In statistics, regression analysis includes many techniques for modeling and analyzing several variables, when the focus is on the relationship between a dependent variable and one or more independent variables. More specifically, regression analysis helps one understand how the typical value of the dependent variable changes when any one of the independent variables is varied, while the other independent variables are held fixed. Most commonly, regression analysis estimates the conditional expectation of the dependent variable given the independent variables — that is, the average value of the dependent variable when the independent variables are held fixed. Less commonly, the focus is on a quantile, or other location parameter of the conditional distribution of the dependent variable given the independent variables. In all cases, the estimation target is a function of the independent variables called the regression function. In regression analysis, it is also of interest to characterize the variation of the dependent variable around the regression function, which can be described by a probability distribution.

Regression analysis is widely used for prediction and forecasting, where its use has substantial overlap with the field of machine learning. Regression analysis is also used to understand which among the independent variables are related to the dependent variable, and to explore the forms of these relationships. In restricted circumstances, regression analysis can be used to infer causal relationships between the independent and dependent variables. However this can lead to illusions or false relationships, so caution is advisable:^[1] see correlation does not imply causation.

A large body of techniques for carrying out regression analysis has been developed. Familiar methods such as linear regression and ordinary least squares regression are parametric, in that the regression function is defined in terms of a finite number of unknown parameters that are estimated from the data. Nonparametric regression refers to techniques that allow the regression function to lie in a specified set of functions, which may be infinite-dimensional.

The performance of regression analysis methods in practice depends on the form of the data generating process, and how it relates to the regression approach being used. Since the true form of the data-generating process is generally not known, regression analysis often depends to some extent on making assumptions about this process. These assumptions are sometimes testable if a large amount of data is available. Regression models for prediction are often useful even when the assumptions are moderately violated, although they may not perform optimally. However, in many applications, especially with small effects or questions of causality based on observational data, regression methods give misleading results.^[2][3]

History[link]

The earliest form of regression was the method of least squares (French: méthode des moindres carrés), which was published by Legendre in 1805,^[4] and by Gauss in 1809.^[5] Legendre and Gauss both applied the method to the problem of determining, from astronomical observations, the orbits of bodies about the Sun. Gauss published a further development of the theory of least squares in 1821,^[6] including a version of the Gauss–Markov theorem.

The term "regression" was coined by Francis Galton in the nineteenth 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).^[7][8] For Galton, regression had only this biological meaning,^[9][10] but his work was later extended by Udny Yule and Karl Pearson to a more general statistical context.^[11][12] 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.^[13][14][15] 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.^[16]

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 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 models[link]

Regression models involve the following variables:

• The unknown parameters, denoted as β, which may represent a scalar or a vector.
• The independent variables, X.
• The dependent variable, Y.

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

A regression model relates Y to a function of X and β.

Failed to parse (Missing texvc executable; please see math/README to configure.): Y \approx f (\mathbf {X}, \boldsymbol{\beta} )

The approximation is usually formalized as E(Y | X) = f(X, β). To carry out regression analysis, the form of the function f must be specified. Sometimes the form of this function is based on knowledge about the relationship between Y and X that does not rely on the data. If no such knowledge is available, a flexible or convenient form for f is chosen.

Assume now that the vector of unknown parameters β is of length k. In order to perform a regression analysis the user must provide information about the dependent variable Y:

• If N data points of the form (Y,X) are observed, where N < k, most classical approaches to regression analysis cannot be performed: since the system of equations defining the regression model is underdetermined, there is not enough data to recover β.
• If exactly N = k data points are observed, and the function f is linear, the equations Y = f(X, β) can be solved exactly rather than approximately. This reduces to solving a set of N equations with N unknowns (the elements of β), which has a unique solution as long as the X are linearly independent. If f is nonlinear, a solution may not exist, or many solutions may exist.
• The most common situation is where N > k data points are observed. In this case, there is enough information in the data to estimate a unique value for β that best fits the data in some sense, and the regression model when applied to the data can be viewed as an overdetermined system in β.

In the last case, the regression analysis provides the tools for:

1. Finding a solution for unknown parameters β that will, for example, minimize the distance between the measured and predicted values of the dependent variable Y (also known as method of least squares).
2. Under certain statistical assumptions, the regression analysis uses the surplus of information to provide statistical information about the unknown parameters β and predicted values of the dependent variable Y.

Necessary number of independent measurements[link]

Consider a regression model which has three unknown parameters, β₀, β₁, and β₂. Suppose an experimenter performs 10 measurements all at exactly the same value of independent variable vector X (which contains the independent variables X₁, X₂, and X₃). In this case, regression analysis fails to give a unique set of estimated values for the three unknown parameters; the experimenter did not provide enough information. The best one can do is to estimate the average value and the standard deviation of the dependent variable Y. Similarly, measuring at two different values of X would give enough data for a regression with two unknowns, but not for three or more unknowns.

If the experimenter had performed measurements at three different values of the independent variable vector X, then regression analysis would provide a unique set of estimates for the three unknown parameters in β.

In the case of general linear regression, the above statement is equivalent to the requirement that matrix X^TX is invertible.

Statistical assumptions[link]

When the number of measurements, N, is larger than the number of unknown parameters, k, and the measurement errors ε_i are normally distributed then the excess of information contained in (N - k) measurements is used to make statistical predictions about the unknown parameters. This excess of information is referred to as the degrees of freedom of the regression.

Underlying assumptions[link]

Classical assumptions for regression analysis include:

• The sample is representative of the population for the inference prediction.
• The error is a random variable with a mean of zero conditional on the explanatory variables.
• The independent variables are measured with no error. (Note: If this is not so, modeling may be done instead using errors-in-variables model techniques).
• The predictors are linearly independent, i.e. it is not possible to express any predictor as a linear combination of the others. See Multicollinearity.
• The errors are uncorrelated, that is, the variance-covariance matrix of the errors is diagonal and each non-zero element is the variance of the error.
• The variance of the error is constant across observations (homoscedasticity). (Note: If not, weighted least squares or other methods might instead be used).

These are sufficient conditions for the least-squares estimator to possess desirable properties, in particular, these assumptions imply that the parameter estimates will be unbiased, consistent, and efficient in the class of linear unbiased estimators. It is important to note that actual data rarely satisfies the assumptions. That is, the method is used even though the assumptions are not true. Variation from the assumptions can sometimes be used as a measure of how far the model is from being useful. Many of these assumptions may be relaxed in more advanced treatments. Reports of statistical analyses usually include analyses of tests on the sample data and methodology for the fit and usefulness of the model.

Assumptions include the geometrical support of the variables^{[clarification needed]} (Cressie, 1996). Independent and dependent variables often refer to values measured at point locations. There may be spatial trends and spatial autocorrelation in the variables that violates statistical assumptions of regression. Geographic weighted regression is one technique to deal with such data (Fotheringham et al., 2002). Also, variables may include values aggregated by areas. With aggregated data the Modifiable Areal Unit Problem can cause extreme variation in regression parameters (Fotheringham and Wong, 1991). When analyzing data aggregated by political boundaries, postal codes or census areas results may be very different with a different choice of units.

Linear regression[link]

In linear regression, the model specification is that the dependent variable, Failed to parse (Missing texvc executable; please see math/README to configure.): 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 Failed to parse (Missing texvc executable; please see math/README to configure.):  n 
data points there is one independent variable: Failed to parse (Missing texvc executable; please see math/README to configure.):  x_i 

, and two parameters, Failed to parse (Missing texvc executable; please see math/README to configure.): \beta_0

and Failed to parse (Missing texvc executable; please see math/README to configure.): \beta_1

straight line: Failed to parse (Missing texvc executable; please see math/README to configure.): y_i=\beta_0 +\beta_1 x_i +\varepsilon_i,\quad i=1,\dots,n.\!

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

Adding a term in x_i² to the preceding regression gives:

parabola: Failed to parse (Missing texvc executable; please see math/README to configure.): y_i=\beta_0 +\beta_1 x_i +\beta_2 x_i^2+\varepsilon_i,\ i=1,\dots,n.\!

This is still linear regression; although the expression on the right hand side is quadratic in the independent variable Failed to parse (Missing texvc executable; please see math/README to configure.): x_i , it is linear in the parameters Failed to parse (Missing texvc executable; please see math/README to configure.): \beta_0 , Failed to parse (Missing texvc executable; please see math/README to configure.): \beta_1

and Failed to parse (Missing texvc executable; please see math/README to configure.): \beta_2.

In both cases, Failed to parse (Missing texvc executable; please see math/README to configure.): \varepsilon_i

is an error term and the subscript Failed to parse (Missing texvc executable; please see math/README to configure.): i
indexes a particular observation. Given a random sample from the population, we estimate the population parameters and obtain the sample linear regression model:

Failed to parse (Missing texvc executable; please see math/README to configure.): \widehat{y_i} = \widehat{\beta}_0 + \widehat{\beta}_1 x_i.

The residual, Failed to parse (Missing texvc executable; please see math/README to configure.): e_i = y_i - \widehat{y}_i , is the difference between the value of the dependent variable predicted by the model, Failed to parse (Missing texvc executable; please see math/README to configure.): \widehat{y_i}

and the true value of the dependent variable Failed to parse (Missing texvc executable; please see math/README to configure.): y_i

. One method of estimation is ordinary least squares. This method obtains parameter estimates that minimize the sum of squared residuals, SSE,^[17][18] also sometimes denoted RSS:

Failed to parse (Missing texvc executable; please see math/README to configure.): SSE=\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, Failed to parse (Missing texvc executable; please see math/README to configure.): \widehat{\beta}_0, \widehat{\beta}_1 .

Illustration of linear regression on a data set.

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

Failed to parse (Missing texvc executable; please see math/README to configure.): \widehat{\beta_1}=\frac{\sum(x_i-\bar{x})(y_i-\bar{y})}{\sum(x_i-\bar{x})^2}\text{ and }\hat{\beta_0}=\bar{y}-\widehat{\beta_1}\bar{x}

where Failed to parse (Missing texvc executable; please see math/README to configure.): \bar{x}

is the mean (average) of the Failed to parse (Missing texvc executable; please see math/README to configure.): x
values and Failed to parse (Missing texvc executable; please see math/README to configure.): \bar{y}
is the mean of the Failed to parse (Missing texvc executable; please see math/README to configure.): y
values. See simple linear regression for a derivation of these formulas and a numerical example.

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

Failed to parse (Missing texvc executable; please see math/README to configure.): \hat{\sigma}^2_\varepsilon = \frac{SSE}{N-2}.\,

This is called the mean square error (MSE) of the regression. The standard errors of the parameter estimates are given by

Failed to parse (Missing texvc executable; please see math/README to configure.): \hat\sigma_{\beta_0}=\hat\sigma_{\varepsilon} \sqrt{\frac{1}{N} + \frac{\bar{x}^2}{\sum(x_i-\bar x)^2}}

Failed to parse (Missing texvc executable; please see math/README to configure.): \hat\sigma_{\beta_1}=\hat\sigma_{\varepsilon} \sqrt{\frac{1}{\sum(x_i-\bar x)^2}}.

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[link]

In the more general multiple regression model, there are p independent variables:

Failed to parse (Missing texvc executable; please see math/README to configure.): y_i = \beta_1 x_{i1} + \beta_2 x_{i2} + \cdots + \beta_p x_{ip} + \varepsilon_i, \,

where x_ij is the i^th observation on the j^th independent variable, and where the first independent variable takes the value 1 for all i (so Failed to parse (Missing texvc executable; please see math/README to configure.): \beta_1

is the regression intercept).

The least squares parameter estimates are obtained from p normal equations. The residual can be written as

Failed to parse (Missing texvc executable; please see math/README to configure.): e_i=y_i - \hat\beta_1 x_{i1} - \cdots - \hat\beta_p x_{ip}.

The normal equations are

Failed to parse (Missing texvc executable; please see math/README to configure.): \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

Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{(X^\top X )\hat{\boldsymbol{\beta}}= {}X^\top Y},\,

where the ij element of X is x_ij, the i element of the column vector Y is y_i, and the j element of Failed to parse (Missing texvc executable; please see math/README to configure.): \hat \beta

is Failed to parse (Missing texvc executable; please see math/README to configure.): \hat \beta_j

. Thus X is n×p, Y is n×1, and Failed to parse (Missing texvc executable; please see math/README to configure.): \hat \beta

is p×1. The solution is

Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{\hat{\boldsymbol{\beta}}= {}(X^\top X )^{-1}X^\top Y}.\,

For a derivation, see linear least squares, and for a numerical example, see linear regression (example).

Regression diagnostics[link]

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 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.

Regression with "limited dependent" variables[link]

The phrase "limited dependent" is used in econometric statistics for categorical and constrained variables.

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 instead.

Interpolation and extrapolation[link]

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.

It is generally advised^{[citation needed]} that when performing extrapolation, one should accompany the estimated value of the dependent variable with a prediction interval that represents the uncertainty. 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.^[19]

However, this does not cover the full set of modelling errors that may be being made: in particular, the assumption of a particular form for the relation between Y and X. 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. Best-practice advice here^{[citation needed]} is that a linear-in-variables and linear-in-parameters relationship should not be chosen simply for computational convenience, but that all available knowledge should be deployed in constructing a regression model. 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).

Nonlinear regression[link]

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

Power and sample size calculations[link]

There are no generally agreed methods for relating the number of observations versus the number of independent variables in the model. One rule of thumb suggested by Good and Hardin is Failed to parse (Missing texvc executable; please see math/README to configure.): N=m^n , where Failed to parse (Missing texvc executable; please see math/README to configure.): N

is the sample size, Failed to parse (Missing texvc executable; please see math/README to configure.): n
is the number of independent variables and Failed to parse (Missing texvc executable; please see math/README to configure.): m
is the number of observations needed to reach the desired precision if the model had only one independent variable.[20] For example, a researcher is building a linear regression model using a dataset that contains 1000 patients (Failed to parse (Missing texvc executable; please see math/README to configure.): N

). If he decides that five observations are needed to precisely define a straight line (Failed to parse (Missing texvc executable; please see math/README to configure.): m ), then the maximum number of independent variables his model can support is 4, because

Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{\log{1000}}{\log{5}}=4.29 .

Other methods[link]

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.^[21]
• 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
• Distance metric learning, which is learned by the search of a meaningful distance metric in a given input space.^[22]

Software[link]

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.

See also[link]

Statistics portal

References[link]

Further reading[link]

• William H. Kruskal and Judith M. Tanur, ed. (1978), "Linear Hypotheses," International Encyclopedia of Statistics. Free Press, v. 1,

Evan J. Williams, "I. Regression," pp. 523–41.

Julian C. Stanley, "II. Analysis of Variance," pp. 541–554.

• Lindley, D.V. (1987). "Regression and correlation analysis," New Palgrave: A Dictionary of Economics, v. 4, pp. 120–23.
• Birkes, David and Dodge, Y., Alternative Methods of Regression. ISBN 0-471-56881-3
• Chatfield, C. (1993) "Calculating Interval Forecasts," Journal of Business and Economic Statistics, 11. pp. 121–135.
• Draper, N.R. and Smith, H. (1998).Applied Regression Analysis Wiley Series in Probability and Statistics
• Fox, J. (1997). Applied Regression Analysis, Linear Models and Related Methods. Sage
• Hardle, W., Applied Nonparametric Regression (1990), ISBN 0-521-42950-1
• Meade, N. and T. Islam (1995) "Prediction Intervals for Growth Curve Forecasts" Journal of Forecasting, 14, pp. 413–430.
• N. Cressie (1996) Change of Support and the Modiable Areal Unit Problem. Geographical Systems 3:159–180.
• A.S. Fotheringham, C. Brunsdon, and M. Charlton. (2002) Geographically weighted regression: the analysis of spatially varying relationships. Wiley.
• T. Strutz: Data Fitting and Uncertainty (A practical introduction to weighted least squares and beyond). Vieweg+Teubner, ISBN 978-3-8348-1022-9.

External links[link]

Wikimedia Commons has media related to: Regression analysis

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