Regression analysis
In statistical modeling, regression analysis is a statistical method for estimating the relationship between a dependent variable and one or more independent variables.
The most common form of regression analysis is linear regression, in which one finds the line that most closely fits the data according to a specific mathematical criterion. For example, the method of ordinary least squares computes the unique line that minimizes the sum of squared differences between the true data and that line. For specific mathematical reasons, this allows the researcher to estimate the conditional expectation 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 or estimate the conditional expectation across a broader collection of non-linear models.
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.
History
The earliest regression form was seen in Isaac Newton's work in 1700 while studying equinoxes, being credited with introducing "an embryonic linear regression analysis" as "not only did he perform the averaging of a set of data, 50 years before Tobias Mayer, but by summing the residuals to zero he forced the regression line to pass through the average point. He also distinguished between two inhomogeneous sets of data and might have thought of an optimal solution in terms of bias, though not in terms of effectiveness." He previously used an averaging method in his 1671 work on Newton's rings, which was unprecedented at the time.The method of least squares 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 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.
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.
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, 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. Modern regression analysis is typically done with statistical and spreadsheet software packages on computers as well as on handheld scientific and graphing calculators.
Regression model
In practice, researchers first select a model they would like to estimate and then use their chosen method 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 .
- 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.
Most regression models propose that is a function of and, with representing an additive error term that may stand in for un-modeled determinants of or random statistical noise:
In the standard regression model, the independent variables are assumed to be free of error. The errors-in-variables model can be used if the independent variables are assumed to contain errors. Other modifications to the standard regression model can be made to account for various scenarios, such as situations involving omitted variables, confounding variables or endogeneity.
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 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 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. However, alternative variants 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 3-dimensional 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
By itself, a regression is just a computation performed on a set of data. In order to interpret the resultant regression as a meaningful statistical model that quantifies 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 without error.
- Deviations from the model have an expected value of zero, conditional on covariates:
- The variance of the residuals is constant across observations.
- The residuals are uncorrelated with one another. Mathematically, the variance–covariance matrix of the errors is diagonal.