# ludwig: Linear regression of U-Pb data with correlated errors, taking... In IsoplotR: Statistical Toolbox for Radiometric Geochronology

 ludwig R Documentation

## Linear regression of U-Pb data with correlated errors, taking into account decay constant uncertainties.

### Description

Implements the maximum likelihood algorithm for Total-Pb/U isochron regression of Ludwig (1998) and extends the underlying methodology to accommodate U-Th-Pb data and initial U-series disequilibrium.

### Usage

ludwig(
x,
model = 1,
anchor = 0,
exterr = FALSE,
type = "joint",
plot = FALSE,
...
)

ludwig(
x,
model = 1,
anchor = 0,
exterr = FALSE,
type = "joint",
plot = FALSE,
...
)


 x an object of class UPb model one of three regression models: 1: fit a discordia_line through the data using the maximum likelihood algorithm of Ludwig (1998), which assumes that the scatter of the data is solely due to the analytical uncertainties. In this case, IsoplotR will either calculate an upper and lower intercept age (for Wetherill concordia), or a lower intercept age and common (^{207}Pb/^{206}Pb)_\circ-ratio intercept (for Tera-Wasserburg). If the p-value for the chi-square test is less than alpha(), then the analytical uncertainties are augmented by a factor \sqrt{MSWD}. 2: fit a discordia_line ignoring the analytical uncertainties 3: fit a discordia_line using a modified maximum likelihood algorithm that includes accounts for any overdispersion by adding a geological (co)variance term. anchor control parameters to fix the intercept age or common Pb composition of the isochron fit. This can be a scalar or a vector. If anchor[1]=0: do not anchor the isochron. If anchor[1]=1: fix the common Pb composition at the values stored in settings('iratio',...). If anchor[1]=2: force the isochron line to intersect the concordia line at an age equal to anchor[2]. If anchor[1]=3: anchor the isochron line to the Stacey-Kramers mantle evolution model. exterr propagate external sources of uncertainty (i.e. decay constants)? type only relevant if x$format>3. Can take on the following values: 'joint' or 0: 3-dimensional isochron regression. 1: 2-dimensional regression of {}^{204}Pb/{}^{206}Pb vs. {}^{238}U/{}^{206}Pb (for U-Pb formats 4, 5 and 6), or of {}^{208}Pb/{}^{206}Pb vs. {}^{238}U/{}^{206}Pb (for U-Pb formats 7 and 8). 2: 2-dimensional regression of {}^{204}Pb/{}^{207}Pb vs. {}^{235}U/{}^{207}Pb (for U-Pb formats 4, 5 and 6), or of {}^{208}Pb/{}^{207}Pb vs. {}^{235}U/{}^{207}Pb (for U-Pb formats 7 and 8). 3: 2-dimensional regression of {}^{206}Pb/{}^{208}Pb vs. {}^{232}Th/{}^{208}Pb (only for U-Pb formats 7 and 8). 4: 2-dimensional regression of {}^{207}Pb/{}^{208}Pb vs. {}^{232}Th/{}^{208}Pb (only for U-Pb formats 7 and 8). plot logical. Only relevant for datasets with measured disequilibrium. If TRUE, plots the posterior distribution of the age and initial activity ratios. ... optional arguments ### Details The 3-dimensional regression algorithm of Ludwig and Titterington (1994) was modified by Ludwig (1998) to fit so-called 'Total Pb-U isochrons'. These are constrained to a radiogenic endmember composition that falls on the concordia line. In its most sophisticated form, this algorithm does not only allow for correlated errors between variables, but also between aliquots. IsoplotR currently uses this algorithm to propagate decay constant uncertainties in the total Pb-U isochron ages. ### Value par a vector with the lower concordia intercept, the common Pb ratios, the dispersion parameter (if model=3), and the initial {}^{234}U/{}^{238}U and {}^{230}Th/{}^{238}U activity ratio (in the presence of initial disequilibrium). cov the covariance matrix of par df the degrees of freedom of the model fit (n-2 if x$format<4 or 2n-3 if x\$format>3, where n is the number of aliquots).

mswd

the mean square of weighted deviates (a.k.a. reduced Chi-square statistic) for the fit.

p.value

p-value of a Chi-square test for the linear fit

### References

Ludwig, K.R., 1998. On the treatment of concordant uranium-lead ages. Geochimica et Cosmochimica Acta, 62(4), pp.665-676.

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.

concordia, titterington, isochron

### Examples

f <- system.file("UPb4.csv",package="IsoplotR")