Model I and Model II Regressions
Results for Model I and Model II Regressions
Here are the results from my MATLAB® shell-script routines for fitting a line to the
example dataset (see the file: data.txt) using the various linear
regression models:
-
Model I
Regressions | M-file | Slope | Intercept | Correlation
Coefficient
|
|---|
| Y-on-X | Lsqfity.m | 1.2931 ± 0.0598 | 9.3670 ± 1.4750 | 0.9014
|
| X-on-Y | Lsqfitx.m | 1.5916 ± 0.0736 | 3.0908 ± 1.8945 | 0.9014
|
| Weighted Y-on-X | Lsqfityw.m ¹ | 1.3322 ± 0.0064 | 4.4067 ± 0.0738 | 0.8973
|
| Weighted Y-on-X | Lsqfityz.m ² | 1.3322 ± 0.0630 | 4.4067 ± 0.7220 | 0.8973
|
Model II
Regressions | M-file | Slope | Intercept | Correlation
Coefficient
|
|---|
| Major axis | Lsqfitma.m | 1.4896 ± 0.0682 | 5.2356 ± 1.6455 | 0.9014
|
| Geometric mean | Lsqfitgm.m | 1.4346 ± 0.0613 | 6.3917 ± 1.5128 | 0.9014
|
| Least-squares-bisector | Lsqbisec.m ³ | 1.4320 ± 0.0613 | 6.4477 ± 1.5114 | 0.9014
|
| Least-squares-cubic | Lsqcubic.m | 1.4948 ± 0.0649 | 5.2594 ± 0.6227 | 0.9583
|
Footnotes to table:
- All parameters were calculated with equations from Bevington and Robinson
(1992). Their standard deviations for the slope and intercept seem very
small when compared to the uncertainties calculated with the other
regression equations.
- Revised weighted model I regression: equations for the slope and intercept
were again taken from Bevington and Robinson (1992), however, the standard
deviations in these parameters are calculated according to equations derived
by York (1966). The uncertainties calculated this way seem more in line with
those calculated by the other regressions.
- The least-squares-bisector algorithm follows the suggestion of Sprent and Dolby
(1980) that in the case of model II regressions, an equally strong case can be
made for the line that bisects the minor angle between the two model I
regressions: Y-on-X and X-on-Y. The uncertainties in the slope and intercept
are calculated according to the equations derived by York (1966) in a manner
analogous to that used for the geometric mean regression.
Also, please note that:
- The answers in the table have not been rounded-off to the proper number of significant
figures. Instead, I have choosen to list the results obtained with Matlab®
in standard precision mode so that one can compare these results to those from
other programs without an undue emphasis on data truncation and round-off errors.
- Equations for the Y-on-X, X-on-Y, major axis, geometric
mean and least-squares-bisector regressions all run through the centroid:
(mean-x,mean-y).
- Likewise, the correlation coefficients for the Y-on-X, X-on-Y,
major axis and geometric mean regressions are all the same. Thus,
the correlation coefficient is properly interpreted as a measure of the linearity
of the data and not how well the line fits the data.
- For the weighted linear regressions and the least squares cubic
the correlation coefficients are different owing to the varying weights applied
to each data point.
References:
- Bevington & Robinson (1992). Data Reduction and Error Analysis for the Physical
Sciences, Second Edition, McGraw-Hill, Inc., New York.
- Sprent and Dolby (1980). The geometric mean functional relationship. Biometrics
36: 547-550.
- York (1966). Least-squares fitting of a straight line. Canad. J. Phys. 44: 1079-1086.
Last updated: Sep. 06, 2007
Questions? Comments? Please contact
Edward Peltzer.
