30
(34)
where
is the standard error of the regression adjusted for the expected measurement error of the
prediction (see Hill, 1994, p. 32; Miller, 1981).
Individual confidence intervals calculated using equation 32 are exact for linear models
with normally distributed residuals, assuming that the model is correct. As these conditions are vi-
olated
to a greater degree, the calculated intervals become progressively less accurate, so that the
actual significance level of the interval can be substantially different than intended. This is of seri-
ous concern for the nonlinear problems considered in this work, as discussed by Donaldson and
Schnabel (1987). Recent publications in the ground-water literature, however, indicate that in
many ground-water flow problems linear intervals are accurate enough to be useful (Christensen
and Cooley, in press). Other types of ground-water problems have not been evaluated.
The calculation of confidence and prediction intervals can (and often needs to) include
more parameters than were included in the regressions performed
for model calibration, as dis-
cussed above in the section ‘Prediction Scaled Sensitivities’ and under guideline 13.
The individual intervals defined above apply when the uncertainty of only one quantitiy is
of interest. When more than one quantity is of interest, different intervals are needed, and these are
called simultaneous intervals. Simultaneous intervals calculated using linear theory are always of
equal size or larger than equivalent linear individual intervals, reflecting the greater uncertainty in-
volved in trying to define intervals which are likely to include the true value of two or more pre-
dictions at the same time. As
more intervals are considered, the intervals tend to become wider.
The largest intervals are calculated when the number of predictions equals the number of parame-
ters included in the uncertainty analysis. Additional predictions do not increase the size of the si-
multaneous intervals.
Simultaneous intervals are difficult to calculate exactly, but can be approximated using the
equations listed in Hill (1994), as discussed by Miller (1981). The equations for simultaneous con-
fidence and prediction intervals are of the same form as equations 32 and 34, respectively,
and dif-
fer only in the critical values used. If the number of intervals considered is represented by k, the
interval limits can be calculated using critical values from a Bonferroni-t distribution or from an F
distribution. The Bonferroni critical value is
t
B
(n,1.0-
α
/2k). (35a)
The F distribution critical value is
[d x F
α
(d,n)]
1/2
,
(35b)
where d equals k or the number of parameters (NP), whichever is less. Intervals calculated with the
z'
l
t n 1.0
α
2
---
–
,
s
z’
l
s
a
+
±
s
a
31
F distribution critical value are called Scheffe intervals. Scheffe intervals are labeled either as
Scheffe d=k or Scheffe d=NP.
Both Bonferroni and Scheffe intervals
are approximate, and tend to be large. Thus, for any
finite value of k, the smaller interval should be used.
In some cases k is not finite. For example, if a prediction of interest is the largest simulated
value over a defined area, the predicted quantity can not be specified exactly before the simulation,
and the number of predictions considered simultaneously needs to be thought of as infinite. In this
circumstance, the only applicable approximate simultaneous interval is the Scheffe d=NP.
As discussed in the section “Prediction Scaled Sensitivities”, in some circumstances the
prediction on interest is the difference between two simulations. Both UCODE and MODFLOWP
can calculate linear confidence and prediction
intervals on differences, as discussed by Hill (1994).
Calculation of linear confidence intervals requires only the sensitivities calculated for the
optimized parameter values and, therefore, takes very little computer execution time.
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