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Power transform

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In <a href="Statistics" title="Statistics">statistics</a>, a power transform is a family of functions applied to create a <a href="Monotonic_function" title="Monotonic function">monotonic transformation</a> of data using <a href="Power_function" class="mw-redirect" title="Power function">power functions</a>. It is a <a href="Data_transformation_(statistics)" title="Data transformation (statistics)">data transformation</a> technique used to <a href="Variance-stabilizing_transformation" title="Variance-stabilizing transformation">stabilize variance</a>, make the data more <a href="Normal_distribution" title="Normal distribution">normal distribution</a>-like, improve the validity of measures of association (such as the <a href="Pearson_product-moment_correlation_coefficient" class="mw-redirect" title="Pearson product-moment correlation coefficient">Pearson correlation</a> between variables), and for other data stabilization procedures.

Power transforms are used in multiple fields, including <a href="Multiresolution_analysis" title="Multiresolution analysis">multi-resolution and wavelet analysis</a>,<a href="#cite_note-1">[1]</a> statistical data analysis, medical research, modeling of physical processes,<a href="#cite_note-2">[2]</a> <a href="Geochemical_modeling" title="Geochemical modeling">geochemical data analysis</a>,<a href="#cite_note-3">[3]</a> <a href="Epidemiology" title="Epidemiology">epidemiology</a><a href="#cite_note-4">[4]</a> and many other clinical, environmental and social research areas.

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

The power transformation is defined as a continuous function of power parameter λ, typically given in piece-wise form that makes it continuous at the point of singularity (λ = 0). For data vectors (y1,..., yn) in which each yi > 0, the power transform is

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<annotation encoding="application/x-tex">{\displaystyle y_{i}^{(\lambda )}={\begin{cases}{\dfrac {y_{i}^{\lambda }-1}{\lambda (\operatorname {GM} (y))^{\lambda -1}}},&{\text{if }}\lambda \neq 0\\[12pt]\operatorname {GM} (y)\ln {y_{i}},&{\text{if }}\lambda =0\end{cases}}}</annotation>

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</math><img src="./8c74fb82826bbd941780fe964a283aef93c84f44.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.433ex; margin-bottom: -0.238ex; width:34.283ex; height:12.509ex;" alt="{\displaystyle y_{i}^{(\lambda )}={\begin{cases}{\dfrac {y_{i}^{\lambda }-1}{\lambda (\operatorname {GM} (y))^{\lambda -1}}},&{\text{if }}\lambda \neq 0\\[12pt]\operatorname {GM} (y)\ln {y_{i}},&{\text{if }}\lambda =0\end{cases}}}" loading="lazy"></dd></dl>

where

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {GM} (y)=\left(\prod _{i=1}^{n}y_{i}\right)^{\frac {1}{n}}={\sqrt[{n}]{y_{1}y_{2}\cdots y_{n}}}\,}</annotation>

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</math><img src="./8e3968a19b1fe6d0881fb8a4df8c0ca96e783630.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:36.528ex; height:8.676ex;" alt="{\displaystyle \operatorname {GM} (y)=\left(\prod _{i=1}^{n}y_{i}\right)^{\frac {1}{n}}={\sqrt[{n}]{y_{1}y_{2}\cdots y_{n}}}\,}" loading="lazy"></dd></dl>

is the <a href="Geometric_mean" title="Geometric mean">geometric mean</a> of the observations y1, ..., yn. The case for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda =0}">

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

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</math><img src="./00c4bba30544017fe76932de5a4e25adb5512d95.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.616ex; height:2.176ex;" alt="{\displaystyle \lambda =0}" loading="lazy"> is the limit as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }">

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

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</math><img src="./b43d0ea3c9c025af1be9128e62a18fa74bedda2a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }" loading="lazy"> approaches 0. To see this, note that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y_{i}^{\lambda }=\exp({\lambda \ln(y_{i})})=1+\lambda \ln(y_{i})+O((\lambda \ln(y_{i}))^{2})}">

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<annotation encoding="application/x-tex">{\displaystyle y_{i}^{\lambda }=\exp({\lambda \ln(y_{i})})=1+\lambda \ln(y_{i})+O((\lambda \ln(y_{i}))^{2})}</annotation>

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</math><img src="./97a79bcaf467bd1d1bb2f15293498d3a4b807a3d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:49.489ex; height:3.343ex;" alt="{\displaystyle y_{i}^{\lambda }=\exp({\lambda \ln(y_{i})})=1+\lambda \ln(y_{i})+O((\lambda \ln(y_{i}))^{2})}" loading="lazy"> - using <a href="Taylor_series" title="Taylor series">Taylor series</a>. Then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dfrac {y_{i}^{\lambda }-1}{\lambda }}=\ln(y_{i})+O(\lambda )}">

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<annotation encoding="application/x-tex">{\displaystyle {\dfrac {y_{i}^{\lambda }-1}{\lambda }}=\ln(y_{i})+O(\lambda )}</annotation>

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</math><img src="./0d947c5e3be37c32fd678fb2c8f033c8ca371744.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:23.754ex; height:6.176ex;" alt="{\displaystyle {\dfrac {y_{i}^{\lambda }-1}{\lambda }}=\ln(y_{i})+O(\lambda )}" loading="lazy">, and everything but <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ln(y_{i})}">

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

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</math><img src="./261f190a2cc6953b2fc663c4298c245b89d439c7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.688ex; height:2.843ex;" alt="{\displaystyle \ln(y_{i})}" loading="lazy"> becomes negligible for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }">

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

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</math><img src="./b43d0ea3c9c025af1be9128e62a18fa74bedda2a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }" loading="lazy"> sufficiently small.

The inclusion of the (λ − 1)th power of the geometric mean in the denominator simplifies the <a href="Dimensional_analysis" title="Dimensional analysis">scientific interpretation of any equation involving</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y_{i}^{(\lambda )}}">

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<annotation encoding="application/x-tex">{\displaystyle y_{i}^{(\lambda )}}</annotation>

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</math><img src="./e1cc57e7109a6767b769eb49df5af6f2274cd8d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.63ex; height:3.676ex;" alt="{\displaystyle y_{i}^{(\lambda )}}" loading="lazy">, because the units of measurement do not change as λ changes.

<a href="George_E._P._Box" title="George E. P. Box">Box</a> and <a href="David_Cox_(statistician)" title="David Cox (statistician)">Cox</a> (1964) introduced the geometric mean into this transformation by first including the <a href="Jacobian_matrix_and_determinant" title="Jacobian matrix and determinant">Jacobian</a> of rescaled power transformation

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<annotation encoding="application/x-tex">{\displaystyle {\frac {y^{\lambda }-1}{\lambda }}.}</annotation>

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</math><img src="./203b257d791996a7bdd330db0fa080d7e3322c19.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:7.837ex; height:6.009ex;" alt="{\displaystyle {\frac {y^{\lambda }-1}{\lambda }}.}" loading="lazy"></dd></dl>

with the likelihood. This Jacobian is as follows:

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<annotation encoding="application/x-tex">{\displaystyle J(\lambda ;y_{1},\ldots ,y_{n})=\prod _{i=1}^{n}|dy_{i}^{(\lambda )}/dy|=\prod _{i=1}^{n}y_{i}^{\lambda -1}=\operatorname {GM} (y)^{n(\lambda -1)}}</annotation>

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</math><img src="./c07e9fdb561b0429498e5703d6ad9166f34d2e85.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:58.01ex; height:6.843ex;" alt="{\displaystyle J(\lambda ;y_{1},\ldots ,y_{n})=\prod _{i=1}^{n}|dy_{i}^{(\lambda )}/dy|=\prod _{i=1}^{n}y_{i}^{\lambda -1}=\operatorname {GM} (y)^{n(\lambda -1)}}" loading="lazy"></dd></dl>

This allows the normal <a href="Maximum_likelihood" class="mw-redirect" title="Maximum likelihood">log likelihood at its maximum</a> to be written as follows:

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\log({\mathcal {L}}({\hat {\mu }},{\hat {\sigma }}))&=(-n/2)(\log(2\pi {\hat {\sigma }}^{2})+1)+n(\lambda -1)\log(\operatorname {GM} (y))\\[5pt]&=(-n/2)(\log(2\pi {\hat {\sigma }}^{2}/\operatorname {GM} (y)^{2(\lambda -1)})+1).\end{aligned}}}</annotation>

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</math><img src="./0b6f6556f73d994f06a6bf97f8cf4f2278d0d9a1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:62.109ex; height:7.843ex;" alt="{\displaystyle {\begin{aligned}\log({\mathcal {L}}({\hat {\mu }},{\hat {\sigma }}))&=(-n/2)(\log(2\pi {\hat {\sigma }}^{2})+1)+n(\lambda -1)\log(\operatorname {GM} (y))\\[5pt]&=(-n/2)(\log(2\pi {\hat {\sigma }}^{2}/\operatorname {GM} (y)^{2(\lambda -1)})+1).\end{aligned}}}" loading="lazy"></dd></dl>

From here, absorbing <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {GM} (y)^{2(\lambda -1)}}">

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {GM} (y)^{2(\lambda -1)}}</annotation>

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</math><img src="./301635ef7d4fff3a77da9ed8890f26953fdc377d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.313ex; height:3.343ex;" alt="{\displaystyle \operatorname {GM} (y)^{2(\lambda -1)}}" loading="lazy"> into the expression for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {\sigma }}^{2}}">

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<annotation encoding="application/x-tex">{\displaystyle {\hat {\sigma }}^{2}}</annotation>

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</math><img src="./1ad9d89160c9e63c0aa4c158282cb75a894de56f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.384ex; height:2.676ex;" alt="{\displaystyle {\hat {\sigma }}^{2}}" loading="lazy"> produces an expression that establishes that minimizing the sum of squares of <a href="Errors_and_residuals_in_statistics" class="mw-redirect" title="Errors and residuals in statistics">residuals</a> from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y_{i}^{(\lambda )}}">

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<annotation encoding="application/x-tex">{\displaystyle y_{i}^{(\lambda )}}</annotation>

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</math><img src="./e1cc57e7109a6767b769eb49df5af6f2274cd8d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.63ex; height:3.676ex;" alt="{\displaystyle y_{i}^{(\lambda )}}" loading="lazy"> is equivalent to maximizing the sum of the normal <a href="Likelihood_function" title="Likelihood function">log likelihood</a> of deviations from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (y^{\lambda }-1)/\lambda }">

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<annotation encoding="application/x-tex">{\displaystyle (y^{\lambda }-1)/\lambda }</annotation>

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</math><img src="./8bddc60b555db553bbec13abce873011a8ac12d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.681ex; height:3.176ex;" alt="{\displaystyle (y^{\lambda }-1)/\lambda }" loading="lazy"> and the log of the Jacobian of the transformation.

The value at Y = 1 for any λ is 0, and the <a href="Derivative" title="Derivative">derivative</a> with respect to Y there is 1 for any λ. Sometimes Y is a version of some other variable scaled to give Y = 1 at some sort of average value.

The transformation is a <a href="Power_(mathematics)" class="mw-redirect" title="Power (mathematics)">power</a> transformation, but done in such a way as to make it <a href="Continuous_function" title="Continuous function">continuous</a> with the parameter λ at λ = 0. It has proved popular in <a href="Regression_analysis" title="Regression analysis">regression analysis</a>, including <a href="Econometrics" title="Econometrics">econometrics</a>.

Box and Cox also proposed a more general form of the transformation that incorporates a shift parameter.

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<annotation encoding="application/x-tex">{\displaystyle \tau (y_{i};\lambda ,\alpha )={\begin{cases}{\dfrac {(y_{i}+\alpha )^{\lambda }-1}{\lambda (\operatorname {GM} (y+\alpha ))^{\lambda -1}}}&{\text{if }}\lambda \neq 0,\\\\\operatorname {GM} (y+\alpha )\ln(y_{i}+\alpha )&{\text{if }}\lambda =0,\end{cases}}}</annotation>

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</math><img src="./98f1eb900802585b0ebefad6d32b99e95ec3383c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.671ex; width:47.953ex; height:12.509ex;" alt="{\displaystyle \tau (y_{i};\lambda ,\alpha )={\begin{cases}{\dfrac {(y_{i}+\alpha )^{\lambda }-1}{\lambda (\operatorname {GM} (y+\alpha ))^{\lambda -1}}}&{\text{if }}\lambda \neq 0,\\\\\operatorname {GM} (y+\alpha )\ln(y_{i}+\alpha )&{\text{if }}\lambda =0,\end{cases}}}" loading="lazy"></dd></dl>

which holds if yi + α > 0 for all i. If τ(Y, λ, α) follows a <a href="Truncated_normal_distribution" title="Truncated normal distribution">truncated normal distribution</a>, then Y is said to follow a <a href="Box%E2%80%93Cox_distribution" title="Box–Cox distribution">Box–Cox distribution</a>.

Bickel and Doksum eliminated the need to use a <a href="Truncated_distribution" title="Truncated distribution">truncated distribution</a> by extending the range of the transformation to all y, as follows:

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<annotation encoding="application/x-tex">{\displaystyle \tau (y_{i};\lambda ,\alpha )={\begin{cases}{\dfrac {\operatorname {sgn} (y_{i}+\alpha )|y_{i}+\alpha |^{\lambda }-1}{\lambda (\operatorname {GM} (y+\alpha ))^{\lambda -1}}}&{\text{if }}\lambda \neq 0,\\\\\operatorname {GM} (y+\alpha )\operatorname {sgn} (y+\alpha )\ln(y_{i}+\alpha )&{\text{if }}\lambda =0,\end{cases}}}</annotation>

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</math><img src="./23d28c7920c0c389293afe5bc52508cb28a70bba.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.838ex; width:59.004ex; height:12.843ex;" alt="{\displaystyle \tau (y_{i};\lambda ,\alpha )={\begin{cases}{\dfrac {\operatorname {sgn} (y_{i}+\alpha )|y_{i}+\alpha |^{\lambda }-1}{\lambda (\operatorname {GM} (y+\alpha ))^{\lambda -1}}}&{\text{if }}\lambda \neq 0,\\\\\operatorname {GM} (y+\alpha )\operatorname {sgn} (y+\alpha )\ln(y_{i}+\alpha )&{\text{if }}\lambda =0,\end{cases}}}" loading="lazy"></dd></dl>

where sgn(.) is the <a href="Sign_function" title="Sign function">sign function</a>. This change in definition has little practical import as long as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }">

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

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</math><img src="./b79333175c8b3f0840bfb4ec41b8072c83ea88d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" loading="lazy"> is less than <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {min} (y_{i})}">

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {min} (y_{i})}</annotation>

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</math><img src="./d07415b7fcec4cc39906f40fbcede8b5bed8e0ec.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.623ex; height:2.843ex;" alt="{\displaystyle \operatorname {min} (y_{i})}" loading="lazy">, which it usually is.<a href="#cite_note-Bickel_and_Doksum-5">[5]</a>

Bickel and Doksum also proved that the parameter estimates are <a href="Consistent_estimator" title="Consistent estimator">consistent</a> and <a href="Local_asymptotic_normality" title="Local asymptotic normality">asymptotically normal</a> under appropriate regularity conditions, though the standard <a href="Cram%C3%A9r%E2%80%93Rao_bound" title="Cramér–Rao bound">Cramér–Rao lower bound</a> can substantially underestimate the variance when parameter values are small relative to the noise variance.<a href="#cite_note-Bickel_and_Doksum-5">[5]</a> However, this problem of underestimating the variance may not be a substantive problem in many applications.<a href="#cite_note-6">[6]</a><a href="#cite_note-7">[7]</a>

<div class="mw-heading mw-heading2"><h2 id="Box–Cox_transformation">Box–Cox transformation</h2></div>

The one-parameter Box–Cox transformations are defined as

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<annotation encoding="application/x-tex">{\displaystyle y_{i}^{(\lambda )}={\begin{cases}{\dfrac {y_{i}^{\lambda }-1}{\lambda }}&{\text{if }}\lambda \neq 0,\\\ln y_{i}&{\text{if }}\lambda =0,\end{cases}}}</annotation>

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</math><img src="./b565ae8f1cce1e4035e2a36213b8c9ce34b5029d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.838ex; width:27.261ex; height:8.843ex;" alt="{\displaystyle y_{i}^{(\lambda )}={\begin{cases}{\dfrac {y_{i}^{\lambda }-1}{\lambda }}&{\text{if }}\lambda \neq 0,\\\ln y_{i}&{\text{if }}\lambda =0,\end{cases}}}" loading="lazy"></dd></dl>

and the two-parameter Box–Cox transformations as

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<annotation encoding="application/x-tex">{\displaystyle y_{i}^{({\boldsymbol {\lambda }})}={\begin{cases}{\dfrac {(y_{i}+\lambda _{2})^{\lambda _{1}}-1}{\lambda _{1}}}&{\text{if }}\lambda _{1}\neq 0,\\\ln(y_{i}+\lambda _{2})&{\text{if }}\lambda _{1}=0,\end{cases}}}</annotation>

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</math><img src="./f0bcf29e7ad0c8261a9f15f4abd9468c9e73cbaf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:37.129ex; height:9.176ex;" alt="{\displaystyle y_{i}^{({\boldsymbol {\lambda }})}={\begin{cases}{\dfrac {(y_{i}+\lambda _{2})^{\lambda _{1}}-1}{\lambda _{1}}}&{\text{if }}\lambda _{1}\neq 0,\\\ln(y_{i}+\lambda _{2})&{\text{if }}\lambda _{1}=0,\end{cases}}}" loading="lazy"></dd></dl>

as described in the original article.<a href="#cite_note-boxcox-8">[8]</a><a href="#cite_note-9">[9]</a> Moreover, the first transformations hold for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y_{i}>0}">

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</math><img src="./2000cdffbebd85d221d8cc0cc18738a220caf555.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.2ex; height:2.509ex;" alt="{\displaystyle y_{i}>0}" loading="lazy">, and the second for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y_{i}>-\lambda _{2}}">

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<annotation encoding="application/x-tex">{\displaystyle y_{i}>-\lambda _{2}}</annotation>

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</math><img src="./bf7caa47cdbf8f18c5d5b43836eb66340fc59b17.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.255ex; height:2.509ex;" alt="{\displaystyle y_{i}>-\lambda _{2}}" loading="lazy">.<a href="#cite_note-boxcox-8">[8]</a>

The parameter <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }">

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

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</math><img src="./b43d0ea3c9c025af1be9128e62a18fa74bedda2a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }" loading="lazy"> is estimated using the <a href="Profile_likelihood" class="mw-redirect" title="Profile likelihood">profile likelihood</a> function and using goodness-of-fit tests.<a href="#cite_note-boxcoxGOF-10">[10]</a>

<div class="mw-heading mw-heading3"><h3 id="Confidence_interval">Confidence interval</h3></div>

Confidence interval for the Box–Cox transformation can be <a href="Confidence_interval#Methods_of_derivation" title="Confidence interval">asymptotically constructed</a> using <a href="Likelihood-ratio_test#Distribution:_Wilks.27s_theorem" title="Likelihood-ratio test">Wilks's theorem</a> on the <a href="Profile_likelihood" class="mw-redirect" title="Profile likelihood">profile likelihood</a> function to find all the possible values of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }">

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

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</math><img src="./b43d0ea3c9c025af1be9128e62a18fa74bedda2a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }" loading="lazy"> that fulfill the following restriction:<a href="#cite_note-11">[11]</a>

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<annotation encoding="application/x-tex">{\displaystyle \ln {\big (}L(\lambda ){\big )}\geq \ln {\big (}L({\hat {\lambda }}){\big )}-{\frac {1}{2}}{\chi ^{2}}_{1,1-\alpha }.}</annotation>

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</math><img src="./39876031edc88303e449b3405053d3fee9689036.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:34.165ex; height:5.176ex;" alt="{\displaystyle \ln {\big (}L(\lambda ){\big )}\geq \ln {\big (}L({\hat {\lambda }}){\big )}-{\frac {1}{2}}{\chi ^{2}}_{1,1-\alpha }.}" loading="lazy"></dd></dl>

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

The BUPA liver data set<a href="#cite_note-12">[12]</a> contains data on liver enzymes <a href="Alanine_transaminase" title="Alanine transaminase">ALT</a> and <a href="Gamma-glutamyl_transpeptidase" class="mw-redirect" title="Gamma-glutamyl transpeptidase">γGT</a>. Suppose we are interested in using log(γGT) to predict ALT. A plot of the data appears in panel (a) of the figure. There appears to be non-constant variance, and a Box–Cox transformation might help.

The log-likelihood of the power parameter appears in panel (b). The horizontal reference line is at a distance of χ12/2 from the maximum and can be used to read off an approximate 95% confidence interval for λ. It appears as though a value close to zero would be good, so we take logs.

Possibly, the transformation could be improved by adding a shift parameter to the log transformation. Panel (c) of the figure shows the log-likelihood. In this case, the maximum of the likelihood is close to zero suggesting that a shift parameter is not needed. The final panel shows the transformed data with a superimposed regression line.

Note that although Box–Cox transformations can make big improvements in model fit, there are some issues that the transformation cannot help with. In the current example, the data are rather heavy-tailed so that the assumption of normality is not realistic and a <a href="Robust_regression" title="Robust regression">robust regression</a> approach leads to a more precise model.

<div class="mw-heading mw-heading3"><h3 id="Econometric_application">Econometric application</h3></div>

Economists often characterize production relationships by some variant of the Box–Cox transformation.<a href="#cite_note-13">[13]</a>

Consider a common representation of production Q as dependent on services provided by a capital stock K and by labor hours N:

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<annotation encoding="application/x-tex">{\displaystyle \tau (Q)=\alpha \tau (K)+(1-\alpha )\tau (N).\,}</annotation>

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</math><img src="./3022b2fbf838b8dcbad83da8c06ae529eeac69fc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.762ex; height:2.843ex;" alt="{\displaystyle \tau (Q)=\alpha \tau (K)+(1-\alpha )\tau (N).\,}" loading="lazy"></dd></dl>

Solving for Q by inverting the Box–Cox transformation we find

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<annotation encoding="application/x-tex">{\displaystyle Q={\big (}\alpha K^{\lambda }+(1-\alpha )N^{\lambda }{\big )}^{1/\lambda },\,}</annotation>

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</math><img src="./2102436e34675e6d69258f0e5e925eafce59885a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:29.161ex; height:3.843ex;" alt="{\displaystyle Q={\big (}\alpha K^{\lambda }+(1-\alpha )N^{\lambda }{\big )}^{1/\lambda },\,}" loading="lazy"></dd></dl>

which is known as the <a href="Constant_elasticity_of_substitution" title="Constant elasticity of substitution">constant elasticity of substitution</a> (CES) <a href="Production_function" title="Production function">production function</a>.

The CES production function is a <a href="Homogeneous_function" title="Homogeneous function">homogeneous function</a> of degree one.

When λ = 1, this produces the linear production function:

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<annotation encoding="application/x-tex">{\displaystyle Q=\alpha K+(1-\alpha )N.\,}</annotation>

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</math><img src="./20bf4b6f8a50453a117f9860594c8a0d2c8b8af1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.728ex; height:2.843ex;" alt="{\displaystyle Q=\alpha K+(1-\alpha )N.\,}" loading="lazy"></dd></dl>

When λ → 0 this produces the famous <a href="Cobb%E2%80%93Douglas" class="mw-redirect" title="Cobb–Douglas">Cobb–Douglas</a> production function:

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<annotation encoding="application/x-tex">{\displaystyle Q=K^{\alpha }N^{1-\alpha }.\,}</annotation>

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</math><img src="./ace9cdb02e22ba846cfd1ef0de12df36c867792f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.856ex; height:3.009ex;" alt="{\displaystyle Q=K^{\alpha }N^{1-\alpha }.\,}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Activities_and_demonstrations">Activities and demonstrations</h3></div>

The <a href="SOCR" class="mw-redirect" title="SOCR">SOCR</a> resource pages contain a number of hands-on interactive activities<a href="#cite_note-14">[14]</a> demonstrating the Box–Cox (power) transformation using Java applets and charts. These directly illustrate the effects of this transform on <a href="Q%E2%80%93Q_plot" title="Q–Q plot">Q–Q plots</a>, X–Y <a href="Scatterplot" class="mw-redirect" title="Scatterplot">scatterplots</a>, <a href="Time-series" class="mw-redirect" title="Time-series">time-series</a> plots and <a href="Histogram" title="Histogram">histograms</a>.

<div class="mw-heading mw-heading2"><h2 id="Yeo–Johnson_transformation">Yeo–Johnson transformation</h2></div>

The Yeo–Johnson transformation<a href="#cite_note-yeojohnson-15">[15]</a>

allows also for zero and negative values of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}">

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</math><img src="./b43d0ea3c9c025af1be9128e62a18fa74bedda2a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }" loading="lazy"> can be any real number, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda =1}">

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</math><img src="./543b4490416437b7c80ea473bbcac0e4ab7a7f11.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.616ex; height:2.176ex;" alt="{\displaystyle \lambda =1}" loading="lazy"> produces the identity transformation.

The transformation law reads:

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<annotation encoding="application/x-tex">{\displaystyle y_{i}^{(\lambda )}={\begin{cases}((y_{i}+1)^{\lambda }-1)/\lambda &{\text{if }}\lambda \neq 0,y\geq 0\\[4pt]\ln(y_{i}+1)&{\text{if }}\lambda =0,y\geq 0\\[4pt]-((-y_{i}+1)^{(2-\lambda )}-1)/(2-\lambda )&{\text{if }}\lambda \neq 2,y<0\\[4pt]-\ln(-y_{i}+1)&{\text{if }}\lambda =2,y<0\end{cases}}}</annotation>

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</math><img src="./37bac8df582b1542485efb80ff6cba029a455348.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.838ex; width:55.954ex; height:14.843ex;" alt="{\displaystyle y_{i}^{(\lambda )}={\begin{cases}((y_{i}+1)^{\lambda }-1)/\lambda &{\text{if }}\lambda \neq 0,y\geq 0\\[4pt]\ln(y_{i}+1)&{\text{if }}\lambda =0,y\geq 0\\[4pt]-((-y_{i}+1)^{(2-\lambda )}-1)/(2-\lambda )&{\text{if }}\lambda \neq 2,y<0\\[4pt]-\ln(-y_{i}+1)&{\text{if }}\lambda =2,y<0\end{cases}}}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Box-Tidwell_transformation">Box-Tidwell transformation</h2></div>

The Box-Tidwell transformation is a statistical technique used to assess and correct non-linearity between predictor variables and the <a href="Logit" title="Logit">logit</a> in a <a href="Generalized_linear_model" title="Generalized linear model">generalized linear model</a>, particularly in <a href="Logistic_regression" title="Logistic regression">logistic regression</a>. This transformation is useful when the relationship between the independent variables and the outcome is non-linear and cannot be adequately captured by the standard model.

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

The Box-Tidwell transformation was developed by <a href="George_E._P._Box" title="George E. P. Box">George E. P. Box</a> and Paul W. Tidwell in 1962 as an extension of Box-Cox transformations, which are applied to the dependent variable. However, unlike the Box-Cox transformation, the Box-Tidwell transformation is applied to the independent variables in regression models. It is often used when the assumption of linearity between the predictors and the outcome is violated.

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

The general idea behind the Box-Tidwell transformation is to apply a power transformation to each independent variable Xi in the regression model:

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</math><img src="./20a42aeb8827c23f2b352573b2dac4d7d0b32148.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.01ex; height:3.176ex;" alt="{\displaystyle X_{i}'=X_{i}^{\lambda }}" loading="lazy">

Where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }">

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</math><img src="./b43d0ea3c9c025af1be9128e62a18fa74bedda2a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }" loading="lazy"> is the parameter estimated from the data. If Box-Tidwell Transformation is significantly different from 1, this indicates a non-linear relationship between Xi and the logit, and the transformation improves the model fit.

The Box-Tidwell test is typically performed by augmenting the regression model with terms like <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{i}\log(X_{i})}">

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</math><img src="./fab1b80b86c7949076758b3abc7dd7019267b06d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.616ex; height:2.843ex;" alt="{\displaystyle X_{i}\log(X_{i})}" loading="lazy"> and testing the significance of the coefficients. If significant, this suggests that a transformation should be applied to achieve a linear relationship between the predictor and the logit.

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

<div class="mw-heading mw-heading3"><h3 id="Stabilizing_Continuous_Predictors">Stabilizing Continuous Predictors</h3></div>

The transformation is beneficial in <a href="Logistic_regression" title="Logistic regression">logistic regression</a> or <a href="Proportional_hazards_models" class="mw-redirect" title="Proportional hazards models">proportional hazards models</a> where non-linearity in continuous predictors can distort the relationship with the dependent variable. It is a flexible tool that allows the researcher to fit a more appropriate model to the data without guessing the relationship's functional form in advance.

<div class="mw-heading mw-heading3"><h3 id="Verifying_Linearity_in_Logistic_Regression">Verifying Linearity in Logistic Regression</h3></div>

In <a href="Logistic_regression" title="Logistic regression">logistic regression</a>, a key assumption is that continuous independent variables exhibit a linear relationship with the logit of the dependent variable. Violations of this assumption can lead to biased estimates and reduced model performance. The Box-Tidwell transformation is a method used to assess and correct such violations by determining whether a continuous predictor requires transformation to achieve linearity with the logit.

<div class="mw-heading mw-heading4"><h4 id="Method_for_Verifying_Linearity">Method for Verifying Linearity</h4></div>

The Box-Tidwell transformation introduces an interaction term between each continuous variable Xi and its natural logarithm <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \log(X_{i})}">

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

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</math><img src="./fab1b80b86c7949076758b3abc7dd7019267b06d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.616ex; height:2.843ex;" alt="{\displaystyle X_{i}\log(X_{i})}" loading="lazy">

This term is included in the logistic regression model to test whether the relationship between Xi and the logit is non-linear. A statistically significant coefficient for this interaction term indicates a violation of the linearity assumption, suggesting the need for a transformation of the predictor. the Box-Tidwell transformation provides an appropriate power transformation to linearize the relationship, thereby improving model accuracy and validity. Conversely, non-significant results support the assumption of linearity.

<div class="mw-heading mw-heading4"><h4 id="Limitations">Limitations</h4></div>

One limitation of the Box-Tidwell transformation is that it only works for positive values of the independent variables. If your data contains negative values, the transformation cannot be applied directly without modifying the variables (e.g., adding a constant).

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

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

<ul><li><cite id="CITEREFBoxCox1964" class="citation journal cs1"><a href="George_E._P._Box" title="George E. P. Box">Box, George E. P.</a>; <a href="David_Cox_(statistician)" title="David Cox (statistician)">Cox, D. R.</a> (1964). "An analysis of transformations". <a href="Journal_of_the_Royal_Statistical_Society%2C_Series_B" class="mw-redirect" title="Journal of the Royal Statistical Society, Series B">Journal of the Royal Statistical Society, Series B</a>. 26 (2): 211–252. <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2984418">2984418</a>. <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0192611">0192611</a>.</cite></li>

<li><cite id="CITEREFCarrollRuppert1981" class="citation journal cs1">Carroll, R. J.; Ruppert, D. (1981). <a rel="nofollow" class="external text" href="http://wiki.stat.ucla.edu/socr/uploads/b/b8/PowerTransformFamily_Biometrica609.pdf">"On prediction and the power transformation family"</a> (PDF). Biometrika. 68 (3): 609–615. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1093%2Fbiomet%2F68.3.609">10.1093/biomet/68.3.609</a>.</cite></li>

<li><cite id="CITEREFDeGroot1987" class="citation journal cs1"><a href="Morris_H._DeGroot" title="Morris H. DeGroot">DeGroot, M. H.</a> (1987). <a rel="nofollow" class="external text" href="https://projecteuclid.org/journals/statistical-science/volume-2/issue-3/A-Conversation-with-George-Box/10.1214/ss/1177013223.pdf">"A Conversation with George Box"</a> (PDF). Statistical Science. 2 (3): 239–258. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1214%2Fss%2F1177013223">10.1214/ss/1177013223</a>.</cite></li>

<li><cite id="CITEREFHandelsman2002" class="citation journal cs1">Handelsman, D. J. (2002). "Optimal Power Transformations for Analysis of Sperm Concentration and Other Semen Variables". Journal of Andrology. 23 (5).</cite></li>

<li><cite id="CITEREFGluzmanYukalov2006" class="citation journal cs1">Gluzman, S.; Yukalov, V. I. (2006). "Self-similar power transforms in extrapolation problems". Journal of Mathematical Chemistry. 39 (1): 47–56. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/cond-mat/0606104">cond-mat/0606104</a>. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2006cond.mat..6104G">2006cond.mat..6104G</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs10910-005-9003-7">10.1007/s10910-005-9003-7</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:118965098">118965098</a>.</cite></li>

<li><cite id="CITEREFHowarthEarle1979" class="citation journal cs1">Howarth, R. J.; Earle, S. A. M. (1979). "Application of a generalized power transformation to geochemical data". Journal of the International Association for Mathematical Geology. 11 (1): 45–62. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF01043245">10.1007/BF01043245</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:121582755">121582755</a>.</cite></li>

<li>Box, G.E.P. and Tidwell, P.W. (1962) Transformation of Independent Variables. Technometrics, 4, 531-550. <a rel="nofollow" class="external free" href="https://doi.org/10.1080/00401706.1962.10490038">https://doi.org/10.1080/00401706.1962.10490038</a> (a.k.a. Box-Tidwell transformation)</li></ul>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li><cite id="CITEREFNishii2001" class="citation cs2">Nishii, R. (2001) [1994], <a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Box–Cox_transformation">"Box–Cox transformation"</a>, <a href="Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a>, <a href="European_Mathematical_Society" title="European Mathematical Society">EMS Press</a></cite> (<a rel="nofollow" class="external text" href="http://www.encyclopediaofmath.org/index.php/Box%E2%80%93Cox_transformation">fixed link</a>)</li>

<li>Sanford Weisberg, <a rel="nofollow" class="external text" href="https://www.stat.umn.edu/arc/yjpower.pdf">Yeo-Johnson Power Transformations</a></li></ul></div><div><div class="zim-footer">

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