Multiple Regression

  • Kunio Takezawa


When data \(\{(x_{i1},x_{i2},\ldots,x_{iq},y_{i})\}\) (1 ≤ i ≤ n) are given, multiple regression derives the values of {a j }(1 ≤ j ≤ q) by minimizing
$$\displaystyle{ \mathit{RSS} =\sum _{ i=1}^{n}{(y_{ i} - a_{0} -\sum _{j=1}^{q}a_{ j}x_{ij})}^{2} =\sum _{ i=1}^{n}e_{ i}^{2}. }$$
Here \(e_{i} = y_{i} - a_{0} -\sum _{j=1}^{q}a_{j}x_{ij}\) are called residuals. The acronym RSS stands for the residual sum of squares. Equation (4.1) yields the regression equation:
$$\displaystyle{ \hat{y} = \hat{a}_{0} +\sum _{ j=1}^{q}\hat{a}_{ j}x_{j}, }$$
where \(\{\hat{a}_{j}\}\) are estimates of the regression coefficients, {x j } the predictor variables, and \(\hat{y}\) the estimates of target variables. Using Eq. (4.2), Eq. (4.1) is transformed into
$$\displaystyle{ \mathit{RSS} =\sum _{ i=1}^{n}{(y_{ i} -\hat{a}_{0} -\sum _{j=1}^{q}\hat{a}_{ j}x_{ij})}^{2}. }$$


Null Hypothesis Regression Coefficient Simulation Data Probability Density Function Alternative Hypothesis 
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Copyright information

© Springer Japan 2014

Authors and Affiliations

  • Kunio Takezawa
    • 1
    • 2
  1. 1.National Agricultural and Food Research OrganizationTsukubaJapan
  2. 2.Graduate School of Life and Environmental SciencesUniversity of TsukubaTsukubaJapan

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