Power for detecting genetic divergence: differences between statistical methods and marker loci

Information on statistical power is critical when planning investigations and evaluating empirical data, but actual power estimates are rarely presented in population genetic studies. We used computer simulations to assess and evaluate power when testing for genetic differentiation at multiple loci through combining test statistics or P values obtained by four different statistical approaches, viz. Pearson's chi-square, the log-likelihood ratio G-test, Fisher's exact test, and an F(ST)-based permutation test. Factors considered in the comparisons include the number of samples, their size, and the number and type of genetic marker loci. It is shown that power for detecting divergence may be substantial for frequently used sample sizes and sets of markers, also at quite low levels of differentiation. The choice of statistical method may be critical, though. For multi-allelic loci such as microsatellites, combining exact P values using Fisher's method is robust and generally provides a high resolving power. In contrast, for few-allele loci (e.g. allozymes and single nucleotide polymorphisms) and when making pairwise sample comparisons, this approach may yield a remarkably low power. In such situations chi-square typically represents a better alternative. The G-test without Williams's correction frequently tends to provide an unduly high proportion of false significances, and results from this test should be interpreted with great care. Our results are not confined to population genetic analyses but applicable to contingency testing in general.     

塔里木河中下游地区荒漠河岸林群落种间关系分析

采用 2× 2列联表, 应用Fisher精确检验法研究了新疆塔里木河中下游荒漠河岸林群落种间关系, 测定了16种植物、共 12 0个种对的种间联结性。研究结果表明 :1) 12 0个种对中有 17个种对分别在不同的样方尺度中表现出显著或极显著的种间联结, 约占总数的 14.2 % ;其中 13个种对为正关联, 4个种对为负关联 ;2 ) 不同取样面积对种间联结性分析的有效性有影响, 不同种对表现出种间联结的最小样方尺度不同 ;3) 随着样方面积的增大, 各种对自有不同的种间联结变化规律, 可归纳为 4种类型 ;4 ) 17个具种间联结的种对以灌木草本和草本草本的种对居多, 占总数的 76.5 % ;主要乔木树种胡杨 (Populuseuphratica) 与灌木之间、灌木和灌木之间趋向独立分布。     

Randomization Modelling of the Crossover Experiment for Clinical Trials

The two-period cross-over experiment for clinical trials has been examined by several writers following a Gaussian linear model approach. Some authors have expressed interest in the “derivation of the finite permutation model” and have pointed out that the randomization approach to modeling the two-period cross-over design “would highlight the importance of randomizing the subjects to the two groups as a basis for inference”. However, in the literature, there is no development of the randomization approach to this important design. In this paper, after a statement of the experimental design and formulation of the observation random variables of the finite population, two additive randomization models—one with residual effects, the other without—which are the analogues of Grizzle's Gaussian models, are derived. Statistical inference is developed for these randomization models and the results are compared with those of the corresponding Gaussian models. Also, exact inference based upon Fischer's approach is presented.     

秦岭太白红杉群落种间关系的数量分析

应用2×2列联表的Fisher精确检验,研究了太白红杉群落中44种植物 共946个种对 的种间关系.结果有33个种对正关联,12个种对负关联 P<0.05或P<0.01 .另外,用Pearson积矩相关系数和Spearman秩相关系数检验刻划了种对间的数量变化关系.Pearson相关系数显示有31个种对呈显著正协变,5个种对呈显著负协变;Spearman秩相关系数显示有30个种对呈显著正协变,5个种对呈显著负协变,并分别计算出了其相关系数.研究结果表明,太白红杉群落种间关系较为简单,但草本层物种间连接相对较为复杂,灌木层次之.乔木层中,太白红杉和巴山冷杉呈显著的负协变效应.     

The Discrete Distribution Used for the Log-rank Test Can Be Inaccurate

When the number of tumors is small, a significance level for the Cox-Mantel (log-rank) test Z is often computed using a discrete approximation to the permutation distribution. For j = 0,…, J let N_j(t) be the number of animals in group j alive and tumor-free at the start of time t. Make a 2 × (1+J) table for each time t of the number of animals R_j(t) with newly palpated tumor out of the total N_j(t) at risk. There are a total of say K tables, one for each distinct time t with observed death or newly palpated tumor. The usual discrete approximation to the permutation distribution of Z is defined by taking tables to be independent with fixed margins N_j(t) and ΣR_j(t) for all t. However, the N_j(t) are random variables for the actual permutation distribution of Z, resulting in dependence among the tables. Calculations for the exact permutation distribution are explained, and examples are given where the exact significance level differs substantially from the usual discrete approximation. The discrepancy arisis primarily because permutations with different Z-scores under the exact distribution can be equal for the discrete approximation, inflating the approximate P-value.