Simple linear regression
Linear regression model with a single explanatory variable / From Wikipedia, the free encyclopedia
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In statistics, simple linear regression (SLR) 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.
It has been suggested that Variance of the mean and predicted responses be merged into this article. (Discuss) Proposed since November 2023. |
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. 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.
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:
where the objective function Q is:
By expanding to get a quadratic expression in and we can derive minimizing values of the function arguments, denoted and :[6]
Here we have introduced
- and as the average of the xi and yi, respectively
- and as the deviations in xi and yi with respect to their respective means.
Expanded formulas
The above equations are efficient to use if the mean of the x and y variables () 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 equations. These expanded equations may be derived from the more general polynomial regression equations[7][8] by defining the regression polynomial to be of order 1, as follows.
The above system of linear equations may be solved directly, or stand-alone equations for 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]
Relationship with the sample covariance matrix
The solution can be reformulated using elements of the covariance matrix:
where
- rxy is the sample correlation coefficient between x and y
- sx and sy are the uncorrected sample standard deviations of x and y
- and are the sample variance and sample covariance, respectively
Substituting the above expressions for and into the original solution 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. This phenomenon is 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.
Interpretation 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.
Interpretation about the intercept
Given with the angle the line makes with the positive x axis, we have
Interpretation 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).