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<table width="100%" summary="page for welch.satterthwaite"><tr><td>welch.satterthwaite</td><td style="text-align: right;">R Documentation</td></tr></table>

<h2>
Welch-Satterthwaite effective degrees of freedom</h2>

<h3>Description</h3>

<p>Provides the Welch-Satterthwaite effective degrees of freedom given standard 
uncertainties and associated degrees of freedom.
</p>
<p>w.s is an alias for welch.satterthwaite. 
</p>


<h3>Usage</h3>

<pre>
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)))
</pre>


<h3>Arguments</h3>

<table summary="R argblock">
<tr valign="top"><td><code>ui</code></td>
<td>
<p>Standard uncertainties</p>
</td></tr>
<tr valign="top"><td><code>df</code></td>
<td>
<p>Degrees of freedom</p>
</td></tr>
<tr valign="top"><td><code>ci</code></td>
<td>
<p>Sensitivity coefficients <i>dy/dx_i</i></p>
</td></tr>
<tr valign="top"><td><code>uc</code></td>
<td>
<p>Combined standard uncertainty</p>
</td></tr>
</table>


<h3>Details</h3>

<p>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 <code>uc</code> is the 
uncertainty in a measurement result <i>y</i>, where <i>y=f(x_1, x_2, &hellip;)</i>, <code>ci</code> are 
the partial derivatives <i>dy/dx[i]</i> and <code>ui</code> is the standard uncertainty associated with <code>xi</code>.
</p>
<p>The implementation assumes that the combined uncertainty <code>uc</code> is equal to
<code>sqrt(sum((ci*ui)^2)</code>. An independent estimate of <code>uc</code> can be provided.
</p>
<p>The <code>ci</code> are 'sensitivity coefficients'; the default is 1, so that the <code>ui</code>
can be given either as standard uncertainties in the values of influence quantities <i>x_i</i>, 
together with the associated <code>ci</code>, or as contributions <code>ci*ui</code> to the uncertainty in <i>y</i>.
</p>
<p>Correlation is not supported, because the Welch-Satterthwaite equation is only valid 
for independent variances.
</p>


<h3>Value</h3>

<p>The calculated effective degrees of freedom associated with <code>uc</code>.
</p>


<h3>Author(s)</h3>

<p>S. L. R. Ellison <a href="mailto:s.ellison@lgc.co.uk">s.ellison@lgc.co.uk</a>
</p>


<h3>References</h3>

<p>JCGM 100 (2008) <em>Evaluation of measurement data - Guide to the expression
of uncertainty in measurement</em>. <a href="http://www.bipm.org/utils/common/documents/jcgm/JCGM_100_2008_E.pdf">http://www.bipm.org/utils/common/documents/jcgm/JCGM_100_2008_E.pdf</a>. 
(JCGM 100:2008 is a public domain copy of ISO/IEC <em>Guide to the expression
of uncertainty in measurement</em> (1995) ). 
</p>
<p>Satterthwaite, F. E. (1946), An Approximate Distribution of Estimates of Variance Components., 
Biometrics Bulletin <b>2</b>, 110-114, doi:10.2307/3002019 
</p>
<p>Welch, B. L. (1947), The generalization of &quot;Student's&quot; problem when several different 
population variances are involved., Biometrika <b>34</b> 28-35
</p>


<h3>See Also</h3>

<p>None, yet.
</p>


<h3>Examples</h3>

<pre>

u &lt;- c(0.1, 0.3, 0.2, 1.1)
ci &lt;- c(1.0, 2.0, 3.0, 0.5)
degfree &lt;- c(Inf,6,8,3)

w.s(ui=u,df=degfree, ci=ci)

</pre>


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