If a model is given, then ci is ignored, and a confidence
interval is calculated using confint() with level set to
0.6914619. This level corresponds to a range of plus or minus one
standard deviation, for the t distribution and a large number of degrees of
freedom (since qt(0.6914619, 100000) is 0.5).
If model is missing, ci must be provided. If it contains 3
elements, then first and third elements are taken as the range of the
confidence interval (which by convention should use the level stated
in the previous paragraph), and the second element is taken as the central
value. Alternatively, if ci has 2 elements, they are taken to be
bounds of the confidence interval and their mean is taken to be the central
value.
In the +/- notation, e.g. \(a \pm b\) indicates that the true
value lies between \(a-b\) and \(a+b\) with a high degree of
certainty. Mills et al. (1993, section 4.1 on page 83) suggest that
\(b\) should be set equal to 2 times the standard uncertainty or
standard deviation. JCGM (2008, section 7.2.2 on pages 25 and 26), however,
suggest that \(b\) should be set to the standard uncertainty, while
also recommending that the \(\pm\) notation (and presumably
the parentheses notation also) be avoided altogether, in favour of
writing sentences that explains uncertainties in clear terms.
The parentheses notation is often called the compact notation. In
it, the digits in parentheses indicate the uncertainty in the corresponding
digits to their left, e.g. 12.34(3) means that the last digit (4) has an
uncertainty of 3. However, as with the \(\pm\) notation, different
authorities offer different advice on defining this uncertainty; Mills et
al. (1993) provide an example in which the
parenthetic value is half the \(\pm\) value, whereas
JCM (2008) suggest using the same values.
The JCM(2008) convention is used by formatCI() for the parentheses
notation, as illustrated in Examples 1 and 2. Note, however, that
if the confidence range exceeds the value, then
a request for parentheses format reverts to +/- format.