## Results

The . 3) the Minitab macro: “SENSLOPE.MAC” (Akritas, C., 2004). 4) StatsDirect (2008), an inexpensive program with many nonparametric tests designed for medical researchers.

This web-page calculates the Theil-Sen slope for a single phase, and also compares slopes from two phases. Significance tests are based on corresponding Kendall’s Tau. Standard Errors are estimated based on transformed Kendall’s Tau standard errors.

**Theil-Sen slope**(Sen, 1968; Theil, 1950), also known as “Kendall’s slope” or the “Nonparametric linear regression slope”, is an alternative to the standard linear regression slope. The Theil-Sen slope is popular in the earth sciences (meteorology, hydrology, ecology, climatology) for measuring over time such phenomena as air and water quality and glacial retreat. The Theil-Sen slope can be represented by a straight line, but, like Koenig’s bi-split (Koenig, 1972) and Tukey’s tri-split slopes (Tukey, 1977), it is “distribution free” and permits use of merely ordinal measurement scales. The Theil-Sen has relatively strong power/precision, greater than both the Koenig and Tukey nonparametric slopes, and (Johnson and Velleman, 1985). When data meet all parametric assumptions, Theil-Sen has about 91% Pitman efficiency of linear regression, and when data are very non-normal and skewed, Theil-Sen efficiency can exceed (1.27 times) that of linear regression Armitage et al, 2002; Helsel & Hirsch, 1992; Sheskin, 2007; Sprent & Smeeton, 2007). The most common significance test for the Theil-Sen slope is that used for Kendall’s Tau Rank Order, calculated from the same data (Hollander & Wolfe, 1999). The Theil-Sen slope is not commonly offered by most statistics packages. Alternatives include: 1) KTRLine Version 1.0, (Granato, 2006), downloadable from the US Geological Survey Office. 2) the free open-source “R” stats program, includes Theil-Sen within its “mblm” package at:This web-page calculates the Theil-Sen slope for a single phase, and also compares slopes from two phases. Significance tests are based on corresponding Kendall’s Tau. Standard Errors are estimated based on transformed Kendall’s Tau standard errors.

- Data are input in up to ten windows at the page top, each headed by a label box and a selection check.
- Type or paste data from just one phase in each input window, and label each phase above, e.g. A1, B3, 2A1, etc.
- You may calculate Theil-Sen slope for a single phase, or may contrast slopes from two phases. Choose one of these options by clicking the Slope button or the Contrast button at top of the page.
- 4. Results are provided, for individual slopes, and, if selected, for their differences.
**Note 1:**The standard errors used in this program are calculated from the Tau/SEtau ratios for each phase. Other Theil-Sen statistics packages do rely on the Tau Z scores and p-values, but do not interpolate the standard error from the Tau ratio, as we do here.**Note 2:**The test of difference between slopes is calculated by student t, with df=N-2. The denominator is the square root of the sum of the two variances: sqrt(Var1+Var2).