Multiple regression analysis (MRA) is a useful method for generating mathematical models where there are several (more than two) variables involved. Some applications are listed at the end of this topic.

In order to appreciate the topic more easily the reader is recommended to read the topic of *Regression and correlation*,
to be found elsewhere on this website.

In that topic one can see that a simple regression model or formula has the following basic form:
*y = a + bx* where a is a constant, fixed value, b is a coefficient of the variable *x* and *y*
is the result to this equation.

This is a simple model describing a situation where there is only *one independent variable*, b
(the dependent variable being *y*, because the value of *y* depends on the value taken by *x*).
The example given in the other topic was:

Time to load a van = 3.4 + (0.3 x number of parcels) *minutes*.

This is simple regression and is in the form of a straight line in this instance.
However, the regression line often is in the form of
a curve of best fit such as a parabola y = bx^{2} or hyperbola or other function.
Where two independent variables are involved we have multiple regression and correlation and the model
will not be a line but a plane.
If more than two independent variables then the model is built up of hyperplanes which cannot be visualized,
but can be represented by a mathematical model, which takes the general form for *n* independent variables as:

Y = a + b_{1}x_{1} + b_{2}x_{2}
+ b_{3}x_{3} + b_{4}x_{4}
+ ... + b_{n}x_{n}

This *could* refine the model. In addition to the number of parcels which mostly are different we could
add (a) total weight carried
in kg. and (b) total volume in cubic metres.

Y = 2.6 + 4.1(number of parcels) + 1.8(weight) + 0.9(volume)

In addition to the coefficients (b_{1}, b_{2}, etc) for the model the computer package
will also provide a full validation
analysis for the results including a value known as the R-squared measure which is an evaluation of
the reliability of the model as predictor
of the Y values.
It is the multiple correlation factor squared.
Other statistical tests are also performed by the program to assess the reliability of each of the coefficients and the constant in the formula.

Other tests include measuring the intercorrelations between any of the variables.

These tests must be considered when evaluating the usefulness and validity of the model in predicting the basic time for work carried out under different circumstances.

- In addition to the application described above multiple regression has other uses in business management.
One is for attitude surveys carried out by a manager for assessing overall ratings of employees,
in the form of marks out of 100 for each attribute. Attributes (the variables) might include
- handling of employee complaints
- assessing employee pay rises based on performance
- how critically poor performance is tolerated
- disciplinary control and
- efficiency of the manager's department.

- Times necessary for the renovating and upgrading of train stations and subway (underground) stations.
- Car traveling times from and to base to various locations with variables such as the amounts of city, urban and rural driving in a journey, types of roads (A class, B class, dual carriageway, motorway, etc), distances on these, weather, and so on.

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