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Instatistics,linear regression is amodel that estimates the relationship between ascalar response (dependent variable) and one or more explanatory variables (regressor orindependent variable). A model with exactly one explanatory variable is asimple linear regression; a model with two or more explanatory variables is amultiple linear regression.[1] This term is distinct frommultivariate linear regression, which predicts multiplecorrelated dependent variables rather than a single dependent variable.[2]
In linear regression, the relationships are modeled usinglinear predictor functions whose unknown modelparameters areestimated from thedata. Most commonly, theconditional mean of the response given the values of the explanatory variables (or predictors) is assumed to be anaffine function of those values; less commonly, the conditionalmedian or some otherquantile is used. Like all forms ofregression analysis, linear regression focuses on theconditional probability distribution of the response given the values of the predictors, rather than on thejoint probability distribution of all of these variables, which is the domain ofmultivariate analysis.
Linear regression is also a type ofmachine learningalgorithm, more specifically asupervised algorithm, that learns from the labelled datasets and maps the data points to the most optimized linear functions that can be used for prediction on new datasets.[3]
Linear regression was the first type of regression analysis to be studied rigorously, and to be used extensively in practical applications.[4] This is because models which depend linearly on their unknown parameters are easier to fit than models which are non-linearly related to their parameters and because the statistical properties of the resulting estimators are easier to determine.
Linear regression has many practical uses. Most applications fall into one of the following two broad categories:
Linear regression models are often fitted using theleast squares approach, but they may also be fitted in other ways, such as by minimizing the "lack of fit" in some othernorm (as withleast absolute deviations regression), or by minimizing a penalized version of the least squarescost function as inridge regression (L2-norm penalty) andlasso (L1-norm penalty). Use of theMean Squared Error (MSE) as the cost on a dataset that has many large outliers, can result in a model that fits the outliers more than the true data due to the higher importance assigned by MSE to large errors. So, cost functions that are robust to outliers should be used if the dataset has many largeoutliers. Conversely, the least squares approach can be used to fit models that are not linear models. Thus, although the terms "least squares" and "linear model" are closely linked, they are not synonymous.
Given adata set ofnstatistical units, a linear regression model assumes that the relationship between the dependent variabley and the vector of regressorsx islinear. This relationship is modeled through adisturbance term orerror variableε—an unobservedrandom variable that adds "noise" to the linear relationship between the dependent variable and regressors. Thus the model takes the formwhereT denotes thetranspose, so thatxiTβ is theinner product betweenvectorsxi andβ.
Often thesen equations are stacked together and written inmatrix notation as
where
Fitting a linear model to a given data set usually requires estimating the regression coefficients such that the error term is minimized. For example, it is common to use the sum of squared errors as a measure of for minimization.
Consider a situation where a small ball is being tossed up in the air and then we measure its heights of ascenthi at various moments in timeti. Physics tells us that, ignoring thedrag, the relationship can be modeled as
whereβ1 determines the initial velocity of the ball,β2 is proportional to thestandard gravity, andεi is due to measurement errors. Linear regression can be used to estimate the values ofβ1 andβ2 from the measured data. This model is non-linear in the time variable, but it is linear in the parametersβ1 andβ2; if we take regressorsxi = (xi1,xi2) = (ti,ti2), the model takes on the standard form
When estimating the parameters of linear regression models with standard estimation techniques such asordinary least squares, it is necessary to make a number of assumptions about the predictor variables, the response variable and their relationship, to get estimators that areunbiased infinite sample. Numerous extensions have been developed that allow each of these assumptions to be relaxed (reduced to a weaker form), and in some cases eliminated entirely. Generally these extensions require more data or modelling assumptions to produce an equally precise model[5].
A fitted linear regression model can be used to identify the relationship between a single predictor variablexj and the response variabley when all the other predictor variables in the model are "held fixed". Specifically, the interpretation ofβj is theexpected change iny for a one-unit change inxj when the other covariates are held fixed—that is, the expected value of thepartial derivative ofy with respect toxj. This is sometimes called theunique effect ofxj ony. In contrast, themarginal effect ofxj ony can be assessed using acorrelation coefficient orsimple linear regression model relating onlyxj toy; this effect is thetotal derivative ofy with respect toxj.
Care must be taken when interpreting regression results, as some of the regressors may not allow for marginal changes (such asdummy variables, or the intercept term), while others cannot be held fixed (recall the example from the introduction: it would be impossible to "holdti fixed" and at the same time change the value ofti2).
It is possible that the unique effect be nearly zero even when the marginal effect is large. This may imply that some other covariate captures all the information inxj, so that once that variable is in the model, there is no contribution ofxj to the variation iny. Conversely, the unique effect ofxj can be large while its marginal effect is nearly zero. This would happen if the other covariates explained a great deal of the variation ofy, but they mainly explain variation in a way that is complementary to what is captured byxj. In this case, including the other variables in the model reduces the part of the variability ofy that is unrelated toxj, thereby strengthening the apparent relationship withxj.
The meaning of the expression "held fixed" may depend on how the values of the predictor variables arise. If the experimenter directly sets the values of the predictor variables according to a study design, the comparisons of interest may literally correspond to comparisons among units whose predictor variables have been "held fixed" by the experimenter. Alternatively, the expression "held fixed" can refer to a selection that takes place in the context of data analysis. In this case, we "hold a variable fixed" by restricting our attention to the subsets of the data that happen to have a common value for the given predictor variable. This is the only interpretation of "held fixed" that can be used in anobservational study.
The notion of a "unique effect" is appealing when studying acomplex system where multiple interrelated components influence the response variable. In some cases, it can literally be interpreted as the causal effect of an intervention that is linked to the value of a predictor variable. However, it has been argued that in many cases multiple regression analysis fails to clarify the relationships between the predictor variables and the response variable when the predictors are correlated with each other and are not assigned following a study design.[6]
Numerous extensions of linear regression have been developed, which allow some or all of the assumptions underlying the basic model to be relaxed.
The simplest case of a singlescalar predictor variablex and a single scalar response variabley is known assimple linear regression. The extension to multiple and/orvector-valued predictor variables (denoted with a capitalX) is known asmultiple linear regression, also known asmultivariable linear regression (not to be confused withmultivariate linear regression).[7]
Multiple linear regression is a generalization ofsimple linear regression to the case of more than one independent variable, and aspecial case of general linear models, restricted to one dependent variable. The basic model for multiple linear regression is
for each observation.
In the formula above we considern observations of one dependent variable andp independent variables. Thus,Yi is theith observation of the dependent variable,Xij isith observation of thejth independent variable,j = 1, 2, ...,p. The valuesβj represent parameters to be estimated, andεi is theith independent identically distributed normal error.
In the more general multivariate linear regression, there is one equation of the above form for each ofm > 1 dependent variables that share the same set of explanatory variables and hence are estimated simultaneously with each other:
for all observations indexed asi = 1, ... ,n and for all dependent variables indexed asj = 1, ... ,m.
Nearly all real-world regression models involve multiple predictors, and basic descriptions of linear regression are often phrased in terms of the multiple regression model. Note, however, that in these cases the response variabley is still a scalar. Another term,multivariate linear regression, refers to cases wherey is a vector, i.e., the same asgeneral linear regression.
Thegeneral linear model considers the situation when the response variable is not a scalar (for each observation) but a vector,yi. Conditional linearity of is still assumed, with a matrixB replacing the vectorβ of the classical linear regression model. Multivariate analogues ofordinary least squares (OLS) andgeneralized least squares (GLS) have been developed. "General linear models" are also called "multivariate linear models". These are not the same as multivariable linear models (also called "multiple linear models").
Various models have been created that allow forheteroscedasticity, i.e. the errors for different response variables may have differentvariances. For example,weighted least squares is a method for estimating linear regression models when the response variables may have different error variances, possibly with correlated errors. (See alsoWeighted linear least squares, andGeneralized least squares.)Heteroscedasticity-consistent standard errors is an improved method for use with uncorrelated but potentially heteroscedastic errors.
TheGeneralized linear model (GLM) is a framework for modeling response variables that are bounded or discrete. This is used, for example:
Generalized linear models allow for an arbitrarylink function,g, that relates themean of the response variable(s) to the predictors:. The link function is often related to the distribution of the response, and in particular it typically has the effect of transforming between the range of the linear predictor and the range of the response variable.
Some common examples of GLMs are:
Single index models[clarification needed] allow some degree of nonlinearity in the relationship betweenx andy, while preserving the central role of the linear predictorβ′x as in the classical linear regression model. Under certain conditions, simply applying OLS to data from a single-index model will consistently estimateβ up to a proportionality constant.[8]
Hierarchical linear models (ormultilevel regression) organizes the data into a hierarchy of regressions, for example whereA is regressed onB, andB is regressed onC. It is often used where the variables of interest have a natural hierarchical structure such as in educational statistics, where students are nested in classrooms, classrooms are nested in schools, and schools are nested in some administrative grouping, such as a school district. The response variable might be a measure of student achievement such as a test score, and different covariates would be collected at the classroom, school, and school district levels.
Errors-in-variables models (or "measurement error models") extend the traditional linear regression model to allow the predictor variablesX to be observed with error. This error causes standard estimators ofβ to become biased. Generally, the form of bias is an attenuation, meaning that the effects are biased toward zero.
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In a multiple linear regression model
parameter of predictor variable represents the individual effect of. It has an interpretation as the expected change in the response variable when increases by one unit with other predictor variables held constant. When is strongly correlated with other predictor variables, it is improbable that can increase by one unit with other variables held constant. In this case, the interpretation of becomes problematic as it is based on an improbable condition, and the effect of cannot be evaluated in isolation.
For a group of predictor variables, say,, a group effect is defined as a linear combination of their parameters
where is a weight vector satisfying. Because of the constraint on, is also referred to as a normalized group effect. A group effect has an interpretation as the expected change in when variables in the group change by the amount, respectively, at the same time with other variables (not in the group) held constant. It generalizes the individual effect of a variable to a group of variables in that () if, then the group effect reduces to an individual effect, and () if and for, then the group effect also reduces to an individual effect.A group effect is said to be meaningful if the underlying simultaneous changes of the variables is probable.
Group effects provide a means to study the collective impact of strongly correlated predictor variables in linear regression models. Individual effects of such variables are not well-defined as their parameters do not have good interpretations. Furthermore, when the sample size is not large, none of their parameters can be accurately estimated by theleast squares regression due to themulticollinearity problem. Nevertheless, there are meaningful group effects that have good interpretations and can be accurately estimated by the least squares regression. A simple way to identify these meaningful group effects is to use an all positive correlations (APC) arrangement of the strongly correlated variables under which pairwise correlations among these variables are all positive, and standardize all predictor variables in the model so that they all have mean zero and length one. To illustrate this, suppose that is a group of strongly correlated variables in an APC arrangement and that they are not strongly correlated with predictor variables outside the group. Let be the centred and be the standardized. Then, the standardized linear regression model is
Parameters in the original model, including, are simple functions of in the standardized model. The standardization of variables does not change their correlations, so is a group of strongly correlated variables in an APC arrangement and they are not strongly correlated with other predictor variables in the standardized model. A group effect of is
and its minimum-variance unbiased linear estimator is
where is the least squares estimator of. In particular, the average group effect of the standardized variables is
which has an interpretation as the expected change in when all in the strongly correlated group increase byth of a unit at the same time with variables outside the group held constant. With strong positive correlations and in standardized units, variables in the group are approximately equal, so they are likely to increase at the same time and in similar amount. Thus, the average group effect is a meaningful effect. It can be accurately estimated by its minimum-variance unbiased linear estimator, even when individually none of the can be accurately estimated by.
Not all group effects are meaningful or can be accurately estimated. For example, is a special group effect with weights and for, but it cannot be accurately estimated by. It is also not a meaningful effect. In general, for a group of strongly correlated predictor variables in an APC arrangement in the standardized model, group effects whose weight vectors are at or near the centre of the simplex () are meaningful and can be accurately estimated by their minimum-variance unbiased linear estimators. Effects with weight vectors far away from the centre are not meaningful as such weight vectors represent simultaneous changes of the variables that violate the strong positive correlations of the standardized variables in an APC arrangement. As such, they are not probable. These effects also cannot be accurately estimated.
Applications of the group effects include (1) estimation and inference for meaningful group effects on the response variable, (2) testing for "group significance" of the variables via testing versus, and (3) characterizing the region of the predictor variable space over which predictions by the least squares estimated model are accurate.
A group effect of the original variables can be expressed as a constant times a group effect of the standardized variables. The former is meaningful when the latter is. Thus meaningful group effects of the original variables can be found through meaningful group effects of the standardized variables.[9]
InDempster–Shafer theory, or alinear belief function in particular, a linear regression model may be represented as a partially swept matrix, which can be combined with similar matrices representing observations and other assumed normal distributions and state equations. The combination of swept or unswept matrices provides an alternative method for estimating linear regression models.
A large number of procedures have been developed forparameter estimation and inference in linear regression. These methods differ in computational simplicity of algorithms, presence of aclosed-form solution,robustness with respect to heavy-tailed distributions, and theoretical assumptions needed to validate desirable statistical properties such asconsistency and asymptoticefficiency.
Some of the more common estimation techniques for linear regression are summarized below.
Assuming that the independent variables are and the model's parameters are, then the model's prediction would be
If is extended to then would become adot product of the parameter and the independent vectors, i.e.
In the least-squares setting, the optimum parameter vector is defined as such that minimizes the sum of mean squared loss:
Now putting the independent and dependent variables in matrices and respectively, the loss function can be rewritten as:
As the loss function isconvex, the optimum solution lies atgradient zero. The gradient of the loss function is (usingDenominator layout convention):
Setting the gradient to zero produces the optimum parameter:
Note: The obtained may indeed be the local minimum, one needs to differentiate once more to obtain theHessian matrix and show that it is positive definite. This is provided by theGauss–Markov theorem.
Linear least squares methods include mainly:
Maximum likelihood estimation can be performed when the distribution of the error terms is known to belong to a certain parametric familyƒθ ofprobability distributions.[12] Whenfθ is a normal distribution with zeromean and variance θ, the resulting estimate is identical to the OLS estimate. GLS estimates are maximum likelihood estimates when ε follows a multivariate normal distribution with a knowncovariance matrix.Let's denote each data point by and the regression parameters as, and the set of all data by and the cost function by.
As shown below the same optimal parameter that minimizes achieves maximum likelihood too.[13] Here the assumption is that the dependent variable is a random variable that follows aGaussian distribution, where the standard deviation is fixed and the mean is a linear combination of:
Now, we need to look for a parameter that maximizes this likelihood function. Since the logarithmic function is strictly increasing, instead of maximizing this function, we can also maximize its logarithm and find the optimal parameter that way.[13]
The optimal parameter is thus equal to:[13]
In this way, the parameter that maximizes is the same as the one that minimizes. This means that in linear regression, the result of the least squares method is the same as the result of the maximum likelihood estimation method.[13]
Ridge regression[14][15][16] and other forms of penalized estimation, such asLasso regression,[17] deliberately introducebias into the estimation ofβ in order to reduce thevariability of the estimate. The resulting estimates generally have lowermean squared error than the OLS estimates, particularly whenmulticollinearity is present or whenoverfitting is a problem. They are generally used when the goal is to predict the value of the response variabley for values of the predictorsx that have not yet been observed. These methods are not as commonly used when the goal is inference, since it is difficult to account for the bias.
Least absolute deviation (LAD) regression is arobust estimation technique in that it is less sensitive to the presence of outliers than OLS (but is lessefficient than OLS when no outliers are present). It is equivalent to maximum likelihood estimation under aLaplace distribution model forε.[18]
If we assume that error terms areindependent of the regressors,, then the optimal estimator is the 2-step MLE, where the first step is used to non-parametrically estimate the distribution of the error term.[19]
Linear regression is widely used in biological, behavioral and social sciences to describe possible relationships between variables. It ranks as one of the most important tools used in these disciplines.
Atrend line represents a trend, the long-term movement intime series data after other components have been accounted for. It tells whether a particular data set (say GDP, oil prices or stock prices) have increased or decreased over the period of time. A trend line could simply be drawn by eye through a set of data points, but more properly their position and slope is calculated using statistical techniques like linear regression. Trend lines typically are straight lines, although some variations use higher degree polynomials depending on the degree of curvature desired in the line.
Trend lines are sometimes used in business analytics to show changes in data over time. This has the advantage of being simple. Trend lines are often used to argue that a particular action or event (such as training, or an advertising campaign) caused observed changes at a point in time. This is a simple technique, and does not require a control group, experimental design, or a sophisticated analysis technique. However, it suffers from a lack of scientific validity in cases where other potential changes can affect the data.
Early evidence relatingtobacco smoking to mortality andmorbidity came fromobservational studies employing regression analysis. In order to reducespurious correlations when analyzing observational data, researchers usually include several variables in their regression models in addition to the variable of primary interest. For example, in a regression model in which cigarette smoking is the independent variable of primary interest and the dependent variable is lifespan measured in years, researchers might include education and income as additional independent variables, to ensure that any observed effect of smoking on lifespan is not due to those othersocio-economic factors. However, it is never possible to include all possibleconfounding variables in an empirical analysis. For example, a hypothetical gene might increase mortality and also cause people to smoke more. For this reason,randomized controlled trials are often able to generate more compelling evidence of causal relationships than can be obtained using regression analyses of observational data. When controlled experiments are not feasible, variants of regression analysis such asinstrumental variables regression may be used to attempt to estimate causal relationships from observational data.
Thecapital asset pricing model uses linear regression as well as the concept ofbeta for analyzing and quantifying the systematic risk of an investment. This comes directly from the beta coefficient of the linear regression model that relates the return on the investment to the return on all risky assets.
Linear regression is the predominant empirical tool ineconomics. For example, it is used to predictconsumption spending,[25]fixed investment spending,inventory investment, purchases of a country'sexports,[26] spending onimports,[26] thedemand to hold liquid assets,[27]labor demand,[28] andlabor supply.[28]
Linear regression finds application in a wide range of environmental science applications such asland use,[29]infectious diseases,[30] andair pollution.[31] For example, linear regression can be used to predict the changing effects of car pollution.[32] One notable example of this application in infectious diseases is theflattening the curve strategy emphasized early in the COVID-19 pandemic, where public health officials dealt with sparse data on infected individuals and sophisticated models of disease transmission to characterize the spread of COVID-19.[33]
Linear regression is commonly used inbuilding science field studies to derive characteristics of building occupants. In athermal comfort field study, building scientists usually ask occupants' thermal sensation votes, which range from -3 (feeling cold) to 0 (neutral) to +3 (feeling hot), and measure occupants' surrounding temperature data. A neutral or comfort temperature can be calculated based on a linear regression between the thermal sensation vote and indoor temperature, and setting the thermal sensation vote as zero. However, there has been a debate on the regression direction: regressing thermal sensation votes (y-axis) against indoor temperature (x-axis) or the opposite: regressing indoor temperature (y-axis) against thermal sensation votes (x-axis).[34]
Linear regression plays an important role in the subfield ofartificial intelligence known asmachine learning. The linear regression algorithm is one of the fundamentalsupervised machine-learning algorithms due to its relative simplicity and well-known properties.[35]
Isaac Newton is credited with inventing "a certain technique known today aslinear regression analysis" in his work on equinoxes in 1700, and wrote down the first of the two normal equations of theordinary least squares method.[36][37] The Least squares linear regression, as a means of finding a good rough linear fit to a set of points was performed byLegendre (1805) andGauss (1809) for the prediction of planetary movement.Quetelet was responsible for making the procedure well-known and for using it extensively in the social sciences.[38]
A simple regression equation has on the right hand side an intercept and an explanatory variable with a slope coefficient. A multiple regression e right hand side, each with its own slope coefficient
Regression analysis ... is probably one of the oldest topics in mathematical statistics dating back to about two hundred years ago. The earliest form of the linear regression was the least squares method, 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.
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| Correlation and dependence | |||||||||
| Regression analysis | |||||||||
| Regression as a statistical model |
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| Decomposition of variance | |||||||||
| Model exploration | |||||||||
| Background | |||||||||
| Design of experiments | |||||||||
| Numericalapproximation | |||||||||
| Applications | |||||||||
