Model I and Model II Regressions

## Summary of Modifications made to Model I and Model II regressions

- I corrected an error in the y-intercept uncertainty calculation for the major axis algorithm:
- The equation was initially derived correctly in Kermack & Haldane (1950), but they introduced a typo when the equation was reprinted in a table of summary formulae at the end of their text. York (1966) copied this typo from the summary formulae and I coded my equations using York's notation. The algorithm now gives the correct uncertainty for the intercept in all four quadrants.
- The Results Table was corrected as required. The single line data plots for the major axis calculations were corrected as well.
- Minor typos in the header information for the lsqcubic were corrected and the revision date was updated.

September 2007 -- revision 3.1

- Additional minor changes were made to the lsqfit algorithms:
- Minor typos in the header information in some of the files were corrected.
- Revision dates on each file were updated.
- Replaced GIF images of line plots with higher resolution JPG files.

April 2007 -- revision 3

- A minor change was made to the names of the Model I regressions:
**Lsqfitxi.m**was renamed lsqfitx.m to make it more consistent with the names of the other algorithms.- Minor changes were made to lsqbisec.m and lsqfitgm so that they are now compatible with this change.
- The headers in each file were expanded to provide more information about each algorithm.

January 2003 -- revision 2

- The m-files for the Model I regressions were renamed to better reflect their applications:
**Lsqfit1.m**was renamed lsqfity.m since this regression minimizes the sum of the squares of the deviations in y.**Lsqfit1i.m**was renamed lsqfitxi.m since this regression minimizes the sum of the squares of the deviations in x and is inverted so that the slope is directly comparable to the slopes from the other regression algorithms.**Lsqfit1w.m**was renamed lsqfityw.m since this regression minimizes the weighted sum of the squares of the deviations in y.- At this time, it was also noticed that the standard deviations for the slope and intercept of the weighted model I regression, lsqfityw.m (formerly known as lsqfit1w.m), were considerably smaller than the standard deviations calculated by the other regression algorithms (see: results).
- The source code was checked against the original and found to be correct.
- The algorithm calculates the correct values according to the example given in Table 6.2 in Bevington and Robinson (1992).
- Until this situation is resolved, an alternate calculation is presented: lsqfityz.m. This algorithm uses the same equations for slope and intercept as used by lsqfit1w.m and lsqfityw.m, but calculates the standard deviations for these terms based upon the more general equations derived by York (1966). The revised values are more in line with the standard deviations for these terms calculated by the other methods.
- The algorithms for the Model II regressions were not changed although the header information for these files was revised for the purposes of clarity and completeness.
- A new Model II regression algorithm was added, known here as the "least squares bisector:" lsqbisec.m.
- Sprent and Dolby (1980) have objected to the
use of the geometric mean regression, because it is not a true unbiased expression for the functional relationship between X and Y.*ad hoc* - They have suggested that an equally strong case can be made for using the line that bisects the minor angle between the two model I regressions.
- While the difference in slope between the geometric mean regression and the "least squares bisector" are small and probably not statistically significant, this new "regression" line is included here for completeness.
- And lastly, the test data file, data.txt, was edited and new results were calculated:
- A simple plot of the original raw data revealed that it was not a bivariate normal distribution. It contained numerous outliers and data clusters. These have been removed to give a cleaner looking dataset.
- The modified file is smaller, 110 points vs 139, and one can easily see the shape of the data ellipse.
- Graphs of the regression lines are now available as well.

January 2000 -- revision 1