Statistical analysis of second harmonic generation experiments: a phenomenological model

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 l-phenylalanine at the air/water interface and toluene.