Simple bootstrap statistical inference using the SAS system.

Nonparametric bootstrap statistical inference is a robust computer intensive method for generating estimates of statistical variability for which formulae are not known or asymptotic assumptions are not met. A SAS macro that implements simple nonparametric bootstrap statistical inference is presented with an example. The program code is easily generalized to any SAS procedure which includes a BY statement, and to cases of clustered data.     

A resampling approach for adjustment in prediction models for covariate measurement error

Recent works on covariate measurement errors focus on the possible biases in model coefficient estimates. Usually, measurement error in a covariate tends to attenuate the coefficient estimate for the covariate, i.e., a bias toward the null occurs. Measurement error in another confounding or interacting variable typically results in incomplete adjustment for that variable. Hence, the coefficient for the covariate of interest may be biased either toward or away from the null. This paper presents a new method based on a resampling technique to deal with covariate measurement errors in the context of prediction modeling. Prediction accuracy is our primary parameter of interest. Prediction accuracy of a model is defined as the success rate of prediction when the model predicts new response. We call our method bootstrap regression calibration (BRC). We study logistic regression with interacting covariates as our prediction model. We measure the prediction accuracy of a model by receiver operating characteristic (ROC) method. Results from simulations show that bootstrap regression calibration offers consistent enhancement over the commonly used regression calibration (RC) method in terms of improving prediction accuracy of the model and reducing bias in the estimated coefficients.     

Error handling strategies in multiphase inverse modeling

Parameter estimation by inverse modeling involves the repeated evaluation of a function of residuals. These residuals represent both errors in the model and errors in the data. In practical applications of inverse modeling of multiphase flow and transport, the error structure of the final residuals often significantly deviates from the statistical assumptions that underlie standard maximum likelihood estimation using the least-squares method. Large random or systematic errors are likely to lead to convergence problems, biased parameter estimates, misleading uncertainty measures, or poor predictive capabilities of the calibrated model. The multiphase inverse modeling code iTOUGH2 supports strategies that identify and mitigate the impact of systematic or non-normal error structures. We discuss these approaches and provide an overview of the error handling features implemented in iTOUGH2.     

Using the hyperbolic distribution to estimate the precision of size distribution percentiles of fluvial gravels

The hyperbolic distribution is fitted to published grain-size data collected from the surface of two gravel-bed rivers (the Fraser and Mamquam River data sets of Rice and Church). Our parametric approach enables calculation of standard errors for estimates of percentiles, as an alternative to the use of the bootstrap for this purpose. For estimation, we have used the statistical package R and the advantages of this software for this type of analysis are highlighted in this paper.     

On a constrained mixture vector autoregressive model

A mixture vector autoregressive model has recently been introduced to the literature. Although this model is a promising candidate for nonlinear multiple time series modeling, high dimensionality of the parameters and lack of method for computing the standard errors of estimates limit its application to real data. The contribution of this paper is threefold. First, a form of parameter constraints is introduced with an efficient EM algorithm for estimation. Second, an accurate method for computing standard errors is presented for the model with and without parameter constraints. Lastly, a hypothesis-testing approach based on likelihood ratio tests is proposed, which aids in the selection of unnecessary parameters and leads to the greater efficiency at the estimation. A case study employing U.S. Treasury constant maturity rates illustrates the applicability of the mixture vector autoregressive model with parameter constraints, and the importance of using a reliable method to compute standard errors.     

Bootstrap control

In this paper, we present a new way to control linear stochastic systems. The method is based on statistical bootstrap techniques. The optimal future control signal is derived in such a way that unknown noise distribution and uncertainties in parameter estimates are taken into account. This is achieved by resampling from existing data when calculating statistical distributions of future process values. The bootstrap algorithm takes care of arbitrary loss functions and unknown noise distribution even for small estimation sets. The efficient way of utilizing data implies that the method is also well suited for slowly time-varying stochastic systems.     

Use of OSWALD for analyzing longitudinal data with informative dropout

OSWALD (Object-oriented Software for the Analysis of Longitudinal Data) is flexible and powerful software written for S-PLUS for the analysis of longitudinal data with dropout for which there is little other software available in the public domain. The implementation of OSWALD is described through analysis of a psychiatric clinical trial that compares antidepressant effects in an elderly depressed sample and a simulation study. In the simulation study, three different dropout mechanisms: completely random dropout (CRD), random dropout (RD) and informative dropout (ID), are considered and the results from using OSWALD are compared across mechanisms. The parameter estimates for ID-simulated data show less bias with OSWALD under the ID missing data assumption than under the CRD or RD assumptions. Under an ID mechanism, OSWALD does not provide standard error estimates. We supplement OSWALD with a bootstrap procedure to derive the standard errors. This report illustrates the usage of OSWALD for analyzing longitudinal data with dropouts and how to draw appropriate conclusions based on the analytic results under different assumptions regarding the dropout mechanism.     

SAS macros for estimation of the cumulative incidence functions based on a Cox regression model for competing risks survival data

When considering competing risks survival data, the cause specific hazard functions are often modelled by the proportional hazards Cox regression model. First, we present how to estimate the parameters in this model when some of the covariates are allowed to have exactly the same effect on several causes of failure. In many cases, the focus is not on the parameter estimates, but rather on the probability of observing a failure from a specific cause for individuals with specified covariate values. These probabilities, the cumulative incidences, are not simple functions of the parameters and they are, so far, not provided by the standard statistical software packages. We present two SAS macros: a SAS macro named CumInc for estimation of the cumulative incidences and a SAS macro named CumIncV for estimation of the cumulative incidences and the variances of the estimated cumulative incidences. The use of the macros is demonstrated through an example.     

Evaluating air quality model performance

The Environmental Protection Agency has developed a computerized system for evaluating the performance of air quality simulation models used for regulatory purposes. The model evaluation process is based on a comprehensive set of statistical calculations accompanied by graphical presentation of key performance measures. The Statistical Analysis System (SAS) was chosen for this application since it provides a wealth of statistical procedures and supports a variety of graphic display options. SAS is ideally suited for model evaluation since new statistical techniques can easily be developed from existing SAS procedures. Code is presented which performs a “bootstrap” analysis of model performance which enables the user to statistically separate the performance of competing air quality simulation models. Examples are provided of SAS listings which perform the statistical calculations and graphic display of selected statistical performance measures.     

Fixing the nonconvergence bug in logistic regression with SPLUS and SAS

When analyzing clinical data with binary outcomes, the parameter estimates and consequently the odds ratio estimates of a logistic model sometimes do not converge to finite values. This phenomenon is due to special conditions in a data set and known as 'separation'. Statistical software packages for logistic regression using the maximum likelihood method cannot appropriately deal with this problem. A new procedure to solve the problem has been proposed by Heinze and Schemper (Stat. Med. 21 (2002) pp. 2409-3419). It has been shown that unlike the standard maximum likelihood method, this method always leads to finite parameter estimates. We developed a SAS macro and an SPLUS library to make this method available from within one of these widely used statistical software packages. Our programs are also capable of performing interval estimation based on profile penalized log likelihood (PPL) and of plotting the PPL function as was suggested by Heinze and Schemper (Stat. Med. 21 (2002) pp. 2409-3419).     

SAS and SPLUS programs to perform Cox regression without convergence problems

When analyzing survival data, the parameter estimates and consequently the relative risk estimates of a Cox model sometimes do not converge to finite values. This phenomenon is due to special conditions in a data set and is known as 'monotone likelihood'. Statistical software packages for Cox regression using the maximum likelihood method cannot appropriately deal with this problem. A new procedure to solve the problem has been proposed by G. Heinze, M. Schemper, A solution to the problem of monotone likelihood in Cox regression, Biometrics 57 (2001). It has been shown that unlike the standard maximum likelihood method, this method always leads to finite parameter estimates. We developed a SAS macro and an SPLUS library to make this method available from within one of these widely used statistical software packages. Our programs are also capable of performing interval estimation based on profile penalized log likelihood (PPL) and of plotting the PPL function as was suggested by G. Heinze, M. Schemper, A solution to the problem of monotone likelihood in Cox regression, Biometrics 57 (2001).     

Comparing the importance of prognostic factors in Cox and logistic regression using SAS

Two SAS macro programs are presented that evaluate the relative importance of prognostic factors in the proportional hazards regression model and in the logistic regression model. The importance of a prognostic factor is quantified by the proportion of variation in the outcome attributable to this factor. For proportional hazards regression, the program %RELIMPCR uses the recently proposed measure V to calculate the proportion of explained variation (PEV). For the logistic model, the R(2) measure based on squared raw residuals is used by the program %RELIMPLR. Both programs are able to compute marginal and partial PEV, to compare PEVs of factors, of groups of factors, and even to compare PEVs of different models. The programs use a bootstrap resampling scheme to test differences of the PEVs of different factors. Confidence limits for P-values are provided. The programs further allow to base the computation of PEV on models with shrinked or bias-corrected parameter estimates. The SAS macros are freely available at www.akh-wien.ac.at/imc/biometrie/relimp     

Composite adaptive dynamic surface control using online recorded data

This paper presents an online recorded data‐based design of composite adaptive dynamic surface control for a class of uncertain parameter strict‐feedback nonlinear systems, where both tracking errors and prediction errors are applied to update parametric estimates. Differing from the traditional composite adaptation that utilizes identification models and linear filters to generate filtered modeling errors as prediction errors, the proposed composite adaptation integrates closed‐loop tracking error equations in a moving time window to generate modified modeling errors as prediction errors. The time‐interval integral operation takes full advantage of online recorded data to improve parameter convergence such that the application of both identification models and linear filters is not necessary. Semiglobal practical asymptotic stability of the closed‐loop system is rigorously established by the time‐scales separation and Lyapunov synthesis. The major contribution of this study is that composite adaptation based on online recorded data is achieved at the presence of mismatched uncertainties. Simulation results have been provided to verify the effectiveness and superiority of this approach. Copyright © 2016 John Wiley & Sons, Ltd.     

Optimal instrumental variable estimation and approximate implementations

The accuracy properties of instrumental variables (IV) methods are investigated. Extensions such as prefiltering of data and use of additional instruments are included in the analysis. The parameter estimates are shown to be asymptotically Gaussian distributed. An explicit expression is given for the covariance matrix of their distribution. The covariance matrix is then taken as a (multivariable) measure of accuracy. It is shown how it can be optimized by an appropriate selection of instruments and prefilter. The so obtained optimal instrumental variable estimates cannot be used directly since the true system and the statistical properties of the disturbance must be known in order to compute the optimal instruments and prefilters. A multistep procedure consisting of three or four simple steps is then proposed as a way to overcome this difficulty. This procedure includes modeling of the disturbance as an ARMA process using a statistically efficient method such as a prediction error method. The statistical properties of the estimates obtained with the multistep procedure are also analyzed. Those estimates are shown to be asymptotically Gaussian distributed as well. The covariance matrix of the estimation errors is compared to that corresponding to a prediction error method. For some model structures these two approaches give the same asymptotic accuracy. The conclusion is that the multistep procedure, which is quite easy to implement and also has nice uniqueness properties, can be viewed as an interesting alternative to prediction error methods.     

Parametric, bootstrap, and jackknife variance estimators for the k-Nearest Neighbors technique with illustrations using forest inventory and satellite image data

Nearest neighbors techniques have been shown to be useful for estimating forest attributes, particularly when used with forest inventory and satellite image data. Published reports of positive results have been truly international in scope. However, for these techniques to be more useful, they must be able to contribute to scientific inference which, for sample-based methods, requires estimates of uncertainty in the form of variances or standard errors. Several parametric approaches to estimating uncertainty for nearest neighbors techniques have been proposed, but they are complex and computationally intensive. For this study, two resampling estimators, the bootstrap and the jackknife, were investigated and compared to a parametric estimator for estimating uncertainty using the k-Nearest Neighbors (k-NN) technique with forest inventory and Landsat data from Finland, Italy, and the USA. The technical objectives of the study were threefold: (1) to evaluate the assumptions underlying a parametric approach to estimating k-NN variances; (2) to assess the utility of the bootstrap and jackknife methods with respect to the quality of variance estimates, ease of implementation, and computational intensity; and (3) to investigate adaptation of resampling methods to accommodate cluster sampling. The general conclusions were that support was provided for the assumptions underlying the parametric approach, the parametric and resampling estimators produced comparable variance estimates, care must be taken to ensure that bootstrap resampling mimics the original sampling, and the bootstrap procedure is a viable approach to variance estimation for nearest neighbor techniques that use very small numbers of neighbors to calculate predictions.     

SAS macros for estimation of direct adjusted cumulative incidence curves under proportional subdistribution hazards models

The cumulative incidence function is commonly reported in studies with competing risks. The aim of this paper is to compute the treatment-specific cumulative incidence functions, adjusting for potentially imbalanced prognostic factors among treatment groups. The underlying regression model considered in this study is the proportional hazards model for a subdistribution function [1]. We propose estimating the direct adjusted cumulative incidences for each treatment using the pooled samples as the reference population. We develop two SAS macros for estimating the direct adjusted cumulative incidence function for each treatment based on two regression models. One model assumes the constant subdistribution hazard ratios between the treatments and the alternative model allows each treatment to have its own baseline subdistribution hazard function. The macros compute the standard errors for the direct adjusted cumulative incidence estimates, as well as the standard errors for the differences of adjusted cumulative incidence functions between any two treatments. Based on the macros’ output, one can assess treatment effects at predetermined time points. A real bone marrow transplant data example illustrates the practical utility of the SAS macros.     

On Fitting Empirical Data under Interval Error

This paper is devoted to the problem of fitting input-output data by a modeling function, linear in its parameters, in the presence of interval-bounded errors in output variable. A method for outlier detection is proposed. Another issue under consideration is the comparative simulation study of the well-known statistical point estimates (least squares, maximum likelihood) and point estimates calculated as the center of interval hull of uncertainty set. The results of the study allow us to draw the conclusion that non-statistical interval based estimation is a competitive alternative to statistical estimation in some cases.     

SAS and R functions to compute pseudo-values for censored data regression

Recently, in a series of papers, a method based on pseudo-values has been proposed for direct regression modeling of the survival function, the restricted mean and cumulative incidence function with right censored data. The models, once the pseudo-values have been computed, can be fit using standard generalized estimating equation software. Here we present SAS macros and R functions to compute these pseudo-values. We illustrate the use of these routines and show how to obtain regression estimates for a study of bone marrow transplant patients.     

A Kalman filtering approach to natural gamma ray spectroscopy in well logging

This paper describes an application of (adaptive) Kalman filtering to a geophysical subsurface estimation problem. The NGT® is a sonde designed to detect the natural gamma rays of various energies emitted from a formation by the radioactive nuclei of potassium (K) and the thorium (Th) and uranium-radium (U) series. Using a minicomputer at the surface, the (Th,U,K) concentrations along the borehole have to be estimated on-line from the detection of the gamma rays in five energy windows. The standard technique in the logging industry has been to compute the elemental concentrations at a given depth using only the observed counting rates at the same depth. The resulting estimates have fairly large statistical errors which have limited the application of the NGT in computer reservoir evaluation. In this paper, it is shown that a Kalman filter based on a dynamical model of the (Th,U,K) vertical variations can produce real-time estimates that are readily usable on a quantitative basis. The paper focuses on the usual critical issues in applying Kalman filtering to real data, namely modeling, adaptivity, and computational aspects.     

Bootstrapping neural networks

Knowledge about the distribution of a statistical estimator is important for various purposes, such as the construction of confidence intervals for model parameters or the determination of critical values of tests. A widely used method to estimate this distribution is the so-called boot-strap, which is based on an imitation of the probabilistic structure of the data-generating process on the basis of the information provided by a given set of random observations. In this article we investigate this classical method in the context of artificial neural networks used for estimating a mapping from input to output space. We establish consistency results for bootstrap estimates of the distribution of parameter estimates.