R: Welch-Satterthwaite approximation to the 'effective degrees...

WelchSatter {propagate} | R Documentation

## 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, ν_i.
`dftot` | an optional known total degrees of freedom for the system, ν_{\mathrm{tot}}. Overrides the internal calculation of ν_{\mathrm{ws}}.
`uc` | the combined uncertainty, u(y).
`alpha` | the significance level for the t-statistic. See 'Details'.

### Details

ν_{\rm{eff}} \approx \frac{u(y)^4}{∑_{i = 1}^n \frac{(c_iu_i)^4}{ν_i}}, \quad k = t(1 - (α/2), ν_{\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)

[Package propagate version 1.0-6 Index]