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Deming regression

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In <a href="Statistics" title="Statistics">statistics</a>, Deming regression, named after <a href="W._Edwards_Deming" title="W. Edwards Deming">W. Edwards Deming</a>, is an <a href="Errors-in-variables_model" title="Errors-in-variables model">errors-in-variables model</a> that tries to find the <a href="Line_of_best_fit" class="mw-redirect" title="Line of best fit">line of best fit</a> for a two-dimensional data set. It differs from the <a href="Simple_linear_regression" title="Simple linear regression">simple linear regression</a> in that it accounts for <a href="Errors_and_residuals_in_statistics" class="mw-redirect" title="Errors and residuals in statistics">errors</a> in observations on both the x- and the y- axis. It is a special case of <a href="Total_least_squares" title="Total least squares">total least squares</a>, which allows for any number of predictors and a more complicated error structure.

Deming regression is equivalent to the <a href="Maximum_likelihood" class="mw-redirect" title="Maximum likelihood">maximum likelihood</a> estimation of an <a href="Errors-in-variables_model" title="Errors-in-variables model">errors-in-variables model</a> in which the errors for the two variables are assumed to be independent and <a href="Normal_distribution" title="Normal distribution">normally distributed</a>, and the ratio of their variances, denoted δ, is known.<a href="#cite_note-FOOTNOTELinnet1993-1">[1]</a> 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 than the <a href="Simple_linear_regression" title="Simple linear regression">simple linear regression</a>. Most statistical software packages used in clinical chemistry offer Deming regression.

The model was originally introduced by <a href="#CITEREFAdcock1878">Adcock (1878)</a> who considered the case δ = 1, and then more generally by <a href="#CITEREFKummell1879">Kummell (1879)</a> with arbitrary δ. However their ideas remained largely unnoticed for more than 50 years, until they were revived by <a href="#CITEREFKoopmans1936">Koopmans (1936)</a> and later propagated even more by <a href="#CITEREFDeming1943">Deming (1943)</a>. The latter book became so popular in <a href="Clinical_chemistry" title="Clinical chemistry">clinical chemistry</a> and related fields that the method was even dubbed Deming regression in those fields.<a href="#cite_note-FOOTNOTECornbleetGochman1979-2">[2]</a>

<div class="mw-heading mw-heading2"><h2 id="Specification">Specification</h2></div>

Assume that the available data (yi, xi) are measured observations of the "true" values (yi*, xi*), which lie on the regression line:

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}y_{i}&=y_{i}^{*}+\varepsilon _{i},\\x_{i}&=x_{i}^{*}+\eta _{i},\end{aligned}}}</annotation>

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</math><img src="./13a38d9ca87e5e34ac51982bca3c36a64cd3628c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:13.806ex; height:6.509ex;" alt="{\displaystyle {\begin{aligned}y_{i}&=y_{i}^{*}+\varepsilon _{i},\\x_{i}&=x_{i}^{*}+\eta _{i},\end{aligned}}}" loading="lazy"></dd></dl>

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

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<annotation encoding="application/x-tex">{\displaystyle \delta ={\frac {\sigma _{\varepsilon }^{2}}{\sigma _{\eta }^{2}}}.}</annotation>

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</math><img src="./9188097de9f9d1d6a481e8715a1bca51742550f0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:8.017ex; height:6.676ex;" alt="{\displaystyle \delta ={\frac {\sigma _{\varepsilon }^{2}}{\sigma _{\eta }^{2}}}.}" loading="lazy"></dd></dl>

In practice, the variances of the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

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</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}">

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</math><img src="./b8a6208ec717213d4317e666f1ae872e00620a0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}" loading="lazy"> parameters are often unknown, which complicates the estimate of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta }">

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<annotation encoding="application/x-tex">{\displaystyle \delta }</annotation>

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</math><img src="./c5321cfa797202b3e1f8620663ff43c4660ea03a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:2.343ex;" alt="{\displaystyle \delta }" loading="lazy">. Note that when the measurement method for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

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</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}">

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</math><img src="./b8a6208ec717213d4317e666f1ae872e00620a0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}" loading="lazy"> is the same, these variances are likely to be equal, so <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta =1}">

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<annotation encoding="application/x-tex">{\displaystyle \delta =1}</annotation>

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</math><img src="./30c5c1a63ccd876384e6cf0a31296e3aee31ac84.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.31ex; height:2.343ex;" alt="{\displaystyle \delta =1}" loading="lazy"> for this case.

We seek to find the line of "best fit"

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<annotation encoding="application/x-tex">{\displaystyle y^{*}=\beta _{0}+\beta _{1}x^{*},}</annotation>

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</math><img src="./529be9ee2b2827d3ffecac88d32919a343248e6e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.924ex; height:2.676ex;" alt="{\displaystyle y^{*}=\beta _{0}+\beta _{1}x^{*},}" loading="lazy"></dd></dl>

such that the weighted sum of squared residuals of the model is minimized:<a href="#cite_note-FOOTNOTEFuller1987Ch._1.3.3-3">[3]</a>

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<annotation encoding="application/x-tex">{\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}</annotation>

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</math><img src="./5ad5a3dbef9a28d8b33eb8e210d77d54bc4a6639.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:90.513ex; height:7.009ex;" alt="{\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}" loading="lazy"></dd></dl>

See <a href="#CITEREFJensen2007">Jensen (2007)</a> for a full derivation.

<div class="mw-heading mw-heading2"><h2 id="Solution">Solution</h2></div>

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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<annotation encoding="application/x-tex">{\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}}\,}</annotation>

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</math><img src="./9af7d8e23a0b3af8f28bba4139f900e29487293c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.598ex; margin-bottom: -0.24ex; width:47.334ex; height:16.843ex;" alt="{\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}}\,}" loading="lazy"></dd></dl>

Finally, the least-squares estimates of model's parameters will be<a href="#cite_note-FOOTNOTEGlaister2001-4">[4]</a>

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<annotation encoding="application/x-tex">{\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}}}</annotation>

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</math><img src="./8519eef26adfaada34c746fbf30266b4ad7c97f3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -9.131ex; margin-bottom: -0.207ex; width:44.096ex; height:19.843ex;" alt="{\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}}}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Orthogonal_regression">Orthogonal regression</h2></div>

For the case of equal error variances, i.e., when <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta =1}">

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</math><img src="./30c5c1a63ccd876384e6cf0a31296e3aee31ac84.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.31ex; height:2.343ex;" alt="{\displaystyle \delta =1}" loading="lazy">, Deming regression becomes orthogonal regression: it minimizes the sum of squared <a href="Distance_from_a_point_to_a_line" title="Distance from a point to a line">perpendicular distances from the data points to the regression line</a>. In this case, denote each observation as a point <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z_{j}=x_{j}+iy_{j}}">

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<annotation encoding="application/x-tex">{\displaystyle z_{j}=x_{j}+iy_{j}}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle i}</annotation>

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</math><img src="./add78d8608ad86e54951b8c8bd6c8d8416533d20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}" loading="lazy"> is the <a href="Imaginary_unit" title="Imaginary unit">imaginary unit</a>). Denote as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S=\sum {(z_{j}-{\overline {z}})^{2}}}">

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<annotation encoding="application/x-tex">{\displaystyle S=\sum {(z_{j}-{\overline {z}})^{2}}}</annotation>

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</math><img src="./386f010c6b40bc1c0539ae8a411fc436189046c7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:17.242ex; height:3.843ex;" alt="{\displaystyle S=\sum {(z_{j}-{\overline {z}})^{2}}}" loading="lazy"> the sum of the squared differences of the data points from the <a href="Centroid" title="Centroid">centroid</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {z}}={\tfrac {1}{n}}\sum z_{j}}">

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<annotation encoding="application/x-tex">{\displaystyle {\overline {z}}={\tfrac {1}{n}}\sum z_{j}}</annotation>

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</math><img src="./e64babd8bb86a048dc3655331c34007a58c2ca14.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:12.249ex; height:3.843ex;" alt="{\displaystyle {\overline {z}}={\tfrac {1}{n}}\sum z_{j}}" loading="lazy"> (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 href="#cite_note-FOOTNOTEMindaPhelps2008Theorem_2.3-5">[5]</a>

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<annotation encoding="application/x-tex">{\displaystyle S=0}</annotation>

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</math><img src="./4ae0fc3192b864f55a46749d8a64e7cf7783d04c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.76ex; height:2.176ex;" alt="{\displaystyle S=0}" loading="lazy">, then every line through the centroid is a line of best orthogonal fit.</li>

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<annotation encoding="application/x-tex">{\displaystyle S\neq 0}</annotation>

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</math><img src="./66a556e414acd1be08ef78c5a296fed171cad0e8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.76ex; height:2.676ex;" alt="{\displaystyle S\neq 0}" loading="lazy">, the orthogonal regression line goes through the centroid and is parallel to the vector from the origin to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {S}}}">

<msqrt>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\sqrt {S}}}</annotation>

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</math><img src="./bff86975b0e7944720b3e635c53c22c032a7a6f1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.435ex; height:3.009ex;" alt="{\displaystyle {\sqrt {S}}}" loading="lazy">.</li></ul>

A <a href="Trigonometry" title="Trigonometry">trigonometric</a> representation of the orthogonal regression line was given by Coolidge in 1913.<a href="#cite_note-FOOTNOTECoolidge1913-6">[6]</a> The <a href="Distance_from_a_point_to_a_line#Another_formula" title="Distance from a point to a line">distance</a> can also be calculated using the more typical equation of a line, given as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=mx+k}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y=mx+k}</annotation>

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</math><img src="./a3f5035c65c601a0d22e4fd5a9104deafe8a222f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.676ex; height:2.509ex;" alt="{\displaystyle y=mx+k}" loading="lazy">.

<div class="mw-heading mw-heading3"><h3 id="Application">Application</h3></div>

In the case of three <a href="Line_(geometry)" title="Line (geometry)">non-collinear</a> points in the plane, the <a href="Triangle" title="Triangle">triangle</a> with these points as its <a href="Vertex_(geometry)" title="Vertex (geometry)">vertices</a> has a unique <a href="Steiner_inellipse" title="Steiner inellipse">Steiner inellipse</a> that is tangent to the triangle's sides at their midpoints. The <a href="Ellipse#Elements_of_an_ellipse" title="Ellipse">major axis of this ellipse</a> falls on the orthogonal regression line for the three vertices.<a href="#cite_note-FOOTNOTEMindaPhelps2008Corollary_2.4-7">[7]</a> The quantification of a biological cell's intrinsic <a href="Cellular_noise" title="Cellular noise">cellular noise</a> can be quantified upon applying Deming regression to the observed behavior of a two reporter <a href="Synthetic_biological_circuit" title="Synthetic biological circuit">synthetic biological circuit</a>.<a href="#cite_note-FOOTNOTEQuarton2020-8">[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 href="#cite_note-9">[9]</a>

<div class="mw-heading mw-heading2"><h2 id="York_regression">York regression</h2></div>

The York regression extends Deming regression by allowing correlated errors in x and y.<a href="#cite_note-10">[10]</a>

<ul><li><a href="Line_fitting" title="Line fitting">Line fitting</a></li>

<li><a href="Regression_dilution" title="Regression dilution">Regression dilution</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

<dl><dt>Notes</dt></dl>

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<li id="cite_note-FOOTNOTELinnet1993-1"><a href="#cite_ref-FOOTNOTELinnet1993_1-0">^</a> <a href="#CITEREFLinnet1993">Linnet 1993</a>.

</li>

<li id="cite_note-FOOTNOTECornbleetGochman1979-2"><a href="#cite_ref-FOOTNOTECornbleetGochman1979_2-0">^</a> <a href="#CITEREFCornbleetGochman1979">Cornbleet & Gochman 1979</a>.

</li>

<li id="cite_note-FOOTNOTEFuller1987Ch._1.3.3-3"><a href="#cite_ref-FOOTNOTEFuller1987Ch._1.3.3_3-0">^</a> <a href="#CITEREFFuller1987">Fuller 1987</a>, Ch. 1.3.3.

</li>

<li id="cite_note-FOOTNOTEGlaister2001-4"><a href="#cite_ref-FOOTNOTEGlaister2001_4-0">^</a> <a href="#CITEREFGlaister2001">Glaister 2001</a>.

</li>

<li id="cite_note-FOOTNOTEMindaPhelps2008Theorem_2.3-5"><a href="#cite_ref-FOOTNOTEMindaPhelps2008Theorem_2.3_5-0">^</a> <a href="#CITEREFMindaPhelps2008">Minda & Phelps 2008</a>, Theorem 2.3.

</li>

<li id="cite_note-FOOTNOTECoolidge1913-6"><a href="#cite_ref-FOOTNOTECoolidge1913_6-0">^</a> <a href="#CITEREFCoolidge1913">Coolidge 1913</a>.

</li>

<li id="cite_note-FOOTNOTEMindaPhelps2008Corollary_2.4-7"><a href="#cite_ref-FOOTNOTEMindaPhelps2008Corollary_2.4_7-0">^</a> <a href="#CITEREFMindaPhelps2008">Minda & Phelps 2008</a>, Corollary 2.4.

</li>

<li id="cite_note-FOOTNOTEQuarton2020-8"><a href="#cite_ref-FOOTNOTEQuarton2020_8-0">^</a> <a href="#CITEREFQuarton2020">Quarton 2020</a>.

</li>

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</style><cite id="CITEREFCiccioneDehaene2021" class="citation journal cs1">Ciccione, Lorenzo; Dehaene, Stanislas (August 2021). <a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.cogpsych.2021.101406">"Can humans perform mental regression on a graph? Accuracy and bias in the perception of scatterplots"</a>. Cognitive Psychology. 128: 101406. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.cogpsych.2021.101406">10.1016/j.cogpsych.2021.101406</a>.</cite>

</li>

<li id="cite_note-10"><a href="#cite_ref-10">^</a> 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, <a rel="nofollow" class="external free" href="https://doi.org/10.1119/1.1632486">https://doi.org/10.1119/1.1632486</a>, 2004.

</li>

</ol></div>

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