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Re: Please h*lp Transformations
Posted:
Oct 23, 1996 5:05 PM


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(x40)^2 + 30 >
Let me see if I can take a crack at a transformation that will give me a linear relation. (I used my TI83, 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 nonlinear 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 dodo  (doodoo?) 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 nonlinear 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.
 Chris Olsen



