Regression analysis
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Regression analysis 

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Background 

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In
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.^{[2]}^{[3]}
History
The earliest form of regression was the
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).^{[7]}^{[8]} For Galton, regression had only this biological meaning,^{}
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 (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 .
 The independent variables, which are observed in data and are often denoted as a vector (where denotes a row of data).
 The dependent variable, which are observed in data and often denoted using the scalar .
 The error terms, which are not directly observed in data and are often denoted using the scalar .
In various fields of application, different terminologies are used in place of dependent and independent variables.
Most regression models propose that is a function (regression function) of and , with representing an additive error term that may stand in for unmodeled determinants of or random statistical noise:
Note that the independent variables are assumed to be free of error. This important assumption is often overlooked, although errorsinvariables models can be used when the independent variables are assumed to contain errors.
The researchers' goal is to estimate the function that most closely fits the data. To carry out regression analysis, the form of the function must be specified. Sometimes the form of this function is based on knowledge about the relationship between and that does not rely on the data. If no such knowledge is available, a flexible or convenient form for is chosen. For example, a simple univariate regression may propose , suggesting that the researcher believes 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 . For example, least squares (including its most common variant, ordinary least squares) finds the value of that minimizes the sum of squared errors . A given regression method will ultimately provide an estimate of , usually denoted 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 for prediction or to assess the accuracy of the model in explaining the data. Whether the researcher is intrinsically interested in the estimate or the predicted value will depend on context and their goals. As described in ordinary least squares, least squares is widely used because the estimated function approximates the conditional expectation .^{[5]} However, alternative variants (e.g., least absolute deviations or quantile regression) are useful when researchers want to model other functions .
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 rows of data with one dependent and two independent variables: . Suppose further that the researcher wants to estimate a bivariate linear model via least squares: . If the researcher only has access to data points, then they could find infinitely many combinations that explain the data equally well: any combination can be chosen that satisfies , all of which lead to 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 equations is to be solved for 3 unknowns, which makes the system underdetermined. Alternatively, one can visualize infinitely many 3dimensional planes that go through fixed points.
More generally, to estimate a least squares model with distinct parameters, one must have distinct data points. If , then there does not generally exist a set of parameters that will perfectly fit the data. The quantity 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 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 is an invertible matrix and therefore that a unique solution exists.
Underlying assumptions
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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 realworld 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:
 The variance of the residuals is constant across observations (homoscedasticity).
 The residuals are uncorrelated with one another. Mathematically, the variance–covariance matrix of the errors is diagonal.
A handful of conditions are sufficient for the leastsquares estimator to possess desirable properties: in particular, the
Linear regression
In linear regression, the model specification is that the dependent variable, is a linear combination of the parameters (but need not be linear in the independent variables). For example, in simple linear regression for modeling data points there is one independent variable: , and two parameters, and :
 straight line:
In multiple linear regression, there are several independent variables or functions of independent variables.
Adding a term in to the preceding regression gives:
 parabola:
This is still linear regression; although the expression on the right hand side is quadratic in the independent variable , it is linear in the parameters , and
In both cases, is an error term and the subscript 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:
The
Minimization of this function results in a set of
In the case of simple regression, the formulas for the least squares estimates are
where is the mean (average) of the values and is the mean of the values.
Under the assumption that the population error term has a constant variance, the estimate of that variance is given by:
This is called the
The
Under the further assumption that the population error term is normally distributed, the researcher can use these estimated standard errors to create
General linear model
In the more general multiple regression model, there are independent variables:
where is the th observation on the th independent variable. If the first independent variable takes the value 1 for all , , then is called the
The least squares parameter estimates are obtained from normal equations. The residual can be written as
The normal equations are
In matrix notation, the normal equations are written as
where the element of is , the element of the column vector is , and the element of is . Thus is , is , and is . The solution is
Diagnostics
Once a regression model has been constructed, it may be important to confirm the
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
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 noncontinuous ("limited" to lie on some subset of the real line). For binary (zero or one) variables, if analysis proceeds with leastsquares linear regression, the model is called the
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 nonlinear least squares.
Prediction (interpolation and extrapolation)
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 modelfitting 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.^{[21]}
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 , where is the sample size, is the number of independent variables and is the number of observations needed to reach the desired precision if the model had only one independent variable.^{[22]} For example, a researcher is building a linear regression model using a dataset that contains 1000 patients (). If the researcher decides that five observations are needed to precisely define a straight line (), then the maximum number of independent variables the model can support is 4, because
 .
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.^{}[23]
 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.^{[24]}
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.
See also
 Anscombe's quartet
 Curve fitting
 Estimation theory
 Forecasting
 Fraction of variance unexplained
 Function approximation
 Generalized linear model
 Kriging (a linear least squares estimation algorithm)
 Local regression
 Modifiable areal unit problem
 Multivariate adaptive regression spline
 Multivariate normal distribution
 Pearson correlation coefficient
 Quasivariance
 Prediction interval
 Regression validation
 Robust regression
 Segmented regression
 Signal processing
 Stepwise regression
 Taxicab geometry
 Linear trend estimation
References
 ^ Necessary Condition Analysis
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 ^ R. Dennis Cook; Sanford Weisberg Criticism and Influence Analysis in Regression, Sociological Methodology, Vol. 13. (1982), pp. 313–361
 ^ A.M. Legendre. Nouvelles méthodes pour la détermination des orbites des comètes, Firmin Didot, Paris, 1805. “Sur la Méthode des moindres quarrés” appears as an appendix.
 ^ ^{a} ^{b} Chapter 1 of: Angrist, J. D., & Pischke, J. S. (2008). Mostly Harmless Econometrics: An Empiricist's Companion. Princeton University Press.
 ^ C.F. Gauss. Theoria combinationis observationum erroribus minimis obnoxiae. (1821/1823)
 ^
Mogull, Robert G. (2004). SecondSemester Applied Statistics. Kendall/Hunt Publishing Company. p. 59. ISBN 9780757511813.
 .
 ^ Francis Galton. "Typical laws of heredity", Nature 15 (1877), 492–495, 512–514, 532–533. (Galton uses the term "reversion" in this paper, which discusses the size of peas.)
 ^ Francis Galton. Presidential address, Section H, Anthropology. (1885) (Galton uses the term "regression" in this paper, which discusses the height of humans.)
 JSTOR 2979746.
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 ^ Rodney Ramcharan. Regressions: Why Are Economists Obessessed with Them? March 2006. Accessed 20111203.
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 McGraw Hill, 1960, page 288.
 ^ Rouaud, Mathieu (2013). Probability, Statistics and Estimation (PDF). p. 60.
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 ^ YangJing Long (2009). "Human age estimation by metric learning for regression problems" (PDF). Proc. International Conference on Computer Analysis of Images and Patterns: 74–82. Archived from the original (PDF) on 20100108.
Further reading
 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.
 New Palgrave: A Dictionary of Economics, v. 4, pp. 120–23.
 Birkes, David and ISBN 0471568813
 Chatfield, C. (1993) "Calculating Interval Forecasts," Journal of Business and Economic Statistics, 11. pp. 121–135.
 Draper, N.R.; Smith, H. (1998). Applied Regression Analysis (3rd ed.). John Wiley. ISBN 9780471170822.
 Fox, J. (1997). Applied Regression Analysis, Linear Models and Related Methods. Sage
 Hardle, W., Applied Nonparametric Regression (1990),
 Meade, Nigel; Islam, Towhidul (1995). "Prediction intervals for growth curve forecasts". Journal of Forecasting. 14 (5): 413–430. .
 A. Sen, M. Srivastava, Regression Analysis — Theory, Methods, and Applications, SpringerVerlag, Berlin, 2011 (4th printing).
 T. Strutz: Data Fitting and Uncertainty (A practical introduction to weighted least squares and beyond). Vieweg+Teubner, .
 Stulp, Freek, and Olivier Sigaud. Many Regression Algorithms, One Unified Model: A Review. Neural Networks, vol. 69, Sept. 2015, pp. 60–79. https://doi.org/10.1016/j.neunet.2015.05.005.
 Malakooti, B. (2013). Operations and Production Systems with Multiple Objectives. John Wiley & Sons.
 Chicco, Davide; Warrens, Matthijs J.; Jurman, Giuseppe (2021). "The coefficient of determination Rsquared is more informative than SMAPE, MAE, MAPE, MSE and RMSE in regression analysis evaluation". PeerJ Computer Science. 7 (e623): e623. PMID 34307865.
External links
 "Regression analysis", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
 Earliest Uses: Regression – basic history and references
 What is multiple regression used for? – Multiple regression
 Regression of Weakly Correlated Data – how linear regression mistakes can appear when Yrange is much smaller than Xrange