Regression
In a cause and effect relationship, 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. Then, the equation
for the regression line would be:
ŷ = b0 + b1x
where b0 is a constant,
b1 is the regression coefficient,
x is the value of the independent variable, and ŷ is the
predicted value of the dependent variable.
Normally, you will
use a computational tool - a software package (e.g., Excel) or a graphing calculator -
to find b0 and b1. You enter the
X and Y values into your program or calculator,
and the tool solves for each parameter.
In the unlikely event that you find yourself 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]
and
b0 = y - b1 * x
where b0 is the constant in the regression equation,
b1 is the regression coefficient,
xi is the X value of observation i,
yi is the Y value of observation i,
and x and y
are the means of X and Y, respectively.