Up until this point, we have considered single predictor variables in the
specification
of linear regression prediction equation. However, in most practical
engineering problems, the independent variables or factors that de termine or
affect
the dependent or the response variable are not often single predictor variables.
If
multiple independent variables affect the response variable, then the analysis
calls for
a model different from that used for the single predictor variable. In a
situation where
more than one independent factor (variable) affects the outcome of a process, a
multiple regression model is used. This is referred to as multiple linear
regression
model or multivariate least squares fitting. Although flexibility is introduced
into the
regression analysis by the existence of multiple predictor variables, the
complexity
added by the use of multiple predictor variables makes this approach most suited
for
computer usage. A simple example problem for which multiple predictor variables
may be required is the conside ration of factors on which the total miles
traveled per
gallon of gas by a car depends. Some of the factors that determine gas usage by
a car
include its speed, its weight and the wind conditions etc. Thus, for its
analysis, a
multiple regression model is used which is often referred to as multiple linear
regression model or multivariate least squares fitting.
Un like the single -variable analysis, the interpretation of the output of a
multivariate least squares fitting is made difficult by the involvement of
several
predictor variables. Hence, even if the data base is sound and correct model
specified,
it is not sufficient and correct to merely examine the magnitudes of the
estimated
coefficients in order to determine which predictor variables most affect the
response.
In the same vein, it is not sound to ignore the interactions of the predictor
variables
when considering the influence of any of the parameters. It is obvious from the
fore
going that modern day computer tools might have solved the computational aspect
of
the multivariate least squares method, but discerning the implications of the
computational result remains a challenge.
The multivariate least squares discussion will be very brief. Consider N
observations on a response y , with m regressors xj , j = 1,2,3,…,m, , the
multiple
linear regression model is written as
In matrix form, we can arrange the data in the fol lowing
form
where 
are the
estimates of the regression coefficients, 
which can be obtained
from the solution of the matrix equation:
Equation 3 is obtained by setting up the sum of squares of
the residuals and
differentiating with respect to each of the unknown coefficients. Similar to the
single
variable regression, the adequacy of the multiple least square regression model
can be
checked by computing the residuals and checking if they are normally
distributed .
1.1 Example
For the model 
, determine
for the data in Table 1
Table 1: Data for multiple least square regression
1.1.1. Solution
he single variable regression, setting up sum of squares of the residuals,
and differentiating with respect to each unknown
coefficient arid equating each
partial derivative to zero ,
we obtain the following matrix expression :
Table 2: Computations for example problem
Using the computed data in Table 2 in eqn. 8 we obtain
Equation 9 is a system of linear algebraic equations and
can be solved by any of
method suitable for solving simultaneous equations including Gauss elimination
or
matrix inversion methods, etc. Using matrix inversion method, the solution to
eqn. 9
gives 
As the number of predictor variables
increase, solving eqn8 becomes more challenging, hence the use of computational
software for the multivariate modeling.
