Regression analysis

In statistics, regression analysis refers to 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 us 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 &mdash; 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.

It is important to note that regression models do not indicate that a causal relationship exists between the variables being tested, and should therefore not be used for prediction. However, regression models can provide information needed to estimate the values of a dependent variable when the values of an independent variable are known, if a relationship does exist between the two variables. Regression models may be used to describe the strength or weakness of this relationship between two variables. Regression models work best when the relationship between two variables exhibits a clear linear trend. 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.

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 infinitely-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 not known, regression analysis depends to some extent on making assumptions about this process. These assumptions are sometimes (but not always) 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 when carrying out inference using regression models, especially involving small effects or questions of causality based on observational data, regression methods must be used cautiously as they can easily give misleading results.

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

The term "regression" was coined by Francis Galton, a cousin of Charles Darwin, 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. For Galton, regression had only this biological meaning, but his work was later extended by Udny Yule and Karl Pearson to a more general statistical context. . In the work of Yule and Pearson, the joint distribution of the response and explanatory variables is assumed to be Gaussian. This assumption was weakened by R.A. Fisher in his works of 1922 and 1925. Fisher assumed that the conditional distribution of the response variable is Gaussian, but the joint distribution need not be. In this respect, Fisher's assumption is closer to Gauss's formulation of 1821.

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.

Underlying assumptions
Classical assumptions for regression analysis include: These are sufficient (but not all necessary) 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. Many of these assumptions may be relaxed in more advanced treatments.
 * The sample must be representative of the population for the inference prediction.
 * The error is assumed to be a random variable with a mean of zero conditional on the explanatory variables.
 * The independent variables are error-free. If this is not so, modeling may be done using errors-in-variables model techniques.
 * The predictors must be linearly independent, i.e. it must not be 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). If not, weighted least squares or other methods might be used.

Assumptions include the geometrical support of the variables (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.

Regression equation
It is convenient to assume an environment in which an experiment is performed: the dependent variable is then outcome of a measurement.

The regression equation deals with the following variables:
 * The unknown parameters denoted as β; this may be a scalar or a vector of length k.
 * The independent variables, X.
 * The dependent variable, Y.

Regression equation is a function of variables X and β.


 * $$Y = f (\mathbf {X}, \boldsymbol{\beta} )$$

The user of regression analysis must make an intelligent guess about this function. Sometimes the form of this function is known, sometimes he must apply a trial and error process.

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 the user performs the measurement N times, where N < k, regression analysis cannot be performed: there is not provided enough information to do so.
 * If the user performs N independent measurements, where N = k, then the problem reduces to solving a set of N equations with N unknowns β.
 * If, on the other hand, the user provides results of N independent measurements, where N > k, regression analysis can be performed. Such a system is also called an overdetermined system;

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.

Independent measurements
Quantitatively, this is explained by the following example: Consider a logistic regression model, which has three unknown parameters, β0, β1, and β2. An experimenter performed 10 measurements all at exactly the same value of independent variable X. In this case, regression analysis fails to give a unique value for the three unknown parameters; the experimenter did not provide enough information. The best one can do is to calculate the average value of the dependent variable Y and its standard deviation. Similarly, measuring at two different values of X would give enough data for a linear or a power regression (two unknowns), but not a logistic (three unknowns) or cubic (four unknowns).

If the experimenter had performed measurements at X1, X2 and X3, where X1, X2, and X3 are different values of X, then regression analysis would provide a unique solution to the unknown parameters β.

In the case of general linear regression, the above statement is equivalent to the requirement that matrix XTX is regular (that is: it has an inverse matrix).

Statistical assumptions
When the number of measurements, N, is larger than the number of unknown parameters, k, and the measurement errors εi are normally distributed then the excess of information contained in (N - k) measurements is used to make statistical predictions about the unknown parameters.

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. For example, adding a term in xi2 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. Given a random sample from the population, we estimate the population parameters and obtain the sample linear regression model:


 * $$ y_i = \widehat{\beta}_0 + \widehat{\beta}_1 X_i + e_i. $$

The term $$ e_i $$ is the residual, $$ e_i = y_i - \widehat{y}_i $$. One method of estimation is ordinary least squares. This method obtains parameter estimates that minimize the sum of squared residuals, SSE:


 * $$SSE=\sum_{i=1}^N e_i^2. \, $$



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, $$\widehat{\beta}_0, \widehat{\beta}_1$$.



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


 * $$\widehat{\beta_1}=\frac{\sum(x_i-\bar{x})(y_i-\bar{y})}{\sum(x_i-\bar{x})^2}\text{ and }\hat{\beta_0}=\bar{y}-\widehat{\beta_1}\bar{x}$$

where $$\bar{x}$$ is the mean (average) of the $$x$$ values and $$\bar{y}$$ is the mean of the $$y$$ values. See linear least squares(straight line fitting) 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:


 * $$ \hat{\sigma}^2_\varepsilon = \frac{SSE}{N-2}.\,$$

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


 * $$\hat\sigma_{\beta_0}=\hat\sigma_{\varepsilon} \sqrt{\frac{1}{N} + \frac{\bar{x}^2}{\sum(x_i-\bar x)^2}}$$


 * $$\hat\sigma_{\beta_1}=\hat\sigma_{\varepsilon} \sqrt{\frac{1}{\sum(x_i-\bar x)^2}}.$$

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


 * $$ y_i = \beta_0 + \beta_1 x_{1i} + \cdots + \beta_p x_{pi} + \varepsilon_i, \, $$

The least square parameter estimates are obtained by p normal equations. The residual can be written as


 * $$e_i=y_i - \hat\beta_0 - \hat\beta_1 x_1 - \cdots - \hat\beta_p x_p.$$

The normal equations are


 * $$\sum_{i=1}^N \sum_{k=1}^p X_{ij}X_{ik}\hat \beta_k=\sum_{i=1}^N X_{ij}y_i,\ j=1,\dots,p.\,$$

Note that for the normal equations depicted above, $$ y_i = \beta_1 x_{1i} + \cdots + \beta_p x_{pi} + \varepsilon_i \, $$

That is, there is no $$ \beta_0 $$. Thus in what follows, $$\boldsymbol \beta = (\beta_1, \beta_2, \dots, \beta_n).$$

In matrix notation, the normal equations are written as


 * $$\mathbf{\left(X^TX\right)\hat \boldsymbol \beta=X^Ty}.\,$$

For a numerical example see linear regression (example).

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

Interpretations of these diagnostic tests rest heavily on the model 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
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 makes it possible to estimate jointly the 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 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, count models like the Poisson regression or the negative binomial model may be used

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 is known as interpolation. Prediction outside the range of the data is known as extrapolation, which is more risky.

Multiple regression


It is often the case in a geographic study that more than one variable impacts the dependent variable in question. In this case, simple linear regression may not provide a satisfactory result. Rogerson states in his book, Statistical Methods for Geography: A Student's Guide that, "It is most often the case that there is more than one variable that is thought to affect the dependent variable." This is where multiple regression may present a more acceptable solution. With the proper assumptions supported and attention to proper data analysis, multiple regression can be a strong tool.

The idea of multiple regression is basically the same as that of linear regression. The main difference is that multiple regression considers two or more independent variables. The figure to the right displays this phenomena. As seen, there are two "horizontal" axes in addition to the standard y-axis. A plane could be used to describe the space of the points in this three-dimensional model. The red dots correspond with one axis and the black dots correspond to another.

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

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
 * 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

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.

GIS&T Body of Knowledge
This topic falls under section AM7-5 of the GIS&T Body of Knowledge.