San Juan National Forest - American Marten Snow Track Index Evaluation.doc 1

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1 San Juan Natonal Forest - Amercan Marten Snow Track Index Evaluaton.doc 1 27 June 2005 Jeffrey L. Beck Department of Zoology and Physology Unversty of Wyomng Larame, WY Phone: Emal: jlbeck@uwyo.edu SECTION I: ESTIMATING THE SNOW TRACK INDEX FOR AMERICAN MARTEN ON THE SAN JUAN NATIONAL FOREST, COLORADO BACKGROUND From 31 December through 22 March, 2005, 18 transects were sampled for Amercan marten tracks on 3 ranger dstrcts of the San Juan Natonal Forest n southern Colorado. Ten transects were sampled on the Columbne Ranger Dstrct, 3 on the Dolores Ranger Dstrct, and 5 on the Pagosa Ranger Dstrct. Transects are the samplng unts n ths samplng desgn, and Amercan marten tracks crossng or wthn 5 meters from transects are the elements measured n the desgn. Transects were stratfed nto 2 strata accordng to sutablty of marten habtat. Randy Ghormley, USFS Enterprse Bologst statoned on the Ro Grande Natonal Forest, summarzed marten habtat avalablty on the San Juan Natonal Forest and found that 88% of marten habtat s sutable and 12% margnal. Paul Morey, wldlfe bologst on the Dolores Ranger Dstrct provded an evaluaton of habtat around each transect by placng a 50-m buffer around each transect and usng the R2 Vegetaton GIS coverage to determne vegetaton and structure classes wthn each buffer. The analyss conducted by Paul Morey showed that of the 18 transects, 2 (11%) were n margnal habtat and 16 (89%) were n sutable habtat, proportonal to the avalablty of marten habtat strata on the forest. ESTIMATING THE ABUNDANCE OF MARTEN WITH THE SNOW TRACK INDEX Transects ranged n length from approxmately 6 to 24 km ( x = 13 km) or equvalently, 4 to 15 mles ( x = 8 mles; Fgure 1). Marten crossngs were talled for each 0.4 km (0.25 mles) segment of transect and summed across the length of each transect. Abundance of marten was calculated by dvdng the sum of track crossngs by the total length (km) of each transect. My statstcal notaton largely follows that of Schaeffer et al. (1996). The estmator for the populaton mean (μ) for a stratfed random sample s: y 1 L st Ny N = 1 = (1) Where, y st = mean estmator for a stratfed random sample, N = number of samplng unts, L = total number of strata, and, y = mean for stratum computed as a smple random sample:

2 San Juan Natonal Forest - Amercan Marten Snow Track Index Evaluaton.doc 2 y = n = 1 In the samplng protocol, tracks are only ncluded n the data set when they cross the samplng unt or approach wthn 5 meters of the road prsm on whch observers travel (Ghormley 2005). Each transect s lnear and the search area around each transect s ll-defned, varyng from 5 meters to the exact pont where tracks cross each transect, wth observers makng an effort to not count tracks that re-cross the transect; thus, transects, or samplng unts are one-dmensonal and results cannot be appled n a densty context. The drect consequence of one-dmensonal samplng unts are that the an nfnte number of samplng unts exst on the landscape, hence obvatng knowledge of N, or the total number of samplng unts. To compute a mean under these crcumstances I modfed the stratfed random samplng estmator for one-dmensonal samplng unts. The area (A) of each stratum s ncorporated n each estmator. Ths type of analyss s aded by computaton of areas of each stratum n a GIS: n y y st A L = y (2) = 1 A Alternatvely, f the proportonate area of each stratum s known, then that proportonate area can be ncluded as a weght (w) for each stratum () n the computaton of a mean: y L = wy (3) st = 1 The varance estmator for yst s: L N ˆ( ) n s V yst = N 2 (4) N = 1 N n Where, n = number of samplng unts n stratum, and s 2 = sample varance for stratum computed as a smple random sample N n Because 0.95 for each stratum, I removed the fnte populaton correcton N N n and N to compute the varance as follows: N s = (5) L 2 2 ˆ( ) V yst w = 1 n

3 San Juan Natonal Forest - Amercan Marten Snow Track Index Evaluaton.doc 3 Standard errors (SE) are computed by takng the square root of the varance estmator above and confdence ntervals are computed as y st ± t α/2, n-1 SE. s = (6) L 2 2 SE( yst ) w = 1 n RESULTS AND DISCUSSION Accordng to the smple random sample estmator, x ± SE Amercan marten tracks/km were 1.3 ± 1.1 for the margnal stratum, and 1.0 ± 0.3 for the sutable stratum. Amercan Marten tracks/km based on the weghted stratfed random estmator for one-dmensonal samplng unts was 1.1 ± 0.3 (95% CI = ; Fgure 2). Please see Excel spreadsheet for calculatons (Amercan Marten Snow Track Index for SJNF.xls). Because there were only 2 samplng unts n margnal habtat, t would be advsable to ncrease the number of transects n an effort to reduce varaton n estmates for ths stratum. Increasng samples n 1 stratum wll thus result n a need to proportonally ncrease the number of samples n the second stratum. Power analyss based on a set sample sze provdes an understandng of the number of transects needed to detect trends n Amercan marten abundance wth the snow track ndex (see Secton II). SECTION II: EVALUATING THE OPTIMAL LENGTH OF SNOW TRACK TRANSECTS AND POWER ANALYSIS TO DETERMINE AN ADEQUATE NUMBER OF TRANSECTS TO MONITOR TO DETECT DECLINE IN AMERICAN MARTEN ON THE SAN JUAN NATIONAL FOREST, COLORADO BACKGROUND In addton to provdng a revsed ndex for Amercan marten snow tracks, Randy Ghormley asked me to: (1) Assess the optmal length of a transect based on the exstng data. I defne optmal length as transects that are long enough to elmnate the bas of zero counts whle achevng hgh precson, as assessed wth the coeffcent of varaton (CV) n marten tracks; and (2) Conduct a power analyss based on the exstng data to estmate the sample sze needed to detect a trend based on the crtera n the protocol. In the San Juan Natonal Forest marten protocol, the use of a snow track ndex s ntended to provde at least 80% power to detect a 10% annual declne n marten tracks over 5 years wth a 20% chance of Type I Errors (α = 0.20; Ghormley 2005). Results from these analyses are to mprove montorng n the future.

4 San Juan Natonal Forest - Amercan Marten Snow Track Index Evaluaton.doc 4 METHODS To more concsely organze the methods I used n ths evaluaton, I have lsted them n bullet format. The frst secton detals methodology used to evaluate the optmal length of transects, whle the second secton descrbes the methods used to evaluate sample sze and power. Evaluatng Optmal Length of Snow Track Transects I defned optmal length of transects as the mnmum number of mles of transect necessary to provde relatvely unbased and precse estmates of the snow track ndex for Amercan marten. Bas was assessed through the proporton of tmes a zero ndex was computed and precson through the CV for 1,000 bootstrapped ndex estmates of Amercan marten tracks per mle I pooled groups of 4 contnuous 0.25-mle segments wthn each transect to sum to 1 mle, and deleted segments from 0.25 to 0.75 mles at the end of transects. I evaluated data only from sutable transects. Each segment was an ndependent observaton because total counts for each segment were not related to counts from other segments To better understand the dstrbuton of the data I summed the total number of 1-mle segments across all transects. I removed those 1-mle segments where the tally of marten tracks was 0 to avod under representng larger values. I then computed the mean and standard devaton (SD) for the remanng 75 transect segments wth non-zero data, and removed all values >3 SDs from the mean. I selected ths cutoff value because for data that are dstrbuted normally, 3 SDs around a mean contan 99.7% of the data (Fowler et el. 1998). The mean number of marten tracks per mle segment was 2.8 and the SD was 2.2. Three SDs above the mean ndex yelded a value of 9.4, thus I removed all transects of at least 9 tracks per mle. Hayward et al. (2002) followed ths procedure n Russa where hgh numbers of track counts ndcated ndvdual Amur tgers (Panthera tgrs altaca) had repeatedly crossed survey routes, weavng on and off the path To represent counts encountered on the San Juan Natonal Forest, I used all transect segments, ncludng those wth 0 counts n bootstrappng. Ths sample ncluded 122, 1- mle long transect segments representng all 16 transects n sutable habtat. I conducted 1,000 bootstrap teratons wth SAS (2001) for subsamples of sze 5, 10, 12, 15, and 20 to produce a new samplng dstrbuton from the orgnal data. The new samplng dstrbuton was equvalent to transects 5, 10, 12, 15, 20 mles n length. I then calculated means for each 1,000 samples wthn each transect length category. From these means I computed SDs, CV s, and proporton of means consstng of zero counts I created box plots to evaluate the dstrbuton of bootstrapped means and SDs. I plotted the mean ndex estmates wth SDs to assess change n estmates and precson for each transect segment sze. I plotted CVs for each transect length and tabulated the change n CV wth each ncreasng segment length to provde a way to assess mprovement n the marten track ndex wth successvely longer transects

5 San Juan Natonal Forest - Amercan Marten Snow Track Index Evaluaton.doc 5 Sample Sze and Power I used Program Montor (Gbbs 1995) to evaluate power to detect trend over tme. I selected a 5-year trend analyss perod because that s the perod of nterest outlned n the San Juan Natonal Forest montorng plan (Ghormley 2005). I also evaluated power based on a 10-year perod, because the farther apart frst and last samplng perods are, the more lkely t s that trends n counts can be detected (Gbbs 1995) To compute power, I selected the 16 transects montored durng the wnter of n the sutable stratum. I only consdered transects n the sutable stratum because they comprsed 89% of all transects. Because Program Montor requres non-zero data to compute power, I provded a value of for 3 transects that dd not have an ndex value. From these 16 transects I computed the snow track ndex for Amercan marten per klometer, whch s equvalent to a mean ndex Program Montor requres entry of SDs of mean counts from each transect to ncorporate temporal varaton wthn transects. Because each transect was only vsted once durng the wnter, I had no measure of SD and thus computed SDs as 5%, 10%, and 25% levels of the mean track ndex. For example the 10% SD was computed as mean ndex To ncorporate spatal varaton, I entered a standard value for SD of 0.10 Power was evaluated n 10,000 teratons and based on an exponental declne. I selected the exponental model because I assumed that declne n each successve observaton would be proportonal to the magntude of the current populaton level (Thompson et al. 1998). For the exponental model, Program Montor generates lognormal sample counts for each survey occason (.e., each year of montorng) that are log-transformed pror to the regresson analyss (Gbbs 1995). Lastly, I tabulated power results accordng to number of years of montorng and temporal varaton level (SD of the mean track ndex) RESULTS AND DISCUSSION Evaluatng Optmal Length of Snow Track Transects Only 1 out of 1,000 bootstrapped samples for transects of 5 mle length were zero, ndcatng that transects of at least 5 mles n length ensure hgh accuracy and detectablty of marten. In comparson, the 3 transects n the sutable strata wth zero counts were 4.0, 6.2, and 7.2 mles n length, and ranked second, ffth, and sxth shortest, respectvely, n transect length. These results ndcate that short transects are most lkely to yeld zero counts of marten tracks. Boxplots of bootstrapped means and SDs for each length of transect show a predctable mprovement n precson as length of transects ncreases (Fgure 3). The mean snow track ndex per mle was about 1.5 for bootstrapped samples of all lengths (Fgure 4). The hghest mprovement n the CV for the mean track ndex occurred as transects ncreased from 5 to 10 mles n length (Table 1; Fgure 5). The CV of the mean marten ndex decreased approxmately 50% from transects of 5 mles n length to those of 20 mles n length.

6 San Juan Natonal Forest - Amercan Marten Snow Track Index Evaluaton.doc 6 Sample Sze and Power Power to detect a 10% declne n the Amercan marten snow track ndex ranged from to for a 5-year montorng effort and from to for a 10-year montorng effort (Table 2). The ntent of montorng abundance of Amercan marten on the San Juan Natonal Forest wth a snow track ndex s to provde at least 80% power to detect a 10% annual declne n marten tracks over 5 years (Ghormley 2005). Power of 80% or hgher to detect a declne was only acheved for montorng over 10 years, and then only prmarly for a 10% declne (Table 2). Montorng over a longer perod of tme such as 10 years s an opton that should be consdered because the farther apart frst and last samplng perods are, the more lkely t s that trends n counts can be detected (Gbbs 1995). However, montorng over a 5-year perod may better meet forest plannng needs. Several steps can be followed to ncrease power to detect a 10% annual declne over 5 years (see Recommendatons). RECOMMENDATIONS Power to detect an annual decrease of 10% over 5 years was not adequate based on the data collected durng wnter Approaches to ncrease power nclude avodng recordng tracks from anmals that weave back and forth across snow transects, montorng over a longer tme perod, addng more transects, and ncreasng the length of some transects to reduce varaton n track counts over tme. My specfc recommendatons to ncrease power are: Tran observers to avod countng multple tracks from ndvdual martens that weave back and forth across the transect path. Counts of multple tracks from the same anmals cause bas n estmates and nflate varaton Estmate varaton after 2 or preferably 3 years of data collecton wthn (temporal varaton) and among (spatal varaton) transects. Enter the SDs obtaned by longer term montorng n Program Montor to counter the use of constraned levels of SD I used based on only 1 year of montorng. More realstc estmates of temporal and spatal varaton should provde a better analyss of power Increase the length of all transects to at least 10 mles to reduce varaton n track counts and to avod 0 counts. Currently, only 4 of 18 transects exceed 10 mles n length. Transects exceedng 10 mles could be truncated to 10 mles to reduce costs, but t s approprate to mantan transects exceedng 10 mles f ths s not logstcally feasble Add at least 5 transects to the sutable stratum and 1 addtonal transect to the margnal stratum to reduce bas and mprove precson Consder evaluatng trend n 10 years. Power analyss for a declne of 10% n the marten ndex ndcated that the present number of transects should provde adequate power to detect a trend n 10 years. Levels of temporal and spatal varaton reman unclear, and these should be ncorporated n the future to reassess power. However, montorng over a 5-year perod may better meet forest plannng needs, so mprovng estmates wll help n obtanng suffcent power to montor a declne n marten snow tracks over 5 years

7 San Juan Natonal Forest - Amercan Marten Snow Track Index Evaluaton.doc 7 LITERATURE CITED Fowler, J., L. Cohen, and P. Jarvs Practcal statstcs for feld bology, second edton. John Wley and Sons, Lmted. Chchester, West Sussex, England. Ghormley, R MIS montorng plan for the Amercan marten on the San Juan Natonal Forest: ntal montorng protocol. San Juan Natonal Forest, Colorado. March Gbbs, J. P Montor verson 6.2 users manual. Exeter Software, Setauket, New York, USA. Hayward, G. D., D. G. Mquelle, E. N. Smrnov, and C. Natons Montorng Amur tger populatons: characterstcs of track surveys n snow. Wldlfe Socety Bulletn 30: SAS Insttute SAS/STAT user s gude, release 8.2. SAS Insttute, Inc., Cary, North Carolna, USA. Scheaffer, R. L., W. Mendenhall III, and R. L. Ott Elementary survey samplng, 5 th ed. Duxbury Press, Boston, MA. Thompson, W. L., G. C. Whte, and C. Gowan Montorng vertebrate populatons. Academc Press, San Dego, Calforna.

8 San Juan Natonal Forest - Amercan Marten Snow Track Index Evaluaton.doc 8 Table 1. Coeffcent of varaton (CV), mprovement (%), or the percentage dfference n CV between each transect length category, and change n CV per mle (%) between each transect length category of mean Amercan marten snow track ndex, San Juan Natonal Forest, wnter Transect length (mle) CV Improvement Change per mle Table 2. Power to detect 10%, 5%, and 3% declne n trend of the mean ndex of Amercan marten tracks per mle across 5 and 10 years. Power analyss based on 16 snow transects sampled n sutable habtat on the San Juan Natonal Forest n wnter The power analyss was one-taled at alpha = 0.20, and ncluded a trend varaton (SD = 0.10) parameter to represent spatal varaton, whch consdered the degree to whch a gven trend vares at random among the 16 transects. To represent temporal varaton n marten counts for each transect, SDs of the mean were computed as 5%, 10%, and 25% of the mean ndex. Power analyss was conducted wth Program Montor (Gbbs 1995). Standard Devaton of the Mean Index Declne n trend 5% 10% 25% 5 years years

9 San Juan Natonal Forest - Amercan Marten Snow Track Index Evaluaton.doc Frequency >15 Transect length (mles) Fgure 1. Frequency hstogram dsplayng the dstrbuton of transect lengths (mles) used to montor Amercan marten wth a snow track ndex, San Juan Natonal Forest, Colorado, wnter Trend n Amercan Marten Snow Track Index - San Juan Natonal Forest Martens/km Year Fgure 2. Amercan marten tracks per klometer ± 95% confdence nterval, San Juan Natonal Forest, Colorado, wnter Ths s an example of a plot that could be used to plot the annual mean ndex wth ts assocated confdence ntervals to depct changes over tme.

10 San Juan Natonal Forest - Amercan Marten Snow Track Index Evaluaton.doc 10 5 Mean Marten Tracks 4 3 mean_tracks subsze Standard Devaton of Marten Tracks 4 3 std_tracks subsze Fgure 3. Box plots of means (upper) and standard devatons (lower) of an ndex representng Amercan marten tracks per mle for subsamples of 5, 10, 12, 15, and 20 mle transects from 1,000 bootstrapped samples. Bootstrappng sampled wth replacement from 122, 1-mle segments pooled from transects sampled on the San Juan Natonal Forest, wnter

11 San Juan Natonal Forest - Amercan Marten Snow Track Index Evaluaton.doc Martens/Mle Mles Fgure 4. Mean ± SD Amercan marten ndex from 1,000 bootstrapped subsamples of 5, 10, 12, 15, and 20 mle transects. Bootstrappng sampled wth replacement from 122, 1-mle segments pooled from transects sampled on the San Juan Natonal Forest, wnter Fgure 5. Coeffcent of varaton n mean counts of Amercan marten snow track ndex from subsamples of 5, 10, 12, 15, and 20 mles derved through bootstrapped samples of 1,000 teratons, San Juan Natonal Forest, wnter