REML location estimate {metRology}

Restricted maximum likelihood estimate of location

Description

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

Usage

	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} \left( \sum_{i=1}^{k}\frac{(x_i-mu)^2}{u_i^2} + \sum_{i=1}^{k}log(u_i^2) + log\left(\sum_{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 [e-mail]

References

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

See Also

loc.est-class

Examples

  #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) )