing
so we are relaxing the usual assumption of a common variance for the
longitudinal residuals. A hazard regression model is considered in addition to
model the time since HIV/AIDS diagnostic until failure, being the coefficients,
accounting for the linking between the longitudinal and survival processes,
time-varying. This flexibility is specified using Penalized Splines and allows
the relationship to vary in time. Because heteroscedasticity may be related
with the survival, the standard deviation is considered as a covariate in the
hazard model, thus enabling to study the effect of the CD4 counts' stability on
the survival. The proposed framework outperforms the most used joint models,
highlighting the importance in correctly taking account the individual
heterogeneity for the measurement errors variance and the evolution of the
disease over time in bringing new insights to better understand this
biomarker-survival relation.Comment: 27 pages, 3 figures, 2 table
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