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My comment “not seeming to be right”
regarding what you originally wrote, comes from being familiar with “Orthogonal
Regression”. I would recommend you read
the article by Carroll and Ruppert, “The Use and Misuse of Orthogonal
Regression in Linear Errors-in-Variables Models” in The American
Statistician, Vol 50, No.1, Feb 1996, p 1. The other article you should
read is the one by Tan and Iglewicz, “Measurement-Methods Comparisons and
Linear Statistical Relationship”, in “Technometrics, Vol 41, No 3, August 1999,
p 192. Both deal with laboratory type measurements. The second deals with your
problem of a second method of measurement in comparison to an older method. The
first article also discusses a method of estimating the variance due to
equation (linear) error. Statistics is nothing more than a scientific
method of determining if some observations and deductive conclusions came about
by pure chance (After R.A. Fisher). The design of experiments (which is what
you should be doing) is a powerful tool to reduce the effects of chance on the
observed data. At some point a model (or several models) that involves the
observations has to be constructed. For example, this would be the linear
regression between two observed variables. One of the models then can be taken
to be true as the null hypothesis, and statistics used to determine if the
model is likely to have occurred by chance or not. Note that statistics does
not tell you which of the many models that fit are true, it only tells you
which ones are probably false. My reference to gurus is about the very many
models and methods of data reduction (and individuals associated with them) to
a statistic, which can be found to not be false. In many cases the models and
associated statistics are completely different, and result in different
numerical results. It could very well be that log transformed data fit just as
well as untransformed data (neither rejected). The designed experiment is
needed to try and more clearly distinguish between the different models. The correlation coefficient between X and Y
can be directly obtained from the covariance matrix. The two diagonals are the
observation variances (not the error variances) and the off diagonal term is
the covariance. The correlation coefficient can be either calculated or
determined graphically by plotting the “cloud” of X-Y points, and drawing the
regression estimate of slope through it. Turner in “A Simple Example
Illustrating a Well-Known Property of the Correlation Coefficient”, The
American Statistician, Vol 51, No 2, May 1997, p 170, shows the form, and the
relationship between the covariance and correlation coefficient. Professor Jan de Leeuw pointed out that Francis Galton observed an elliptical cloud of points on male height data (Father and Son) back in the 1880’s and J.D. Hamilton Dickson reduced it to a correlation coefficient in 1886. So the technique is not new. The
methods of Total Least Squares which a set of very power-full analytical tools have
been investigated by the numerical analysis community. The statistical
communities have not yet worked out the distributional properties of the
resulting error estimates. For example you can do multivariate regressions
involving many simultaneous Y values (such as absorption (Y) versus wavelengths
and specimen compositions (X)). DAHeiser |
