Home
<hr align="left" color="#aca899" noshade="noshade" size="2" width="85%">
The goal of regression analysis is to determine the values of parameters for
a function that cause the function to best fit a set of data observations that
you provide. In linear regression, the function is a linear (straight-line)
equation. For example, if we assume the value of an automobile decreases by a
constant amount each year after its purchase, and for each mile it is driven,
the following linear function would predict its value (the dependent variable
on the left side of the equal sign) as a function of the two independent
variables which are age and miles:
value = price + depage*age + depmiles*miles
where value, the dependent variable, is the value of
the car, age is the age of the car, and miles is the number of
miles that the car has been driven. The regression analysis performed by NLREG
will determine the best values of the three parameters, price, the
estimated value when age is 0 (i.e., when the car was new), depage, the
depreciation that takes place each year, and depmiles, the depreciation
for each mile driven. The values of depage and depmiles will be
negative because the car loses value as age and miles increase.
For an analysis such as this car depreciation example, you must provide a
data file containing the values of the dependent and independent variables for
a set of observations. In this example each observation data record would
contain three numbers: value, age, and miles, collected from used car ads for
the same model car. The more observations you provide, the more accurate will
be the estimate of the parameters. The NLREG statements to perform this
regression are shown below:
Variables value,age,miles;Parameters price,depage,depmiles;Function value = price + depage*age + depmiles*miles;Data;{data values go here}
Once the values of the parameters are determined by NLREG,
you can use the formula to predict the value of a car based on its age and
miles driven. For example, if NLREG computed a value of 16000 for price,
-1000 for depage, and -0.15 for depmiles, then the function
value = 16000 - 1000*age - 0.15*miles
could be used to estimate the value of a car with a known
age and number of miles.
If a perfect fit existed between the function and the actual data, the
actual value of each car in your data file would exactly equal the predicted
value. Typically, however, this is not the case, and the difference between the
actual value of the dependent variable and its predicted value for a particular
observation is the error of the estimate which is known as the "deviation''
or "residual''. The goal of regression analysis is to determine the
values of the parameters that minimize the sum of the squared residual values
for the set of observations. This is known as a "least squares''
regression fit.
Here is a plot of a linear function fitted to a set of data values. The
actual data points are marked with ''x''. The red line between a point and the
fitted line represents the residual for the observation.
Diagram RA-1
https://i.servimg.com/u/f40/11/10/02/04/ra-110.jpg
NLREG is a very powerful regression analysis program. Using it you can
perform multivariate, linear, polynomial, exponential, logistic, and general
nonlinear regression. What this means is that you specify the form of the
function to be fitted to the data, and the function may include nonlinear terms
such as variables raised to powers and library functions such as log,
exponential, sine, etc. For complex analyses, NLREG allows you to specify
function models using conditional statements (if, else), looping
(for, do, while), work variables, and arrays. NLREG uses a
state-of-the-art regression algorithm that works as well, or better, than any
you are likely to find in any other, more expensive, commercial statistical
packages.
As an example of nonlinear regression, consider another depreciation
problem. The value of a used airplane decreases for each year of its age.
Assuming the value of a plane falls by the same amount each year, a linear
function relating value to age is:
value = p0 + p1*Age
Where p0 and p1 are the parameters whose
values are to be determined. However, it is a well-known fact that planes (and
automobiles) lose more value the first year than the second, and more the
second than the third, etc. This means that a linear (straight-line) function
cannot accurately model this situation. A better, nonlinear, function is:
value = p0 + p1*exp(-p2*Age)
Where the ''exp'' function is the value of e
(2.7182818...) raised to a power. This type of function is known as
"negative exponential" and is appropriate for modeling a value whose
rate of decrease is proportional to the difference between the value and some
base value. Here is a plot of a negative exponential function fitted to a set
of data values.
Diagram RA-2
https://i.servimg.com/u/f40/11/10/02/04/ra-210.jpg
Much of the convenience of NLREG comes from the fact that you
can enter complicated functions using ordinary algebraic notation. Examples of
functions that can be handled with NLREG include:
Linear: Y = p0 + p1*XQuadratic: Y = p0 + p1*X + p2*X^2Multivariate: Y = p0 + p1*X + p2*Z + p3*X*ZExponential: Y = p0 + p1*exp(X)Periodic: Y = p0 + p1*sin(p2*X)Misc: Y = p0 + p1*Y + p2*exp(Y) + p3*sin(Z)
In other words, the function is a general expression
involving one dependent variable (on the left of the equal sign), one or more
independent variables, and one or more parameters whose values are to be
estimated. NLREG can handle up to 500 variables and 500 parameters.
Because of its generality, NLREG can perform all of the regressions handled
by ordinary linear or multivariate regression programs as well as nonlinear
regression.
Some other regression programs claim to perform nonlinear regression but
actually do it by transforming the values of the variables such that the
function is converted to linear form. They then perform a linear regression on
the transformed function. This technique has a major flaw: it determines the
values of the parameters that minimize the squared residuals for the
transformed, linearized function rather than the original function. This is
different than minimizing the squared residuals for the actual function and the
estimated values of the parameters may not produce the best fit of the original
function to the data. NLREG uses a true nonlinear regression technique that
minimizes the squared residuals for the actual function. Also, NLREG can handle
functions that cannot be transformed to a linear form. < --> Error:
#include file specification missing closing quote <-- /tbody>
» Video for our MPH colleagues. Must watch
» Salam
» Feeling Sad
» Look here. Its 2020 and this is what we found
» Sad News
» Pakistan Demographic Profile 2018
» Good evening all fellows
» Urdu Poetry