weightedmean: Calculate the weighted mean age

In pvermees/IsoplotR: Statistical Toolbox for Radiometric Geochronology

Calculate the weighted mean age

Description

Averages heteroscedastic data either using the ordinary weighted mean, or using a random effects model with two sources of variance. Computes the MSWD of a normal fit without overdispersion. Implements a modified Chauvenet criterion to detect and reject outliers. 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

weightedmean(x, ...)

## Default S3 method:
weightedmean(
  x,
  from = NA,
  to = NA,
  random.effects = FALSE,
  detect.outliers = TRUE,
  plot = TRUE,
  levels = NA,
  clabel = "",
  rect.col = c("#00FF0080", "#FF000080"),
  outlier.col = "#00FFFF80",
  sigdig = 2,
  oerr = 3,
  ranked = FALSE,
  hide = NULL,
  omit = NULL,
  omit.col = NA,
  ...
)

## S3 method for class 'other'
weightedmean(
  x,
  from = NA,
  to = NA,
  random.effects = FALSE,
  detect.outliers = TRUE,
  plot = TRUE,
  levels = NA,
  clabel = "",
  rect.col = c("#00FF0080", "#FF000080"),
  outlier.col = "#00FFFF80",
  sigdig = 2,
  oerr = 3,
  ranked = FALSE,
  hide = NULL,
  omit = NULL,
  omit.col = NA,
  ...
)

## S3 method for class 'UPb'
weightedmean(
  x,
  random.effects = FALSE,
  detect.outliers = TRUE,
  plot = TRUE,
  from = NA,
  to = NA,
  levels = NA,
  clabel = "",
  rect.col = c("#00FF0080", "#FF000080"),
  outlier.col = "#00FFFF80",
  sigdig = 2,
  type = 4,
  cutoff.76 = 1100,
  oerr = 3,
  cutoff.disc = discfilter(),
  exterr = FALSE,
  ranked = FALSE,
  common.Pb = 0,
  hide = NULL,
  omit = NULL,
  omit.col = NA,
  ...
)

## S3 method for class 'PbPb'
weightedmean(
  x,
  random.effects = FALSE,
  detect.outliers = TRUE,
  plot = TRUE,
  from = NA,
  to = NA,
  levels = NA,
  clabel = "",
  rect.col = c("#00FF0080", "#FF000080"),
  outlier.col = "#00FFFF80",
  sigdig = 2,
  oerr = 3,
  exterr = FALSE,
  common.Pb = 2,
  ranked = FALSE,
  hide = NULL,
  omit = NULL,
  omit.col = NA,
  ...
)

## S3 method for class 'ThU'
weightedmean(
  x,
  random.effects = FALSE,
  detect.outliers = TRUE,
  plot = TRUE,
  from = NA,
  to = NA,
  levels = NA,
  clabel = "",
  rect.col = c("#00FF0080", "#FF000080"),
  outlier.col = "#00FFFF80",
  sigdig = 2,
  oerr = 3,
  ranked = FALSE,
  Th0i = 0,
  hide = NULL,
  omit = NULL,
  omit.col = NA,
  ...
)

## S3 method for class 'ArAr'
weightedmean(
  x,
  random.effects = FALSE,
  detect.outliers = TRUE,
  plot = TRUE,
  from = NA,
  to = NA,
  levels = NA,
  clabel = "",
  rect.col = c("#00FF0080", "#FF000080"),
  outlier.col = "#00FFFF80",
  sigdig = 2,
  oerr = 3,
  exterr = FALSE,
  ranked = FALSE,
  i2i = FALSE,
  hide = NULL,
  omit = NULL,
  omit.col = NA,
  ...
)

## S3 method for class 'KCa'
weightedmean(
  x,
  random.effects = FALSE,
  detect.outliers = TRUE,
  plot = TRUE,
  from = NA,
  to = NA,
  levels = NA,
  clabel = "",
  rect.col = c("#00FF0080", "#FF000080"),
  outlier.col = "#00FFFF80",
  sigdig = 2,
  oerr = 3,
  exterr = FALSE,
  ranked = FALSE,
  i2i = FALSE,
  hide = NULL,
  omit = NULL,
  omit.col = NA,
  ...
)

## S3 method for class 'ThPb'
weightedmean(
  x,
  random.effects = FALSE,
  detect.outliers = TRUE,
  plot = TRUE,
  from = NA,
  to = NA,
  levels = NA,
  clabel = "",
  rect.col = c("#00FF0080", "#FF000080"),
  outlier.col = "#00FFFF80",
  sigdig = 2,
  oerr = 3,
  exterr = FALSE,
  ranked = FALSE,
  i2i = TRUE,
  hide = NULL,
  omit = NULL,
  omit.col = NA,
  ...
)

## S3 method for class 'ReOs'
weightedmean(
  x,
  random.effects = FALSE,
  detect.outliers = TRUE,
  plot = TRUE,
  from = NA,
  to = NA,
  levels = NA,
  clabel = "",
  rect.col = c("#00FF0080", "#FF000080"),
  outlier.col = "#00FFFF80",
  sigdig = 2,
  oerr = 3,
  exterr = FALSE,
  ranked = FALSE,
  i2i = TRUE,
  hide = NULL,
  omit = NULL,
  omit.col = NA,
  ...
)

## S3 method for class 'SmNd'
weightedmean(
  x,
  random.effects = FALSE,
  detect.outliers = TRUE,
  plot = TRUE,
  from = NA,
  to = NA,
  levels = NA,
  clabel = "",
  rect.col = c("#00FF0080", "#FF000080"),
  outlier.col = "#00FFFF80",
  sigdig = 2,
  oerr = 3,
  exterr = FALSE,
  ranked = FALSE,
  i2i = TRUE,
  hide = NULL,
  omit = NULL,
  omit.col = NA,
  ...
)

## S3 method for class 'RbSr'
weightedmean(
  x,
  random.effects = FALSE,
  detect.outliers = TRUE,
  plot = TRUE,
  from = NA,
  to = NA,
  levels = NA,
  clabel = "",
  rect.col = c("#00FF0080", "#FF000080"),
  outlier.col = "#00FFFF80",
  sigdig = 2,
  oerr = 3,
  exterr = FALSE,
  i2i = TRUE,
  ranked = FALSE,
  hide = NULL,
  omit = NULL,
  omit.col = NA,
  ...
)

## S3 method for class 'LuHf'
weightedmean(
  x,
  random.effects = FALSE,
  detect.outliers = TRUE,
  plot = TRUE,
  from = NA,
  to = NA,
  levels = NA,
  clabel = "",
  rect.col = c("#00FF0080", "#FF000080"),
  outlier.col = "#00FFFF80",
  sigdig = 2,
  oerr = 3,
  exterr = FALSE,
  i2i = TRUE,
  ranked = FALSE,
  hide = NULL,
  omit = NULL,
  omit.col = NA,
  ...
)

## S3 method for class 'UThHe'
weightedmean(
  x,
  random.effects = FALSE,
  detect.outliers = TRUE,
  plot = TRUE,
  from = NA,
  to = NA,
  levels = NA,
  clabel = "",
  rect.col = c("#00FF0080", "#FF000080"),
  outlier.col = "#00FFFF80",
  sigdig = 2,
  oerr = 3,
  ranked = FALSE,
  hide = NULL,
  omit = NULL,
  omit.col = NA,
  ...
)

## S3 method for class 'fissiontracks'
weightedmean(
  x,
  random.effects = FALSE,
  detect.outliers = TRUE,
  plot = TRUE,
  from = NA,
  to = NA,
  levels = NA,
  clabel = "",
  rect.col = c("#00FF0080", "#FF000080"),
  outlier.col = "#00FFFF80",
  sigdig = 2,
  oerr = 3,
  exterr = FALSE,
  ranked = FALSE,
  hide = NULL,
  omit = NULL,
  omit.col = NA,
  ...
)

Arguments

x	a two column matrix of values (first column) and their standard errors (second column) OR an object of class UPb, PbPb, ThPb, ArAr, KCa, ReOs, SmNd, RbSr, LuHf, ThU, fissiontracks or UThHe
...	optional arguments
from	minimum y-axis limit. Setting from=NA scales the plot automatically.
to	maximum y-axis limit. Setting to=NA scales the plot automatically.
random.effects	if TRUE, computes the weighted mean using a random effects model with two parameters: the mean and the dispersion. This is akin to a ‘model-3’ isochron regression. if FALSE, attributes any excess dispersion to an underestimation of the analytical uncertainties. This akin to a ‘model-1’ isochron regression.
detect.outliers	logical flag indicating whether outliers should be detected and rejected using Chauvenet's Criterion.
plot	logical flag indicating whether the function should produce graphical output or return numerical values to the user.
levels	a vector with additional values to be displayed as different background colours of the plot symbols.
clabel	label of the colour legend
rect.col	Fill colour for the measurements or age estimates. This can either be a single colour or multiple colours to form a colour ramp (to be used if levels!=NA): a single colour: rgb(0,1,0,0.5), '#FF000080', 'white', etc.; multiple colours: c(rbg(1,0,0,0.5), rgb(0,1,0,0.5)), c('#FF000080','#00FF0080'), c('blue','red'), c('blue','yellow','red'), etc.; a colour palette: rainbow(n=100), topo.colors(n=100,alpha=0.5), etc.; or a reversed palette: rev(topo.colors(n=100,alpha=0.5)), etc. For empty boxes, set rect.col=NA
outlier.col	if detect.outliers=TRUE, the outliers are given a different colour.
sigdig	the number of significant digits of the numerical values reported in the title of the graphical output.
oerr	indicates whether the analytical uncertainties of the output are reported in the plot title as: 1: 1\sigma absolute uncertainties. 2: 2\sigma absolute uncertainties. 3: absolute (1-\alpha)% confidence intervals, where \alpha equales the value that is stored in settings('alpha'). 4: 1\sigma relative uncertainties (\%). 5: 2\sigma relative uncertainties (\%). 6: relative (1-\alpha)% confidence intervals, where \alpha equales the value that is stored in settings('alpha').
ranked	plot the aliquots in order of increasing age?
hide	vector with indices of aliquots that should be removed from the weighted mean plot.
omit	vector with indices of aliquots that should be plotted but omitted from the weighted mean calculation.
omit.col	colour that should be used for the omitted aliquots.
type	scalar indicating whether to plot the ^{207}Pb/^{235}U age (type=1), the ^{206}Pb/^{238}U age (type=2), the ^{207}Pb/^{206}Pb age (type=3), the ^{207}Pb/^{206}Pb-^{206}Pb/^{238}U age (type=4), the concordia_age (type=5), or the ^{208}Pb/^{232}Th age (type=6).
cutoff.76	the age (in Ma) below which the ^{206}Pb/^{238}U age and above which the ^{207}Pb/^{206}Pb age is used. This parameter is only used if type=4.
cutoff.disc	discordance cutoff filter. This is an object of class discfilter
exterr	propagate decay constant uncertainties?
common.Pb	common lead correction: 0: none 1: use the Pb-composition stored in settings('iratio','Pb207Pb206') (if x has class UPb and x$format<4); settings('iratio','Pb206Pb204') and settings('iratio','Pb207Pb204') (if x has class PbPb or x has class UPb and 3<x$format<7); or settings('iratio','Pb206Pb208') and settings('iratio','Pb207Pb208') (if x has class UPb and x$format=7 or 8). 2: remove the common Pb by projecting the data along an inverse isochron. Note: choosing this option introduces a degree of circularity in the weighted age calculation. In this case the weighted mean plot just serves as a way to visualise the residuals of the data around the isochron, and one should be careful not to over-interpret the numerical output. 3: use the Stacey-Kramers two-stage model to infer the initial Pb-composition (only applicable if x has class UPb)
Th0i	initial ^{230}Th correction. 0: no correction 1: project the data along an isochron fit 2: if x$format is 1 or 2, correct the data using the measured present day ^{230}Th/^{238}U, ^{232}Th/^{238}U and ^{234}U/^{238}U activity ratios in the detritus. If x$format is 3 or 4, correct the data using the measured ^{238}U/^{232}Th activity ratio of the whole rock, as stored in x by the read.data() function. 3: correct the data using an assumed initial ^{230}Th/^{232}Th-ratio for the detritus (only relevant if x$format is 1 or 2).
i2i	‘isochron to intercept’: calculates the initial (aka ‘inherited’, ‘excess’, or ‘common’) ^{40}Ar/^{36}Ar, ^{40}Ca/^{44}Ca, ^{207}Pb/^{204}Pb, ^{87}Sr/^{86}Sr, ^{143}Nd/^{144}Nd, ^{187}Os/^{188}Os, ^{230}Th/^{232}Th, ^{176}Hf/^{177}Hf or ^{204}Pb/^{208}Pb ratio from an isochron fit. Setting i2i to FALSE uses the default values stored in settings('iratio',...). Note that choosing this option introduces a degree of circularity in the weighted age calculation. In this case the weighted mean plot just serves as a way to visualise the residuals of the data around the isochron, and one should be careful not to over-interpret the numerical output.

Details

Let \{t_1, ..., t_n\} be a set of n age estimates determined on different aliquots of the same sample, and let \{s[t_1], ..., s[t_n]\} be their analytical uncertainties. IsoplotR then calculates the weighted mean of these data using one of two methods:

1. The ordinary error-weighted mean:

   \mu = \sum(t_i/s[t_i]^2)/\sum(1/s[t_i]^2)

2. A random effects model with two sources of variance:

   \log[t_i] \sim N(\log[\mu], \sigma^2 = (s[t_i]/t_i)^2 + \omega^2 )

   where \mu is the mean, \sigma^2 is the total variance and \omega is the 'overdispersion'. This equation can be solved for \mu and \omega by the method of maximum likelihood.

IsoplotR uses a modified version of Chauvenet's criterion for outlier detection:

1. Compute the error-weighted mean (\mu) of the n age determinations t_i using their analytical uncertainties s[t_i]

2. For each t_i, compute the probability p_i that that |t-\mu|>|t_i-\mu| for t \sim N(\mu, s[t_i]^2 MSWD) (ordinary weighted mean) or \log[t] \sim N(\log[\mu],s[t_i]^2+\omega^2) (random effects model)

3. Let p_j \equiv \min(p_1, ..., p_n). If p_j<0.05/n, then reject the j^{th} date, reduce n by one (i.e., n \rightarrow n-1) and repeat steps 1 through 3 until the surviving dates pass the third step.

If the analytical uncertainties are small compared to the scatter between the dates (i.e. if \omega \gg s[t] for all i), then this generalised algorithm reduces to the conventional Chauvenet criterion. If the analytical uncertainties are large and the data do not exhibit any overdispersion, then the heuristic outlier detection method is equivalent to Ludwig (2003)'s ‘2-sigma’ method.

The uncertainty budget of the weighted mean does not include the uncertainty of the initial daughter correction (if any). This uncertainty is neither a purely systematic nor a purely random uncertainty and cannot easily be propagated with conventional geochronological data processing algorithms. This caveat is especially pertinent to chronometers whose initial daughter composition is determined by isochron regression. You may note that the uncertainties of the weighted mean are usually much smaller than those of the isochron. In this case the isochron errors are more meaningful, and the weighted mean plot should just be used to inspect the residuals of the data around the isochron.

Value

Returns a list with the following items:

mean

a two or three element vector with:

t: the weighted mean. An asterisk is added to the plot title if the initial daughter correction is based on an isochron regression, to mark the circularity of using an isochron to compute a weighted mean.

s[t]: the standard error of the weighted mean, excluding the uncertainty of the initial daughter correction. This is because this uncertainty is neither purely random nor purely systematic.

disp

a two-element vector with the (over)dispersion and its standard error.

mswd

the Mean Square of the Weighted Deviates (a.k.a. ‘reduced Chi-square’ statistic)

df

the number of degrees of freedom of the Chi-square test for homogeneity (df=n-1, where n is the number of samples).

p.value

the p-value of a Chi-square test with df degrees of freedom, testing the null hypothesis that the underlying population is not overdispersed.

valid

vector of logical flags indicating which steps are included into the weighted mean calculation

plotpar

list of plot parameters for the weighted mean diagram, including mean (the mean value), ci (a grey rectangle with the (1 s.e., 2 s.e. or 100[1-\alpha]%, depending on the value of oerr) confidence interval ignoring systematic errors), ci.exterr (a grey rectangle with the confidence interval including systematic errors), dash1 and dash2 (lines marking the confidence interval augmented by \sqrt{mswd} overdispersion if random.effects=FALSE), and marking the confidence limits of a normal distribution whose standard deviation equals the overdispersion parameter if random.effects=TRUE).

See Also

central

Examples

ages <- c(251.9,251.59,251.47,251.35,251.1,251.04,250.79,250.73,251.22,228.43)
errs <- c(0.28,0.28,0.63,0.34,0.28,0.63,0.28,0.4,0.28,0.33)
weightedmean(cbind(ages,errs))

attach(examples)
weightedmean(LudwigMean)

pvermees/IsoplotR documentation built on Nov. 18, 2024, 1:02 a.m.