Gotelli Nicholas J, Entsminger Gary L
Department of Biology, University of Vermont, 05405, Burlington, VT, USA.
Acquired Intelligence Inc., 99 Schillhammer Road, 05465, Jericho, VT, USA.
Oecologia. 2001 Oct;129(2):281-291. doi: 10.1007/s004420100717. Epub 2001 Oct 1.
Community assembly rules are often inferred from patterns in presence-absence matrices. A challenging problem in the analysis of presence-absence matrices has been to devise a null model algorithm to produce random matrices with fixed row and column sums. Previous studies by Roberts and Stone [(1990) Oecologia 83:560-567] and Manly [(1995) Ecology 76:1109-1115] used a "Sequential Swap" algorithm in which submatrices are repeatedly swapped to produce null matrices. Sanderson et al. [(1998) Oecologia 116:275-283] introduced a "Knight's Tour" algorithm that fills an empty matrix one cell at a time. In an analysis of the presence-absence matrix for birds of the Vanuatu islands, Sanderson et al. obtained different results from Roberts and Stone and concluded that "results from previous studies are generally flawed". However, Sanderson et al. did not investigate the statistical properties of their algorithm. Using simple probability calculations, we demonstrate that their Knight's Tour is biased and does not sample all unique matrices with equal frequency. The bias in the Knight's Tour arises because the algorithm samples exhaustively at each step before retreating in sequence. We introduce an unbiased Random Knight's Tour that tests only a small number of cells and retreats by removing a filled cell from anywhere in the matrix. This algorithm appears to sample unique matrices with equal frequency. The Random Knight's Tour and Sequential Swap algorithms generate very similar results for the large Vanuatu matrix, and for other presence-absence matrices we tested. As a further test of the Sequential Swap, we constructed a set of 100 random matrices derived from the Vanuatu matrix, analyzed them with the Sequential Swap, and found no evidence that the algorithm is prone to Type I errors (rejecting the null hypothesis too frequently). These results support the original conclusions of Roberts and Stone and are consistent with Gotelli's [(2000) Ecology 81:2606-2621] Type I and Type II error tests for the Sequential Swap. In summary, Sanderson et al.'s Knight's Tourgenerates large variances and does not sample matrices equiprobably. In contrast, the Sequential Swap generates results that are very similar to those of an unbiased Random Knight's Tour, and is not overly prone to Type I or Type II errors. We suggest that the statistical properties of proposed null model algorithms be examined carefully, and that their performance judged by comparisons with artificial data sets of known structure. In this way, Type I and Type II error frequencies can be quantified, and different algorithms and indices can be compared meaningfully.
群落组装规则通常是从存在 - 缺失矩阵中的模式推断出来的。在分析存在 - 缺失矩阵时，一个具有挑战性的问题是设计一种空模型算法，以生成具有固定行和列总和的随机矩阵。罗伯茨和斯通（1990年，《生态学》83卷：560 - 567页）以及曼利（1995年，《生态学》76卷：1109 - 1115页）之前的研究使用了一种“顺序交换”算法，其中子矩阵被反复交换以生成空矩阵。桑德森等人（1998年，《生态学》116卷：275 - 283页）引入了一种“骑士巡游”算法，该算法一次填充一个空矩阵的单元格。在对瓦努阿图群岛鸟类的存在 - 缺失矩阵进行分析时，桑德森等人得到了与罗伯茨和斯通不同的结果，并得出结论“先前研究的结果通常存在缺陷”。然而，桑德森等人没有研究他们算法的统计特性。通过简单的概率计算，我们证明他们的骑士巡游算法存在偏差，并且不会以相等的频率对所有唯一矩阵进行采样。骑士巡游算法中的偏差源于该算法在依次回撤之前在每个步骤进行了详尽的采样。我们引入了一种无偏差的随机骑士巡游算法，该算法只测试少量单元格，并通过从矩阵中的任何位置移除一个已填充的单元格来回撤。这种算法似乎以相等的频率对唯一矩阵进行采样。对于大型的瓦努阿图矩阵以及我们测试的其他存在 - 缺失矩阵，随机骑士巡游算法和顺序交换算法产生的结果非常相似。作为对顺序交换算法的进一步测试，我们构建了一组从瓦努阿图矩阵派生的100个随机矩阵，用顺序交换算法对它们进行分析，没有发现该算法容易出现I型错误（过于频繁地拒绝零假设）的证据。这些结果支持了罗伯茨和斯通的原始结论，并且与戈特利（2000年，《生态学》81卷：2606 - 2621页）对顺序交换算法的I型和II型错误测试一致。总之，桑德森等人的骑士巡游算法产生的方差很大，并且不会等概率地对矩阵进行采样。相比之下，顺序交换算法产生的结果与无偏差的随机骑士巡游算法非常相似，并且不太容易出现I型或II型错误。我们建议仔细检查所提出的空模型算法的统计特性，并通过与已知结构的人工数据集进行比较来判断它们的性能。通过这种方式，可以量化I型和II型错误频率，并且可以有意义地比较不同的算法和指标。
