1 Robust Regression Analysis: Some Popular Statistical Package Options
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Robust Statistics Part 3: Regression Analysis
Robust Statistics Part 3: Regression analysis Peter Rousseeuw LARS-IASC School, May 2019 Peter Rousseeuw Robust Statistics, Part 3: Regression LARS-IASC School, May 2019 p. 1 Linear regression Linear regression: Outline 1 Classical regression estimators 2 Classical outlier diagnostics 3 Regression M-estimators 4 The LTS estimator 5 Outlier detection 6 Regression S-estimators and MM-estimators 7 Regression with categorical predictors 8 Software Peter Rousseeuw Robust Statistics, Part 3: Regression LARS-IASC School, May 2019 p. 2 Linear regression Classical estimators The linear regression model The linear regression model says: yi = β0 + β1xi1 + ... + βpxip + εi ′ = xiβ + εi 2 ′ ′ with i.i.d. errors εi ∼ N(0,σ ), xi = (1,xi1,...,xip) and β =(β0,β1,...,βp) . ′ Denote the n × (p + 1) matrix containing the predictors xi as X =(x1,..., xn) , ′ ′ the vector of responses y =(y1,...,yn) and the error vector ε =(ε1,...,εn) . Then: y = Xβ + ε Any regression estimate βˆ yields fitted values yˆ = Xβˆ and residuals ri = ri(βˆ)= yi − yˆi . Peter Rousseeuw Robust Statistics, Part 3: Regression LARS-IASC School, May 2019 p. 3 Linear regression Classical estimators The least squares estimator Least squares estimator n ˆ 2 βLS = argmin ri (β) β i=1 X If X has full rank, then the solution is unique and given by ˆ ′ −1 ′ βLS =(X X) X y The usual unbiased estimator of the error variance is n 1 σˆ2 = r2(βˆ ) LS n − p − 1 i LS i=1 X Peter Rousseeuw Robust Statistics, Part 3: Regression LARS-IASC School, May 2019 p. 4 Linear regression Classical estimators Outliers in regression Different types of outliers: vertical outlier good leverage point • • y • • • regular data • ••• • •• ••• • • • • • • • • • bad leverage point • • •• • x Peter Rousseeuw Robust Statistics, Part 3: Regression LARS-IASC School, May 2019 p. -
Generalized Least Squares and Weighted Least Squares Estimation
REVSTAT – Statistical Journal Volume 13, Number 3, November 2015, 263–282 GENERALIZED LEAST SQUARES AND WEIGH- TED LEAST SQUARES ESTIMATION METHODS FOR DISTRIBUTIONAL PARAMETERS Author: Yeliz Mert Kantar – Department of Statistics, Faculty of Science, Anadolu University, Eskisehir, Turkey [email protected] Received: March 2014 Revised: August 2014 Accepted: August 2014 Abstract: • Regression procedures are often used for estimating distributional parameters because of their computational simplicity and useful graphical presentation. However, the re- sulting regression model may have heteroscedasticity and/or correction problems and thus, weighted least squares estimation or alternative estimation methods should be used. In this study, we consider generalized least squares and weighted least squares estimation methods, based on an easily calculated approximation of the covariance matrix, for distributional parameters. The considered estimation methods are then applied to the estimation of parameters of different distributions, such as Weibull, log-logistic and Pareto. The results of the Monte Carlo simulation show that the generalized least squares method for the shape parameter of the considered distri- butions provides for most cases better performance than the maximum likelihood, least-squares and some alternative estimation methods. Certain real life examples are provided to further demonstrate the performance of the considered generalized least squares estimation method. Key-Words: • probability plot; heteroscedasticity; autocorrelation; generalized least squares; weighted least squares; shape parameter. 264 Yeliz Mert Kantar Generalized Least Squares and Weighted Least Squares 265 1. INTRODUCTION Regression procedures are often used for estimating distributional param- eters. In this procedure, the distribution function is transformed to a linear re- gression model. Thus, least squares (LS) estimation and other regression estima- tion methods can be employed to estimate parameters of a specified distribution. -
Generalized and Weighted Least Squares Estimation
LINEAR REGRESSION ANALYSIS MODULE – VII Lecture – 25 Generalized and Weighted Least Squares Estimation Dr. Shalabh Department of Mathematics and Statistics Indian Institute of Technology Kanpur 2 The usual linear regression model assumes that all the random error components are identically and independently distributed with constant variance. When this assumption is violated, then ordinary least squares estimator of regression coefficient looses its property of minimum variance in the class of linear and unbiased estimators. The violation of such assumption can arise in anyone of the following situations: 1. The variance of random error components is not constant. 2. The random error components are not independent. 3. The random error components do not have constant variance as well as they are not independent. In such cases, the covariance matrix of random error components does not remain in the form of an identity matrix but can be considered as any positive definite matrix. Under such assumption, the OLSE does not remain efficient as in the case of identity covariance matrix. The generalized or weighted least squares method is used in such situations to estimate the parameters of the model. In this method, the deviation between the observed and expected values of yi is multiplied by a weight ω i where ω i is chosen to be inversely proportional to the variance of yi. n 2 For simple linear regression model, the weighted least squares function is S(,)ββ01=∑ ωii( yx −− β0 β 1i) . ββ The least squares normal equations are obtained by differentiating S (,) ββ 01 with respect to 01 and and equating them to zero as nn n ˆˆ β01∑∑∑ ωβi+= ωiixy ω ii ii=11= i= 1 n nn ˆˆ2 βω01∑iix+= βω ∑∑ii x ω iii xy. -
Department of Geography
Department of Geography UNIVERSITY OF FLORIDA, SPRING 2019 GEO 4167c section #09A6 / GEO 6161 section # 09A9 (3.0 credit hours) Course# 15235/15271 Intermediate Quantitative Methods Instructor: Timothy J. Fik, Ph.D. (Associate Professor) Prerequisite: GEO 3162 / GEO 6160 or equivalent Lecture Time/Location: Tuesdays, Periods 3-5: 9:35AM-12:35PM / Turlington 3012 Instructor’s Office: 3137 Turlington Hall Instructor’s e-mail address: [email protected] Formal Office Hours Tuesdays -- 1:00PM – 4:30PM Thursdays -- 1:30PM – 3:00PM; and 4:00PM – 4:30PM Course Materials (Power-point presentations in pdf format) will be uploaded to the on-line course Lecture folder on Canvas. Course Overview GEO 4167x/GEO 6161 surveys various statistical modeling techniques that are widely used in the social, behavioral, and environmental sciences. Lectures will focus on several important topics… including common indices of spatial association and dependence, linear and non-linear model development, model diagnostics, and remedial measures. The lectures will largely be devoted to the topic of Regression Analysis/Econometrics (and the General Linear Model). Applications will involve regression models using cross-sectional, quantitative, qualitative, categorical, time-series, and/or spatial data. Selected topics include, yet are not limited to, the following: Classic Least Squares Regression plus Extensions of the General Linear Model (GLM) Matrix Algebra approach to Regression and the GLM Join-Count Statistics (Dacey’s Contiguity Tests) Spatial Autocorrelation / Regression -
Lecture 24: Weighted and Generalized Least Squares 1
Lecture 24: Weighted and Generalized Least Squares 1 Weighted Least Squares When we use ordinary least squares to estimate linear regression, we minimize the mean squared error: n 1 X MSE(b) = (Y − X β)2 (1) n i i· i=1 th where Xi· is the i row of X. The solution is T −1 T βbOLS = (X X) X Y: (2) Suppose we minimize the weighted MSE n 1 X W MSE(b; w ; : : : w ) = w (Y − X b)2: (3) 1 n n i i i· i=1 This includes ordinary least squares as the special case where all the weights wi = 1. We can solve it by the same kind of linear algebra we used to solve the ordinary linear least squares problem. If we write W for the matrix with the wi on the diagonal and zeroes everywhere else, then W MSE = n−1(Y − Xb)T W(Y − Xb) (4) 1 = YT WY − YT WXb − bT XT WY + bT XT WXb : (5) n Differentiating with respect to b, we get as the gradient 2 r W MSE = −XT WY + XT WXb : b n Setting this to zero at the optimum and solving, T −1 T βbW LS = (X WX) X WY: (6) But why would we want to minimize Eq. 3? 1. Focusing accuracy. We may care very strongly about predicting the response for certain values of the input | ones we expect to see often again, ones where mistakes are especially costly or embarrassing or painful, etc. | than others. If we give the points near that region big weights, and points elsewhere smaller weights, the regression will be pulled towards matching the data in that region. -
The Evolution of Econometric Software Design: a Developer's View
Journal of Economic and Social Measurement 29 (2004) 205–259 205 IOS Press The evolution of econometric software design: A developer’s view Houston H. Stokes Department of Economics, College of Business Administration, University of Illinois at Chicago, 601 South Morgan Street, Room 2103, Chicago, IL 60607-7121, USA E-mail: [email protected] In the last 30 years, changes in operating systems, computer hardware, compiler technology and the needs of research in applied econometrics have all influenced econometric software development and the environment of statistical computing. The evolution of various representative software systems, including B34S developed by the author, are used to illustrate differences in software design and the interrelation of a number of factors that influenced these choices. A list of desired econometric software features, software design goals and econometric programming language characteristics are suggested. It is stressed that there is no one “ideal” software system that will work effectively in all situations. System integration of statistical software provides a means by which capability can be leveraged. 1. Introduction 1.1. Overview The development of modern econometric software has been influenced by the changing needs of applied econometric research, the expanding capability of com- puter hardware (CPU speed, disk storage and memory), changes in the design and capability of compilers, and the availability of high-quality subroutine libraries. Soft- ware design in turn has itself impacted applied econometric research, which has seen its horizons expand rapidly in the last 30 years as new techniques of analysis became computationally possible. How some of these interrelationships have evolved over time is illustrated by a discussion of the evolution of the design and capability of the B34S Software system [55] which is contrasted to a selection of other software systems. -
International Journal of Forecasting Guidelines for IJF Software Reviewers
International Journal of Forecasting Guidelines for IJF Software Reviewers It is desirable that there be some small degree of uniformity amongst the software reviews in this journal, so that regular readers of the journal can have some idea of what to expect when they read a software review. In particular, I wish to standardize the second section (after the introduction) of the review, and the penultimate section (before the conclusions). As stand-alone sections, they will not materially affect the reviewers abillity to craft the review as he/she sees fit, while still providing consistency between reviews. This applies mostly to single-product reviews, but some of the ideas presented herein can be successfully adapted to a multi-product review. The second section, Overview, is an overview of the package, and should include several things. · Contact information for the developer, including website address. · Platforms on which the package runs, and corresponding prices, if available. · Ancillary programs included with the package, if any. · The final part of this section should address Berk's (1987) list of criteria for evaluating statistical software. Relevant items from this list should be mentioned, as in my review of RATS (McCullough, 1997, pp.182- 183). · My use of Berk was extremely terse, and should be considered a lower bound. Feel free to amplify considerably, if the review warrants it. In fact, Berk's criteria, if considered in sufficient detail, could be the outline for a review itself. The penultimate section, Numerical Details, directly addresses numerical accuracy and reliality, if these topics are not addressed elsewhere in the review. -
Quantile Regression for Overdispersed Count Data: a Hierarchical Method Peter Congdon
Congdon Journal of Statistical Distributions and Applications (2017) 4:18 DOI 10.1186/s40488-017-0073-4 RESEARCH Open Access Quantile regression for overdispersed count data: a hierarchical method Peter Congdon Correspondence: [email protected] Abstract Queen Mary University of London, London, UK Generalized Poisson regression is commonly applied to overdispersed count data, and focused on modelling the conditional mean of the response. However, conditional mean regression models may be sensitive to response outliers and provide no information on other conditional distribution features of the response. We consider instead a hierarchical approach to quantile regression of overdispersed count data. This approach has the benefits of effective outlier detection and robust estimation in the presence of outliers, and in health applications, that quantile estimates can reflect risk factors. The technique is first illustrated with simulated overdispersed counts subject to contamination, such that estimates from conditional mean regression are adversely affected. A real application involves ambulatory care sensitive emergency admissions across 7518 English patient general practitioner (GP) practices. Predictors are GP practice deprivation, patient satisfaction with care and opening hours, and region. Impacts of deprivation are particularly important in policy terms as indicating effectiveness of efforts to reduce inequalities in care sensitive admissions. Hierarchical quantile count regression is used to develop profiles of central and extreme quantiles according to specified predictor combinations. Keywords: Quantile regression, Overdispersion, Poisson, Count data, Ambulatory sensitive, Median regression, Deprivation, Outliers 1. Background Extensions of Poisson regression are commonly applied to overdispersed count data, focused on modelling the conditional mean of the response. However, conditional mean regression models may be sensitive to response outliers. -
Weighted Least Squares
Weighted Least Squares 2 ∗ The standard linear model assumes that Var("i) = σ for i = 1; : : : ; n. ∗ As we have seen, however, there are instances where σ2 Var(Y j X = xi) = Var("i) = : wi ∗ Here w1; : : : ; wn are known positive constants. ∗ Weighted least squares is an estimation technique which weights the observations proportional to the reciprocal of the error variance for that observation and so overcomes the issue of non-constant variance. 7-1 Weighted Least Squares in Simple Regression ∗ Suppose that we have the following model Yi = β0 + β1Xi + "i i = 1; : : : ; n 2 where "i ∼ N(0; σ =wi) for known constants w1; : : : ; wn. ∗ The weighted least squares estimates of β0 and β1 minimize the quantity n X 2 Sw(β0; β1) = wi(yi − β0 − β1xi) i=1 ∗ Note that in this weighted sum of squares, the weights are inversely proportional to the corresponding variances; points with low variance will be given higher weights and points with higher variance are given lower weights. 7-2 Weighted Least Squares in Simple Regression ∗ The weighted least squares estimates are then given as ^ ^ β0 = yw − β1xw P w (x − x )(y − y ) ^ = i i w i w β1 P 2 wi(xi − xw) where xw and yw are the weighted means P w x P w y = i i = i i xw P yw P : wi wi ∗ Some algebra shows that the weighted least squares esti- mates are still unbiased. 7-3 Weighted Least Squares in Simple Regression ∗ Furthermore we can find their variances 2 ^ σ Var(β1) = X 2 wi(xi − xw) 2 3 1 2 ^ xw 2 Var(β0) = 4X + X 25 σ wi wi(xi − xw) ∗ Since the estimates can be written in terms of normal random variables, the sampling distributions are still normal. -
Robust Linear Regression: a Review and Comparison Arxiv:1404.6274
Robust Linear Regression: A Review and Comparison Chun Yu1, Weixin Yao1, and Xue Bai1 1Department of Statistics, Kansas State University, Manhattan, Kansas, USA 66506-0802. Abstract Ordinary least-squares (OLS) estimators for a linear model are very sensitive to unusual values in the design space or outliers among y values. Even one single atypical value may have a large effect on the parameter estimates. This article aims to review and describe some available and popular robust techniques, including some recent developed ones, and compare them in terms of breakdown point and efficiency. In addition, we also use a simulation study and a real data application to compare the performance of existing robust methods under different scenarios. arXiv:1404.6274v1 [stat.ME] 24 Apr 2014 Key words: Breakdown point; Robust estimate; Linear Regression. 1 Introduction Linear regression has been one of the most important statistical data analysis tools. Given the independent and identically distributed (iid) observations (xi; yi), i = 1; : : : ; n, 1 in order to understand how the response yis are related to the covariates xis, we tradi- tionally assume the following linear regression model T yi = xi β + "i; (1.1) where β is an unknown p × 1 vector, and the "is are i.i.d. and independent of xi with E("i j xi) = 0. The most commonly used estimate for β is the ordinary least square (OLS) estimate which minimizes the sum of squared residuals n X T 2 (yi − xi β) : (1.2) i=1 However, it is well known that the OLS estimate is extremely sensitive to the outliers. -
Estimating Regression Models for Categorical Dependent Variables Using SAS, Stata, LIMDEP, and SPSS*
© 2003-2008, The Trustees of Indiana University Regression Models for Categorical Dependent Variables: 1 Estimating Regression Models for Categorical Dependent Variables Using SAS, Stata, LIMDEP, and SPSS* Hun Myoung Park (kucc625) This document summarizes regression models for categorical dependent variables and illustrates how to estimate individual models using SAS 9.1, Stata 10.0, LIMDEP 9.0, and SPSS 16.0. 1. Introduction 2. The Binary Logit Model 3. The Binary Probit Model 4. Bivariate Logit/Probit Models 5. Ordered Logit/Probit Models 6. The Multinomial Logit Model 7. The Conditional Logit Model 8. The Nested Logit Model 9. Conclusion 1. Introduction A categorical variable here refers to a variable that is binary, ordinal, or nominal. Event count data are discrete (categorical) but often treated as continuous variables. When a dependent variable is categorical, the ordinary least squares (OLS) method can no longer produce the best linear unbiased estimator (BLUE); that is, OLS is biased and inefficient. Consequently, researchers have developed various regression models for categorical dependent variables. The nonlinearity of categorical dependent variable models (CDVMs) makes it difficult to fit the models and interpret their results. 1.1 Regression Models for Categorical Dependent Variables In CDVMs, the left-hand side (LHS) variable or dependent variable is neither interval nor ratio, but rather categorical. The level of measurement and data generation process (DGP) of a dependent variable determines the proper type of CDVM. Binary responses (0 or 1) are modeled with binary logit and probit regressions, ordinal responses (1st, 2nd, 3rd, …) are formulated into (generalized) ordered logit/probit regressions, and nominal responses are analyzed by multinomial logit, conditional logit, or nested logit models depending on specific circumstances. -
A Guide to Robust Statistical Methods in Neuroscience
A GUIDE TO ROBUST STATISTICAL METHODS IN NEUROSCIENCE Authors: Rand R. Wilcox1∗, Guillaume A. Rousselet2 1. Dept. of Psychology, University of Southern California, Los Angeles, CA 90089-1061, USA 2. Institute of Neuroscience and Psychology, College of Medical, Veterinary and Life Sciences, University of Glasgow, 58 Hillhead Street, G12 8QB, Glasgow, UK ∗ Corresponding author: [email protected] ABSTRACT There is a vast array of new and improved methods for comparing groups and studying associations that offer the potential for substantially increasing power, providing improved control over the probability of a Type I error, and yielding a deeper and more nuanced understanding of data. These new techniques effectively deal with four insights into when and why conventional methods can be unsatisfactory. But for the non-statistician, the vast array of new and improved techniques for comparing groups and studying associations can seem daunting, simply because there are so many new methods that are now available. The paper briefly reviews when and why conventional methods can have relatively low power and yield misleading results. The main goal is to suggest some general guidelines regarding when, how and why certain modern techniques might be used. Keywords: Non-normality, heteroscedasticity, skewed distributions, outliers, curvature. 1 1 Introduction The typical introductory statistics course covers classic methods for comparing groups (e.g., Student's t-test, the ANOVA F test and the Wilcoxon{Mann{Whitney test) and studying associations (e.g., Pearson's correlation and least squares regression). The two-sample Stu- dent's t-test and the ANOVA F test assume that sampling is from normal distributions and that the population variances are identical, which is generally known as the homoscedastic- ity assumption.