Simple linear regression: Difference between revisions

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=== Interpretation about the slope ===
By multiplying all members of the summation in the numerator by : <math>\begin{align}\frac{(x_i - \bar{x})}{(x_i - \bar{x})} = 1\end{align}</math> (thereby not changing it):
 
<math display="block">\begin{align}
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\end{align}</math>
 
We can see that the slope (tangent of angle) of the regression line is the weighted average of <math>\frac{(y_i - \bar{y})}{(x_i - \bar{x})}</math> that is the slope (tangent of angle) of the line that connects the i-th point to the average of all points, weighted by <math>(x_i - \bar{x})^2</math> 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 ===
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| The residuals and {{mvar|x}} values are uncorrelated (whether or not there is an intercept term in the model), meaning:
<math display="block">\sum_{i=1}^n x_i \widehat{\varepsilon}_i \;=\; 0</math>
| The relationship between <math>\rho_{xy}</math> (the [[Pearson_correlation_coefficient#For_a_population|correlation coefficient for the population]]) and the population variances of <math>y</math> (<math>\sigma_y^2</math>) and the error term of <math>\epsilonvarepsilon</math> (<math>\sigma_\epsilonvarepsilon^2</math>) is:<ref name = "Valliant2013">Valliant, Richard, Jill A. Dever, and Frauke Kreuter. Practical tools for designing and weighting survey samples. New York: Springer, 2013.</ref>{{rp|401}}
<math display="block">\sigma_\epsilonvarepsilon^2 = (1-\rho_{xy}^2)\sigma_y^2</math>
For extreme values of <math>\rho_{xy}</math> this is self evident. Since when <math>\rho_{xy} = 0</math> then <math>\sigma_\epsilonvarepsilon^2 = \sigma_y^2</math>. And when <math>\rho_{xy} = 1</math> then <math>\sigma_\epsilonvarepsilon^2 = 0</math>.
}}
 
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==See also==
* [[{{slink|Design matrix#Simple linear regression]]}}
* [[Linear trend estimation]]
* [[Segmented regression|Linear segmented regression]]