Fast computation of spatially adaptive kernel estimates

Kernel smoothing of spatial point data can often be improved using an adaptive, spatially varying bandwidth instead of a fixed bandwidth. However, computation with a varying bandwidth is much more demanding, especially when edge correction and bandwidth selection are involved. This paper proposes several new computational methods for adaptive kernel estimation from spatial point pattern data. A key idea is that a variable-bandwidth kernel estimator for d-dimensional spatial data can be represented as a slice of a fixed-bandwidth kernel estimator in \((d+1)\)-dimensional scale space, enabling fast computation using Fourier transforms. Edge correction factors have a similar representation. Different values of global bandwidth correspond to different slices of the scale space, so that bandwidth selection is greatly accelerated. Potential applications include estimation of multivariate probability density and spatial or spatiotemporal point process intensity, relative risk, and regression functions. The new methods perform well in simulations and in two real applications concerning the spatial epidemiology of primary biliary cirrhosis and the alarm calls of capuchin monkeys.     

Simultaneous estimation of quantile curves using quantile sheets

The results of quantile smoothing often show crossing curves, in particular, for small data sets. We define a surface, called a quantile sheet, on the domain of the independent variable and the probability. Any desired quantile curve is obtained by evaluating the sheet for a fixed probability. This sheet is modeled by $P$ -splines in form of tensor products of $B$ -splines with difference penalties on the array of coefficients. The amount of smoothing is optimized by cross-validation. An application for reference growth curves for children is presented.     

Semiparametric analysis for case-control studies: a partial smoothing spline approach

Case-control data are often used in medical-related applications, and most studies have applied parametric logistic regression to analyze such data. In this study, we investigated a semiparametric model for the analysis of case-control data by relaxing the linearity assumption of risk factors by using a partial smoothing spline model. A faster computation method for the model by extending the lower-dimensional approximation approach of Gu and Kim 4 Gu, C. and Kim, Y.-J. 2002. Penalized likelihood regression: General formulation and efficient approximation. Canad. J. Statist., 30: 619–628. [Crossref], [Web of Science ®] [Google Scholar] developed in penalized likelihood regression is considered to apply to case-control studies. Simulations were conducted to evaluate the performance of the method with selected smoothing parameters and to compare the method with existing methods. The method was applied to Korean gastric cancer case-control data to estimate the nonparametric probability function of age and regression parameters for other categorical risk factors simultaneously. The method could be used in preliminary studies to identify whether there is a flexible function form of risk factors in the semiparametric logistic regression analysis involving a large data set.     

Non-parametric regression on compositional covariates using Bayesian P-splines

Methods to perform regression on compositional covariates have recently been proposed using isometric log-ratios (ilr) representation of compositional parts. This approach consists of first applying standard regression on ilr coordinates and second, transforming the estimated ilr coefficients into their contrast log-ratio counterparts. This gives easy-to-interpret parameters indicating the relative effect of each compositional part. In this work we present an extension of this framework, where compositional covariate effects are allowed to be smooth in the ilr domain. This is achieved by fitting a smooth function over the multidimensional ilr space, using Bayesian P-splines. Smoothness is achieved by assuming random walk priors on spline coefficients in a hierarchical Bayesian framework. The proposed methodology is applied to spatial data from an ecological survey on a gypsum outcrop located in the Emilia Romagna Region, Italy.     

Some characteristics on the selection of spline smoothing parameter

The smoothing spline method is used to fit a curve to a noisy data set, where selection of the smoothing parameter is essential. An adaptive C_p criterion (Chen and Huang 2011 Chen, C. S., and H. C. Huang. 2011. An improved C_p criterion for spline smoothing. Journal of Statistical Planning and Inference 141:445–52.[Crossref], [Web of Science ®] , [Google Scholar]) based on the Stein’s unbiased risk estimate has been proposed to select the smoothing parameter, which not only considers the usual effective degrees of freedom but also takes into account the selection variability. The resulting fitted curve has been shown to be superior and more stable than commonly used selection criteria and possesses the same asymptotic optimality as C_p. In this paper, we further discuss some characteristics on the selection of smoothing parameter, especially for the selection variability.     

Automatic Smoothing for Poisson Regression

Abstract

Adaptive choice of smoothing parameters for nonparametric Poisson regression (O'Sullivan et al., 1986 O'Sullivan , F. , Yandell , B. S. , Raynor , W. J., Jr. ( 1986 ). Automatic smoothing of regression functions in generalized linear models . J. Amer. Statist. Assoc. 81 : 96 – 103 . [CSA] [Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) is considered in this article. A computable approximation of the unbiased risk estimate (AUBR) for Poisson regression is introduced. This approximation can be used to automatically tune the smoothing parameter for the penalized likelihood estimator. An alternative choice is the generalized approximate cross validation (GACV) proposed by Xiang and Wahba (1996 Xiang , D. , Wahba , G. ( 1996 ). A generalized approximate cross validation for smoothing splines with non-Gaussian data . Statist. Sinica 6 (3): 675 – 692 .[Web of Science ®] , [Google Scholar]). Although GACV enjoys a great success in practice when applying for nonparametric logisitic regression, its performance for Poisson regression is not clear. Numerical simulations have been conducted to evaluate the GACV and AUBR based tuning methods. We found that GACV has a tendency to oversmooth the data when the intensity function is small. As a consequence, we suggest tuning the smoothing parameter using AUBR in practice.     

P-splines with derivative based penalties and tensor product smoothing of unevenly distributed data

The P-splines of Eilers and Marx (Stat Sci 11:89–121, 1996) combine a B-spline basis with a discrete quadratic penalty on the basis coefficients, to produce a reduced rank spline like smoother. P-splines have three properties that make them very popular as reduced rank smoothers: (i) the basis and the penalty are sparse, enabling efficient computation, especially for Bayesian stochastic simulation; (ii) it is possible to flexibly ‘mix-and-match’ the order of B-spline basis and penalty, rather than the order of penalty controlling the order of the basis as in spline smoothing; (iii) it is very easy to set up the B-spline basis functions and penalties. The discrete penalties are somewhat less interpretable in terms of function shape than the traditional derivative based spline penalties, but tend towards penalties proportional to traditional spline penalties in the limit of large basis size. However part of the point of P-splines is not to use a large basis size. In addition the spline basis functions arise from solving functional optimization problems involving derivative based penalties, so moving to discrete penalties for smoothing may not always be desirable. The purpose of this note is to point out that the three properties of basis-penalty sparsity, mix-and-match penalization and ease of setup are readily obtainable with B-splines subject to derivative based penalization. The penalty setup typically requires a few lines of code, rather than the two lines typically required for P-splines, but this one off disadvantage seems to be the only one associated with using derivative based penalties. As an example application, it is shown how basis-penalty sparsity enables efficient computation with tensor product smoothers of scattered data.     

Determination of the Selection Statistics and Best Significance Level in Backward Stepwise Logistic Regression

The weighted generalized estimating equation (WGEE), an extension of the generalized estimating equation (GEE) method, is a method for analyzing incomplete longitudinal data. An inappropriate specification of the working correlation structure results in the loss of efficiency of the GEE estimation. In this study, we evaluated the efficiency of WGEE estimation for incomplete longitudinal data when the working correlation structure was misspecified. As a result, we found that the efficiency of the WGEE estimation was lower when an improper working correlation structure was selected, similar to the case of the GEE method. Furthermore, we modified the criterion proposed by Gosho et al. (2011 Gosho, M., Hamada, C. and Yoshimura, I. 2011. Criterion for the selection of a working correlation structure in the generalized estimating equation approach for longitudinal balanced data. Communications in Statistics -Theory and Methods, 40: 3839–3856. [Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]) for selecting a working correlation structure, such that the GEE and WGEE methods can be applied to incomplete longitudinal data, and we investigated the performance of the modified criterion. The results revealed that when the modified criterion was adopted, the proportion that the true correlation structure was selected was likely higher than that in the case of adopting other competing approaches.     

Limiting bias-reduced Amoroso kernel density estimators for non-negative data

The Amoroso kernel density estimator (Igarashi and Kakizawa 2017 Igarashi, G., and Y. Kakizawa. 2017. Amoroso kernel density estimation for nonnegative data and its bias reduction. Department of Policy and Planning Sciences Discussion Paper Series No. 1345, University of Tsukuba. [Google Scholar]) for non-negative data is boundary-bias-free and has the mean integrated squared error (MISE) of order O(n^{? 4/5}), where n is the sample size. In this paper, we construct a linear combination of the Amoroso kernel density estimator and its derivative with respect to the smoothing parameter. Also, we propose a related multiplicative estimator. We show that the MISEs of these bias-reduced estimators achieve the convergence rates n^{? 8/9}, if the underlying density is four times continuously differentiable. We illustrate the finite sample performance of the proposed estimators, through the simulations.     

A skew-normal random effects model for longitudinal ordinal categorical responses with missing data

Missing values are common in longitudinal data studies. The missing data mechanism is termed non-ignorable (NI) if the probability of missingness depends on the non-response (missing) observations. This paper presents a model for the ordinal categorical longitudinal data with NI non-monotone missing values. We assumed two separate models for the response and missing procedure. The response is modeled as ordinal logistic, whereas the logistic binary model is considered for the missing process. We employ these models in the context of so-called shared-parameter models, where the outcome and missing data models are connected by a common set of random effects. It is commonly assumed that the random effect follows the normal distribution in longitudinal data with or without missing data. This can be extremely restrictive in practice, and it may result in misleading statistical inferences. In this paper, we instead adopt a more flexible alternative distribution which is called the skew-normal distribution. The methodology is illustrated through an application to Schizophrenia Collaborative Study data [19 D. Hedeker, Generalized linear mixed models, in Encyclopedia of Statistics in Behavioral Science, B. Everitt and D. Howell, eds., John Wiley, London, 2005, pp. 729–738. [Google Scholar]] and a simulation.     

Analysis of Longitudinal Data in the Case–Control Studies via Empirical Likelihood

The case–control studies using longitudinal data are cost and time efficient when the disease is rare and assessing the exposure level of risk factors is difficult. Instead of GEE method, the method of using a prospective logistic model for analyzing case–control longitudinal data was proposed and the semiparametric inference procedure was explored by Park and Kim (2004 Park , E. , Kim , Y. (2004). Analysis of longitudinal data in case–control studies. Biometrika 91:321–330. [Google Scholar]). In this article, we apply an empirical likelihood ratio method to derive limiting distribution of the empirical likelihood ratio and find one likelihood-ratio based confidence region for the unknown vector of regression parameters. We compare empirical likelihood method with normal approximation based method. Simulation results show that the proposed empirical likelihood ratio method performs well in terms of coverage probability and interval width.     

The Identification of Potential Longitudinal Biomarkers and Measurements of Effectiveness for Biomarkers as Surrogates in Multivariate Survival Data

A proposed method based on frailty models is used to identify longitudinal biomarkers or surrogates for a multivariate survival. This method is an extention of earlier models by Wulfsohn and Tsiatis (1997 Wulfsohn , M. S. , Tsiatis , A. A. ( 1997 ). A joint model for survival and longitudinal data measured with error . Biometrics 53 ( 1 ): 330 – 339 .[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]) and Song et al. (2002 Song , X. , Davidian , M. , Tsiatis , A. A. ( 2002 ). A Semiparametric likelihood approach to joint modeling of longitudinal and time-to-event data . Biometrics 58 ( 4 ): 742 – 753 .[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]). In this article, similar to Henderson et al. (2002 Henderson , R. , Diggle , P. J. , Dobson , A. ( 2002 ). Identification and efficacy of longitudinal markers for survival . Biostatistics 3 ( 1 ): 33 – 50 .[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]), a joint likelihood function combines the likelihood functions of the longitudinal biomarkers and the multivariate survival times. We use simulations to explore how the number of individuals, the number of time points per individual and the functional form of the random effects from the longitudianl biomarkers influence the power to detect the association of a longitudinal biomarker and the multivariate survival time. The proposed method is illustrate by using the gastric cancer data.