Response Curve for Negative Beta
Could you please clarify page 12 of the GLM text where they describe the response curves for logged variables when beta is <0?
It seems to me like the response curve for any negative beta would decrease at a decreasing rate, being of the form c*x^(-b) for some constant "base AOI" c, and some beta b. What am I missing here? Thank you!
Comments
This is a matter of calculus. Let's quickly abstract to make it easier to see.
I'll write b for beta and use a subscript _B to indicate the base level. We'll assume b < 0 and write b = -|b|.
We'll assume the AOI variable is the only variable in a log-link GLM. We'll write mu for the response curve.
Then ln(mu) = b* ln( AOI / AOI_B ) which becomes mu = (AOI / AOI_B)^b = (AOI / AOI_B)^(-|b|) = (AOI_B / AOI)^|b| = c * AOI^(-|b|) where c = AOI_B ^|b| is a positive constant.
Taking the first derivative with respect to AOI yields -c|b| AOI ^ -(|b|+1)
Since AOI > 0 the first derivative is negative. So the response curve is a decreasing function.
Taking the second derivative with respect to AOI yields c|b|(|b|+1) AOI ^ -(|b|+2)
This is positive if and only if |b|+1 > 0.
So if 0 < |b| < 1 the function decreases at a decreasing rate, while if |b| > 1 then it decreases at an increasing rate.
I am with you up to the second derivatives:
Won't |b|+1 > 0 be true for any b, meaning we'll always have a positive second derivative?
The book uses beta = (-1.2) as an example of a function that will decrease at an increasing rate, but it seems like this can't be if it has a positive second derivative.
I appreciate your help with this concept!
H L Mencken said "For every complex problem there is a solution which is clear, simple and wrong."
In this case, I clearly needed more coffee as |b| + 1 is always greater than 0.
The text is right but the language used is tripping us up.
Let's recreate their example with labelling.
If we negate the beta values we get
These are all curves which we say "decrease at a decreasing rate" despite the second derivative being positive. This is because "decreasing at a decreasing rate" is a relative concept whereas the second derivative is absolute in referring to the original function. In other words, "decreasing at a decreasing rate" means the ratio of the second derivative to the first derivative is negative.
We can see that using the algebra in the prior post; the ratio is -(|b|+1)*AOI^(-1) < 0.
So what is the text trying to show? The key is the authors say "a coefficient of -1.0 would
indicate a direct inverse relationship" which means instead of looking at the relationship with AOI, we should be thinking about replacing AOI with the new variable 1/AOI, or in our case 1000 / AOI since the authors rebase their example to 1,000.
Graphing this yields
So -1 < beta < 0 yield a decreasing rate while beta < -1 yields an increasing rate.