Fig 1.
Two ways of displaying a 2×2 table of co-occurrences and occurrences of two entities with fixed margins.
The example illustrates the distribution of Species A and B occupying mA and mB sites, respectively, out of a total N sites. The number of sites where both species co-occur (top-left quadrant), where only Species A exists (bottom-left quadrant), where only Species B exists (top-right quadrant), and where none of the species exist (bottom-right quadrant) typically appear in popular scientific literature with one set of notations (Fig 1a) but often appear in statistical literature with different notations (Fig 1b).
Fig 2.
Model development for affinity between two species.
The three stages illustrate how the notion of affinity was formulated mathematically (a). The null model for co-occurrence counts in the situation of zero affinity emerges as the Hypergeometric distribution (b) and extends in the case of nonzero affinity to the Extended Hypergeometric distribution, leading to an estimable parameter α that governs association in the 2×2 table.
Fig 3.
Workflow of CooccurrenceAffinity.
The functions are grouped by color based on their role, as indicated in the box. A function at the base of an arrow feeds its output to the function at the tip of the arrow. Not all functions of the package are shown here.
Fig 4.
An example of syntax and output of AlphInts and ML.Alpha.
Fig 5.
The log cross-product ratio of the example in Fig 4.
Fig 6.
Analysis of a presence-absence matrix for computing several pairwise analyses for row or column.
The top panel shows a portion of the dataset. The code block shows how easily affinity between species (rows) or between islands (columns) can be computed using affinity, and how the results can be visualized using plotgg.
Fig 7.
Intervals and MLE for α for all possible X values (co-occurrence) in an example of mA = 50, mB = 70, N = 150.
(a) MLE, median interval and Blaker CI (95%), (b) ratio of the length of Blaker CI (95%) to that of Clopper-Pearson type CI (95%).
Fig 8.
Functions F(x, α) plotted for selected pairs x, x − 1 and all α in the range (−3, 3.3) for the same example of mA = 50, mB = 70, N = 150.
The colored sigmoid lines are the curves F(X, ⋅), and immediately to the left of each such curve is another (with the same color) for X − 1. Horizontal dashed lines are the quantile levels 0.025, 0.50, 0.975. Black solid circles are plotted for the X values in each curve-pair at (, 0.5). Colored solid circles encircled by black circles are plotted for the upper and lower bounds of Clopper-Pearson 95% confidence intervals. The median-interval of α’s for each X is the colored horizontal bar connecting the curve-pair at height 0.5.
Fig 9.
True coverage probability of the four types of confidence intervals for the example presented in Fig 7 with α values shown from -4 to 4, computed as CovrgPlot(c(50,70,150), lev = 0.95).
The true coverage probability of the 95% Clopper-Pearson (a) or Blaker (c) CI’s are always at least 95% (dashed line) for all co-occurrences resulting in a histogram with its entire mass above 0.95 (b,d). Plots of true coverage probability are also shown for midQ (e,f) and midP (g,h) CIs.