Lenth method

Ye, K. Q. and Hamada, M. (2000). Critical values of the Lenth method for unreplicated factorial designs. Journal of Quality Technology, 32, 57-66. 

Saunders V R and Van Lenthe J H 1983 The direst Cl method a detailed analysis Mol. Rhys. 48 923-54... 

E J.H. van Lenthe, J.G.C.M. van Duijneveldt-van de Rijdt and F.B. van Duijneveldt, Weakly bonded systems, Ab initio methods in Quantum Chemistry, vol. II, in Advances in Chemical Physics, vol. LXIX, Wiley, 1987. 

ZORA Method. Very recently, we began applying a new relativistic approach, the ZORA method of van Lenthe and co-workers (14-16). The ZORA Hamiltonian is given by... 

J. H. van Lenthe, G. G. Balint-Kurti, Chem. Phys. Lett. 76, 138 (1980). The Valence-Bond SCF (VB SCF) Method. Synopsis of Theory and Test Calculations of OH Potential Energy Curve. 

The Valence Bond Self-Consistent Field (VBSCF) method has been devised by Balint-Kurti and van Lenthe (32), and was further modified by Verbeek (6,33) who also developed an efficient implementation in a package called TURTLE (11). Basically, the VBSCF method is a multiconfiguration SCF procedure that allows the use of nonorthogonal orbitals of any type. The wave function is given as a linear combination of VB structures, (Eq. 9.7). 

J. H. van Lenthe, J. Verbeek, P. Pulay, Mol. Phys. 73, 1159 (1991). Convergence and Efficiency of the Valence Bond Self-Consistent Field Method. 

J.H. van Lenthe and G.G. Balint-Kurti, VBSCF The optimisation of non-orthogonal orbitals in a general (Valence Bond) wavefunction, in 5th seminar on Computational Methods in Quantum Chemistry (Groningen, 1981). 

Technically, the simultaneous optimization of orbitals and coefficients for a multistructure VB wave function can be done with the VBSCF method due to Balint-Kurti and van Lenthe [21,22], The VBSCF method has the same format as the classical VB method with an important difference. While the classical VB method uses orbitals that are optimized for the separate atoms, the VBSCF method uses a variational optimization of the atomic orbitals in the molecular wave function. In this manner the atomic orbitals adapt themselves to the molecular environment with a resulting significant improvement in the total energy and other computed properties. 

Gutowski M, Van Duijneveldt-Van der Rijdt JGCM, Van Lenthe JH, Van Duijneveldt FB (1993) Accuracy of the Boys and Bernardi function counterpoise method. J Chem Phys 98 4728-4737... 

Figure 3 shows a half-normal plot for hue from the full experiment. Two effects clearly stand out and these are the main effects of factors A and F. All the other effects appear consistent with a null hypothesis of no effect. These conclusions are reinforced by other analyses. Fitting a model with all main effects and two-factor interactions results in highly significant effects for factors A and F. No other effects are significant at the 5% level, but the main effect of B and the AC interaction are both quite close, with values less than 0.075. Analysis by Lenth s (1989) method, discussed in Chapter 12, also finds that the only significant effects are those for A and F. 

The most influential method of analysis of orthogonal saturated designs yet proposed is the robust adaptive method of Lenth (1989). The quick and easy method that he proposed is based on the following estimator of the standard deviation, op, of the effect estimators Pi. This estimator is robust and adaptive , concepts which are explained in detail after the following description of the method. 

Critical values for individual tests and confidence intervals are based on the null distribution of i /aL, that is, on the distribution of this statistic when all effects Pi are zero. Lenth proposed a /-distribution approximation to the null distribution, whereas Ye and Hamada (2000) obtained exact critical values by simulation of Pi /ai under the null distribution. From their tables of exact critical values, the upper 0.05 quantile of the null distribution of Pi /aL is CL = 2.156. On applying Lenth s method for the plasma etching experiment and using a = 0.05 for individual inferences, the minimum significant difference for each estimate is calculated to be cl x l = 60.24. Hence, the effects A, AB, and E are declared to be nonzero, based on individual 95% confidence intervals. 

From empirical comparisons of various proposed methods of analysis of orthogonal saturated designs (Hamada and Balakrishnan, 1998 Wang and Voss, 2003), Lenth s method can be shown to have competitive power over a variety of parameter configurations. It remains an open problem to prove that the null case is the least favourable parameter configuration. 

We now discuss what it means for a method of analysis to be robust or adaptive . Lenth s method is adaptive because of the two-stage procedure used to obtain the pseudo standard error. The pseudo standard error i is computed from... 

Consider now robustness. If the estimators A are computed from independent response variables then, as noted in Section 1, the estimators have equal variances and are usually at least approximately normal. Thus the usual assumptions, that estimators are normally distributed with equal variances, are approximately valid and we say that there is inherent robustness to these assumptions. However, the notion of robust methods of analysis for orthogonal saturated designs refers to something more. When making inferences about any effect A, all of the other effects At (k i) are regarded as nuisance parameters and robust means that the inference procedures work well, even when several of the effects ft are large in absolute value. Lenth s method is robust because the pseudo standard error is based on the median absolute estimate and hence is not affected by a few large absolute effect estimates. The method would still be robust even if one used the initial estimate 6 of op, rather than the adaptive estimator 6L, for the same reason. 

In summary, Lenth s method is robust in the sense that it maintains good power as long as there is effect sparsity and it is adaptive to the degree of effect sparsity, using a pseudo standard error that attempts to involve only the estimates of negligible effects. 

Like many methods of analysis of orthogonal saturated designs proposed in the literature, the critical values for Lenth s method are obtained in the null case (all A zero), assuming this is sufficient to control the Type I error rates. This raises the question can one establish analytically that Lenth s and other proposed methods do indeed provide the claimed level of confidence or significance under standard model assumptions The rest of this chapter concerns methods for which the answer is yes. ... 

For any of the seven largest estimates, expression (3) gives y = 191.59, so the minimum significant difference is Cyoy = 70.37. From Table 1, we see that three effects have estimates larger than 70.37 in absolute value, namely, A, AB, and E. Hence, individual 95% confidence intervals for these three effects do not include zero, so these effects are declared to be nonzero. No other effects can be declared to be nonzero using this method. These results match those obtained using Lenth s individual 95% confidence intervals. 

The main disadvantage of the method is its dependence on the zero point of the electrostatic potential, i.e. gauge dependence. This occurs because the potential enters nonlinearly (in the denominator of the operator for the energy), so that a constant shift of the potential does not lead to a constant shift in the energy. This deficiency can, however, be approximately remedied by suitable means (van Lenthe et al. 1994 van Wiillen 1998). 

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