Testing the performance of the Poisson iREM for animals with

As discussed in Section 2.7.1 Rowcliffe et al. (2008) used day range to derive an approx- imate estimator of speed. Ten (10) focal watches, for each species, distributed evenly between 08:00-18:00 were conducted to monitor movement, and control for variation in movement patterns. Arbitrary animals were selected and followed around for 30 minutes during each focal watch, recording the total distance travelled during that time as the sum of all straight-line movements. Day range was then calculated as the mean across all focal watches for that species. Hence, the distribution of speeds over this short-time period was then used as the distribution of speeds over the entire trapping period. It was noted that during the period the animals were followed around, some of them were observed to not move, hence, there are records of zero speeds. We expect that these observed zero speeds would have an effect on estimated density. Note that in REM density is estimated as ˆ D = λ t π (2 + θ)r¯v, (3.7.2)

where r and θ are the detection distance and detection angle of the camera trap, re- spectively, and t is the camera trap time period. These parameters are held fixed. The parameter ¯v is the mean speed. So including the observed zero speeds would result in a smaller estimate in the average speed, and hence, a larger estimate of the density compared with the density estimated density from excluding the observed zero speeds. For example, suppose we have m = 4 speed observations such that v = {0.453, 0, 1.865, 0}. Let ¯v = 0.5795 (km/day−1), the mean speed including the observed zeros, and ¯v1 = 1.159 (km/day−1), the mean speed excluding the observed zeros, therefore, we would expect the density ˆD, estimated using ¯v to be larger com- pared with the density ˆD1 estimated using ¯v1.

In this section we explore how a zero-adjusted gamma model works in estimating the density. As a demonstration we fit a Poisson iREM model to the encounter data. We also examine the fit of the gamma model with the observed speeds, excluding the zeros, and we compare estimates from iREM with estimates from REM. First, we test the ZAGA

model using large and small values of the density; D = 100 (km2_{) or D = 20 (km}2_{) with} a probability of the zero-response category set to w = 0.30, and sampling effort of the camera trap days and speed observations set to n = 40; m = 40 or n = 100; m = 100. The number of speed observations, m, is considered to be reasonably large, that is, greater than 30. The results are given in Table 3.7.8. The simulation results (3.7.8) indicate that the model works well in estimating the density for the given set of para- meter values and sample sizes. Also, estimation precision improves when sample sizes increase. REM and iREM gave similar estimates of the density but REM gave smaller estimates of the standard errors and larger RMSEs.

Second, we examine the fit of the ZAGA model and the gamma for parameter values and sample sizes that reflect our WWAP data set. In Table 3.7.9 we give the results from fitting iREM and REM where animal speed is assumed to follow a ZAGA model or a gamma model. The density D = 468 (km2) or D = 119 (km2); the expected speed is µx = 0.71 (km/day−1) or µx = 1.17 km/day−1
, respectively, and the probability of the zero-response category is w = 0.21 or w = 0.40, respectively. For each scenario, we set the number of the speed observations to m = 10, which is the sample size of the observed speeds for the WWAP data. We compare estimates of the density for larger sample sizes of the speed observations: m = 40 or m = 70. The number of camera trapping days, n = 42, which is the number of camera trapping days for the WWAP survey, is held fixed. The results (Table 3.7.9) show that for the ZAGA model, a pos- itive bias is introduced for all scenarios and this bias is substantial for smaller sample sizes of the speed observations, in particular, when the density is large (D = 468 km2). But for the gamma model the bias introduced when the sample size is small is much smaller, and the model appears to provide better estimates of the density compared with the ZAGA model. It is worth reiterating that including the observed zeros would give smaller estimates of the mean speed and, hence, inflate the density. This implies that the estimated mean speed from excluding the zeros would be larger, and hence, provide a smaller estimate of density. We give the sample sizes of the speed observa- tions, excluding the zeros, for the first 6 simulation runs in Table 3.7.11. For example when D = 119 (km2) (Table 3.7.9), the first sample size simulated for the speed ob- servations, excluding the zeros, and used in the gamma model is 4 (Table 3.7.11, Case

(4)), which suggests that 6 out of the 10 simulated speeds are zeros. A sample size of 4 is a rather small sample size, therefore, we would expect a larger estimate of the mean speed, hence, a smaller estimate of the density, compared with the estimated mean speed for this same scenario from a ZAGA model which would provide a larger estimated density. In Table 3.7.9 average estimated densities with the true sample size, m = 10 used in the simulation process are: ˆD = 166.08 (0.27) (km2) for the ZAGA model and ˆD = 92.68 (39.23) (km2_{) for the gamma (average estimated standard errors} in parentheses). On the other hand, the evidence shows that the ZAGA model provides reasonably accurate estimates of the density when m increases but a negative bias is introduced from the gamma model, which increases with sample size (see Table 3.7.9).

Finally, we examine the fit of the ZAGA model and the gamma model for a small fixed sample size of the speed observations m = 10 and increasing probabilities of the zero- response category. We set D = 68 (km2) or D = 468 (km2), which reflects the density from the census for two species at WWAP. The probability of the zero-response category is set to w = 10, w = 20, or w = 30. The number of camera trap days is set to n = 42. The results are given in Table 3.7.10. The simulation results show that for small m and small w (= 0.10) the bias introduced from a ZAGA model when D (= 68 km2) is much smaller (and positive) compared with larger values of the density (D = 468 km2). Also, the bias increases with increasing values of w. For the gamma model, when the density is small a negative bias is introduced, which increases with increasing values of w, and the bias is larger compared with the bias from a ZAGA model. On the other hand, when the density is large the bias introduced from the gamma model is smaller (and positive) compared with the bias produced by the ZAGA model, in particular, when w = 0.20. Note that we expect to have more non-zero speed observations in the sample for reasonably small values of w (see for e.g., sample Case (11) in Table 3.7.11).

Based on these simulation results we can conclude that the size of the density, the sample size of the speed observations and the size of the probability of the zero-response category would have an effect on estimated density. If there are observed zeros in the speed data, it is important to model these but we would recommend, where possible to increase survey effort, particularly for the speed observations and, if there are more zeros than

expected. One important point to note is that the WWAP survey is a special case where independent estimates of the speed and average group size were obtained by following the animals around, and during this time period some animals were observed to not move. As discussed in Section 2.1 the speed of movement of an animal is assumed to be constant but this speed may vary between animals. However, whether an animal varies it speeds or move at a fixed speed (or stop altogether) is not as important as the total distance covered by the individual over the trapping period, since this is what determines the probability of encounter. And since we are interested in the mean across animals of the mean speed over time for each individual, we need to include the observed zero speeds in estimating that mean, to avoid bias. But as shown, at least 60 observations are required to provide reasonably accurate estimates of the density when the probability of the zero-response category is high, and when the density is large. But for small density values, and if the probability of observing zero speeds is low then smaller sample sizes can provide reasonably accurate estimates of the density.

Table 3.7.8: Average parameter estimates for animals with observed zero speed of movement (average standard errors in parentheses). The following are the true values; µx = 0.71 (km/day−1) and w = 0.30. The Standard deviation (Sd) and Root Mean Square Error (RMSE) are also given.

Poisson iREM Poisson REM ˆ D µˆx wˆ Dˆ D = 100, n = m = 40 estimate 103 (31.79) 0.73 (0.17) 0.29 (0.10) 105 (21.79) Sd 29.27 0.16 0.07 32.47 RMSE 866.81 0.03 0.005 1079.09 D = 100, n = m = 100 estimate 101 (19.61) 0.72 (0.10) 0.30 (0.07) 102 (13.28) Sd 17.12 0.09 0.04 18.43 RMSE 294.20 0.01 0.02 342.58 D = 20, n = m = 40 estimate 21 (10.96) 0.74 (0.17) 0.29 (0.10) 22 (10.71) Sd 8.07 0.17 0.06 11.28 RMSE 65.77 0.03 0.004 129.76 D = 20 n = m = 100 estimate 20 (6.65) 0.72 (0.10) 0.30 (0.07) 20 (6.11) Sd 5.33 0.10 0.04 6.12 RMSE 28.37 0.01 0.002 37.54

Table 3.7.9: Average parameter estimates from fitting a ZAGA model and a gamma model to the speed data where the density is set to D = 468 (km2) or D = 119 (km2) (average standard errors in parentheses), and where sample size of the speed data, m increases. The number of camera trapping days is n = 42. The chosen parameter values and sample sizes reflect the WWAP data set. The Standard deviation (Sd) and Root Mean Square Error (RMSE) are also given.

ZAGA model gamma model

Poisson iREM Poisson REM Poisson iREM Poisson REM

ˆ D µˆx wˆ Dˆ Dˆ µˆx Dˆ D = 468, m = 10, w = 0.21, µx= 0.71 estimate 608.93 (283.27) 0.69 (0.68) 0.21 (0.16) 608.94 (56.98) 472.20 (204.13) 0.88 (0.37) 472.20 (44.19) Sd 380.06 0.31 0.13 380.98 306.00 0.38 306.00 RMSE 405.35 0.10 0.13 405.35 306.02 0.42 306.02 D = 468, m = 30, w = 0.21, µx= 0.71 estimate 512.63 (147.64) 0.71 (0.53) 0.21 (0.10) 512.63 (47.54) 400.20 (108.71) 0.90 (0.23) 400.20 (37.12) Sd 173.46 0.19 0.07 125.21 142.39 0.30 125.21 RMSE 179.11 0.04 0.07 179.11 142.39 0.29 142.39 D = 468, m = 60, w = 0.21, µx= 0.71 estimate 479.54 (102.67) 0.72 (0.41) 0.21 (0.07) 479.53 (44.73) 376.92 (76.63) 0.91 (0.16) 376.92 (35.16) Sd 105.45 0.15 0.05 105.45 78.43 0.17 78.43 RMSE 106.08 0.02 0.05 106.08 120.19 0.26 120.19 D = 119, m = 10, w = 0.40, µx= 1.17 estimate 166.08 (0.27) 1.16 (0.72) 0.39 (0.27) 166.08 (24.08) 92.68 (39.23) 1.89 (0.75) 92.68 (13.44) Sd 123.73 0.62 0.14 123.73 54.00 0.87 54.00 RMSE 132.39 0.38 0.14 132.39 60.07 1.13 60.07 D = 119, m = 30, w = 0.40, µx= 1.17 estimate 133.42 (44.10) 1.15 (0.76) 0.40 (0.15) 133.42 (19.21) 77.96 (22.83) 1.92 (0.49) 77.96 (11.23) Sd 43.73 0.35 0.09 133.42 23.43 0.49 23.43 RMSE 46.05 0.13 0.09 46.05 47.26 0.90 47.26 D = 119, m = 60, w = 0.40, µx= 1.17 estimate 122.26 (31.45) 1.17 (0.64) 0.40 (0.11) 122.26 (17.82) 72.00 (16.83) 1.97 (0.36) 72.00 (10.49) Sd 33.22 0.24 0.06 33.22 16.63 0.35 16.63 RMSE 33.38 0.06 0.06 33.38 49.86 0.87 49.86 85

Table 3.7.10: Average parameter estimates from fitting a ZAGA model and a gamma model to the speed data where the probability of the zero-response category, w increases (average standard errors in parentheses). The chosen parameter values and sample sizes reflect the WWAP data set. Here we set the density, D = 68 (km2) or D = 468 (km2). The Standard deviation (Sd) and Root Mean Square Error (RMSE) are also given.

ZAGA model gamma model

Poisson iREM Poisson REM Poisson iREM Poisson REM

ˆ D µˆx wˆ Dˆ Dˆ µˆx Dˆ D = 68, m = 10, w = 0.10, µx= 2.56 estimate 71.78 (24.28) 2.62 (0.31) 0.10 (0.06) 71.77 (9.35) 63.79 (20.42) 2.91 (0.85) 63.79 (8.31) Sd 24.27 0.31 0.06 24.83 20.59 0.38 20.59 RMSE 25.11 0.64 0.09 25.11 21.02 0.92 21.02 D = 68, m = 10, w = 0.20, µx= 2.56 estimate 75.64 (26.94) 2.62 (0.36) 0.19 (0.14) 75.64 (9.72) 59.34 (18.84) 3.24 (0.93) 59.34 (7.63) Sd 29.86 0.90 0.12 29.86 20.53 0.98 20.53 RMSE 179.11 0.04 0.07 179.11 142.39 0.29 142.39 D = 68, m = 10, w = 0.30, µx= 2.56 estimate 77.52 (28.26) 2.66 (0.41) 0.28 (0.20) 77.52 (9.72) 53.11 (15.83) 3.69 (0.98) 53.11 (6.84) Sd 39.98 0.97 0.14 39.98 20.53 0.98 21.92 RMSE 179.11 0.04 0.07 179.11 142.39 0.29 142.39 D = 468, m = 10, w = 0.10, µx= 0.71 estimate 575.23 (245.16) 0.74 (0.66) 0.09 (0.05) 575.23 (53.68) 512.39 (211.85) 0.81 (0.33) 512.19 (47.80) Sd 412.21 0.33 0.09 412.21 322.91 0.36 321.29 RMSE 405.35 0.10 0.13 405.35 306.02 0.42 306.02 D = 468, m = 10, w = 0.20, µx= 0.71 estimate 600.77 (269.94) 0.74 (0.65) 0.19 (0.13) 600.77 (56.06) 455.28 (190.12) 0.92 (0.37) 455.28 (42.48) Sd 514.74 0.35 0.14 514.74 278.88 0.40 514.74 RMSE 405.35 0.10 0.13 405.35 306.02 0.42 306.02 D = 468, m = 10, w = 0.30, µ = 0.71 86

Table 3.7.11: Sample sizes of speed observations, excluding the zero speeds, for the first 6 simulation runs for the simulation scenarios examined.

Sample sizes for scenarios in Table 3.7.9 Sample sizes for scenarios in Table 3.7.10

D = 468, m = 10, w = 0.21, µx= 0.71 D = 68, m = 10, w = 0.10, µx= 2.56 Case (1) 8 8 5 9 9 10 Case (7) 8 10 10 10 8 9 D = 468, m = 30, w = 0.21, µx= 0.71 D = 68, m = 10, w = 0.10, µx= 2.56 Case (2) 23 24 25 22 21 22 Case (8) 10 8 7 8 7 9 D = 468, m = 60, w = 0.21, µx= 0.71 D = 68, m = 10, w = 0.10, µx= 2.56 Case (3) 46 49 53 51 50 47 Case (9) 6 5 5 7 9 5 D = 119, m = 10, w = 0.40, µx= 1.17 D = 468, m = 10, w = 0.10, µx= 2.56 Case (4) 4 6 6 6 4 7 Case (10) 10 9 9 9 8 9 D = 119, m = 30, w = 0.40, µx= 1.17 D = 468, m = 10, w = 0.20, µx= 2.56 Case (5) 20 18 14 21 18 17 Case (11) 10 7 7 9 8 8 D = 119, m = 60, w = 0.40, µx= 1.17 D = 468, m = 10, w = 0.30, µx= 2.56 Case (6) 36 39 38 31 36 33 Case (12) 4 4 7 5 7 6 87