37,501 research outputs found
Bayesian uncertainty quantification in linear models for diffusion MRI
Diffusion MRI (dMRI) is a valuable tool in the assessment of tissue
microstructure. By fitting a model to the dMRI signal it is possible to derive
various quantitative features. Several of the most popular dMRI signal models
are expansions in an appropriately chosen basis, where the coefficients are
determined using some variation of least-squares. However, such approaches lack
any notion of uncertainty, which could be valuable in e.g. group analyses. In
this work, we use a probabilistic interpretation of linear least-squares
methods to recast popular dMRI models as Bayesian ones. This makes it possible
to quantify the uncertainty of any derived quantity. In particular, for
quantities that are affine functions of the coefficients, the posterior
distribution can be expressed in closed-form. We simulated measurements from
single- and double-tensor models where the correct values of several quantities
are known, to validate that the theoretically derived quantiles agree with
those observed empirically. We included results from residual bootstrap for
comparison and found good agreement. The validation employed several different
models: Diffusion Tensor Imaging (DTI), Mean Apparent Propagator MRI (MAP-MRI)
and Constrained Spherical Deconvolution (CSD). We also used in vivo data to
visualize maps of quantitative features and corresponding uncertainties, and to
show how our approach can be used in a group analysis to downweight subjects
with high uncertainty. In summary, we convert successful linear models for dMRI
signal estimation to probabilistic models, capable of accurate uncertainty
quantification.Comment: Added results from a group analysis and a comparison with residual
bootstra
The vector floor and ceiling model
This paper motivates and develops a nonlinear extension of the Vector Autoregressive model which we call the Vector Floor and Ceiling model. Bayesian and classical methods for estimation and testing are developed and compared in the context of an application involving U.S. macroeconomic data. In terms of statistical significance both classical and Bayesian methods indicate that the (Gaussian) linear model is inadequate. Using impulse response functions we investigate the economic significance of the statistical analysis. We find evidence of strong nonlinearities in the contemporaneous relationships between the variables and milder evidence of nonlinearity in the conditional mean
Dynamic Bayesian Nonlinear Calibration
Statistical calibration where the curve is nonlinear is important in many
areas, such as analytical chemistry and radiometry. Especially in radiometry,
instrument characteristics change over time, thus calibration is a process that
must be conducted as long as the instrument is in use. We propose a dynamic
Bayesian method to perform calibration in the presence of a curvilinear
relationship between the reference measurements and the response variable. The
dynamic calibration approach adequately derives time dependent calibration
distributions in the presence of drifting regression parameters. The method is
applied to microwave radiometer data and simulated spectroscopy data based on
work by Lundberg and de Mar\'{e} (1980)
An approach for jointly modeling multivariate longitudinal measurements and discrete time-to-event data
In many medical studies, patients are followed longitudinally and interest is
on assessing the relationship between longitudinal measurements and time to an
event. Recently, various authors have proposed joint modeling approaches for
longitudinal and time-to-event data for a single longitudinal variable. These
joint modeling approaches become intractable with even a few longitudinal
variables. In this paper we propose a regression calibration approach for
jointly modeling multiple longitudinal measurements and discrete time-to-event
data. Ideally, a two-stage modeling approach could be applied in which the
multiple longitudinal measurements are modeled in the first stage and the
longitudinal model is related to the time-to-event data in the second stage.
Biased parameter estimation due to informative dropout makes this direct
two-stage modeling approach problematic. We propose a regression calibration
approach which appropriately accounts for informative dropout. We approximate
the conditional distribution of the multiple longitudinal measurements given
the event time by modeling all pairwise combinations of the longitudinal
measurements using a bivariate linear mixed model which conditions on the event
time. Complete data are then simulated based on estimates from these pairwise
conditional models, and regression calibration is used to estimate the
relationship between longitudinal data and time-to-event data using the
complete data. We show that this approach performs well in estimating the
relationship between multivariate longitudinal measurements and the
time-to-event data and in estimating the parameters of the multiple
longitudinal process subject to informative dropout. We illustrate this
methodology with simulations and with an analysis of primary biliary cirrhosis
(PBC) data.Comment: Published in at http://dx.doi.org/10.1214/10-AOAS339 the Annals of
Applied Statistics (http://www.imstat.org/aoas/) by the Institute of
Mathematical Statistics (http://www.imstat.org
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