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

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

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.

thielsen returns an object of `class`

"thielsen" with components

the intercept and slope

residuals from the fitted line

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

number of data points

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

the terms object corresponding to the formula

na.action information, if applicable

a copy of the call to the function

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.

```
# NOT RUN {
afit1 <- thielsen(aes ~ aas, symmetric=TRUE, data= arsenate)
afit2 <- thielsen(aas ~ aes, symmetric=TRUE, data= arsenate)
rbind(coef(afit1), coef(afit2)) # symmetric results
1/coef(afit1)[2]
# }
```

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