Thiel-Sen regression is a robust regression method for two variables. The symmetric option gives a variant that is symmentric in x and y.

```
theilsen(formula, data, subset, weights, na.action, conf=.95,
nboot = 0, symmetric=FALSE, eps=sqrt(.Machine$double.eps),
x = FALSE, y = FALSE, model = TRUE)
```

theilsen returns an object of `class`

"theilsen" with components

- coefficients
the intercept and slope

- residuals
residuals from the fitted line

- angle
if the symmetric option is chosen, this contains all of the solutions for the angle of the regression line

- n
number of data points

- model,x,y
optional componets as specified by the x, y, and model arguments

- terms
the terms object corresponding to the formula

- na.action
na.action information, if applicable

- call
a copy of the call to the function

The generic accessor functions

`coef`

, `residuals`

, and `terms`

extract the relevant components.

- formula
a model formula with a single continuous response on the left and a single continuous predictor on the right.

- data
an optional data frame, list or environment containing the variables in the model.

- subset
an optional vector specifying a subset of observations to be used in the fitting process.

- weights
an optional vector of weights to be used in the fitting process.

- na.action
a function which indicates what should happen when the data contain NAs. The default is set by the na.action setting of

`options`

.- conf
the width of the computed confidence limit.

- nboot
number of bootstrap samples used to compute standard errors and/or confidence limits. If this is 0 or missing then an asypmtotic formula is used.

- symmetric
compute an estimate whose slope is symmetric in x and y.

- eps
the tolerance used to detect tied values in x and y

- x,y, model
logicals. If TRUE the corresponding components of the fit (the model frame, the model matrix, or the response) is returned.

Terry Therneau

One way to characterize the slope of an ordinary least squares line is that \(\rho(x, r)\) =0, where where \(\rho\) is the correlation coefficient and r is the vector of residuals from the fitted line. Thiel-Sen regression replaces \(\rho\) with Kendall's \(\tau\), a non-parametric alternative. It it resistant to outliers while retaining good statistical efficiency.

The symmetric form of the estimate is based on solving the inverse equation: find that rotation of the original data such that \(\tau(x,y)=0\) for the rotated data. (In a similar fashion,the rotation such the least squares slope is zero yields Deming regression.) In this case it is possible to have multiple solutions, i.e., slopes that yeild a 0 correlation, although this is rare unless the deviations from the fitted line are large.

The default confidence interval estimate is based on the result of Sen, which is in turn based on the relationship to Kendall's tau and is essentially an inversion of the confidence interval for tau. The argument does not extend to the symmetric case, for which we recommend using a bootstrap confidence interval based on 500-1000 replications.

Thiel, H. (1950), A rank-invariant method of linear and polynomial regression analysis. I, II, III, Nederl. Akad. Wetensch., Proc. 53: 386-392, 521-525, 1397-1412.

Sen, P.B. (1968), Estimates of the regression coefficient based on Kendall's tau, Journal of the American Statistical Association 63: 1379-1389.

`deming`

, `pbreg`

```
afit1 <- theilsen(aes ~ aas, symmetric=TRUE, data= arsenate)
afit2 <- theilsen(aas ~ aes, symmetric=TRUE, data= arsenate)
rbind(coef(afit1), coef(afit2)) # symmetric results
1/coef(afit1)[2]
```

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