# reml_loc: Restricted maximum likelihood estimate of location In metRology: Support for Metrological Applications

## Description

Calculates REML estimate of location, with standard error, assuming a random-effects model

## Usage

 1 2 3 4 5  reml.loc(x, ..., na.rm = FALSE) ## Default S3 method: reml.loc(x, s, n = NULL, groups = NULL, na.rm = FALSE, tol=.Machine\$double.eps^0.5, REML=TRUE, ...) 

## Arguments

 x numeric vector of mean values for groups, or (if groups is given) of individual observations s numeric vector of length length(x) of standard deviations or standard uncertainties associated with the values x. n integer giving the number of observations in each group. May be a vector of length length(x). If n is NULL, s is interpreted as a vector of 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 s and n ignored, if present, with a warning. na.rm logical: if TRUE, NA values are removed before processing. tol numeric tolerance for convergence, used by optimize(). REML logical: if TRUE (the default), the function optimises the REML criterion (see Details). If FALSE, the maximum likelihood criterion is used. ... Further parameters passed to optimize().

## Details

reml.loc finds an excess variance tau^2 and location mu that maximise the restricted maximum likelihood criterion.

The estimator assumes a model of the form

x[i]=mu+b[i]+e[i]

in which b[i] is drawn from N(0, tau^2) and e[i] is drawn from N(0, sigma[i]^2).

By default the function maximises the data-dependent part of the negative log restricted likelihood:

\frac{1}{2} ≤ft( ∑_{i=1}^{k}\frac{(x_i-mu)^2}{u_i^2} + ∑_{i=1}^{k}log(u_i^2) + log≤ft(∑_{i=1}^{k}(1/u_i^2)\right) \right)

where u[i]=s[i]^2 + tau^2 and k is the number of mean values. If REML=FALSE, the final term is omitted to give the maximum likelihood criterion.

This implementation permits input in the form of:

• means x and standard errors s, in which case neither n nor groups are supplied;

• means x, standard deviations s and group size(s) n, standard errors then being calculated as s/sqrt(n)

• individual observations x with a groupinf factor groups, in which case standard errors are calculated from the groups using tapply.

## Value

A loc.est object; see loc.est for details. In the returned object, individual values xi are always input means (calculated from groups and n as necessary); method.details is returned as a list containing:

mu

The estimated location.

s

The standard error in the location.

tau

The excess variance (as a standard deviation).

REML

Logical, giving the value of REML used.

## Author(s)

S L R Ellison s.ellison@lgc.co.uk

## References

None, but see documentation for the metafor package for a more general implementation of REML.

loc.est-class
 1 2 3 4 5 6  #PCB measurements in a sediment from Key Comparison CCQM-K25 #s are reported standard uncertainties pcb105 <- data.frame(x=c(10.21, 10.9, 10.94, 10.58, 10.81, 9.62, 10.8), s=c(0.381, 0.250, 0.130, 0.410, 0.445, 0.196, 0.093)) with( pcb105, reml.loc(x, s) )