# Hypothesis testing: likelihood of alternative models

• NOTE 1: Hypothesis testing is only meaningful if the model fits the data well!
• NOTE 2: Hypothesis testing does not require knowledge of confidence intervals.

Akaike's Information Criterion (AIC) is used to assess relative likelihood of alternative models. My normalized sum of squares is essentially SS/N term in AICc definition. Priorities applied to different parts in sum of squares calculation are essentially uniformly scaling the uncertainties used to weigh sum of squares of residuals. When statistical testing is performed we use the same data with the same priorities and assess models effect on the total sum of squares with respect to the number of variable parameters in the models. For more information on AICc see [1-4].

NOTE: There are some theoretical aspects that need to be clarified in the original publications:

1. here we are using WEIGHTED sum of squares instead of just sum of squares as asked by the the original test, 2. we use dataset PRIORITIES to re-weight contributions of different pieces of the data to the final sum of squares of residuals. The number of points (outside of the log()) from these datasets has to (probably) also be weighted by these priorities but that needs a theoretical proof.

Intuitively, it should not make difference because the models being compared utilize the same data and the same formula for the sum of squares. However, the sensitivity of the test is likely to be unequal to contributions of different datasets. Probably, the most accurate results will be achieved if one (1) does not use the priorities and (2) includes data with similar total sizes. Therefore if you only fit a bunch of line shapes AICc calculation will be accurate. But if you mix in the CPMG datasets or ITC data, which have much FEWER points relatively to line shapes, which are (normally) oversampled you will probably bias model selection to what line shapes dictate. In such cases the recipie will be to remove any additional zero-filling from spectral data and keep baseline regions to a minimum: all to reduce total number of data points in the line shape datasets to a minimum and balance total data size with data sizes of other types in the analysis.

For a specific usage see TotalFit.m

REFERENCES

1. Bozdogan, H., Model selection and Akaike's Information Criterion (AIC): The general theory and its analytical extensions. Psychometrika, 1987. 52(3): p. 345-370.
2. Akaike, H., Information theory and an extension of the maximum likelihood principle, in Second International Symposium on Information Theory, B.N. Petrov and B.F. Csaki, Editors. 1973, Academiai Kiado: Budapest. p. 267-281.
3. Akaike, H., Likelihood of a model and information criteria. Journal of Econometrics, 1981. 16: p. 3-14.
4. Motulsky, H. and Christopoulos, A., Fitting Models to Biological Data Using Linear and Nonlinear Regression: A Practical Guide to Curve Fitting. 1 edition ed. 2004: Oxford University Press, USA. 352