Linear regression

Example of linear regression with one independent variable.

In statistics, linear regression is any approach to modeling the relationship between a scalar variable y and one or more variables denoted X. In linear regression, models of the unknown parameters are estimated from the data using linear functions. Such models are called “linear models.” Most commonly, linear regression refers to a model in which the conditional mean of y given the value of X is an affine function of X. Less commonly, linear regression could refer to a model in which the median, or some other quantile of the conditional distribution of y given X is expressed as a linear function of X. Like all forms of regression analysis, linear regression focuses on the conditional probability distribution of y given X, rather than on the joint probability distribution of y and X, which is the domain of multivariate analysis.

Linear regression was the first type of regression analysis to be studied rigorously, and to be used extensively in practical applications. This is because models which depend linearly on their unknown parameters are easier to fit than models which are non-linearly related to their parameters and because the statistical properties of the resulting estimators are easier to determine.

Linear regression has many practical uses. Most applications of linear regression fall into one of the following two broad categories:

• If the goal is prediction, or forecasting, linear regression can be used to fit a predictive model to an observed data set of y and X values. After developing such a model, if an additional value of X is then given without its accompanying value of y, the fitted model can be used to make a prediction of the value of y.
• Given a variable y and a number of variables X₁, ..., X_p that may be related to y, then linear regression analysis can be applied to quantify the strength of the relationship between y and the X_j, to assess which X_j may have no relationship with y at all, and to identify which subsets of the X_j contain redundant information about y, thus once one of them is known, the others are no longer informative.

Linear regression models are often fitted using the least squares approach, but they may also be fitted in other ways, such as by minimizing the “lack of fit” in some other norm, or by minimizing a penalized version of the least squares loss function as in ridge regression. Conversely, the least squares approach can be used to fit models that are not linear models. Thus, while the terms “least squares” and linear model are closely linked, they are not synonymous.

Introduction to linear regression

Given a data set of n statistical units, a linear regression model assumes that the relationship between the dependent variable y_i and the p-vector of regressors x_i is approximately linear. This approximate relationship is modeled through a so-called “disturbance term” ε_i — an unobserved random variable that adds noise to the linear relationship between the dependent variable and regressors. Thus the model takes form

where ′ denotes the transpose, so that x_i′β is the inner product between vectors x_i and β.

Often these n equations are stacked together and written in vector form as

where

Some remarks on terminology and general use:

• is called the regressand, dependent variable, endogenous variable, response variable, or measured variable. (see dependent and independent variables.) The decision as to which variable in a data set is modeled as the dependent variable and which are modeled as the independent variables may be based on a presumption that the value of one of the variables is caused by, or directly influenced by the other variables. Alternatively, there may be an operational reason to model one of the variables in terms of the others, in which case there need be no presumption of causality.
• are called regressors, exogenous variables, explanatory variables, covariates, input variables, predictor variables, or independent variables (see dependent and independent variables, but not to be confused with independent random variables). The matrix is sometimes called the design matrix.
  • Usually a constant is included as one of the regressors. For example we can take x_i1 = 1 for i = 1, ..., n. The corresponding element of β is called the intercept. Many statistical inference procedures for linear models require an intercept to be present, so it is often included even if theoretical considerations suggest that its value should be zero.
  • Sometimes one of the regressors can be a non-linear function of another regressor or of the data, as in polynomial regression and segmented regression. The model remains linear as long as it is linear in the parameter vector β.
  • The regressors x_i may be viewed either as random variables, which we simply observe, or they can be considered as predetermined fixed values which we can choose. Both interpretations may be appropriate in different cases, and they generally lead to the same estimation procedures; however different approaches to asymptotic analysis are used in these two situations.
• is a p-dimensional parameter vector. Its elements are also called effects, or regression coefficients. Statistical estimation and inference in linear regression focuses on β.
• is called the error term, disturbance term, or noise. This variable captures all other factors which influence the dependent variable y_i other than the regressors x_i . The relationship between the error term and the regressors, for example whether they are correlated, is a crucial step in formulating a linear regression model, as it will determine the method to use for estimation.

Example. Consider a situation where a small ball is being tossed up in the air and then we measure its heights of ascent h_i at various moments in time t_i. Physics tells us that, ignoring the drag, the relationship can be modeled as

where β₁ determines the initial velocity of the ball, β₂ is proportional to the standard gravity, and ε_i is due to measurement errors. Linear regression can be used to estimate the values of β₁ and β₂ from the measured data. This model is non-linear in the time variable, but it is linear in the parameters β₁ and β₂; if we take regressors x_i = (x_i1, x_i2)  = (t_i, t_i²), the model takes on the standard form

Assumptions

Two key assumptions are common to all estimation methods used in linear regression analysis:

• The design matrix X must have full column rank p. For this property to hold, we must have n > p, where n is the sample size (this is a necessary but not a sufficient condition). If this condition fails this is called the multicollinearity in the regressors. In this case the parameter vector β will be not identifiable — at most we will be able to narrow down its value to some linear subspace of R^p.
  Methods for fitting linear models with multicollinearity have been developed,^[1][2][3][4] but require additional assumptions such as “effect sparsity” — that a large fraction of the effects are exactly zero.

A simpler statement of this is that there must be enough data available compared to the number of parameters to be estimated. If there is too little data, then you end up with a system of equations with no unique solution. See partial least squares regression.

• The regressors x_i are assumed to be error-free, that is they are not contaminated with measurement errors. Although not realistic in many settings, dropping this assumption leads to significantly more difficult errors-in-variables models.

Beyond these two assumptions, several other statistical properties of the data strongly influence the performance of different estimation methods:

• Some estimation methods are based on a lack of correlation, among the n observations . Statistical independence of the observations is not needed, although it can be exploited if it is known to hold.
• The statistical relationship between the error terms and the regressors plays an important role in determining whether an estimation procedure has desirable sampling properties such as being unbiased and consistent.
• The variances of the error terms may be equal across the n units (termed homoscedasticity) or not (termed heteroscedasticity). Some linear regression estimation methods give less precise parameter estimates and misleading inferential quantities such as standard errors when substantial heteroscedasticity is present.
• The arrangement, or probability distribution of the predictor variables x has a major influence on the precision of estimates of β. Sampling and design of experiments are highly-developed subfields of statistics that provide guidance for collecting data in such a way to achieve a precise estimate of β.

Interpretation

A fitted linear regression model can be used to identify the relationship between a single predictor variable x_j and the response variable y when all the other predictor variables in the model are “held fixed”. Specifically, the interpretation of β_j is the expected change in y for a one-unit change in x_j when the other covariates are held fixed. This is sometimes called the unique effect of x_j on y. In contrast, the marginal effect of x_j on y can be assessed using a correlation coefficient or simple linear regression model relating x_j to y.

Care must be taken when interpreting regression results, as some of the regressors may not allow for marginal changes (such as dummy variables, or the intercept term), while others cannot be held fixed (recall the example from the introduction: it would be impossible to “hold t_i fixed” and at the same time change the value of t_i²).

It is possible that the unique effect can be nearly zero even when the marginal effect is large. This may imply that some other covariate captures all the information in x_j, so that once that variable is in the model, there is no contribution of x_j to the variation in y. Conversely, the unique effect of x_j can be large while its marginal effect is nearly zero. This would happen if the other covariates explained a great deal of the variation of y, but they mainly explain variation in a way that is complementary to what is captured by x_j. In this case, including the other variables in the model reduces the part of the variability of y that is unrelated to x_j, thereby strengthening the apparent relationship with x_j.

The meaning of the expression “held fixed” may depend on how the values of the predictor variables arise. If the experimenter directly sets the values of the predictor variables according to a study design, the comparisons of interest may literally correspond to comparisons among units whose predictor variables have been “held fixed” by the experimenter. Alternatively, the expression “held fixed” can refer to a selection that takes place in the context of data analysis. In this case, we “hold a variable fixed” by restricting our attention to the subsets of the data that happen to have a common value for the given predictor variable. This is the only interpretation of “held fixed” that can be used in an observational study.

The notion of a “unique effect” is appealing when studying a complex system where multiple interrelated components influence the response variable. In some cases, it can literally be interpreted as the causal effect of an intervention that is linked to the value of a predictor variable. However, it has been argued that in many cases multiple regression analysis fails to clarify the relationships between the predictor variables and the response variable when the predictors are correlated with each other and are not assigned following a study design.^[5]

Estimation methods

Numerous procedures have been developed for parameter estimation and inference in linear regression. These methods differ in computational simplicity of algorithms, presence of a closed-form solution, robustness with respect to heavy-tailed distributions, and theoretical assumptions needed to validate desirable statistical properties such as consistency and asymptotic efficiency.

Some of the more common estimation techniques for linear regression are summarized below.

• Ordinary least squares (OLS) is the simplest and thus very common estimator. It is conceptually simple and computationally straightforward. OLS estimates are commonly used to analyze both experimental and observational data.
  The OLS method minimizes the sum of squared residuals, and leads to a closed-form expression for the estimated value of the unknown parameter β:

  

  The estimator is unbiased and consistent if the errors have finite variance and are uncorrelated with the regressors^[6]

  

  It is also efficient under the assumption that the errors have finite variance and are homoscedastic, meaning that E[ε_i²|x_i] does not depend on i. The condition that the errors are uncorrelated with the regressors will generally be satisfied in an experiment, but in the case of observational data, it is difficult to exclude the possibility of an omitted covariate z that is related to both the observed covariates and the response variable. The existence of such a covariate will generally lead to a correlation between the regressors and the response variable, and hence to an inconsistent estimator of β. The condition of homoscedasticity can fail with either experimental or observational data. If the goal is either inference or predictive modeling, the performance of OLS estimates can be poor if multicollinearity is present, unless the sample size is large.
  In simple linear regression, where there is only one regressor (with a constant), the OLS coefficient estimates have a simple form that is closely related to the correlation coefficient between the covariate and the response.
• Generalized least squares (GLS) is an extension of the OLS method, that allows efficient estimation of β when either heteroscedasticity, or correlations, or both are present among the error terms of the model, as long as the form of heteroscedasticity and correlation is known independently of the data. To handle heteroscedasticity when the error terms are uncorrelated with each other, GLS minimizes a weighted analogue to the sum of squared residuals from OLS regression, where the weight for the i^th case is inversely proportional to var(ε_i). This special case of GLS is called “weighted least squares”. The GLS solution to estimation problem is

  

  where Ω is the covariance matrix of the errors. GLS can be viewed as applying a linear transformation to the data so that the assumptions of OLS are met for the transformed data. For GLS to be applied, the covariance structure of the errors must be known up to a multiplicative constant.
• Iteratively reweighted least squares (IRLS) is used when heteroscedasticity, or correlations, or both are present among the error terms of the model, but where little is known about the covariance structure of the errors independently of the data^[7]. In the first iteration, OLS, or GLS with a provisional covariance structure is carried out, and the residuals are obtained from the fit. Based on the residuals, an improved estimate of the covariance structure of the errors can usually be obtained. A subsequent GLS iteration is then performed using this estimate of the error structure to define the weights. The process can be iterated to convergence, but in many cases, only one iteration is sufficient to achieve an efficient estimate of β.^[8][9]
• Instrumental variables regression (IV) can be performed when the regressors are correlated with the errors. In this case, we need the existence of some auxiliary instrumental variables z_i such that E[z_iε_i] = 0. If Z is the matrix of instruments, then the estimator can be given in closed form as

  

• Optimal instruments regression is an extension of classical IV regression to the situation where E[ε_i|z_i] = 0.
• Least absolute deviation (LAD) regression is a robust estimation technique in that it is less sensitive to the presence of outliers than OLS (but is less efficient than OLS when no outliers are present). It is equivalent to maximum likelihood estimation under a Laplace distribution model for ε^[10].
• Quantile regression focuses on the conditional quantiles of y given X rather than the conditional mean of y given X. Linear quantile regression models a particular conditional quantile, often the conditional median, as a linear function β′x of the predictors.
• Maximum likelihood estimation can be performed when the distribution of the error terms is known to belong to a certain parametric family ƒ_θ of probability distributions^[11]. When f_θ is a normal distribution with mean zero and variance θ, the resulting estimate is identical to the OLS estimate. GLS estimates are maximum likelihood estimates when ε follows a multivariate normal distribution with a known covariance matrix.
• Adaptive estimation. If we assume that error terms are independent from the regressors , the optimal estimator is the 2-step MLE, where the first step is used to non-parametrically estimate the distribution of the error term.^[12]
• Mixed models are widely used to analyze linear regression relationships involving dependent data when the dependencies have a known structure. Common applications of mixed models include analysis of data involving repeated measurements, such as longitudinal data, or data obtained from cluster sampling. They are generally fit as parametric models, using maximum likelihood or Bayesian estimation. In the case where the errors are modeled as normal random variables, there is a close connection between mixed models and generalized least squares^[13]. Fixed effects estimation is an alternative approach to analyzing this type of data.
• Principal component regression (PCR) ^[3][4] is used when the number of predictor variables is large, or when strong correlations exist among the predictor variables. This two-stage procedure first reduces the predictor variables using principal component analysis then uses the reduced variables in an OLS regression fit. While it often works well in practice, there is no general theoretical reason that the most informative linear function of the predictor variables should lie among the dominant principal components of the multivariate distribution of the predictor variables. The partial least squares regression is the extension of the PCR method which does not suffer from the mentioned deficiency.
• Total least squares (TLS) ^[14] is an approach to least squares estimation of the linear regression model that treats the covariates and response variable in a more geometrically symmetric manner than OLS. It is one approach to handling the "errors in variables" problem, and is sometimes used when the covariates are assumed to be error-free..
• Ridge regression^[15][16][17], and other forms of penalized estimation such as the Lasso^[1], deliberately introduce bias into the estimation of β in order to reduce the variability of the estimate. The resulting estimators generally have lower mean squared error than the OLS estimates, particularly when multicollinearity is present. They are generally used when the goal is to predict the value of the response variable y for values of the predictors x that have not yet been observed. These methods are not as commonly used when the goal is inference, since it is difficult to account for the bias.
• Least angle regression ^[2] is an estimation procedure for linear regression models that was developed to handle high-dimensional covariate vectors, potentially with more covariates than observations.
• Other robust estimation techniques, including the α-trimmed mean approach, and L-, M-, S-, and R-estimators have been introduced.

Extensions

• General linear model considers the situation when the response variable y is not a scalar but a vector. Conditional linearity of E(y|x) = Bx is still assumed, with a matrix B replacing the vector β of the classical linear regression model. Multivariate analogues of OLS and GLS have been developed.
• Generalized linear models are a framework for modeling a response variable y in the form g(β′x)  +  ε, where g is an arbitrary link function. Single index models allow some degree of nonlinearity in the relationship between x and y, while preserving the central role of the linear predictor β′x as in the classical linear regression model. Under certain conditions, simply applying OLS to data from a single-index model will consistently estimate β up to a proportionality constant ^[18].
• Hierarchical linear models (or multilevel regression) organizes the data into a hierarchy of regressions, for example where A is regressed on B, and B is regressed on C. It is often used where the data have a natural hierarchical structure such as in educational statistics, where students are nested in classrooms, classrooms are nested in schools, and schools are nested in some administrative grouping such as a school district. The response variable might be a measure of student achievement such as a test score, and different covariates would be collected at the classroom, school, and school district levels.
• Errors-in-variables models (or “measurement error models”) extend the traditional linear regression model to allow the predictor variables X to be observed with error. This error causes standard estimators of β to become biased. Generally, the form of bias is an attenuation, meaning that the effects are biased toward zero.
• In Dempster–Shafer theory, or a linear belief function in particular, a linear regression model may be represented as a partially swept matrix, which can be combined with similar matrices representing observations and other assumed normal distributions and state equations. The combination of swept or unswept matrices provides an alternative method for estimating linear regression models.

Applications of linear regression

Linear regression is widely used in biological, behavioral and social sciences to describe possible relationships between variables. It ranks as one of the most important tools used in these disciplines.

Trend line

For trend lines as used in technical analysis, see Trend lines (technical analysis)

A trend line represents a trend, the long-term movement in time series data after other components have been accounted for. It tells whether a particular data set (say GDP, oil prices or stock prices) have increased or decreased over the period of time. A trend line could simply be drawn by eye through a set of data points, but more properly their position and slope is calculated using statistical techniques like linear regression. Trend lines typically are straight lines, although some variations use higher degree polynomials depending on the degree of curvature desired in the line.

Trend lines are sometimes used in business analytics to show changes in data over time. This has the advantage of being simple. Trend lines are often used to argue that a particular action or event (such as training, or an advertising campaign) caused observed changes at a point in time. This is a simple technique, and does not require a control group, experimental design, or a sophisticated analysis technique. However, it suffers from a lack of scientific validity in cases where other potential changes can affect the data.

Epidemiology

Early evidence relating tobacco smoking to mortality and morbidity came from observational studies employing regression analysis. In order to reduce spurious correlations when analyzing observational data, researchers usually include several variables in their regression models in addition to the variable of primary interest. For example, suppose we have a regression model in which cigarette smoking is the independent variable of interest, and the dependent variable is lifespan measured in years. Researchers might include socio-economic status as an additional dependent variable, to ensure that any observed effect of smoking on lifespan is not due to some effect of education or income. However, it is never possible to include all possible confounding variables in an empirical analysis. For example, a hypothetical gene might increase mortality and also cause people to smoke more. For this reason, randomized controlled trials are often able to generate more compelling evidence of causal relationships than can be obtained using regression analyses of observational data. When controlled experiments are not feasible, variants of regression analysis such as instrumental variables regression may be used to attempt to estimate causal relationships from observational data.

Finance

The capital asset pricing model uses linear regression as well as the concept of Beta for analyzing and quantifying the systematic risk of an investment. This comes directly from the Beta coefficient of the linear regression model that relates the return on the investment to the return on all risky assets.

Regression may not be the appropriate way to estimate beta in finance given that it is supposed to provide the volatility of an investment relative to the volatility of the market as a whole. This would require that both these variables be treated in the same way when estimating the slope. Whereas regression treats all variability as being in the investment returns variable, i.e. it only considers residuals in the dependent variable.^[19]

Environmental science

Linear regression finds application in a wide range of environmental science applications. In Canada, the Environmental Effects Monitoring Program uses statistical analyses on fish and benthic surveys to measure the effects of pulp mill or metal mine effluent on the aquatic ecosystem.^[20]

See also

• ANOVA, or analysis of variance, is historically a precursor to the development of linear models. Here the model parameters themselves are not computed, but X column contributions and their significance are identified using the ratios of within-group variances to the error variance and applying the F test.
• Anscombe's quartet
• Cross-sectional regression
• Curve fitting
• Empirical Bayes methods
• M-estimator
• Nonlinear regression
• Nonparametric regression
• Multivariate adaptive regression splines
• Lack-of-fit sum of squares
• Truncated regression model
• Censored regression model
• Simple linear regression
• Segmented linear regression

Further reading

• Pedhazur, Elazar J (1982). Multiple regression in behavioral research: Explanation and prediction (2nd ed.). New York: Holt, Rinehart and Winston. ISBN 0-03-041760-0 

Notes

References

• Cohen, J., Cohen P., West, S.G., & Aiken, L.S. (2003). Applied multiple regression/correlation analysis for the behavioral sciences. (2nd ed.) Hillsdale, NJ: Lawrence Erlbaum Associates
• Charles Darwin. The Variation of Animals and Plants under Domestication. (1868) (Chapter XIII describes what was known about reversion in Galton's time. Darwin uses the term "reversion".)
• Draper, N.R. and Smith, H. Applied Regression Analysis Wiley Series in Probability and Statistics (1998)
• Francis Galton. "Regression Towards Mediocrity in Hereditary Stature," Journal of the Anthropological Institute, 15:246-263 (1886). (Facsimile at: [1])
• Robert S. Pindyck and Daniel L. Rubinfeld (1998, 4h ed.). Econometric Models and Economic Forecasts,, ch. 1 (Intro, incl. appendices on Σ operators & derivation of parameter est.) & Appendix 4.3 (mult. regression in matrix form).
• Kaw, Autar; Kalu, Egwu (2008). "Numerical Methods with Applications". [2]. , Chapter 6 deals with linear and non-linear regression.

External links

• Open source implementations
  • ALGLIB has an implementation of Ordinary Least Squares in C++ / C# / Pascal / VBA.
  • GSL has an implementation of Ordinary Least Squares regression algorithm (C/C++).
• Online tutorials and simulators
  • Online regression by eye (simulation).
• Papers and articles
  • Restricted regression - Lecture in the Department of Statistics, University of Udine
  • Linear Regression - Notes, PPT, Videos, Mathcad, Matlab, Mathematica, Maple at Numerical Methods for STEM undergraduate