R help - Apr 2003 - Measure of linearity between two variables?

Hello,

I am looking for a measure of linearity in the relationship between
two variables.

Specifically, I have two variables for which the relationship is
reasonably linear over a certain range of values, and then diverges
from linearity at either end of the range, as one or other variable
"saturates" at a maximum or minimum value. I want to identify the
region of linearity, where neither variable has saturated.

This is a problem that will be repeated many times so I want a
programmatic solution.  I am intending to implement some kind of
search over the central range of values, expanding out and testing
for linearity over each incrementally increased range. However,
I need a measure if linearity.

So far, I have thought of doing a regression on x ~ y + y^2, and
using the absolute value of the ratio of coefficients of the squared
and linear terms. Does anyone have any better ideas, either for a
linearity measure or a different approach to finding the region of
linearity between the two variables ?

Thanks,

Luke Whitaker

Maybe I'm missing something, but why not use the Pearson Product
Moment Correlation Coefficient (r) ?  It directly measures the strength
of the linear relationship between two variables.  A simple approach
would be the following:

(1) fix a percentage p of the data you are interested in
(2) fix one of your two variables (x,y) as a reference - call
	it x
(3) subset your data.frame down to those pairs (x*,y*) 
	corresponding to the middle p percent of x
(4) calculate r for the pairs (x*,y*)

By doing (1) through (4) many time for increasing values of p
I think you'll get what you want.

Best, 
  david paul


-----Original Message-----
From: Luke Whitaker [mailto:luke at inpharmatica.co.uk] 
Sent: Thursday, April 17, 2003 12:03 PM
To: r-help at stat.math.ethz.ch
Subject: [R] Measure of linearity between two variables?



Hello,

I am looking for a measure of linearity in the relationship between two
variables.

Specifically, I have two variables for which the relationship is reasonably
linear over a certain range of values, and then diverges from linearity at
either end of the range, as one or other variable "saturates" at a
maximum
or minimum value. I want to identify the region of linearity, where neither
variable has saturated.

This is a problem that will be repeated many times so I want a programmatic
solution.  I am intending to implement some kind of search over the central
range of values, expanding out and testing for linearity over each
incrementally increased range. However, I need a measure if linearity.

So far, I have thought of doing a regression on x ~ y + y^2, and using the
absolute value of the ratio of coefficients of the squared and linear terms.
Does anyone have any better ideas, either for a linearity measure or a
different approach to finding the region of linearity between the two
variables ?

Thanks,

Luke Whitaker

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