central {IsoplotR}

Fits random effects models to overdispersed datasets

Description

Computes the logratio mean composition of a continuous mixture of fission track or U-Th-He data and returns the corresponding age and fitting parameters. Only propagates the systematic uncertainty associated with decay constants and calibration factors after computing the weighted mean isotopic composition. Does not propagate the uncertainty of any initial daughter correction, because this is neither a purely random or purely systematic uncertainty.

Usage

central(x, ...)

## Default S3 method:
central(x, ...)

## S3 method for class 'UThHe'
central(x, compositional = FALSE, model = 1, ...)

## S3 method for class 'fissiontracks'
central(x, exterr = FALSE, ...)

Arguments

x

an object of class UThHe or fissiontracks, OR a 2-column matrix with (strictly positive) values and uncertainties

...

optional arguments

compositional

logical. If TRUE, calculates the 'barycentric' U-Th-He, age, i.e. the age corresponding to the weighted mean logratio composition.

model

only relevant if compositional is TRUE. If the scatter between the data points is solely caused by the analytical uncertainty, then the MSWD value should be approximately equal to one. There are three strategies to deal with the case where MSWD>1.choose one of the following statistical models:

1: assume that the analytical uncertainties have been underestimated by a factor \sqrt{MSWD}.

2: ignore the analytical uncertainties.

3: attribute any excess dispersion to the presence of geological uncertainty, which manifests itself as an added (co)variance term.

exterr

include the zeta or decay constant uncertainty into the error propagation for the central age?

Details

The central age assumes that the observed age distribution is the combination of two sources of scatter: analytical uncertainty and true geological dispersion.

1. For fission track data, the analytical uncertainty is assumed to obey Binomial counting statistics and the geological dispersion is assumed to follow a lognormal distribution.

2. For U-Th-He data, the U-Th-(Sm)-He compositions and uncertainties are assumed to follow a logistic normal distribution.

3. For all other data types, both the analytical uncertainties and the true ages are assumed to follow lognormal distributions.

The difference between the central age and the weighted mean age is usually small unless the data are imprecise and/or strongly overdispersed.

The uncertainty budget of the central age does not include the uncertainty of the initial daughter correction (if any), for the same reasons as discussed under the weightedmean function.

Value

If x has class UThHe and compositional is TRUE, returns a list containing the following items:

uvw

(if the input data table contains Sm) or uv (if it does not): the mean log[U/He], log[Th/He] (, and log[Sm/He]) composition.

covmat

the covariance matrix of uvw or uv.

mswd

the reduced Chi-square statistic of data concordance, i.e. mswd=SS/df, where SS is the sum of squares of the log[U/He]-log[Th/He] compositions.

model

the fitting model.

df

the degrees of freedom (2n-2) of the fit (only reported if model=1).

p.value

the p-value of a Chi-square test with df degrees of freedom (only reported if model=1.)

age

a two- or three-element vector with:
t: the 'barycentric' age, i.e. the age corresponding to uvw.
s[t]: the standard error of t.
disp[t]: the standard error of t enhanced by a factor of \sqrt{mswd} (only reported if model=1).

w

the geological overdispersion term. If model=3, this is a two-element vector with the standard deviation of the (assumedly) Normal dispersion and its standard error. w=0 if model<3.

OR, otherwise:

age

a two-element vector with the central age and its standard error.

disp

a two-element vector with the overdispersion (standard deviation) of the excess scatter, and its standard error.

mswd

the reduced Chi-square statistic of data concordance, i.e. mswd=X^2/df, where X^2 is a Chi-square statistic of the EDM data or ages

df

the degrees of freedom (n-2)

p.value

the p-value of a Chi-square test with df degrees of freedom

References

Galbraith, R.F. and Laslett, G.M., 1993. Statistical models for mixed fission track ages. Nuclear Tracks and Radiation Measurements, 21(4), pp.459-470.

Vermeesch, P., 2008. Three new ways to calculate average (U-Th)/He ages. Chemical Geology, 249(3), pp.339-347.

See Also

weightedmean, radialplot, helioplot

Examples

attach(examples)
print(central(UThHe)$age)