Matlab shell-scripts for linear regression analysis

Model I regressions

• lsqfity – Standard Model I linear regression
  • [Y-on-X regression]
  • Sometimes referred to as ordinary least squares (OLS).
  • Fit line by minimizing y-residuals only
  • All data points are given equal weight
  • Use whenever X is controlled and Y is measured
• lsqfitx – Alternate or Reversed Model I regression
  • [X-on-Y regression]
  • Fit line by minimizing x-residuals only
  • All data points are given equal weight
• lsqfityw – Weighted Model I regression
  • [wY-on-X regression]
  • Fit line by minimizing y-residuals only
  • Data points are given varying weight
  • Use whenever X is controlled and Y is measured with varying uncertainty
  • Please note that:
    • This m-file gives the correct results for all parameters (See table 6.2 in Bevington and Robinson, 2003); but,
    • the uncertainties in the slope and intercept are much smaller than the same uncertainties calculated by lsqfity.m or lsqfitx.m (see the results table).
    • Also note that if the values for the uncertainties in the y-data (sY) are constant, then the slope and intercept will be the same as that calculated with lsqfity.m, but the uncertainties in these parameters (sm & sb) will be different.
• lsqfityz – Weighted Model I regression – revised Jan 2000
  • [zY-on-X regression]
  • This is the same regression as lsqfityw.m, except that:
    • York’s (1966) equations for the uncertainties in weighted slope and intercept are used.
    • The revised uncertainties are more in line with those from lsqfity.m.

Model II regressions

• lsqfitma – Pearson’s Major Axis
  • aka: first principal component
  • [Correlation of X & Y]
  • Line is fit by minimizing BOTH x- and y-residuals simultaneously
  • All data are given equal weight
  • Use when units and range of X and Y are the same or very similar
• lsqfitgm – Geometric Mean regression
  • aka: reduced major axis or standard major axis
  • [Correlation of X & Y]
  • Slope of line is the geometric mean of the two slopes determined by regressing Y-on-X and X-on-Y
  • All data are given equal weight
  • Use when units or range of X and Y are different
  • Please note that:
    • This algorithm uses both lsqfity.m and lsqfitx.m for determining the slope.
    • The uncertainties in the slope and intercept are estimated by analogy with the symmetrical uncertainty limits for a model I regression following the treatment by Ricker (1973).
• lsqbisec – Least Squares Bisector
  • [Regression/correlation of X & Y]
  • Slope of line is determined by bisecting the minor angle between the two model I regressions:
    Y-on-X and X-on-Y
  • All data are given equal weight
  • Use when units or range of X and Y are different
  • Please note that:
    • This algorithm uses both lsqfity.m and lsqfitx.m for determining the slope.
    • The uncertainties in the slope and intercept are estimated by analogy with the symmetrical uncertainty limits for a model I regression following the treatment by Ricker (1973).
• lsqcubic – Least-Squares-Cubic
  • [Regression/correlation of wX & wY.]
  • Line is fit by minimizing both x- and y-residuals simultaneously for WEIGHTED data points.
  • Each data point can be given its own weight either as the inverse-square of the actual measurement precision or as the inverse-square of the product of the relative measurement precision for the method times concentration.
  • Use when the measurement error of X and/or Y varies.
  • Please note that:
    • This algorithm uses lsqfitma.m for the first estimate of the slope.
    • Iteration proceeds until the change in the slope is less than the user defined limit.

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