Multiple Regression
Forrest Young's Notes
Copyright © 19979 by Forrest W. Young.
Overview of Multiple Regression

Multiple Regression

Multiple Regression is a statistical technique that is used to measure
and describe the function relating two (or more) predictor (independent)
variables to a single response (dependent) variable.


Multiple Correlation Coefficient

The Multiple Correlation Coefficient tells us about the strength of the
relationship of a combination of the several predictor variables with the
response variable. The square of the multiple correlation tells us the
proportion of the response variable's variance that is shared with the
predictor variables variance.

Multiple Regression

Multiple regression tells us about the nature of the function that relates
a set of predictor variables to a single response variable. OLS
(ordinary least squares) multiple regression assumes the function is alinear
function (straight line, flat plane, etc), and fits the function so
that the relationship is is a least squares relationship. The function
is called the regression function.

The Regression Line:

Multiple Regression Analysis fits a linear function to the relationship
between the two variables. The line that is fit to the relationship has
certain properties:

The regression function is as close as possible, in a specific average
least
squares sense, to all of the points.

The regression function identifies the "central tendency" of the relationship
between the two variables, just as the mean identifies the "central tendency"
of a single variable.

The regression function provides a simplified description  a model
 of the relationship between the two variables.

The regression function gives us a way to predict values for the
response variable from values of the predictor variable.

Regression Equation for a Linear Relationship

A linear relationship of n predictor variables, denoted

X1, X2,
... Xn

to a single response variable, denote

Y

is described by the linear equation involving several variables. The general
linear equation is:

Y = a + b1X1
+ b2X2 + ... +
bnXn

This equation shows that any linear relationship can be described
by its:

Intercept: The linear combination of the X's is zero. Usually we
don't interpret the intercept.

Slopes: The slope for each predictor variable is denoted by the
various b's in the equation. The slope specifies how much the variable
Y will change when the particular X changes by one unit.
