Instatistics,simple linear regression (SLR) is alinear regression model with a singleexplanatory variable.^{[1][2][3][4][5]} That is, it concerns two-dimensional sample points withone independent variable and one dependent variable (conventionally, thex andy coordinates in aCartesian coordinate system) and finds a linear function (a non-verticalstraight line) that, as accurately as possible, predicts the dependent variable values as a function of the independent variable.The adjectivesimple refers to the fact that the outcome variable is related to a single predictor.

It is common to make the additional stipulation that theordinary least squares (OLS) method should be used: the accuracy of each predicted value is measured by its squaredresidual (vertical distance between the point of the data set and the fitted line), and the goal is to make the sum of these squared deviations as small as possible. In this case, the slope of the fitted line is equal to thecorrelation betweeny andx corrected by the ratio of standard deviations of these variables. The intercept of the fitted line is such that the line passes through the center of mass(x,y) of the data points.

Consider themodel function$${\displaystyle y=\alpha +\beta x,}$$which describes a line with slopeβ andy-interceptα. In general, such a relationship may not hold exactly for the largely unobserved population of values of the independent and dependent variables; we call the unobserved deviations from the above equation theerrors. Suppose we observen data pairs and call them{(x_i,y_i),i = 1, ...,n}. We can describe the underlying relationship betweeny_i andx_i involving this error termε_i by

$${\displaystyle y_i=\alpha +\beta x_i+{\epsilon }_i.}$$

This relationship between the true (but unobserved) underlying parametersα andβ and the data points is called a linear regression model.

The goal is to find estimated values${\displaystyle \hat{\alpha }}$ and${\displaystyle \hat{\beta }}$ for the parametersα andβ which would provide the "best" fit in some sense for the data points. As mentioned in the introduction, in this article the "best" fit will be understood as in theleast-squares approach: a line that minimizes thesum of squared residuals (see alsoErrors and residuals)${\displaystyle {\hat{\epsilon }}_i}$ (differences between actual and predicted values of the dependent variabley), each of which is given by, for any candidate parameter values${\displaystyle \alpha }$ and${\displaystyle \beta }$,

$${\displaystyle {\hat{\epsilon }}_i=y_i-\alpha -\beta x_i.}$$

In other words,${\displaystyle \hat{\alpha }}$ and${\displaystyle \hat{\beta }}$ solve the followingminimization problem:

$${\displaystyle (\hat{\alpha },\hat{\beta })=\mathrm{argmin}\left(Q(\alpha ,\beta )\right),}$$where theobjective functionQ is:$${\displaystyle Q(\alpha ,\beta )=\sum _{i=1}^n{\hat{\epsilon }}_i^2=\sum _{i=1}^n(y_i-\alpha -\beta x_i{)}^2.}$$

By expanding to get a quadratic expression in${\displaystyle \alpha }$ and${\displaystyle \beta ,}$ we can derive minimizing values of the function arguments, denoted${\displaystyle \hat{\alpha }}$ and${\displaystyle \hat{\beta }}$:^[6]

Here we have introduced

The above equations are efficient to use if the mean of the x and y variables (${\displaystyle \overline{x}\text{ and }\overline{y}}$) are known. If the means are not known at the time of calculation, it may be more efficient to use the expanded version of the${\displaystyle \hat{\alpha }\text{ and }\hat{\beta }}$ equations. These expanded equations may be derived from the more generalpolynomial regression equations^[7][8] by defining the regression polynomial to be of order 1, as follows.

The abovesystem of linear equations may be solved directly, or stand-alone equations for${\displaystyle \hat{\alpha }\text{ and }\hat{\beta }}$ may be derived by expanding the matrix equations above. The resultant equations are algebraically equivalent to the ones shown in the prior paragraph, and are shown below without proof.^[9][7]

The solution can be reformulated using elements of thecovariance matrix:$${\displaystyle \hat{\beta }=\frac{s_{x,y}}{s_x^2}=r_{xy}\frac{s_y}{s_x}}$$

where

Substituting the above expressions for${\displaystyle \hat{\alpha }}$ and${\displaystyle \hat{\beta }}$ into the original solution yields

$${\displaystyle \frac{y-\overline{y}}{s_y}=r_{xy}\frac{x-\overline{x}}{s_x}.}$$

This shows thatr_xy is the slope of the regression line of thestandardized data points (and that this line passes through the origin). Since${\displaystyle -1\le r_{xy}\le 1}$ then we get that if x is some measurement and y is a followup measurement from the same item, then we expect that y (on average) will be closer to the mean measurement than it was to the original value of x. This phenomenon is known asregressions toward the mean.

Generalizing the${\displaystyle \overline{x}}$ notation, we can write a horizontal bar over an expression to indicate the average value of that expression over the set of samples. For example:

$${\displaystyle \overline{xy}=\frac{1}{n}\sum _{i=1}^n{x}_i{y}_i.}$$

This notation allows us a concise formula forr_xy:

$${\displaystyle r_{xy}=\frac{\overline{xy}-\overline{x}\overline{y}}{\sqrt{\left(\overline{x^2}-{\overline{x}}^2\right)\left(\overline{y^2}-{\overline{y}}^2\right)}}.}$$

Thecoefficient of determination ("R squared") is equal to${\displaystyle r_{xy}^2}$ when the model is linear with a single independent variable. Seesample correlation coefficient for additional details.

By multiplying all members of the summation in the numerator by :${\displaystyle \frac{x_i-\overline{x}}{x_i-\overline{x}}=1}$ (thereby not changing it):

We can see that the slope (tangent of angle) of the regression line is the weighted average of${\displaystyle \frac{y_i-\overline{y}}{x_i-\overline{x}}}$ that is the slope (tangent of angle) of the line that connects the i-th point to the average of all points, weighted by${\displaystyle (x_i-\overline{x}{)}^2}$ because the further the point is the more "important" it is, since small errors in its position will affect the slope connecting it to the center point more.

The parameter${\displaystyle \hat{\alpha }}$ is the intercept of the linear function. Therefore, the${\displaystyle y}$-intercept of the function found with simple linear regression is

$${\displaystyle y_{\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{c}\mathrm{e}\mathrm{p}\mathrm{t}}=\hat{\alpha }=\overline{y}-\hat{\beta }\overline{x}}$$.

Because${\displaystyle \hat{\beta }}$ is the slope of the linear function,${\displaystyle \hat{\beta }=\tan (\theta )}$. Therefore, the angle${\displaystyle \theta }$ the graph of the function makes with the${\displaystyle x}$ axis is equal to

$${\displaystyle \theta =\arctan (\hat{\beta })}$$.

In the above formulation, notice that each${\displaystyle x_i}$ is a constant ("known upfront") value, while the${\displaystyle y_i}$ are random variables that depend on the linear function of${\displaystyle x_i}$ and the random term${\displaystyle {\epsilon }_i}$. This assumption is used when deriving the standard error of the slope and showing that it isunbiased.

In this framing, when${\displaystyle x_i}$ is not actually arandom variable, what type of parameter does the empirical correlation${\displaystyle r_{xy}}$ estimate? The issue is that for each value i we'll have:${\displaystyle E(x_i)=x_i}$ and${\displaystyle Var(x_i)=0}$. A possible interpretation of${\displaystyle r_{xy}}$ is to imagine that${\displaystyle x_i}$ defines a random variable drawn from theempirical distribution of the x values in our sample. For example, if x had 10 values from thenatural numbers: [1,2,3...,10], then we can imagine x to be aDiscrete uniform distribution. Under this interpretation all${\displaystyle x_i}$ have the same expectation and some positive variance. With this interpretation we can think of${\displaystyle r_{xy}}$ as the estimator of thePearson's correlation between the random variable y and the random variable x (as we just defined it).

Description of the statistical properties of estimators from the simple linear regression estimates requires the use of astatistical model. The following is based on assuming the validity of a model under which the estimates are optimal. It is also possible to evaluate the properties under other assumptions, such asinhomogeneity, but this is discussed elsewhere.^{[clarification needed]}

The estimators${\displaystyle \hat{\alpha }}$ and${\displaystyle \hat{\beta }}$ areunbiased.

To formalize this assertion we must define a framework in which these estimators are random variables. We consider the residualsε_i as random variables drawn independently from some distribution with mean zero. In other words, for each value ofx, the corresponding value ofy is generated as a mean responseα +βx plus an additional random variableε called theerror term, equal to zero on average. Under such interpretation, the least-squares estimators${\displaystyle \hat{\alpha }}$ and${\displaystyle \hat{\beta }}$ will themselves be random variables whose means will equal the "true values"α andβ. This is the definition of an unbiased estimator.

Since the data in this context is defined to be (x,y) pairs for every observation, themean response at a given value ofx, sayx_d, is an estimate of the mean of they values in the population at thex value ofx_d, that is${\displaystyle \hat{E}(y\mid x_d)\equiv {\hat{y}}_d}$. The variance of the mean response is given by:^[11]

$${\displaystyle \mathrm{Var}\left(\hat{\alpha }+\hat{\beta }{x}_d\right)=\mathrm{Var}\left(\hat{\alpha }\right)+\left(\mathrm{Var}\hat{\beta }\right)x_d^2+2x_d\mathrm{Cov}\left(\hat{\alpha },\hat{\beta }\right).}$$

This expression can be simplified to

$${\displaystyle \mathrm{Var}\left(\hat{\alpha }+\hat{\beta }{x}_d\right)={\sigma }^2\left(\frac{1}{m}+\frac{{\left(x_d-\overline{x}\right)}^2}{\sum (x_i-\overline{x}{)}^2}\right),}$$

wherem is the number of data points.

To demonstrate this simplification, one can make use of the identity

$${\displaystyle \sum _i(x_i-\overline{x}{)}^2=\sum _i{x}_i^2-\frac{1}{m}{\left(\sum _i{x}_i\right)}^2.}$$

Thepredicted response distribution is the predicted distribution of the residuals at the given pointx_d. So the variance is given by

The second line follows from the fact that${\displaystyle \mathrm{Cov}\left(y_d,\left[\hat{\alpha }+\hat{\beta }{x}_d\right]\right)}$ is zero because the new prediction point is independent of the data used to fit the model. Additionally, the term${\displaystyle \mathrm{Var}\left(\hat{\alpha }+\hat{\beta }{x}_d\right)}$ was calculated earlier for the mean response.

Since${\displaystyle \mathrm{Var}(y_d)={\sigma }^2}$ (a fixed but unknown parameter that can be estimated), the variance of the predicted response is given by

The formulas given in the previous section allow one to calculate thepoint estimates ofα andβ — that is, the coefficients of the regression line for the given set of data. However, those formulas do not tell us how precise the estimates are, i.e., how much the estimators${\displaystyle \hat{\alpha }}$ and${\displaystyle \hat{\beta }}$ vary from sample to sample for the specified sample size.Confidence intervals were devised to give a plausible set of values to the estimates one might have if one repeated the experiment a very large number of times.

The standard method of constructing confidence intervals for linear regression coefficients relies on the normality assumption, which is justified if either:

1. the errors in the regression arenormally distributed (the so-calledclassic regression assumption), or
2. the number of observationsn is sufficiently large, in which case the estimator is approximately normally distributed.

The latter case is justified by thecentral limit theorem.

Under the first assumption above, that of the normality of the error terms, the estimator of the slope coefficient will itself be normally distributed with meanβ and variance${\sigma }^2/\sum _i(x_i-\overline{x}{)}^2,$ whereσ² is the variance of the error terms (seeProofs involving ordinary least squares). At the same time the sum of squared residualsQ is distributed proportionally toχ² withn − 2 degrees of freedom, and independently from${\displaystyle \hat{\beta }}$. This allows us to construct at-value

$${\displaystyle t=\frac{\hat{\beta }-\beta }{s_{\hat{\beta }}}\sim t_{n-2},}$$

where

$${\displaystyle s_{\hat{\beta }}=\sqrt{\frac{\frac{1}{n-2}\sum _{i=1}^n{\hat{\epsilon }}_i^2}{\sum _{i=1}^n(x_i-\overline{x}{)}^2}}}$$

is the unbiasedstandard error estimator of the estimator${\displaystyle \hat{\beta }}$.

Thist-value has aStudent'st-distribution withn − 2 degrees of freedom. Using it we can construct a confidence interval forβ:

$${\displaystyle \beta \in \left[\hat{\beta }-s_{\hat{\beta }}{t}_{n-2}^{\ast },\hat{\beta }+s_{\hat{\beta }}{t}_{n-2}^{\ast }\right],}$$

at confidence level(1 −γ), where${\displaystyle t_{n-2}^{\ast }}$ is the${\displaystyle \left(1-\frac{\gamma }{2}\right)\text{-th}}$ quantile of thet_n−2 distribution. For example, ifγ = 0.05 then the confidence level is 95%.

Similarly, the confidence interval for the intercept coefficientα is given by

$${\displaystyle \alpha \in \left[\hat{\alpha }-s_{\hat{\alpha }}{t}_{n-2}^{\ast },\hat{\alpha }+s_{\hat{\alpha }}{t}_{n-2}^{\ast }\right],}$$

at confidence level (1 −γ), where

$${\displaystyle s_{\hat{\alpha }}=s_{\hat{\beta }}\sqrt{\frac{1}{n}\sum _{i=1}^n{x}_i^2}=\sqrt{\frac{1}{n(n-2)}\left(\sum _{i=1}^n{\hat{\epsilon }}_i^2\right)\frac{\sum _{i=1}^n{x}_i^2}{\sum _{i=1}^n(x_i-\overline{x}{)}^2}}}$$

The confidence intervals forα andβ give us the general idea where these regression coefficients are most likely to be. For example, in theOkun's law regression shown here the point estimates are

$${\displaystyle \hat{\alpha }=0.859,\hat{\beta }=-\mathrm{1.817.}}$$

The 95% confidence intervals for these estimates are

$${\displaystyle \alpha \in \left[0.76,0.96\right],\beta \in \left[-2.06,-1.58\right].}$$

In order to represent this information graphically, in the form of the confidence bands around the regression line, one has to proceed carefully and account for the joint distribution of the estimators. It can be shown^[12] that at confidence level (1 − γ) the confidence band has hyperbolic form given by the equation

$${\displaystyle (\alpha +\beta \xi )\in \left[\hat{\alpha }+\hat{\beta }\xi \pm t_{n-2}^{\ast }\sqrt{\left(\frac{1}{n-2}\sum {\hat{\epsilon }}_i^2\right)\cdot \left(\frac{1}{n}+\frac{(\xi -\overline{x}{)}^2}{\sum (x_i-\overline{x}{)}^2}\right)}\right].}$$

When the model assumed the intercept is fixed and equal to 0 (${\displaystyle \alpha =0}$), the standard error of the slope turns into:

$${\displaystyle s_{\hat{\beta }}=\sqrt{\frac{1}{n-1}\frac{\sum _{i=1}^n{\hat{\epsilon }}_i^2}{\sum _{i=1}^n{x}_i^2}}}$$

With:${\displaystyle {\hat{\epsilon }}_i=y_i-{\hat{y}}_i}$

The alternative second assumption states that when the number of points in the dataset is "large enough", thelaw of large numbers and thecentral limit theorem become applicable, and then the distribution of the estimators is approximately normal. Under this assumption all formulas derived in the previous section remain valid, with the only exception that the quantilet*_n−2 ofStudent'st distribution is replaced with the quantileq* of thestandard normal distribution. Occasionally the fraction⁠1/n−2⁠ is replaced with⁠1/n⁠. Whenn is large such a change does not alter the results appreciably.

This data set gives average masses for women as a function of their height in a sample of American women of age 30–39. Although theOLS article argues that it would be more appropriate to run a quadratic regression for this data, the simple linear regression model is applied here instead.

${\displaystyle i}$	${\displaystyle x_i}$	${\displaystyle y_i}$	${\displaystyle x_i^2}$	${\displaystyle x_i{y}_i}$	${\displaystyle y_i^2}$
1	1.47	52.21	2.1609	76.7487	2725.8841
2	1.50	53.12	2.2500	79.6800	2821.7344
3	1.52	54.48	2.3104	82.8096	2968.0704
4	1.55	55.84	2.4025	86.5520	3118.1056
5	1.57	57.20	2.4649	89.8040	3271.8400
6	1.60	58.57	2.5600	93.7120	3430.4449
7	1.63	59.93	2.6569	97.6859	3591.6049
8	1.65	61.29	2.7225	101.1285	3756.4641
9	1.68	63.11	2.8224	106.0248	3982.8721
10	1.70	64.47	2.8900	109.5990	4156.3809
11	1.73	66.28	2.9929	114.6644	4393.0384
12	1.75	68.10	3.0625	119.1750	4637.6100
13	1.78	69.92	3.1684	124.4576	4888.8064
14	1.80	72.19	3.2400	129.9420	5211.3961
15	1.83	74.46	3.3489	136.2618	5544.2916
${\displaystyle \mathrm{\Sigma }}$	24.76	931.17	41.0532	1548.2453	58498.5439

There aren = 15 points in this data set. Hand calculations would be started by finding the following five sums:

These quantities would be used to calculate the estimates of the regression coefficients, and their standard errors.

The 0.975 quantile of Student'st-distribution with 13 degrees of freedom ist^*₁₃ = 2.1604, and thus the 95% confidence intervals forα andβ are

Theproduct-moment correlation coefficient might also be calculated:

$${\displaystyle \hat{r}=\frac{n{S}_{xy}-S_x{S}_y}{\sqrt{(n{S}_{xx}-S_x^2)(n{S}_{yy}-S_y^2)}}=0.9946}$$

In SLR, there is an underlying assumption that only the dependent variable contains measurement error; if the explanatory variable is also measured with error, then simple regression is not appropriate for estimating the underlying relationship because it will be biased due toregression dilution.

Other estimation methods that can be used in place of ordinary least squares includeleast absolute deviations (minimizing the sum of absolute values of residuals) and theTheil–Sen estimator (which chooses a line whoseslope is themedian of the slopes determined by pairs of sample points).

Deming regression (total least squares) also finds a line that fits a set of two-dimensional sample points, but (unlike ordinary least squares, least absolute deviations, and median slope regression) it is not really an instance of simple linear regression, because it does not separate the coordinates into one dependent and one independent variable and could potentially return a vertical line as its fit. can lead to a model that attempts to fit the outliers more than the data.

This section is an excerpt fromLine fitting.[edit]

This page is aprimary topic and an article should be written about it. One or more editors believe it holds the title of abroad-concept article. The article may be written here ordrafted elsewhere first. Related titles should be described here, while unrelated titles should be moved toSimple linear regression (disambiguation). Relevant discussion may be found on thetalk page.(May 2019)

Line fitting is the process of constructing astraight line that has the best fit to a series of data points.

Several methods exist, considering:

• Vertical distance: Simple linear regression
• Resistance tooutliers:Robust simple linear regression
• Perpendicular distance:Orthogonal regression (this is not scale-invariant i.e. changing the measurement units leads to a different line.)
• Weighted geometric distance:Deming regression
• Scale invariant approach:Major axis regression This allows for measurement error in both variables, and gives an equivalent equation if the measurement units are altered.

Sometimes it is appropriate to force the regression line to pass through the origin, becausex andy are assumed to be proportional. For the model without the intercept term,y =βx, the OLS estimator forβ simplifies to

$${\displaystyle \hat{\beta }=\frac{\sum _{i=1}^n{x}_i{y}_i}{\sum _{i=1}^n{x}_i^2}=\frac{\overline{xy}}{\overline{x^2}}}$$

Substituting(x −h,y −k) in place of(x,y) gives the regression through(h,k):

where Cov and Var refer to the covariance and variance of the sample data (uncorrected for bias).The last form above demonstrates how moving the line away from the center of mass of the data points affects the slope.

• Design matrix § Simple linear regression
• Linear trend estimation
• Linear segmented regression
• Proofs involving ordinary least squares—derivation of all formulas used in this article in general multidimensional case
• Newey–West estimator

• Wolfram MathWorld's explanation of Least Squares Fitting, and how to calculate it
• Mathematics of simple regression (Robert Nau, Duke University)

• Curve fitting
• Regression analysis
• Parametric statistics

• Articles with short description
• Short description matches Wikidata
• Wikipedia articles needing clarification from October 2015
• Articles with excerpts