Linear Regression

How It Works

Suppose that
you measure a response, such as absorbance, for several different levels
of a factor, such as the concentration of Cu^{2+}. If the data are
linear, then you should be able to model the data using the equation

absorbance = slope*[Cu^{2+}] + intercept

If you assume that errors affecting the concentrations
of Cu^{2+} are
insignificant,
then any difference between an experimental data point and the model
is due to an error in measuring the absorbance. For each data
point, the difference between the experimental absorbance, A_{expt},
and the predicted absorbance, A_{pred}, is a residual error, RE.

Because these residual errors
can be positive or negative, the individual values are first squared and
then summed to give a total residual error, RE_{tot} (note: this
is the reason that a linear regression is sometimes called a "least-squares" analysis).

Different values for the slope and intercept lead to different total residual errors. The best values for the slope and intercept, therefore, are those that lead to the smallest total residual error.

This applet provides an excellent visualization of how the slope and intercept affect the total residual error. Give it a try and see if you can achieve a total residual error that is lower than my best effort of 843.

When you are done, use the link on the left to proceed to Problem 1.