Supplementary to:
EstimatingtheNumberofSharedSpeciesbya JackknifeProcedure
Chia-JuiChuang, Tsung-JenShenandWen-HanHwang
S1. Proof of Theorem 1
S2. Evaluating the relative asymptotic bias of D and
S3. Jackknife estimators for the number of shared species in three communities
S4. Comparison of various estimators in Empirical Test
S5. Simulation study
S1. Proof of Theorem 1
This theorem is proved by the mathematical induction. First, we show that the formulae of and are correct as . Recall that and Note that which is theobserved number of shared species when the -th individual is deleted from sample I. The event “”stands for that the -th individual belongs to the observed shared species with one individual in sample I and at least one individual in sample II. In addition, the event “” represents the -th individual belongs to the observed shared species with exactly i individuals and j individuals in sample I and II, respectively. Hence
Since the coefficients and as or , Equation (7) is true for . In symmetry, Equation (8) is also true as .
Next consider and delete the m-th individualfrom the sample II fora shared species, that follows . Following the proposed procedure in Section 3.1, the second order jackknife estimator is
By noting that , the equation (9) holds for .
Suppose that Equations (7) to (9) are true when where is a positive integer, we are going to prove the statement when . Using the induction hypothesis for , the estimators and have been well-defined as shown in (7) to (9) with . That follows
Therefore, the estimator of can be expressed as
For simplicity, the above expression was rearranged into three components, say A1,B1
andC1 in order, involving the frequencies, ; , ; and the remaining frequencies. These three terms were compiled as follows:
and
As a result, integrating three terms into one follows
which has the same expression as Equation (7) with . In symmetry, the formula of can be shown as well in the parallel way. Simply taking the weighted average of and givens
To demonstrate the result of , we need the result below
Thus,
Again, for simplicity the above expression was rearranged into three components, say A2,B2,and C2 in order, involving the frequencies , ; , ; and the remaining frequencies. These three terms were compiled as follows:
,
and
To sum up these three terms along with the observed number of shared species (D) follows
which has the same expression as Equation (9) with . Therefore, the proof with the mathematical induction is complete.
S2. Evaluating the relative asymptotic bias of D and
Let () be the average of observed number of individuals for the shared species in Community I (Community II). We define the coefficients of variation (CVs) as, , and .Using the Taylor expansion around and , the asymptotic biases of D and , denoted ABias(D) and ABias(), respectively, can be derived as:
and
respectively.
Web Figure 1:(a) (,,) = (0,0,0); (b) (,,) = (0.5,0.5,0.5); (c)(,,) = (0.5,2,1); (d) (,,) = (2,2,0); (e) (,,) = (3,2,1); (f) (,,) = (5,5,5). The gray shadow indicates the range of andwhere the relative asymptotic bias of is smaller than the initial estimator D.
S3. Jackknife estimators for the number of shared species in three communities
Consider that there is another community, say community III, in addition to communities I and II. A random samplewith size is accordingly drawn from communityIII,stand for the observed frequencies where M3 is the number of species in community III.For simplicity, S represents the number of shared species in the three communities. Let Dbe the observed number of shared species from three samples andbe the observed number of shared species with exactly r, s, and tindividuals observed in sample I, II, and III in order. For an estimatorof S, denotes that the given estimator is computed using the full data but with the -thobservation moved out from sample I, see Section 3.1 for the scenario of two samples.In addition, some other alternative notation can also be defined straightforwardly. Nextan algorithm was put forward to illustrate the jackknife estimators for the parameterS. Though a general result can be established for three communities as well, itis more complicated and would not be reported here. Instead, the explicit formula ofjackknife estimators up to sixth order is displayed in Web Table 1when the sample sizesare sufficiently large.
Step0. Initializeν=0.
Step1. Let,define. Moreover, and can be defined with the similar way. The k-th,order jackknife estimator is
Step2. Letand a similar way for and . The (k+1)-th,, order jackknife estimator is
Step3. The (k + 2)-th,, order jackknife estimator is
Step4. Update the integer to be and return to Step 1.
Web Table 1: The explicit expressions of jackknife estimators, , for thescenario of three communities when the sample sizes are large. Let for , , and .
S4. Comparison of various estimators in Empirical Test
In Web Tables 2-4, we reported the performance of various estimators for the number of shared species in Pasoh and Lambir plots. We considered three quadratsizes(5×5m,10×10m,and20×20m)against eightsampling proportions (0.5%,1%,3%,5%,10%,20%, 33%≅1/3, and 50%=1/2). For eachcombination of quadrat size and sampling proportion, 2,000 two-samples of quadrats were randomly selected with replacement from the Pasoh and Lambir plots.
For each generated data, we computed the following estimators: the sample coverage estimator(),the Laplace approximation estimator(),the lower bound estimator(),the jackknife estimatorsand its resulting estimatorbytheorderselectionprocedureinSection3.2withKu=6. To determine the selected order to, we set the significance levelα=0.1.The resulting estimatesalong with their SEs were averaged to give the “Estimate” andin Web Tables 2-4. The sample SE (denoted byσ)and sample root mean squared error (RMSE) were also presented. We computed the percentage (denoted by CP)of the 2,000 simulated datasets in which the 95% confidence intervals covered the true number of shared species. We also recorded the averaged p-values (denoted by p-value) that used in the order selection for. The proportions of selected order are also presented (denoted by “Selection”).
S1
Web Table 2: Comparison of various estimators for the number of shared species in Pasoh and Lambir plots where the sampling quadrat size is m.
Estimate / 82.0 / 111.9 / 138.4 / 171.1 / 206.2 / 330.9 / 146.9 / 122.4 / 112.4 / 112/ 7.9 / 12.9 / 23.8 / 47.7 / 104.4 / 271.6 / 45.2 / 28.2 / 26.7 / 25.6
/ 6.0 / 12.3 / 23.6 / 48.1 / 116.5 / 299.1 / 39.5 / 32.4 / 35.2 / 29.0
RMSE / 127.2 / 97.9 / 74.6 / 60.9 / 104.4 / 297.7 / 76.8 / 91.1 / 100.2 / 100.3
CP / 0.00 / 0.00 / 0.36 / 0.99 / 0.93 / 0.84 / 0.79 / 0.55 / 0.45 / 0.36
P-value / < 10-3 / < 10-3 / 0.125 / 0.333 / 0.619 / 0.713 / − / − / − / −
Selection / 0.00 / 0.34 / 0.45 / 0.22 / 0.00 / 0.00 / − / − / − / −
Estimate / 115.5 / 144.8 / 164.4 / 187.2 / 217.4 / 342.5 / 174.4 / 135.1 / 135.1 / 135
/ 7.7 / 12.1 / 22.5 / 44.3 / 94.3 / 263.5 / 59.7 / 17.2 / 19.3 / 19.3
/ 6.2 / 11.8 / 22.3 / 43.8 / 105.8 / 281.6 / 48.9 / 17.5 / 23.1 / 20.7
RMSE / 93.8 / 65.4 / 49.9 / 49.3 / 94.6 / 295.3 / 68.9 / 75.9 / 76.4 / 76.5
CP / 0.00 / 0.01 / 0.68 / 0.98 / 0.91 / 0.80 / 0.79 / 0.19 / 0.37 / 0.31
P-value / < 10-3 / < 10-3 / 0.225 / 0.385 / 0.649 / 0.731 / − / − / − / −
Selection / 0.00 / 0.54 / 0.27 / 0.14 / 0.04 / 0.00 / − / − / − / −
Estimate / 154.3 / 173.9 / 184.0 / 195.9 / 221.3 / 307.5 / 190.3 / 151.6 / 165.1 / 163.7
/ 6.4 / 9.7 / 17.7 / 31.8 / 69.8 / 186.7 / 50.7 / 8.1 / 14.9 / 13.9
/ 5.2 / 9.4 / 17.1 / 31.2 / 73.6 / 192.6 / 46.8 / 7.2 / 18.9 / 15.0
RMSE / 55.1 / 36.4 / 30.6 / 34.4 / 70.9 / 211.1 / 53.9 / 58.0 / 46.4 / 47.4
CP / 0.00 / 0.15 / 0.86 / 0.96 / 0.87 / 0.77 / 0.81 / 0.01 / 0.56 / 0.43
P-value / < 10-3 / < 10-3 / 0.357 / 0.461 / 0.645 / 0.726 / − / − / − / −
Selection / 0.00 / 0.73 / 0.15 / 0.08 / 0.04 / 0.00 / − / − / − / −
Estimate / 167.4 / 183.6 / 192.8 / 203.7 / 224.9 / 293.0 / 200.5 / 162.9 / 178.7 / 175.9
/ 5.8 / 8.5 / 15.2 / 26.6 / 55.7 / 145.2 / 57.8 / 6.84 / 15.9 / 13.0
/ 4.72 / 8.4 / 15.4 / 27.7 / 63.9 / 159.9 / 56.6 / 6.0 / 20.9 / 15.1
RMSE / 41.9 / 26.8 / 22.2 / 27.1 / 57.9 / 167.7 / 58.4 / 46.6 / 34.2 / 35.6
CP / 0.00 / 0.35 / 0.94 / 0.96 / 0.87 / 0.77 / 0.88 / 0.03 / 0.76 / 0.62
P-value / < 10-3 / < 10-3 / 0.347 / 0.437 / 0.639 / 0.716 / − / − / − / −
Selection / 0.00 / 0.74 / 0.14 / 0.06 / 0.04 / 0.01 / − / − / − / −
Estimate / 182.5 / 195.3 / 202.8 / 211.4 / 228.4 / 284.9 / 208.8 / 179.3 / 193.1 / 189.7
/ 5.2 / 7.6 / 13.4 / 22.9 / 46.9 / 125.1 / 49.6 / 7.0 / 16.8 / 11.9
/ 4.2 / 7.5 / 13.4 / 23.6 / 54.1 / 133.4 / 52.9 / 7.7 / 21.3 / 14.5
RMSE / 27.0 / 15.6 / 14.7 / 22.9 / 50.8 / 146.3 / 49.5 / 30.5 / 23.1 / 22.7
CP / 0.01 / 0.72 / 0.98 / 0.93 / 0.86 / 0.77 / 0.91 / 0.37 / 0.92 / 0.83
P-value / < 10-3 / < 10-3 / 0.372 / 0.459 / 0.661 / 0.726 / − / − / − / −
Selection / 0.00 / 0.78 / 0.13 / 0.04 / 0.04 / 0.01 / − / − / − / −
Estimate / 194.4 / 204.2 / 209.8 / 215.9 / 229.6 / 273.9 / 213.5 / 190.8 / 203.5 / 199.6
/ 4.5 / 6.3 / 11.0 / 18.3 / 37.7 / 99.6 / 34.2 / 5.9 / 13.7 / 9.9
/ 3.7 / 6.4 / 11.4 / 19.1 / 42.2 / 104.2 / 47.3 / 10.7 / 19.5 / 13.1
RMSE / 15.3 / 7.9 / 11.0 / 19.5 / 42.9 / 118.9 / 34.5 / 19.2 / 14.7 / 13.6
CP / 0.21 / 0.94 / 0.94 / 0.87 / 0.80 / 0.72 / 0.95 / 0.66 / 0.97 / 0.93
P-value / < 10-3 / < 10-3 / 0.400 / 0.474 / 0.662 / 0.725 / − / − / − / −
Selection / 0.00 / 0.80 / 0.10 / 0.05 / 0.04 / 0.01 / − / − / − / −
Estimate / 200.9 / 208.5 / 211.6 / 215.2 / 225.0 / 264.7 / 210.5 / 196.3 / 207.0 / 203.7
/ 3.9 / 5.3 / 9.0 / 14.3 / 29.5 / 83.9 / 9.5 / 4.0 / 11.2 / 8.6
/ 3.3 / 5.6 / 9.7 / 15.5 / 32.7 / 80.3 / 10.9 / 6.3 / 15.0 / 10.2
RMSE / 9.0 / 5.3 / 9.4 / 15.6 / 33.5 / 100.8 / 9.6 / 13.4 / 11.4 / 10.1
CP / 0.65 / 0.96 / 0.91 / 0.85 / 0.79 / 0.70 / 0.94 / 0.62 / 0.98 / 0.96
P-value / 0.001 / 0.001 / 0.478 / 0.519 / 0.701 / 0.706 / − / − / − / −
Selection / 0.00 / 0.88 / 0.07 / 0.05 / 0.00 / 0.00 / − / − / − / −
Estimate / 204.8 / 210.7 / 212.1 / 214.1 / 220.5 / 246.1 / 211.7 / 200.8 / 208.7 / 205.6
/ 3.3 / 4.5 / 7.5 / 11.2 / 21.8 / 59.1 / 6.8 / 3.3 / 7.9 / 5.8
/ 2.9 / 4.8 / 8.3 / 12.7 / 25.1 / 57.6 / 8.5 / 4.9 / 11.4 / 7.9
RMSE / 5.3 / 4.8 / 8.1 / 12.3 / 24.6 / 69.8 / 7.4 / 8.9 / 7.9 / 6.6
CP / 0.90 / 0.90 / 0.87 / 0.82 / 0.80 / 0.71 / 0.95 / 0.63 / 0.97 / 0.96
P-value / 0.002 / 0.003 / 0.532 / 0.576 / 0.740 / 0.730 / − / − / − / −
Selection / 0.00 / 0.94 / 0.03 / 0.03 / 0.00 / 0.00 / − / − / − / −
S1
Web Table 3: Comparison of various estimators for the number of shared species in Pasoh and Lambir plots where the quadrat size ism.
Estimate / 78.2 / 107.5 / 135.3 / 170.5 / 206.5 / 322.1 / 148.1 / 133.0 / 109.3 / 109.0/ 8.2 / 13.2 / 23.8 / 46.8 / 101.4 / 256.2 / 45.9 / 36.0 / 26.7 / 25.8
/ 6.0 / 12.1 / 23.2 / 46.9 / 112.5 / 283.5 / 39.7 / 36.1 / 35.7 / 29.4
RMSE / 131.1 / 102.3 / 77.4 / 60.6 / 101.4 / 279.9 / 76.2 / 84.1 / 103.2 / 103.3
CP / 0.00 / 0.00 / 0.32 / 0.98 / 0.94 / 0.84 / 0.80 / 0.64 / 0.45 / 0.36
P-value / < 10-3 / < 10-3 / 0.097 / 0.290 / 0.596 / 0.700 / − / − / − / −
Selection / 0.00 / 0.26 / 0.46 / 0.28 / 0.00 / 0.00 / − / − / − / −
Estimate / 111.8 / 141.2 / 161.2 / 183.8 / 209.8 / 323.6 / 167.9 / 136.7 / 131.4 / 131.3
/ 7.9 / 12.4 / 22.9 / 44.6 / 94.0 / 258.7 / 51.8 / 18.4 / 19.6 / 19.4
/ 6.13 / 11.8 / 22.2 / 43.6 / 104.1 / 267.4 / 46.4 / 19.1 / 22.6 / 20.6
RMSE / 97.5 / 68.9 / 53.0 / 51.2 / 94.0 / 282.9 / 66.1 / 74.6 / 80.1 / 80.1
CP / 0.00 / 0.00 / 0.63 / 0.98 / 0.92 / 0.83 / 0.80 / 0.26 / 0.34 / 0.28
P-value / < 10-3 / < 10-3 / 0.219 / 0.400 / 0.662 / 0.744 / − / − / − / −
Selection / 0.00 / 0.53 / 0.31 / 0.14 / 0.02 / 0.00 / − / − / − / −
Estimate / 153.1 / 173.6 / 184.0 / 196.7 / 221.7 / 311.6 / 188.0 / 150.9 / 163.5 / 162.4
/ 6.6 / 9.8 / 17.8 / 31.9 / 68.3 / 188.1 / 43.0 / 8.3 / 14.6 / 13.4
/ 5.3 / 9.6 / 17.5 / 32.2 / 76.4 / 202.5 / 45.5 / 7.4 / 17.9 / 14.6
RMSE / 56.2 / 36.7 / 30.7 / 34.2 / 69.5 / 214.3 / 47.8 / 58.7 / 47.8 / 48.5
CP / 0.00 / 0.16 / 0.87 / 0.96 / 0.88 / 0.78 / 0.83 / 0.01 / 0.52 / 0.4
P-value / < 10-3 / < 10-3 / 0.353 / 0.452 / 0.645 / 0.720 / − / − / − / −
Selection / 0.00 / 0.75 / 0.13 / 0.09 / 0.03 / 0.00 / − / − / − / −
Estimate / 166.7 / 183.4 / 192.4 / 203.3 / 223.7 / 294.8 / 198.3 / 162.5 / 176.8 / 174.5
/ 5.9 / 8.8 / 15.8 / 27.7 / 58.3 / 154.2 / 52.9 / 7.1 / 14.8 / 12.5
/ 4.8 / 8.6 / 15.7 / 28.3 / 65.2 / 162.3 / 54.6 / 6.1 / 19.6 / 14.4
RMSE / 42.7 / 27.1 / 22.9 / 28.3 / 60.1 / 176.4 / 53.9 / 47.1 / 35.4 / 36.7
CP / 0.00 / 0.36 / 0.93 / 0.96 / 0.87 / 0.77 / 0.86 / 0.03 / 0.73 / 0.57
P-value / < 10-3 / < 10-3 / 0.360 / 0.452 / 0.648 / 0.721 / − / − / − / −
Selection / 0.00 / 0.76 / 0.13 / 0.06 / 0.04 / 0.01 / − / − / − / −
Estimate / 182.2 / 195.4 / 203.0 / 211.7 / 229.5 / 288.5 / 210.2 / 179.1 / 193.4 / 189.8
/ 5.35 / 7.8 / 14.2 / 24.3 / 50.6 / 131.8 / 52.8 / 6.9 / 16.9 / 13.1
/ 4.3 / 7.6 / 13.6 / 24.0 / 54.3 / 136.1 / 52.4 / 7.5 / 21.9 / 14.9
RMSE / 27.3 / 15.7 / 15.4 / 24.5 / 54.6 / 153.9 / 52.8 / 30.6 / 23.0 / 23.2
CP / 0.00 / 0.72 / 0.98 / 0.92 / 0.84 / 0.76 / 0.91 / 0.34 / 0.90 / 0.81
P-value / < 10-3 / < 10-3 / 0.371 / 0.466 / 0.659 / 0.724 / − / − / − / −
Selection / 0.00 / 0.77 / 0.12 / 0.05 / 0.05 / 0.01 / − / − / − / −
Estimate / 194.2 / 204.3 / 209.7 / 215.8 / 229.4 / 277.4 / 215.1 / 190.6 / 203 / 199.4
/ 4.6 / 6.5 / 11.5 / 19.0 / 38.4 / 105.7 / 41.3 / 6.0 / 13.7 / 10.1
/ 3.8 / 6.5 / 11.5 / 19.3 / 42.2 / 107.2 / 45.6 / 9.2 / 19.3 / 12.9
RMSE / 15.5 / 8.1 / 11.5 / 20.2 / 43.5 / 125.8 / 41.7 / 19.4 / 14.9 / 13.9
CP / 0.22 / 0.93 / 0.94 / 0.86 / 0.80 / 0.72 / 0.94 / 0.61 / 0.96 / 0.92
P-value / < 10-3 / < 10-3 / 0.407 / 0.471 / 0.676 / 0.722 / − / − / − / −
Selection / 0.00 / 0.81 / 0.10 / 0.04 / 0.04 / 0.01 / − / − / − / −
Estimate / 193.4 / 201.0 / 208.9 / 212.4 / 216.4 / 226.1 / 211.3 / 196.3 / 207.5 / 203.9
/ 3.9 / 5.5 / 9.8 / 15.5 / 30.0 / 79.8 / 10.4 / 4.1 / 11.0 / 8.1
/ 3.4 / 5.7 / 9.9 / 16.0 / 33.4 / 79.8 / 11.7 / 5.9 / 15.7 / 10.5
RMSE / 8.9 / 5.5 / 10.4 / 17.2 / 34.5 / 94.7 / 10.7 / 13.3 / 11.1 / 9.5
CP / 0.66 / 0.94 / 0.88 / 0.82 / 0.79 / 0.74 / 0.94 / 0.62 / 0.98 / 0.96
P-value / 0.001 / 0.001 / 0.459 / 0.509 / 0.707 / 0.740 / − / − / − / −
Selection / 0.00 / 0.86 / 0.08 / 0.06 / 0.00 / 0.00 / − / − / − / −
Estimate / 204.7 / 210.7 / 212.4 / 214.9 / 223.0 / 250.0 / 211.9 / 200.7 / 209.4 / 206.1
/ 3.3 / 4.5 / 7.8 / 11.7 / 23.1 / 60.0 / 7.6 / 7.3 / 8.9 / 6.1
/ 2.9 / 4.9 / 8.4 / 12.9 / 26.1 / 60.4 / 8.8 / 4.9 / 12.6 / 8.5
RMSE / 5.5 / 4.8 / 8.5 / 13.1 / 27.0 / 72.7 / 8.2 / 11.0 / 8.9 / 6.8
CP / 0.88 / 0.90 / 0.86 / 0.81 / 0.76 / 0.67 / 0.94 / 0.64 / 0.97 / 0.97
P-value / 0.002 / 0.002 / 0.514 / 0.565 / 0.712 / 0.711 / − / − / − / −
Selection / 0.00 / 0.92 / 0.03 / 0.05 / 0.00 / 0.00 / − / − / − / −
S1
Web Table 4: Comparison of various estimators for the number of shared species in Pasoh and Lambir plots where the quadrat size ism.
Estimate / 66.3 / 92.1 / 121.8 / 161.3 / 194.2 / 266.8 / 144.3 / 131.8 / 96.9 / 96.6/ 8.5 / 13.1 / 22.0 / 40.5 / 84.5 / 184.5 / 41.3 / 41.0 / 26.0 / 25.4
/ 5.43 / 10.9 / 20.5 / 39.5 / 90.4 / 207.5 / 37.2 / 50.1 / 35.1 / 28.5
RMSE / 142.9 / 117.7 / 89.9 / 62.6 / 85.8 / 193.3 / 76.8 / 87.4 / 115.0 / 115.2
CP / 0.00 / 0.00 / 0.10 / 0.90 / 0.93 / 0.84 / 0.79 / 0.82 / 0.39 / 0.28
P-value / < 10-3 / < 10-3 / 0.027 / 0.146 / 0.518 / 0.657 / − / − / − / −
Selection / 0.00 / 0.09 / 0.41 / 0.50 / 0.00 / 0.00 / − / − / − / −
Estimate / 105.0 / 135.0 / 159.3 / 187.4 / 211.1 / 307.8 / 173.2 / 147.7 / 125.8 / 125.8
/ 8.6 / 13.3 / 23.4 / 44.2 / 91.9 / 228.8 / 55.1 / 24.3 / 20.0 / 20.0
/ 6.1 / 11.9 / 22.1 / 42.9 / 102.1 / 249.9 / 45.9 / 23.2 / 22.7 / 20.8
RMSE / 104.4 / 75.2 / 54.9 / 49.2 / 91.9 / 249.1 / 65.7 / 66.0 / 85.6 / 85.6
CP / 0.00 / 0.00 / 0.59 / 0.98 / 0.92 / 0.84 / 0.83 / 0.46 / 0.30 / 0.25
P-value / < 10-3 / < 10-3 / 0.132 / 0.325 / 0.626 / 0.729 / − / − / − / −
Selection / 0.00 / 0.34 / 0.40 / 0.23 / 0.03 / 0.00 / − / − / − / −
Estimate / 150.8 / 173 / 184.6 / 198.5 / 223.6 / 319.0 / 188.7 / 150.7 / 161.3 / 160.7
/ 7.2 / 10.5 / 18.6 / 33.7 / 72.3 / 196.6 / 44.1 / 9.3 / 14.9 / 14.3
/ 5.5 / 9.9 / 18.2 / 33.5 / 79.2 / 206.2 / 44.6 / 8.2 / 17.4 / 14.9
RMSE / 58.6 / 37.5 / 30.7 / 35.3 / 73.7 / 225.3 / 48.5 / 59.1 / 49.9 / 50.4
CP / 0.00 / 0.17 / 0.87 / 0.97 / 0.88 / 0.77 / 0.82 / 0.01 / 0.47 / 0.39
P-value / < 10-3 / < 10-3 / 0.338 / 0.431 / 0.641 / 0.716 / − / − / − / −
Selection / 0.00 / 0.72 / 0.15 / 0.10 / 0.03 / 0.00 / − / − / − / −
Estimate / 165.2 / 183.2 / 192.5 / 204.0 / 224.8 / 300.2 / 198.4 / 161.2 / 174.7 / 173.1
/ 6.2 / 9.1 / 16.1 / 28.3 / 59.7 / 160.3 / 55.1 / 7.2 / 13.8 / 12.7
/ 5.0 / 9.0 / 16.3 / 29.6 / 68.7 / 172.4 / 52.0 / 6.2 / 18.2 / 14.2
RMSE / 44.3 / 27.3 / 23.0 / 28.7 / 61.8 / 184.4 / 56.1 / 48.4 / 37.0 / 38.1
CP / 0.00 / 0.38 / 0.94 / 0.96 / 0.87 / 0.77 / 0.87 / 0.01 / 0.66 / 0.54
P-value / < 10-3 / < 10-3 / 0.362 / 0.456 / 0.652 / 0.722 / − / − / − / −
Selection / 0.00 / 0.77 / 0.13 / 0.06 / 0.03 / 0.01 / − / − / − / −
Estimate / 180.9 / 194.9 / 202.6 / 211.8 / 231.4 / 294.9 / 209.0 / 178.2 / 191.2 / 188.3
/ 5.6 / 8.2 / 14.7 / 25.4 / 52.8 / 137.7 / 50.3 / 7.2 / 15.5 / 12.2
/ 4.4 / 7.8 / 14.2 / 25.3 / 57.9 / 144.6 / 54.6 / 7.0 / 20.5 / 14.4
RMSE / 28.6 / 16.3 / 16.0 / 25.6 / 57.3 / 162.3 / 50.3 / 31.7 / 23.6 / 24.1
CP / 0.01 / 0.73 / 0.98 / 0.92 / 0.83 / 0.76 / 0.91 / 0.28 / 0.88 / 0.79
P-value / < 10-3 / < 10-3 / 0.378 / 0.459 / 0.653 / 0.722 / − / − / − / −
Selection / 0.00 / 0.77 / 0.13 / 0.04 / 0.05 / 0.01 / − / − / − / −
Estimate / 193.7 / 204.2 / 209.7 / 215.9 / 229.7 / 280.1 / 213.4 / 190.4 / 202.3 / 199.0
/ 4.7 / 6.7 / 11.7 / 19.3 / 39.2 / 108.8 / 36.9 / 5.8 / 13.0 / 9.8
/ 3.8 / 6.6 / 11.8 / 20.0 / 44.3 / 113.4 / 45.3 / 7.7 / 19.4 / 12.9
RMSE / 16.0 / 8.2 / 11.7 / 20.5 / 44.3 / 129.9 / 37.2 / 19.5 / 14.6 / 14.0
CP / 0.19 / 0.94 / 0.94 / 0.87 / 0.81 / 0.74 / 0.95 / 0.61 / 0.95 / 0.91
P-value / < 10-3 / < 10-3 / 0.409 / 0.482 / 0.688 / 0.735 / − / − / − / −
Selection / 0.00 / 0.81 / 0.11 / 0.04 / 0.03 / 0.01 / − / − / − / −
Estimate / 200.6 / 208.8 / 212.4 / 216.4 / 227.0 / 265.7 / 211.3 / 196.0 / 207.2 / 203.9
/ 4.1 / 5.7 / 9.8 / 15.4 / 30.3 / 83.7 / 10.8 / 4.2 / 11.3 / 8.4
/ 3.4 / 5.8 / 10.1 / 16.2 / 34.4 / 86.3 / 11.7 / 5.5 / 15.8 / 10.8
RMSE / 9.3 / 5.7 / 10.4 / 17.1 / 35.2 / 101.1 / 11.0 / 13.6 / 11.4 / 9.9
CP / 0.63 / 0.95 / 0.88 / 0.82 / 0.78 / 0.71 / 0.94 / 0.60 / 0.98 / 0.96
P-value / < 10-3 / 0.001 / 0.460 / 0.506 / 0.694 / 0.725 / − / − / − / −
Selection / 0.00 / 0.87 / 0.07 / 0.06 / 0.00 / 0.00 / − / − / − / −
Estimate / 204.7 / 210.9 / 212.5 / 214.8 / 222.1 / 248.7 / 212.2 / 200.4 / 209.2 / 206.0
/ 3.4 / 4.6 / 8.0 / 12.0 / 23.4 / 61.6 / 8.0 / 3.1 / 9.6 / 6.5
/ 3.0 / 4.9 / 8.6 / 13.1 / 26.3 / 60.0 / 9.0 / 4.2 / 12.5 / 8.3
RMSE / 5.5 / 5.0 / 8.7 / 13.3 / 26.8 / 73.3 / 8.6 / 9.2 / 9.6 / 7.2
CP / 0.88 / 0.90 / 0.86 / 0.82 / 0.80 / 0.71 / 0.94 / 0.62 / 0.97 / 0.97
P-value / 0.002 / 0.002 / 0.514 / 0.574 / 0.728 / 0.724 / − / − / − / −
Selection / 0.00 / 0.92 / 0.05 / 0.03 / 0.00 / 0.00 / − / − / − / −
S1
S5. Simulation Study
In order to evaluate performance of the jackknife estimators in finite samples, we further conducted a simulation study in which samples of abundance data and incidence data were generated from postulated communities. We considered six communities where each community consists of 500 species. Let denote theprobability of the i-th species can be observed/detectedfor a chosen individual (abundance data) or in a quadrat (incidence data). The settings of were considered as follows:
C1: with CV=0.
C2: is a random sample from a uniform distribution with CV=0.55.
C3: is a random sample from a beta distribution with CV=1.06.
C4: with CV=1.45
C5: with CV=2.45
C6: with CV=4.10
c is a normalizing factor so that for the abundance-based data and c = 1 for the incidence-based data.
For each data type, we consideredall 21 combined pairs out of the six communities, namely, C1vs.C1,C1vs.C2, …,C6vs.C6.To make the observed number of shared species sufficient for estimation of shared species richness, we considered taking 20 quadrats for C1 to C3 and 40 quadrats for C4 to C6 for incidence data while, in the scenario of abundance data, we sampled 500 individuals for each sample. For each paired communities, the first 200 species were assumed to be the shared species, and independently generated1,000 datasets.
As in Empirical Test, we calculated the jackknife estimators, andaswell as the existing methods,and. All simulation results are displayed in Web Table 5 (abundance-based data) and Web Table 6 (incidence-based data) of Appendix S4. In Web Table 5, the proposed jackknife estimator showed a positive bias when one of the communities was a homogeneous population (C1). Nevertheless, it worked reasonably well in most other cases too. Overall, when neither community is homogeneous, all estimators exhibit similar performance characteristics that we found in the empirical study.
S1
Web Table 5: Comparison of various estimators for the number of shared species when the sample size is of 500 individuals.
C1 vs. C1 ()Estimate / 126.04 / 199.26 / 272.5 / 371.97 / 394.57 / 571.25 / 271.22 / 272.4 / 209.46 / 209.7
/ 10.84 / 19.53 / 38.33 / 82.33 / 188.85 / 459.37 / 40.5 / 107.58 / 45.33 / 47.13
/ 9.09 / 19.95 / 39.69 / 84.12 / 208.5 / 508.16 / 43.31 / 203.9 / 50.69 / 50.17
RMSE / 74.75 / 19.53 / 82 / 190.64 / 271.08 / 590.45 / 81.92 / 129.63 / 46.28 / 48.09
CP / 0.00 / 0.95 / 0.40 / 0.24 / 0.63 / 0.70 / 0.46 / 0.88 / 0.97 / 0.97
P-value / < 10-3 / < 10-3 / 0.010 / 0.234 / 0.583 / 0.720 / − / − / − / −
Selection / 0.00 / 0.03 / 0.97 / 0.00 / 0.00 / 0.00 / − / − / − / −
C1 vs. C2 ()
Estimate / 109.39 / 170.35 / 230.36 / 311.87 / 344.45 / 519.43 / 227.85 / 228.21 / 177.48 / 177.72
/ 10.32 / 18.40 / 35.34 / 75.28 / 174.14 / 439.50 / 38.25 / 91.46 / 39.45 / 40.39
/ 8.32 / 18.13 / 35.97 / 76.00 / 189.30 / 467.35 / 39.71 / 123.68 / 47.00 / 45.58
RMSE / 91.20 / 34.88 / 46.58 / 134.82 / 226.19 / 543.14 / 47.30 / 95.66 / 45.41 / 46.11
CP / 0.00 / 0.72 / 0.80 / 0.49 / 0.69 / 0.73 / 0.84 / 0.95 / 0.93 / 0.92
P-value / < 10-3 / < 10-3 / 0.021 / 0.272 / 0.575 / 0.719 / − / − / − / −
Selection / 0.00 / 0.07 / 0.93 / 0.00 / 0.00 / 0.00 / − / − / − / −
C1 vs. C3 ()
Estimate / 83.91 / 128.62 / 172.67 / 232.01 / 266.81 / 419.73 / 169.18 / 173.75 / 136.99 / 137.28
/ 8.84 / 15.75 / 30.47 / 64.21 / 142.05 / 359.92 / 34.42 / 72.72 / 35.59 / 36.12
/ 7.16 / 15.44 / 30.54 / 64.19 / 156.99 / 387.45 / 33.58 / 105.91 / 47.54 / 42.67
RMSE / 116.43 / 73.10 / 40.92 / 71.72 / 156.91 / 421.54 / 46.19 / 77.28 / 72.36 / 72.36
CP / 0.00 / 0.03 / 0.91 / 0.86 / 0.82 / 0.78 / 0.89 / 0.97 / 0.76 / 0.72
P-value / < 10-3 / < 10-3 / 0.048 / 0.222 / 0.605 / 0.719 / − / − / − / −
Selection / 0.00 / 0.16 / 0.84 / 0.00 / 0.00 / 0.00 / − / − / − / −
C1vs. C4 ()
Estimate / 134.25 / 199.14 / 257.79 / 331.46 / 348.29 / 502.13 / 255.07 / 260.70 / 197.06 / 197.06
/ 10.59 / 18.52 / 35.52 / 73.53 / 163.50 / 413.98 / 38.90 / 76.74 / 37.26 / 37.26
/ 8.73 / 18.44 / 36.08 / 75.40 / 190.39 / 470.70 / 39.38 / 108.48 / 40.17 / 39.60
RMSE / 66.60 / 18.53 / 67.83 / 150.61 / 220.67 / 512.33 / 67.41 / 97.82 / 37.36 / 37.36
CP / 0.00 / 0.95 / 0.52 / 0.34 / 0.71 / 0.74 / 0.58 / 0.98 / 0.95 / 0.96
P-value / < 10-3 / < 10-3 / 0.026 / 0.227 / 0.619 / 0.749 / − / − / − / −
Selection / 0.00 / 0.09 / 0.91 / 0.00 / 0.00 / 0..00 / − / − / − / −
C1vs. C5 ()
Estimate / 123.58 / 185.89 / 246.88 / 326.16 / 355.31 / 516.27 / 245.66 / 271.37 / 193.15 / 193.33
/ 10.53 / 18.53 / 35.59 / 74.16 / 168.92 / 413.48 / 37.85 / 95.68 / 40.37 / 41.18
/ 8.52 / 18.14 / 35.71 / 74.68 / 185.95 / 456.36 / 39.04 / 169.55 / 45.32 / 44.60
RMSE / 77.14 / 23.28 / 58.85 / 146.32 / 229.41 / 520.41 / 59.30 / 119.33 / 40.93 / 41.70
CP / 0.00 / 0.90 / 0.64 / 0.38 / 0.68 / 0.71 / 0.69 / 0.96 / 0.95 / 0.95
P-value / < 10-3 / < 10-3 / 0.018 / 0.271 / 0.580 / 0.722 / − / − / − / −
Selection / 0.00 / 0.05 / 0.95 / 0.00 / 0.00 / 0.00 / − / − / − / −
C1vs. C6 ()
Estimate / 112.84 / 172.31 / 233.82 / 316.68 / 354.30 / 512.28 / 232.96 / 264.13 / 187.59 / 187.84
/ 10.04 / 17.80 / 34.43 / 71.92 / 161.77 / 389.39 / 35.96 / 100.14 / 44.08 / 44.46
/ 8.28 / 17.81 / 35.26 / 73.87 / 183.78 / 461.26 / 38.48 / 171.03 / 52.15 / 49.60
RMSE / 87.73 / 32.91 / 48.25 / 137.04 / 223.50 / 498.99 / 48.77 / 118.87 / 45.77 / 46.07
CP / 0.00 / 0.76 / 0.79 / 0.42 / 0.68 / 0.72 / 0.82 / 0.93 / 0.96 / 0.95
P-value / < 10-3 / < 10-3 / 0.017 / 0.240 / 0.579 / 0.701 / − / − / − / −
Selection / 0.00 / 0.04 / 0.96 / 0.00 / 0.00 / 0.00 / − / − / − / −
C2vs. C2 ()
Estimate / 109.25 / 164.14 / 209.73 / 274.87 / 300.15 / 483.31 / 203.00 / 165.93 / 151.95 / 151.95
/ 9.58 / 17.19 / 33.84 / 71.85 / 155.17 / 408.20 / 39.67 / 46.62 / 30.45 / 30.46
/ 7.94 / 17.11 / 33.76 / 71.19 / 174.35 / 450.69 / 37.38 / 104.58 / 33.81 / 33.38
RMSE / 91.25 / 39.76 / 35.19 / 103.74 / 184.62 / 496.71 / 39.76 / 57.73 / 56.88 / 56.88
CP / 0.00 / 0.58 / 0.93 / 0.66 / 0.78 / 0.76 / 0.93 / 0.99 / 0.74 / 0.75
P-value / < 10-3 / < 10-3 / 0.070 / 0.229 / 0.628 / 0.722 / − / − / − / −
Selection / 0.00 / 0.25 / 0.75 / 0.00 / 0.00 / 0..00 / − / − / − / −
C2vs. C3 ()
Estimate / 72.26 / 109.35 / 144.76 / 192.79 / 231.95 / 387.78 / 139.96 / 142.31 / 114.85 / 114.74
/ 8.18 / 14.38 / 27.75 / 58.36 / 130.50 / 350.57 / 31.82 / 57.97 / 33.03 / 32.30
/ 6.54 / 14.02 / 27.64 / 57.97 / 140.49 / 357.66 / 30.25 / 99.19 / 44.99 / 38.43
RMSE / 128.00 / 91.78 / 61.81 / 58.77 / 134.29 / 397.54 / 67.95 / 81.76 / 91.33 / 91.17
CP / 0.00 / 0.00 / 0.68 / 0.96 / 0.88 / 0.79 / 0.72 / 0.98 / 0.61 / 0.58
P-value / < 10-3 / < 10-3 / 0.081 / 0.272 / 0.609 / 0.707 / − / − / − / −
Selection / 0.00 / 0.25 / 0.75 / 0.00 / 0.00 / 0..00 / − / − / − / −
C2vs. C4 ()
Estimate / 116.50 / 170.81 / 218.21 / 277.78 / 302.09 / 464.87 / 214.16 / 222.65 / 166.75 / 166.83
/ 9.20 / 16.07 / 31.48 / 66.49 / 150.93 / 384.28 / 36.07 / 68.04 / 31.84 / 32.05
/ 8.01 / 16.83 / 32.83 / 68.48 / 170.42 / 423.97 / 36.11 / 116.58 / 36.96 / 35.90
RMSE / 84.01 / 33.31 / 36.35 / 102.30 / 182.15 / 466.56 / 38.73 / 71.68 / 46.02 / 46.11
CP / 0.00 / 0.70 / 0.90 / 0.62 / 0.77 / 0.75 / 0.90 / 0.86 / 0.86 / 0.85
P-value / < 10-3 / < 10-3 / 0.049 / 0.254 / 0.607 / 0.732 / − / − / − / −
Selection / 0.00 / 0.16 / 0.84 / 0.00 / 0.00 / 0..00 / − / − / − / −
C2vs. C5 ()
Estimate / 106.88 / 159.00 / 209.06 / 275.07 / 309.73 / 468.00 / 206.17 / 224.26 / 163.62 / 163.96
/ 9.73 / 17.12 / 33.03 / 68.81 / 154.05 / 387.62 / 36.38 / 81.67 / 37.20 / 37.90
/ 7.80 / 16.55 / 32.52 / 67.83 / 169.62 / 421.31 / 35.90 / 138.63 / 42.87 / 40.53
RMSE / 93.63 / 44.43 / 34.23 / 101.81 / 189.08 / 471.09 / 36.89 / 85.16 / 52.02 / 52.28
CP / 0.00 / 0.45 / 0.93 / 0.66 / 0.76 / 0.74 / 0.92 / 0.97 / 0.87 / 0.86
P-value / < 10-3 / < 10-3 / 0.037 / 0.199 / 0.584 / 0.719 / − / − / − / −
Selection / 0.00 / 0.11 / 0.89 / 0.00 / 0.00 / 0..00 / − / − / − / −
C2vs. C6 ()
Estimate / 97.44 / 147.20 / 198.45 / 266.34 / 305.43 / 447.04 / 196.45 / 232.48 / 162.22 / 162.88
/ 9.61 / 16.76 / 31.57 / 64.18 / 146.17 / 362.32 / 34.29 / 92.70 / 40.85 / 42.09
/ 7.59 / 146.24 / 32.08 / 67.03 / 167.54 / 412.89 / 35.13 / 146.60 / 52.33 / 46.97
RMSE / 103.00 / 55.39 / 31.59 / 92.28 / 180.16 / 438.37 / 34.46 / 98.18 / 55.63 / 56.10
CP / 0.00 / 0.23 / 0.96 / 0.72 / 0.77 / 0.77 / 0.94 / 0.94 / 0.89 / 0.87
P-value / < 10-3 / < 10-3 / 0.026 / 0.209 / 0.573 / 0.715 / − / − / − / −
Selection / 0.00 / 0.08 / 0.92 / 0.00 / 0.00 / 0..00 / − / − / − / −
C3vs. C3 ()
Estimate / 82.21 / 118.04 / 145.60 / 189.94 / 225.98 / 398.52 / 135.59 / 105.93 / 106.26 / 106.32
/ 7.56 / 13.66 / 26.91 / 56.29 / 121.55 / 343.98 / 30.47 / 24.93 / 22.45 / 22.55
/ 6.42 / 13.78 / 27.23 / 57.11 / 137.12 / 370.17 / 27.73 / 30.14 / 27.22 / 25.91
RMSE / 118.04 / 83.09 / 60.69 / 57.16 / 124.23 / 397.01 / 71.25 / 97.32 / 96.39 / 96.36
CP / 0.00 / 0.00 / 0.70 / 0.97 / 0.90 / 0.80 / 0.57 / 0.46 / 0.34 / 0.33
P-value / < 10-3 / < 10-3 / 0.174 / 0.289 / 0.631 / 0.706 / − / − / − / −
Selection / 0.00 / 0.55 / 0.45 / 0.00 / 0.00 / 0..00 / − / − / − / −
C3vs. C4 ()
Estimate / 88.38 / 127.59 / 163.55 / 210.37 / 247.25 / 391.99 / 158.57 / 154.61 / 129.72 / 129.87
/ 8.57 / 14.79 / 28.20 / 57.57 / 129.51 / 333.61 / 32.72 / 48.13 / 31.75 / 31.90
/ 6.80 / 14.26 / 27.88 / 57.83 / 143.20 / 360.61 / 30.41 / 66.17 / 39.35 / 35.00
RMSE / 111.95 / 73.90 / 46.08 / 58.46 / 137.80 / 384.76 / 52.78 / 66.14 / 77.11 / 77.04
CP / 0.00 / 0.02 / 0.85 / 0.92 / 0.85 / 0.79 / 0.81 / 0.93 / 0.65 / 0.61
P-value / < 10-3 / < 10-3 / 0.079 / 0.280 / 0.607 / 0.714 / − / − / − / −
Selection / 0.00 / 0.26 / 0.74 / 0.00 / 0.00 / 0..00 / − / − / − / −
C3vs. C5 ()
Estimate / 81.35 / 118.53 / 154.50 / 201.50 / 240.60 / 370.81 / 150.06 / 152.48 / 125.23 / 125.33
/ 8.53 / 14.81 / 28.17 / 58.17 / 127.21 / 315.19 / 32.38 / 54.70 / 33.73 / 33.35
/ 6.60 / 13.92 / 27.30 / 56.70 / 137.29 / 343.65 / 29.79 / 83.85 / 44.82 / 37.18
RMSE / 118.96 / 82.80 / 53.51 / 58.16 / 133.47 / 358.36 / 59.51 / 72.44 / 82.02 / 81.77
CP / 0.00 / 0.00 / 0.76 / 0.94 / 0.86 / 0.80 / 0.77 / 0.96 / 0.67 / 0.62
P-value / < 10-3 / < 10-3 / 0.072 / 0.268 / 0.607 / 0.720 / − / − / − / −
Selection / 0.00 / 0.24 / 0.76 / 0.00 / 0.00 / 0..00 / − / − / − / −
C3vs. C6 ()
Estimate / 74.30 / 109.60 / 145.47 / 193.85 / 235.63 / 372.02 / 141.46 / 153.47 / 119.87 / 119.79
/ 7.95 / 13.75 / 25.97 / 54.17 / 120.46 / 303.03 / 29.88 / 63.51 / 34.10 / 33.39
/ 6.41 / 13.61 / 26.83 / 55.81 / 135.07 / 337.07 / 29.41 / 92.03 / 49.68 / 39.64
RMSE / 125.95 / 91.44 / 60.39 / 54.49 / 125.56 / 348.32 / 65.72 / 78.70 / 87.07 / 86.88
CP / 0.00 / 0.00 / 0.68 / 0.96 / 0.89 / 0.79 / 0.74 / 0.97 / 0.69 / 0.61
P-value / < 10-3 / < 10-3 / 0.059 / 0.237 / 0.584 / 0.702 / − / − / − / −
Selection / 0.00 / 0.20 / 0.80 / 0.00 / 0.00 / 0..00 / − / − / − / −
C4vs. C4 ()
Estimate / 146.37 / 202.83 / 243.45 / 302.98 / 325.88 / 501.83 / 234.14 / 210.80 / 179.54 / 179.64
/ 9.42 / 17.03 / 33.61 / 70.70 / 154.13 / 425.75 / 39.26 / 33.07 / 25.94 / 26.22
/ 8.09 / 17.30 / 34.0 / 71.59 / 173.35 / 452.09 / 36.61 / 28.01 / 28.15 / 27.77
RMSE / 54.45 / 17.26 / 54.93 / 124.90 / 198.94 / 521.71 / 52.02 / 34.77 / 33.03 / 33.19
CP / 0.00 / 0.94 / 0.64 / 0.49 / 0.74 / 0.71 / 0.75 / 0.89 / 0.89 / 0.89
P-value / < 10-3 / < 10-3 / 0.119 / 0.270 / 0.648 / 0.744 / − / − / − / −
Selection / 0.00 / 0.39 / 0.62 / 0.00 / 0.00 / 0..00 / − / − / − / −
C4vs. C5 ()
Estimate / 135.90 / 190.25 / 234.63 / 300.66 / 332.91 / 512.82 / 227.36 / 202.28 / 174.81 / 174.91
/ 9.48 / 17.13 / 33.14 / 68.93 / 153.01 / 401.74 / 38.97 / 34.79 / 28.37 / 28.46
/ 7.90 / 17.03 / 33.67 / 740.82 / 176.85 / 459.67 / 37.06 / 30.21 / 31.78 / 31.13
RMSE / 64.80 / 19.70 / 47.92 / 121.98 / 202.62 / 509.01 / 47.60 / 34.85 / 37.93 / 37.93
CP / 0.00 / 0.92 / 0.74 / 0.50 / 0.71 / 0.71 / 0.80 / 0.92 / 0.89 / 0.89
P-value / < 10-3 / < 10-3 / 0.080 / 0.212 / 0.602 / 0.712 / − / − / − / −
Selection / 0.00 / 0.28 / 0.72 / 0.00 / 0.00 / 0..00 / − / − / − / −
C4vs. C6 ()
Estimate / 125.20 / 176.68 / 221.85 / 288.64 / 331.04 / 510.84 / 216.12 / 195.41 / 169.06 / 169.59
/ 9.03 / 16.28 / 31.85 / 65.68 / 147.71 / 400.11 / 37.44 / 37.85 / 31.64 / 32.73
/ 7.68 / 16.59 / 32.88 / 69.09 / 170.21 / 440.68 / 36.31 / 32.34 / 35.59 / 34.31
RMSE / 75.34 / 28.44 / 38.61 / 110.31 / 197.41 / 506.51 / 40.74 / 38.10 / 44.25 / 44.66
CP / 0.00 / 0.79 / 0.85 / 0.56 / 0.70 / 0.70 / 0.88 / 0.91 / 0.86 / 0.86
P-value / < 10-3 / < 10-3 / 0.064 / 0.203 / 0.599 / 0.700 / − / − / − / −
Selection / 0.00 / 0.18 / 0.82 / 0.00 / 0.00 / 0..00 / − / − / − / −
C5vs. C5 ()
Estimate / 126.82 / 179.25 / 225.72 / 296.77 / 341.67 / 522.81 / 219.26 / 199.65 / 168.77 / 169.13
/ 9.32 / 16.94 / 32.95 / 69.86 / 153.61 / 395.36 / 38.31 / 39.94 / 29.42 / 30.52
/ 7.72 / 16.79 / 33.39 / 70.29 / 172.90 / 453.31 / 37.08 / 31.97 / 34.25 / 33.50
RMSE / 73.77 / 26.78 / 41.79 / 119.33 / 208.91 / 510.25 / 42.86 / 39.92 / 42.89 / 43.40
CP / 0.00 / 0.83 / 0.83 / 0.51 / 0.69 / 0.68 / 0.86 / 0.90 / 0.88 / 0.87
P-value / < 10-3 / < 10-3 / 0.060 / 0.288 / 0.592 / 0.696 / − / − / − / −
Selection / 0.00 / 0.23 / 0.77 / 0.00 / 0.00 / 0..00 / − / − / − / −
C5vs. C6 ()
Estimate / 117.33 / 167.11 / 213.55 / 284.89 / 331.94 / 504.92 / 208.38 / 191.95 / 161.84 / 162.09
/ 9.00 / 16.37 / 31.90 / 66.11 / 148.00 / 377.60 / 37.04 / 39.49 / 31.23 / 31.56
/ 7.51 / 16.39 / 32.69 / 68.79 / 170.60 / 442.99 / 36.36 / 33.40 / 37.37 / 35.76
RMSE / 83.16 / 36.74 / 34.64 / 107.58 / 198.22 / 485.19 / 37.95 / 40.29 / 49.30 / 49.32
CP / 0.00 / 0.62 / 0.91 / 0.62 / 0.72 / 0.70 / 0.92 / 0.90 / 0.85 / 0.85
P-value / < 10-3 / < 10-3 / 0.061 / 0.268 / 0.582 / 0.694 / − / − / − / −
Selection / 0.00 / 0.19 / 0.81 / 0.00 / 0.00 / 0..00 / − / − / − / −
C6vs. C6 ()
Estimate / 109.01 / 156.76 / 203.69 / 278.26 / 335.25 / 525.17 / 196.87 / 185.13 / 155.88 / 156.22
/ 8.81 / 16.08 / 31.51 / 67.11 / 158.02 / 402.57 / 37.37 / 42.26 / 34.05 / 34.69
/ 7.33 / 16.10 / 32.27 / 67.88 / 166.81 / 433.20 / 36.11 / 34.94 / 41.83 / 38.78
RMSE / 91.41 / 46.14 / 31.71 / 103.07 / 207.94 / 517.34 / 37.48 / 44.75 / 55.72 / 55.85
CP / 0.00 / 0.39 / 0.96 / 0.64 / 0.68 / 0.67 / 0.95 / 0.89 / 0.84 / 0.83
P-value / < 10-3 / < 10-3 / 0.043 / 0.254 / 0.536 / 0.666 / − / − / − / −
Selection / 0.00 / 0.22 / 0.78 / 0.00 / 0.00 / 0..00 / − / − / − / −
Web Table 6: Comparison of various estimators for the number of shared species when the sample size is 20 quadrats for C1-C3 and 40 quadrats for C4-C6.
C1 vs. C1 ()Estimate / 39.34 / 69.83 / 119.87 / 205.75 / 327.46 / 529.11 / 144.50 / 33.37 / 335.41 / 190.00
/ 7.373 / 14.25 / 27.11 / 53.98 / 117.74 / 268.25 / 67.70 / 32.76 / 306.69 / 118.24
/ 5.63 / 12.96 / 26.84 / 55.48 / 123.57 / 289.29 / 60.58 / 40.26 / 352.93 / 118.08
RMSE / 160.95 / 130.95 / 84.58 / 54.26 / 173.49 / 424.50 / 87.51 / 169.82 / 335.11 / 118.61
CP / 0.00 / 0.00 / 0.36 / 0.96 / 0.65 / 0.50 / 0.94 / 0.30 / 0.91 / 0.92
P-value / < 10-3 / < 10-3 / 0.002 / 0.020 / 0.196 / 0.364 / − / − / − / −
Selection / 0.00 / 0.35 / 0.05 / 0.60 / 0.00 / 0.00 / − / − / − / −
C1 vs. C2 ()
Estimate / 35.43 / 62.41 / 105.52 / 178.22 / 276.76 / 442.27 / 133.00 / 34.87 / 259.46 / 157.01
/ 7.21 / 13.26 / 25.39 / 51.44 / 115.06 / 258.93 / 60.05 / 34.35 / 225.00 / 104.56
/ 5.30 / 12.14 / 25.03 / 51.76 / 116.34 / 270.86 / 55.45 / 40.11 / 274.99 / 98.83
RMSE / 164.73 / 138.23 / 97.82 / 55.84 / 138.27 / 354.50 / 89.96 / 168.66 / 232.62 / 113.00
CP / 0.00 / 0.00 / 0.17 / 0.97 / 0.79 / 0.61 / 0.92 / 0.33 / 0.94 / 0.86
P-value / < 10-3 / < 10-3 / 0.005 / 0.039 / 0.267 / 0.436 / − / − / − / −
Selection / 0.00 / 0.28 / 0.10 / 0.62 / − / − / − / −
C1 vs. C3 ()
Estimate / 76.99 / 108.20 / 150.41 / 197.25 / 258.86 / 360.57 / 189.37 / 210.48 / 206.63 / 192.36
/ 9.21 / 14.05 / 23.96 / 44.16 / 103.67 / 242.40 / 50.83 / 102.84 / 101.55 / 84.18
/ 6.28 / 12.05 / 22.67 / 43.33 / 107.46 / 266.19 / 46.76 / 122.48 / 135.60 / 90.21
RMSE / 123.35 / 92.86 / 55.07 / 44.22 / 119.17 / 290.66 / 51.90 / 103.32 / 101.71 / 84.49
CP / 0.00 / 0.00 / 0.57 / 0.95 / 0.81 / 0.75 / 0.93 / 0.86 / 0.98 / 0.96
P-value / < 10-3 / < 10-3 / 0.002 / 0.128 / 0.439 / 0.618 / − / − / − / −
Selection / 0.00 / 0.00 / 0.34 / 0.66 / 0.00 / 0.00 / − / − / − / −
C1vs. C4 ()
Estimate / 43.01 / 74.45 / 118.79 / 196.11 / 286.12 / 453.34 / 178.31 / 67.00 / 230.13 / 171.10
/ 7.92 / 14.36 / 26.91 / 54.36 / 126.58 / 292.39 / 62.90 / 66.15 / 198.48 / 118.04
/ 5.66 / 12.96 / 26.58 / 55.00 / 130.61 / 303.89 / 64.89 / 60.39 / 240.95 / 106.79
RMSE / 157.18 / 126.36 / 85.55 / 54.47 / 153.04 / 386.77 / 66.51 / 148.52 / 200.66 / 121.47
CP / 0.00 / 0.00 / 0.35 / 0.97 / 0.78 / 0.66 / 0.97 / 0.42 / 0.95 / 0.91
P-value / < 10-3 / < 10-3 / 0.011 / 0.043 / 0.354 / 0.490 / − / − / − / −
Selection / 0.00 / 0.10 / 0.10 / 0.80 / 0.00 / 0.00 / − / − / − / −
C1vs. C5 ()
Estimate / 45.81 / 78.02 / 121.87 / 197.70 / 283.55 / 440.87 / 187.98 / 81.50 / 225.42 / 167.21
/ 8.4 / 15.14 / 28.55 / 56.46 / 128.62 / 281.13 / 63.78 / 76.48 / 180.48 / 90.18
/ 5.68 / 12.96 / 26.38 / 54.50 / 129.20 / 298.97 / 64.54 / 60.36 / 234.40 / 99.06
RMSE / 154.42 / 122.92 / 83.18 / 56.48 / 153.32 / 370.10 / 64.88 / 141.02 / 182.17 / 95.91
CP / 0.00 / 0.00 / 0.39 / 0.94 / 0.79 / 0.66 / 0.97 / 0.41 / 0.96 / 0.92
P-value / < 10-3 / < 10-3 / 0.014 / 0.049 / 0.370 / 0.507 / − / − / − / −
Selection / 0.00 / 0.07 / 0.09 / 0.84 / 0.00 / 0.00 / − / − / − / −
C1vs. C6 ()
Estimate / 45.84 / 77.99 / 122.01 / 199.77 / 291.59 / 463.65 / 190.51 / 84.81 / 235.12 / 173.72
/ 8.05 / 14.54 / 27.57 / 55.36 / 127.42 / 289.90 / 62.06 / 77.48 / 227.80 / 103.16
/ 5.66 / 12.94 / 26.34 / 54.35 / 128.39 / 302.93 / 64.55 / 60.62 / 246.59 / 107.13
RMSE / 154.37 / 122.87 / 82.71 / 55.34 / 156.87 / 391.75 / 62.76 / 138.80 / 230.38 / 106.40
CP / 0.00 / 0.00 / 0.38 / 0.95 / 0.75 / 0.62 / 0.96 / 0.43 / 0.95 / 0.90
P-value / < 10-3 / < 10-3 / 0.013 / 0.041 / 0.358 / 0.479 / − / − / − / −
Selection / 0.00 / 0.06 / 0.09 / 0.85 / 0.00 / 0.00 / − / − / − / −
C2vs. C2 ()
Estimate / 39.93 / 68.72 / 111.95 / 182.91 / 269.23 / 418.41 / 160.91 / 57.77 / 193.05 / 144.03
/ 7.33 / 13.25 / 24.97 / 50.60 / 114.18 / 266.81 / 59.86 / 52.16 / 158.90 / 84.67
/ 5.51 / 12.50 / 25.58 / 52.97 / 120.63 / 287.68 / 60.67 / 53.12 / 211.92 / 93.24
RMSE / 160.24 / 131.95 / 91.52 / 53.38 / 133.48 / 344.71 / 71.47 / 151.48 / 158.97 / 101.46
CP / 0.00 / 0.00 / 0.24 / 0.98 / 0.82 / 0.69 / 0.97 / 0.53 / 0.95 / 0.87
P-value / < 10-3 / < 10-3 / 0.001 / 0.054 / 0.344 / 0.512 / − / − / − / −
Selection / 0.00 / 0.12 / 0.13 / 0.75 / 0.00 / 0.00 / − / − / − / −
C2vs. C3 ()
Estimate / 67.82 / 94.43 / 128.97 / 167.54 / 21.89 / 301.48 / 159.20 / 164.93 / 153.73 / 143.59
/ 8.64 / 13.22 / 22.55 / 41.43 / 93.67 / 210.64 / 46.95 / 79.37 / 74.46 / 56.58
/ 5.80 / 11.11 / 20.87 / 39.88 / 97.77 / 236.92 / 42.38 / 101.80 / 94.25 / 63.87
RMSE / 132.46 / 106.40 / 74.52 / 52.61 / 94.80 / 233.72 / 62.18 / 86.73 / 87.63 / 79.87
CP / 0.00 / 0.00 / 0.24 / 0.93 / 0.91 / 0.80 / 0.88 / 0.94 / 0.91 / 0.83
P-value / < 10-3 / < 10-3 / 0.010 / 0.169 / 0.468 / 0.648 / − / − / − / −
Selection / 0.00 / 0.03 / 0.39 / 0.58 / 0.00 / 0.00 / − / − / − / −
C2vs. C4 ()
Estimate / 40.16 / 68.56 / 108.11 / 174.75 / 252.52 / 398.91 / 161.03 / 70.42 / 184.37 / 136.07
/ 7.81 / 14.18 / 26.76 / 53.39 / 118.29 / 263.91 / 60.33 / 62.80 / 164.55 / 71.05
/ 5.43 / 12.30 / 25.17 / 52.03 / 122.51 / 287.67 / 59.90 / 56.18 / 198.67 / 85.29
RMSE / 160.03 / 132.20 / 95.71 / 59.03 / 129.38 / 330.37 / 71.80 / 143.98 / 165.21 / 95.55
CP / 0.00 / 0.00 / 0.22 / 0.97 / 0.84 / 0.70 / 0.95 / 0.51 / 0.95 / 0.83
P-value / < 10-3 / < 10-3 / 0.021 / 0.074 / 0.394 / 0.528 / − / − / − / −
Selection / 0.00 / 0.09 / 0.15 / 0.76 / 0.00 / 0.00 / − / − / − / −
C2vs. C5 ()
Estimate / 42.51 / 71.58 / 110.62 / 177.47 / 253.63 / 404.61 / 167.81 / 90.60 / 171.18 / 138.87
/ 7.77 / 13.94 / 25.94 / 51.58 / 117.66 / 269.50 / 59.86 / 73.23 / 136.84 / 74.22
/ 5.44 / 12.33 / 25.09 / 51.79 / 123.46 / 290.63 / 60.41 / 56.60 / 181.48 / 88.58
RMSE / 157.69 / 129.18 / 93.07 / 56.26 / 129.25 / 338.27 / 67.94 / 131.63 / 139.77 / 96.12
CP / 0.00 / 0.00 / 0.23 / 0.98 / 0.84 / 0.70 / 0.97 / 0.56 / 0.95 / 0.84
P-value / < 10-3 / < 10-3 / 0.020 / 0.066 / 0.399 / 0.527 / − / − / − / −
Selection / 0.00 / 0.06 / 0.15 / 0.78 / 0.00 / 0.00 / − / − / − / −
C2vs. C6 ()
Estimate / 42.93 / 71.91 / 110.30 / 175.44 / 250.20 / 393.69 / 165.99 / 89.42 / 176.06 / 135.99
/ 7.53 / 13.44 / 25.13 / 50.67 / 113.45 / 257.64 / 58.93 / 73.41 / 166.32 / 70.78
/ 5.43 / 12.28 / 24.90 / 51.39 / 120.71 / 283.60 / 59.43 / 58.75 / 178.12 / 81.54
RMSE / 157.25 / 128.79 / 93.15 / 56.29 / 124.01 / 322.23 / 68.02 / 132.71 / 167.95 / 95.41
CP / 0.00 / 0.00 / 0.236 / 0.97 / 0.84 / 0.72 / 0.96 / 0.56 / 0.95 / 0.83
P-value / < 10-3 / < 10-3 / 0.025 / 0.080 / 0.411 / 0.541 / − / − / − / −
Selection / 0.00 / 0.08 / 0.14 / 0.78 / 0.00 / 0.00 / − / − / − / −
C3vs. C3 ()
Estimate / 159.71 / 183.30 / 189.03 / 204.07 / 229.09 / 356.62 / 188.05 / 163.19 / 163.24 / 163.19
/ 5.81 / 10.06 / 20.07 / 38.57 / 80.37 / 250.40 / 26.50 / 7.96 / 10.59 / 10.45
/ 5.31 / 10.73 / 20.34 / 40.38 / 90.31 / 254.56 / 27.25 / 7.61 / 11.47 / 11.94
RMSE / 40.71 / 19.50 / 22.86 / 38.76 / 85.44 / 295.24 / 13.18 / 37.66 / 38.25 / 38.26
CP / 0.00 / 0.79 / 0.98 / 0.92 / 0.86 / 0.76 / 0.93 / 0.38 / 0.42 / 0.42
P-value / < 10-3 / < 10-3 / 0.483 / 0.527 / 0.731 / 0.737 / − / − / − / −
Selection / 0.00 / 0.89 / 0.06 / 0.05 / 0.00 / 0.00 / − / − / − / −
C3vs. C4 ()
Estimate / 80.90 / 105.15 / 147.17 / 178.93 / 232.19 / 322.67 / 165.23 / 149.88 / 138.53 / 133.70
/ 8.98 / 12.99 / 23.48 / 41.53 / 96.82 / 224.64 / 43.44 / 57.24 / 46.31 / 41.44
/ 6.75 / 11.60 / 23.12 / 42.21 / 103.34 / 253.30 / 40.94 / 72.37 / 64.57 / 44.80
RMSE / 119.44 / 95.74 / 57.81 / 46.55 / 101.99 / 255.86 / 55.61 / 76.07 / 76.96 / 78.17
CP / 0.00 / 0.00 / 0.58 / 0.97 / 0.87 / 0.79 / 0.88 / 0.92 / 0.83 / 0.73
P-value / < 10-3 / < 10-3 / 0.008 / 0.274 / 0.453 / 0.666 / − / − / − / −
Selection / 0.00 / 0.01 / 0.66 / 0.33 / 0.00 / 0.00 / − / − / − / −
C3vs. C5 ()
Estimate / 85.55 / 110.04 / 150.41 / 181.24 / 233.06 / 328.31 / 167.08 / 161.22 / 136.78 / 133.97
/ 8.98 / 13.08 / 24.16 / 43.23 / 101.00 / 239.39 / 44.22 / 55.42 / 38.10 / 35.92
/ 6.73 / 11.57 / 22.95 / 41.85 / 102.81 / 257.41 / 40.46 / 60.52 / 56.07 / 42.11
RMSE / 114.80 / 90.91 / 55.16 / 47.10 / 106.22 / 271.50 / 55.11 / 67.62 / 73.80 / 75.16
CP / 0.00 / 0.00 / 0.58 / 0.96 / 0.85 / 0.78 / 0.88 / 0.90 / 0.80 / 0.69
P-value / < 10-3 / < 10-3 / 0.015 / 0.284 / 0.465 / 0.665 / − / − / − / −
Selection / 0.00 / 0.04 / 0.64 / 0.32 / 0.00 / 0.00 / − / − / − / −
C3vs. C6 ()
Estimate / 87.04 / 111.82 / 152.11 / 182.76 / 233.45 / 324.33 / 169.31 / 165.41 / 137.80 / 134.27
/ 8.78 / 12.67 / 23.37 / 42.11 / 97.91 / 225.72 / 43.49 / 55.32 / 39.44 / 34.99
/ 6.74 / 11.62 / 23.03 / 42.05 / 102.52 / 257.41 / 41.15 / 61.00 / 55.63 / 40.98
RMSE / 113.30 / 89.08 / 53.28 / 45.48 / 103.42 / 257.60 / 53.21 / 65.22 / 73.64 / 74.46
CP / 0.00 / 0.00 / 0.61 / 0.98 / 0.87 / 080 / 0.90 / 0.91 / 0.79 / 0.69
P-value / < 10-3 / < 10-3 / 0.012 / 0.283 / 0.470 / 0.674 / − / − / − / −
Selection / 0.00 / 0.03 / 0.62 / 0.34 / 0.00 / 0.00 / − / − / − / −
C4vs. C4 ()
Estimate / 59.21 / 92.99 / 133.79 / 201.60 / 276.72 / 455.64 / 190.89 / 116.88 / 118.11 / 115.77
/ 8.06 / 14.64 / 28.41 / 58.55 / 132.27 / 331.09 / 68.84 / 55.23 / 69.02 / 48.29
/ 6.07 / 13.51 / 27.34 / 56.95 / 135.33 / 350.21 / 65.30 / 73.22 / 79.96 / 55.10
RMSE / 141.02 / 108.01 / 72.04 / 58.54 / 152.85 / 418.16 / 69.41 / 99.78 / 107.08 / 97.07
CP / 0.00 / 0.00 / 0.53 / 0.96 / 0.82 / 0.72 / 0.93 / 0.94 / 0.79 / 0.71
P-value / < 10-3 / < 10-3 / 0.036 / 0.110 / 0.446 / 0.588 / − / − / − / −
Selection / 0.00 / 0.10 / 0.18 / 0.72 / 0.00 / 0.00 / − / − / − / −
C4vs. C5 ()
Estimate / 62.90 / 96.63 / 137.16 / 203.69 / 273.94 / 433.97 / 192.14 / 124.82 / 118.78 / 118.23
/ 7.94 / 14.41 / 27.89 / 57.58 / 131.32 / 322.57 / 69.03 / 51.90 / 49.90 / 48.13
/ 6.08 / 13.48 / 27.24 / 56.64 / 134.78 / 341.33 / 64.41 / 60.71 / 74.61 / 50.97
RMSE / 137.33 / 10.37 / 68.75 / 57.67 / 150.65 / 398.36 / 69.44 / 91.34 / 95.31 / 94.88
CP / 0.00 / 0.00 / 0.56 / 0.95 / 0.80 / 0.72 / 0.94 / 0.88 / 0.78 / 0.69
P-value / < 10-3 / < 10-3 / 0.034 / 0.116 / 0.458 / 0.603 / − / − / − / −
Selection / 0.00 / 0.09 / 0.23 / 0.69 / 0.00 / 0.00 / − / − / − / −
C4vs. C6 ()
Estimate / 63.56 / 97.00 / 137.21 / 203.60 / 278.59 / 462.92 / 192.17 / 126.44 / 118.53 / 117.92
/ 7.86 / 14.39 / 28.30 / 58.55 / 132.20 / 327.88 / 69.26 / 52.44 / 47.99 / 45.02
/ 6.06 / 13.40 / 27.07 / 56.21 / 132.66 / 336.83 / 64.21 / 67.17 / 73.05 / 50.82
RMSE / 136.66 / 104.00 / 68.87 / 58.63 / 153.74 / 420.15 / 69.67 / 90.32 / 94.54 / 93.61
CP / 0.00 / 0.00 / 0.57 / 0.94 / 0.78 / 0.66 / 0.93 / 0.90 / 0.77 / 0.66
P-value / < 10-3 / < 10-3 / 0.0368 / 0.120 / 0.443 / 0.570 / − / − / − / −
Selection / 0.00 / 0.11 / 0.19 / 0.70 / 0.00 / 0.00 / − / − / − / −
C5vs. C5 ()
Estimate / 66.67 / 100.05 / 139.31 / 203.26 / 268.05 / 422.87 / 190.25 / 141.98 / 117.46 / 117.27
/ 7.6 / 13.78 / 26.32 / 53.55 / 123.88 / 315.45 / 66.01 / 47.48 / 39.52 / 38.58
/ 6.05 / 13.42 / 27.09 / 56.46 / 135.79 / 344.89 / 64.57 / 41.38 / 66.86 / 48.61
RMSE / 133.55 / 100.89 / 66.15 / 53.62 / 141.29 / 386.11 / 66.70 / 74.95 / 91.50 / 91.27
CP / 0.00 / 0.00 / 0.61 / 0.96 / 0.83 / 0.75 / 0.96 / 0.75 / 0.77 / 0.69
P-value / < 10-3 / < 10-3 / 0.035 / 0.119 / 0.484 / 0.629 / − / − / − / −
Selection / 0.00 / 0.10 / 0.24 / 0.66 / 0.00 / 0.00 / − / − / − / −
C5vs. C6 ()
Estimate / 67.36 / 100.80 / 140.57 / 206.41 / 277.88 / 448.78 / 194.14 / 139.54 / 121.69 / 120.36
/ 7.83 / 14.29 / 27.88 / 57.12 / 132.47 / 331.58 / 69.22 / 52.22 / 50.87 / 44.09
/ 6.05 / 13.43 / 27.16 / 56.54 / 135.04 / 343.57 / 64.45 / 43.79 / 71.54 / 50.61
RMSE / 132.87 / 100.22 / 65.63 / 57.45 / 153.61 / 414.40 / 69.43 / 79.88 / 93.36 / 91.02
CP / 0.00 / 0.00 / 0.60 / 0.95 / 0.78 / 0.69 / 0.94 / 0.74 / 0.78 / 0.68
P-value / < 10-3 / < 10-3 / 0.039 / 0.120 / 0.454 / 0.593 / − / − / − / −
Selection / 0.00 / 0.10 / 0.23 / 0.67 / 0.00 / 0.00 / − / − / − / −
C6vs. C6 ()
Estimate / 69.03 / 102.98 / 143.73 / 211.28 / 288.31 / 468.94 / 199.54 / 144.95 / 125.34 / 124.66
/ 8.07 / 14.65 / 28.48 / 59.43 / 134.96 / 334.17 / 70.90 / 47.32 / 51.73 / 47.36
/ 6.09 / 13.54 / 27.39 / 57.04 / 134.65 / 347.28 / 64.90 / 42.13 / 75.56 / 53.56
RMSE / 131.22 / 98.12 / 63.06 / 60.46 / 161.23 / 428.82 / 70.86 / 72.58 / 90.81 / 88.98
CP / 0.00 / 0.00 / 0.64 / 0.92 / 0.76 / 0.67 / 0.92 / 0.77 / 0.80 / 0.72
P-value / < 10-3 / < 10-3 / 0.036 / 0.121 / 0.447 / 0.582 / − / − / − / −
Selection / 0.00 / 0.10 / 0.21 / 0.69 / 0.00 / 0.00 / − / − / − / −
S1