Substitutional Analysis (Dover Books on Mathematics)

Fundamental Concepts of Abstract Algebra. A Course in Linear Algebra. An Introduction to Linear Algebra. Introduction to Linear Algebra and Differential Equations.

An Introduction to Linear Algebra and Tensors. Lectures on Linear Algebra. Linear Algebra and Geometry: These results tend to justify the assumptions behind the subtree-power testing procedure. The fact that topologies were not inferred in these studies reflects the reality that, in practice, much is usually known about topology in advance.

The goal is now to maximize the likelihood-ratio test's power over subsets of size k chosen from the 21 species, for various values of k. This entails searching for the maximal-power family subtree, or k -most-powerful Steiner subtree k -MPSS , among the 21 k subtrees with k leaves. A Steiner subtree on k leaves is the unique smallest subtree rooted at their last common ancestor. Finding the k -MPSS is a combinatorial optimization problem, which we solve in small to moderate-sized cases by evaluating the power of the likelihood-ratio test corresponding to every candidate Steiner subtree.

We can also solve the problem for larger k , by constraining the species at many of the leaves in the subtree. The power computation for a particular subtree is described in General Phylogenies below. Table 1 shows the k -MPSS asterisks in comparison to the subtree on k leaves with largest additive divergence the k -most-divergent Steiner subtree, or k -MDSS, daggers.

The latter has been the focus of previous work 4 , 10 , These two subtree selection criteria do not coincide. The t statistic on the difference in power is 2. The disagreement at larger values of k , where subtree topology becomes more complicated, highlights the importance of including a realistic phylogenetic topology in the species selection procedure. Results are across 10 repetitions of the Monte Carlo power estimation procedure.

The last three columns display the average power and standard error , the t statistic for the power difference between the k -MDSS and the k -MPSS in cases where they differ , and the average power ranking among all subtrees.

Сведения о продавце

We carried out a similar comparison, under the constraint that the nine completely or partially sequenced vertebrates in the data set appear in the subtree Table 2. This procedure reveals the species whose addition to the current sequencing mix would most improve the power to detect single-site conservation. As in Table 1 , the most-powerful and most-divergent subtrees generally differ. The differences in power are smaller than in Table 1. This decrease may be because forcing half of the phylogeny's leaves to appear in the subtree limits the possible power increase from any subtree selection method.

The scheme of the table is the same as that of Table 1. Table 1 exhibits similar properties. Table 2 reveals that the single most beneficial species to sequence next is the dunnart improving power by a relative Our fitted phylogenetic topology differs slightly from estimates based on considerations of large-scale indel mutations and morphology, for example in its placement of the chicken and platypus.

At issue here, however, is its suitability for a single-site power analysis under a substitutional mutation model. We chose our tree estimation procedure to obtain a phylogeny directed to this goal. Here we derive properties of the FOSST likelihood-ratio test, using the notion of a monotone likelihood ratio Fix k and t.

It is now a standard result of monotone likelihood-ratio theory that the likelihood-ratio test is uniformly most powerful. As a further consequence of the monotone likelihood ratio, the likelihood-ratio test is equivalent to rejecting for large values of n x 0 , x. This is an intuitive procedure, which declares conservation when few descendant bases have mutated. In this section we employ the Jukes—Cantor substitution process. The notation in Eq. Each descendant nucleotide X i has probability d r, t of differing from X 0 , independent of all other descendants.

Thus n X 0 , X is a binomial random variable with k trials and success probability d r, t. The kinks in each power curve correspond to values of t at which the critical value of the likelihood-ratio test changes. The locations of the kinks are easily determined, and the power curves are smooth between kinks. We use the Jukes—Cantor substitution process in this section also. Here, the likelihood-ratio statistic has the form. This equation is more difficult to deal with than Eq.

It is clear that Eq. This invariance means that there are only as many distinct values of Eq. The number of leaf configurations corresponding to each integer partition is the combinatorial quantity. We can generate all of the required integer partitions quickly, even for k in the hundreds. The section Empirical Power Analysis uses a general topology, with the Felsenstein substitution process.

The likelihood-ratio statistic based on the leaf subset has the form. The numerator and denominator can be computed efficiently by using the Felsenstein pruning algorithm To compute the power of a test based on Eq. This simulation induced null and alternative empirical distributions on the leaves of every possible subtree. From these we obtained approximations to the true null and alternative distributions of the likelihood-ratio statistic. Littlewood, The skeleton key of mathematics , Hutchinson Univ. Library, London, Reprinted in French translation: Macdonald, On the degrees of the irreducible representations of finite Coxeter groups , J.

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