Parameter estimates geometric interpretation

The uncertainty in the estimate of Pi is smaller (s, = s x5.04), and the uncertainty in the estimate of Pn is smaller still (Sj, = s x0.04). The geometric interpretation of the parameters Pi and Ph in this model is not straightforward, but Pi essentially moves the apex of the parabola away from Xi = 0, and pn is a measure of the steepness of curvature. The geometric interpretation of the associated uncertainties in the parameter estimates is also not straightforward (for example, P, Pi, and p,i are expressed in different units). We will simply note that such uncertainties do exist, and note also that there is covariance between bg and hi, between and b,i, and between hi and hn. 

It is to be stressed, however, that the geometric interpretation of the parameter estimates obtained using coded factor levels is usually different from the interpretation of those parameter estimates obtained using uncoded factor levels. As an illustration, pj (the intercept in the coded system) represents the response at the center of the experimental design, whereas Pq (the intercept in the uncoded system) represents the response at the origin of the original coordinate system the two estimates (Pq and Pq) are usually quite different numerically. This difficulty will not be important in the remainder of this book, and we will feel equally free to use either coded or uncoded factor levels as the examples require. Later, in Section 11.5, we will show how to translate coded parameter estimates back into uncoded parameter estimates. 

Fitting this non-Kramers doublet model to the data in Fig. 12 provides an experimental estimate of 8 and for the ground state doublet in Fig. 13 of 4 cm and 8.9, respectively. The ground state parameters can now be interpreted in terms of the geometric and electronic structure of the metal site using ligand field theory as described in the next section. 

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