Notes 4. ROBUST REGRESSION for the LINEAR MODEL L-Estimators
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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. -
Power Comparisons of Five Most Commonly Used Autocorrelation Tests
Pak.j.stat.oper.res. Vol.16 No. 1 2020 pp119-130 DOI: http://dx.doi.org/10.18187/pjsor.v16i1.2691 Pakistan Journal of Statistics and Operation Research Power Comparisons of Five Most Commonly Used Autocorrelation Tests Stanislaus S. Uyanto School of Economics and Business, Atma Jaya Catholic University of Indonesia, [email protected] Abstract In regression analysis, autocorrelation of the error terms violates the ordinary least squares assumption that the error terms are uncorrelated. The consequence is that the estimates of coefficients and their standard errors will be wrong if the autocorrelation is ignored. There are many tests for autocorrelation, we want to know which test is more powerful. We use Monte Carlo methods to compare the power of five most commonly used tests for autocorrelation, namely Durbin-Watson, Breusch-Godfrey, Box–Pierce, Ljung Box, and Runs tests in two different linear regression models. The results indicate the Durbin-Watson test performs better in the regression model without lagged dependent variable, although the advantage over the other tests reduce with increasing autocorrelation and sample sizes. For the model with lagged dependent variable, the Breusch-Godfrey test is generally superior to the other tests. Key Words: Correlated error terms; Ordinary least squares assumption; Residuals; Regression diagnostic; Lagged dependent variable. Mathematical Subject Classification: 62J05; 62J20. 1. Introduction An important assumption of the classical linear regression model states that there is no autocorrelation in the error terms. Autocorrelation of the error terms violates the ordinary least squares assumption that the error terms are uncorrelated, meaning that the Gauss Markov theorem (Plackett, 1949, 1950; Greene, 2018) does not apply, and that ordinary least squares estimators are no longer the best linear unbiased estimators. -
Multiple Regression
ICPSR Blalock Lectures, 2003 Bootstrap Resampling Robert Stine Lecture 4 Multiple Regression Review from Lecture 3 Review questions from syllabus - Questions about underlying theory - Math and notation handout from first class Review resampling in simple regression - Regression diagnostics - Smoothing methods (modern regr) Methods of resampling – which to use? Today Multiple regression - Leverage, influence, and diagnostic plots - Collinearity Resampling in multiple regression - Ratios of coefficients - Differences of two coefficients - Robust regression Yes! More t-shirts! Bootstrap Resampling Multiple Regression Lecture 4 ICPSR 2003 2 Locating a Maximum Polynomial regression Why bootstrap in least squares regression? Question What amount of preparation (in hours) maximizes the average test score? Quadmax.dat Constructed data for this example Results of fitting a quadratic Scatterplot with added quadratic fit (or smooth) 0 5 10 15 20 HOURS Least squares regression of the model Y = a + b X + c X2 + e gives the estimates a = 136 b= 89.2 c = –4.60 So, where’s the maximum and what’s its CI? Bootstrap Resampling Multiple Regression Lecture 4 ICPSR 2003 3 Inference for maximum Position of the peak The maximum occurs where the derivative of the fit is zero. Write the fit as f(x) = a + b x + c x2 and then take the derivative, f’(x) = b + 2cx Solving f’(x*) = 0 for x* gives b x* = – 2c ≈ –89.2/(2)(–4.6)= 9.7 Questions - What is the standard error of x*? - What is the 95% confidence interval? - Is the estimate biased? Bootstrap Resampling Multiple Regression Lecture 4 ICPSR 2003 4 Bootstrap Results for Max Location Standard error and confidence intervals (B=2000) Standard error is slightly smaller with fixed resampling (no surprise)… Observation resampling SE* = 0.185 Fixed resampling SE* = 0.174 Both 95% intervals are [9.4, 10] Bootstrap distribution The kernel looks pretty normal. -
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. -
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. -
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. -
Robust Bayesian General Linear Models ⁎ W.D
www.elsevier.com/locate/ynimg NeuroImage 36 (2007) 661–671 Robust Bayesian general linear models ⁎ W.D. Penny, J. Kilner, and F. Blankenburg Wellcome Department of Imaging Neuroscience, University College London, 12 Queen Square, London WC1N 3BG, UK Received 22 September 2006; revised 20 November 2006; accepted 25 January 2007 Available online 7 May 2007 We describe a Bayesian learning algorithm for Robust General Linear them from the data (Jung et al., 1999). This is, however, a non- Models (RGLMs). The noise is modeled as a Mixture of Gaussians automatic process and will typically require user intervention to rather than the usual single Gaussian. This allows different data points disambiguate the discovered components. In fMRI, autoregressive to be associated with different noise levels and effectively provides a (AR) modeling can be used to downweight the impact of periodic robust estimation of regression coefficients. A variational inference respiratory or cardiac noise sources (Penny et al., 2003). More framework is used to prevent overfitting and provides a model order recently, a number of approaches based on robust regression have selection criterion for noise model order. This allows the RGLM to default to the usual GLM when robustness is not required. The method been applied to imaging data (Wager et al., 2005; Diedrichsen and is compared to other robust regression methods and applied to Shadmehr, 2005). These approaches relax the assumption under- synthetic data and fMRI. lying ordinary regression that the errors be normally (Wager et al., © 2007 Elsevier Inc. All rights reserved. 2005) or identically (Diedrichsen and Shadmehr, 2005) distributed. In Wager et al. -
Sketching for M-Estimators: a Unified Approach to Robust Regression
Sketching for M-Estimators: A Unified Approach to Robust Regression Kenneth L. Clarkson∗ David P. Woodruffy Abstract ing algorithm often results, since a single pass over A We give algorithms for the M-estimators minx kAx − bkG, suffices to compute SA. n×d n n where A 2 R and b 2 R , and kykG for y 2 R is specified An important property of many of these sketching ≥0 P by a cost function G : R 7! R , with kykG ≡ i G(yi). constructions is that S is a subspace embedding, meaning d The M-estimators generalize `p regression, for which G(x) = that for all x 2 R , kSAxk ≈ kAxk. (Here the vector jxjp. We first show that the Huber measure can be computed norm is generally `p for some p.) For the regression up to relative error in O(nnz(A) log n + poly(d(log n)=")) d time, where nnz(A) denotes the number of non-zero entries problem of minimizing kAx − bk with respect to x 2 R , of the matrix A. Huber is arguably the most widely used for inputs A 2 Rn×d and b 2 Rn, a minor extension of M-estimator, enjoying the robustness properties of `1 as well the embedding condition implies S preserves the norm as the smoothness properties of ` . 2 of the residual vector Ax − b, that is kS(Ax − b)k ≈ We next develop algorithms for general M-estimators. kAx − bk, so that a vector x that makes kS(Ax − b)k We analyze the M-sketch, which is a variation of a sketch small will also make kAx − bk small. -
On Robust Regression with High-Dimensional Predictors
On robust regression with high-dimensional predictors Noureddine El Karoui ∗, Derek Bean ∗ , Peter Bickel ∗ , Chinghway Lim y, and Bin Yu ∗ ∗University of California, Berkeley, and yNational University of Singapore Submitted to Proceedings of the National Academy of Sciences of the United States of America We study regression M-estimates in the setting where p, the num- pointed out a surprising feature of the regime, p=n ! κ > 0 ber of covariates, and n, the number of observations, are both large for LSE; fitted values were not asymptotically Gaussian. He but p ≤ n. We find an exact stochastic representation for the dis- was unable to deal with this regime otherwise, see the discus- Pn 0 tribution of β = argmin p ρ(Y − X β) at fixed p and n b β2R i=1 i i sion on p.802 of [6]. under various assumptions on the objective function ρ and our sta- In this paper we intend to, in part heuristically and with tistical model. A scalar random variable whose deterministic limit \computer validation", analyze fully what happens in robust rρ(κ) can be studied when p=n ! κ > 0 plays a central role in this regression when p=n ! κ < 1. We do limit ourselves to Gaus- representation. Furthermore, we discover a non-linear system of two deterministic sian covariates but present grounds that the behavior holds equations that characterizes rρ(κ). Interestingly, the system shows much more generally. We also investigate the sensitivity of that rρ(κ) depends on ρ through proximal mappings of ρ as well as our results to the geometry of the design matrix. -
Regression Diagnostics and Model Selection
Regression diagnostics and model selection MS-C2128 Prediction and Time Series Analysis Fall term 2020 MS-C2128 Prediction and Time Series Analysis Regression diagnostics and model selection Week 2: Regression diagnostics and model selection 1 Regression diagnostics 2 Model selection MS-C2128 Prediction and Time Series Analysis Regression diagnostics and model selection Content 1 Regression diagnostics 2 Model selection MS-C2128 Prediction and Time Series Analysis Regression diagnostics and model selection Regression diagnostics Questions: Does the model describe the dependence between the response variable and the explanatory variables well 1 contextually? 2 statistically? A good model describes the dependencies as well as possible. Assessment of the goodness of a regression model is called regression diagnostics. Regression diagnostics tools: graphics diagnostic statistics diagnostic tests MS-C2128 Prediction and Time Series Analysis Regression diagnostics and model selection Regression model selection In regression modeling, one has to select 1 the response variable(s) and the explanatory variables, 2 the functional form and the parameters of the model, 3 and the assumptions on the residuals. Remark The first two points are related to defining the model part and the last point is related to the residuals. These points are not independent of each other! MS-C2128 Prediction and Time Series Analysis Regression diagnostics and model selection Problems in defining the model part (i) Linear model is applied even though the dependence between the response variable and the explanatory variables is non-linear. (ii) Too many or too few explanatory variables are chosen. (iii) It is assumed that the model parameters are constants even though the assumption does not hold. -
Robust Regression in Stata
The Stata Journal (yyyy) vv, Number ii, pp. 1–23 Robust Regression in Stata Vincenzo Verardi University of Namur (CRED) and Universit´eLibre de Bruxelles (ECARES and CKE) Rempart de la Vierge, 8, B-5000 Namur. E-mail: [email protected]. Vincenzo Verardi is Associated Researcher of the FNRS and gratefully acknowledges their finacial support Christophe Croux K.U.Leuven, Faculty of Business and Economics Naamsestraat 69. B-3000, Leuven E-mail: [email protected]. Abstract. In regression analysis, the presence of outliers in the data set can strongly distort the classical least squares estimator and lead to un- reliable results. To deal with this, several robust-to-outliers methods have been proposed in the statistical literature. In Stata, some of these methods are available through the commands rreg and qreg. Unfortunately, these methods only resist to some specific types of outliers and turn out to be ineffective under alternative scenarios. In this paper we present more ef- fective robust estimators that we implemented in Stata. We also present a graphical tool that allows recognizing the type of detected outliers. KEYWORDS: S-estimators, MM-estimators, Outliers, Robustness JEL CLASSIFICATION: C12, C21, C87 c yyyy StataCorp LP st0001 2 Robust Regression in Stata 1 Introduction The objective of linear regression analysis is to study how a dependent variable is linearly related to a set of regressors. In matrix notation, the linear regression model is given by: y = Xθ + ε (1) where, for a sample of size n, y is the (n × 1) vector containing the values for the dependent variable, X is the (n × p) matrix containing the values for the p regressors and ε is the (n × 1) vector containing the error terms. -
Outlier Detection Based on Robust Parameter Estimates
International Journal of Applied Engineering Research ISSN 0973-4562 Volume 12, Number 23 (2017) pp.13429-13434 © Research India Publications. http://www.ripublication.com Outlier Detection based on Robust Parameter Estimates Nor Azlida Aleng1, Nyi Nyi Naing2, Norizan Mohamed3 and Kasypi Mokhtar4 1,3 School of Informatics and Applied Mathematics, Universiti Malaysia Terengganu, 21030 Kuala Terengganu, Terengganu, Malaysia. 2 Institute for Community (Health) Development (i-CODE), Universiti Sultan Zainal Abidin, Gong Badak Campus, 21300 Kuala Terengganu, Malaysia. 4 School of Maritime Business and Management, Universiti Malaysia Terengganu, 21030 Kuala Terengganu, Terengganu, Malaysia. Orcid; 0000-0003-1111-3388 Abstract evidence that the outliers can lead to model misspecification, incorrect analysis result and can make all estimation procedures Outliers can influence the analysis of data in various different meaningless, (Rousseeuw and Leroy, 1987; Alma, 2011; ways. The outliers can lead to model misspecification, incorrect Zimmerman, 1994, 1995, 1998). Outliers in the response analysis results and can make all estimation procedures variable represent model failure. Outliers in the regressor meaningless. In regression analysis, ordinary least square variable values are extreme in X-space are called leverage estimation is most frequently used for estimation of the points. parameters in the model. Unfortunately, this estimator is sensitive to outliers. Thus, in this paper we proposed some Rousseeuw and Lorey (1987) defined that outliers in three statistics for detection of outliers based on robust estimation, types; 1) vertical outliers, 2) bad leverage point, 3) good namely least trimmed squares (LTS). A simulation study was leverage point. Vertical outlier is an observation that has performed to prove that the alternative approach gives a better influence on the error term of y-dimension (response) but is results than OLS estimation to identify outliers.