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In statistics, linear regression is a form of regression analysis in which the relationship between one or more independent variables and another variable, called dependent variable, is modeled by a least squares function, called linear regression equation. This function is a linear combination of one or more model parameters, called regression coefficients. A linear regression equation with one independent variable represents a straight line. The results are subject to statistical analysis.

In general there are m parameters to be determined, $beta\_1,\; ldots,\; beta\_m$. The model is a linear combination of these parameters,

- $y\_i\; =\; sum\_\{j=1\}^\{m\}\; X\_\{ij\}\; beta\_j\; +\; varepsilon\_i.$

In simple linear regression the data model represents a straight line and can be written as

- $y\_i\; =\; beta\_1\; +\; x\_i\; beta\_2\; +\; varepsilon\_i$

A linear regression model need not be a linear function of the independent variable: linear in this context means that the conditional mean of y is linear in the parameters $beta$. For example, the model $y=beta\_1+beta\_2\; x+beta\_3\; x^2+varepsilon$ is linear in the parameters, $beta\_1,\; beta\_2$ and $beta\_3,$, but it is not linear in x, as $X\_\{i3\}=x\_i^2$, a nonlinear function of x_{i}. An illustration of this model is shown in the example, below.

If there is any linear dependence amongst the columns of X, then the parameters β cannot be estimated by least squares unless β is constrained, as, for example, by requiring the sum of some of its components to be 0. However, some linear combinations of the components of β may still be uniquely estimable in such cases. For example, the model

- $y\_i=beta\_1+x\_ibeta\_2+2x\_ibeta\_3+varepsilon\_i,$

- $y\_i=beta\_1+x\_i(beta\_2+2beta\_3)+varepsilon\_i,$

Classical assumptions for linear regression include the assumptions that the sample is selected at random from the population of interest, that the dependent variable is continuous on the real line, and that the error terms follow identical and independent normal distributions, that is, that the errors are i.i.d. and Gaussian. Note that these assumptions imply that the error term does not statistically depend on the values of the independent variables, that is, that $epsilon$ is statistically independent of the predictor variables X. This article adopts these assumptions unless otherwise stated. Note that in more advanced treatments all of these assumptions may be relaxed. In particular note that the assumption that the error terms are normally distributed is of no consequence unless the sample is very small because central limit theorems imply that, so long as the error terms have finite variance and are not too strongly correlated, the parameter estimates will be approximately normally distributed even when the underlying errors are not.

An equivalent, under these assumptions, formulation of simple linear regression that explicitly shows the linear regression as a model of conditional expectation can be given as

- $mbox\{E\}(y|x)\; =\; alpha\; +\; beta\; x.,$

The conditional distribution of y given x is a linear transformation of the distribution of the error term. Note that this expression follows from the assumption that the mean of $varepsilon$ is zero conditional on x.

- Scalars and vectors are denoted by lower case letters.
- Matrices are denoted by upper case letters.
- Parameters are denoted by Greek letters.
- Vectors and matrices are denoted by bold letters. Using the symbols defined above, for example, vector y is the column vector of y
_{1}, y_{2}, ..., y_{n}:

- $bold\; y\; =$

y_1

y_2

vdots

y_nend{pmatrix} .

- A parameter with a hat, such as $hatbeta\_j$, denotes a parameter estimator.

The first objective of regression analysis is to best-fit the data by adjusting the parameters of the model. Of the different criteria that can be used to define what constitutes a best fit, the least squares criterion is a very powerful one. The objective function, S, is defined as a sum of squared residuals, r_{i}

- $S=sum\_\{i=1\}^\{n\}r\_i^2\; quad\; bigg[text\; \{in\; matrix\; notation:\; \}\; S\; =\; bold\; \{r^T\; r\}\; bigg],$

where each residual is the difference between the observed value and the value calculated by the model:

- $r\_i\; =\; y\_i-sum\_\{j=1\}^\{m\}\; X\_\{ij\}\; hatbeta\_j$

From the theory of linear least squares, the parameter estimators are found by solving the normal equations

- $sum\_\{i=1\}^\{n\}sum\_\{k=1\}^\{m\}\; X\_\{ij\}X\_\{ik\}hat\; beta\_k=sum\_\{i=1\}^\{n\}\; X\_\{ij\}y\_i,~~~\; j=1,ldots,m.$

- $mathbf\{left(X^TXright)\; boldsymbol\; \{hat\; beta\}=X^Ty\}$,

And thus when the matrix $X^TX$ is non singular:

- $boldsymbol\; \{hat\; beta\}\; =\; mathbf\{left(X^TXright)^\{-1\}\; X^Ty\}$,

Denote by $sigma^2$ the variance of the error term $varepsilon$ (recall we assume that $varepsilon\_i\; sim\; N(0,sigma^2),$ for every $i=1,ldots,n$). An unbiased estimate of $sigma\; ^2$ is given by

- $hat\; sigma^2\; =\; frac\; \{S\}\; \{n-m\}.$

- $hatsigma^2\; cdot\; frac\{n-m\}\{sigma^2\}\; sim\; chi\_\{n-m\}^2$

The solution to the normal equations can be written as

- $hatboldsymbolbeta=(mathbf\{X^TX)^\{-1\}X^Ty\}.$

- $hatbeta\; sim\; N\; (beta,\; sigma^2\; (X^TX)^\{-1\}\; )$

where $N(cdot)$ denotes the multivariate normal distribution.

The standard error of a parameter estimator is given by

- $hatsigma\_j=sqrt\{\; frac\{S\}\{n-m\}left[mathbf\{(X^TX)\}^\{-1\}right]\_\{jj\}\}.$

- $hat\; beta\_j\; pm\; t\_\{frac\{alpha\; \}\{2\},n\; -\; m\}\; hat\; sigma\_j.$

The residuals can be expressed as

- $mathbf\{hat\; r\; =\; y-X\; hatboldsymbolbeta=\; y-X(X^TX)^\{-1\}X^Ty\}.,$

Given a value of the independent variable, x_{d}, the predicted response is calculated as

- $y\_d=sum\_\{j=1\}^\{m\}X\_\{dj\}hatbeta\_j.$

- $mathbf\{z^Thatboldsymbolbeta\}\; pm\; t\_\{\; frac\{alpha\; \}\{2\}\; ,n-m\}\; hat\; sigma\; sqrt\; \{mathbf\{\; z^T(X^T\; X)^\{-\; 1\}z\; \}\}.$

The $100(1-alpha)\%$ predicted response confidence intervals for the data are given by:

- $mathbf\; z^T\; hatboldsymbolbeta\; pm\; t\_\{frac\{alpha\; \}\{2\},n-m\}\; hat\; sigma\; sqrt\; \{1\; +\; mathbf\{z^T(X^TX)^\{-1\}z\}\}.$

In the case that the formula to be fitted is a straight line, $y=alpha+beta\; x!$, the normal equations are

- $begin\{array\}\{lcl\}$

where all summations are from i = 1 to i = n. In this case, β is called the regression coefficient.

Thence, by Cramer's rule,

- $hatbeta\; =\; frac\; \{n\; sum\; x\_iy\_i\; -\; sum\; x\_i\; sum\; y\_i\}\; \{Delta\}$

- $hatalpha\; =\; frac\; \{sum\; x\_i^2\; sum\; y\_i\; -\; sum\; x\_i\; sum\; x\_iy\_i\}\; \{Delta\}=\; bar\; y-bar\; x\; hatbeta$

- $S\; =\; sum\; (y\_i-hat\; y\_i)^2$

- $hat\; sigma^2\; =\; frac\; \{S\}\; \{n-2\}.$

where

- $Delta\; =\; nsum\; x\_i^2\; -\; left(sum\; x\_iright)^2$

The covariance matrix is

- $frac\{1\}\{Delta\}begin\{pmatrix\}$

sum x_i^2 & -sum x_i

-sum x_i & nend{pmatrix}

The mean response confidence interval is given by

- $y\_d\; =\; (alpha+hatbeta\; x\_d)\; pm\; t\_\{\; frac\{alpha\; \}\{2\}\; ,n-2\}\; hat\; sigma\; sqrt\; \{frac\{1\}\{n\}\; +\; frac\{(x\_d\; -\; bar\{x\})^2\}\{sum\; (x\_i\; -\; bar\{x\})^2\}\}$

The predicted response confidence interval is given by

- $y\_d\; =\; (alpha+hatbeta\; x\_d)\; pm\; t\_\{\; frac\{alpha\; \}\{2\}\; ,n-2\}\; hat\; sigma\; sqrt\; \{1+frac\{1\}\{n\}\; +\; frac\{(x\_d\; -\; bar\{x\})^2\}\{sum\; (x\_i\; -\; bar\{x\})^2\}\}$

The "total (corrected) sum of squares" is

- $text\{SST\}\; =\; sum\_\{i=1\}^n\; (y\_i\; -\; bar\; y)^2,$

where

- $bar\; y\; =\; frac\{1\}\{n\}\; sum\_i\; y\_i$

("corrected" means $scriptstylebar\; y,$ has been subtracted from each y-value).

The total sum of squares is partitioned as the sum of the "regression sum of squares" SSReg (or RSS, also called the "explained sum of squares") and the "error sum of squares" SSE, which is the sum of squares of residuals.

The regression sum of squares is

- $text\{SSReg\}\; =\; sum\; left(hat\; y\_i\; -\; bar\; y\; right)^2$

where u is an n-by-1 vector in which each element is 1. Note that

- $mathbf\{y^T\; u\}\; =\; mathbf\{u^T\; y\}\; =\; sum\_i\; y\_i,$

and

- $frac\{1\}\{n\}\; mathbf\{y^T\; u\; u^T\; y\}\; =\; frac\{1\}\{n\}left(sum\_i\; y\_iright)^2.,$

The error (or "unexplained") sum of squares SSE, which is the sum of square of residuals, is given by

- $text\{SSE\}\; =\; sum\_i\; \{left(\{y\_i\; -\; hat\; y\_i\}\; right)^2\; \}$

The total sum of squares SST is

- $text\{SST\}\; =\; sum\_i\; left(y\_i-bar\; y\; right)^2\; =\; mathbf\{\; y^T\; y\}-frac\{1\}\{n\}left(mathbf\{y^Tuu^Ty\}right)=text\{SSReg\}+\; text\{SSE\}.$

Pearson's coefficient of regression, R^{ 2} is then given as

- $R^2\; =\; frac\{text\{SSReg\}\}\}$

If the errors are independent and normally distributed with expected value 0 and they all have the same variance, then under the null hypothesis that all of the elements in β = 0 except the constant, the statistic

- $frac\{\; R^2\; /\; (m-1\; )\; \}\{\; (1\; -\; R^2\; )/(n-m)\}$

follows an F-distribution with (m-1) and (n-m) degrees of freedom. If that statistic is too large, then one rejects the null hypothesis. How large is too large depends on the level of the test, which is the tolerated probability of type I error; see statistical significance.

A plot of weight against height (see below) shows that it cannot be modeled by a straight line, so a regression is performed by modeling the data by a parabola.Height/ m 1.47 1.5 1.52 1.55 1.57 1.60 1.63 1.65 1.68 1.7 1.73 1.75 1.78 1.8 1.83 Weight/kg 52.21 53.12 54.48 55.84 57.2 58.57 59.93 61.29 63.11 64.47 66.28 68.1 69.92 72.19 74.46 - $y\; =\; beta\_0\; +\; beta\_1\; x+\; beta\_2\; x^2\; +varepsilon!$

- $X=$

The values of the parameters are found by solving the normal equations

- $mathbf\{(X^TX)boldsymbolhatbeta=X^Ty\}$

- $X\_\{ij\}=sum\_\{k=1\}^\{k=15\}\; X\_\{ki\}X\_\{kj\}$

- $left(mathbf\{X^Ty\}right)\_i=sum\; X\_\{ki\}\; y\_k$

Thus, the normal equations are

- $begin\{bmatrix\}$

begin{bmatrix} hatbeta_0 hatbeta_1 hatbeta_2 end{bmatrix} = begin{bmatrix} 931 1548 2586 end{bmatrix}

- $hatbeta\_0=129\; pm\; 16$ (value $pm$ standard deviation)

- $hatbeta\_1=-143\; pm\; 20$

- $hatbeta\_2=62\; pm\; 6$

The calculated values are given by

- $y^\{calc\}\_i\; =\; hatbeta\_0\; +\; hatbeta\_1\; x\_i+\; hatbeta\_2\; x^2\_i$

The confidence intervals are computed using:

- $[hat\{beta\_j\}-sigma\_j\; t\_\{m-n;1-frac\{alpha\}\{2\}\};hat\{beta\_j\}+sigma\_j\; t\_\{m-n;1-frac\{alpha\}\{2\}\}]$

with $alpha$=5%, $t\_\{m-n;1-frac\{alpha\}\{2\}\}$ = 2.2. Therefore, we can say that the 95% confidence intervals are:

- $beta\_0in[92.9,164.7]$

- $beta\_1in[-186.8,-99.5]$

- $beta\_2in[48.7,75.2]$

## Examining results of regression models

### Checking model assumptions

Some of the model assumptions can be evaluated by calculating the residuals and plotting or otherwise analyzing them. The following plots can be constructed to test the validity of the assumptions:- Residuals against the explanatory variables in the model, as illustrated above. The residuals should have no relation to these variables (look for possible non-linear relations) and the spread of the residuals should be the same over the whole range.
- Residuals against explanatory variables not in the model. Any relation of the residuals to these variables would suggest considering these variables for inclusion in the model.
- Residuals against the fitted values, $hatmathbf\; y,$.
- A time series plot of the residuals, that is, plotting the residuals as a function of time.
- Residuals against the preceding residual.
- A normal probability plot of the residuals to test normality. The points should lie along a straight line.

There should not be any noticeable pattern to the data in all but the last plot.

### Assessing goodness of fit

- The coefficient of determination gives what fraction of the observed variance of the response variable can be explained by the given variables.
- Examine the observational and prediction confidence intervals. In most contexts, the smaller they are the better.

## Other procedures

### Generalized least squares

Generalized least squares, which includes weighted least squares as a special case, can be used when the observational errors have unequal variance or serial correlation.### Errors-in-variables model

Errors-in-variables model or total least squares when the independent variables are subject to error### Generalized linear model

Generalized linear model is used when the distribution function of the errors is not a Normal distribution. Examples include exponential distribution, gamma distribution, inverse Gaussian distribution, Poisson distribution, binomial distribution, multinomial distribution

### Robust regression

A host of alternative approaches to the computation of regression parameters are included in the category known as robust regression. One technique minimizes the mean absolute error, or some other function of the residuals, instead of mean squared error as in linear regression. Robust regression is much more computationally intensive than linear regression and is somewhat more difficult to implement as well. While least squares estimates are not very sensitive to breaking the normality of the errors assumption, this is not true when the variance or mean of the error distribution is not bounded, or when an analyst that can identify outliers is unavailable.

Among Stata users, Robust regression is frequently taken to mean linear regression with Huber-White standard error estimates due to the naming conventions for regression commands. This procedure relaxes the assumption of homoscedasticity for variance estimates only; the predictors are still ordinary least squares (OLS) estimates. This occasionally leads to confusion; Stata users sometimes believe that linear regression is a robust method when this option is used, although it is actually not robust in the sense of outlier-resistance.

### Instrumental variables and related methods

The assumption that the error term in the linear model can be treated as uncorrelated with the independent variables will frequently be untenable, as omitted-variables bias, "reverse" causation, and errors-in-variables problems can generate such a correlation. Instrumental variable and other methods can be used in such cases.## Applications of linear regression

Linear regression is widely used in biological, behavioral and social sciences to describe relationships between variables. It ranks as one of the most important tools used in these disciplines.### The trend line

- For trend lines as used in technical analysis, see Trend lines (technical analysis)

A trend line represents a trend, the long-term movement in time 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.

### Epidemiology

As one example, early evidence relating tobacco smoking to mortality and morbidity came from studies employing regression. Researchers usually include several variables in their regression analysis in an effort to remove factors that might produce spurious correlations. For the cigarette smoking example, researchers might include socio-economic status in addition to smoking to ensure that any observed effect of smoking on mortality is not due to some effect of education or income. However, it is never possible to include all possible confounding variables in a study employing regression. For the smoking 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 correlational analysis using linear regression. When controlled experiments are not feasible, variants of regression analysis such as instrumental variables and other methods may be used to attempt to estimate causal relationships from observational data.### Finance

The capital asset pricing model uses linear regression as well as the concept of Beta 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.Regression may not be the appropriate way to estimate beta in finance given that it is supposed to provide the volatility of an investment relative to the volatility of the market as a whole. This would require that both these variables be treated in the same way when estimating the slope. Whereas regression treats all variability as being in the investment returns variable, i.e. it only considers residuals in the dependent variable.

## See also

- Anscombe's quartet
- Cross-sectional regression
- Econometrics
- Empirical Bayes methods
- Hierarchical linear modeling
- Instrumental variable
- Least-squares estimation of linear regression coefficients
- M-estimator
- Nonlinear regression
- Nonparametric regression
- Multivariate adaptive regression splines
- Segmented regression
- Ridge regression
- Robust regression
- Lack-of-fit sum of squares

## References

### Additional sources

- Cohen, J., Cohen P., West, S.G., & Aiken, L.S. (2003). Applied multiple regression/correlation analysis for the behavioral sciences. (2nd ed.) Hillsdale, NJ: Lawrence Erlbaum Associates
- Charles Darwin. The Variation of Animals and Plants under Domestication. (1869) (Chapter XIII describes what was known about reversion in Galton's time. Darwin uses the term "reversion".)
- Draper, N.R. and Smith, H. Applied Regression Analysis Wiley Series in Probability and Statistics (1998)
- Francis Galton. "Regression Towards Mediocrity in Hereditary Stature," Journal of the Anthropological Institute, 15:246-263 (1886). (Facsimile at: )
- Robert S. Pindyck and Daniel L. Rubinfeld (1998, 4h ed.). Econometric Models and Economic Forecasts,, ch. 1 (Intro, incl. appendices on Σ operators & derivation of parameter est.) & Appendix 4.3 (mult. regression in matrix form).

## External links

- http://homepage.mac.com/nshoffner/nsh/CalcBookAll/Chapter%201/1functions.html
- Investment Volatility: A Critique of Standard Beta Estimation and a Simple Way Forward, C.TofallisDownloadable version of paper, subsequently published in the European Journal of Operational Research 2008.
- Scale-adaptive nonparametric regression (with Matlab software).
- Visual Least Squares: An interactive, visual flash demonstration of how linear regression works.
- In Situ Adaptive Tabulation: Combining many linear regressions to approximate any nonlinear function.
- Earliest Known uses of some of the Words of Mathematics See: for "error", for "Gauss-Markov theorem", for "method of least squares", and for "regression".
- Online linear regression calculator.
- Perpendicular Regression Of a Line at MathPages
- Online regression by eye (simulation).
- Leverage Effect Interactive simulation to show the effect of outliers on the regression results
- Linear regression as an optimisation problem
- Visual Statistics with Multimedia
- Multiple Regression by Elmer G. Wiens. Online multiple and restricted multiple regression package.
- ZunZun.com Online curve and surface fitting.
- CAUSEweb.org Many resources for teaching statistics including Linear Regression.
- Multivariate Regression Python, Smalltalk & Java Implementation of Linear Regression Calculation.
- Matlab SUrrogate MOdeling Toolbox - SUMO Toolbox - Matlab code for Active Learning + Model Selection + Surrogate Model Regression
- "Mahler's Guide to Regression"

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This article is licensed under the GNU Free Documentation License.

Last updated on Thursday October 09, 2008 at 17:43:46 PDT (GMT -0700)

View this article at Wikipedia.org - Edit this article at Wikipedia.org - Donate to the Wikimedia Foundation

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