Marbled Murrelet Inland Survey Protocol Statistical Analyses. Final Recommendations from the Steering Committee By Danielle Prenzlow Escene

Transcription

1 Marbled Murrelet Inland Survey Protocol Statistical Analyses Final Recommendations from the Steering Committee By A 1-year protocol, a 2-year protocol, or a 3-year protocol? (Or investigations of site-status changes) We recommend a 2-year protocol. In his write-up, Investigating the possibility of changes in status at a site, dated November 21, 2001, Jim Baldwin reports the proportion of sites that change occupancy status of the sites that are occupied in at least one of two years, for seven pairs of consecutive years from 1991 to This ratio addresses the question, Of those sites where occupancy occurs, what proportion change status? The results range from in to in The remaining 5 pairs of years are between and The weighted average is 0.393; however, because the actual sites sampled can differ between sets of years, it is not clear how to interpret such an average. Nevertheless, these ratios do give us a general sense of the relative amount of occupied sites that change status in a two-year time frame. Jim also found that status is not independent between years. The above analysis required a one-year definition of occupancy status, or annual site status, and calculated an estimate of the probability of detecting occupancy on a single visit given the site is occupied (1!q) based on this one-year definition of occupancy. The analysis also used only those sites sampled with the binomial model, sites that are visited a set number of times regardless of the behaviors observed. Years Ratio of sites that change occupancy status to sites occupied in at least one of two years Number of qualifying sites In his write-up, Assessing the 2-year Protocol for Marbled Murrelets, dated May 15, 2002, (the final Markov chain report), Jim reports the estimates of the probabilities of sites having occupancy, presence-only, or neither in a particular year, (P 2, P 1, and P 0, respectively). He also reports the estimates of probabilities of sites changing from status i in year one to status j the next year, (M ij ; i and j equal 2 for occupancy, 1 for presence-only, and 0 for neither). The estimate of the ratio of sites that change occupancy status to the sites that are occupied in at least 1

2 one of two years would be (P 0 M 02 + P 1 M 12 + P 2 (M 20 + M 21 )) / (P 0 M 02 + P 1 M 12 + P 2 (M 20 + M 21 + M 22 )). This value is However, again, because the actual sites sampled differ between years, it is difficult to defend the estimates of parameters for the proportion of sites with a particular annual status. Even so, the estimate does, again, give us a general sense of the relative amount of sites that change status. Jim explains that the estimation of q accounts for the variation in survey effort in the analyses; thus all levels of survey effort are taken into account. These results provide some perspective for status changes over 2-year time frames. It was the general opinion of the Committee and the statisticians that the data set would not be adequate to draw a picture of the average situation for status changes over more than 2 years. Thus, based on the above information the Committee recommends a 2-year protocol. Given a site is occupied, how many visits should be conducted to determine occupancy? (Or investigations of detection probabilities) We recommend a minimum of 5 visits be conducted in each of 2 years. In year one, if there are no detections observed in the first 5 visits, visits may be terminated for that year. If presence behaviors are observed, we recommend an additional 4 visits be conducted for a total of 9 visits to determine occupancy. In year two, for the sites with presence behaviors but no occupancy behaviors observed in year one, we recommend a total of 9 year-two visits be conducted for a two-year total of 18 visits to determine occupancy. For the sites with no detections in year one, a minimum of 5 visits should be conducted in year two. If there are no detections observed in the first 5 visits to those sites, visits may be terminated for a two-year total of 10 visits. If presence behaviors are first observed during year two, an additional 4 visits should be conducted for a total of 9 year-two visits to determine occupancy, and for a two-year total of 14 visits. Of course, in all instances described above, visits may be terminated once occupancy behaviors are observed. The above recommendation is based on the estimates of q for presence and occupancy (given below), the estimates of q 0, q 1, and q 2 for the set of occupied sites (given below), the assumption of 0.40 for the proportion of sites that change status between years of the sites that are occupied in at least one of two years (supporting results given in the previous section), and an overall error rate of 5% or less, and also assuming the set of sites to be surveyed are similar to the set of sites in the 3-state database. In his write-up, Assessing the 2-year Protocol for Marbled Murrelets, dated July 16, 2002, Jim reports the following estimates of detection probabilities for presence and occupancy. Given that a site has presence in a year, the probability of not observing a presence behavior on a single visit during that year is (q for presence). Thus, within a one-year time frame, a site with a true annual status of presence would need to be visited 5 times (1!q 5 ) to reach a probability of detecting presence during that year (see table below). Or stated another way, a site with a true annual status of presence would need to be visited 5 times (q 5 ) to have a probability of not detecting presence during that year. Given that a site is occupied in a year, the probability of not observing an occupied behavior on a single visit during that year is (q 2

3 for occupancy). Thus, a site with a true annual status of occupancy would need to be visited 12 times (q 12 ) to have a probability of not detecting occupancy during that year. These above estimates of q for presence and occupancy were calculated using the entire database. Probability of detecting presence and occupancy based on a one-year definition of status and a one-year time frame Number Presence Occupancy of visits In order to recommend a two-stage effort as described in the first paragraph of this section, we needed to determine for occupied sites not only the probability of not observing an occupied behavior but also the probability of not observing even presence behaviors on a single visit. In his write-up, Assessing the 2-year Protocol for Marbled Murrelets, dated July 16, 2002, Jim reports the appropriate detection probabilities for the set of occupied sites. Given that a site is occupied in a year, the probability of observing no detections on a single visit during that year is (q 0 ). Given that a site is occupied in a year, the probability of observing presenceonly behaviors on a single visit during that year is (q 1 ). Given that a site is occupied in a year, the probability of observing occupied behaviors on a single visit during that year is (q 2 ). The above estimates of q 0, q 1, and q 2 were calculated from the set of sites that have either occupied detections or no detections; however, note that q 0 + q 1 = , which is very close, within 2.5%, to the estimate of q for occupancy of , calculated from the entire database. The sum of q 0, q 1, and q 2 is 1. The probability of observing occupancy using a two-stage effort as described above, is 1! (q 0 + q 1 ) s! q 0 s* (1! (q 0 + q 1 ) s!s* ), where s* is the number of minimum visits conducted in one year (in our recommendation s* = 5), and s is the number of total visits conducted in one year to determine occupancy, given presence is observed during the first s* visits (in our recommendation s = 9). See the derivation in Appendix A. Thus, within a one-year time frame, a site with a true annual status of occupancy could be visited 5 times with no detections OR 12 times with presence behaviors observed on one or more of the first 5 visits to reach a probability of detecting occupancy during that year (see table below), or a probability of not detecting occupancy during that year. 3

4 Probability of detecting occupancy using a two-stage effort and a one-year time frame s s* = 4 s* = 5 s* = 6 no two-stage For a two-year time frame, for a site with a true annual status of occupancy in both of the two years, the probability of observing occupancy using a two-stage effort is 1! (q 0 + q 1 ) 2s! q 0 s* (1! (q 0 + q 1 ) s!s* )( q 0 s* + (q 0 + q 1 ) s ). Again, the derivation is in Appendix A. Thus, within a two-year time frame, a site with a true annual status of occupancy in both of two years could be visited 4 times each year with no detections OR 6 times in year one with presence behaviors observed during one or more of the first 4 visits and 6 times in year two OR 4 times in year one with no detections and 6 times in year two with presence behaviors observed during one or more of the first 4 visits to reach a probability of detecting occupancy some time during the two years (see table below), or a probability of not detecting occupancy during the two years. Probability of detecting occupancy using a two-stage effort and a two-year time frame, for a site with a true annual status of occupancy in both of 2 years s s* = 4 s* = 5 s* = 6 no two-stage

5 To calculate the average probability of detecting occupancy using a two-stage effort and a two-year time frame, considering the fact that some sites have a true annual status of occupancy in only 1 of 2 years and some sites have a true annual status of occupancy in both of two years, calculate the weighted average of the probabilities of observing occupancy for each case. The table below assigns Q and QN to the formulas for probability of observing occupancy for the case of occupied in one of two years and two of two years, respectively. The table also assigns the variables A, B, and C to the proportions of sites in the various combinations of true annual site status over two years, conditional on observing at least one observation of occupancy. Since at least one year must have a true annual site status of occupancy, A + B + C = 1. True Annual Site Status Probability of Observing Proportion Year 1 Year 2 Occupancy Absence Absence 0 0 Absence Occupancy Q = 1! (q 0 + q 1 ) s! q 0 s* (1! (q 0 + q 1 ) s!s* ) A Occupancy Absence Q = 1! (q 0 + q 1 ) s! q 0 s* (1! (q 0 + q 1 ) s!s* ) B Occupancy Occupancy QN = 1! (q 0 + q 1 ) 2s! q 0 s* (1! (q 0 + q 1 ) s!s* )( q 0 s* + (q 0 + q 1 ) s ) C The average probability of detecting occupancy is AQ + BQ + CQN which equals (A + B)Q + CQN which equals (1! C)Q + CQN. The following table lists the average probabilities of detecting occupancy given a site is occupied in at least one of two years. These probabilities are dependent on 1) the estimates of q 0 and q 1, given above, 2) the specified number of survey visits (s and s*), and 3) an average of forty percent of occupied sites being occupied in only one of two years (A=0.2, B=0.2, and C=0.6). See Appendix B for additional tables for the case of C=0.3, C=0.4, C=0.5, and C=0.7. Probability of detecting occupancy using a two-stage effort and a two-year time frame, when, on average, 40 percent of occupied sites have a true annual status of occupancy in only 1 of 2 years s s* = 4 s* = 5 s* = 6 no two-stage

6 Thus, within a two-year time frame, when 40 percent of occupied sites have a true annual status of occupancy in only one of two years, the sites could be visited 5 times each year with no detections OR 9 times in year one with presence behaviors observed during one or more of the first 5 visits and 9 times in year two OR 5 times in year one with no detections and 9 times in year two with presence behaviors observed during one or more of the first 4 visits to reach a average probability of detecting occupancy some time during the two years, or a probability of not detecting occupancy during the two years. Hence, the Committee s recommendation, stated at the beginning of this section. See Appendix C for an actual-numbers illustration of the case above where, s*=5, s=9, and C=0.6. Moreover, using the above formulas, one could calculate the average probability using different values of q 0 and q 1, using different values of s and s*, and/or using a different assumption of the proportion of occupied sites that change status between years, 1!C. These different values and assumptions might be derived from data from a particular geographic area of interest; however, assistance from a qualified statistician to determine the sample size needed and to help with the derivations would be necessary. What are the seasonal limitations of distributing the survey visits? (Investigations of seasonal variability in detection probabilities) The following recommendations are similar to the current recommendations in the 2000 protocol with only slight modifications. We recommend that the 5 minimum visits be conducted at regular intervals throughout the survey season. Surveys should be scheduled to begin within the first two to three weeks of the survey season. To help maintain even distribution, surveyors should aim for a minimum of 6 and a maximum of 30 days between these 5 survey visits. We recommend that when increased survey effort is needed to determine occupancy (total of 9 visits), survey visits be spaced as evenly as possible throughout the breeding season, aiming for a minimum of 2 days between visits. We recommend that at least 2 of the 5 minimum visits and at least 4 of the 9 visits for occupancy occur after June 30, and that half of these visits occur within the first 3 weeks of July. If the first observation of presence is detected on the 5 th visit after July 17, or on the 6 th visit after July 21 st, or on the 7 th visit after July 24 th, or on the 8 th visit after July 29 th, in either the first or second year of surveys, we recommend a third year of surveys be added to determine occupancy. We make the previous recommendation because, with presence being first detected late in the year, additional survey visits to detect occupancy in that year would have to be conducted in too short an amount of time. Here s an example: 3 visits scheduled before July 1 and 2 visits scheduled between June 30 and July 18, so that if presence is detected on the 5 th visit in July, adequate time remains to conduct an additional 4 visits to determine occupancy. In his draft document AInvestigation of Seasonal Heterogeneity of Detecting Presence and Occupancy Status for Marbled Murrelets, dated September 10, Jim reports that detection rates of murrelet presence and occupancy vary within the 16-week survey season. For presence, detection rates in a 2-week period in the middle of July are higher than the rest of the season. A 5-week period from the beginning/middle of May to the beginning/middle of June has lower presence detection rates. For occupancy, detection rates are low through the season until about a 1-week period in the middle of July. 6

7 Week ID Dates Week ID Dates 1 April 15 April 21 9 June 10 June 16 2 April 22 April June 17 June 23 3 April 29 May 5 11 June 24 June 30 4 May 6 May July 1 July 7 5 May 13 May July 8 July 14 6 May 20 May July 15 July 21 7 May 27 June 2 15 July 22 July 28 8 June 3 June 9 16 July 29 August 5 7

8 In his addendum to the report on Test 1, Investigation of Seasonal Heterogeneity of Detecting Presence and Occupancy Status for Marbled Murrelets, dated October 15, Jim reports how the range in seasonal (weekly) q values translates into the probability of detecting presence or occupancy after 10 and 16 visits. The following q s are all-year q s. For occupancy, the average q, 0.825, translates to 85 and 95 percent probability after an all-year total of 10 and 16 visits, respectively. The smallest weekly q, 0.684, in the 14 th week of the season (July 15 to July 21) translates to 98 and 100 percent probability after a total of 10 and 18 visits, respectively. The largest weekly q, 0.871, in the 14 th week of the season (April 29 - May 5) translates to 75 and 89 percent probability after a total of 10 and 16 visits, respectively. Is there heterogeneity in probability of detecting marbled murrelet presence and occupancy among different sites? (Or investigations of spatial variability in detection probabilities) In their report, Heterogeneity in Probability of Detecting Marbled Murrelet Presence and Occupancy Among Different Sites, dated June 7, 2002, Chris Nations and Bryan Manley, from Western EcoSystems Technology, conclude that there is strong evidence that the detection probabilities for both presence and occupancy vary among the sites for all years pooled and in every year individually, except Presence Occupancy Year T P-value T P-value All years The Committee recognizes that, for sites that truly have murrelet presence or occupancy, detecting murrelets is easier at some sites than at others. This protocol recommends a survey effort to handle the average situation. However, perhaps a protocol that accounts for the fact that some sites have higher detection probabilities and also accounts for the fact that some sites have higher initial probabilities of occupancy would be more efficient in terms of survey effort. It just makes sense that if initially we knew that one set of sites was different from another set of sites, we might approach the survey effort at those sets of sites differently. Unfortunately, we don t yet have the tool we would need to determine how those approaches might differ. 8

9 Appendix A. Q = probability of detecting occupancy in a single year at a site with a true status of occupancy Q = 1! probability of not detecting occupancy in a single year at a site with a true status of occupancy Q = 1! ( no detections in the first s* visits + {presence detections in the first s* visits with no occupancy detections in the additional s!s* visits} ) Q = 1! ( q 0 s* + { [ (q 0 + q 1 ) s*! q 0 s* ] H (q 0 + q 1 ) s!s* } ) Q = 1! ( q 0 s* + { (q 0 + q 1 ) s*+ s!s*! q 0 s* (q 0 + q 1 ) s!s* } ) Q = 1! ( q 0 s* + (q 0 + q 1 ) s! q 0 s* (q 0 + q 1 ) s!s* ) Q = 1! ( (q 0 + q 1 ) s + q 0 s* (1! (q 0 + q 1 ) s!s* ) ) Q = 1! (q 0 + q 1 ) s! q 0 s* (1! (q 0 + q 1 ) s!s* ) QN = probability of detecting occupancy in either year at a site with a true status of occupancy in both years QN = 1! probability of not detecting occupancy in either year at a site with a true status of occupancy in both years QN = 1! ( {no detections in Year-1 first s* visits with no detections in Year-2 first s* visits} + {presence detections in Year-1 first s* visits with no occupancy detections in the additional Year-1 s!s* visits with no occupancy detections in Year-2 s visits} + {no detections in Year-1 first s* visits with presence detections in Year-2 first s* visits with no occupancy detections in the additional Year-2 s!s* visits} ) QN = 1! ({q 0 s* H q 0 s* } + {[(q 0 + q 1 ) s*! q 0 s* ] H (q 0 + q 1 ) s!s* H (q 0 + q 1 ) s }+ {q 0 s* H [ (q 0 + q 1 ) s*! q 0 s* ] H (q 0 + q 1 ) s!s* } ) QN = 1! q 0 2s*! [ (q 0 + q 1 ) s*! q 0 s* ] H (q 0 + q 1 ) s!s* H ( (q 0 + q 1 ) s + q 0 s* ) QN = 1! q 0 2s*! [ (q 0 + q 1 ) s! q 0 s* (q 0 + q 1 ) s!s* ] H ( (q 0 + q 1 ) s + q 0 s* ) QN = 1! q 0 2s*! [ (q 0 + q 1 ) s! q 0 s* (q 0 + q 1 ) s!s* ] H ( (q 0 + q 1 ) s + q 0 s* ) QN = 1! q 0 2s*! [ (q 0 + q 1 ) 2s! q 0 s* (q 0 + q 1 ) 2s!s* + q 0 s* (q 0 + q 1 ) s! q 0 2s* (q 0 + q 1 ) s!s* ] QN = 1! q 0 2s*! (q 0 + q 1 ) 2s + q 0 s* (q 0 + q 1 ) 2s!s*! q 0 s* (q 0 + q 1 ) s + q 0 2s* (q 0 + q 1 ) s!s* QN = 1! (q 0 + q 1 ) 2s! q 0 s* [ q 0 s*! (q 0 + q 1 ) 2s!s* + (q 0 + q 1 ) s! q 0 s* (q 0 + q 1 ) s!s* ] QN = 1! (q 0 + q 1 ) 2s! q 0 s* ( 1! (q 0 + q 1 ) s!s* ) ( q 0 s* + (q 0 + q 1 ) s ) 9

10 Appendix B. The following tables list the average probabilities of detecting occupancy given a site is occupied in at least one of two years. These probabilities are dependent on 1) estimates of q 0 and q 1, given in the text, 2) the specified number of survey visits (s and s*), and 3) specified proportions of various combinations of true annual site status over two years (A, B, and C). Thirty percent of occupied sites have a true annual status of occupancy in only 1 of 2 years. A=0.15, B=0.15, C=0.7 s s* = 4 s* = 5 s* = 6 no two-stage Sixty percent of occupied sites have a true annual status of occupancy in only 1 of 2 years. A=0.3, B=0.3, C=0.4 s s* = 4 s* = 5 s* = 6 no two-stage Fifty percent of occupied sites have a true annual status of occupancy in only 1 of 2 years. A=0.25, B=0.25, C=0.5 s s* = 4 s* = 5 s* = 6 no two-stage Seventy percent of occupied sites have a true annual status of occupancy in only 1 of 2 years. A=0.35, B=0.35, C=0.3 s s* = 4 s* = 5 s* = 6 no two-stage

11 Appendix C. Scenario: Here is the objective for the scenario: For 1000 sites that have a true annual status of occupied sometime during a 2-year period ( given that a site is occupied... ), determine for a specified effort the predicted percentage of sites at which occupancy will be detected. Here is the assumption for the scenario: These sites match the larger 3-state database; thus the estimated parameters representing detection probabilities and the proportion of sites changing status are applicable. Here are the parameters for the scenario: Given that a site is occupied in a year, the probability of observing no detections on a single visit during that year is q 0, which = Given that a site is occupied in a year, the probability of observing presence detections, but not observing occupied detections, on a single visit during that year is q 1, which = Given that a site is occupied in a year, the probability of observing occupied detections on a single visit during that year is q 2, which = Forty percent of these sites are occupied in only 1 of the 2 years. Sixty percent of the sites are occupied in both of the 2 years. Thus, of the 1000 sites, 600 are occupied in year 1 and in year 2; 200 are occupied in year 1 and not occupied in year 2; 200 are not occupied in year 1 and are occupied in year 2. Here is the effort for the scenario: Year 1: A minimum of 5 visits will be conducted in year 1. If there are no detections observed in the first 5 visits, visits are terminated for that year. If during the first 5 visits, presence detections are observed, an additional 4 visits are conducted for a total of 9 visits. Year 2: For the sites with presence detections but no occupancy detections observed in year 1, a total of 9 year-2 visits are conducted. For the sites with no detections in year 1, a minimum of 5 visits will be conducted in year 2. If there are no detections observed in the first 5 year-2 visits to those sites, visits are terminated for year 2. If during the first 5 year-2 visits, presence detections are observed, an additional 4 visits are conducted for a total of 9 year-2 visits. 11

12 Year sites occupied only in Year sites occupied in both years = 800 occupied sites. a. First 5 visits 800(1! ( ) 5 ) sites with occupied detections = 589 occupied sites 800( ) sites with no detections = 11 no detection sites (200 / 800)(11) are not occupied in Year 2 = 3 errors after 5 visits b. Additional 4 visits (800! 589! 11) sites with presence-only detections = 200 presence-only sites after 5 visits 200(1! ( ) 4 ) sites with occupied detections = 131 occupied sites 200( ) 4 sites with no occupied detections = 69 presence only sites after 9 visits (200 / 800)(69) are not occupied in Year 2 = 17 errors after 9 visits Year sites occupied only in Year 2 + (600/800)11 sites with no detections in Year 1 = 208 sites with no previous detections (600 / 800)(69) presence-only sites from Year 1 = 52 Year-1 presence-only sites a. First 5 visits to 208 no-detection sites 208(1! ( ) 5 ) sites with occupied detections = 153 occupied sites 208( ) sites with no detections = 3 no detection sites = 3 errors after 5 or 10 total visits b. Additional 4 visits to above sites (208!153!3) sites with presence-only detections = 52 presence-only sites after 5 Year-2 visits 52(1! ( ) 4 ) sites with occupied detections = 34 occupied sites 52( ) 4 sites with no occupied detections = 18 presence-only sites after 9 Year-2 visits = 18 errors after 9 or 18 total visits c. Nine visits to 52 Year-1 presence-only sites 52(1! ( ) 9 ) sites with occupied detections = 47 occupied sites 52( ) 9 sites with no occupied detections = 5 presence only sites after 9 Year-2 visits = 5 errors after 18 total visits Predicted percentage of occupied sites at which occupancy will be detected sites at which occupied detections were observed / 1000 total occupied sites Occupancy will be detected at 95.4 percent of occupied sites. 12