# Robust regression

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In robust statistics, robust regression is a form of regression analysis designed to overcome some limitations of traditional parametric and non-parametric methods. Regression analysis seeks to find the relationship between one or more independent variables and a dependent variable. Certain widely used methods of regression, such as ordinary least squares, have favourable properties if their underlying assumptions are true, but can give misleading results if those assumptions are not true; thus ordinary least squares is said to be not robust to violations of its assumptions. Robust regression methods are designed to be not overly affected by violations of assumptions by the underlying data-generating process.

## Contents

In particular, least squares estimates for regression models are highly sensitive to outliers. While there is no precise definition of an outlier, outliers are observations that do not follow the pattern of the other observations. This is not normally a problem if the outlier is simply an extreme observation drawn from the tail of a normal distribution, but if the outlier results from non-normal measurement error or some other violation of standard ordinary least squares assumptions, then it compromises the validity of the regression results if a non-robust regression technique is used.

## Applications

### Heteroscedastic errors

One instance in which robust estimation should be considered is when there is a strong suspicion of heteroscedasticity. In the homoscedastic model, it is assumed that the variance of the error term is constant for all values of x. Heteroscedasticity allows the variance to be dependent on x, which is more accurate for many real scenarios. For example, the variance of expenditure is often larger for individuals with higher income than for individuals with lower incomes. Software packages usually default to a homoscedastic model, even though such a model may be less accurate than a heteroscedastic model. One simple approach (Tofallis, 2008) is to apply least squares to percentage errors, as this reduces the influence of the larger values of the dependent variable compared to ordinary least squares.

### Presence of outliers

Another common situation in which robust estimation is used occurs when the data contain outliers. In the presence of outliers that do not come from the same data-generating process as the rest of the data, least squares estimation is inefficient and can be biased. Because the least squares predictions are dragged towards the outliers, and because the variance of the estimates is artificially inflated, the result is that outliers can be masked. (In many situations, including some areas of geostatistics and medical statistics, it is precisely the outliers that are of interest.)

Although it is sometimes claimed that least squares (or classical statistical methods in general) are robust, they are only robust in the sense that the type I error rate does not increase under violations of the model. In fact, the type I error rate tends to be lower than the nominal level when outliers are present, and there is often a dramatic increase in the type II error rate. The reduction of the type I error rate has been labelled as the conservatism of classical methods.

## History and unpopularity of robust regression

Despite their superior performance over least squares estimation in many situations, robust methods for regression are still not widely used. Several reasons may help explain their unpopularity (Hampel et al. 1986, 2005). One possible reason is that there are several competing methods [ citation needed ] and the field got off to many false starts. Also, computation of robust estimates is much more computationally intensive than least squares estimation; in recent years, however, this objection has become less relevant, as computing power has increased greatly. Another reason may be that some popular statistical software packages failed to implement the methods (Stromberg, 2004). The belief of many statisticians that classical methods are robust may be another reason[ citation needed ].

Although uptake of robust methods has been slow, modern mainstream statistics text books often include discussion of these methods (for example, the books by Seber and Lee, and by Faraway; for a good general description of how the various robust regression methods developed from one another see Andersen's book). Also, modern statistical software packages such as R, Statsmodels, Stata and S-PLUS include considerable functionality for robust estimation (see, for example, the books by Venables and Ripley, and by Maronna et al.).

## Methods for robust regression

### Least squares alternatives

The simplest methods of estimating parameters in a regression model that are less sensitive to outliers than the least squares estimates, is to use least absolute deviations. Even then, gross outliers can still have a considerable impact on the model, motivating research into even more robust approaches.

In 1964, Huber introduced M-estimation for regression. The M in M-estimation stands for "maximum likelihood type". The method is robust to outliers in the response variable, but turned out not to be resistant to outliers in the explanatory variables (leverage points). In fact, when there are outliers in the explanatory variables, the method has no advantage over least squares.

In the 1980s, several alternatives to M-estimation were proposed as attempts to overcome the lack of resistance. See the book by Rousseeuw and Leroy for a very practical review. Least trimmed squares (LTS) is a viable alternative and is currently (2007) the preferred choice of Rousseeuw and Ryan (1997, 2008). The Theil–Sen estimator has a lower breakdown point than LTS but is statistically efficient and popular. Another proposed solution was S-estimation. This method finds a line (plane or hyperplane) that minimizes a robust estimate of the scale (from which the method gets the S in its name) of the residuals. This method is highly resistant to leverage points and is robust to outliers in the response. However, this method was also found to be inefficient.

MM-estimation attempts to retain the robustness and resistance of S-estimation, whilst gaining the efficiency of M-estimation. The method proceeds by finding a highly robust and resistant S-estimate that minimizes an M-estimate of the scale of the residuals (the first M in the method's name). The estimated scale is then held constant whilst a close by M-estimate of the parameters is located (the second M).

### Parametric alternatives

Another approach to robust estimation of regression models is to replace the normal distribution with a heavy-tailed distribution. A t-distribution with 4–6 degrees of freedom has been reported to be a good choice in various practical situations. Bayesian robust regression, being fully parametric, relies heavily on such distributions.

Under the assumption of t-distributed residuals, the distribution is a location-scale family. That is, ${\displaystyle x\leftarrow (x-\mu )/\sigma }$. The degrees of freedom of the t-distribution is sometimes called the kurtosis parameter. Lange, Little and Taylor (1989) discuss this model in some depth from a non-Bayesian point of view. A Bayesian account appears in Gelman et al. (2003).

An alternative parametric approach is to assume that the residuals follow a mixture of normal distributions (Daemi et al. 2019); in particular, a contaminated normal distribution in which the majority of observations are from a specified normal distribution, but a small proportion are from a normal distribution with much higher variance. That is, residuals have probability ${\displaystyle 1-\varepsilon }$ of coming from a normal distribution with variance ${\displaystyle \sigma ^{2}}$, where ${\displaystyle \varepsilon }$ is small, and probability ${\displaystyle \varepsilon }$ of coming from a normal distribution with variance ${\displaystyle c\sigma ^{2}}$ for some ${\displaystyle c>1}$:

${\displaystyle e_{i}\sim (1-\varepsilon )N(0,\sigma ^{2})+\varepsilon N(0,c\sigma ^{2}).}$

Typically, ${\displaystyle \varepsilon <0.1}$. This is sometimes called the ${\displaystyle \varepsilon }$-contamination model.

Parametric approaches have the advantage that likelihood theory provides an "off-the-shelf" approach to inference (although for mixture models such as the ${\displaystyle \varepsilon }$-contamination model, the usual regularity conditions might not apply), and it is possible to build simulation models from the fit. However, such parametric models still assume that the underlying model is literally true. As such, they do not account for skewed residual distributions or finite observation precisions.

### Unit weights

Another robust method is the use of unit weights (Wainer & Thissen, 1976), a method that can be applied when there are multiple predictors of a single outcome. Ernest Burgess (1928) used unit weights to predict success on parole. He scored 21 positive factors as present (e.g., "no prior arrest" = 1) or absent ("prior arrest" = 0), then summed to yield a predictor score, which was shown to be a useful predictor of parole success. Samuel S. Wilks (1938) showed that nearly all sets of regression weights sum to composites that are very highly correlated with one another, including unit weights, a result referred to as Wilks' theorem (Ree, Carretta, & Earles, 1998). Robyn Dawes (1979) examined decision making in applied settings, showing that simple models with unit weights often outperformed human experts. Bobko, Roth, and Buster (2007) reviewed the literature on unit weights and concluded that decades of empirical studies show that unit weights perform similar to ordinary regression weights on cross validation.

## Example: BUPA liver data

The BUPA liver data have been studied by various authors, including Breiman (2001). The data can be found at the classic data sets page, and there is some discussion in the article on the Box–Cox transformation. A plot of the logs of ALT versus the logs of γGT appears below. The two regression lines are those estimated by ordinary least squares (OLS) and by robust MM-estimation. The analysis was performed in R using software made available by Venables and Ripley (2002).

The two regression lines appear to be very similar (and this is not unusual in a data set of this size). However, the advantage of the robust approach comes to light when the estimates of residual scale are considered. For ordinary least squares, the estimate of scale is 0.420, compared to 0.373 for the robust method. Thus, the relative efficiency of ordinary least squares to MM-estimation in this example is 1.266. This inefficiency leads to loss of power in hypothesis tests and to unnecessarily wide confidence intervals on estimated parameters.

### Outlier detection

Another consequence of the inefficiency of the ordinary least squares fit is that several outliers are masked because the estimate of residual scale is inflated; the scaled residuals are pushed closer to zero than when a more appropriate estimate of scale is used. The plots of the scaled residuals from the two models appear below. The variable on the x axis is just the observation number as it appeared in the data set. Rousseeuw and Leroy (1986) contains many such plots.

The horizontal reference lines are at 2 and −2, so that any observed scaled residual beyond these boundaries can be considered to be an outlier. Clearly, the least squares method leads to many interesting observations being masked.

Whilst in one or two dimensions outlier detection using classical methods can be performed manually, with large data sets and in high dimensions the problem of masking can make identification of many outliers impossible. Robust methods automatically detect these observations, offering a serious advantage over classical methods when outliers are present.

## Related Research Articles

Econometrics is the application of statistical methods to economic data in order to give empirical content to economic relationships. More precisely, it is "the quantitative analysis of actual economic phenomena based on the concurrent development of theory and observation, related by appropriate methods of inference". An introductory economics textbook describes econometrics as allowing economists "to sift through mountains of data to extract simple relationships". The first known use of the term "econometrics" was by Polish economist Paweł Ciompa in 1910. Jan Tinbergen is one of the two founding fathers of econometrics. The other, Ragnar Frisch, also coined the term in the sense in which it is used today.

The method of least squares is a standard approach in regression analysis to approximate the solution of overdetermined systems by minimizing the sum of the squares of the residuals made in the results of every single equation.

In statistics, a studentized residual is the quotient resulting from the division of a residual by an estimate of its standard deviation. It is a form of a Student's t-statistic, with the estimate of error varying between points.

In statistics, a vector of random variables is heteroscedastic if the variability of the random disturbance is different across elements of the vector. Here, variability could be quantified by the variance or any other measure of statistical dispersion. Thus heteroscedasticity is the absence of homoscedasticity. A typical example is the set of observations of income in different cities.

In statistical modeling, regression analysis is a set of statistical processes for estimating the relationships between a dependent variable and one or more independent variables. The most common form of regression analysis is linear regression, in which one finds the line that most closely fits the data according to a specific mathematical criterion. For example, the method of ordinary least squares computes the unique line that minimizes the sum of squared differences between the true data and that line. For specific mathematical reasons, this allows the researcher to estimate the conditional expectation of the dependent variable when the independent variables take on a given set of values. Less common forms of regression use slightly different procedures to estimate alternative location parameters or estimate the conditional expectation across a broader collection of non-linear models.

In statistics, the coefficient of determination, denoted R2 or r2 and pronounced "R squared", is the proportion of the variance in the dependent variable that is predictable from the independent variable(s).

In statistics, ordinary least squares (OLS) is a type of linear least squares method for estimating the unknown parameters in a linear regression model. OLS chooses the parameters of a linear function of a set of explanatory variables by the principle of least squares: minimizing the sum of the squares of the differences between the observed dependent variable in the given dataset and those predicted by the linear function of the independent variable.

In statistics, simple linear regression is a linear regression model with a single explanatory variable. That is, it concerns two-dimensional sample points with one independent variable and one dependent variable and finds a linear function that, as accurately as possible, predicts the dependent variable values as a function of the independent variable. The adjective simple refers to the fact that the outcome variable is related to a single predictor.

In econometrics, the seemingly unrelated regressions (SUR) or seemingly unrelated regression equations (SURE) model, proposed by Arnold Zellner in (1962), is a generalization of a linear regression model that consists of several regression equations, each having its own dependent variable and potentially different sets of exogenous explanatory variables. Each equation is a valid linear regression on its own and can be estimated separately, which is why the system is called seemingly unrelated, although some authors suggest that the term seemingly related would be more appropriate, since the error terms are assumed to be correlated across the equations.

Robust statistics are statistics with good performance for data drawn from a wide range of probability distributions, especially for distributions that are not normal. Robust statistical methods have been developed for many common problems, such as estimating location, scale, and regression parameters. One motivation is to produce statistical methods that are not unduly affected by outliers. Another motivation is to provide methods with good performance when there are small departures from parametric distribution. For example, robust methods work well for mixtures of two normal distributions with different standard-deviations; under this model, non-robust methods like a t-test work poorly.

In statistics, the Breusch–Pagan test, developed in 1979 by Trevor Breusch and Adrian Pagan, is used to test for heteroskedasticity in a linear regression model. It was independently suggested with some extension by R. Dennis Cook and Sanford Weisberg in 1983. Derived from the Lagrange multiplier test principle, it tests whether the variance of the errors from a regression is dependent on the values of the independent variables. In that case, heteroskedasticity is present.

In statistics, generalized least squares (GLS) is a technique for estimating the unknown parameters in a linear regression model when there is a certain degree of correlation between the residuals in a regression model. In these cases, ordinary least squares and weighted least squares can be statistically inefficient, or even give misleading inferences. GLS was first described by Alexander Aitken in 1936.

Bootstrapping is any test or metric that uses random sampling with replacement, and falls under the broader class of resampling methods. Bootstrapping assigns measures of accuracy to sample estimates. This technique allows estimation of the sampling distribution of almost any statistic using random sampling methods.

In statistics, polynomial regression is a form of regression analysis in which the relationship between the independent variable x and the dependent variable y is modelled as an nth degree polynomial in x. Polynomial regression fits a nonlinear relationship between the value of x and the corresponding conditional mean of y, denoted E(y |x). Although polynomial regression fits a nonlinear model to the data, as a statistical estimation problem it is linear, in the sense that the regression function E(y | x) is linear in the unknown parameters that are estimated from the data. For this reason, polynomial regression is considered to be a special case of multiple linear regression.

The following outline is provided as an overview of and topical guide to regression analysis:

Least trimmed squares (LTS), or least trimmed sum of squares, is a robust statistical method that fits a function to a set of data whilst not being unduly affected by the presence of outliers. It is one of a number of methods for robust regression.

Linear least squares (LLS) is the least squares approximation of linear functions to data. It is a set of formulations for solving statistical problems involved in linear regression, including variants for ordinary (unweighted), weighted, and generalized (correlated) residuals. Numerical methods for linear least squares include inverting the matrix of the normal equations and orthogonal decomposition methods.

In non-parametric statistics, the Theil–Sen estimator is a method for robustly fitting a line to sample points in the plane by choosing the median of the slopes of all lines through pairs of points. It has also been called Sen's slope estimator, slope selection, the single median method, the Kendall robust line-fit method, and the Kendall–Theil robust line. It is named after Henri Theil and Pranab K. Sen, who published papers on this method in 1950 and 1968 respectively, and after Maurice Kendall because of its relation to the Kendall tau rank correlation coefficient.

In statistics, linear regression is a linear approach to modelling the relationship between a scalar response and one or more explanatory variables. The case of one explanatory variable is called simple linear regression; for more than one, the process is called multiple linear regression. This term is distinct from multivariate linear regression, where multiple correlated dependent variables are predicted, rather than a single scalar variable.

Robust Regression and Outlier Detection is a book on robust statistics, particularly focusing on the breakdown point of methods for robust regression. It was written by Peter Rousseeuw and Annick M. Leroy, and published in 1987 by Wiley.

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