Simple linear regression

Last updated
Okun's law in macroeconomics is an example of the simple linear regression. Here the dependent variable (GDP growth) is presumed to be in a linear relationship with the changes in the unemployment rate. Okuns law quarterly differences.svg
Okun's law in macroeconomics is an example of the simple linear regression. Here the dependent variable (GDP growth) is presumed to be in a linear relationship with the changes in the unemployment rate.

In statistics, simple linear regression is a linear regression model with a single explanatory variable. [1] [2] [3] [4] [5] That is, it concerns two-dimensional sample points with one independent variable and one dependent variable (conventionally, the x and y coordinates in a Cartesian coordinate system) and finds a linear function (a non-vertical straight line) that, as accurately as possible, predicts the dependent variable values as a function of the independent variable. The adjective simple refers to the fact that the outcome variable is related to a single predictor.

Contents

It is common to make the additional stipulation that the ordinary least squares (OLS) method should be used: the accuracy of each predicted value is measured by its squared residual (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. Other regression methods that can be used in place of ordinary least squares include least absolute deviations (minimizing the sum of absolute values of residuals) and the Theil–Sen estimator (which chooses a line whose slope is the median 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.

The remainder of the article assumes an ordinary least squares regression. In this case, the slope of the fitted line is equal to the correlation between y and x 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.

Fitting the regression line

Consider the model function

which describes a line with slope β and y-intercept α. 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 the errors. Suppose we observe n data pairs and call them {(xi, yi), i = 1, ..., n}. We can describe the underlying relationship between yi and xi involving this error term εi by

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

The goal is to find estimated values and for the parameters α and β 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 the least-squares approach: a line that minimizes the sum of squared residuals (see also Errors and residuals) (differences between actual and predicted values of the dependent variable y), each of which is given by, for any candidate parameter values and ,

In other words, and solve the following minimization problem:

By expanding to get a quadratic expression in and we can derive values of and that minimize the objective function Q (these minimizing values are denoted and ): [6]

Here we have introduced

Substituting the above expressions for and into

yields

This shows that rxy is the slope of the regression line of the standardized data points (and that this line passes through the origin). Since 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. A phenomena known as regressions toward the mean.

Generalizing the 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:

This notation allows us a concise formula for rxy:

The coefficient of determination ("R squared") is equal to when the model is linear with a single independent variable. See sample correlation coefficient for additional details.

Intuition about the slope

By multiplying all members of the summation in the numerator by : (thereby not changing it):

We can see that the slope (tangent of angle) of the regression line is the weighted average of that is the slope (tangent of angle) of the line that connects the i-th point to the average of all points, weighted by 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.

Intuition about the intercept

Given with the angle the line makes with the positive x axis, we have

Intuition about the correlation

In the above formulation, notice that each is a constant ("known upfront") value, while the are random variables that depend on the linear function of and the random term . This assumption is used when deriving the standard error of the slope and showing that it is unbiased.

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

Simple linear regression without the intercept term (single regressor)

Sometimes it is appropriate to force the regression line to pass through the origin, because x and y are assumed to be proportional. For the model without the intercept term, y = βx, the OLS estimator for β simplifies to

Substituting (xh, yk) 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.

Numerical properties

  1. The regression line goes through the center of mass point, , if the model includes an intercept term (i.e., not forced through the origin).
  2. The sum of the residuals is zero if the model includes an intercept term:
  3. The residuals and x values are uncorrelated (whether or not there is an intercept term in the model), meaning:
  4. The relationship between (the correlation coefficient for the population) and the population variances of () and the error term of () is: [7] :401
    For extreme values of this is self evident. Since when then . And when then .

Model-based properties

Description of the statistical properties of estimators from the simple linear regression estimates requires the use of a statistical 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 as inhomogeneity, but this is discussed elsewhere.[ clarification needed ]

Unbiasedness

The estimators and are unbiased.

To formalize this assertion we must define a framework in which these estimators are random variables. We consider the residuals εi as random variables drawn independently from some distribution with mean zero. In other words, for each value of x, the corresponding value of y is generated as a mean response α + βx plus an additional random variable ε called the error term, equal to zero on average. Under such interpretation, the least-squares estimators and will themselves be random variables whose means will equal the "true values" α and β. This is the definition of an unbiased estimator.

Confidence intervals

The formulas given in the previous section allow one to calculate the point estimates of α and β — that is, the coefficients of the regression line for the given set of data. However, those formulas don't tell us how precise the estimates are, i.e., how much the estimators and 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 are normally distributed (the so-called classic regression assumption), or
  2. the number of observations n is sufficiently large, in which case the estimator is approximately normally distributed.

The latter case is justified by the central limit theorem.

Normality assumption

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 mean β and variance where σ2 is the variance of the error terms (see Proofs involving ordinary least squares). At the same time the sum of squared residuals Q is distributed proportionally to χ2 with n − 2 degrees of freedom, and independently from . This allows us to construct a t-value

where

is the standard error of the estimator .

This t-value has a Student's t-distribution with n − 2 degrees of freedom. Using it we can construct a confidence interval for β:

at confidence level (1 − γ), where is the quantile of the tn−2 distribution. For example, if γ = 0.05 then the confidence level is 95%.

Similarly, the confidence interval for the intercept coefficient α is given by

at confidence level (1 − γ), where

The US "changes in unemployment - GDP growth" regression with the 95% confidence bands. Okuns law with confidence bands.svg
The US "changes in unemployment – GDP growth" regression with the 95% confidence bands.

The confidence intervals for α and β give us the general idea where these regression coefficients are most likely to be. For example, in the Okun's law regression shown here the point estimates are

The 95% confidence intervals for these estimates are

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 [8] that at confidence level (1  γ) the confidence band has hyperbolic form given by the equation

Asymptotic assumption

The alternative second assumption states that when the number of points in the dataset is "large enough", the law of large numbers and the central 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 quantile t*n−2 of Student's t distribution is replaced with the quantile q* of the standard normal distribution. Occasionally the fraction 1/n−2 is replaced with 1/n. When n is large such a change does not alter the results appreciably.

Numerical example

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 the OLS 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.

Height (m), xi1.471.501.521.551.571.601.631.651.681.701.731.751.781.801.83
Mass (kg), yi52.2153.1254.4855.8457.2058.5759.9361.2963.1164.4766.2868.1069.9272.1974.46
11.4752.212.160976.74872725.8841
21.5053.122.250079.68002821.7344
31.5254.482.310482.80962968.0704
41.5555.842.402586.55203118.1056
51.5757.202.464989.80403271.8400
61.6058.572.560093.71203430.4449
71.6359.932.656997.68593591.6049
81.6561.292.7225101.12853756.4641
91.6863.112.8224106.02483982.8721
101.7064.472.8900109.59904156.3809
111.7366.282.9929114.66444393.0384
121.7568.103.0625119.17504637.6100
131.7869.923.1684124.45764888.8064
141.8072.193.2400129.94205211.3961
151.8374.463.3489136.26185544.2916
24.76931.1741.05321548.245358498.5439

There are n = 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.

Graph of points and linear least squares lines in the simple linear regression numerical example OLS example weight vs height fitted linear.svg
Graph of points and linear least squares lines in the simple linear regression numerical example

The 0.975 quantile of Student's t-distribution with 13 degrees of freedom is t*13 = 2.1604, and thus the 95% confidence intervals for α and β are

The product-moment correlation coefficient might also be calculated:

This example also demonstrates that sophisticated calculations will not overcome the use of badly prepared data. The heights were originally given in inches, and have been converted to the nearest centimetre. Since the conversion has introduced rounding error, this is not an exact conversion. The original inches can be recovered by Round(x/0.0254) and then re-converted to metric without rounding: if this is done, the results become

Thus a seemingly small variation in the data has a real effect.

See also

Related Research Articles

Exponential distribution Probability distribution

In probability theory and statistics, the exponential distribution is the probability distribution of the time between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant average rate. It is a particular case of the gamma distribution. It is the continuous analogue of the geometric distribution, and it has the key property of being memoryless. In addition to being used for the analysis of Poisson point processes it is found in various other contexts.

Least squares Approximation method in statistics

The method of least squares is a standard approach in regression analysis to approximate the solution of overdetermined systems by minimizing the sum of the squares of the residuals made in the results of each individual equation.

In statistics, the Gauss–Markov theorem states that the ordinary least squares (OLS) estimator has the lowest sampling variance within the class of linear unbiased estimators, if the errors in the linear regression model are uncorrelated, have equal variances and expectation value of zero. The errors do not need to be normal, nor do they need to be independent and identically distributed. The requirement that the estimator be unbiased cannot be dropped, since biased estimators exist with lower variance. See, for example, the James–Stein estimator, ridge regression, or simply any degenerate estimator.

Deming regression

In statistics, Deming regression, named after W. Edwards Deming, is an errors-in-variables model which tries to find the line of best fit for a two-dimensional dataset. It differs from the simple linear regression in that it accounts for errors in observations on both the x- and the y- axis. It is a special case of total least squares, which allows for any number of predictors and a more complicated error structure.

In statistics, a studentized residual is the quotient resulting from the division of a residual by an estimate of its standard deviation. It is a form of a Student's t-statistic, with the estimate of error varying between points.

Regression analysis Set of statistical processes for estimating the relationships among variables

In statistical modeling, regression analysis is a set of statistical processes for estimating the relationships 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.

Coefficient of determination Indicator for how well data points fit a line or curve

In statistics, the coefficient of determination, denoted R2 or r2 and pronounced "R squared", is the proportion of the variation in the dependent variable that is predictable from the independent variable(s).

In statistics, a probit model is a type of regression where the dependent variable can take only two values, for example married or not married. The word is a portmanteau, coming from probability + unit. The purpose of the model is to estimate the probability that an observation with particular characteristics will fall into a specific one of the categories; moreover, classifying observations based on their predicted probabilities is a type of binary classification model.

In statistics, ordinary least squares (OLS) is a type of linear least squares method for estimating the unknown parameters in a linear regression model. OLS chooses the parameters of a linear function of a set of explanatory variables by the principle of least squares: minimizing the sum of the squares of the differences between the observed dependent variable in the given dataset and those predicted by the linear function of the independent variable.

In statistics, the residual sum of squares (RSS), also known as the sum of squared residuals (SSR) or the sum of squared estimate of errors (SSE), is the sum of the squares of residuals. It is a measure of the discrepancy between the data and an estimation model, such as a linear regression. A small RSS indicates a tight fit of the model to the data. It is used as an optimality criterion in parameter selection and model selection.

Difference in differences is a statistical technique used in econometrics and quantitative research in the social sciences that attempts to mimic an experimental research design using observational study data, by studying the differential effect of a treatment on a 'treatment group' versus a 'control group' in a natural experiment. It calculates the effect of a treatment on an outcome by comparing the average change over time in the outcome variable for the treatment group to the average change over time for the control group. Although it is intended to mitigate the effects of extraneous factors and selection bias, depending on how the treatment group is chosen, this method may still be subject to certain biases.

In statistics, a fixed effects model is a statistical model in which the model parameters are fixed or non-random quantities. This is in contrast to random effects models and mixed models in which all or some of the model parameters are random variables. In many applications including econometrics and biostatistics a fixed effects model refers to a regression model in which the group means are fixed (non-random) as opposed to a random effects model in which the group means are a random sample from a population. Generally, data can be grouped according to several observed factors. The group means could be modeled as fixed or random effects for each grouping. In a fixed effects model each group mean is a group-specific fixed quantity.

In linear regression, mean response and predicted response are values of the dependent variable calculated from the regression parameters and a given value of the independent variable. The values of these two responses are the same, but their calculated variances are different.

In statistics, principal component regression (PCR) is a regression analysis technique that is based on principal component analysis (PCA). More specifically, PCR is used for estimating the unknown regression coefficients in a standard linear regression model.

In statistics, a sum of squares due to lack of fit, or more tersely a lack-of-fit sum of squares, is one of the components of a partition of the sum of squares of residuals in an analysis of variance, used in the numerator in an F-test of the null hypothesis that says that a proposed model fits well. The other component is the pure-error sum of squares.

In statistics and in particular in regression analysis, leverage is a measure of how far away the independent variable values of an observation are from those of the other observations. High-leverage points, if any, are outliers with respect to the independent variables. That is, high-leverage points have no neighboring points in space, where is the number of independent variables in a regression model. This makes the fitted model likely to pass close to a high leverage observation. Hence high-leverage points have the potential to cause large changes in the parameter estimates when they are deleted i.e., to be influential points. Although an influential point will typically have high leverage, a high leverage point is not necessarily an influential point. The leverage is typically defined as the diagonal elements of the hat matrix.

The purpose of this page is to provide supplementary materials for the ordinary least squares article, reducing the load of the main article with mathematics and improving its accessibility, while at the same time retaining the completeness of exposition.

Errors-in-variables models Regression models accounting for possible errors in independent variables

In statistics, errors-in-variables models or measurement error models are regression models that account for measurement errors in the independent variables. In contrast, standard regression models assume that those regressors have been measured exactly, or observed without error; as such, those models account only for errors in the dependent variables, or responses.

In statistics, Tukey's test of additivity, named for John Tukey, is an approach used in two-way ANOVA to assess whether the factor variables are additively related to the expected value of the response variable. It can be applied when there are no replicated values in the data set, a situation in which it is impossible to directly estimate a fully general non-additive regression structure and still have information left to estimate the error variance. The test statistic proposed by Tukey has one degree of freedom under the null hypothesis, hence this is often called "Tukey's one-degree-of-freedom test."

In statistics, particularly regression analysis, the Working–Hotelling procedure, named after Holbrook Working and Harold Hotelling, is a method of simultaneous estimation in linear regression models. One of the first developments in simultaneous inference, it was devised by Working and Hotelling for the simple linear regression model in 1929. It provides a confidence region for multiple mean responses, that is, it gives the upper and lower bounds of more than one value of a dependent variable at several levels of the independent variables at a certain confidence level. The resulting confidence bands are known as the Working–Hotelling–Scheffé confidence bands.

References

  1. Seltman, Howard J. (2008-09-08). Experimental Design and Analysis (PDF). p. 227.
  2. "Statistical Sampling and Regression: Simple Linear Regression". Columbia University. Retrieved 2016-10-17. When one independent variable is used in a regression, it is called a simple regression;(...)
  3. Lane, David M. Introduction to Statistics (PDF). p. 462.
  4. Zou KH; Tuncali K; Silverman SG (2003). "Correlation and simple linear regression". Radiology. 227 (3): 617–22. doi:10.1148/radiol.2273011499. ISSN   0033-8419. OCLC   110941167. PMID   12773666.
  5. Altman, Naomi; Krzywinski, Martin (2015). "Simple linear regression". Nature Methods. 12 (11): 999–1000. doi:10.1038/nmeth.3627. ISSN   1548-7091. OCLC   5912005539. PMID   26824102.
  6. Kenney, J. F. and Keeping, E. S. (1962) "Linear Regression and Correlation." Ch. 15 in Mathematics of Statistics, Pt. 1, 3rd ed. Princeton, NJ: Van Nostrand, pp. 252–285
  7. Valliant, Richard, Jill A. Dever, and Frauke Kreuter. Practical tools for designing and weighting survey samples. New York: Springer, 2013.
  8. Casella, G. and Berger, R. L. (2002), "Statistical Inference" (2nd Edition), Cengage, ISBN   978-0-534-24312-8, pp. 558–559.