P.Mean: Deciphering a non-linear curve (created 2013-04-19). News: Sign up for "The Monthly Mean," the newsletter that dares to call itself average, www.pmean.com/news.

I was attending a research presentation and one of the speakers presented a non-linear equation. I thought it would be good to pick apart that equation and understand a bit better how it might behave. Here is the equation.

It is a bit of a hint that the dependent variable is represented by a p rather than by another letter. This often means that you are modeling a probability and the first thing to check is that the function is always between 0 and 1. It was also obvious from the context of the talk that the independent variable (X) is non-negative.

When you look at a function like this, you should figure out what it looks like at the extremes. Then look at the first and possibly second derivatives to understand better how it behaves.

I had mentioned that the indpendent variable (X) was known to be non-negative ffrom the context, but a quick look at the function should remind you why this is so. You are raising X to a power, and if that power is something like 1/2 (for a square root), it would create some rather difficult to explain values for the function. For the same reason, you should assume that the value of nu should also be positive. Sure, you could nitpick and say "what about the case where both X and nu are negative, but this is just being silly.

If beta is zero, the function simplifies too much and becomes one for any value of X. So let's eliminate the possibility that beta could be zero. Once you eliminate this possibility, then it seems logical to assume that it can only take on positive values.

When X is as small as it could possibly be (0), then the function simplifies again to 1. As X increases, the denominator increases, making the entire fraction smaller. You can nitpick again and point out for negative values of beta that the denominator would actually increase as X increases, but I warned you already to stop nitpicking.

As X increases without bound, the denominator also increases without bound. The fraction slowly sinks away to zero leaving you with just a value of delta. So we see here that function starts at 1 when X starts at zero. As X increases, the function decreases. It doesn't go any lower than delta, no matter how big X gets.

Now this sort of function makes sense if you have something that starts out "perfect" (probability of 1) and declines, but not all the way to zero. We see another important constraint in this model--the value of delta has to be somewhere between 0 and 1. A value of exactly one leads to a function that is constant for any value of X.

So if you fit this model, you want to make sure that you avoid coefficients that lead to a constant function, such as beta=0 of delta=1.

Let's compute a derivative.

This derivative is negative for any value of X (with the possible exception of X=0. It can never equal zero, so the maximum and/or minimum have to appear at the extremes. We knew this already, so it was not too helpful to compute the first derivative. The second derivative, however, provides a bit more information.

This simplifies to

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