Results for Model I and Model II regressions

Here are the results from the MATLAB® shell-script routines for fitting a line to the example datasets from Bevington and Robinson (2003):

Model I
Regressions
Data File M-file Slope Intercept R
Y-on-X [1] BR3tab61 Lsqfity.m 0.02622 ± 0.00034 0.07139 ± 0.01917 0.99941
X-on-Y [1] BR3tab61 Lsqfitx.m 0.02625 ± 0.00034 0.06984 ± 0.02000 0.99941
Weighted Y-on-X [1,2] BR3tab62 Lsqfityw.m 30.6979 ± 1.0341 119.4963 ± 7.5676 0.99385
Weighted Y-on-X [1,3] BR3tab62 Lsqfityz.m 30.6979 ± 1.2096 119.4963 ± 8.8522 0.99385

Here are the results from my MATLAB® shell-script routines for fitting a line to the example dataset (data.txt) using the various linear regression models:

Model I
Regressions
M-file Slope Intercept R
Y-on-X [1] Lsqfity.m    1.2931 ± 0.0598    9.3670 ± 1.4750 0.9014
X-on-Y [1] Lsqfitx.m 1.5916 ± 0.0736 3.0908 ± 1.8945 0.9014
Weighted Y-on-X [1,2] Lsqfityw.m 1.3322 ± 0.0064 4.4067 ± 0.0738 0.8973
Weighted Y-on-X [1,3] Lsqfityz.m 1.3322 ± 0.0630 4.4067 ± 0.7220 0.8973
Model II
Regressions
M-file Slope Intercept R
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 [4] Lsqbisec.m 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:

  1. All parameters were calculated with equations from Bevington and Robinson (1992).
  2. The standard deviations for the slope and intercept of the weighted regression (lsqfityw.m) seem small when compared to the uncertainties calculated with the other regression equations.
  3. 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.
  4. 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.
  • Bevington & Robinson (2003). Data Reduction and Error Analysis for the Physical Sciences, Third Edition, McGraw-Hill, Inc., New York.

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