Provides the Welch-Satterthwaite effective degrees of freedom given standard uncertainties and associated degrees of freedom.
w.s is an alias for welch.satterthwaite.
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)))
Standard uncertainties
Degrees of freedom
Sensitivity coefficients \(dy/dx_i\)
Combined standard uncertainty
The calculated effective degrees of freedom associated with uc
.
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.
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
None, yet.
# 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)
# }
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