propagate (version 1.0-4)

WelchSatter: 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).

Usage

WelchSatter(ufinal, usamp, df, alpha = 0.05)

Arguments

ufinal

the propagated uncertainty of \(y\).

usamp

the uncertainties of the samples, \(x_i\).

df

the degrees of freedom of the samples, \(\nu_i\).

alpha

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

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.

Details

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

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

# NOT RUN {
## Taken from GUM H.1.6, 4).
WelchSatter(32, c(25, 9.7, 2.9, 16.6), c(18, 25.6, 50, 2), alpha = 0.01)
# }