|
|
| (8 intermediate revisions by 3 users not shown) |
| Line 1: |
Line 1: |
| ===Rank Regression on Y===
| | #REDIRECT [[The Normal Distribution]] |
| Performing rank regression on Y requires that a straight line be fitted to a set of data points such that the sum of the squares of the vertical deviations from the points to the line is minimized.
| |
| | |
| The least squares parameter estimation method (regression analysis) was discussed in Chapter 3 and the following equations for regression on Y were derived: | |
| | |
| ::<math>\begin{align}\hat{a}= & \bar{b}-\hat{b}\bar{x} \\
| |
| =& \frac{\sum_{i=1}^N y_{i}}{N}-\hat{b}\frac{\sum_{i=1}^{N}x_{i}}{N}\\
| |
| \end{align}
| |
| </math>
| |
| | |
| :and:
| |
| | |
| ::<math>\hat{b}=\frac{\underset{i=1}{\overset{N}{\mathop{\sum }}}\,{{x}_{i}}{{y}_{i}}-\tfrac{\underset{i=1}{\overset{N}{\mathop{\sum }}}\,{{x}_{i}}\underset{i=1}{\overset{N}{\mathop{\sum }}}\,{{y}_{i}}}{N}}{\underset{i=1}{\overset{N}{\mathop{\sum }}}\,x_{i}^{2}-\tfrac{{{\left( \underset{i=1}{\overset{N}{\mathop{\sum }}}\,{{x}_{i}} \right)}^{2}}}{N}}</math>
| |
| | |
| In the case of the normal distribution, the equations for <math>{{y}_{i}}</math> and <math>{{x}_{i}}</math> are:
| |
| | |
| ::<math>{{y}_{i}}={{\Phi }^{-1}}\left[ F({{T}_{i}}) \right]</math>
| |
| | |
| :and:
| |
| | |
| ::<math>{{x}_{i}}={{T}_{i}}</math>
| |
| | |
| | |
| where the values for <math>F({{T}_{i}})</math> are estimated from the median ranks. Once <math>\widehat{a}</math> and <math>\widehat{b}</math> are obtained, <math>\widehat{\sigma }</math> and <math>\widehat{\mu }</math> can easily be obtained from Eqns. (an) and (bn).
| |
| | |
| ====The Correlation Coefficient====
| |
| The estimator of the sample correlation coefficient, <math>\hat{\rho }</math> , is given by:
| |
| | |
| ::<math>\hat{\rho }=\frac{\underset{i=1}{\overset{N}{\mathop{\sum }}}\,({{x}_{i}}-\overline{x})({{y}_{i}}-\overline{y})}{\sqrt{\underset{i=1}{\overset{N}{\mathop{\sum }}}\,{{({{x}_{i}}-\overline{x})}^{2}}\cdot \underset{i=1}{\overset{N}{\mathop{\sum }}}\,{{({{y}_{i}}-\overline{y})}^{2}}}}</math>
| |