Deming regression

<h2 id="specification">Specification</h2>
Assume that the available data (yi, xi) are measured observations of the "true" values (yi*, xi*), which lie on the regression line:

y
                  
                    i
                  
                
              
              
                
                =
                
                  y
                  
                    i
                  
                  
                    ∗
                  
                
                +
                
                  ε
                  
                    i
                  
                
                ,
              
            
            
              
                
                  x
                  
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                =
                
                  x
                  
                    i
                  
                  
                    ∗
                  
                
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                  η
                  
                    i
                  
                
                ,
              
            
          
        
      
    
    {\displaystyle {\begin{aligned}y_{i}&=y_{i}^{*}+\varepsilon _{i},\\x_{i}&=x_{i}^{*}+\eta _{i},\end{aligned}}}

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

δ
        =
        
          
            
              σ
              
                ε
              
              
                2
              
            
            
              σ
              
                η
              
              
                2
              
            
          
        
        .
      
    
    {\displaystyle \delta ={\frac {\sigma _{\varepsilon }^{2}}{\sigma _{\eta }^{2}}}.}

In practice, the variances of the 
 
 
 
 x
 
 
 {\displaystyle x}
 
 and 
 
 
 
 y
 
 
 {\displaystyle y}
 
 parameters are often unknown, which complicates the estimate of 
 
 
 
 δ
 
 
 {\displaystyle \delta }
 
. Note that when the measurement method for 
 
 
 
 x
 
 
 {\displaystyle x}
 
 and 
 
 
 
 y
 
 
 {\displaystyle y}
 
 is the same, these variances are likely to be equal, so 
 
 
 
 δ
 =
 1
 
 
 {\displaystyle \delta =1}
 
 for this case.
We seek to find the line of "best fit"

y
          
            ∗
          
        
        =
        
          β
          
            0
          
        
        +
        
          β
          
            1
          
        
        
          x
          
            ∗
          
        
        ,
      
    
    {\displaystyle y^{*}=\beta _{0}+\beta _{1}x^{*},}

such that the weighted sum of squared residuals of the model is minimized:<a class="footnote-ref" id="fnref:3" href="#fn:3">3</a>

S
        S
        R
        =
        
          ∑
          
            i
            =
            1
          
          
            n
          
        
        
          
            (
          
        
        
          
            
              ε
              
                i
              
              
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        +
        
          
            
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                i
              
              
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            )
          
        
        =
        
          
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          ∑
          
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            =
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          y
          
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        →
         
        
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              β
              
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        S
        S
        R
      
    
    {\displaystyle SSR=\sum _{i=1}^{n}{\bigg (}{\frac {\varepsilon _{i}^{2}}{\sigma _{\varepsilon }^{2}}}+{\frac {\eta _{i}^{2}}{\sigma _{\eta }^{2}}}{\bigg )}={\frac {1}{\sigma _{\epsilon }^{2}}}\sum _{i=1}^{n}{\Big (}(y_{i}-\beta _{0}-\beta _{1}x_{i}^{*})^{2}+\delta (x_{i}-x_{i}^{*})^{2}{\Big )}\ \to \ \min _{\beta _{0},\beta _{1},x_{1}^{*},\ldots ,x_{n}^{*}}SSR}

See Jensen (2007) for a full derivation. 

<h2 id="solution">Solution</h2>
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):

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    {\displaystyle {\begin{aligned}{\overline {x}}&={\tfrac {1}{n}}\sum x_{i}&{\overline {y}}&={\tfrac {1}{n}}\sum y_{i},\\s_{xx}&={\tfrac {1}{n}}\sum (x_{i}-{\overline {x}})^{2}&&={\overline {x^{2}}}-{\overline {x}}^{2},\\s_{xy}&={\tfrac {1}{n}}\sum (x_{i}-{\overline {x}})(y_{i}-{\overline {y}})&&={\overline {xy}}-{\overline {x}}\,{\overline {y}},\\s_{yy}&={\tfrac {1}{n}}\sum (y_{i}-{\overline {y}})^{2}&&={\overline {y^{2}}}-{\overline {y}}^{2}.\end{aligned}}\,}

Finally, the least-squares estimates of model's parameters will be<a class="footnote-ref" id="fnref:4" href="#fn:4">4</a>

β
                        ^
                      
                    
                  
                  
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    {\displaystyle {\begin{aligned}&{\hat {\beta }}_{1}={\frac {s_{yy}-\delta s_{xx}+{\sqrt {(s_{yy}-\delta s_{xx})^{2}+4\delta s_{xy}^{2}}}}{2s_{xy}}},\\&{\hat {\beta }}_{0}={\overline {y}}-{\hat {\beta }}_{1}{\overline {x}},\\&{\hat {x}}_{i}^{*}=x_{i}+{\frac {{\hat {\beta }}_{1}}{{\hat {\beta }}_{1}^{2}+\delta }}(y_{i}-{\hat {\beta }}_{0}-{\hat {\beta }}_{1}x_{i}).\end{aligned}}}

<h2 id="orthogonal-regression">Orthogonal regression</h2>
For the case of equal error variances, i.e., when 
 
 
 
 δ
 =
 1
 
 
 {\displaystyle \delta =1}
 
, Deming regression becomes orthogonal regression: it minimizes the sum of squared <a href="/facts/Distance_from_a_point_to_a_line/4mcAB59F">perpendicular distances from the data points to the regression line</a>. In this case, denote each observation as a point 
 
 
 
 
 z
 
 j
 
 
 =
 
 x
 
 j
 
 
 +
 i
 
 y
 
 j
 
 
 
 
 {\displaystyle z_{j}=x_{j}+iy_{j}}
 
 in the complex plane (i.e., the point 
 
 
 
 (
 
 x
 
 j
 
 
 ,
 
 y
 
 j
 
 
 )
 
 
 {\displaystyle (x_{j},y_{j})}
 
 where 
 
 
 
 i
 
 
 {\displaystyle i}
 
 is the <a href="/facts/Imaginary_unit/1SCllQlU">imaginary unit</a>). Denote as 
 
 
 
 S
 =
 ∑
 
 (
 
 z
 
 j
 
 
 −
 
 
 z
 ¯
 
 
 
 )
 
 2
 
 
 
 
 
 {\displaystyle S=\sum {(z_{j}-{\overline {z}})^{2}}}
 
 the sum of the squared differences of the data points from the <a href="/facts/Centroid/H6dDbg0h">centroid</a> 
 
 
 
 
 
 z
 ¯
 
 
 =
 
 
 
 1
 n
 
 
 
 ∑
 
 z
 
 j
 
 
 
 
 {\displaystyle {\overline {z}}={\tfrac {1}{n}}\sum z_{j}}
 
 (also denoted in complex coordinates), which is the point whose horizontal and vertical locations are the averages of those of the data points. Then:<a class="footnote-ref" id="fnref:5" href="#fn:5">5</a>

<ul><li>If 
 
 
 
 S
 =
 0
 
 
 {\displaystyle S=0}
 
, then every line through the centroid is a line of best orthogonal fit.</li>
<li>If 
 
 
 
 S
 ≠
 0
 
 
 {\displaystyle S\neq 0}
 
, the orthogonal regression line goes through the centroid and is parallel to the vector from the origin to 
 
 
 
 
 
 S
 
 
 
 
 {\displaystyle {\sqrt {S}}}
 
.</li></ul>
A <a href="/facts/Trigonometry/K4hfaC6F">trigonometric</a> representation of the orthogonal regression line was given by Coolidge in 1913.<a class="footnote-ref" id="fnref:6" href="#fn:6">6</a>

<h3>Application</h3>
In the case of three <a href="/facts/Line_(geometry)/gJqsr4Gj">non-collinear</a> points in the plane, the <a href="/facts/Triangle/cRXUwsMn">triangle</a> with these points as its <a href="/facts/Vertex_(geometry)/ID3cea9z">vertices</a> has a unique <a href="/facts/Steiner_inellipse/IFJ4pLGr">Steiner inellipse</a> that is tangent to the triangle's sides at their midpoints. The <a href="/facts/Ellipse/1wUDnGxY">major axis of this ellipse</a> falls on the orthogonal regression line for the three vertices.<a class="footnote-ref" id="fnref:7" href="#fn:7">7</a> The quantification of a biological cell's intrinsic <a href="/facts/Cellular_noise/USAIPFHe">cellular noise</a> can be quantified upon applying Deming regression to the observed behavior of a two reporter <a href="/facts/Synthetic_biological_circuit/a0VFeqqe">synthetic biological circuit</a>.<a class="footnote-ref" id="fnref:8" href="#fn:8">8</a>
When humans are asked to draw a linear regression on a scatterplot by guessing, their answers are closer to orthogonal regression than to ordinary least squares regression.<a class="footnote-ref" id="fnref:9" href="#fn:9">9</a>

<h2 id="york-regression">York regression</h2>
The York regression extends Deming regression by allowing correlated errors in x and y.<a class="footnote-ref" id="fnref:10" href="#fn:10">10</a>

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Line_fitting/0uHwdadB">Line fitting</a></li>
<li><a href="/facts/Regression_dilution/yshVFGzE">Regression dilution</a></li></ul>

Notes

Bibliography
<ul><li>Adcock, R. J. (1878). <a href="https://doi.org/10.2307%2F2635758">"A problem in least squares"</a>. The Analyst. 5 (2): 53–54. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.2307%2F2635758">10.2307/2635758</a>. <a href="/facts/JSTOR_(identifier)/YTeVmaJ7">JSTOR</a> <a href="https://www.jstor.org/stable/2635758">2635758</a>.</li>
<li>Coolidge, J. L. (1913). "Two geometrical applications of the mathematics of least squares". The American Mathematical Monthly. 20 (6): 187–190. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.2307%2F2973072">10.2307/2973072</a>. <a href="/facts/JSTOR_(identifier)/YTeVmaJ7">JSTOR</a> <a href="https://www.jstor.org/stable/2973072">2973072</a>.</li>
<li>Cornbleet, P.J.; Gochman, N. (1979). <a href="https://doi.org/10.1093%2Fclinchem%2F25.3.432">"Incorrect Least–Squares Regression Coefficients"</a>. Clinical Chemistry. 25 (3): 432–438. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1093%2Fclinchem%2F25.3.432">10.1093/clinchem/25.3.432</a>. <a href="/facts/PMID_(identifier)/JlHAvMHt">PMID</a> <a href="https://pubmed.ncbi.nlm.nih.gov/262186">262186</a>.</li>
<li><a href="/facts/W._Edwards_Deming/ytaxg6sv">Deming, W. E.</a> (1943). Statistical adjustment of data. Wiley, NY (Dover Publications edition, 1985). <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-486-64685-8. {{cite book}}: ISBN / Date incompatibility (help)</li>
<li>Fuller, Wayne A. (1987). Measurement error models. John Wiley & Sons, Inc. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-471-86187-1.</li>
<li>Glaister, P. (2001). "Least squares revisited". <a href="/facts/The_Mathematical_Gazette/2PoBAS9b">The Mathematical Gazette</a>. 85: 104–107. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.2307%2F3620485">10.2307/3620485</a>. <a href="/facts/JSTOR_(identifier)/YTeVmaJ7">JSTOR</a> <a href="https://www.jstor.org/stable/3620485">3620485</a>. <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:125949467">125949467</a>.</li>
<li>Jensen, Anders Christian (2007). <a href="http://staff.pubhealth.ku.dk/~bxc/MethComp/Deming.pdf">"Deming regression, MethComp package"</a> (PDF). Gentofte, Denmark: Steno Diabetes Center.</li>
<li>Koopmans, T. C. (1936). Linear regression analysis of economic time series. DeErven F. Bohn, Haarlem, Netherlands.</li>
<li>Kummell, C. H. (1879). <a href="https://doi.org/10.2307%2F2635646">"Reduction of observation equations which contain more than one observed quantity"</a>. The Analyst. 6 (4): 97–105. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.2307%2F2635646">10.2307/2635646</a>. <a href="/facts/JSTOR_(identifier)/YTeVmaJ7">JSTOR</a> <a href="https://www.jstor.org/stable/2635646">2635646</a>.</li>
<li>Linnet, K. (1993). <a href="http://www.clinchem.org/cgi/reprint/39/3/424">"Evaluation of regression procedures for method comparison studies"</a>. Clinical Chemistry. 39 (3): 424–432. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1093%2Fclinchem%2F39.3.424">10.1093/clinchem/39.3.424</a>. <a href="/facts/PMID_(identifier)/JlHAvMHt">PMID</a> <a href="https://pubmed.ncbi.nlm.nih.gov/8448852">8448852</a>.</li>
<li><a href="/facts/David_Minda/8Bh25MV8">Minda, D.</a>; Phelps, S. (2008). "Triangles, ellipses, and cubic polynomials". <a href="/facts/American_Mathematical_Monthly/Te6CHvkg">American Mathematical Monthly</a>. 115 (8): 679–689. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1080%2F00029890.2008.11920581">10.1080/00029890.2008.11920581</a>. <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=2456092">2456092</a>. <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:15049234">15049234</a>.</li>
<li>Quarton, T. G. (2020). <a href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC7498316">"Uncoupling gene expression noise along the central dogma using genome engineered human cell lines"</a>. Nucleic Acids Research. 48 (16): 9406–9413. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1093%2Fnar%2Fgkaa668">10.1093/nar/gkaa668</a>. <a href="/facts/PMC_(identifier)/dX1zMt71">PMC</a> <a href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC7498316">7498316</a>. <a href="/facts/PMID_(identifier)/JlHAvMHt">PMID</a> <a href="https://pubmed.ncbi.nlm.nih.gov/32810265">32810265</a>.</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Linnet 1993. - Linnet, K. (1993). "Evaluation of regression procedures for method comparison studies". Clinical Chemistry. 39 (3): 424–432. doi:10.1093/clinchem/39.3.424. PMID 8448852. <a href="http://www.clinchem.org/cgi/reprint/39/3/424" target="_blank">http://www.clinchem.org/cgi/reprint/39/3/424</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Cornbleet & Gochman 1979. - Cornbleet, P.J.; Gochman, N. (1979). "Incorrect Least–Squares Regression Coefficients". Clinical Chemistry. 25 (3): 432–438. doi:10.1093/clinchem/25.3.432. PMID 262186. <a href="https://doi.org/10.1093%2Fclinchem%2F25.3.432" target="_blank">https://doi.org/10.1093%2Fclinchem%2F25.3.432</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Fuller 1987, Ch. 1.3.3. - Fuller, Wayne A. (1987). Measurement error models. John Wiley & Sons, Inc. ISBN 0-471-86187-1. <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Glaister 2001. - Glaister, P. (2001). "Least squares revisited". The Mathematical Gazette. 85: 104–107. doi:10.2307/3620485. JSTOR 3620485. S2CID 125949467. <a href="https://doi.org/10.2307%2F3620485" target="_blank">https://doi.org/10.2307%2F3620485</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">Minda & Phelps 2008, Theorem 2.3. - Minda, D.; Phelps, S. (2008). "Triangles, ellipses, and cubic polynomials". American Mathematical Monthly. 115 (8): 679–689. doi:10.1080/00029890.2008.11920581. MR 2456092. S2CID 15049234. <a href="https://doi.org/10.1080%2F00029890.2008.11920581" target="_blank">https://doi.org/10.1080%2F00029890.2008.11920581</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">Coolidge 1913. - Coolidge, J. L. (1913). "Two geometrical applications of the mathematics of least squares". The American Mathematical Monthly. 20 (6): 187–190. doi:10.2307/2973072. JSTOR 2973072. <a href="https://doi.org/10.2307%2F2973072" target="_blank">https://doi.org/10.2307%2F2973072</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">Minda & Phelps 2008, Corollary 2.4. - Minda, D.; Phelps, S. (2008). "Triangles, ellipses, and cubic polynomials". American Mathematical Monthly. 115 (8): 679–689. doi:10.1080/00029890.2008.11920581. MR 2456092. S2CID 15049234. <a href="https://doi.org/10.1080%2F00029890.2008.11920581" target="_blank">https://doi.org/10.1080%2F00029890.2008.11920581</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
<li id="fn:8">Quarton 2020. - Quarton, T. G. (2020). "Uncoupling gene expression noise along the central dogma using genome engineered human cell lines". Nucleic Acids Research. 48 (16): 9406–9413. doi:10.1093/nar/gkaa668. PMC 7498316. PMID 32810265. <a href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC7498316" target="_blank">https://www.ncbi.nlm.nih.gov/pmc/articles/PMC7498316</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></li>
<li id="fn:9">Ciccione, Lorenzo; Dehaene, Stanislas (August 2021). "Can humans perform mental regression on a graph? Accuracy and bias in the perception of scatterplots". Cognitive Psychology. 128: 101406. doi:10.1016/j.cogpsych.2021.101406. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></li>
<li id="fn:10">York, D., Evensen, N. M., Martınez, M. L., and Delgado, J. D. B.: Unified equations for the slope, intercept, and standard errors of the best straight line, Am. J. Phys., 72, 367–375, https://doi.org/10.1119/1.1632486, 2004. <a href="https://doi.org/10.1119/1.1632486" target="_blank">https://doi.org/10.1119/1.1632486</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></li>
</ol>

Deming regression open-in-new

Deming regression