Hypothesis testing and earthquake prediction.

Requirements for testing include advance specification of the conditional rate density (probability per unit time, area, and magnitude) or, alternatively, probabilities for specified intervals of time, space, and magnitude. Here I consider testing fully specified hypotheses, with no parameter adjustments or arbitrary decisions allowed during the test period. Because it may take decades to validate prediction methods, it is worthwhile to formulate testable hypotheses carefully in advance. Earthquake prediction generally implies that the probability will be temporarily higher than normal. Such a statement requires knowledge of "normal behavior"--that is, it requires a null hypothesis. Hypotheses can be tested in three ways: (i) by comparing the number of actual earth-quakes to the number predicted, (ii) by comparing the likelihood score of actual earthquakes to the predicted distribution, and (iii) by comparing the likelihood ratio to that of a null hypothesis. The first two tests are purely self-consistency tests, while the third is a direct comparison of two hypotheses. Predictions made without a statement of probability are very difficult to test, and any test must be based on the ratio of earthquakes in and out of the forecast regions.

Inferring identify from DNA profile evidence.

The controversy over the interpretation of DNA profile evidence in forensic identification can be attributed in part to confusion over the mode(s) of statistical inference appropriate to this setting. Although there has been substantial discussion in the literature of, for example, the role of population genetics issues, few authors have made explicit the inferential framework which underpins their arguments. This lack of clarity has led both to unnecessary debates over ill-posed or inappropriate questions and to the neglect of some issues which can have important consequences. We argue that the mode of statistical inference which seems to underlie the arguments of some authors, based on a hypothesis testing framework, is not appropriate for forensic identification. We propose instead a logically coherent framework in which, for example, the roles both of the population genetics issues and of the nonscientific evidence in a case are incorporated. Our analysis highlights several widely held misconceptions in the DNA profiling debate. For example, the profile frequency is not directly relevant to forensic inference. Further, very small match probabilities may in some settings be consistent with acquittal. Although DNA evidence is typically very strong, our analysis of the coherent approach highlights situations which can arise in practice where alternative methods for assessing DNA evidence may be misleading.

Bootstrap confidence levels for phylogenetic trees

Evolutionary trees are often estimated from DNA or RNA sequence data. How much confidence should we have in the estimated trees? In 1985, Felsenstein [Felsenstein, J. (1985) Evolution 39, 783–791] suggested the use of the bootstrap to answer this question. Felsenstein’s method, which in concept is a straightforward application of the bootstrap, is widely used, but has been criticized as biased in the genetics literature. This paper concerns the use of the bootstrap in the tree problem. We show that Felsenstein’s method is not biased, but that it can be corrected to better agree with standard ideas of confidence levels and hypothesis testing. These corrections can be made by using the more elaborate bootstrap method presented here, at the expense of considerably more computation.

Patterned library analysis: A method for the quantitative assessment of hypotheses concerning the determinants of protein structure

Site-directed mutagenesis and combinatorial libraries are powerful tools for providing information about the relationship between protein sequence and structure. Here we report two extensions that expand the utility of combinatorial mutagenesis for the quantitative assessment of hypotheses about the determinants of protein structure. First, we show that resin-splitting technology, which allows the construction of arbitrarily complex libraries of degenerate oligonucleotides, can be used to construct more complex protein libraries for hypothesis testing than can be constructed from oligonucleotides limited to degenerate codons. Second, using eglin c as a model protein, we show that regression analysis of activity scores from library data can be used to assess the relative contributions to the specific activity of the amino acids that were varied in the library. The regression parameters derived from the analysis of a 455-member sample from a library wherein four solvent-exposed sites in an α-helix can contain any of nine different amino acids are highly correlated (P < 0.0001, R2 = 0.97) to the relative helix propensities for those amino acids, as estimated by a variety of biophysical and computational techniques.

Bootstrap confidence levels for phylogenetic trees.

Evolutionary trees are often estimated from DNA or RNA sequence data. How much confidence should we have in the estimated trees? In 1985, Felsenstein [Felsenstein, J. (1985) Evolution 39, 783-791] suggested the use of the bootstrap to answer this question. Felsenstein's method, which in concept is a straightforward application of the bootstrap, is widely used, but has been criticized as biased in the genetics literature. This paper concerns the use of the bootstrap in the tree problem. We show that Felsenstein's method is not biased, but that it can be corrected to better agree with standard ideas of confidence levels and hypothesis testing. These corrections can be made by using the more elaborate bootstrap method presented here, at the expense of considerably more computation.

Testing the Cambrian explosion hypothesis by using a molecular dating technique

Molecular studies have the potential to shed light on the origin of the animal phyla by providing independent estimates of the divergence times, but have been criticized for failing to account adequately for variation in rate of evolution. A method of dating divergence times from molecular data addresses the criticisms of earlier studies and provides more realistic, but wider, confidence intervals. The data are not compatible with the Cambrian explosion hypothesis as an explanation for the origin of metazoan phyla, and provide additional support for an extended period of Precambrian metazoan diversification.

Estimation of evolutionary distances under stationary and nonstationary models of nucleotide substitution

Estimation of evolutionary distances has always been a major issue in the study of molecular evolution because evolutionary distances are required for estimating the rate of evolution in a gene, the divergence dates between genes or organisms, and the relationships among genes or organisms. Other closely related issues are the estimation of the pattern of nucleotide substitution, the estimation of the degree of rate variation among sites in a DNA sequence, and statistical testing of the molecular clock hypothesis. Mathematical treatments of these problems are considerably simplified by the assumption of a stationary process in which the nucleotide compositions of the sequences under study have remained approximately constant over time, and there now exist fairly extensive studies of stationary models of nucleotide substitution, although some problems remain to be solved. Nonstationary models are much more complex, but significant progress has been recently made by the development of the paralinear and LogDet distances. This paper reviews recent studies on the above issues and reports results on correcting the estimation bias of evolutionary distances, the estimation of the pattern of nucleotide substitution, and the estimation of rate variation among the sites in a sequence.