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Replies: 0

 Jim Swift Posts: 42 Registered: 12/6/04
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(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 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.

-- Chris Olsen