Robust Estimation of a Spatiotemporal Model with Structural Change

A spatiotemporal model is postulated and estimated using a procedure that infuses the forward search algorithm and maximum likelihood estimation into the backfitting framework. The forward search algorithm filters the effect of temporary structural change in the estimation of covariate and spatial parameters. Simulation studies illustrate capability of the method in producing robust estimates of the parameters even in the presence of structural change. The method provides good model fit even for small sample sizes in short time series data and good predictions for a wide range of lengths of contamination periods and levels of severity of contamination.     

Robust estimation of a dynamic spatio-temporal model with structural change

We postulate a dynamic spatio-temporal model with constant covariate effect but with varying spatial effect over time and varying temporal effect across locations. To mitigate the effect of temporary structural change, the model can be estimated using the backfitting algorithm embedded with forward search algorithm and bootstrap. A simulation study is designed to evaluate structural optimality of the model with the estimation procedure. The fitted model exhibit superior predictive ability relative to the linear model. The proposed algorithm also consistently produced lower relative bias and standard errors for the spatial parameter estimates. While additional neighbourhoods do not necessarily improve predictive ability of the model, it trims down relative bias on the parameter estimates, specially for spatial parameter. Location of the temporary structural change along with the degree of structural change contributes to lower relative bias of parameter estimates and in better predictive ability of the model. The estimation procedure is able to produce parameter estimates that are robust to the occurrence of temporary structural change.     

Robust methods for generalized linear models with nonignorable missing covariates

The EM algorithm is often used for finding the maximum likelihood estimates in generalized linear models with incomplete data. In this article, the author presents a robust method in the framework of the maximum likelihood estimation for fitting generalized linear models when nonignorable covariates are missing. His robust approach is useful for downweighting any influential observations when estimating the model parameters. To avoid computational problems involving irreducibly high‐dimensional integrals, he adopts a Metropolis‐Hastings algorithm based on a Markov chain sampling method. He carries out simulations to investigate the behaviour of the robust estimates in the presence of outliers and missing covariates; furthermore, he compares these estimates to the classical maximum likelihood estimates. Finally, he illustrates his approach using data on the occurrence of delirium in patients operated on for abdominal aortic aneurysm.     

Bootstrap procedures in a spatial-temporal model

Two bootstrap procedures are introduced into the hybrid of the backfitting algorithm and the Cochrane–Orcutt procedure in the estimation of a spatial-temporal model. The use of time blocks of consecutive observations in resampling steps proved to be optimal in terms of stability and efficiency of estimates. Between iterations, there were minimal changes in the empirical distributions of the parameter estimates associated with the covariate and temporal effects indicating convergence of the algorithm. Crop yield data are used to illustrate the proposed methods.

The simulation study indicated that prediction error from the fitted model (estimated from either Method 1 or Method 2) is very low. Also, the prediction error is relatively robust to the number of spatial units and the number of time points.     

Instrumental variable estimation in ordinal probit models with mismeasured predictors

Researchers in the medical, health, and social sciences routinely encounter ordinal variables such as self‐reports of health or happiness. When modelling ordinal outcome variables, it is common to have covariates, for example, attitudes, family income, retrospective variables, measured with error. As is well known, ignoring even random error in covariates can bias coefficients and hence prejudice the estimates of effects. We propose an instrumental variable approach to the estimation of a probit model with an ordinal response and mismeasured predictor variables. We obtain likelihood‐based and method of moments estimators that are consistent and asymptotically normally distributed under general conditions. These estimators are easy to compute, perform well and are robust against the normality assumption for the measurement errors in our simulation studies. The proposed method is applied to both simulated and real data. The Canadian Journal of Statistics 47: 653–667; 2019 © 2019 Statistical Society of Canada     

Analysis of hypoglycemic events using negative binomial models

Negative binomial regression is a standard model to analyze hypoglycemic events in diabetes clinical trials. Adjusting for baseline covariates could potentially increase the estimation efficiency of negative binomial regression. However, adjusting for covariates raises concerns about model misspecification, in which the negative binomial regression is not robust because of its requirement for strong model assumptions. In some literature, it was suggested to correct the standard error of the maximum likelihood estimator through introducing overdispersion, which can be estimated by the Deviance or Pearson Chi‐square. We proposed to conduct the negative binomial regression using Sandwich estimation to calculate the covariance matrix of the parameter estimates together with Pearson overdispersion correction (denoted by NBSP). In this research, we compared several commonly used negative binomial model options with our proposed NBSP. Simulations and real data analyses showed that NBSP is the most robust to model misspecification, and the estimation efficiency will be improved by adjusting for baseline hypoglycemia. Copyright © 2013 John Wiley & Sons, Ltd.     

Robust Smoothed Rank Estimation Methods for Accelerated Failure Time Model Allowing Clusters

Smoothed Gehan rank estimation methods are widely used in accelerated failure time (AFT) models with/without clusters. However, most methods are sensitive to outliers in the covariates. In order to solve this problem, we propose robust approaches based on the smoothed Gehan rank estimation methods for the AFT model, allowing for clusters by employing two different weight functions. Simulation studies show that the proposed methods outperform existing smoothed rank estimation methods regarding their biases and standard deviations when there are outliers in the covariates. The proposed methods are also applied to a real dataset from the “Major cardiovascular interventions” study.     

A Method for Bayesian Monotonic Multiple Regression

Abstract. When applicable, an assumed monotonicity property of the regression function w.r.t. covariates has a strong stabilizing effect on the estimates. Because of this, other parametric or structural assumptions may not be needed at all. Although monotonic regression in one dimension is well studied, the question remains whether one can find computationally feasible generalizations to multiple dimensions. Here, we propose a non‐parametric monotonic regression model for one or more covariates and a Bayesian estimation procedure. The monotonic construction is based on marked point processes, where the random point locations and the associated marks (function levels) together form piecewise constant realizations of the regression surfaces. The actual inference is based on model‐averaged results over the realizations. The monotonicity of the construction is enforced by partial ordering constraints, which allows it to asymptotically, with increasing density of support points, approximate the family of all monotonic bounded continuous functions.     

Modelling a non-stationary BINAR(1) Poisson process

ABSTRACT

Non-stationarity in bivariate time series of counts may be induced by a number of time-varying covariates affecting the bivariate responses due to which the innovation terms of the individual series as well as the bivariate dependence structure becomes non-stationary. So far, in the existing models, the innovation terms of individual INAR(1) series and the dependence structure are assumed to be constant even though the individual time series are non-stationary. Under this assumption, the reliability of the regression and correlation estimates is questionable. Besides, the existing estimation methodologies such as the conditional maximum likelihood (CMLE) and the composite likelihood estimation are computationally intensive. To address these issues, this paper proposes a BINAR(1) model where the innovation series follow a bivariate Poisson distribution under some non-stationary distributional assumptions. The method of generalized quasi-likelihood (GQL) is used to estimate the regression effects while the serial and bivariate correlations are estimated using a robust moment estimation technique. The application of model and estimation method is made in the simulated data. The GQL method is also compared with the CMLE, generalized method of moments (GMM) and generalized estimating equation (GEE) approaches where through simulation studies, it is shown that GQL yields more efficient estimates than GMM and equally or slightly more efficient estimates than CMLE and GEE.     

Cure rate survival models with missing covariates: a simulation study

In this paper we study the cure rate survival model involving a competitive risk structure with missing categorical covariates. A parametric distribution that can be written as a sequence of one-dimensional conditional distributions is specified for the missing covariates. We consider the missing data at random situation so that the missing covariates may depend only on the observed ones. Parameter estimates are obtained by using the EM algorithm via the method of weights. Extensive simulation studies are conducted and reported to compare estimates efficiency with and without missing data. As expected, the estimation approach taking into consideration the missing covariates presents much better efficiency in terms of mean square errors than the complete case situation. Effects of increasing cured fraction and censored observations are also reported. We demonstrate the proposed methodology with two real data sets. One involved the length of time to obtain a BS degree in Statistics, and another about the time to breast cancer recurrence.     

Evaluation of Bayesian multiple stage estimation under spatial CAR model variants

In this study, an evaluation of Bayesian hierarchical models is made based on simulation scenarios to compare single-stage and multi-stage Bayesian estimations. Simulated datasets of lung cancer disease counts for men aged 65 and older across 44 wards in the London Health Authority were analysed using a range of spatially structured random effect components. The goals of this study are to determine which of these single-stage models perform best given a certain simulating model, how estimation methods (single- vs. multi-stage) compare in yielding posterior estimates of fixed effects in the presence of spatially structured random effects, and finally which of two spatial prior models – the Leroux or ICAR model, perform best in a multi-stage context under different assumptions concerning spatial correlation. Among the fitted single-stage models without covariates, we found that when there is low amount of variability in the distribution of disease counts, the BYM model is relatively robust to misspecification in terms of DIC, while the Leroux model is the least robust to misspecification. When these models were fit to data generated from models with covariates, we found that when there was one set of covariates – either spatially correlated or non-spatially correlated, changing the values of the fixed coefficients affected the ability of either the Leroux or ICAR model to fit the data well in terms of DIC. When there were multiple sets of spatially correlated covariates in the simulating model, however, we could not distinguish the goodness of fit to the data between these single-stage models. We found that the multi-stage modelling process via the Leroux and ICAR models generally reduced the variance of the posterior estimated fixed effects for data generated from models with covariates and a UH term compared to analogous single-stage models. Finally, we found the multi-stage Leroux model compares favourably to the multi-stage ICAR model in terms of DIC. We conclude that the mutli-stage Leroux model should be seriously considered in applications of Bayesian disease mapping when an investigator desires to fit a model with both fixed effects and spatially structured random effects to Poisson count data.     

Robust inference for generalized partially linear mixed models that account for censored responses and missing covariates – an application to Arctic data analysis

In this article, we propose a family of bounded influence robust estimates for the parametric and non-parametric components of a generalized partially linear mixed model that are subject to censored responses and missing covariates. The asymptotic properties of the proposed estimates have been looked into. The estimates are obtained by using Monte Carlo expectation–maximization algorithm. An approximate method which reduces the computational time to a great extent is also proposed. A simulation study shows that performances of the two approaches are similar in terms of bias and mean square error. The analysis is illustrated through a study on the effect of environmental factors on the phytoplankton cell count.     

A SEMIPARAMETRIC REGRESSION MODEL WITH MISSING COVARIATES IN CONTINUOUS-TIME CAPTURE-RECAPTURE STUDIES

Covariate data were missing when a semiparametric regression model was used to study bird abundance in the Mai Po Sanctuary, Hong Kong. This paper proposes an EM‐type algorithm to estimate the regression parameters for that study. Analytical calculation of the expectation in the EM method is difficult, or even impossible, especially when missing covariates are continuous. A Monte Carlo method is used in the EM algorithm to ease the calculation complexity. Asymptotic variances of the parameter estimates are also derived. Properties of the proposed estimators are assessed through numerical simulations and a real example.     

Efficient initial designs for binary response data

In this paper we introduce a binary search algorithm that efficiently finds initial maximum likelihood estimates for sequential experiments where a binary response is modeled by a continuous factor. The problem is motivated by switching measurements on superconducting Josephson junctions. In this quantum mechanical experiment, the current is the factor controlled by the experimenter and a binary response indicating the presence or the absence of a voltage response is measured. The prior knowledge on the model parameters is typically poor, which may cause the common approaches of initial estimation to fail. The binary search algorithm is designed to work reliably even when the prior information is very poor. The properties of the algorithm are studied in simulations and an advantage over the initial estimation with equally spaced factor levels is demonstrated. We also study the cost-efficiency of the binary search algorithm and find the approximately optimal number of measurements per stage when there is a cost related to the number of stages in the experiment.     

Latent Variable Models for Mixed Discrete and Continuous Outcomes

We propose a latent variable model for mixed discrete and continuous outcomes. The model accommodates any mixture of outcomes from an exponential family and allows for arbitrary covariate effects, as well as direct modelling of covariates on the latent variable. An EM algorithm is proposed for parameter estimation and estimates of the latent variables are produced as a by-product of the analysis. A generalized likelihood ratio test can be used to test the significance of covariates affecting the latent outcomes. This method is applied to birth defects data, where the outcomes of interest are continuous measures of size and binary indicators of minor physical anomalies. Infants who were exposed in utero to anticonvulsant medications are compared with controls.     

Point and Interval Estimations of Marginal Risk Difference by Logistic Model

We use logistic model to get point and interval estimates of the marginal risk difference in observational studies and randomized trials with dichotomous outcome. We prove that the maximum likelihood estimate of the marginal risk difference is unbiased for finite sample and highly robust to the effects of dispersing covariates. We use approximate normal distribution of the maximum likelihood estimates of the logistic model parameters to get approximate distribution of the maximum likelihood estimate of the marginal risk difference and then the interval estimate of the marginal risk difference. We illustrate application of the method by a real medical example.     

Semiparametric multinomial logit models for analysing consumer choice behaviour

The multinomial logit model (MNL) is one of the most frequently used statistical models in marketing applications. It allows one to relate an unordered categorical response variable, for example representing the choice of a brand, to a vector of covariates such as the price of the brand or variables characterising the consumer. In its classical form, all covariates enter in strictly parametric, linear form into the utility function of the MNL model. In this paper, we introduce semiparametric extensions, where smooth effects of continuous covariates are modelled by penalised splines. A mixed model representation of these penalised splines is employed to obtain estimates of the corresponding smoothing parameters, leading to a fully automated estimation procedure. To validate semiparametric models against parametric models, we utilise different scoring rules as well as predicted market share and compare parametric and semiparametric approaches for a number of brand choice data sets.     

Obtaining least absolute value estimates for a two-way classification model

The importance of the two-way classification model is well known, but the standard method of analysis is least squares. Often, the data of the model calls for a more robust estimation technique. This paper demonstrates the equivalence between the problem of obtaining least absolute value estimates for the two-way classification model and a capacitated transportation problem. A special purpose primal algorithm is developed to provide the least absolute value estimates. A computational comparison is made between an implementation of this specialized algorithm and a standard capacitated transportation code.     

A linear model-based test for the heterogeneity of conditional correlations

Current methods of testing the equality of conditional correlations of bivariate data on a third variable of interest (covariate) are limited due to discretizing of the covariate when it is continuous. In this study, we propose a linear model approach for estimation and hypothesis testing of the Pearson correlation coefficient, where the correlation itself can be modeled as a function of continuous covariates. The restricted maximum likelihood method is applied for parameter estimation, and the corrected likelihood ratio test is performed for hypothesis testing. This approach allows for flexible and robust inference and prediction of the conditional correlations based on the linear model. Simulation studies show that the proposed method is statistically more powerful and more flexible in accommodating complex covariate patterns than the existing methods. In addition, we illustrate the approach by analyzing the correlation between the physical component summary and the mental component summary of the MOS SF-36 form across a fair number of covariates in the national survey data.     

Robust signed-rank estimation and variable selection for semi-parametric additive partial linear models

A fully nonparametric model may not perform well or when the researcher wants to use a parametric model but the functional form with respect to a subset of the regressors or the density of the errors is not known. This becomes even more challenging when the data contain gross outliers or unusual observations. However, in practice the true covariates are not known in advance, nor is the smoothness of the functional form. A robust model selection approach through which we can choose the relevant covariates components and estimate the smoothing function may represent an appealing tool to the solution. A weighted signed-rank estimation and variable selection under the adaptive lasso for semi-parametric partial additive models is considered in this paper. B-spline is used to estimate the unknown additive nonparametric function. It is shown that despite using B-spline to estimate the unknown additive nonparametric function, the proposed estimator has an oracle property. The robustness of the weighted signed-rank approach for data with heavy-tail, contaminated errors, and data containing high-leverage points are validated via finite sample simulations. A practical application to an economic study is provided using an updated Canadian household gasoline consumption data.