Smoothed quantile regression for partially functional linear models in high dimensions

Practitioners of current data analysis are regularly confronted with the situation where the heavy-tailed skewed response is related to both multiple functional predictors and high-dimensional scalar covariates. We propose a new class of partially functional penalized convolution-type smoothed quantile regression to characterize the conditional quantile level between a scalar response and predictors of both functional and scalar types. The new approach overcomes the lack of smoothness and severe convexity of the standard quantile empirical loss, considerably improving the computing efficiency of partially functional quantile regression. We investigate a folded concave penalized estimator for simultaneous variable selection and estimation by the modified local adaptive majorize-minimization (LAMM) algorithm. The functional predictors can be dense or sparse and are approximated by the principal component basis. Under mild conditions, the consistency and oracle properties of the resulting estimators are established. Simulation studies demonstrate a competitive performance against the partially functional standard penalized quantile regression. A real application using Alzheimer's Disease Neuroimaging Initiative data is utilized to illustrate the practicality of the proposed model.     

Asynchronous functional linear regression models for longitudinal data in reproducing kernel Hilbert space

Motivated by the analysis of longitudinal neuroimaging studies, we study the longitudinal functional linear regression model under asynchronous data setting for modeling the association between clinical outcomes and functional (or imaging) covariates. In the asynchronous data setting, both covariates and responses may be measured at irregular and mismatched time points, posing methodological challenges to existing statistical methods. We develop a kernel weighted loss function with roughness penalty to obtain the functional estimator and derive its representer theorem. The rate of convergence, a Bahadur representation, and the asymptotic pointwise distribution of the functional estimator are obtained under the reproducing kernel Hilbert space framework. We propose a penalized likelihood ratio test to test the nullity of the functional coefficient, derive its asymptotic distribution under the null hypothesis, and investigate the separation rate under the alternative hypotheses. Simulation studies are conducted to examine the finite-sample performance of the proposed procedure. We apply the proposed methods to the analysis of multitype data obtained from the Alzheimer's Disease Neuroimaging Initiative (ADNI) study, which reveals significant association between 21 regional brain volume density curves and the cognitive function. Data used in preparation of this paper were obtained from the ADNI database (adni.loni.usc.edu).     

Bivariate line-fitting methods for allometry

Fitting a line to a bivariate dataset can be a deceptively complex problem, and there has been much debate on this issue in the literature. In this review, we describe for the practitioner the essential features of line-fitting methods for estimating the relationship between two variables: what methods are commonly used, which method should be used when, and how to make inferences from these lines to answer common research questions. A particularly important point for line-fitting in allometry is that usually, two sources of error are present (which we call measurement and equation error), and these have quite different implications for choice of line-fitting method. As a consequence, the approach in this review and the methods presented have subtle but important differences from previous reviews in the biology literature. Linear regression, major axis and standardised major axis are alternative methods that can be appropriate when there is no measurement error. When there is measurement error, this often needs to be estimated and used to adjust the variance terms in formulae for line-fitting. We also review line-fitting methods for phylogenetic analyses. Methods of inference are described for the line-fitting techniques discussed in this paper. The types of inference considered here are testing if the slope or elevation equals a given value, constructing confidence intervals for the slope or elevation, comparing several slopes or elevations, and testing for shift along the axis amongst several groups. In some cases several methods have been proposed in the literature. These are discussed and compared. In other cases there is little or no previous guidance available in the literature. Simulations were conducted to check whether the methods of inference proposed have the intended coverage probability or Type I error. We identified the methods of inference that perform well and recommend the techniques that should be adopted in future work.     

Correcting for measurement error in fractional polynomial models using Bayesian modelling and regression calibration,with an application to alcohol and mortality

Exposure measurement error can result in a biased estimate of the association between an exposure and outcome. When the exposure–outcome relationship is linear on the appropriate scale (e.g. linear, logistic) and the measurement error is classical, that is the result of random noise, the result is attenuation of the effect. When the relationship is non‐linear, measurement error distorts the true shape of the association. Regression calibration is a commonly used method for correcting for measurement error, in which each individual's unknown true exposure in the outcome regression model is replaced by its expectation conditional on the error‐prone measure and any fully measured covariates. Regression calibration is simple to execute when the exposure is untransformed in the linear predictor of the outcome regression model, but less straightforward when non‐linear transformations of the exposure are used. We describe a method for applying regression calibration in models in which a non‐linear association is modelled by transforming the exposure using a fractional polynomial model. It is shown that taking a Bayesian estimation approach is advantageous. By use of Markov chain Monte Carlo algorithms, one can sample from the distribution of the true exposure for each individual. Transformations of the sampled values can then be performed directly and used to find the expectation of the transformed exposure required for regression calibration. A simulation study shows that the proposed approach performs well. We apply the method to investigate the relationship between usual alcohol intake and subsequent all‐cause mortality using an error model that adjusts for the episodic nature of alcohol consumption.     

Extending conventional priors for testing general hypotheses in linear models

We consider that observations come from a general normal linearmodel and that it is desirable to test a simplifying null hypothesisabout the parameters. We approach this problem from an objectiveBayesian, model-selection perspective. Crucial ingredients forthis approach are ‘proper objective priors’ to beused for deriving the Bayes factors. Jeffreys-Zellner-Siow priorshave good properties for testing null hypotheses defined byspecific values of the parameters in full-rank linear models.We extend these priors to deal with general hypotheses in generallinear models, not necessarily of full rank. The resulting priors,which we call ‘conventional priors’, are expressedas a generalization of recently introduced ‘partiallyinformative distributions’. The corresponding Bayes factorsare fully automatic, easily computed and very reasonable. Themethodology is illustrated for the change-point problem andthe equality of treatments effects problem. We compare the conventionalpriors derived for these problems with other objective Bayesianproposals like the intrinsic priors. It is concluded that bothpriors behave similarly although interesting subtle differencesarise. We adapt the conventional priors to deal with nonnestedmodel selection as well as multiple-model comparison. Finally,we briefly address a generalization of conventional priors tononnormal scenarios.     

Optimal sufficient dimension reduction for the conditional mean in multivariate regression

The aim of this article is to develop optimal sufficient dimensionreduction methodology for the conditional mean in multivariateregression. The context is roughly the same as that of a relatedmethod by Cook & Setodji (2003), but the new method hasseveral advantages. It is asymptotically optimal in the sensedescribed herein and its test statistic for dimension alwayshas a chi-squared distribution asymptotically under the nullhypothesis. Additionally, the optimal method allows tests ofpredictor effects. A comparison of the two methods is provided.     

A comparison of two methods for estimating measurement repeatability in morphometric studies

1. Measurement repeatability is often reported in morphometric studies as an index of the contribution of measurement error to trait measurements. However, the common method of remeasuring a mounted specimen fails to capture some components of measurement error and could therefore yield inflated repeatability estimates. Remounting specimens between successive measurements is likely to provide more realistic estimates of repeatability, particularly for structures that are difficult to measure.
2. Using measurements of 22 somatic and genitalic traits of the neriid fly Telostylinus angusticollis, we compared repeatability estimates obtained via remeasurement of a specimen that is mounted once (single‐mounted method) versus remeasurement of a specimen that is remounted between measurements (remounted method). We also asked whether the difference in repeatability estimates obtained via the two methods depends on trait size, trait type (somatic vs. genitalic), sclerotization, or sex.
3. Repeatability estimates obtained via the remounted method were lower than estimates obtained via the single‐mounted method for each of the 22 traits, and the difference between estimates obtained via the two methods was generally greater for small structures (such as genitalic traits) than for large structures (such as legs and wings). However, the difference between estimates obtained via the two methods did not depend on trait type (genitalic or somatic), tissue type (soft or sclerotized) or sex.
4. Remounting specimens between successive measurements can provide more accurate estimates of measurement repeatability than remeasuring from a single mount, especially for small structures that are difficult to measure.

     

On power and sample size calculations for likelihood ratio tests in generalized linear models

A direct extension of the approach described in Self, Mauritsen, and Ohara (1992, Biometrics 48, 31-39) for power and sample size calculations in generalized linear models is presented. The major feature of the proposed approach is that the modification accommodates both a finite and an infinite number of covariate configurations. Furthermore, for the approximation of the noncentrality of the noncentral chi-square distribution for the likelihood ratio statistic, a simplification is provided that not only reduces substantial computation but also maintains the accuracy. Simulation studies are conducted to assess the accuracy for various model configurations and covariate distributions.     

Differential measurement errors in zero-truncated regression models for count data

Measurement errors in covariates may result in biased estimates in regression analysis. Most methods to correct this bias assume nondifferential measurement errors-i.e., that measurement errors are independent of the response variable. However, in regression models for zero-truncated count data, the number of error-prone covariate measurements for a given observational unit can equal its response count, implying a situation of differential measurement errors. To address this challenge, we develop a modified conditional score approach to achieve consistent estimation. The proposed method represents a novel technique, with efficiency gains achieved by augmenting random errors, and performs well in a simulation study. The method is demonstrated in an ecology application.     

Individual toluene exposure in rotary printing: increasing accuracy of estimation by linear models based on protocols of daily activity and other measures

Industrial exposure varies distinctly both between persons and for each person over time. It is often not possible to measure individual exposure repeatedly due to high costs. Therefore, a method for assessment of exposure is needed that accounts for inter- and intraindividual variability. We consider a strategy suggested by Preller et al. (1995, Scandinavian Journal of Work, Environment, and Health 21, 504-512), the idea of which is to predict exposure on several days via a linear model using additional variables as regressors. Those additional variables are easier to obtain than exposure measurements and are assumed to influence exposure. The paper gives a theoretical proof of the use of this method. An example is given using toluene exposure data from a study in a rotogravure printing plant.     

Conditional estimation for generalized linear models when covariates are subject-specific parameters in a mixed model for longitudinal measurements

The relationship between a primary endpoint and features of longitudinal profiles of a continuous response is often of interest, and a relevant framework is that of a generalized linear model with covariates that are subject-specific random effects in a linear mixed model for the longitudinal measurements. Naive implementation by imputing subject-specific effects from individual regression fits yields biased inference, and several methods for reducing this bias have been proposed. These require a parametric (normality) assumption on the random effects, which may be unrealistic. Adapting a strategy of Stefanski and Carroll (1987, Biometrika74, 703-716), we propose estimators for the generalized linear model parameters that require no assumptions on the random effects and yield consistent inference regardless of the true distribution. The methods are illustrated via simulation and by application to a study of bone mineral density in women transitioning to menopause.     

Correction for Covariate Measurement Error in Nonparametric Longitudinal Regression

Summary We introduce a correction for covariate measurement error in nonparametric regression applied to longitudinal binary data arising from a study on human sleep. The data have been surveyed to investigate the association of some hormonal levels and the probability of being asleep. The hormonal effect is modeled flexibly while we account for the error‐prone measurement of its concentration in the blood and the longitudinal character of the data. We present a fully Bayesian treatment utilizing Markov chain Monte Carlo inference techniques, and also introduce block updating to improve sampling and computational performance in the binary case. Our model is partly inspired by the relevance vector machine with radial basis functions, where usually very few basis functions are automatically selected for fitting the data. In the proposed approach, we implement such data‐driven complexity regulation by adopting the idea of Bayesian model averaging. Besides the general theory and the detailed sampling scheme, we also provide a simulation study for the Gaussian and the binary cases by comparing our method to the naive analysis ignoring measurement error. The results demonstrate a clear gain when using the proposed correction method, particularly for the Gaussian case with medium and large measurement error variances, even if the covariate model is misspecified.