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

## Description

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

## Usage

 ```1 2 3 4 5 6 7``` ```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) ```

## Arguments

 `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

## Details

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.

## Value

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

## Author(s)

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

## References

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

## Examples

 ``` 1 2 3 4 5 6 7 8 9 10 11``` ```## 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)) ```

metRology documentation built on May 2, 2019, 12:20 p.m.