Linear Regression
Linear regression is a statistical method used to model the relationship between a dependent variable 𝑦 and one or more independent variables 𝑥 by fitting a linear equation of the form:
Types of Lines of Regression:
- Line of Regression of y on x
- Line of Regression of x on y
Line of Regression of y on x
| Equation | y – ȳ = byx(x – x̄) |
| (i) Mean of x (x̄) | = ∑x/n |
| (ii) Mean of y (ȳ) | = ∑y/n |
| (iii) byx | = ∑xy - {1\over n}∑x∑y\over ∑x^2 - {1\over n}(∑x)^2 |
| = cov(x,y)\over var (x) | |
| = cov(x, y)\over σ_x | |
| = rσ_y\over σ_x | |
| = ∑(x - x̄)(y - ȳ)\over ∑(x - x̄)^2 | |
| = ∑uv - {1\over n}∑u∑v\over ∑u^2 - {1\over n}(∑u)^2 |
Line of Regression of x on y
| Equation | x – x̄ = byx(y – ȳ) |
| (i) Mean of x (x̄) | = ∑x/n |
| (ii) Mean of y (ȳ) | = ∑y/n |
| (iii) byx | = ∑xy - {1\over n}∑x∑y\over ∑y^2 - {1\over n}(∑y)^2 |
| = cov(x,y)\over var (y) | |
| = cov(x, y)\over σ_y | |
| = rσ_x\over σ_y | |
| = ∑(x - x̄)(y - ȳ)\over ∑(y - ȳ)^2 | |
| = ∑uv - {1\over n}∑u∑v\over ∑v^2 - {1\over n}(∑v)^2 |
Properties of Lines of Regression
- bxy, byx and r or ρ (x, y) are of same sign
- Square of Coefficient of correlation (r2) = bxy. byx
- 0 ≤ bxy. byx ≤ 1 or 0 ≤ r2≤ 1
- If one of bxy. byx is numerically greater than one, the other is numerically smaller than one.
- The two regression lines intersect at (x̄, ȳ)
- If r = 0, then the lines of regression are parallel to the coordinate axes.
- The angle between two regression lines (θ) is given by tan θ = |{1 - r^2\over b_{xy} + b_{yx}}|
