mpaule: Mandel-Paule calculation
In metRology: Support for Metrological Applications

Description Usage Arguments Details Value Author(s) References Examples

Calculate a weighted mean, between-group standard deviation and standard error on the weighted mean using the Mandel-Paule algorithm.

mpaule(x, ..., tol=.Machine$double.eps^0.25, maxiter=25) 

## Default S3 method:
mpaule(x, u=NULL, n=NULL, groups=NULL, 
                   tol=.Machine$double.eps^0.25, maxiter=25, ...)

mandel.paule(x, ..., tol=.Machine$double.eps^0.25, maxiter=25)

`x`	numeric vector of mean values for groups, or (if `groups` is given) of individual observations
`u`	numeric vector of standard deviations or standard uncertainties associated with the values `x`
`n`	integer vector of numbers in each group. If `NULL`, `u` are interpreted as standard uncertainties or standard errors. `n` is recycled to `length(x)`.
`groups`	factor, or vetor which can be coerced to factor, of groups. If present, `x` is interpreted as a vector of individual observations and `u` and `n` ignored.
`...`	Additional parameters passed to other functions.
`tol`	numeric tolerance; iteration stops when the variance step size drops below `tol*var(x)`
`maxiter`	numeric maximum number of iterations

The Mandel-Paule algorithm finds the between-method variance by iteratively solving the equation relating the weighted mean to the weighting factor applied. The weighting factor is the inverse of the sum of the standard error in x and the between-group variance.

If the iterative procedure produces a negative estimate for the between-group variance, the between-group variance is set to zero.

For the default method, if u is present and n=NULL, u is interpreted as a vector of standard uncertainties or standard errors. If n is not NULL, u is interpreted as a vector of standard deviations and standard errors are calculated as u/sqrt(n). If groups is not NULL, x is interpreted as a vector of individual observations grouped by groups, and the algorithm is applied to the corresponding group means and standard errors.

If maxiter is set less than 1, no iterations are performed and the consensus mean is returned as NA.

mandel.paule is an alias for mpaule retained for backward compatibility.

A loc.est object; see loc.est for details. In the returned object, df is set to n-1 where n is the number of non-NA observations or group means as appropriate, and method.details is returned as :

`var.between`	the estimated between-group variance)
`iter`	the number of iterations taken
`converged`	`converged` indicates the convergence status. `0L` indicates failure to converge (`maxiter` reached before step size drops below tolerance); `1L` indicates normal convergence; `2L` indicates that the final step size resulted in a negative between-group variance, at which point the variance and step size are set to 0.0

S. Cowen simon.cowen@lgc.co.uk with amendments by S. L. R. Ellison.

Paule, R. C. and Mandel, J. (1982), J Res Nat Bur Stand, 87, (5) 377-385

## the second example in the paper cited above
x <- c(201.533, 216.55)
s <- c(0.154, 0.25)
n <- c(6, 2)

mpaule(x, s/sqrt(n))

## Cd heat of vapourisation example from the paper cited above
x2<-c(27.044, 26.022, 26.340, 26.787, 26.796)
v<-c(3, 76, 464, 3, 14)*1e-3
mpaule(x2, sqrt(v))