WelchSatter {propagate}

Welch-Satterthwaite approximation to the 'effective degrees of freedom'

Description

Calculates the Welch-Satterthwaite approximation to the 'effective degrees of freedom' by using the samples' uncertainties and degrees of freedoms, as described in Welch (1947) and Satterthwaite (1946). External sensitivity coefficients can be supplied optionally.

Usage

WelchSatter(ui, ci = NULL, df = NULL, dftot = NULL, uc = NULL, alpha = 0.05)

Arguments

ui

the uncertainties u_i for each variable x_i.

ci

the sensitivity coefficients c_i = \partial y/\partial x_i.

df

the degrees of freedom for the samples, \nu_i.

dftot

an optional known total degrees of freedom for the system, \nu_{\mathrm{tot}}. Overrides the internal calculation of \nu_{\mathrm{ws}}.

uc

the combined uncertainty, u(y).

alpha

the significance level for the t-statistic. See 'Details'.

Details

\nu_{\rm{eff}} \approx \frac{u(y)^4}{\sum_{i = 1}^n \frac{(c_iu_i)^4}{\nu_i}}, \quad k = t(1 - (\alpha/2), \nu_{\rm{eff}}), \quad u_{\rm{exp}} = ku(y)

Value

A list with the following items:

ws.df

the 'effective degrees of freedom'.

k

the coverage factor for calculating the expanded uncertainty.

u.exp

the expanded uncertainty u_{\rm{exp}}.

Author(s)

Andrej-Nikolai Spiess

References

An Approximate Distribution of Estimates of Variance Components.
Satterthwaite FE.
Biometrics Bulletin (1946), 2: 110-114.

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

Examples

## Taken from GUM H.1.6, 4).
WelchSatter(ui = c(25, 9.7, 2.9, 16.6), df = c(18, 25.6, 50, 2), uc = 32, alpha = 0.01)