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First, springs are the machine of fit for a continuous response model (Farebrother 1987). Suppose that you have n points and that you want to know the expected value (mean) of the points. Envision what happens when you lay the points out on a scale and connect them to a common junction with springs (see Figure A.7). When you let go, the springs wiggle the junction point up and down and then bring it to rest at the mean. This is what must happen according to physics.
Figure A.7 Connect Springs to Data Points
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 in Figure A.8, 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 that you need. That energy is the main ingredient in the F test for the hypothesis that tests whether the means are the same.
Figure A.8 A Oneway Plot for a Continuous Response Variable
Figure A.9 shows what the situation looks like for a single category such as the medium size cars (see the mosaic column from Carpoll.jmp labeled medium in Figure A.10). Suppose there are thirteen responses (cars). The first level (American) has six responses, the next has two, and the last has five responses. The response probabilities become 6/13, 2/13, and 5/13, respectively, as the pressure against the response partitions balances out to minimize the total energy.
Figure A.9 Effect of Pressure Cylinders in Partitions
Figure A.10 A Mosaic Plot for Categorical Data

Help created on 3/19/2020