# titterington: Linear regression of X,Y,Z-variables with correlated errors In IsoplotR: Statistical Toolbox for Radiometric Geochronology

## Description

Implements the maximum likelihood algorithm of Ludwig and Titterington (1994) for linear regression of three dimensional data with correlated uncertainties.

## Usage

 `1` ```titterington(x, alpha = 0.05) ```

## Arguments

 `x` an `[nx9]` matrix with the following columns: `X, sX,` `Y, sY,` `Z, sZ`, `rhoXY,` `rhoXZ,` `rhoYZ`. `alpha` cutoff value for confidence intervals

## Details

Ludwig and Titterington (1994)'s 3-dimensional linear regression algorithm for data with correlated uncertainties is an extension of the 2-dimensional algorithm by Titterington and Halliday (1979), which itself is equivalent to the algorithm of York et al. (2004). Given n triplets of (approximately) collinear measurements X_i, Y_i and Z_i (for 1 ≤q i ≤q n), their uncertainties s[X_i], s[Y_i] and s[Z_i], and their covariances cov[X_i,Y_i], cov[X_i,Z_i] and cov[Y_i,Z_i], the `titterington` function fits two slopes and intercepts with their uncertainties. It computes the MSWD as a measure of under/overdispersion. Overdispersed datasets (MSWD>1) can be dealt with in the same three ways that are described in the documentation of the `isochron` function.

## Value

A four-element list of vectors containing:

par

4-element vector `c(a,b,A,B)` where `a` is the intercept of the `X-Y` regression, `b` is the slope of the `X-Y` regression, `A` is the intercept of the `X-Z` regression, and `B` is the slope of the `X-Z` regression.

cov

`[4x4]`-element covariance matrix of `par`

mswd

the mean square of the residuals (a.k.a 'reduced Chi-square') statistic

p.value

p-value of a Chi-square test for linearity

df

the number of degrees of freedom for the Chi-square test (2n-4)

tfact

the 100(1-α/2)\% percentile of the t-distribution with (n-2k+1) degrees of freedom

## References

Ludwig, K.R. and Titterington, D.M., 1994. Calculation of ^{230}Th/U isochrons, ages, and errors. Geochimica et Cosmochimica Acta, 58(22), pp.5031-5042.

Titterington, D.M. and Halliday, A.N., 1979. On the fitting of parallel isochrons and the method of maximum likelihood. Chemical Geology, 26(3), pp.183-195.

York, D., Evensen, N.M., Martinez, M.L. and De Basebe Delgado, J., 2004. Unified equations for the slope, intercept, and standard errors of the best straight line. American Journal of Physics, 72(3), pp.367-375.

`york`, `isochron`, `ludwig`

## Examples

 ```1 2 3 4 5 6 7 8 9``` ```d <- matrix(c(0.1677,0.0047,1.105,0.014,0.782,0.015,0.24,0.51,0.33, 0.2820,0.0064,1.081,0.013,0.798,0.015,0.26,0.63,0.32, 0.3699,0.0076,1.038,0.011,0.819,0.015,0.27,0.69,0.30, 0.4473,0.0087,1.051,0.011,0.812,0.015,0.27,0.73,0.30, 0.5065,0.0095,1.049,0.010,0.842,0.015,0.27,0.76,0.29, 0.5520,0.0100,1.039,0.010,0.862,0.015,0.27,0.78,0.28), nrow=6,ncol=9) colnames(d) <- c('X','sX','Y','sY','Z','sZ','rXY','rXZ','rYZ') titterington(d) ```

### Example output

```\$par
a          b          A          B
0.6523258 -1.1448293  0.6287848 -0.8317666

\$cov
a            b             A            B
a -0.0014659472  0.002534783 -0.0002053377  0.000577532
b  0.0025347834 -0.029356631 -0.0025913430  0.006835091
A -0.0002053377 -0.002591343  0.0015645768 -0.003932355
B  0.0005775320  0.006835091 -0.0039323548  0.010804817

\$df
 8

\$mswd
b
-19.35144

\$p.value
 1

\$type
 "titterington"
```

IsoplotR documentation built on July 10, 2021, 1:06 a.m.