Modeling and Multivariate Methods - SAS

676 Statistical Details Appendix A
Key Statistical Concepts
Figure A.6 Connect Springs to Data Points
data value
junction
point, mean
That is how you fit one mean or fit several means. That is how you fit a line, or a plane, or a hyperplane.
That is how you fit almost any model to continuous data. You measure the energy or uncertainty by the sum
of squares of the distances you must stretch the springs.
Statisticians put faith in the normal distribution because it is the one that requires the least faith. It is, in a
sense, the most random. It has the most non-informative shape for a distribution. It is the one distribution
that has the most expected uncertainty for a given variance. It is the distribution whose uncertainty is
measured in squared distance. In many cases it is the limiting distribution when you have a mixture of
distributions or a sum of independent quantities. It is the distribution that leads to test statistics that can be
measured fairly easily.
When the fit is constrained by hypotheses, you test the hypotheses by measuring this same spring energy.
Suppose you have responses from four different treatments in an experiment, and you want to test if the
means are significantly different. First, envision your data plotted in groups as shown here, but with springs
connected to a separate mean for each treatment. Then exert pressure against the spring force to move the
individual means to the common mean. Presto! The amount of energy that constrains the means to be the
same is the test statistic you need. That energy is the main ingredient in the F-test for the hypothesis that
tests whether the means are the same.