metRology (version 0.9-28-1)

welch.satterthwaite: Welch-Satterthwaite effective degrees of freedom

Description

Provides the Welch-Satterthwaite effective degrees of freedom given standard uncertainties and associated degrees of freedom.

w.s is an alias for welch.satterthwaite.

Usage

w.s(ui, df, ci = rep(1, length(ui)), uc=sqrt(sum((ci*ui)^2)))
welch.satterthwaite(ui, df, ci = rep(1, length(ui)), 
                       uc=sqrt(sum((ci*ui)^2)))

Arguments

ui

Standard uncertainties

df

Degrees of freedom

ci

Sensitivity coefficients \(dy/dx_i\)

uc

Combined standard uncertainty

Value

The calculated effective degrees of freedom associated with uc.

Details

Implements the Welch-Satterthwaite equation as provided in the ISO Guide to the expression of uncertainty in measurement (1995) (See JCGM 100:2008). This assumes that uc is the uncertainty in a measurement result \(y\), where \(y=f(x_1, x_2, \dots)\), ci are the partial derivatives \(\partial y/\partial x_i\) and ui is the standard uncertainty associated with xi.

The implementation assumes that the combined uncertainty uc is equal to sqrt(sum((ci*ui)^2). An independent estimate of uc can be provided.

The ci are 'sensitivity coefficients'; the default is 1, so that the ui can be given either as standard uncertainties in the values of influence quantities \(x_i\), together with the associated ci, or as contributions ci*ui to the uncertainty in \(y\).

Correlation is not supported, because the Welch-Satterthwaite equation is only valid for independent variances.

References

JCGM 100 (2008) Evaluation of measurement data - Guide to the expression of uncertainty in measurement. http://www.bipm.org/utils/common/documents/jcgm/JCGM_100_2008_E.pdf. (JCGM 100:2008 is a public domain copy of ISO/IEC Guide to the expression of uncertainty in measurement (1995) ).

Satterthwaite, F. E. (1946), An Approximate Distribution of Estimates of Variance Components., Biometrics Bulletin 2, 110-114, doi:10.2307/3002019

Welch, B. L. (1947), The generalization of "Student's" problem when several different population variances are involved., Biometrika 34 28-35

See Also

None, yet.

Examples

# NOT RUN {
u <- c(0.1, 0.3, 0.2, 1.1)
ci <- c(1.0, 2.0, 3.0, 0.5)
degfree <- c(Inf,6,8,3)

w.s(ui=u,df=degfree, ci=ci)

# }