Methods and guidelines for effective model calibration

Standard Deviations, Linear Confidence Intervals, and Coefficients of Variation
Parameter standard deviations equal the square root of the parameter variances. Parameter 
standard deviations are perhaps most useful when used to calculate two other statistics: confidence 
intervals for parameter values and coefficients of variation. Linear confidence intervals calculated 
as described by Hill (1994) and references cited therein require trivial amounts of execution time 
and are calculated and printed by UCODE and MODFLOWP. The more accurate nonlinear confi-
dence intervals of Vecchia and Cooley (1987) and Christensen and Cooley (1996) discussed below 
in section 
“
Nonlinear Confidence and Prediction Intervals
”
require substantial execution time and 
are not calculated by the current versions of UCODE or MODFLOWP.
A linear confidence interval for each parameter is calculated as
(28)
β
j
b
j
t n 1.0
α
2
---
–
,




sb
j
±

---

27
where
is the Student-t statistic for n degrees of freedom and a significance level of 
α
; and
is the standard deviation of the jth parameter. 
Confidence intervals are referred to in a way that can be confusing, and that is derived from their 
definition. Technically, a confidence interval is a range that has a stated probability of containing 
the true value. As such, confidence intervals are referred to using the true, unknown value that is 
being estimated. Thus, equation 28 is said to be the confidence interval for , the true, unknown 
jth parameter value, and the width of the confidence interval can be thought of as a measure of the 
likely precision of the estimate. Narrow intervals indicate greater precision. If the model correctly 
represents the system, the interval also can be thought of as a measure of the likely accuracy of the 
estimate. This was discussed in more detail by Hill (1994).
The derivation of equation 28 requires an assumption that is not needed to perform the re-
gression -- that is, the assumption that the true errors and, therefore, for a linear problem, the pa-
rameter estimates, be normally distributed. For further discussion, see the section “Normal 
Probability Graphs and the Correlation Coefficient R
2
N
.”
When plotted on graphs with the related estimated values, linear confidence intervals pro-
vide a vivid graphical image of the precision with which parameters are estimated using the data 
included as observations in the regression, given the constructed model. 
The coefficient of variation for each parameter equals the standard deviation divided by the 
parameter value and provides a dimensionless number with which the relative accuracy of different 
parameter estimates can be compared.
For log-transformed parameters, confidence intervals and coefficients of variation of the 
transformed parameters can be difficult to interpret. In UCODE and MODFLOWP the confidence 
intervals are untransformed by taking the exponential of the confidence interval limits, and these 
are printed. The coefficients of variation are untransformed by untransforming the parameter vari-
ance, (
Slogb
)
2
as:
(29)
where the exponentials and logs are in base 10, b is the native parameter, and logb is the estimated 
log-transformed parameter. The coefficient of variation of the native parameter is calculated by di-
t n 1.0
α
2
---
–
,




s
b
j
β
j
s
b
2
2.3 s
b
log




2
2
b
log
+
2.3 s
b
log




2




1.
–
exp
exp
=

---

28
viding the square root of its variance by b. It should be noted that the linear confidence intervals 
for the true, unknown native parameters are symmetric when plotted on a log scale, but are not 
symmetric when plotted on an arithmetic scale.