1,919 research outputs found
Covariance Estimation: The GLM and Regularization Perspectives
Finding an unconstrained and statistically interpretable reparameterization
of a covariance matrix is still an open problem in statistics. Its solution is
of central importance in covariance estimation, particularly in the recent
high-dimensional data environment where enforcing the positive-definiteness
constraint could be computationally expensive. We provide a survey of the
progress made in modeling covariance matrices from two relatively complementary
perspectives: (1) generalized linear models (GLM) or parsimony and use of
covariates in low dimensions, and (2) regularization or sparsity for
high-dimensional data. An emerging, unifying and powerful trend in both
perspectives is that of reducing a covariance estimation problem to that of
estimating a sequence of regression problems. We point out several instances of
the regression-based formulation. A notable case is in sparse estimation of a
precision matrix or a Gaussian graphical model leading to the fast graphical
LASSO algorithm. Some advantages and limitations of the regression-based
Cholesky decomposition relative to the classical spectral (eigenvalue) and
variance-correlation decompositions are highlighted. The former provides an
unconstrained and statistically interpretable reparameterization, and
guarantees the positive-definiteness of the estimated covariance matrix. It
reduces the unintuitive task of covariance estimation to that of modeling a
sequence of regressions at the cost of imposing an a priori order among the
variables. Elementwise regularization of the sample covariance matrix such as
banding, tapering and thresholding has desirable asymptotic properties and the
sparse estimated covariance matrix is positive definite with probability
tending to one for large samples and dimensions.Comment: Published in at http://dx.doi.org/10.1214/11-STS358 the Statistical
Science (http://www.imstat.org/sts/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Fitting Linear Mixed-Effects Models using lme4
Maximum likelihood or restricted maximum likelihood (REML) estimates of the
parameters in linear mixed-effects models can be determined using the lmer
function in the lme4 package for R. As for most model-fitting functions in R,
the model is described in an lmer call by a formula, in this case including
both fixed- and random-effects terms. The formula and data together determine a
numerical representation of the model from which the profiled deviance or the
profiled REML criterion can be evaluated as a function of some of the model
parameters. The appropriate criterion is optimized, using one of the
constrained optimization functions in R, to provide the parameter estimates. We
describe the structure of the model, the steps in evaluating the profiled
deviance or REML criterion, and the structure of classes or types that
represents such a model. Sufficient detail is included to allow specialization
of these structures by users who wish to write functions to fit specialized
linear mixed models, such as models incorporating pedigrees or smoothing
splines, that are not easily expressible in the formula language used by lmer.Comment: 51 pages, including R code, and an appendi
Fixed effects selection in the linear mixed-effects model using adaptive ridge procedure for L0 penalty performance
This paper is concerned with the selection of fixed effects along with the
estimation of fixed effects, random effects and variance components in the
linear mixed-effects model. We introduce a selection procedure based on an
adaptive ridge (AR) penalty of the profiled likelihood, where the covariance
matrix of the random effects is Cholesky factorized. This selection procedure
is intended to both low and high-dimensional settings where the number of fixed
effects is allowed to grow exponentially with the total sample size, yielding
technical difficulties due to the non-convex optimization problem induced by L0
penalties. Through extensive simulation studies, the procedure is compared to
the LASSO selection and appears to enjoy the model selection consistency as
well as the estimation consistency
Covariance pattern mixture models for the analysis of multivariate heterogeneous longitudinal data
We propose a novel approach for modeling multivariate longitudinal data in
the presence of unobserved heterogeneity for the analysis of the Health and
Retirement Study (HRS) data. Our proposal can be cast within the framework of
linear mixed models with discrete individual random intercepts; however,
differently from the standard formulation, the proposed Covariance Pattern
Mixture Model (CPMM) does not require the usual local independence assumption.
The model is thus able to simultaneously model the heterogeneity, the
association among the responses and the temporal dependence structure. We focus
on the investigation of temporal patterns related to the cognitive functioning
in retired American respondents. In particular, we aim to understand whether it
can be affected by some individual socio-economical characteristics and whether
it is possible to identify some homogenous groups of respondents that share a
similar cognitive profile. An accurate description of the detected groups
allows government policy interventions to be opportunely addressed. Results
identify three homogenous clusters of individuals with specific cognitive
functioning, consistent with the class conditional distribution of the
covariates. The flexibility of CPMM allows for a different contribution of each
regressor on the responses according to group membership. In so doing, the
identified groups receive a global and accurate phenomenological
characterization.Comment: Published at http://dx.doi.org/10.1214/15-AOAS816 in the Annals of
Applied Statistics (http://www.imstat.org/aoas/) by the Institute of
Mathematical Statistics (http://www.imstat.org
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A Mixed-Effects Location Scale Model for Dyadic Interactions.
We present a mixed-effects location scale model (MELSM) for examining the daily dynamics of affect in dyads. The MELSM includes person and time-varying variables to predict the location, or individual means, and the scale, or within-person variances. It also incorporates a submodel to account for between-person variances. The dyadic specification can accommodate individual and partner effects in both the location and the scale components, and allows random effects for all location and scale parameters. All covariances among the random effects, within and across the location and the scale are also estimated. These covariances offer new insights into the interplay of individual mean structures, intra-individual variability, and the influence of partner effects on such factors. To illustrate the model, we use data from 274 couples who provided daily ratings on their positive and negative emotions toward their relationship - up to 90 consecutive days. The model is fit using Hamiltonian Monte Carlo methods, and includes subsets of predictors in order to demonstrate the flexibility of this approach. We conclude with a discussion on the usefulness and the limitations of the MELSM for dyadic research
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