Model I and Model II Regressions
Rules of Thumb
My "rules-of-thumb" for choosing which regression to use are as follows:
- For situations where the X-parameter is controlled, as in making-up standards for instrument calibration or doing laboratory experiments where only one variable is changed, then the standard model-I regressions are required.
- When all data points are given equal weight, use lsqfity.
- When the data points are individually weighted, use lsqfityw or lsqfityz.
- To decide between using lsqfityw.m or lsqfityz.m, see the footnote #2 to my "Results for Model I and Model II Regressions Table.
- One word of caution: the slope of major axis is sensitive to changes in scale (See Kermack and Haldane, 1950, for an explanation.
- If this is a concern (as in cases where the choice of measurement units is arbitrary), then the geometric mean regression [lsqfitgm] may be used.
- The reduced major axis method is both difficult and tedious to compute, so I recommend that the geometric mean regression [lsqfitgm] be used in this case as this gives the identical line (see Ricker, 1973, for the derivation). I have used the symmetrical limits for a model I regression to estimate the uncertainty in the slope and intercept of the geometric mean regression following Ricker's (1973) treatment.
- Sprent and Dolby (1980) have taken exception to the ad hoc use of the geometric mean regression in model II cases. They recommend the use of the least squares bisector [lsqbisec], the line that bisects the angle between the two model I regressions. Unfortunately, they did not present a statistical treatment for the estimation of the uncertainty limits for the least squares bisector slope, or intercept.
- The method developed by York called "the least squares cubic" [lsqcubic] can be used in this case.
- This method allows the uncertainty in each X and Y measurement be specified in order to appropriately weight the data.