Approximations to the expectations and variances of ratios of tree properties under the coalescent.
Approximations to the expectations and variances of ratios of tree properties under the coalescent.
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DOI：
10.1093/g3journal/jkac205
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发表时间：
2022-09-30
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--
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基因谱系的特性，如树高(H)、总枝长(L)、外枝和内枝总长度(E)、基枝平均长度(B)和底层聚结时间(T)，可用于研究群体遗传过程和开发群体遗传模型的统计检验。在统计测试中使用树状特征通常依赖于依赖于这些特征之间的成对关系的预测。对于聚结下的系谱，我们提供了对所有15对变量的期望值和比值方差的泰勒近似的精确表达式，考虑n个叶片和。对于比率的期望值，近似值与基于经验模拟的值密切匹配。对方差的近似不那么准确，但随着n的增加，它们的趋势通常与模拟相匹配。尽管在极限情况下，En的期望为2，Hn的期望为2，但极限期望的近似不是1，而是等于。新的近似增加了关于系谱树形状的凝聚理论的基本结果。
Properties of gene genealogies such as tree height (H), total branch length (L), total lengths of external (E) and internal (I) branches, mean length of basal branches (B), and the underlying coalescence times (T) can be used to study population-genetic processes and to develop statistical tests of population-genetic models. Uses of tree features in statistical tests often rely on predictions that depend on pairwise relationships among such features. For genealogies under the coalescent, we provide exact expressions for Taylor approximations to expected values and variances of ratios , for all 15 pairs among the variables , considering n leaves and . For expected values of the ratios, the approximations match closely with empirical simulation-based values. The approximations to the variances are not as accurate, but they generally match simulations in their trends as n increases. Although En has expectation 2 and Hn has expectation 2 in the limit as , the approximation to the limiting expectation for is not 1, instead equaling . The new approximations augment fundamental results in coalescent theory on the shapes of genealogical trees.
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