Methods and guidelines for effective model calibration

Correlation Coefficients
Correlation coefficients are calculated as the covariance between two parameters divided 
by the product of their standard deviations. Using the notation of equation (27), the correlation be-
tween the ith and jth parameter is calculated as:
(30)
Correlation coefficients range in value from -1.0 to 1.0, with values close to -1.0 and 1.0 indicative 
of parameter values that cannot be uniquely estimated with the observations used in the regression. 
Poeter and Hill (1997) provide a good description of correlation coefficients calculated for a simple 
test case; Anderman and others (1996) show how they can be used to evaluate the worth of different 
kinds of observations. Correlation coefficients are typically displayed as a matrix, such as:
1.0 0.96 0.05
0.96 1.0 0.46
(31)
0.05 0.46 1.0
Correlation coefficient matrices are always symmetric and the diagonal elements always equal 1.0. 
Correlation coefficients can be calculated using any of the variations of the parameter variance-
covariance matrix discussed above. Correlation coefficients calculated using the parameter vari-
ance-covariance matrix with all parameters are called correlation coefficients for all parameters; 
correlation coefficients calculated using the parameter varience-covariance matrix with predictions 
are called prediction correlation coefficients.
Influence Statistics
While dimensionless scaled sensitivities indicate the importance of an observation to the 
estimation of a parameter, the actual effect of the observation in the regression also depends on cal-
culated residuals. The Cook’s D and DFBETA influence statistics incorporate this effect. The 
Cook’s D statistics can be calculated for each observation as described by Cook and Weisberg 
(1982) and Helsel and Hirsch (1992). DFBETAs are calculated for each parameter j and each ob-
servation i .
Anderman (1996) and Yager (in press) show how the DFBETA influence statistic can be 
cor i j
,
( )
cov i j
,
( )
var i
( )
1 2
⁄
var j
( )
1 2
⁄
----------------------------------------------------
=

---

29
used to identify the interaction between the observations and estimated parameter values in 
ground-water problems.