Regression Analysis
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In statistical modeling, regression analysis is a set of statistical processes for
estimating Estimation (or estimating) is the process of finding an estimate or approximation, which is a value that is usable for some purpose even if input data may be incomplete, uncertain, or unstable. The value is nonetheless usable because it is der ...
the relationships between a dependent variable (often called the 'outcome' or 'response' variable, or a 'label' in machine learning parlance) and one or more independent variables (often called '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 In statistics, ordinary least squares (OLS) is a type of linear least squares method for choosing the unknown parameters in a linear regression model (with fixed level-one effects of a linear function of a set of explanatory variables) by the ...
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 In probability theory, the conditional expectation, conditional expected value, or conditional mean of a random variable is its expected value – the value it would take “on average” over an arbitrarily large number of occurrences – give ...
(or population
average value In colloquial, ordinary language, an average is a single number taken as representative of a list of numbers, usually the sum of the numbers divided by how many numbers are in the list (the arithmetic mean). For example, the average of the numbers ...
) 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 Quantile regression is a type of regression analysis used in statistics and econometrics. Whereas the method of least squares estimates the conditional ''mean'' of the response variable across values of the predictor variables, quantile regress ...
or Necessary Condition Analysis) or estimate the conditional expectation across a broader collection of non-linear models (e.g.,
nonparametric regression Nonparametric regression is a category of regression analysis in which the predictor does not take a predetermined form but is constructed according to information derived from the data. That is, no parametric form is assumed for the relationship ...
). Regression analysis is primarily used for two conceptually distinct purposes. First, regression analysis is widely used for prediction and
forecasting Forecasting is the process of making predictions based on past and present data. Later these can be compared (resolved) against what happens. For example, a company might estimate their revenue in the next year, then compare it against the actual ...
, where its use has substantial overlap with the field of
machine learning Machine learning (ML) is a field of inquiry devoted to understanding and building methods that 'learn', that is, methods that leverage data to improve performance on some set of tasks. It is seen as a part of artificial intelligence. Machine ...
. Second, in some situations regression analysis can be used to infer
causal relationships Causality (also referred to as causation, or cause and effect) is influence by which one event, process, state, or object (''a'' ''cause'') contributes to the production of another event, process, state, or object (an ''effect'') where the cau ...
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 In fields such as epidemiology, social sciences, psychology and statistics, an observational study draws inferences from a sample to a population where the independent variable is not under the control of the researcher because of ethical concern ...
.


History

The earliest form of regression was the
method of least squares The method of least squares is a standard approach in regression analysis to approximate the solution of overdetermined systems (sets of equations in which there are more equations than unknowns) by minimizing the sum of the squares of the res ...
, which was published by Legendre in 1805, 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.
and by
Gauss Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes refer ...
in 1809.Chapter 1 of: Angrist, J. D., & Pischke, J. S. (2008). ''Mostly Harmless Econometrics: An Empiricist's Companion''. Princeton University Press. 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,C.F. Gauss
''Theoria combinationis observationum erroribus minimis obnoxiae''
(1821/1823)
including a version of the
Gauss–Markov theorem In statistics, the Gauss–Markov theorem (or simply Gauss theorem for some authors) states that the ordinary least squares (OLS) estimator has the lowest sampling variance within the class of linear unbiased estimators, if the errors in 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). For Galton, regression had only this biological meaning, but his work was later extended by
Udny Yule George Udny Yule FRS (18 February 1871 – 26 June 1951), usually known as Udny Yule, was a British statistician, particularly known for the Yule distribution. Personal life Yule was born at Beech Hill, a house in Morham near Haddingto ...
and Karl Pearson to a more general statistical context. In the work of Yule and Pearson, the
joint distribution Given two random variables that are defined on the same probability space, the joint probability distribution is the corresponding probability distribution on all possible pairs of outputs. The joint distribution can just as well be considered ...
of the response and explanatory variables is assumed to be
Gaussian Carl Friedrich Gauss (1777–1855) is the eponym of all of the topics listed below. There are over 100 topics all named after this German mathematician and scientist, all in the fields of mathematics, physics, and astronomy. The English eponymo ...
. This assumption was weakened by
R.A. Fisher Sir Ronald Aylmer Fisher (17 February 1890 – 29 July 1962) was a British polymath who was active as a mathematician, statistician, biologist, geneticist, and academic. For his work in statistics, he has been described as "a genius who ...
in his works of 1922 and 1925. Fisher assumed that the
conditional distribution In probability theory and statistics, given two jointly distributed random variables X and Y, the conditional probability distribution of Y given X is the probability distribution of Y when X is known to be a particular value; in some cases the ...
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 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 involving correlated responses such as
time series In mathematics, a time series is a series of data points indexed (or listed or graphed) in time order. Most commonly, a time series is a sequence taken at successive equally spaced points in time. Thus it is a sequence of discrete-time data. Exa ...
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 Nonparametric regression is a category of regression analysis in which the predictor does not take a predetermined form but is constructed according to information derived from the data. That is, no parametric form is assumed for the relationship ...
,
Bayesian Thomas Bayes (/beɪz/; c. 1701 – 1761) was an English statistician, philosopher, and Presbyterian minister. Bayesian () refers either to a range of concepts and approaches that relate to statistical methods based on Bayes' theorem, or a followe ...
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 In statistics, ordinary least squares (OLS) is a type of linear least squares method for choosing the unknown parameters in a linear regression model (with fixed level-one effects of a linear function of a set of explanatory variables) by the ...
) to estimate the parameters of that model. Regression models involve the following components: *The unknown parameters, often denoted as a
scalar Scalar may refer to: *Scalar (mathematics), an element of a field, which is used to define a vector space, usually the field of real numbers * Scalar (physics), a physical quantity that can be described by a single element of a number field such ...
or
vector Vector most often refers to: *Euclidean vector, a quantity with a magnitude and a direction *Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism Vector may also refer to: Mathematic ...
\beta. *The independent variables, which are observed in data and are often denoted as a vector X_i (where i denotes a row of data). *The dependent variable, which are observed in data and often denoted using the scalar Y_i. *The error terms, which are ''not'' directly observed in data and are often denoted using the scalar e_i. In various fields of application, different terminologies are used in place of dependent and independent variables. Most regression models propose that Y_i is a function of X_i and \beta, with e_i representing an additive error term that may stand in for un-modeled determinants of Y_i or random statistical noise: :Y_i = f (X_i, \beta) + e_i The researchers' goal is to estimate the function f(X_i, \beta) that most closely fits the data. 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_i and X_i that does not rely on the data. If no such knowledge is available, a flexible or convenient form for f is chosen. For example, a simple univariate regression may propose f(X_i, \beta) = \beta_0 + \beta_1 X_i, suggesting that the researcher believes 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 \beta . For example, least squares (including its most common variant,
ordinary least squares In statistics, ordinary least squares (OLS) is a type of linear least squares method for choosing the unknown parameters in a linear regression model (with fixed level-one effects of a linear function of a set of explanatory variables) by the ...
) finds the value of \beta that minimizes the sum of squared errors \sum_i (Y_i - f(X_i, \beta))^2. A given regression method will ultimately provide an estimate of \beta, usually denoted \hat 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'' \hat = f(X_i,\hat) for prediction or to assess the accuracy of the model in explaining the data. Whether the researcher is intrinsically interested in the estimate \hat or the predicted value \hat will depend on context and their goals. As described in
ordinary least squares In statistics, ordinary least squares (OLS) is a type of linear least squares method for choosing the unknown parameters in a linear regression model (with fixed level-one effects of a linear function of a set of explanatory variables) by the ...
, least squares is widely used because the estimated function f(X_i, \hat) approximates the
conditional expectation In probability theory, the conditional expectation, conditional expected value, or conditional mean of a random variable is its expected value – the value it would take “on average” over an arbitrarily large number of occurrences – give ...
E(Y_i, X_i). However, alternative variants (e.g.,
least absolute deviations Least absolute deviations (LAD), also known as least absolute errors (LAE), least absolute residuals (LAR), or least absolute values (LAV), is a statistical optimality criterion and a statistical optimization technique based minimizing the ''sum o ...
or
quantile regression Quantile regression is a type of regression analysis used in statistics and econometrics. Whereas the method of least squares estimates the conditional ''mean'' of the response variable across values of the predictor variables, quantile regress ...
) are useful when researchers want to model other functions 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 N rows of data with one dependent and two independent variables: (Y_i, X_, X_). Suppose further that the researcher wants to estimate a bivariate linear model via least squares: Y_i = \beta_0 + \beta_1 X_ + \beta_2 X_ + e_i. If the researcher only has access to N=2 data points, then they could find infinitely many combinations (\hat_0, \hat_1, \hat_2) that explain the data equally well: any combination can be chosen that satisfies \hat_i = \hat_0 + \hat_1 X_ + \hat_2 X_, all of which lead to \sum_i \hat_i^2 = \sum_i (\hat_i - (\hat_0 + \hat_1 X_ + \hat_2 X_))^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 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 N=2 fixed points. More generally, to estimate a least squares model with k distinct parameters, one must have N \geq k distinct data points. If N > k, then there does not generally exist a set of parameters that will perfectly fit the data. The quantity k-N 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 (X_, X_, ..., X_) must be
linearly independent In the theory of vector spaces, a set of vectors is said to be if there is a nontrivial linear combination of the vectors that equals the zero vector. If no such linear combination exists, then the vectors are said to be . These concepts are ...
: 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 In statistics, ordinary least squares (OLS) is a type of linear least squares method for choosing the unknown parameters in a linear regression model (with fixed level-one effects of a linear function of a set of explanatory variables) by the ...
, this condition ensures that X^X is an
invertible matrix In linear algebra, an -by- square matrix is called invertible (also nonsingular or nondegenerate), if there exists an -by- square matrix such that :\mathbf = \mathbf = \mathbf_n \ where denotes the -by- identity matrix and the multiplicati ...
and therefore that a unique solution \hat 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: E(e_i , X_i) = 0 *The variance of the residuals e_i is constant across observations (
homoscedasticity In statistics, a sequence (or a vector) of random variables is homoscedastic () if all its random variables have the same finite variance. This is also known as homogeneity of variance. The complementary notion is called heteroscedasticity. The s ...
). * The residuals e_i are
uncorrelated In probability theory and statistics, two real-valued random variables, X, Y, are said to be uncorrelated if their covariance, \operatorname ,Y= \operatorname Y- \operatorname \operatorname /math>, is zero. If two variables are uncorrelated, there ...
with one another. Mathematically, the variance–covariance matrix of the errors is
diagonal In geometry, a diagonal is a line segment joining two vertices of a polygon or polyhedron, when those vertices are not on the same edge. Informally, any sloping line is called diagonal. The word ''diagonal'' derives from the ancient Greek δ ...
. 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 Bias is a disproportionate weight ''in favor of'' or ''against'' an idea or thing, usually in a way that is closed-minded, prejudicial, or unfair. Biases can be innate or learned. People may develop biases for or against an individual, a group, ...
,
consistent In classical deductive logic, a consistent theory is one that does not lead to a logical contradiction. The lack of contradiction can be defined in either semantic or syntactic terms. The semantic definition states that a theory is 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 The topic of heteroskedasticity-consistent (HC) standard errors arises in statistics and econometrics in the context of linear regression and time series analysis. These are also known as heteroskedasticity-robust standard errors (or simply robust ...
allow the variance of e_i to change across values of 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 e_i within geographic units can have important consequences. The subfield of
econometrics Econometrics is the application of statistical methods to economic data in order to give empirical content to economic relationships. M. Hashem Pesaran (1987). "Econometrics," '' The New Palgrave: A Dictionary of Economics'', v. 2, p. 8 p. 8 ...
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

In linear regression, the model specification is that the dependent variable, 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 n data points there is one independent variable: x_i , and two parameters, \beta_0 and \beta_1: :straight line: 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_i^2 to the preceding regression gives: :parabola: 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 x_i, it is linear in the parameters \beta_0, \beta_1 and \beta_2. In both cases, \varepsilon_i is an error term and the subscript 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: : \widehat_i = \widehat_0 + \widehat_1 x_i. The residual, e_i = y_i - \widehat_i , is the difference between the value of the dependent variable predicted by the model, \widehat_i, and the true value of the dependent variable, y_i. One method of estimation is
ordinary least squares In statistics, ordinary least squares (OLS) is a type of linear least squares method for choosing the unknown parameters in a linear regression model (with fixed level-one effects of a linear function of a set of explanatory variables) by the ...
. This method obtains parameter estimates that minimize the sum of squared residuals, SSR: :SSR=\sum_^n e_i^2. \, Minimization of this function results in a set of
normal equations In statistics, ordinary least squares (OLS) is a type of linear least squares method for choosing the unknown parameters in a linear regression model (with fixed level-one effects of a linear function of a set of explanatory variables) by the p ...
, a set of simultaneous linear equations in the parameters, which are solved to yield the parameter estimators, \widehat_0, \widehat_1. In the case of simple regression, the formulas for the least squares estimates are :\widehat_1=\frac :\widehat_0=\bar-\widehat_1\bar where \bar is the
mean There are several kinds of mean in mathematics, especially in statistics. Each mean serves to summarize a given group of data, often to better understand the overall value (magnitude and sign) of a given data set. For a data set, the '' ari ...
(average) of the x values and \bar is the mean of the y values. Under the assumption that the population error term has a constant variance, the estimate of that variance is given by: : \hat^2_\varepsilon = \frac.\, 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, (n-p) for p regressors or (n-p-1) if an intercept is used. In this case, p=1 so the denominator is n-2. The
standard error The standard error (SE) of a statistic (usually an estimate of a parameter) is the standard deviation of its sampling distribution or an estimate of that standard deviation. If the statistic is the sample mean, it is called the standard error ...
s of the parameter estimates are given by :\hat\sigma_=\hat\sigma_ \sqrt :\hat\sigma_=\hat\sigma_\varepsilon \sqrt=\hat\sigma_ \sqrt. 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 test A statistical hypothesis test is a method of statistical inference used to decide whether the data at hand sufficiently support a particular hypothesis. Hypothesis testing allows us to make probabilistic statements about population parameters. ...
s about the
population parameter In statistics, as opposed to its general use in mathematics, a parameter is any measured quantity of a statistical population that summarises or describes an aspect of the population, such as a mean or a standard deviation. If a population e ...
s.


General linear model

In the more general multiple regression model, there are p independent variables: : y_i = \beta_1 x_ + \beta_2 x_ + \cdots + \beta_p x_ + \varepsilon_i, \, where x_ is the i-th observation on the j-th independent variable. If the first independent variable takes the value 1 for all i, x_ = 1, then \beta_1 is called the
regression intercept In statistics, linear regression is a linear approach for modelling the relationship between a scalar response and one or more explanatory variables (also known as dependent and independent variables). The case of one explanatory variable is cal ...
. The least squares parameter estimates are obtained from p normal equations. The residual can be written as :\varepsilon_i=y_i - \hat\beta_1 x_ - \cdots - \hat\beta_p x_. The normal equations are :\sum_^n \sum_^p x_x_\hat \beta_k=\sum_^n x_y_i,\ j=1,\dots,p.\, In matrix notation, the normal equations are written as :\mathbf,\, where the ij element of \mathbf X is x_, the i element of the column vector Y is y_i, and the j element of \hat \boldsymbol \beta is \hat \beta_j. Thus \mathbf X is n \times p, Y is n \times 1, and \hat \boldsymbol \beta is p \times 1. The solution is :\mathbf.\,


Diagnostics

Once a regression model has been constructed, it may be important to confirm the
goodness of fit The goodness of fit of a statistical model describes how well it fits a set of observations. Measures of goodness of fit typically summarize the discrepancy between observed values and the values expected under the model in question. Such measure ...
of the model and the statistical significance of the estimated parameters. Commonly used checks of goodness of fit include the
R-squared In statistics, the coefficient of determination, denoted ''R''2 or ''r''2 and pronounced "R squared", is the proportion of the variation in the dependent variable that is predictable from the independent variable(s). It is a statistic used ...
, analyses of the pattern of residuals and hypothesis testing. Statistical significance can be checked by an
F-test An ''F''-test is any statistical test in which the test statistic has an ''F''-distribution under the null hypothesis. It is most often used when comparing statistical models that have been fitted to a data set, in order to identify the model ...
of the overall fit, followed by
t-test A ''t''-test is any statistical hypothesis test in which the test statistic follows a Student's ''t''-distribution under the null hypothesis. It is most commonly applied when the test statistic would follow a normal distribution if the value of ...
s 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 A ''t''-test is any statistical hypothesis test in which the test statistic follows a Student's ''t''-distribution under the null hypothesis. It is most commonly applied when the test statistic would follow a normal distribution if the value of ...
or
F-test An ''F''-test is any statistical test in which the test statistic has an ''F''-distribution under the null hypothesis. It is most often used when comparing statistical models that have been fitted to a data set, in order to identify the model ...
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 In probability theory, the central limit theorem (CLT) establishes that, in many situations, when independent random variables are summed up, their properly normalized sum tends toward a normal distribution even if the original variables themsel ...
can be invoked such that hypothesis testing may proceed using asymptotic approximations.


Limited dependent variables

Limited dependent variable A limited dependent variable is a variable whose range of possible values is "restricted in some important way." In econometrics, the term is often used when estimation of the relationship between the ''limited'' dependent variable of interest ...
s, which are response variables that are categorical variables or are variables constrained to fall only in a certain range, often arise in
econometrics Econometrics is the application of statistical methods to economic data in order to give empirical content to economic relationships. M. Hashem Pesaran (1987). "Econometrics," '' The New Palgrave: A Dictionary of Economics'', v. 2, p. 8 p. 8 ...
. 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 In statistics, a linear probability model (LPM) is a special case of a binary regression model. Here the dependent variable for each observation takes values which are either 0 or 1. The probability of observing a 0 or 1 in any one case is treated ...
. Nonlinear models for binary dependent variables include the
probit In probability theory and statistics, the probit function is the quantile function associated with the standard normal distribution. It has applications in data analysis and machine learning, in particular exploratory statistical graphics and s ...
and
logit model In statistics, the logistic model (or logit model) is a statistical model that models the probability of an event taking place by having the log-odds for the event be a linear combination of one or more independent variables. In regression ana ...
. 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 In statistics, multinomial logistic regression is a classification method that generalizes logistic regression to multiclass problems, i.e. with more than two possible discrete outcomes. That is, it is a model that is used to predict the prob ...
. For
ordinal variable Ordinal data is a categorical, statistical data type where the variables have natural, ordered categories and the distances between the categories are not known. These data exist on an ordinal scale, one of four levels of measurement described b ...
s with more than two values, there are the
ordered logit In statistics, the ordered logit model (also ordered logistic regression or proportional odds model) is an ordinal regression model—that is, a regression model for ordinal dependent variables—first considered by Peter McCullagh. For exampl ...
and
ordered probit In statistics, ordered probit is a generalization of the widely used probit analysis to the case of more than two outcomes of an ordinal dependent variable (a dependent variable for which the potential values have a natural ordering, as in poor, ...
models.
Censored regression model Censored regression models are a class of models in which the dependent variable is censored above or below a certain threshold. A commonly used likelihood-based model to accommodate to a censored sample is the Tobit model, but quantile and nonp ...
s may be used when the dependent variable is only sometimes observed, and
Heckman correction The Heckman correction is a statistical technique to correct bias from non-randomly selected samples or otherwise incidentally truncated dependent variables, a pervasive issue in quantitative social sciences when using observational data. Concep ...
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 In statistics, polychoric correlation{{Cite web, url=https://support.sas.com/documentation/cdl/en/procstat/65543/HTML/default/viewer.htm#procstat_corr_details14.htm, title=Base SAS(R) 9.3 Procedures Guide: Statistical Procedures, Second Edition, we ...
(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 In statistics, Poisson regression is a generalized linear model form of regression analysis used to model count data and contingency tables. Poisson regression assumes the response variable ''Y'' has a Poisson distribution, and assumes the logari ...
or the
negative binomial In probability theory and statistics, the negative binomial distribution is a discrete probability distribution that models the number of failures in a sequence of independent and identically distributed Bernoulli trials before a specified (non-r ...
model may be used.


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.


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 model-fitting is known informally as interpolation. Prediction ''outside'' this range of the data is known as
extrapolation In mathematics, extrapolation is a type of estimation, beyond the original observation range, of the value of a variable on the basis of its relationship with another variable. It is similar to interpolation, which produces estimates between know ...
. 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 that when performing extrapolation, one should accompany the estimated value of the dependent variable with a
prediction interval In statistical inference, specifically predictive inference, a prediction interval is an estimate of an interval in which a future observation will fall, with a certain probability, given what has already been observed. Prediction intervals are ...
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. However, this does not cover the full set of modeling errors that may be 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 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).


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 N=m^n, where N is the sample size, n is the number of independent variables and 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 (N). If the researcher decides that five observations are needed to precisely define a straight line (m), then the maximum number of independent variables the model can support is 4, because : \frac=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 method Bayesian inference is a method of statistical inference in which Bayes' theorem is used to update the probability for a hypothesis as more evidence or information becomes available. Bayesian inference is an important technique in statistics, and e ...
s, e.g.
Bayesian linear regression Bayesian linear regression is a type of conditional modeling in which the mean of one variable is described by a linear combination of other variables, with the goal of obtaining the posterior probability of the regression coefficients (as well ...
* Percentage regression, for situations where reducing ''percentage'' errors is deemed more appropriate. *
Least absolute deviations Least absolute deviations (LAD), also known as least absolute errors (LAE), least absolute residuals (LAR), or least absolute values (LAV), is a statistical optimality criterion and a statistical optimization technique based minimizing the ''sum o ...
, which is more robust in the presence of outliers, leading to
quantile regression Quantile regression is a type of regression analysis used in statistics and econometrics. Whereas the method of least squares estimates the conditional ''mean'' of the response variable across values of the predictor variables, quantile regress ...
*
Nonparametric regression Nonparametric regression is a category of regression analysis in which the predictor does not take a predetermined form but is constructed according to information derived from the data. That is, no parametric form is assumed for the relationship ...
, requires a large number of observations and is computationally intensive *
Scenario optimization The scenario approach or scenario optimization approach is a technique for obtaining solutions to robust optimization and chance-constrained optimization problems based on a sample of the constraints. It also relates to inductive reasoning in mod ...
, leading to
interval predictor model In regression analysis, an interval predictor model (IPM) is an approach to regression where bounds on the function to be approximated are obtained. This differs from other techniques in machine learning, where usually one wishes to estimate point v ...
s * Distance metric learning, which is learned by the search of a meaningful distance metric in a given input space.


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 A spreadsheet is a computer application for computation, organization, analysis and storage of data in tabular form. Spreadsheets were developed as computerized analogs of paper accounting worksheets. The program operates on data entered in cel ...
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 Curve fitting is the process of constructing a curve, or mathematical function, that has the best fit to a series of data points, possibly subject to constraints. Curve fitting can involve either interpolation, where an exact fit to the data i ...
* Estimation theory *
Forecasting Forecasting is the process of making predictions based on past and present data. Later these can be compared (resolved) against what happens. For example, a company might estimate their revenue in the next year, then compare it against the actual ...
*
Fraction of variance unexplained In statistics, the fraction of variance unexplained (FVU) in the context of a regression task is the fraction of variance of the regressand (dependent variable) ''Y'' which cannot be explained, i.e., which is not correctly predicted, by the e ...
*
Function approximation In general, a function approximation problem asks us to select a function among a that closely matches ("approximates") a in a task-specific way. The need for function approximations arises in many branches of applied mathematics, and comput ...
* Generalized linear model *
Kriging In statistics, originally in geostatistics, kriging or Kriging, also known as Gaussian process regression, is a method of interpolation based on Gaussian process governed by prior covariances. Under suitable assumptions of the prior, kriging giv ...
(a linear least squares estimation algorithm) *
Local regression Local regression or local polynomial regression, also known as moving regression, is a generalization of the moving average and polynomial regression. Its most common methods, initially developed for scatterplot smoothing, are LOESS (locally e ...
*
Modifiable areal unit problem __NOTOC__ The modifiable areal unit problem (MAUP) is a source of statistical bias that can significantly impact the results of statistical hypothesis tests. MAUP affects results when point-based measures of spatial phenomena are aggregated into ...
*
Multivariate adaptive regression splines In statistics, multivariate adaptive regression splines (MARS) is a form of regression analysis introduced by Jerome H. Friedman in 1991. It is a non-parametric regression technique and can be seen as an extension of linear models that automaticall ...
*
Multivariate normal distribution In probability theory and statistics, the multivariate normal distribution, multivariate Gaussian distribution, or joint normal distribution is a generalization of the one-dimensional ( univariate) normal distribution to higher dimensions. One ...
*
Pearson correlation coefficient In statistics, the Pearson correlation coefficient (PCC, pronounced ) ― also known as Pearson's ''r'', the Pearson product-moment correlation coefficient (PPMCC), the bivariate correlation, or colloquially simply as the correlation coefficient ...
* Quasi-variance *
Prediction interval In statistical inference, specifically predictive inference, a prediction interval is an estimate of an interval in which a future observation will fall, with a certain probability, given what has already been observed. Prediction intervals are ...
*
Regression validation In statistics, regression validation is the process of deciding whether the numerical results quantifying hypothesized relationships between variables, obtained from regression analysis, are acceptable as descriptions of the data. The validation ...
*
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 ...
*
Segmented regression Segmented regression, also known as piecewise regression or broken-stick regression, is a method in regression analysis in which the independent variable is partitioned into intervals and a separate line segment is fit to each interval. Segmented r ...
*
Signal processing Signal processing is an electrical engineering subfield that focuses on analyzing, modifying and synthesizing ''signals'', such as audio signal processing, sound, image processing, images, and scientific measurements. Signal processing techniq ...
* Stepwise regression *
Taxicab geometry A taxicab geometry or a Manhattan geometry is a geometry whose usual distance function or metric of Euclidean geometry is replaced by a new metric in which the distance between two points is the sum of the absolute differences of their Cartesian c ...
*
Trend estimation Linear trend estimation is a statistical technique to aid interpretation of data. When a series of measurements of a process are treated as, for example, a sequences or time series, trend estimation can be used to make and justify statements abou ...


References


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 Julian Cecil Stanley (July 9, 1918 – August 12, 2005) was an American psychologist. He was an advocate of accelerated education for academically gifted children. He founded the Johns Hopkins University Center for Talented Youth (CTY), as wel ...
, "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''. * Chatfield, C. (1993)
Calculating Interval Forecasts
" ''Journal of Business and Economic Statistics,'' 11. pp. 121–135. * * Fox, J. (1997). ''Applied Regression Analysis, Linear Models and Related Methods.'' Sage * Hardle, W., ''Applied Nonparametric Regression'' (1990), * * A. Sen, M. Srivastava, ''Regression Analysis — Theory, Methods, and Applications'', Springer-Verlag, 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.


External links

*

– basic history and references
What is multiple regression used for?
– Multiple regression

– how linear regression mistakes can appear when Y-range is much smaller than X-range {{DEFAULTSORT:Regression Analysis Actuarial science Curve fitting Estimation theory