Declared Sum of Squares

This graphic shows the sum of squares decomposition , i.e. H. the decomposition of the total square sum into the explained square sum and the residual square sum . The sum of the green squares is the explained sum of squares.

In the statistics the (by which is regression ) explained sum of squares , and declared deviation square sum , short SQE for S umme the Q of uadrate E rklärten deviations ( English sum of squared explained 'Deviations shortly SSE or explained' sum of squares , short ESS ) S umme the A bweichungs q uadrate of values ${\displaystyle {\hat {y}}}$ , in short , or SAQ _explained , often model sum of squares or regression sum of squares , the square sum of the estimates or regress values. It is calculated as the sum of the squares of the centered estimated values ​​and can be interpreted as the “total variation of the estimated values ” (“explained variation”). There is no international agreement on the exact name and its abbreviations. In German-language literature, the German name is sometimes used with English abbreviations. ${\displaystyle SAQ_{\hat {y}}}$ ${\displaystyle \{{\hat {y}}_{i}\}}$

definition

The explained (deviation) sum of squares or regression sum of squares is defined as the sum of squares of the deviations explained by the regression function : ${\displaystyle ({\hat {y}}_{i}-{\overline {y}})}$

${\displaystyle SQE=SQ_{\mathrm {Erkl{\ddot {a}}rt} }\equiv \sum _{i=1}^{n}({\hat {y}}_{i}-{\overline {y}})^{2}=\sum _{i=1}^{n}({\hat {y}}_{i}-{\overline {\hat {y}}})^{2}}$

Sometimes there is also the abbreviation or . This term, however, can easily be with the " residual sum " ( English sum of squared residuals be confused) also with abbreviated. ${\displaystyle SQR}$ ${\displaystyle SQ_{\text{Regression}}}$ ${\displaystyle SQR}$

If the underlying linear model contains an absolute term other than zero , the empirical mean value of the estimated values agrees with that of the observed measured values , i.e. (for a proof in the multiple case, see coefficient of determination # matrix notation ). The explained sum of squares measures the spread of the estimated values around their mean . The ratio of the sum of squares explained by the regression to the total sum of squares is called the coefficient of determination of the regression. ${\displaystyle \beta _{0}}$ ${\displaystyle {\hat {y}}_{i}}$ ${\displaystyle y_{i}}$ ${\displaystyle \textstyle {\overline {\hat {y}}}={\tfrac {1}{n}}\sum \nolimits _{i=1}^{n}{\hat {y_{i}}}={\overline {y}}}$ ${\displaystyle {\hat {y}}_{i}}$ ${\displaystyle {\overline {\hat {y}}}={\overline {y}}}$

Simple linear regression

In simple linear regression (model with only one explanatory variable) the explained sum of squares can also be expressed as follows: ${\displaystyle y_{i}=\beta _{0}+x_{i}\beta _{1}+\varepsilon _{i}}$

${\displaystyle SQE=\sum _{i=1}^{n}({\hat {y}}_{i}-{\overline {y}})^{2}=\sum _{i=1}^{n}({\hat {y}}_{i}-({\hat {\beta }}_{0}+{\hat {\beta }}_{1}{\overline {x}}))^{2}}$.

Here, the represent the predicted values ​​and are the estimate of the absolute term and the estimate of the slope parameter . From this notation it can be seen that the explained sum of squares can also be represented as the product of the square of the Bravais-Pearson correlation coefficient and the  total sum of squares : ${\displaystyle {\hat {y}}_{i}={\hat {\beta }}_{0}+{\hat {\beta }}_{1}x}$ ${\displaystyle {\hat {\beta }}_{0}}$ ${\displaystyle {\hat {\beta }}_{1}}$ ${\displaystyle r_{xy}^{2}}$ ${\displaystyle SQT}$

${\displaystyle SQE=SQT\cdot r_{xy}^{2}}$,

where the least squares estimator for the slope is the quotient of the product sum of and and the square sum of . To show this, it must first be shown that if the underlying linear model contains an absolute term other than zero , the empirical mean value of the estimated values corresponds to that of the observed measured values . This is true because of ${\displaystyle {\hat {\beta }}_{1}}$ ${\displaystyle {\hat {\beta }}_{1}=SP_{xy}/SQ_{x}}$ ${\displaystyle x}$ ${\displaystyle y}$ ${\displaystyle x}$ ${\displaystyle \beta _{0}}$ ${\displaystyle {\hat {y}}_{i}}$ ${\displaystyle y_{i}}$

${\displaystyle {\overline {\hat {y}}}={\frac {1}{n}}\sum _{i=1}^{n}{\hat {y_{i}}}={\frac {1}{n}}\sum _{i=1}^{n}({\hat {\beta }}_{0}+{\hat {\beta }}_{1}x)={\hat {\beta }}_{0}+{\hat {\beta }}_{1}{\overline {x}}={\overline {y}}-{\hat {\beta }}_{1}{\overline {x}}+{\hat {\beta }}_{1}{\overline {x}}={\overline {y}}}$

and therefore

${\displaystyle {\begin{aligned}SQE&=\sum \nolimits _{i=1}^{n}({\hat {y}}_{i}-{\overline {y}})^{2}=\displaystyle \sum \nolimits _{i=1}^{n}({\hat {y}}_{i}-{\overline {\hat {y}}})^{2}\\&=\displaystyle \sum \nolimits _{i=1}^{n}(({\hat {\beta }}_{0}+{\hat {\beta }}_{1}x_{i})-({\hat {\beta }}_{0}+{\hat {\beta }}_{1}{\overline {x}}))^{2}\\&=\displaystyle \sum \nolimits _{i=1}^{n}({\hat {\beta }}_{1}(x_{i}-{\overline {x}}))^{2}\\&={\hat {\beta }}_{1}^{2}\displaystyle \sum \nolimits _{i=1}^{n}\left(x_{i}-{\overline {x}}\right)^{2}\\&={\hat {\beta }}_{1}^{2}SQ_{x}\\&=SQT\cdot r_{xy}^{2}\end{aligned}}}$,

where the last step follows from the fact that it can also be written as: ${\displaystyle {\hat {\beta }}_{1}}$

${\displaystyle {\hat {\beta }}_{1}={\frac {{\sqrt {\displaystyle \sum \nolimits _{i=1}^{n}\left(x_{i}-{\bar {x}}\right)^{2}}}{\sqrt {\displaystyle \sum \nolimits _{i=1}^{n}\left(y_{i}-{\bar {y}}\right)^{2}}}}{\displaystyle \sum \nolimits _{i=1}^{n}\left(x_{i}-{\bar {x}}\right)^{2}}}r_{xy}}$.

By the sum of squares decomposition or may be obtained by replacing by in this way also the following representation for the residual sum find: ${\displaystyle SQT=SQE+SQR}$ ${\displaystyle SQE=SQT-SQR}$ ${\displaystyle SQE=SQT-SQR}$ ${\displaystyle SQE=SQT\cdot r_{xy}^{2}}$ ${\displaystyle SQR}$

${\displaystyle SQR=SQT\cdot (1-r_{xy}^{2})}$.

Matrix notation

In matrix notation , the explained sum of squares can be expressed as follows

${\displaystyle SQE=\left({\hat {\mathbf {y} }}-{\overline {\mathbf {y} }}\right)^{\top }\left({\hat {\mathbf {y} }}-{\overline {\mathbf {y} }}\right)={\hat {\mathbf {y} }}^{\top }{\hat {\mathbf {y} }}-n{\overline {\hat {y}}}^{2}=\mathbf {b} ^{\top }\mathbf {X} ^{\top }\mathbf {y} -n{\overline {y}}^{2}}$.

Here is a vector with the elements and is defined by , where represents the least squares estimation vector and the data matrix. ${\displaystyle {\overline {\mathbf {y} }}}$ ${\displaystyle {\overline {y}}}$ ${\displaystyle {\hat {\mathbf {y} }}}$ ${\displaystyle {\hat {\mathbf {y} }}=\mathbf {X} \mathbf {b} }$ ${\displaystyle \mathbf {b} =\left(\mathbf {X} ^{\top }\mathbf {X} \right)^{-1}\mathbf {X} ^{\top }\mathbf {y} }$ ${\displaystyle \mathbf {X} }$

Individual evidence

1. ↑ ^{a b} Jeffrey Marc Wooldridge : Introductory econometrics: A modern approach. 4th edition. Nelson Education, 2015, p. 39.
2. ↑ Moosmüller, Gertrud. Methods of empirical economic research. Pearson Deutschland GmbH, 2008. p. 239.
3. ↑ Werner Timischl : Applied Statistics. An introduction for biologists and medical professionals. 2013, 3rd edition, p. 315.
4. ^ Ludwig Fahrmeir , Rita artist, Iris Pigeot , Gerhard Tutz : Statistics. The way to data analysis. 8., revised. and additional edition. Springer Spectrum, Berlin / Heidelberg 2016, ISBN 978-3-662-50371-3 , p. 151.