Simple Linear Regression
The independent variable is the cause, and the dependent variable is the effect. Least squares linear regression is a method for predicting the value of a dependent variable Y, based on the value of an independent variable X.
linear regression finds the straight line, called the least squares regression line or LSRL, that best represents observations in a bivariate data set. Suppose Y is a dependent variable, and X is an independent variable. The population regression line is:
Y = Β0 + Β1X
where Β0 is a constant, Β1 is the regression coefficient, X is the value of the independent variable, and Y is the value of the dependent variable.
Given a random sample of observations, the population regression line is estimated by:
ŷ = b0 + b1x
where b0 is a constant, b1 is the regression coefficient, x is the value of the independent variable, and ŷ is thepredicted value of the dependent variable.
On a desert island without a computer or a graphing calculator, you can solve for b0 and b1 "by hand". Here are the equations.
b1 = Σ [ (xi - x)(yi - y) ] / Σ [ (xi - x)2]
b1 = r * (sy / sx).
b0 = y - b1 * x
where b0 is the constant in the regression equation, b1 is the regression coefficient, r is the correlation between x and y, xi is the X value of observation i, yi is the Y value of observation i, x is the mean of X, y is the mean of Y, sx is the standard deviation of X, and sy is the standard deviation of Y.
