Chapter 9 Multiple Regression 353
Regression Models with Multiple Parameters
In Chapter 8, you used simple linear regression to predict a dependent vari-
able (y) from a single independent variable (x, a predictor variable). In mul-
tiple regression, you predict a dependent variable from several independent
variables. For three predictors, x 1 , x 2 , and x 3 , the multiple regression model
takes the form:
y 5 b 01 b 1 x 11 b 2 x 21 b 3 x 3 1e
where the coeffi cients are unknown parameter values that you can estimate
and e is random error, which follows a normal distribution with mean 0 and
variance s^2. Note that the predictors can also be functions of variables. The
following are also examples of models whose parameters you can estimate
with multiple regression:
Polynomial: y5b 0 1b 1 x1b 2 x^2 1b 3 x^3 1e
Trigonometric: y5b 0 1b 1 sin x1b 2 cos x1e
Logarithmic: y5b 0 1b 1 log x 11 b 2 log x 2 1e
Note that all of these equations are examples of linear models, even
though they use various trigonometric and logarithmic functions. The linear
in linear model refers to the error term e and the parameters bi. The equa-
tions are linear in those terms. For example, one could create new variables
l 5 sin x and k 5 cos x, and then the second model is the linear equation
y5b 0 1b 1 l1b 2 k1e.
After computing estimated values for the b coefficients, you can plug
them into the equation to get predicted values for y. The estimated regres-
sion model is expressed as
y 5 b 01 b 1 x 11 b 2 x 21 b 3 x 3
where the bi’s are the estimated parameter values, and the residuals cor-
respond to the error term e.
CONCEPT TUTORIALS
The F distribution
The F distribution is basic to regression and analysis of variance as studied
in this chapter and the next. An example of the F distribution is shown in
the Distributions workbook.
To view the F distribution:
1 Open the Distributions workbook located in the Explore folder of
your Student fi les. Enable the macros in the workbook.
