# [R] Regression versus functional/structural relationship?

Thu Oct 30 17:14:47 CET 2008

```Hi Leif,

Thank you for your response.  I went back and carefully read a classic paper
by Dennis Lindley (JRSSB 1947).  He makes clear the distinction between
regression and functional relationship.  Here I quote his summary from his
paper:

"(A) The functional relationship is required for the statement of laws in th
eempirical sciences which would hold if no errors existed.
(B) The regression line is required for prediction of either true or
observed values of one variate from observation of of the other whether or
not the latter is in error; it being understood that the x from which y is
to be predicted comes from the same population as those x's used in the
estimation of the regression line."

Thus, according to Lindley, for my purposes, which is the prediction of most
likely (or expected) values of one assay given the observed values (with or
without error) of the another assay, a simple linear regression of y on x is
what is needed.  Note that more advance methods such as orthogonal
regression (and your suggestions) would be required, if we are interested in
the functional relationships between true values, X and Y.

Best,
Ravi.

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Assistant Professor, The Center on Aging and Health

Division of Geriatric Medicine and Gerontology

Johns Hopkins University

Ph: (410) 502-2619

Fax: (410) 614-9625

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-----Original Message-----
From: r-help-bounces at r-project.org [mailto:r-help-bounces at r-project.org] On
Behalf Of Leif Peterson
Sent: Wednesday, October 29, 2008 9:19 PM
To: r-help at r-project.org
Subject: Re: [R] Regression versus functional/structural relationship?

The two test outcomes will have correlated results, so you will need to look
at either bivariate probit regression or seemingly unrelated regression.
For either of these two methods, you will need to constrain all independent
variable coefficients to be equal, or you will have difficulty making sense
of the results.  Stata has biprobit and sureg, and also a constraint
command.  (Also bivariate probit requires binary dependents, so you will
need to apply a "clinically interesting" cutpoint of (+)/(-) test results.

If you can't find anything like these in R you will likely need to perform
quantile normalization of both dependents (x,y) before regression.  Look at
the qnorm package in bioconductor, by Bolstad.  LP

-----Original Message-----
From: r-help-bounces at r-project.org [mailto:r-help-bounces at r-project.org] On
Sent: Wednesday, October 29, 2008 6:01 PM
To: r-help at stat.math.ethz.ch
Subject: [R] Regression versus functional/structural relationship?

Hi,

I am dealing with the following problem.  There are two biochemical assays,
say A and B, available for analyzing blood samples.  Half the samples have
been analyzed with A.  Now, for some insurmountable logistic reasons, we
have to use B to analyze the remaining samples.  However, we can do a
comparative study on a small number of samples where we can obtain
concentrations using both A and B.  This gives us the data of the form (x,
y), where x are values from A and y from B.  Now, my question:  Can we
simply use the regression equation from regressing y on x, to convert all
the x values for which only method A was used?  Or do we need to obtain the
functional (or structural) relationship between X and Y (the true values
without measurement error) and use that to do this conversion.  It seems to
me that since we can only observe error-prone x, and we should be predicting
the expected value of error-prone y (i.e E[y | x]).  Therefore, we can
simply use the ordinary regression equation.  However, I have seen papers
using the Deming's orthogonal regression or something equivalent in the
clinical chemistry literature to address this problem.  Deming's method
would make sense if I am interested in obtaining the functional relationship
between X and Y (the true values of two assays), but I don't see why I
should care about that. Am I right?

I would appreciate any clarifying thoughts on this.  I apologize for posting
this methodological, non-R question.

Thank you,
Ravi.
----------------------------------------------------------------------------
-------

Assistant Professor, The Center on Aging and Health

Division of Geriatric Medicine and Gerontology

Johns Hopkins University

Ph: (410) 502-2619

Fax: (410) 614-9625

----------------------------------------------------------------------------
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