doi:10.1016/j.chemolab.2004.05.002 
Copyright © 2004 Elsevier B.V. All rights reserved.
Statistical analysis of second harmonic generation experiments: a phenomenological model
a Centre for Mathematics and its Applications, The Australian National University, ACT 0200, Canberra, Australia
b School of Mathematics, University of Southampton, Southampton, SO17 1BJ, UK
c School of Chemistry, University of Southampton, Southampton, SO17 1BJ, UK
Available online 25 June 2004.
References and further reading may be available for this article. To view references and further reading you must
purchase this article.
Abstract
We discuss issues arising in fitting theoretically derived nonlinear models with complex coefficients to data from surface Second Harmonic Generation (SHG) experiments conducted at the air/liquid interface. We explore different parametrisations for the complex parameters and show that the Euler (magnitude and phase angle) parametrisation is preferable to the real and imaginary parts parametrisation, both theoretically and empirically. We emphasise the importance and value of diagnostic plots for evaluating the quality of model fit. We derive approximate standard errors for the parameter estimates and discuss issues of making inference about ratios of parameters. We consider approximate confidence intervals (using the approximate standard errors), profile likelihood intervals, Fieller's method and bootstrap intervals. Fieller's method (and the bootstrap intervals) provide useful information on the value of the simpler approximate confidence intervals. We also propose and implement a likelihood ratio test to assess whether a common model can be fitted to several independent data sets. Finally, the methods are applied to data sets obtained from SHG experiments on 
-phenylalanine at the air/water interface and toluene.
Author Keywords: Complex coefficients; Confidence intervals; Likelihood ratio test; Nonlinear least squares; Ratios of parameters
Fig. 1. Profile plots for the phenylalanine data, for the two parametrisations. (a) Euler parametrisation fitted simultaneously. (b) Real and imaginary parametrisation fitted simultaneously.
Fig. 2. Polirisation dependence of the SHG signal generated for S, P and 45° output angles as a function of the output polarisation angle for 5 mM phenylalanine solution. Fitted model included in the graph.
Fig. 3. Residual and normal probability pots for the model fitted to the phenylalanine data. (a) Euler parametrisation fitted simultaneously. (b) Real and imaginary parametrisation fitted simultaneously.
Fig. 4. Residual plots for the models fitted to the toluene data.
Table 1. Values of RMS curvature for the two parametrisations, based on the phenylalanine data of 05-11-2001
Table 2. Parameters estimates and their approximate standard errors for the simultaneous and sequential fitting methods applied to the phenylalanine data
Table 3. Ratios of parameters A, B, and C computed for the Euler and real and imaginary parametrisations. Approximate variance are also shown (in parenthesis), as well as 95% confidence intervals (C.I.) of the ratios calculated by Fieller's method
Table 4. Estimates of ratios of parameters from the model using the profile approach
Table 5. Bootstrap estimates for the parameters and ratios of parameters from the model
Table 6. Parameter estimates and standard errors (in parenthesis) for the models fitted to the ten toluene data sets. There are multiple experiments performed on two of the days
Table 7. Ratios of parameters for the toluene models. Confidence intervals are given for the ratios of the magnitudes using both the delta and Fieller method. Some of the phase angles (φR) are zero because the fitted models were simplified when the phase angles were considered negligible
Abstract
Purchase PDF (601 K)
E-mail Article
Add to my Quick Links
Cited By in Scopus (1)
Purchase PDF (408 K)
