Deming Regression

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Deming regression 1

Deming regression

Deming regression. The red lines show the error

in bothx andy. This is different from the

traditional least squares method which measures

error parallel to they axis. The case shown, with

deviations measured perpendicularly, arises when

x andy have equal variances.

In statistics, Deming regression, named after W. Edwards Deming, is

an errors-in-variables model which tries to find the line of best fit for a

two-dimensional dataset. It differs from the simple linear regression in

that it accounts for errors in observations on both thex- and they- axis.

It is a special case of total least squares, which allows for any number

of predictors and a more complicated error structure.

Deming regression is equivalent to the maximum likelihood estimation

of an errors-in-variables model in which the errors for the two

variables are assumed to be independent and normally distributed, and

the ratio of their variances, denoted , is known. In practice, this ratio

might be estimated from related data-sources; however the regression

procedure takes no account for possible errors in estimating this ratio.

The Deming regression is only slightly more difficult to compute

compared to the simple linear regression. Many software packages

used in clinical chemistry, such as Analyse-it, EP Evaluator, MedCalc,

R, S-PLUS and StatsDirect offer Deming regression.

The model was originally introduced by Adcock (1878) who considered the case = 1, and then more generally by

Kummell (1879) with arbitrary . However their ideas remained largely unnoticed for more than 50 years, until they

were revived by Koopmans (1937) and later propagated even more by Deming (1943). The latter book became so

popular in clinical chemistry and related fields that the method was even dubbed Deming regression in those

fields.[1]

Specification

Assume that the available data (yi,x

i) are measured observations of the "true" values (y

i*,x

i*):

where errors and are independent and the ratio of their variances is assumed to be known:

In practice the variance of the and parameters is often unknown which complicates the estimate of but

where the measurement method for and is the same they are likely to be equal so that for this case.

We seek to find the line of "best fit"

such that the weighted sum of squared residuals of the model is minimized:[2]
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Deming regression 2

Solution

The solution can be expressed in terms of the second-degree sample moments. That is, we first calculate the

following quantities (all sums go from i= 1 to n):

Finally, the least-squares estimates of model's parameters will be[3]

The case of equal error variances

When , Deming regression becomes orthogonal regression: it minimizes the sum of squared perpendicular

distances from the data points to the regression line. In this case, denote each observation as a pointzj

in the complex

plane (i.e., the point (xj,y

j) is written asz

j=x

j+ iy

jwhere i is the imaginary unit). Denote asZ the sum of the squared

differences of the data points from the centroid (also denoted in complex coordinates), which is the point whose

horizontal and vertical locations are the averages of those of the data points. Then:[4]

IfZ = 0, then every line through the centroid is a line of best orthogonal fit.

IfZ 0, the orthogonal regression line goes through the centroid and is parallel to the vector from the origin to

.

A trigonometric representation of the orthogonal regression line was given by Coolidge in 1913.[5]

Application

In the case of three non-collinear points in the plane, the triangle with these points as its vertices has a unique Steiner

inellipse that is tangent to the triangle's sides at their midpoints. The major axis of this ellipse falls on the orthogonal

regression line for the three vertices.[6]

Notes

[1][1] Cornbleet, Gochman (1979)

[2][2] Fuller, ch.1.3.3

[3][3] Glaister (2001)

[4][4] Minda and Phelps (2008), Theorem 2.3.

[5][5] Coolidge, J. L. (1913).

[6][6] Minda and Phelps (2008), Corollary 2.4.
http://en.wikipedia.org/w/index.php?title=Ellipse%23Elements_of_an_ellipsehttp://en.wikipedia.org/w/index.php?title=Steiner_inellipsehttp://en.wikipedia.org/w/index.php?title=Steiner_inellipsehttp://en.wikipedia.org/w/index.php?title=Vertex_%28geometry%29http://en.wikipedia.org/w/index.php?title=Trianglehttp://en.wikipedia.org/w/index.php?title=Line_%28geometry%29http://en.wikipedia.org/w/index.php?title=Trigonometryhttp://en.wikipedia.org/w/index.php?title=Centroidhttp://en.wikipedia.org/w/index.php?title=Imaginary_unithttp://en.wikipedia.org/w/index.php?title=Orthogonal_regression


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Deming regression 3

References

Adcock, R. J. (1878). "A problem in least squares". The Analyst (Annals of Mathematics) 5 (2): 5354. doi:

10.2307/2635758 (http://dx.doi.org/10.2307/2635758). JSTOR 2635758 (http://www.jstor.org/stable/

2635758).

Coolidge, J. L. (1913). "Two geometrical applications of the mathematics of least squares". The American

Mathematical Monthly20 (6): 187190. doi: 10.2307/2973072 (http://dx.doi.org/10.2307/2973072).

Cornbleet, P.J.; Gochman, N. (1979). "Incorrect LeastSquares Regression Coefficients". Clin. Chem.25 (3):

432438. PMID 262186 (http://www.ncbi.nlm.nih.gov/pubmed/262186).

Deming, W. E. (1943). Statistical adjustment of data. Wiley, NY (Dover Publications edition, 1985).

ISBN 0-486-64685-8.

Fuller, Wayne A. (1987).Measurement error models. John Wiley & Sons, Inc. ISBN 0-471-86187-1.

Glaister, P. (March 2001). "Least squares revisited". The Mathematical Gazette 85: 104-107.

Koopmans, T. C. (1937).Linear regression analysis of economic time series. DeErven F. Bohn, Haarlem,

Netherlands.

Kummell, C. H. (1879). "Reduction of observation equations which contain more than one observed quantity".

The Analyst (Annals of Mathematics) 6 (4): 97105. doi: 10.2307/2635646 (http://dx.doi.org/10.2307/

2635646). JSTOR 2635646 (http://www.jstor.org/stable/2635646).

Linnet, K. (1993). "Evaluation of regression procedures for method comparison studies" (http://www.clinchem.

org/cgi/reprint/39/3/424). Clinical Chemistry39 (3): 424432. PMID 8448852 (http://www.ncbi.nlm. nih.

gov/pubmed/8448852).

Minda, D.; Phelps, S. (2008). "Triangles, ellipses, and cubic polynomials" (http://www.geogebra.org/en/

upload/files/english/steve_phelps/minda phelps.pdf).American Mathematical Monthly115 (8): 679689. MR

2456092 (http://www.ams.org/mathscinet-getitem?mr=2456092).
http://www.ams.org/mathscinet-getitem?mr=2456092http://en.wikipedia.org/w/index.php?title=Mathematical_Reviewshttp://en.wikipedia.org/w/index.php?title=American_Mathematical_Monthlyhttp://www.geogebra.org/en/upload/files/english/steve_phelps/minda%20phelps.pdfhttp://www.geogebra.org/en/upload/files/english/steve_phelps/minda%20phelps.pdfhttp://en.wikipedia.org/w/index.php?title=David_Mindahttp://www.ncbi.nlm.nih.gov/pubmed/8448852http://www.ncbi.nlm.nih.gov/pubmed/8448852http://en.wikipedia.org/w/index.php?title=PubMed_Identifierhttp://www.clinchem.org/cgi/reprint/39/3/424http://www.clinchem.org/cgi/reprint/39/3/424http://www.jstor.org/stable/2635646http://en.wikipedia.org/w/index.php?title=JSTORhttp://dx.doi.org/10.2307%2F2635646http://dx.doi.org/10.2307%2F2635646http://en.wikipedia.org/w/index.php?title=Digital_object_identifierhttp://en.wikipedia.org/w/index.php?title=The_Mathematical_Gazettehttp://en.wikipedia.org/w/index.php?title=Special:BookSources/0-471-86187-1http://en.wikipedia.org/w/index.php?title=International_Standard_Book_Numberhttp://en.wikipedia.org/w/index.php?title=Special:BookSources/0-486-64685-8http://en.wikipedia.org/w/index.php?title=International_Standard_Book_Numberhttp://en.wikipedia.org/w/index.php?title=W._Edwards_Deminghttp://www.ncbi.nlm.nih.gov/pubmed/262186http://en.wikipedia.org/w/index.php?title=PubMed_Identifierhttp://dx.doi.org/10.2307%2F2973072http://en.wikipedia.org/w/index.php?title=Digital_object_identifierhttp://en.wikipedia.org/w/index.php?title=American_Mathematical_Monthlyhttp://en.wikipedia.org/w/index.php?title=American_Mathematical_Monthlyhttp://www.jstor.org/stable/2635758http://www.jstor.org/stable/2635758http://en.wikipedia.org/w/index.php?title=JSTORhttp://dx.doi.org/10.2307%2F2635758http://en.wikipedia.org/w/index.php?title=Digital_object_identifier


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Article Sources and ContributorsDeming regression Source: http://en.wikipedia.org/w/index.php?oldid=611197829 Contributors: A. Parrot, Arthur Rubin, Btyner, David Eppstein, Den fjttrade ankan, Docu, Dstivers,

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