8. Spectral Analysis  7. Hypothesis testing  7.4 Use of the

7.5 Hypothesis testing by likelihood ratios

In cases where statistical inference is difficult or unwieldy by other means, likelihood ratios often prove useful because of their general applicability. A likelihood ratio can be defined as

$$\lambda = {{L(A)}\over{L(B)}}$$ (7.29)

 

where A represents the multidimensional volume corresponding to a particular hypothesis and B represents the union of volumes corresponding to that hypothesis and its alternative. In the case where there are n parameters, the hypothesis might be that x₁=c with no restriction on other parameters. To test the hypothesis:

1.

Calculate L(A), the maximum likelihood with x₁=c. That is, find the set of parameters $\{x_i\}$, $i\ne 1$, that give the maximum likelihood while x₁ is constrained to be c.

2.

Calculate L(B), the usual maximum likelihood without any restriction on x₁.

3.

Calculate $\lambda$ from (7.29).

4.

A value of $\lambda$ near unity indicates that the hypothesis should be accepted, while a small value of $\lambda$ suggests rejection.

5.

Specifically, for large sample sizes, the parameter v=-2ln($\lambda$) will be distributed as $\chi^2$ for f_B-f_Adegrees of freedom, where f_B(f_A) is the number of degrees of freedom for hypothesis B (A).

Tests based on likelihood ratios are notoriously difficult to interpret, and are often misleading. The reason is that, if the functional form is wrong or the fit is poor, large likelihood ratios will result, and the resulting large changes in values of the likelihood when parameters change by small amounts can lead an analyst to think that the best-fit values are determined with great accuracy when instead the correct interpretation is that the fit is not adequate to represent the data. Confidence limits should be determined from likelihood ratios only in those cases where there is evidence that the fit is adequate.
SOURCES AND FURTHER READING

Abramowitz, M. and I. A. Stegun, 1972: Handbook of Mathematical Functions. Dover Publications, New York, 1046 pp.

Anderson, V. L., and R. A. McLean, 1974: Design of Experiments. Marcel Dekker, Inc., New York, 418 pp.

Box, G. E. P., W. G. Hunter, and J. S. Hunter, 1978: Statistics for Experimenters. John Wiley and Sons, New York, 653 pp.

Brownlee, K. A., 1965: Statistical Theory and Methodology in Science and Engineering. John Wiley and Sons, New York, 590 pp.

Murphy, A. H., and R. W. Katz, 1985: Probability, Statistics, and Decision Making in the Atmospheric Sciences. Westview Press, Boulder, Colorado, 545 pp.

Panofsky, H. A., and G. W. Brier, 1968: Some applications of Statistics to Meteorology. Pennsylvania State University, 224 pp.

   
 8. Spectral Analysis  7. Hypothesis testing  7.4 Use of the