I am forwarding this reply from Chris Olsen. It was "bounced" to me because of the word h*lp in the subject line or in the first few lines of the message. That's what happens if certain words like h*lp and s*bscr*be are used in the beginning of a message.
Apologies for the delay in spotting this bounce Chris. I dont examine that folder on a daily basis!!!!
Hello Al and All --
Let me see if I can leap into this with what passes for help on a Sunday evening. I must say that transformations is (verb agreement??) one of those things that I always found troublesome, so someone who really knows please jump in and point out my errors of thought and pen!
> The AP Syllabus includes: > > (I.D.5) Transformations to achieve linearity: logarithmic > and power transformations. > > I assume that means tranforming data which fits functions such as: > > y = alog x and y = a b^x. > > Where I am having problems is with my students' collections of real data. If > a student collects a set of data for which an approximate model might be: > > y = 3log(x - 40) + 30 or even y = 3(x-40)^2 + 30 >
Let me see if I can take a crack at a transformation that will give me a linear relation. (I used my TI-83, so I can blame it if this is screwed up!)
I put 100, 110, ... 180 in List 1 Then I zapped 3*LN(L1 - 40) + 30 into List 2 Now, I graph L1 vs L2 and I see a non-linear plot.
OK -- now lets take our List2 and see if it can straighten out. (My method will be immediately transparent!!)
(L2 - 30)/3 --> L3 EXP(L3) --> L4 L4 + 40 --> L5
Now, if I plot my original X (L1) values vs my Transformed Y's (L5) I get a linear plot!
With a similar approach (i.e. simply solving for x) I should be able to get a transformation that will allow me to get a straight line plot.
> Am I right that there is no easy transformation which will "straighten" these > "curves"??? If so what should we do with the data when trying to model it? >
Having successfully(??) transformed these plots to linear, however, I am not sure what I have. I don't know if I could do all this with real data and real dots!
Firstly, there seem to be a lot of parameters (constants) in these models that would not be recoverable in a real data analysis --
1. In the first model, I might get lucky and think it "looks" like a log function,
y = a + b*ln(x),
and work my algebraic magic to get some sort of transformation that would give me a straight line. But at most, I have two estimated parameters.
2. Similarly if I get lucky and guess that it looks quadratic and similarly I have only two estimated parameters. But if my original model is of the form,
y = a(x - h)^p + k,
I think I am in deep statistical do-do -- (doo-doo?) in that my mere linear regression, no matter what transformation, cannot recover all those a, h, p, and k.
So I guess my response would be that if your models have only two parameters, these particular ones could be "straightened out." Easily, I'm not sure about -- let your students judge that question.
I WOULD like to know, however, what data your students are coming up with that would lead to the opinion that these are approximate models. I consider myself lucky if my students come up with any models at all, let alone these!! :) :) :)
Somebody please discuss this soon -- the next section in my stat class is non-linear regression, and this is your last chance to stop me before I teach again...
I'm hoping that
y = a*e^x and y = a*ln(x) and y = a*x^p
are enough to get the idea across, though I may to show my biology types a logistic regression for information only.