19
;
(15)
where,
n is the degrees of freedom, here equal to ND+NPR-NP (See equation 1 for definitions);
χ
u
2
is the upper tail value of a chi-square distribution with n degrees of freedom, with the area to
the right equal to one-half the significance level of the confidence interval (the signifi-
cance level is 0.05 for a 95-percent interval);
χ
L
2
is the lower tail value of a chi-square distribution with n degrees of freedom with the area to
the left equal to one-half the significance level.
The calculated standard error can be evaluated similarly by taking the square root of the limits of
equation 5. Equivalently, the test can be conducted using a
χ
2
test statistic, as presented by Ott
(1993, p.234).
Values of the calculated error variance and the standard error are typically greater than 1.0
in practice, reflecting the presence of model error as well as the measurement error typically rep-
resented in the weighting, or larger than expected measurement error (see Guideline 8).
When the weight matrix is defined as suggested in Guideline 4, the calculated error vari-
ance and standard error are dimensionless.The dimensionless standard error is not a very intuitively
informative measure of goodness of fit. A more intuitive measure is the product of the calculated
standard error and the statistics used to calculate the weights (generally standard deviations and co-
efficients of variation; see the discussion for guideline 4). Such products are called fitted standard
deviations and fitted coefficients of variation by Hill and others (1998) and in general can be called
fitted error statistics. These statistics clearly represent model fit both to modelers and resource
managers. For example, if a standard deviation of 0.3 m is used to calculate the weights for most
of the hydraulic-head observations and the calculated standard error is 3.0, the fitted standard error
of 0.9 m accurately represents the overall fit achieved for these hydraulic heads. If a coefficient of
variation of 0.25 (25 percent) is used to calculate weights for a set of springflow observations and
the calculated standard error is 2.0, the fitted coefficient of variation of 0.50 (50 percent) accurately
represents the overall fit achieved to these springflows. Generally this approach applies only if the
fitted error statistic summarizes the fit to a fairly large number of observations. Application to a
single observation can produce misleading results.
Do'stlaringiz bilan baham: