# mandel.k: Calculate Mandel's k statistics for replicate observations In metRology: Support for Metrological Applications

## Description

mandel.k calculates Mandel's k statistics for replicate observations. Mandel's k an indicator of precision compared to the pooled standard deviation across all groups.

## Usage

  1 2 3 4 5 6 7 8 9 10  mandel.k(x, g = NULL, m = NULL, na.rm = T, rowname = NULL, method=c("classical", "robust"), n = NA, ...) ## Default S3 method: mandel.k(x, g = NULL, m = NULL, na.rm = T, rowname = NULL, method=c("classical", "robust"), n = NA, ...) ## S3 method for class 'ilab' mandel.k(x, g = NULL, m = NULL, na.rm = T, rowname = NULL, method=c("classical", "robust"), n = NA, ...) 

## Arguments

 x An R object (see Details below), which contains replicate observations or, if g is absent, means or standard deviations. g A primary grouping factor, usually corresponding to Laboratory in an inter-laboratory study. If not present, x is taken as a set of means or standard deviations (depending on whether type is "h" or "k". m A secondary grouping factor, usually corresponding to test item or measured quantity. m is ignored if x has more than one column. na.rm A logical value indicating whether 'NA' values should be stripped before the computation proceeds. Passed to functions such as mean and sd. rowname A single character label for the primary grouping factor (e.g. "Lab", "Organisation"). method Character scalar giving the calculation method. "classical" gives the traditional calculation; "robust" gives a robust variant (see Details). n scalar number of observations per group. Required only if x consists of calculated standard deviations. ... Additional parameters passed to other methods. Currently not implemented.

## Details

mandel.k is a convenience wrapper for mandel.kh(..., type="k"). It is generic, with methods for numeric vectors, arrays, data frames, matrices and objects of class 'ilab'. All parameters are passed to mandel.kh.

Mandel's k is an indicator of relative dispersion for grouped sets of observations. Given a set of observations x[i,j,l] where i, j, l denotes observation l, l=1, 2, ... n for measurand or test item j and group (usually laboratory) i, i=1, 2, ... p, Mandel's k is given by:

k=√{\frac{s_{ij}^2}{∑_{i=1}^p{s_{ij}^2/p}}}

where s_{ij} is the standard deviation of values x_{ijk} over k=1, 2, ..., n.

If x is a vector, one-dimensional array or single-column matrix, values are aggregated by g and, if present, by m. If x is a data frame or matrix, each column is aggregated by g and m silently ignored if present. In all cases, if g is NULL or missing, each row (or value, if a vector) in x is taken as a pre-calculated mean (for Mandel's h) or standard deviation (for Mandel's k).

If x is an object of class 'ilab', g defaults to '$org' and m to $measurand.

The returned object includes a label ('grouped.by') for the primary grouping factor. For the 'ilab' method, this is "Organisation". For other methods, If rowname is non-null, rowname is used. If rowname is NULL, the default is deparse(substitute(g)); if g is also NULL or missing, "Row" is used.

If method="robust", Mandel's k is calculated by replacing the classical pooled standard deviation with the robust pooled standard deviation calculated by algorithm S (see algS).

## Value

mandel.k returns an object of class "mandel.kh", which is a data frame consisting of the required Mandel's statistics and in which each row corresponds to a level of g and each column to a level of m or (if x was a matrix or data frame) to the corresponding column in x. In addition to the class, the object has attributes:

'mandel.type'

"h" or "k"

'grouped.by'

Character scalar giving the label used for the grouping factor g; see Details above for the defaults.

'n'

Number of observations per group (n if specified

## Author(s)

S Ellison s.ellison@lgc.co.uk

## References

Accuracy (trueness and precision) of measurement methods and results – Part 2: Basic method for the determination of repeatability and reproducibility of a standard measurement method. ISO, Geneva (1994).

mandel.h, mandel.kh; pmandelh, pmandelk for probabilities, quantiles etc.; plot.mandel.kh, barplot.mandel.kh for plotting methods.

## Examples

  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24  data(RMstudy) #Data frame examples: note no secondary grouping factor h <- with(RMstudy, mandel.k(RMstudy[2:9], g=Lab)) plot(h, las=2) #Vector variant RMstk <- stack(RMstudy[,2:9]) names(RMstk) <- c("x", "meas") #names replace 'values' and 'ind' RMstk$Lab <- rep(RMstudy$Lab, 8) h2 <- with(RMstk, mandel.k(x, g=Lab, m=meas, rowname="Laboratory")) #Note use of rowname to override g plot(h2, las=2) #ilab method RM.ilab <- with(RMstk, construct.ilab(org=Lab, x=x, measurand=meas, item=factor(rep("CRM", nrow(RMstk))) ) ) plot(mandel.k(RM.ilab)) #Robust variant krob <- with(RMstudy, mandel.kh(RMstudy[2:9], g=Lab, type="k", method="robust")) plot(krob, las=2) 

### Example output

Attaching package: 'metRology'

The following objects are masked from 'package:base':

cbind, rbind


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