18
cate model fit. It is rarely used for more formal comparisons because its value nearly always de-
creases as more parameters are added, and the negative aspect of adding parameters is not
reflected. The negative aspect of adding parameter values is that as the
data available for the esti-
mation get spread over more and more parameter values the certainty with which the parameter
values are estimated decreases. The measures presented below more effectively account for this
circumstance.
Calculated Error Variance and Standard Error
A commonly used indicator of the overall magnitude of the weighted residuals is the cal-
culated error variance,
, which equals:
(14)
where
is the weighted least-squares objective function value of equation 1 or 2 and the other
variables are defined after equation 1. The square root of
the calculated error variance, s, is called
the standard error of the regression and also is used to indicate model fit. Smaller values of both
the calculated error variance and the standard error indicate a closer fit to the observations, and
smaller values are preferred as long as the weighted residuals do not indicate model error (see be-
low).
If the fit achieved by regression is consistent with the data accuracy as reflected in the
weighting, the expected value of both the calculated error variance and the standard error is 1.0.
This can be proven by substituting equation 2 into equation 4 and taking the expected value. For
non-statisticians, it may be more convincing to perform a similar calculation using generated ran-
dom numbers instead of residuals. Assuming a diagonal weight matrix,
this can be accomplished
using any software package that can generate random numbers and perform basic calculations.
Simply do the following: (1) Generate n random numbers using any distribution (such as normal,
uniform, and so on). These are equivalent to the residuals of equation 1 or 2. (2) Square each ran-
dom number. (3) Divide each squared number by the variance of the distribution used. If weights
are defined to be one divided by the variances, these numbers are equivalent to squared weighted
residuals. (4) Sum the numbers from (3) and divide by n. (5) Compare this value to 1.0. As n in-
creases, the value should approach 1.0.
Significant deviations of the calculated error variance or the standard error from 1.0 indi-
cate that the fit is inconsistent with the weighting. For
the calculated error variance, significant de-
viations from 1.0 are indicated if the value 1.0 falls outside a confidence interval constructed using
the calculated variance. The confidence interval limits can be calculated as (Ott, 1993, p.332 ):
s
2
s
2
S b
( )
ND
NPR
NP
–
+
(
)
----------------------------------------------
=
S b
( )
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.
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