Regression analysis

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Regression_analysis

Regression line for 50 random points in a Gaussian distribution around the line y=1.5x+2 (not shown)

In statistical modeling, regression analysis is a set of statistical processes for estimating the relationships between a dependent variable (often called the outcome or response variable, or a label in machine learning parlance) and one or more error-free independent variables (often called regressors, predictors, covariates, explanatory variables or features). The most common form of regression analysis is linear regression, in which one finds the line (or a more complex linear combination) that most closely fits the data according to a specific mathematical criterion. For example, the method of ordinary least squares computes the unique line (or hyperplane) that minimizes the sum of squared differences between the true data and that line (or hyperplane). For specific mathematical reasons (see linear regression), this allows the researcher to estimate the conditional expectation (or population average value) 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 (e.g., quantile regression or Necessary Condition Analysis) or estimate the conditional expectation across a broader collection of non-linear models (e.g., nonparametric regression).

Regression analysis is primarily used for two conceptually distinct purposes. First, regression analysis is widely used for prediction and forecasting, where its use has substantial overlap with the field of machine learning. Second, in some situations regression analysis can be used to infer causal relationships between the independent and dependent variables. Importantly, regressions by themselves only reveal relationships between a dependent variable and a collection of independent variables in a fixed dataset. To use regressions for prediction or to infer causal relationships, respectively, a researcher must carefully justify why existing relationships have predictive power for a new context or why a relationship between two variables has a causal interpretation. The latter is especially important when researchers hope to estimate causal relationships using observational data.

History

The earliest form of regression was the method of least squares, which was published by Legendre in 1805, and by Gauss in 1809. Legendre and Gauss both applied the method to the problem of determining, from astronomical observations, the orbits of bodies about the Sun (mostly comets, but also later the then newly discovered minor planets). Gauss published a further development of the theory of least squares in 1821, including a version of the Gauss–Markov theorem.

The term "regression" was coined by Francis Galton in the 19th century to describe a biological phenomenon. The phenomenon was that the heights of descendants of tall ancestors tend to regress down towards a normal average (a phenomenon also known as regression toward the mean). For Galton, regression had only this biological meaning, but his work was later extended by Udny Yule and Karl Pearson to a more general statistical context. In the work of Yule and Pearson, the joint distribution of the response and explanatory variables is assumed to be Gaussian. This assumption was weakened by R.A. Fisher in his works of 1922 and 1925. Fisher assumed that the conditional distribution of the response variable is Gaussian, but the joint distribution need not be. In this respect, Fisher's assumption is closer to Gauss's formulation of 1821.

In the 1950s and 1960s, economists used electromechanical desk calculators to calculate regressions. Before 1970, it sometimes took up to 24 hours to receive the result from one regression.

Regression methods continue to be an area of active research. In recent decades, new methods have been developed for robust regression, regression involving correlated responses such as time series and growth curves, regression in which the predictor (independent variable) or response variables are curves, images, graphs, or other complex data objects, regression methods accommodating various types of missing data, nonparametric regression, Bayesian methods for regression, regression in which the predictor variables are measured with error, regression with more predictor variables than observations, and causal inference with regression.

Regression model

In practice, researchers first select a model they would like to estimate and then use their chosen method (e.g., ordinary least squares) to estimate the parameters of that model. Regression models involve the following components:

• The unknown parameters, often denoted as a scalar or vector ${\displaystyle \beta }$.
• The independent variables, which are observed in data and are often denoted as a vector ${\displaystyle X_i}$ (where ${\displaystyle i}$ denotes a row of data).
• The dependent variable, which are observed in data and often denoted using the scalar ${\displaystyle Y_i}$.
• The error terms, which are not directly observed in data and are often denoted using the scalar ${\displaystyle e_i}$.

In various fields of application, different terminologies are used in place of dependent and independent variables.

Most regression models propose that ${\displaystyle Y_i}$ is a function (regression function) of ${\displaystyle X_i}$ and ${\displaystyle \beta }$, with ${\displaystyle e_i}$ representing an additive error term that may stand in for un-modeled determinants of ${\displaystyle Y_i}$ or random statistical noise:

${\displaystyle Y_i=f(X_i,\beta )+e_i}$

Note that the independent variables ${\displaystyle X_i}$ are assumed to be free of error. This important assumption is often overlooked, although errors-in-variables models can be used when the independent variables are assumed to contain errors.

The researchers' goal is to estimate the function ${\displaystyle f(X_i,\beta )}$ that most closely fits the data. To carry out regression analysis, the form of the function ${\displaystyle f}$ must be specified. Sometimes the form of this function is based on knowledge about the relationship between ${\displaystyle Y_i}$ and ${\displaystyle X_i}$ that does not rely on the data. If no such knowledge is available, a flexible or convenient form for ${\displaystyle f}$ is chosen. For example, a simple univariate regression may propose ${\displaystyle f(X_i,\beta )={\beta }_0+{\beta }_1{X}_i}$, suggesting that the researcher believes ${\displaystyle Y_i={\beta }_0+{\beta }_1{X}_i+e_i}$ to be a reasonable approximation for the statistical process generating the data.

Once researchers determine their preferred statistical model, different forms of regression analysis provide tools to estimate the parameters ${\displaystyle \beta }$. For example, least squares (including its most common variant, ordinary least squares) finds the value of ${\displaystyle \beta }$ that minimizes the sum of squared errors ${\displaystyle \sum _i(Y_i-f(X_i,\beta ){)}^2}$. A given regression method will ultimately provide an estimate of ${\displaystyle \beta }$, usually denoted ${\displaystyle \hat{\beta }}$ to distinguish the estimate from the true (unknown) parameter value that generated the data. Using this estimate, the researcher can then use the fitted value ${\displaystyle \widehat{Y_i}=f(X_i,\hat{\beta })}$ for prediction or to assess the accuracy of the model in explaining the data. Whether the researcher is intrinsically interested in the estimate ${\displaystyle \hat{\beta }}$ or the predicted value ${\displaystyle \widehat{Y_i}}$ will depend on context and their goals. As described in ordinary least squares, least squares is widely used because the estimated function ${\displaystyle f(X_i,\hat{\beta })}$ approximates the conditional expectation ${\displaystyle E(Y_i|X_i)}$. However, alternative variants (e.g., least absolute deviations or quantile regression) are useful when researchers want to model other functions ${\displaystyle f(X_i,\beta )}$.

It is important to note that there must be sufficient data to estimate a regression model. For example, suppose that a researcher has access to ${\displaystyle N}$ rows of data with one dependent and two independent variables: ${\displaystyle (Y_i,X_{1i},X_{2i})}$. Suppose further that the researcher wants to estimate a bivariate linear model via least squares: ${\displaystyle Y_i={\beta }_0+{\beta }_1{X}_{1i}+{\beta }_2{X}_{2i}+e_i}$. If the researcher only has access to ${\displaystyle N=2}$ data points, then they could find infinitely many combinations ${\displaystyle ({\hat{\beta }}_0,{\hat{\beta }}_1,{\hat{\beta }}_2)}$ that explain the data equally well: any combination can be chosen that satisfies ${\displaystyle {\hat{Y}}_i={\hat{\beta }}_0+{\hat{\beta }}_1{X}_{1i}+{\hat{\beta }}_2{X}_{2i}}$, all of which lead to ${\displaystyle \sum _i{\hat{e}}_i^2=\sum _i({\hat{Y}}_i-({\hat{\beta }}_0+{\hat{\beta }}_1{X}_{1i}+{\hat{\beta }}_2{X}_{2i}){)}^2=0}$ and are therefore valid solutions that minimize the sum of squared residuals. To understand why there are infinitely many options, note that the system of ${\displaystyle N=2}$ equations is to be solved for 3 unknowns, which makes the system underdetermined. Alternatively, one can visualize infinitely many 3-dimensional planes that go through ${\displaystyle N=2}$ fixed points.

More generally, to estimate a least squares model with ${\displaystyle k}$ distinct parameters, one must have ${\displaystyle N\ge k}$ distinct data points. If ${\displaystyle N>k}$, then there does not generally exist a set of parameters that will perfectly fit the data. The quantity ${\displaystyle N-k}$ appears often in regression analysis, and is referred to as the degrees of freedom in the model. Moreover, to estimate a least squares model, the independent variables ${\displaystyle (X_{1i},X_{2i},...,X_{ki})}$ must be linearly independent: one must not be able to reconstruct any of the independent variables by adding and multiplying the remaining independent variables. As discussed in ordinary least squares, this condition ensures that ${\displaystyle X^{T}X}$ is an invertible matrix and therefore that a unique solution ${\displaystyle \hat{\beta }}$ exists.

Underlying assumptions

By itself, a regression is simply a calculation using the data. In order to interpret the output of regression as a meaningful statistical quantity that measures real-world relationships, researchers often rely on a number of classical assumptions. These assumptions often include:

• The sample is representative of the population at large.
• The independent variables are measured with no error.
• Deviations from the model have an expected value of zero, conditional on covariates: ${\displaystyle E(e_i|X_i)=0}$
• The variance of the residuals ${\displaystyle e_i}$ is constant across observations (homoscedasticity).
• The residuals ${\displaystyle e_i}$ are uncorrelated with one another. Mathematically, the variance–covariance matrix of the errors is diagonal.

A handful of conditions are sufficient for the least-squares estimator to possess desirable properties: in particular, the Gauss–Markov assumptions imply that the parameter estimates will be unbiased, consistent, and efficient in the class of linear unbiased estimators. Practitioners have developed a variety of methods to maintain some or all of these desirable properties in real-world settings, because these classical assumptions are unlikely to hold exactly. For example, modeling errors-in-variables can lead to reasonable estimates independent variables are measured with errors. Heteroscedasticity-consistent standard errors allow the variance of ${\displaystyle e_i}$ to change across values of ${\displaystyle X_i}$. Correlated errors that exist within subsets of the data or follow specific patterns can be handled using clustered standard errors, geographic weighted regression, or Newey–West standard errors, among other techniques. When rows of data correspond to locations in space, the choice of how to model ${\displaystyle e_i}$ within geographic units can have important consequences. The subfield of econometrics is largely focused on developing techniques that allow researchers to make reasonable real-world conclusions in real-world settings, where classical assumptions do not hold exactly.

Linear regression

Main article: Linear regression

See simple linear regression for a derivation of these formulas and a numerical example

In linear regression, the model specification is that the dependent variable, ${\displaystyle y_i}$ is a linear combination of the parameters (but need not be linear in the independent variables). For example, in simple linear regression for modeling ${\displaystyle n}$ data points there is one independent variable: ${\displaystyle x_i}$, and two parameters, ${\displaystyle {\beta }_0}$ and ${\displaystyle {\beta }_1}$:

straight line: ${\displaystyle y_i={\beta }_0+{\beta }_1{x}_i+{\epsilon }_i,i=1,\dots ,n.}$

In multiple linear regression, there are several independent variables or functions of independent variables.

Adding a term in ${\displaystyle x_i^2}$ to the preceding regression gives:

parabola: ${\displaystyle y_i={\beta }_0+{\beta }_1{x}_i+{\beta }_2{x}_i^2+{\epsilon }_i,i=1,\dots ,n.}$

This is still linear regression; although the expression on the right hand side is quadratic in the independent variable ${\displaystyle x_i}$, it is linear in the parameters ${\displaystyle {\beta }_0}$, ${\displaystyle {\beta }_1}$ and ${\displaystyle {\beta }_2.}$

In both cases, ${\displaystyle {\epsilon }_i}$ is an error term and the subscript ${\displaystyle i}$ indexes a particular observation.

Returning our attention to the straight line case: Given a random sample from the population, we estimate the population parameters and obtain the sample linear regression model:

${\displaystyle {\hat{y}}_i={\hat{\beta }}_0+{\hat{\beta }}_1{x}_i.}$

The residual, ${\displaystyle e_i=y_i-{\hat{y}}_i}$, is the difference between the value of the dependent variable predicted by the model, ${\displaystyle {\hat{y}}_i}$, and the true value of the dependent variable, ${\displaystyle y_i}$. One method of estimation is ordinary least squares. This method obtains parameter estimates that minimize the sum of squared residuals, SSR:

${\displaystyle SSR=\sum _{i=1}^n{e}_i^2}$

Minimization of this function results in a set of normal equations, a set of simultaneous linear equations in the parameters, which are solved to yield the parameter estimators, ${\displaystyle {\hat{\beta }}_0,{\hat{\beta }}_1}$.

Illustration of linear regression on a data set

In the case of simple regression, the formulas for the least squares estimates are

${\displaystyle {\hat{\beta }}_1=\frac{\sum (x_i-\overline{x})(y_i-\overline{y})}{\sum (x_i-\overline{x}{)}^2}}$

${\displaystyle {\hat{\beta }}_0=\overline{y}-{\hat{\beta }}_1\overline{x}}$

where ${\displaystyle \overline{x}}$ is the mean (average) of the ${\displaystyle x}$ values and ${\displaystyle \overline{y}}$ is the mean of the ${\displaystyle y}$ values.

Under the assumption that the population error term has a constant variance, the estimate of that variance is given by:

${\displaystyle {\hat{\sigma }}_{\epsilon }^2=\frac{SSR}{n-2}}$

This is called the mean square error (MSE) of the regression. The denominator is the sample size reduced by the number of model parameters estimated from the same data, ${\displaystyle (n-p)}$ for ${\displaystyle p}$ regressors or ${\displaystyle (n-p-1)}$ if an intercept is used. In this case, ${\displaystyle p=1}$ so the denominator is ${\displaystyle n-2}$.

The standard errors of the parameter estimates are given by

${\displaystyle {\hat{\sigma }}_{{\beta }_1}={\hat{\sigma }}_{\epsilon }\sqrt{\frac{1}{\sum (x_i-\overline{x}{)}^2}}}$

${\displaystyle {\hat{\sigma }}_{{\beta }_0}={\hat{\sigma }}_{\epsilon }\sqrt{\frac{1}{n}+\frac{{\overline{x}}^2}{\sum (x_i-\overline{x}{)}^2}}={\hat{\sigma }}_{{\beta }_1}\sqrt{\frac{\sum x_i^2}{n}}.}$

Under the further assumption that the population error term is normally distributed, the researcher can use these estimated standard errors to create confidence intervals and conduct hypothesis tests about the population parameters.

General linear model

For a derivation, see linear least squares

For a numerical example, see linear regression

In the more general multiple regression model, there are ${\displaystyle p}$ independent variables:

${\displaystyle y_i={\beta }_1{x}_{i1}+{\beta }_2{x}_{i2}+\cdots +{\beta }_p{x}_{ip}+{\epsilon }_i,}$

where ${\displaystyle x_{ij}}$ is the ${\displaystyle i}$-th observation on the ${\displaystyle j}$-th independent variable. If the first independent variable takes the value 1 for all ${\displaystyle i}$, ${\displaystyle x_{i1}=1}$, then ${\displaystyle {\beta }_1}$ is called the regression intercept.

The least squares parameter estimates are obtained from ${\displaystyle p}$ normal equations. The residual can be written as

${\displaystyle {\epsilon }_i=y_i-{\hat{\beta }}_1{x}_{i1}-\cdots -{\hat{\beta }}_p{x}_{ip}.}$

The normal equations are

${\displaystyle \sum _{i=1}^n\sum _{k=1}^p{x}_{ij}{x}_{ik}{\hat{\beta }}_k=\sum _{i=1}^n{x}_{ij}{y}_i,j=1,\dots ,p.}$

In matrix notation, the normal equations are written as

${\displaystyle \mathbf{(}{\mathbf{X}}^{\mathrm{\top }}\mathbf{X}\mathbf{)}\hat{\mathit{\beta }}\mathbf{=}{\mathbf{X}}^{\mathrm{\top }}\mathbf{Y},}$

where the ${\displaystyle ij}$ element of ${\displaystyle \mathbf{X}}$ is ${\displaystyle x_{ij}}$, the ${\displaystyle i}$ element of the column vector ${\displaystyle Y}$ is ${\displaystyle y_i}$, and the ${\displaystyle j}$ element of ${\displaystyle \hat{\mathit{\beta }}}$ is ${\displaystyle {\hat{\beta }}_j}$. Thus ${\displaystyle \mathbf{X}}$ is ${\displaystyle n\times p}$, ${\displaystyle Y}$ is ${\displaystyle n\times 1}$, and ${\displaystyle \hat{\mathit{\beta }}}$ is ${\displaystyle p\times 1}$. The solution is

${\displaystyle \hat{\mathit{\beta }}\mathbf{=}\mathbf{(}{\mathbf{X}}^{\mathrm{\top }}\mathbf{X}{\mathbf{)}}^{\mathbf{-}\mathbf{1}}{\mathbf{X}}^{\mathrm{\top }}\mathbf{Y}.}$

Diagnostics

Main article: Regression diagnostics

See also: Category:Regression diagnostics

Once a regression model has been constructed, it may be important to confirm the goodness of fit of the model and the statistical significance of the estimated parameters. Commonly used checks of goodness of fit include the R-squared, analyses of the pattern of residuals and hypothesis testing. Statistical significance can be checked by an F-test of the overall fit, followed by t-tests of individual parameters.

Interpretations of these diagnostic tests rest heavily on the model's assumptions. Although examination of the residuals can be used to invalidate a model, the results of a t-test or F-test are sometimes more difficult to interpret if the model's assumptions are violated. For example, if the error term does not have a normal distribution, in small samples the estimated parameters will not follow normal distributions and complicate inference. With relatively large samples, however, a central limit theorem can be invoked such that hypothesis testing may proceed using asymptotic approximations.

Limited dependent variables

Limited dependent variables, which are response variables that are categorical variables or are variables constrained to fall only in a certain range, often arise in econometrics.

The response variable may be non-continuous ("limited" to lie on some subset of the real line). For binary (zero or one) variables, if analysis proceeds with least-squares linear regression, the model is called the linear probability model. Nonlinear models for binary dependent variables include the probit and logit model. The multivariate probit model is a standard method of estimating a joint relationship between several binary dependent variables and some independent variables. For categorical variables with more than two values there is the multinomial logit. For ordinal variables with more than two values, there are the ordered logit and ordered probit models. Censored regression models may be used when the dependent variable is only sometimes observed, and Heckman correction type models may be used when the sample is not randomly selected from the population of interest. An alternative to such procedures is linear regression based on polychoric correlation (or polyserial correlations) between the categorical variables. Such procedures differ in the assumptions made about the distribution of the variables in the population. If the variable is positive with low values and represents the repetition of the occurrence of an event, then count models like the Poisson regression or the negative binomial model may be used.

Nonlinear regression

Main article: Nonlinear regression

When the model function is not linear in the parameters, the sum of squares must be minimized by an iterative procedure. This introduces many complications which are summarized in Differences between linear and non-linear least squares.

Prediction (interpolation and extrapolation)

Further information: Predicted response and Prediction interval

In the middle, the fitted straight line represents the best balance between the points above and below this line. The dotted straight lines represent the two extreme lines, considering only the variation in the slope. The inner curves represent the estimated range of values considering the variation in both slope and intercept. The outer curves represent a prediction for a new measurement.

Regression models predict a value of the Y variable given known values of the X variables. Prediction within the range of values in the dataset used for model-fitting is known informally as interpolation. Prediction outside this range of the data is known as extrapolation. Performing extrapolation relies strongly on the regression assumptions. The further the extrapolation goes outside the data, the more room there is for the model to fail due to differences between the assumptions and the sample data or the true values.

A prediction interval that represents the uncertainty may accompany the point prediction. Such intervals tend to expand rapidly as the values of the independent variable(s) moved outside the range covered by the observed data.

For such reasons and others, some tend to say that it might be unwise to undertake extrapolation.

Model selection

Further information: Model selection

The assumption of a particular form for the relation between Y and X is another source of uncertainty. A properly conducted regression analysis will include an assessment of how well the assumed form is matched by the observed data, but it can only do so within the range of values of the independent variables actually available. This means that any extrapolation is particularly reliant on the assumptions being made about the structural form of the regression relationship. If this knowledge includes the fact that the dependent variable cannot go outside a certain range of values, this can be made use of in selecting the model – even if the observed dataset has no values particularly near such bounds. The implications of this step of choosing an appropriate functional form for the regression can be great when extrapolation is considered. At a minimum, it can ensure that any extrapolation arising from a fitted model is "realistic" (or in accord with what is known).

Power and sample size calculations

There are no generally agreed methods for relating the number of observations versus the number of independent variables in the model. One method conjectured by Good and Hardin is ${\displaystyle N=m^n}$, where ${\displaystyle N}$ is the sample size, ${\displaystyle n}$ is the number of independent variables and ${\displaystyle m}$ is the number of observations needed to reach the desired precision if the model had only one independent variable. For example, a researcher is building a linear regression model using a dataset that contains 1000 patients (${\displaystyle N}$). If the researcher decides that five observations are needed to precisely define a straight line (${\displaystyle m}$), then the maximum number of independent variables the model can support is 4, because

${\displaystyle \frac{\log 1000}{\log 5}\approx 4.29}$.

Other methods

Although the parameters of a regression model are usually estimated using the method of least squares, other methods which have been used include:

• Bayesian methods, e.g. Bayesian linear regression
• Percentage regression, for situations where reducing percentage errors is deemed more appropriate.
• Least absolute deviations, which is more robust in the presence of outliers, leading to quantile regression
• Nonparametric regression, requires a large number of observations and is computationally intensive
• Scenario optimization, leading to interval predictor models
• Distance metric learning, which is learned by the search of a meaningful distance metric in a given input space.

Software

For a more comprehensive list, see List of statistical software.

All major statistical software packages perform least squares regression analysis and inference. Simple linear regression and multiple regression using least squares can be done in some spreadsheet applications and on some calculators. While many statistical software packages can perform various types of nonparametric and robust regression, these methods are less standardized. Different software packages implement different methods, and a method with a given name may be implemented differently in different packages. Specialized regression software has been developed for use in fields such as survey analysis and neuroimaging.

Robust regression

From Wikipedia, the free encyclopedia
https://en.wikipedia.org/wiki/Robust_regression

In robust statistics, robust regression seeks to overcome some limitations of traditional regression analysis. A regression analysis models the relationship between one or more independent variables and a dependent variable. Standard types of regression, such as ordinary least squares, have favourable properties if their underlying assumptions are true, but can give misleading results otherwise (i.e. are not robust to assumption violations). Robust regression methods are designed to limit the effect that violations of assumptions by the underlying data-generating process have on regression estimates.

For example, least squares estimates for regression models are highly sensitive to outliers: an outlier with twice the error magnitude of a typical observation contributes four (two squared) times as much to the squared error loss, and therefore has more leverage over the regression estimates. The Huber loss function is a robust alternative to standard square error loss that reduces outliers' contributions to the squared error loss, thereby limiting their impact on regression estimates.

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  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). Perhaps the most important reason for the unpopularity of robust regression methods is that when the error variance is quite large or does not exist, for any given dataset any estimate, robust or otherwise, of the regression coefficients will likely be practically worthless unless the sample is quite large.

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^[vague] 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 ${\displaystyle 1-\epsilon }$ of coming from a normal distribution with variance ${\displaystyle {\sigma }^2}$, where ${\displaystyle \epsilon }$ is small, and probability ${\displaystyle \epsilon }$ of coming from a normal distribution with variance ${\displaystyle c{\sigma }^2}$ for some ${\displaystyle c>1}$:

${\displaystyle e_i\sim (1-\epsilon )N(0,{\sigma }^2)+\epsilon N(0,c{\sigma }^2).}$

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