🚀 go-pugleaf

Here is a linear regression problem over which two of us are arguing.  Any
help would be appreciated.  

We are examining the performance of a Passive ozone (pollutant) sampling
Device (P).  We have "true" ozone concentrations from a Continuous
analyzer device (C) with known precision and accuracy co-located with P. 
It is suspected that P differs from the ozone concentration (C) according
to other environmental conditions (e.g. temp, time of P exposure, etc.) and
we want to get a handle on these other explantory variables.  

The OTHER GUY says that we should model the DIFFERENCE, P-C, as a function of
these other explanatory variables.  I say that we should be modeling P as a
function of C and other explantory variables.  I claim that OTHER GUY's
model will be misspecified unless (1) C is uncorrelated with the
explanatory variables and (2) the regression coefficient corresponding to C
(if C were to be included as an explanatory variable) is 1.

i.e. I claim the following (parentheses denote subscripts):

      suppose true model is P = b(0) + b(1) C + X(2)'B(2) + î(1)        (1)
      where B(2) and X(2) are vectors of parameters and explantory variables.

      We fit model  P-C = b(3) + X(2)'B(4) + î(2)
             ====>  P   = b(3) + 1 C + X(2)'B(4) + î(2)                 (2)
      
      The difference between (1) and (2) is the misspecification:

       b(0) - b(3) + [b(1)-1]C + X(2)'[B(2) - B(4)] + î(1)-î(2)         (3)

It seems to me that by fitting his model, we are forcing b(1) = 1, and that
there will be biases resulting from 1) deviation of b(1) from 1, and 2)
correlation between C and X(2).  I don't think there's any way of saying
anything about î(1)-î(2), except that this quantity differs from î(1).

Am I essentially correct or am I missing a concept?

thanks.

Chip Harvey
charvey@csn.org