It is a common job in many fields. You have a set of data (x,y) from an experiment, for example. You make a graph with this set and you get a resulting curve joining the points on the graph with lines. Then, according to what you see, your next step is to derive an algebraic relationship for the curve you got from the data. Maybe you see that the points lay very close to a straight line or maybe a parable so you work on speciphic procedures (curve fitting) in order to have a mathematical relationship that approachs the curve.
That's it and maybe you want to state an hypothesis from your results that will bear you to a scientific law or just simplify your tasks.

The r-square value is a measure of fit. It tells you how much dispersion occurs around the semi-log function that has been proposed as the true mathematical representation (functional form) of the hypothesized relationship.

By way of further explanation consider the following:

Suppose that height and weight are positively related. A mathematical expression that would express this idea is W = f(H) where f'(H)>0 -- namely, W increases with positive increases in H. Unfortunately, W = f(H) is not something you can measure empirically, because you do not know the true nature of f. So, you must hypothesize what that true nature is -- namely, you must propose a functional form.

A straight line of the form W = aH + m, where a > 0, would probably be inappropriate, because short fat people can be heavier than tall thin people. Nevertheless, still you believe that taller people are generally heavier than short people. So, you might hypothesize a different functional form such as W = a•log(H) + m, which, although expressed as a line, actually forms an upward sloping curve when drawn on a two-dimensional sheet of paper.

Both W = aH + m and W = a•log(H) + m, where a > 0, are functional forms of the same mathematical expression W = f(H), where f'(H) >0.

Plot the height (H) and weight (W) for each individual on a pair of coordinate (X,Y) axes, where H and W are X and Y, respectively. What do you obtain? A scatter plot.

Perform two statistical runs that estimate the values of a and m for each of your two functional forms. Plot each of the resulting two curves (one will be a straight line) on your scatter plot. The curve (line) that matches the scatter plot best demonstrates the best fit! It will have the highest r-square value.

R is the correlation coefficient and r-square is the coefficient of determination. The latter tells you the percent of total variation explained by your model -- i.e., selected functional form.





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Note added at 2 hrs 57 mins (2004-11-25 04:45:55 GMT)
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