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Evaluation of aerial surveys of ptarmigan (Lagopus spp.) Pelletier, Luc 1996

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E V A L U A T I O N OF AERIAL SURVEYS OF PTARMIGAN {Lagopus spp.). by Luc Pelletier B.Sc, The University of Ottawa, 1994 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR T H E DEGREE OF MASTER OF SCIENCE in THE F A C U L T Y OF G R A D U A T E STUDIES Department of Zoology We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA August 1996 © L u c Pelletier, 1996 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of The University of British Columbia Vancouver, Canada DE-6 (2/88) ABSTRACT Ptarmigan (Lagopus spp.) range over most of the Canadian arctic and alpine; thus, a technique is needed to track their densities over a large scale, which current techniques do not. I evaluated a program of aerial surveys done from 1990 to 1996 at Kluane, Yukon, which could do so. The indices of density obtained from aerial counts were subject to many types of errors. Of these, the counting bias, probably the largest source of bias, could have been corrected partially by using the simultaneous 2-sample capture-recapture method. However, the correction factors were highly imprecise due to the low number of double-counts. The variation of total survey bias (with 5 replicates of the survey over 2 weeks in 1996) was low in comparison to the variability of the indices between years; thus, the technique is repeatable. About 70% of the variability of the index in a given year could be attributed to daily factors. I also calibrated the index using ground counts (line transect estimates based on perpendicular distances). A test of accuracy of the line-transect method on a 0.77 km 2 grid showed line-transect estimates had low bias (-3 to -7%). The calibration showed a positive and linear relationship between the aerial index and ptarmigan density. However, this calibration held only across areas in the same year. The 1996 index was significantly lower than the 1995 index but the density was higher in 1996. This could be explained by a reduced flushing behaviour in 1996, perhaps caused by increased avian predation pressure. Other indices of abundance of the Kluane ptarmigan population from winter encounter rates are also unreliable as they also predicted a drop in abundance for 1996 while the density actually increased. I recommend the use of line-transects to wildlife managers since the technique provided accurate and fairly precise (%cv = 15-25%) estimates of density. n ABSTRACT TABLE OF CONTENTS II TABLE OF CONTENTS III LIST OF FIGURES VIII LIST OF TABLES VI ACKNOWLEDGEMENTS X 1. INTRODUCTION. 1 2. AERIAL SURVEY PROTOCOL. 5 2.2.1 Sampling frame. 5 2.2.2 Counting. 6 2.2.3 Index of density - the ratio estimator. 7 2.2.4 Index of density - frequency of transects having at least one sighting. 12 2.2.5 Comparing the index across years. 13 3. TOTAL SURVEY ERROR. 14 3.1 THEORETICAL ANALYSIS OF ERRORS. 14 3.1.1 Sources of errors. 14 3.1.2 Estimator bias. 15 3.1.3 Sampling bias. 17 3.1.4 Counting bias. 18 UJ 3.1.5 Random-sampling error. 27 3.1.6 Conclusion. 28 3.2 REPEATABILITY. 28 3.3 RESULTS. 30 3.4 DISCUSSION. 35 3.4.1 Validity of the replication of the 1996 aerial survey. 35 3.4.2 Repeatability. 36 3.5 CONCLUSION. 36 4. LINE TRANSECT METHODS FOR GROUND COUNTS. 37 4.1 INTRODUCTION. 37 4.2 THEORY. 38 4.2.1 Estimators based on perpendicular distances. 38 4.2.2 Estimators based on sighting angles and distances. 40 4.3 METHODS. 41 4.3.1 Sampling protocol. 41 4.3.2 Analysis of data. 41 4.3.3 Test of accuracy. 42 4.4 RESULTS. 43 4.4.1 Line-transect estimates based on perpendicular distances. 43 4.4.2 Estimates based on sighting distances and angles. 51 IV 4.5 DISCUSSION. 54 4.5.1 Mobility prior to detection. 54 4.5.2 Accuracy and precision. 54 4.5.3 Estimators based on sighting distances and angles. 55 4.6 CONCLUSION. 55 5. CALIBRATION. 57 5.1 INTRODUCTION. 57 5.2 METHODS. 58 5.3 RESULTS. 59 5.4 DISCUSSION. 64 5.4.1 Calibration. 64 5.4.2 Ratio estimator vs. frequency estimator for the aerial index. 72 5.4.3 Comparison with trends from other techniques. 75 6.5 CONCLUSION. 78 7. CONCLUSIONS. 79 BIBLIOGRAPHY 80 V LIST OF TABLES Table 1. Comparison of the accuracy, relative cost and scale using different techniques to obtain indices of density for ptarmigan. 3 Table 2. Examples of uses of aerial surveys in wildlife management over the past 15 years. 3 Table 3. Effect of a bias B on the probability of an a error greater than 1.96a (from Cochran 1977) 17 Table 4. Some of the factors suspected of having an effect on the aerial counts of ptarmigan at Kluane. 19 Table 5. Visibility correction factors (vcf) for five aerial surveys of manatees (from Packard et al. 1985). 22 Table 6. Cost and type of bias that should be corrected by different methods for aerial surveys of ptarmigan. 23 Table 7. Aerial survey indices of density for 1990-1996 and transect efforts. 31 Table 8. Correction factors for the perception bias a) for each of the 1995 and 1996 aerial surveys b) for each of the areas in 1995 and 1996. 34 Table 9. One-tailed variance ratio test of Ho: s 1990-1996 ^ s 1995 o r 1996', using the aerial indices of Table 7. 35 Table 10. Transect effort and sightings by area for the ground line transects of ptarmigan. 45 Table 11. Estimates of density D (males/ km2) by area based on perpendicular distances line-transects estimators. 45 Table 12. Estimation summary of the probability detection functions of Figure 4. 46 Table 13. Results of the tests of accuracy of the line-transect methods on the grid. 49 VI Table 14. Line-transect estimates of density D (males/km2) based on sighting distances, using either the Hayne's (H) or the generalized Hayne's (GH) estimator, for each of the areas surveyed on ground in 1995 and 1996. 51 Table 15. Relative bias of estimates of density D (males/km2) on the grid based on sighting distances and angles. 52 Table 16. Comparison of the means of aerial indices per year between 1995 and 1996 with 2-tailed t-tests. 67 Table 17. Comparison of the mean proportions of ptarmigan seen on the ground out of total per survey between 1995 and 1996 with a 2-tailed t-test. 67 v n LIST OF FIGURES Figure 1. 1957-1993 population trends of willow ptarmigan in 4 areas of northern British Columbia and Yukon based on ground censuses (territories/km ). 4 Figure 2. Correlation between the counts of ptarmigan per transect in aerial surveys and the length of the transects. 8 Figure 3. Trends of the aerial indices from 1990-1996. 32 Figure 4. Distribution of perpendicular distances and probability detection functions for line-transects of male ptarmigan. 47 Figure 5. Precision (%cv) of line-transect estimates in relation to the number of observations n.50 Figure 6. Correlation of the line-transect estimates of density (males/km2) based on sighting distances (Dh) with the estimates based on perpendicular distances (Dp). 53 Figure 7. Correlation of the Hayne's line-transect estimates of density (DH) (males/km2) with the estimates based on perpendicular distances (Dp). 56 Figure 8. Central trend line of the geometric mean regression ± 9 5 % confidence limits for the calibration of the aerial index per area vs. the density. 61 Figure 9. Comparison of the density of ptarmigan between 1995 and 1996 in areas 1 and 2 and on the grid. 68 Figure 10. Central trend line of the geometric mean regression ± 9 5 % confidence limits of the aerial index RQ based on the number of ptarmigan flushed/ km of transect vs. the density. 70 Figure 11. Non-linear correlation between the frequency estimates/(% of transects with > 1 ptarmigan) and the ratio estimates RQ (ptarmigan/km of transect) used as aerial indices. 73 Figure 12. Comparison of trends of different indices of density for the Kluane ptarmigan population from 1989-1996. The indices are from different techniques and are ratios of i) v m aerial counts (# ptarmigan/ km of transect), ii) encounter rates on foot (# ptarmigan/ hour in the field), and iii) encounter rates on snowmobile (# ptarmigan/ hour in the field). 76 IX ACKNOWLEDGEMENTS I am in debt to my supervisor Charles Krebs for having let me worked on a crazy project in spite that in my first year of master (and maybe for even longer), I looked as if I had no idea of what I was doing. Charley always granted me immediate help when I needed so and this is a major factor that permitted me to complete my master in 2 years. I want to thank Jamie Smith, Wendy Choquenot, and Charley for reviewing my early drafts. In addition to those people, I want to thank Frank Doyle, Kathy Martin, David Choquenot, David Hik and Tony Sinclair for precious comments and ideas about my work. My work was greatly helped by Jocylyn McDowell, head technician of the CSP at Kluane, the Williams, managers of the Arctic Institute field station at Kluane, and Irene Wingate, head technician of the CSP in Vancouver. Je tiens a remercier mon assistante de terrain Julie Senecal pour son aide hors de 1'ordinaire. Julie et ma famille m'ont fourni un support dont je n'aurais pu me passer. This project was funded by a special collaborative project research grant from the Natural Sciences and Engineering Research Council of Canada (NSERC), granted to Charles Krebs, and by a Northern Scientific Training Program grant, a NSERC graduate scholarship, and a Quebec second language scholarship. X 1. INTRODUCTION. Ptarmigan (Lagopus spp.) range over the Canadian arctic and alpine and have a special cultural importance to people of the North. For the management and conservation of these species, it is imperative to have a tool that permits us to detect changes in population sizes on a large scale. Such a tool would permit evaluation of the success or failure of management plans, and provide information on the natural variability of population densities. For instance, on some small areas of northern B.C and Yukon the density of willow ptarmigan was shown to follow a 10-year cycle (Mossop 1994, Hannon (in Boutin et al. 1995)) (Figure 1). Sampling techniques are needed to track population trends of ptarmigan on a large scale. Present methods of estimation of density of ptarmigan, typically censuses, apply only to small areas (< 2 km2), and it is not known whether trends from those small areas of optimal habitat represent what is happening at a much broader scale (hundreds of km2) and in less optimal habitat. A census is a complete enumeration (or total count) of the population while a survey is only a sample of that population (Cochran 1977). Surveys have a large advantage over censuses: reduced cost, greater speed, greater scope, and even greater accuracy in some cases (higher quality data can be obtained when the volume of work is reduced) (Cochran 1977). Accuracy is the nearness of a measurement to the actual value of the variable being measured, whereas precision refers to the closeness to each other of repeated measurements of the same quantity (Zar 1984). The most accurate census method is considered to be to band all the birds in a given area and to map the territories (e.g. Mossop 1987; Hannon 1983 in Boutin et al. 1995) (Table 1). However, it is impossible to use this technique on a large scale. For instance, Hannon (in Boutin et al. 1995) used one 57 ha plot from 1980 to 1992. Transect censusing by counting every bird in an area using a dog (e.g. Mossop 1994) is faster and more practical, but may be less accurate. Transect sampling (developed for ptarmigan in this study) can be applied reliably on even larger 1 areas (about 5-10 km2) for about the same effort as transect censusing, but may be more accurate than the latter. The line-transect method (Buckland et al. 1993) is an efficient way to sample elusive populations such as ptarmigan (see section on line-transect methods). Aerial surveys could provide a way to estimate densities of ptarmigan over large, remote, and inaccessible areas at a cost similar to ground censuses of small areas. Aerial surveys are widely used in wildlife management (Table 2). However, aerial counts can be highly inaccurate. Given this inaccuracy, the method could still be used to provide indices of population trends instead of estimates of density. An estimate tries to provide a number close to the true value of density (i.e. it has to be accurate). It is expressed in the same units, e.g. number of birds per square kilometre. On the other hand, an index tries to provide a number that reflects changes in the true value. Indices do not need to be expressed in the same units as the values they try to reflect, e.g. encounter rates. Indices may be sufficient when only population trends are needed. For the aerial surveys of ptarmigan conducted at Kluane, south-western Yukon, Canada, from 1990 to 1996, it became even questionable if high aerial counts corresponded to higher densities of ptarmigan. An evaluation of the technique was necessary to establish the reliability of the existing trend data and of the technique itself. Chapter 2 summarizes the aerial survey protocol. Chapter 3 is about the total survey error. The first part showing that there are components of the total error that could not be reduced or estimated, and the second part assessing the variability of the total bias and its effect on the repeatability of the technique. Chapter 4 is about the line-transect methods that I used for ground counts in a calibration of the aerial survey indices of density, chapter 5. 2 Table 1. Comparison of the accuracy, relative cost and scale using different techniques to obtain indices of density for ptarmigan (+ = good, 0 = medium, - = bad). Techniques accuracy cost scale mapping territories census transects sampling transects aerial counts ++ 0 0 + * + + - + ++ * requires at least 40 sightings to get line-transect estimates to be accurate Table 2. Examples of uses of aerial surveys in wildlife management over the past 15 years. BIG GAME BIRDS bighorn sheep (Bodie et al. 1995) great blue heron nests (Dodd & Murphy 1995) mule deer (Kufeld et al. 1980; White et al. wood stork nests (Rodgers et al. 1995) 1989) white-tailed deer (Samuel et al. 1987) bald eagles nests (Grieretal. 1981) bison (Wolfe & Kimball 1989) emus (Caughley & Grice 1982) feral equids (Graham & Bell 1989) black ducks (Conroy et al. 1988) elk (Bear etal. 1989) mallards (Reinecke et al. 1992) moose (Thompson & Schachow 1981; Peterson waterfowl (Broome 1985; Butler et al. 1995; & Page 1993; Bowden & Kufeld 1995) Prenzlow & Lovvorn 1996) kangaroos (Barnes et al. 1986; Choquenot 1995a) ptarmigan (this study) feral pigs (Choquenot 1995b) AQUATIC MAMMALS caiman (Mouraoetal. 1994) harp seals (Myers & Bowen 1989) pronghorn (Firchow et al. 1990; Pojar et al. manatees (Packard et al. 1985) 1995; Kraft etal. 1995) dugongs (Marsh & Sinclair 1989) dolphins (Holt & Cologne 1987) 3 giu^/sauoiujai 2. AERIAL SURVEY PROTOCOL. 2.2.1 Sampling frame. Each year around 1 May, from 1990 to 1996, the Kluane Boreal Forest Ecosystem Project (Boutin et al. 1995; Krebs et al. 1995) conducted aerial surveys of the breeding population of ptarmigan in the subalpine region of Kluane. The objective of the survey was to track population trends of the breeding population of ptarmigan over years. At least two species of ptarmigan breed within the boundaries surveyed: the common willow ptarmigan (Lagopus lagopus) and the rarer rock ptarmigan (Lagopus mutus). Few white-tailed ptarmigan (Lagopus leucurus) occur and there was no indication that they were breeding within the areas surveyed. The survey occurred during the peak of male territorial behaviour, about 3 weeks before the start of egg laying (which started around May 23 (n = 8 clutches) in 1995). In the Chilkat Pass, at least 70% of the males were present on territories at the end of April (Mossop 1987). Non-territorial males should also be in the area, but in the tall and dense willow (Mossop 1987) which was also surveyed. The vegetation of the Kluane subalpine is quite homogeneous. Three major plant species provide cover for the birds: willow (Salix spp.), dwarf birch (Betula glandulosa), and white spruce (Picea glauca). The vegetation is mostly low (< 2 m) and with the snow pack did not provide much cover. Snow cover varied from year to year, and ranged from 25 to 100%. The subalpine zone surrounding the Kluane valley was divided in 7 distinct areas. Natural features such as ridges and streams were used to delineate the areas. The size of the areas varied from 3 to 13.5 km 2. Not all of the areas were sampled each year. Two to 5 transects, 1.5 to 10 km long, were run parallel to the contour lines in each area. The details will be given in Table 7 (chapter 3). The transects are the sampling units. The areas were not used as strata to increase precision because the sample size per area was too low. A pseudo-systematic sampling procedure was used since the transects were more or less evenly spaced out by the naked eye. The distance 5 between transects was about 300-500 m. The number of transects flown per area was adjusted roughly to the width of the areas. There was no randomization of the location of the first sampling unit. They tried to survey the same transects over years, but this could not be done precisely without having flagged the transects on the ground. The section on the sampling bias deals with ways to improve the sampling frame. 2.2.2 Counting. Three observers counted ptarmigan in the Heliocourier aircraft: one at the front at the right hand side of the pilot, and two on the back seats, one on each side. The pilot did not take part in counts. Speed and elevation (30-50 m) were kept as constant and low as possible, but still varied because of wind gusts. Even in fine weather, the wind could become gusty fast in the alpine. Counts were unbounded. Some ptarmigan tended to flush when the plane was over them, which made them more visible. They flushed in pairs or alone. Only in 1991 were birds still seen in flocks. It was sometimes possible to detect birds on the ground or perched on small spruce. For each transect, each observer recorded i) the number of birds seen, if they were ii) singles, in a pair or in a flock, and iii) flushed or on the ground. Until 1995, ii) and iii) were not recorded systematically. The counts are expressed as encounter rates, i.e. number of observations per km of transect, and as the frequency of transects having at least one sighting. They are not expressed as estimates of density because the strip widths were too variable and too many birds were missed. Since the rates of encounter were low, a simple method was used to avoid double-counting by the two observers sitting on the right hand side of the plane. In the plane, observers were linked by intercom. Whenever the back seat observer detected a ptarmigan, he/she spoke into the microphone about 1 second later. The front seat observer noted whether or not he made the same observation. The number of double-counts were kept. This lapse of 1 second was short 6 enough to make sure that the front seat observer counted as a distinct individual another bird seen a few seconds after or before. Graham and Bell (1989) used a similar method for feral equids (Equus caballus) and donkeys (E. asinus). More complex survey procedures have been used for higher densities or when more observers could be used (e.g. Marsh and Sinclair 1989). We started using this procedure to determine the number of double-counts in 1995. Before 1995, the double-counts were determined after the end of each transect, which may have been inaccurate. 2.2.3 Index of density - the ratio estimator. Since the transects were of unequal lengths, the number of observations per transect was weighed by the length of the transects. I used the ratio method to weigh the observations. With the ratio method, an auxiliary variate x, (in this case, the length of transect i), correlated with y, (the number of observations in transect i), is obtained for each unit in the sample. Figure 2 shows that the relationship between the y, and the JC, fulfils the conditions so that the ratio estimator be a valid linear unbiased estimator: i) straight line relation passing through the origin, and ii) the variance of the points y, about the line seems roughly proportional to JC,-. There is no need to transform the data. Since the ratio estimator for systematic sampling is not available yet, I had to use the ratio estimator for simple random sampling and for stratified random sampling. It is assumed that both y and x are random variables. In the survey, y is random but x is not. The theory presented here is from Cochran (1977). Thompson (1992) and Rao (1988) also provide reviews of the ratio estimator. 7 Fig 2. Correlation between the counts of ptarmigan per transect and the length of the transect, a) 1991 survey, n = 19 40 0) (00 c CS Q. Q5 1_ f o 3 Z 5 0 -| H 1 —• • -0 1 2 3 4 5 6 7 8 9 10 Length of transect (km) b) 1992 survey, n = 20 14 C 1 2 O o CO c IO-CS O) E 8 cs o I_ -Q 4 4, • 3 • • • 0 • 0 1 2 . 3 4 5 6 7 8 9 10 Length of transect (km) c) 1993 survey, n = 20 4 j Oi 0) W 3 c (0 O) 2,5 E >_ « 2 Q. H— O 1,5 i-• 0) 1 ' 2 0,5-• — i < 1 • 0 ( 3 • — 1 2 •» • 1 • • 1 ' 3 4 5 6 7 Length of transect (km) 8 9 10 d) 1995a survey, n = 18 12 -i § 10 -0) to c flj 8 CD 1 <S 6 O. o i- 4 V • • Numbe - -• • • • 1 1 0 0 1 +-1 2 3 4 5 6 7 Length of transect (km) 8 9 10 9 RATIO ESTIMATOR FOR SIMPLE R A N D O M SAMPLING The ratio estimator for simple random sampling is: X*,. X X where y, x = sample totals of the yi, x\ y , x = sample means of the y\, Xj The sample estimate of the variance is: varl n - 1 where /= n/N = sampling fraction n = number of units sampled N = total number of sampling units The total number of possible sampling units, N, is unknown because there are no real boundaries to the sampling units. Hence, a conservative approach is to assume N as infinite. Therefore, the sampling fraction/is set equal to zero. The estimated standard error is: Confidence limits may be obtained using the normal approximation. The normal approximation and x are both less than 0.1. When these conditions do not apply, the formula for var(R) tends to holds reasonably well if the sample size is at least 30 and is large enough so that the cv's of y 10 give values that are too low and the positive skewness in the distribution of R may become noticeable. A more conservative approach is to use the Student's distribution: fi±'«(2),n-1A/var(/?) RATIO ESTIMATOR FOR STRATIFIED R A N D O M SAMPLING Stratification can help to increase the precision of a sample. There are two types of stratified ratio estimate. One is the separate ratio estimate: h h Xh where Rh = ratio estimate for stratum h yh, xh = 2>/,i, 2>m for stratum h An alternative is the combined ratio estimate: Xst ^ NhXh where Nh = total number of sampling units in stratum h ~ size of area Unless Rh, the ratio estimate for stratum h, is constant from stratum to stratum, the use of a separate ratio estimate in each stratum is likely to be more precise than the regular ratio estimate if the sample in each stratum is large enough. With only a small sample in each stratum, the combined estimate is recommended (Cochran 1977, p.167). The formulas for variance estimation need to be derived. 11 JACKKNIFE M E T H O D The jackknife (or Quenouille's) estimate RQ is worth consideration for moderate or small samples since it reduces the bias of the estimation method from 1/n to 1/n (Cochran 1977, p. 175-180). With g groups, n = mg, and the sampling fraction/negligible, RQ is the average of the g quantities Rj=gR-(g-l)Rj where R. = £ y / £ x , computed after omitting group j. The choice m = 1, g = n, seems best with the jackknife in small samples. If the R/ could be regarded as g independent estimates of R then, with simple random sampling, an unbiased estimate of the variance would be v a r f e U ( 1 f>> ^ 2.2.4 Index of density - frequency of transects having at least one sighting. A problem with the data is the preponderance of zeroes. No observations were recorded in half of the sampling units on average (range between 26% and 80% with average of 49%). An alternative to the ratio estimator is to use the frequency of transects having at least one sighting (see Caughley 1977b, p.20). Such a technique assumes that the transects are of equal length, which was not the case. In order to standardize the length of the aerial transects to about 5 km, transects shorter than 4 km were recombined randomly two by two. I calculated the frequency from that new set of recombined and standardized transects. I repeated the process 1000 times for each survey to remove the effect of randomness on the estimates. By doing so, the maximum variation of a frequency was of 9.1 % (%cv). The biggest problem with this estimator is that with its discontinuous nature it is not precise given the small sample sizes used. A minor problem is that it does not provide any estimate of random error. 12 2.2.5 Comparing the index across years. To determine if the difference between 2 independent ratios was significant, I used a t-test where v a r - R2) = var(fl,) + var(fl2) (Cochran 1977 p. 180) so that For the frequency estimator, I used the same test but on the arcsin transformed frequencies . 13 3. TOTAL SURVEY ERROR. Many errors can affect the aerial survey indices of density. Usually wildlife managers focus on factors that can affect the counts themselves, but I show in a theoretical analysis of errors that many more factors are part of the total survey error and affect the final estimate. Since I expected from this analysis that the total error be large, I assessed its importance empirically to determine how repeatable the technique is. 3.1 THEORETICAL ANAL YSIS OF ERRORS. 3.1.1 Sources of errors. The total error of a survey estimate is made of non-accidental errors or biases, of accidental errors or random errors, and of a rounding error (Bjerhammar 1973). There are 4 sources of non-accidental errors or biases: (i) consistent errors in counting (counting bias or errors of measurements) (Jolly 1969; Cochran 1977) (ii) faulty methods of sampling (sampling bias) (Jolly 1969) (iii) methods of obtaining the final estimate from the sample counts (estimator bias) (Jolly 1969) (iv) failure to measure some of the units in the chosen sample (nonresponse bias) (Cochran 1977, p.359) There are at least 2 sources of accidental errors: (i) from sampling only a portion of the total population (random-sampling error) (Cochran 1977) (ii) errors introduced in editing, coding and tabulating the results ("human" errors) (Cochran 1977) 14 To summarize, the total error is made of: Total error - [counting bias + sampling bias + estimator bias + non-response bias] My goal was to reduce or eliminate each component of the total survey error. When I could not eliminate an error, I tried to assess its effect on the total error. I did not analyse the effects of the human errors and of the rounding error. I tried to minimize human errors by checking the data twice and by using a computer for calculations and data storage. For the rounding error, I tried to be as precise as possible in every measurement. 3.1.2 Estimator bias. The first component of the total error I could eliminate was the estimator bias. Errors in estimation come from faulty methods of analyzing the results. They also come to a lesser degree from the limitations of some estimators, especially with low sample sizes. A method of estimation is said to be unbiased when the average value of the estimate, taken over all possible samples of given size n, is exactly equal to the true population value (Cochran 1977, p.22). For this survey, the ratio estimator was probably the best available for the data (Figure 2). However, the low sample size causes the ratio estimator to be biased. The bias of the ratio estimate comes from its positively-skewed distribution (Cochran 1977, p. 162), the skewness decreasing with an increase in the sample size. In general, its bias is of order 1/n. Since the standard error of the estimate is of order iNn, the quantity (bias/^) is also of order 1/Vn and becomes negligible as n becomes large. In practice, this quantity is usually unimportant in samples of moderate size (> 30), but can be of interest in small samples. The bias can be assessed by: + [random-sampling error + human errors] + rounding error -covl (R,x) ,Jvar(/?)var(» X x 15 an equivalent expression to assess the importance of the bias in relation to the standard error is: \ bias in R • Zcvs sk If the cvx < 0.1, the bias may safely be regarded as negligible in relation to the standard error. Otherwise, the estimator bias distorts the probability of error a (Cochran 1977, p. 13-14). Table 3 shows that for a normal distribution and a = 0.05, the real probability of an error greater than 1.96a is increased as the ratio of the bias on the standard deviation increases. This shows that the combined ratio estimate, for stratified random sampling, should not be used for this survey since the bias in rc is very large relative to its standard error. For example, the first survey of 1995 had a cvZu = 17. Therefore, there is no gain at all from using the combined ratio estimate since the real probability of error a will be almost 1 (if the tendency is the same as in Table 3). The cvlsl were very large since the longest transects were in the biggest areas. It shows that the ratio estimator R also distorts the a error, but to a lesser degree. The cvj ranged from 0.33 to 0.62 from 1990 to 1996; thus, the use of R would have doubled the a error in some years. The jackknife, or Quenouille's, method applied on R reduces the bias of the estimation method from 1/n to 1/n2. Hence, I used the jackknife method for every ratio estimate in the thesis. The estimator bias of the frequency estimator is unknown at present because there does not exist similar theory to that for the ratio estimator. 16 Table 3. Effect of a bias B on the probability of an a error greater than 1.96a (from Cochran 1977, p. 14). For the ratio estimator, cvx > B/a. Hence, cvx replaces B/a to assess the effect of the bias. B/a a error 0.02 0.0500 0.10 0.0511 0.20 0.0546 0.40 0.0685 0.60 0.0921 0.80 0.1259 1.00 0.1700 3.1.3 Sampling bias. Unlike the estimator bias, the sampling bias could not be totally eliminated because of the use of an airplane. Its effect on the total error could not be estimated either. A sampling bias arises from faulty sampling methods, e.g. faulty selection of sampling units, lack of randomization, etc. For this survey, there was a sampling bias because: (i) the sampling procedure used was pseudo-systematic: in a given area, the sampling units were about equally separated but there was no randomization of the location of the first sampling unit; and the estimator used was not designed for this sampling procedure. (ii) the transects were run along the stratification of the habitat instead of across. (iii) some transects did not fit in the sampling procedure, e.g. one transect around the periphery of one of the areas. I corrected factor (iii) by cutting the peripheral transects in two so they would be parallel to the other ones. It was also more precise to have two transects of 5 km than a single one of 10 km. It increased the sample size. Also to increase the precision, the allocation of transect effort was set proportional to the size of the areas. Factors (i) and (ii) could not be corrected. An helicopter is necessary to do so. 17 3.1.4 Counting bias. 3.1.4.1 Introduction. The counting bias, or visibility bias, is usually the main source of error in aerial surveys (along with the random-sampling error), especially in surveys of elusive organisms such as ptarmigan. The counting bias is a consistent error in counting. It is always negative in aerial surveys and results in an underestimation of the actual population size (review of 17 studies by Caughley 1974; LeResche and Rausch 1974; Wolfe and Kimball 1989). The counting bias is often defined (e.g. in Choquenot 1995a, b) as the systematic underestimation of true abundance during aerial surveys, but it is not unless one proceeds to a total count. For surveys, the latter definition applies only to a given sample (e.g. one transect) and not to the survey estimate. Please refer to the equation of total error. The counting bias has two components (Marsh and Sinclair 1989): 1. an availability bias, i.e. animals are concealed by other animals or the environment, e.g. whales that are deep underwater when the plane flies overhead or moose hiding under trees are not "available". 2. a perception bias, i.e. animals are potentially visible to observers but are not seen. In other words, observers miss available animals. In aerial surveys of ptarmigan, the availability bias is mainly due to the fact that many birds do not flush or flush very late when the plane flies over them. Ptarmigan that stay on the ground are much harder to detect. The perception bias is due to observers missing available birds. The ptarmigan are not easy to detect as they are mostly white with a black tail over a snow background, and the time-frame during which they can be detected is short (2-3 seconds). This perception bias can depend on visibility conditions and ability of the observers. Both types of biases may vary across surveys. 18 Many factors are known to affect aerial counts. They can be divided into two broad categories. First, some factors can be standardized, e.g. aircraft factors (Caughley et al. 1976) and observer factors (Caughley 1974; LeResche and Rausch 1974). In the Kluane surveys, few of these were controlled. Flying conditions in the alpine depend on wind. Gusts of wind often forced the aircraft to fly 50 m higher and faster. Some areas could not be sampled in some years. Two of the observers were the same through all the years, but the third varied. In addition, observers sometimes got airsick. The second category includes factors that can not be standardized, e.g. group size of animals, distribution, individual behaviour, heterogeneity of vegetation (and snow) cover, habitat and terrain (Packard et al. 1984; Samuel et al. 1987). Table 4 lists factors I suspected specifically of affecting the counting bias. Since I could not standardize all the factors, the bias was variable across surveys. I needed a method to correct for it. Table 4. Some of the factors suspected of having an effect on the aerial counts of ptarmigan at Kluane. speed and height of plane (depend on wind) vegetation and snow cover areas sampled % of population present on breeding grounds sunny/overcast sky flushing behaviour of the birds 3rd observer air sickness of observers 3.1.4.2 Correction methods for counting bias. Most of the literature on aerial surveys in the past 20 years has been devoted to methods to correct for the counting bias. Four broad classes of methods have been proposed to correct parts of or the totality of the counting bias: i) double-sampling, ii) double counts, iii) marked subpopulation, and iv) line transects (reviews by Pollock and Kendall 1987 and Seber 1992). I reviewed those methods to choose one based on: applicability, cost, accuracy and precision. All of the methods presented could theoretically be applied to this survey. However, in terms of cost 19 and effort needed not all of these methods can be applied to this survey, especially since a new correction factor is needed for every survey (as recommended by Haramis et al. 1985). It is difficult to discuss the performance of correction methods in terms of accuracy and precision because evaluations of correction methods using known population sizes are quite rare (e.g. White et al. 1989; Packard et al. 1985) and results may be species specific. Instead I focused on the types of biases the methods could correct. Double-sampling. For some samples, accurate ground counts (y) can be done in addition to the aerial counts (x). The correction factor is the ratio (section 2.2.3) (Jolly 1969; Eberhardt & Simmons 1987). For many species (e.g. marine mammals), accurate and precise ground counts may be impossible or too hard to conduct (Broome 1985; Packard et al. 1985). For ptarmigan, ground counts are possible and were conducted in 1995 and 1996 on many of the areas (section on calibration). However, in the long run the technique requires a lot of effort. Accurate and precise ground counts in a few of the areas would be needed every year. Double counts. In situations where observed objects can be mapped and individually identified by both the aerial and ground observer (e.g. bald eagle (Haliaeetus leucocephalus) nests (Grier et al. 1981) ), the ground count can be used without assuming that it is a complete count (Pollock and Kendall 1987). The analysis uses an adaptation of the capture-recapture Petersen estimate (Seber 1982) . The double-count method was extended to free-ranging animals by Cook and Jacobson (1979) and is called the simultaneous 2-sample capture-recapture. Caughley and Grice (1982) used the method to adjust aerial counts of emus (Dromaeus novaehollandiae) and Graham and Bell (1989) of feral horses (Equus caballus) and donkeys (E. asinus). To cope with moving 20 targets, two observers sit on the same side in the plane and count the animals independently. A procedure must be planned to permit the observers to agree on which animals they both saw. It works for low densities of animals. When both observers have the same vantage points, only the perception bias is corrected. The correction factor obtained by this method can be used to see how visibility varies depending on the habitat or species (Choquenot 1995a, b), or on the time of sampling. It has the additional advantage that it does not add any cost to the survey. Marked subpopulation. Part of the population can be tagged and released in the survey area. The proportion tagged of those seen from the air then provides a correction factor for the total count (Seber 1984). Bear et al. (1989) evaluated the method on an elk population using coloured tags. They concluded that the method provided precise (%cv = 4.4-10%) but biased estimates of population size. Radio collars can also be used to develop models for sightability (Floyd et al. 1979; Samuel et al. 1987; Packard et al. 1985). Packard et al. (1985) described the use of radio telemetry to estimate visibility bias for manatees (Trichechus manatus). Because their estimates of visibility varied between standardized surveys, they cautioned against the use of a general factor for correcting counting bias in aerial surveys of manatees. I found that their results (Table 5) raise serious doubts on the use of radio collars to provide a visibility correction factor for manatees. In most cases, the correction factor obtained from radioed manatees does not reflect at all the existing bias (compare the two last columns of Table 5). Hence, the technique should correct for the counting bias, but as the results of Packard et al. (1985) showed, the correction factor obtained from marked animals does not always apply to the whole population. What is more, this method cannot be applied to the Kluane survey since the cost is prohibitive: a marked subpopulation would be needed every year to provide a correction factor. 21 Table 5. Visibility correction factors (vcf) for five aerial surveys of manatees (from Packard et al. 1985). Survey Total manatees Observeda In survey area Visibility correction factor (vcf) True vcf vcf from radioed manatees 1 2 3 4 5 8 11 10 9 8 24 23 19 20 14 0.33 0.48 0.53 0.45 0.57 0.75 0.00 0.33 0.33 0.50 a number of radioed and non-radioed manatees sighted on aerial survey of study area b number of radioed and non-radioed manatees present in the study area c Observed divided by in survey area d number of radioed manatees sighted in study area divided by number of radioed manatees located in area Line transect methods. The line transect method has the advantage of requiring minimal effort to establish, just like the simultaneous double-counts. It relies on the estimated distances of detected animals from the flight centreline, and on the principle that the probability of sighting an animal decreases as distance from the centreline increases (Burnham et al. 1980; Buckland et al. 1993). The line transect sampling model can be viewed as a method of adjusting for counting bias as it increases with distance away from the transect line (Pollock et al. 1987). The method is fully described in the section on line-transect methods since it was used for ground counts. However, it cannot be used for the Kluane aerial surveys. A critical assumption is that every animal on the centreline is sighted. This is a problem since most ptarmigan do not flush. Ptarmigan who sit on the ground are hard to detect since they are white on snow. In addition, the plane flies too fast to allow enough time to detect the original location of the birds and to estimate that distance from the centreline. To apply the line-transect method to the Kluane aerial survey, it would be necessary to use a helicopter instead of an airplane, making the method twice as expensive. 22 Conclusion on the choice of a correction method. The best correction methods for this survey, based on cost and bias corrected (Table 6), seem to be double-sampling and line-transects. I did a variation of double-sampling in 1995 and 1996 (see calibration). However, the method can not be used for future years because it requires too much effort. Line-transects with an helicopter would then be a good alternative (1995 and 1996 surveys still had to be done with an airplane to evaluate the 1990-1994 data). The mark-recapture method is too expensive if we consider the trapping and the equipment necessary. The simultaneous double-counts corrects only for the perception bias but since it could be applied to the Kluane survey at no extra cost and effort I tried it. Table 6. Cost and type of bias that should be corrected by different methods for aerial surveys of ptarmigan. ($$$ = most expensive) Method Cost Types of bias corrected double-sampling $ counting bias double-counts * - perception bias mark-recapture $$$ counting bias line-transect $ ** counting bias * simultaneous 2-sample capture-recapture ** only possible with helicopters 3.1.4.3 Double-counts: the simultaneous 2-sample capture-recapture correction method. The simultaneous 2-sample capture-recapture method provides a survey-specific correction for the perception bias. An estimate of the perception bias is useful as visibility conditions and the sighting ability of the observers vary between surveys. A few capture-recapture models are available (Pollock and Kendal 1987; Seber 1982): the Petersen model, the removal model and the Cook and Jacobson model. Of the 3 models, only the Petersen model is applicable. 23 Removal model (y\-yi) where Y = the estimate of population total, yi = the number of animals observed by the first observer, y>2 = the number of animals observed by the second observer, but not by the first. Pollock & Kendall (1987) did not see this approach as being as useful as the Petersen model because it does not use the number of animals seen by both observers. Also it assumes that both observers have equal sighting probabilities. The value of Y depends radically on the decision of who will be the first observer and who will be the second, which is irrelevant when the observations are simultaneous. The problem is obvious when having two observers of different skills. A higher skilled observer as observer 1 results in a high estimate, while as observer 2, it results in a very low estimate (below the number of animals actually seen!). Cook and Jacobsen model To offset the possible difference in sighting skill between the two observers, the design calls for them to switch roles (i.e. to switch places) halfway through the survey (Cook and Jacobsen 1979). This is not possible in a Heliocourier airplane with only 1.5 hour of survey to do. Also, it is not as efficient as the Petersen or removal methods (Seber 1982). Petersen model The Petersen model has an advantage over the other models in that it allows for 2 observers to have different sighting probabilities (i.e. observers can be of different skills) (Seber 1982). The Kluane survey meets most of the assumptions: 24 i) it is clear which objects are seen by one or both observers ii) sightings of different objects and by different observers occur independently iii) the population size is constant (not a concern since the observations are simultaneous) iv) the probability of spotting each object is the same for each particular observer, but can vary between observers. This is not true if we include birds on the ground. Their "catchability" is lower. v) the 2 observers gain experience at the same rate. This is unknown. Since the Petersen estimator can produce large biases for small samples, I used the modified version of the Petersen estimator (Krebs 1989, p. 16-27): Y _ (y, + O f o +1) i where Yc = the corrected population total estimate for the perception bias ya = the number seen by the first observer yb = the number seen by the second observer z = the number seen by both observers The large sample estimated variance of the estimator is: (Z + f / ( z + 2) The modified Petersen estimator is unbiased if (ya + y0) > Y, and nearly unbiased if there are at least 7 recaptures (z > 7) (Krebs 1989). For the present surveys the estimates were certainly biased since (ya + yb) < Y and z ranged from 0-6 (Table 9). Another way to get 1^  is with the perception correction factor IIP (Magnusson et al. 1978; Graham and Bell 1989): 25 where Y = population total estimate P = probability of a bird being seen by > 1 of the 2 observers. (y f l + 0(y* + 0 - ( z + 1) where all terms are defined above. The large sample variance estimate for P is: v ( p ) _ p y {ya + y»- 2^ha - zfob - z) ybl This correction is not only correcting for the variable visibility conditions, it also correct for the offset in ability between the 2 right seat observers. It is specific to the counts of the 2 right seat observers and cannot be applied to the counts of the left observer. I could not correct the surveys prior to 1995 since the numbers of double-counts were not reported. The perception correction factor IIP can be a very useful tool. For instance, assuming that the 2 observers gain experience at the same rate through surveys, one could use the correction factor to determine how visibility conditions vary through surveys. 3.1.4.4 Conclusion. Because factors affecting the counting bias could not all be standardized across surveys, a survey-specific correction factor was needed. Based mainly on cost and easiness of use, I chose the simultaneous 2-sample capture-recapture method (using the modified Petersen estimator) as the method to correct part of the counting bias, the perception bias, in 1995 and 1996. The availability bias remained uncorrected. I also used the double-sampling method (section on calibration). 26 3.1.5 Random-sampling error. Random-sampling error is due to chance differences between those members of the population who are surveyed and those who are not (Stopher and Meyburg 1978). The random-sampling error cannot be eliminated unless the whole population is sampled, but its effect on the total error can be assessed theoretically by making assumptions about its distribution. The random-sampling error is approximately proportional to (i - „//v )/4n , where n is the number of units sampled and N the total number of sampling units. Therefore, to reduce the random-sampling error and to increase the sampling precision, I simply need to increase the sample size n. The sampling precision (or precision of a sample statistic) refers to the closeness with which it estimates the population parameter. It is equal to the width of the confidence interval (Zar 1984). For a given sample size, the sampling precision depends both on the method by which the estimate is calculated from the sample data (i.e. which estimator is used) and on the plan of sampling (Cochran 1977). For instance, the sampling procedure can be refined by stratification (Siniff and Skoog 1964), or choosing between quadrat vs. transect as sampling units (Caughley 1977; Kraft et al. 1995)). The sampling sensitivity is another measure of sampling precision. The sensitivity, or detection threshold, is how small a change or difference a given study reliably detects (Eberhardt 1978). For a one-sample hypothesis, the sampling sensitivity of an estimate is: ((1 - a)%UCL - (1 - a)% LCL) sampling sensitivity = - - X 100% estimate where (1 - a)%UCL = upper confidence limit at the a level (1 - a)%LCL = lower confidence limit at the a level I increased the sample size to 28 transects in 1996 to reduce the random-sampling error. 27 3.1.6 Conclusion. I tried to reduce or to quantify the effect of each components of the total survey error. The estimator bias of the ratio estimator was reduced by using the jackknife method. The sampling bias was slightly reduced by improving the sampling procedure (making it more systematic), but it cannot be completely removed with the use of an aircraft. The counting bias could be corrected partially by using the simultaneous 2-sample capture-recapture method. However, this method only corrects for the perception bias, not for the availability bias. Additionally, the counts of the third observer can not be corrected. The effect of the random-sampling error could be assessed by using the standard variance estimate of the ratio estimator. Since the counting and sampling biases could not be eliminated or assessed quantitatively, and since they could have a large effect on the value of the index, I assessed the repeatability of the technique by evaluating empirically the variability of the total bias. 3.2 REPEATABILITY. I established repeatability by testing if the absolute precision of a given year was high compared to the variability of the index over years. In this case, I used the concept of precision of a measurement which is a measure of repeatability and refers to the nearness of repeated measurements to each other (Zar 1984). Each survey index is a measurement. This type of precision is an indication of the variability of the total bias (combined with the effect of random error). It is not a direct measure of total error since it does not detect a constant bias across measurements. The variance is a measure of the precision of a measurement (Zar 1984). To estimate the precision of the technique, I replicated the aerial survey twice in 1995 and 5 times in 1996 between 24 April - 8 May. The number of survey replicates was cost limited. The replication procedure was two-layered. In each survey, we surveyed 6 of the 7 areas once and the other area, twice. From survey to survey, a different area was sampled twice. This multi-level of 28 replicates provided estimates of the empirical variance of the index i) over 2 weeks (s 199$) and ii) on the same day (s i996-sameday)- The estimates of variances allowed me to evaluate i) the repeatability of the technique in a given year, and ii) the importance of daily variability to the value of the index. If the technique is repeatable, the absolute precision (i.e. variance) between replicate surveys in 1996 (s 1996) should be lower than the variance between 1990 and 1996 (s 1990-1996)-1 tested H 0 : s2i99o-i996 ^ s2i996 with a one-tailed variance ratio test where F = s2i99o-i996/ s2i996 (a = 0.05) (Zar 1984, p. 125). I did not transform the ratio indices logarithmically for normality, even though the distribution of ratios is positively skewed (Cochran 1977). The log transformation would have changed the test to the difference between coefficients of variation (i.e. relative precision) as shown in Zar (1984). I used the arcsin transformation for the frequency indices (binomial distribution) (Zar 1984). If H 0 is not rejected, the technique is not repeatable because there is as much absolute variability within the replicate indices of a given year as there is between 7 years of counts. This could mean i) that the population size fluctuated as much if not more in the space of 2 weeks than over 7 years; therefore, the timing of the survey could be so critical that it would make the method unusable, or ii) that the total bias varies too much between replicates. From 1990 to 1994 the annual index came from only one aerial survey done on only one day per year; thus, daily factors (e.g. wind, sunny/overcast sky, snow cover, biological timing) could have a big effect on the value of the index. To extract the relative variability coming from daily factors (cvdaily factors), I subtracted the relative precision of replicate counts on the same areas during the same day (cvi996.sameday) from the relative precision of counts from replicate surveys done over a few days (cvi996). So that cvdauy factors = CV1996 - cv m^ame day and that the % of total variability depending on daily factors = cvdaily factor/cv 199$. 29 3.3 RESULTS. Figure 3 shows the trends of the aerial indices from 1990 to 1996. Increasing the sample size to 28 transects in 1996 increased the sensitivity from an average of 143% prior to 1996 to an average of 79% (Table 7). This enabled the estimator to detect differences about twice as small based on the random-sampling error. I could not correct the aerial indices of 1995 and 1996 (Table 7) for the perception bias since the correction factors per survey (Table 8a) were highly imprecise in many cases (%cv up to 116%) and probably highly biased. The number of double-counts was always < 3 except in one case where it was 6. Low numbers of double-counts make the modified Petersen estimator not robust (Krebs 1989). The real variance is probably higher than reported in Table 8 since I had to use a large sample variance estimator. I judged it preferable not to use the correction factors to correct for the changes in perception through surveys nor to draw conclusions on the variability of visibility conditions from them. The same considerations applied to the correction factors per area (Table 8b). Areas 6 and 7 were almost the only areas that had double-counts, and even those 2 areas had less than 7 double-counts total. The variance between the 5 survey indices of 1996 was significantly lower than the variance between 1990-1996 (Table 9). I did not need to log transform the ratios since a Kolmogorov-Smirnov normality test failed to detect a departure from normality (p > 0.20). This permitted me to keep the meaning of the test to a difference between 2 variances. About 70% of the variability of RQ between replicate surveys in 1996, and 55% for/, could be attributed to daily factors. The variance between the 2 survey indices of 1995 was also low, but the power of the test was too low with only 2 replicates in 1995 to detect a significant difference (Table 9). The 1990-1996 variance of RQ could have been 5 times smaller than its present value before reaching the critical value where the 1996 variance would not be significantly smaller anymore (given a = 30 0.05 and df = 6, 4). The 1990-1996 variance of/could be only 1.4 times lower than the present value. Table 7. Aerial survey indices of density for 1990-1996 and transect efforts. The indices are i) RQ the number of ptarmigan seen/ km of transect, and ii)/the frequency of transects having one or more observations. The cv given for / i s not for the random-sampling error, but for the variation between the estimates after standardizing the lengths of the transects (section 2.2.4). The cvx are the coefficients of variation of the transect lengths, ns is the number of sightings. The sampling sensitivity is based on the 95% confidence limits. Also given are the means of the replicate surveys for 1995 and 1996; the estimates of precision of those means are based on the variance between survey estimates. year survey date #of total length of cvx ns f %cv %cv % transects transects sampling (km) sensitivity 1990 8 May 9 25.5 0.33 17 0.802 8.2 0.677 35.5 132 1991 27, 29 April 19 65.7 0.59 132 0.941 4.8 2.003 35.5 123 1992 30 April 19 67.7 0.56 76 0.939 4.6 1.135 16.7 58 1993 6 May 20 63.9 0.62 9 0.34 9.1 0.140 50.0 173 1994 2 May 10 33 0.38 14 0.267 0.2 0.440 74.8 274 1 29 April 19 64.3 0.58 53 0.675 5.8 0.824 33.5 116 1995 2 1 May 12 48 0.56 34 0.762 0.4 0.708 34.6 124 mean 0.719 4.4 0.766 10.7 1 24 April 28 93.5 0.41 25 0.509 5.0 0.268 28.1 96 2 27 April 28 93.5 0.40 37 0.731 5.8 0.394 22.1 75 1996 3 3 May 28 92.5 0.37 44 0.754 5.2 0.478 22.9 78 4 6 May 28 94.5 0.35 17 0.599 6.7 0.180 22 75 5 8 May 28 95.5 0.35 33 0.672 5.1 0.346 20.8 71 mean 0.653 4.5 0.333 34.4 31 Figure 3. Trends of the aerial indices from 1990-1996. The indices are a) RQ , the number of ptarmigan seen/ km of transect, and b)/the frequency of transects having one or more observations. Transect efforts and data in Table 8. Only the means of the replicate surveys are shown for 1995 and 1996. 32 Fig 3. a) aerial index RQ, the number of observations/ km nf tranctprt 4- R F — o DC X a> TJ c ra < 1,5 + 1 + 0,5 0 -I 1990 —I— 1991 1992 1993 Year 1994 1995 1996 X <D TJ C < Fig 3 b) aerial index / , the frequency of transects having at least one observation. 0,75 + 0,25 + 1996 33 Table 8. Correction factors for the perception bias a) for each of the 1995 and 1996 aerial surveys b) for each of the areas in 1995 and 1996. The correction factor 1/P is obtained with the modified Petersen estimator, simultaneous 2-sample capture-recapture method, using the observations of the 2 right seat observers (ya = total number of observations from back seat observer, yD = total number of observations from front seat observer, z - number of double-counts). P = probability of a bird being seen by > 1 of the 2 observers. a) year survey ya Yb z P %cv 1/P 1 7 5 0 0.255 116 3.92 2 9 13 0 0.158 109 6.33 1996 3 7 39 3 0.544 38 1.84 4 4 14 3 0.845 22 1.18 5 3 9 1 0.579 62 1.73 1995 1 6 12 2 0.545 45 1.83 2 7 15 6 0.926 9 1.08 b) year area ya Yb z P %cv 1/P 1 3 7 0 0.32 121 3.10 2 7 5 0 0.26 116 3.92 3 2 4 0 0.43 131 2.33 1996 4 0 1 0 1 0 1.00 5 1 11 0 0.52 144 1.92 6 4 9 2 0.70 37 1.42 7 13 43 5 0.50 30 1.99 1 0 0 0 0 0 a 2 6 6 3 0.80 23 1.25 3 0 4 0 1 0 1.00 1995 4 0 0 0 0 0 a 5 0 0 0 0 0 a 6 0 0 0 0 0 a 7 ' 7 17 4 0.72 25 1.39 34 Table 9. One-tailed variance ratio test of Ho: s 2 i 9 9 0 . i 9 9 6 < s 2i995 o r 1996; using the aerial indices o f Table 7. d f i 9 9 6 = 6, 4 and d f 1995 = 6, 1. year estimator 'A S 90-96 0 * 0 year transformation F P RQ 0.396 0.0131 none 30.2 0.005 > p > 0.0025 * 1996 0.806** 0.344** log 5.23 0.10 > p > 0.05 f 0.0728 0.0101 arcsin 8.61 0.05 > p > 0.025 * Ro 0.396 0.00673 none 58.9 0.10 > p > 0.05 1995 0.806** 0.107** log 64.9 0.10 > p > 0.05 f 0.0728 0.00379 arcsin 20.3 0.25>p>0.10 * Significant difference (a = 0.05) ** these values are the coefficients of variation instead of the variances since the log transformation changes the test from a difference between variances to a difference between coefficients of variation. 3.4 DISCUSSION. 3.4.1 Validity of the replication of the 1996 aerial survey. Two factors may have had some impact on the validity of the replication procedure. Ptarmigan may become habituated to the airplane, making them less prone to flush, and the 2 observers that did the 5 surveys may have improved their counting skills during the surveys. The repetition of the aerial surveys over a short time could have "habituated" the ptarmigan to the airplane, making them less prone to respond (i.e. flush) to the airplane. This would have potentially decreased the availability of the birds and increased the variability between replicate counts. There was no evidence that habituation occurred. The 5 surveys were spaced by 2-6 days and the location of transects varied by up to 100 m, so different birds could be flushed from time to time. Only one transect was static (marked with orange boards on the ground) and on this transect, the number of birds flushed increased through surveys (0-0-0-0-4). Ground transect observations showed that birds flushed once were just as easy to flush a second or even a third time. Ptarmigan did not even seem to habituate on a short time-scale since the relative variance between replicates done on the same day was only 10% (cv). 35 The 2 observers who did the 5 surveys could become more skilled through surveys, but there was little time for it. The total number of hours of aerial surveying amounted to about 6 hours in 1996, split into 5 surveys. There was still a possible gain in experience throughout surveys, but it probably had less effect on the counts than the natural detection ability of the observers. At worst, the 2 factors could have increased the variance between replicate surveys, resulting in a conservative approach. 3.4.2 Repeatability. Since the variance between the 5 survey indices of 1996 was lower than the variance between 1990-1996, the technique is repeatable. However, the technique is more repeatable using RQ than /since the 1990-1996 variance of RQ could have been 5 times smaller than its present value before reaching the critical value where the 1996 variance would not be significantly ) smaller anymore, while it could have been only 1.4 times smaller using/. It could also mean that RQ could be more powerful than/since it can detect trends under a lower variability of the index over years. Under the present conditions,/could be working about at its limit of precision and detectability. 3.5 CONCLUSION. In this chapter, I analysed the total survey error theoretically and its effect empirically. By repeating the survey in 1996,1 showed that the technique was repeatable. However, there remain the questions: How do aerial indices relate to density? Are they telling us the good story in terms of population trends? The next 2 chapters deal with those questions with a calibration of the aerial indices. 36 4. LINE TRANSECT METHODS FOR GROUND COUNTS. 4.1 INTRODUCTION. I used ground line-transects to estimate densities of ptarmigan in 6 of the areas for the calibration of aerial indices. I present the calibration in details in chapter 5. This chapter deals only with the line-transect techniques (Burnham et al. 1980; Buckland et al. 1993) applied to ptarmigan. The size of the 6 areas ranged from 3 to 13.5 km 2; therefore, I needed a technique that could provide me accurate estimates of densities on such a scale. As mentioned in the general introduction, the common techniques to estimate density of ptarmigan, mapping territories and census transects using dogs, could hardly be applied to a scale larger than 2 km . Line-transect sampling could do so. Hayne (1949) provided the first line-transect estimator that has a rigorous justification in statistical theory (Buckland et al. 1993); his technique was developed for flushing animals such as grouse (Phasianidae family) and was based on sighting distances and angles. Nowadays, robust line-transect estimators exist based on perpendicular distances (Buckland et al. 1993). The technique is used widely in wildlife management, in aerial (e.g. Holt and Cologne 1987; White et al. 1989) and ground surveys (e.g. Kelley 1996; Sherman et al. 1996). Line-transects have been used on ptarmigan in Sweden, but using only the concept of effective strip width as defined in Gates (1979) (Brittas 1995, pers. comm.). The efficiency of the robust estimators had yet to be evaluated for ptarmigan, which I did. I conducted a basic test of accuracy of the method and evaluated the precision given different sample size of observations. I also compared estimates based on sighting angles and distances with estimates based on perpendicular distances since there are no known field comparison of the 2 types of estimators (Anderson and Burnham 1996, pers. comm.). 37 4.2 THEORY. 4.2.1 Estimators based on perpendicular distances. Since a complete review of the theory is given in Burnham et al. (1980) and in Buckland et al. (1993), only a brief synopsis will be presented here (taken from Anderson and Burnham 1995). Central to the concept of distance sampling is the detection function g(y) where g(y) = the probability of detecting an object, given that it is at perpendicular distance y from the random line. Generally, the detection function decreases with increasing distance, but 0 ^ g(y) < 1 always. In general, the estimator of density from distance data can be expressed as: a • C • K 'go where n = sample size (the number of detections). E(s) = an estimate of cluster size (if detections represent individuals, then E(s) = 1); s represents cluster size. a = surveyed area = 2wL for line transects where w = width and L = length of transect c = proportion of area searched = 1 Pa = an estimate of the unconditional detection probability of objects in the strip surveyed. go = an estimate of the probability of detection on the centreline (usually go =1). The line transects are a special case of this estimator. The estimator of density for line transect sampling uses a special form for Pa that is a function of the distances, The true detection function g(y) is not known. Furthermore, it varies due to numerous factors. Therefore, it is important that strong assumptions about the shape of the detection function be avoided; flexible or robust models for g(y) are essential. This class of models also excludes those 38 that have restricted shapes, or have inefficient estimators. The models are selected a priori, and without particular reference to the given data set. The modelling process can be conceptualized in two steps. First, a key function is selected as a starting point, possibly based on visual inspection of the histogram of distances, after truncation of obvious outliers. Second, a series expansion is used to adjust the key function, using perhaps one or two more parameters, to improve the fit of the model to the distance data. Conceptually, the detection function is modelled in the general form: g(y) = key(y)[l + series(y)] These series-expansion models are semi-parametric in the sense that the number of parameters used is data-dependent. Of all the robust models recommended by Buckland et al. (1993), I considered the following ones: the half-normal key with cosine or Hermite adjustment, and the hazard-rate key with cosine or polynomial adjustment. I selected the model with the lowest AIC (Akaike's information criterion), as implemented in the program DISTANCE v2.1 (Laake et al. 1994) used for calculations. Buckland et al. (1993) listed 3 critical assumptions in achieving reliable estimates of density from line-transect sampling. These assumptions are given roughly in order of importance from most to least critical. All three assumptions can be relaxed under certain circumstances. 1. Objects directly on the line or point are always detected 2. Objects are detected at their initial location, prior to any movement 3. Distances are measured accurately. Assumption 3 is not a problem since we used 30 m metal or polyurethane measuring tapes to measure distances to the closest 25 cm. Assumption 1 is not a problem in ground counts unless assumption 2 fails. If a substantial part of the population moves further from the line prior to detection, this movement will often be apparent from examination of the histogram of the 39 distance data (Buckland et al. 1993, p.33 & 337). Also, if evasive movement prior to detection occurs at a high level, the estimator will be biased low. 4.2.2 Estimators based on sighting angles and distances. The Hayne 's estimator is based on the concept that detection is due to the animal flushing in response to the observer distances (Burnham et al. 1980, p.39, 79-88, 147-149, 171-3). The basic assumption of the Hayne's estimator is that the flushing distance, r, is fixed. The Hayne's estimator of density is: " 2L ^ i1 where DH = Hayne's estimate of density n = number of birds seen flushing L = length of transect Ti = sighting distance to each bird / This estimator has only 1 critical assumption: that sin(9), 8 being the sighting angle, is a uniform random variable on the interval [0, 1]. This assumption implies that the expected average sighting angle is 32.7°. With the test statistic z = Vn"(0-32.7) 2156 reject H 0 : E(Q) = 32.7° if I z I > 1.96 (a = 0.05) and try the generalized Hayne's estimator. The generalized Hayne's estimator is an extension of the fixed flushing radius model and allows an elliptical flushing curve. This model assumes an average sighting angle of 45°. However, these models based on sighting distances are not as robust as the ones based on perpendicular distances (Burnham et al. 1980). Therefore, I provided the Hayne's estimates not for the calibration but 40 simply to compare them with the perpendicular distances estimates, as there are no reports in the literature of a field comparison of the behaviour of the 2 kinds of estimators. 4.3 METHODS. 4.3.1 Sampling protocol. With the help of a second observer, I sampled 2 of the 7 subalpine areas by ground line-transects in 1995, and 6 in 1996. This provided an estimate of breeding density of male ptarmigan for each of these areas. We sampled areas one at a time. I determined the order of sampling areas by accessibility. We did transects perpendicular to the stratification of the habitat (from the treeline to the alpine, and then vice-versa) to prevent a sampling bias. For simplicity of use in the field and to get an even coverage of the study regions, I used a systematic sampling procedure, except that the location of the first transect was not random. We recorded the length of each transects, and for each observation of male ptarmigan 1) the perpendicular distance d of the original location of the bird from the centreline, and 2) the flushing distance r from the observer if the bird flushed. The sighting angle 0 could be calculated afterward with 0 = arcsin (d /r). I used 2 different methods to estimate the length of transects 1) a topographic map and 2) a pedometer. The pedometer gave the upper limit of transect length, and the topographic map the lower limit. I used 30 m polyurethane or metal measuring tapes to measure distances of males from the line and from the observer. I did not use dogs on the transects because the snow was too deep in early May. 4.3.2 Analysis of data. I used the program DISTANCE v.2.1 (Laake et al. 1994) for the estimation of density based on perpendicular distances. To account for the change in visibility conditions during the ground line transects, I pooled the data in 2 main sets for the calculation of the detection 41 function. Perpendicular distances estimators used here are pooling robust (Buckland et al. 1993, p.74 - Burnham et al. 1980, p.45). A model is pooling robust if it is robust to variation in detection probability for any given distance y (Buckland et al. 1993, p.74). But I still decided to create 2 sets of data as visibility conditions changed drastically in the space of one week (around 15 May in 1995 and 20 May in 1996): willow and birch started to leaf-out, the snow cover completely disappeared and egg laying started. So I pooled in one set observations from the areas sampled before this change and in another set, observations from areas sampled afterwards. Many other factors could have influenced the detection function and to make sure that the pooling was justified (e.g. for observations from the 2 observers), I controlled it visually by comparing plots of detection probability. The precision had to be high enough so that the estimates could be used for the calibration. The precision of line transect estimates depended on the number of observations and on the number of transects. For the estimation of density based on sighting distances and angles, I used the program T R A N S E C T II (Laake et al. 1979). For each area, observations had to be pooled in only one transect because of the reduced sample size for this estimator (only the observations of flushed males could be used). So the variance estimation was based only on the number of observations and not on the number of transects. 4.3.3 Test of accuracy. The line-transect estimates were used in place of the true density for the calibration, so they had to be accurate and precise. I performed a basic field test of the accuracy to show that the estimates were valid approximates of the true density. This permitted me to test if the possible violation of some assumptions (e.g. mobility of the animals) produced a large bias. In 1995,1 set up a grid at Hayden Lake. The grid was 0.7 km x 1.1 km (77 ha). To help run the transects and define accurately this small area, I divided the grid by parallel lines of flags 42 100 m apart, flags being spaced by 100 m on the line. On this grid I used the same protocol for line-transects as for the large areas. I did line-transects once in 1995 and twice in 1996.1 then compared these estimates of density for the grid to the 'true' density and calculated the relative bias. The relative bias = (estimate - true density)/ true density x 100%. The best estimate available of the 'true' density was a total count. In both years, once I finished the line-transects on the grid, I proceeded to a total count of the males present in the area using a human chain (first week of June in both years). I used 5 observers in 1995 and 8 in 1996. The observers walked in parallel 10-20 m apart. The transects were from one side of the grid to the other side. I recorded every observation of a male. To prevent counting the same males twice when they flushed to an area that was not yet sampled, an observer was left behind with binoculars to keep an eye on them and to see if they flew back to their original location. The grid was particular in that it permitted this kind of observation from a long distance due to the concave topography of the area. In parts where the willow was thick and males could hide, one observer would go in and try to scare the birds out while other observers would watch from the side. Territorial mapping on small parts of the grid in early May (when the males were highly territorial and visibility conditions were really good) corroborated the total counts. 4.4 RESULTS. 4.4.1 Line-transect estimates based on perpendicular distances. The estimates of density per area (Table 11) were calculated with the probability detection functions (pdf) of Figure 4. A separate pooling was necessary for data gathered before (Table 12a, Figure 4a) and after (Table 12b, Figure 4b) the leaf-out of willow and complete 43 melting of snow since f(0) was significantly higher and ESW lower after (using the 95% confidence intervals) (Table 12). We counted 19 male ptarmigan on the 0.77 km 2 grid in 1995 (24.7 males/km2), and 50 in 1996 (64.9 males/km2). The first estimate of 1996 was highly overbiased (82%) (Table 13). I attribute this to possible multiple double-counts of the same birds on some transects during the 29 April. Double counts occurred because of the really high activity of the ptarmigan on that day (flying back and forth over the observers) and high density. If I do not include the transects done on that day, it reduces the bias to -7%. Including the two other surveys with biases of -5% and -3%, the range of the bias is of 4%. The precision (%cv) of line-transect estimates based on perpendicular distances reached a plateau of 15-25% when at least about 20 observations of males could be recorded in an area (Figure 5). 44 Table 10. Transect effort and sightings by area for the ground line transects of ptarmigan. year area size of area #of total length of dates # males # males (km2) transects transects (km) seen flushed 1 5 12 26.0 24 April, 4, 12 May 23 17 2 13.5 16 45.9 28-30 May, 2 June 39 34 1996 3 3 8 12.0 4 June 9 8 4 4 8 12.0 11 June 0 0 5 4.5 not sampled 6 5 12 26.3 20, 21, 24 May 26 21 7 12 16 36.4 9, 10, 15, 16 May 67 41 1995 1 5 4 20.0 19-20 May 9 7 2 13.5 6 17.6 23, 26 May 5 4 Table 11. Estimates of density D (males/ km2) by area based on perpendicular distances line-transects estimators. The transect efforts in Table 12. Year area D SE %cv df 95%LCL 95%UCL % sampling sensitivity 1 18.0 3.37 18.7 16 12.2 26.7 81 2 25.7 3.7 14.5 37 19.2 34.4 59 1996 3 22.6 9.1 40.2 8 9.3 55.3 204 4 0 6 20.2 4.86 24.1 13 12.1 33.7 107 7 37.5 7.3 19.3 21 25.2 55.9 82 1995 1 13.6 4.69 34.5 3 4.7 39.5 256 2 8.6 4.24 49.4 5 2.6 28.5 301 45 Table 12. Estimation summary of the probability detection functions of Figure 4. m = number of parameters in the model, AIC = Akaike's information criterion, %2 = probability for %2 goodness-of-ftt test, f(0) = value of probability density function at zero for line transects, p = probability of observing an object in defined area, ESW = effective strip width (m). Sampling effort given in Tables 10 and 13. a) summary for Figure 4a: half-normal/cosine model estimate %cv df 95%LCL 95%UCL m 2 AIC 1155.2 x2 0.68292 f(0) 0.040793 7.51 146 0.035216 0.047253 P 0.32255 7.51 146 0.27846 0.37363 ESW 24.514 7.51 146 21.163 28.396 b) summary for Figure 4b: hazard rate/cosine model estimate %cv df 95%LCL 95%UCL m 2 AIC 610.42 0.89756 f(0) 0.060393 8.98 92 0.050544 0.072161 P 0.36312 8.98 92 0.3039 0.43388 ESW 16.558 8.98 92 13.858 19.785 46 Figure 4. Distribution of perpendicular distances and probability detection functions for line-transects of male ptarmigan. Details of the estimation procedure in Table 12 and transect efforts in Tables 10 and 13. a) pooled data from areas 1, 6, 7, and grid 1* of 1996 (n = 148), and b) pooled data from areas 1, 2, and grid of 1995, plus areas 2, 3,4, and grid 2 of 1996 (n = 94). F i g 4 a) l+-1.0974 + 1 f f f 1 .9636 f 1 * * * * * f | * * f | * * f. | * * * * * £ * * * j D .8297 + * * * f * e 1 * * * f* | t * * * | e 1 * * * f j c * * • * f j t . 6959 + * * * * i * * * f * * * * * | o * * * * f * j n * * * * * * * * j * * * * f * | P .5621 + * * * * * f * r 1 * • * * * • | o * * * * * f* | b * * * * * f j a * * * * * j b .4283 + * * * * * * f i 1 * • * * * * f * * * * * 1 1 * * * * * * f * * i i * * * * * * * * * £ * * I t * * * * * * * * * * * y .2944 + * * * * * * * f * * + 1 * * * * * * f f * | * * * * * * * f * * * • * * * * * f * * • * * * * * * f f | .1606 + * * * * * * * * * f f f **** + 1 * * * * * * * * f f f f * * | * * * * * * * * * * * * * * * * f f f f f f f f f f f * * * • • * * * * * * * * * * * * * * f f f f f f f j * * * * * * * * * * * * * * * * f f f f f f f .0268 + * * * * * * * * * * * * * * * * * * * * * * * f f f f + | * * * * * 1 | +- - - + - - - - — - + - - _ + _ + + + + + + + - - + | 000 16. 889 29. 556 42.222 54.889 67.556 4 222 21. I l l 33.778 46.444 59.111 71.778 Perpendicular distance i n Meters 47 F i g 4 b) 1 + 1.1303 + * * * * * * * * * * * * .9925 + * f f f f f f f f f f f f * * 1 * * * * * * f* * * * * f * * * * * * * * * * f * D .8546 + * * * * * e 1 * * * * f * t 1 * * * * * * * * * e * * * * * c 1 * * * * f * t .7168 + * * * * * i 1 * * * * f* o * * * * * n * * * * * * * * * f P .5789 + * * * * * r 1 * * * * * f o * * * * * b 1 * * * * * * f * * a j * * * * * * b .4411 + * * * * * f* i 1 * * * * * * 1 * * * * * f i * * * * * * t * * * * * * f y .3033 + * * * * * * f 1 * * * * * * * * £ * * 1 * * * * * * f* * * * * * * f * * * * * * * f .1654 + * * * * * * * f f * 1 * * * * * * f f 1 * * * * * * * * * * * * * * * * * * * * * * * * * * * * .0276 + * * * * * * * * + + + + + + l + + , 000 3 .257 f f f * f f f * * * f f f f f f f * * * * * ******! * * * f f f f f f f f f f f f f f f f f ********** f f j —+ +—+ + +—+ +—+ + +—+ +1 9.771 16.286 22.800 29.314 35.829 42.343 13.029 19.543 26.057 32.571 39.086 45.600 Perpendicular distance i n Meters 48 Table 13. Results of the tests of accuracy of the line-transect methods on the grid. a) Transect effort of the 1995 and 1996 line-transect surveys on the grid including the number of _ i „ „ A . u „ ^ „ „ „ u . . r-» /™„i,.„/i.~2\ c .„»„i „ „ »„ Year survey dates #of total length #of # D transects of transects observations flushed (km) 1 29 April, 9 May 8 8.8 51 36 1996 1* 9 May 4 4.4 13 11 64.9 2 8 June 8 8.8 18 9 1995 1 30 May, 6-7 8 17.6 14 11 24.7 June * this survey is the same as the 1st one of 1996 but without the transects of April where I suspect that some birds were counted more than once during some transects due to the really high activity and density on that day. b) relative bias of estimates of density D (males/km2) based on perpendicular distances Year survey D SE %cv df 95%LCL 95%UCL sensitivity %bias 1 118.2 29.6 25.0 8 66.9 208.8 120 82.0 1996 1* 60.3 14.6 24.3 4 31 117.1 143 -7.1 2 61.8 15.5 25.1 9 35.4 107.9 117 -4.8 1995 1 24.0 7.97 33.2 8 11.4 50.6 163 -2.7 49 5 Precision {%cv) of line-transect estimates (from Tables 11 and 13) in relation to the number of observations n. 10 20 30 40 50 60 50 4.4.2 Estimates based on sighting distances and angles. In 7 out of 10 cases, I had to use the generalized Hayne's estimator instead of the Hayne's estimator because the average sighting angle was closer to 45° than 32.7° (Table 14). In one case I could not use either estimators because the average sighting angle was too high. The only generalized Hayne's estimate tested for accuracy (not including the first survey of 1996) was highly biased (-98%) (Table 15). The Hayne's estimate had a bias of 13%. The fit between the estimates of Table 14 based on sighting distances and angles and the estimates of Table 11 based on perpendicular distances was poor (coefficient of correlation r = 0.18) (Figure 6). Table 14. Line-transect estimates of density D (males/km2) based on sighting distances, using either the Hayne's (H) or the generalized Hayne's (GH) estimator, for each of the areas surveyed on ground in 1995 and 1996. Transect efforts in Tables 12 and 15. n is the number of males flushed (Only observations where male ptarmigan flushed were used with those estimators), mean angle is the average angle from the transect line from which male ptarmigan flushed. P32.7 and P45 are respectively the probabilities of the Z-test that H 0 : mean angle = 32.7° (for H) or 45° (for GH). If P < 0.05, the estimator can not be used; H is preferred over G H since it is more robust. year area n mean P32.7 P45 estimator D %cv angle used 1 15 45.4 < 0.001 0.13 GH 11.7 4.7 2 34 51.8 < 0.001 0.06 GH 13.7 4.0 1996 3 8 43.1 0.16 0.57 GH 18.4 10.9 4 0 none 0 6 21 48.3 0.005 0.29 GH 9.1 3.2 7 41 50 < 0.001 0.08 GH 13.7 3.7 1995 1 7 30.7 0.60 0.97 H 15.9 6.8 2 4 25 0.70 0.91 H 17.6 9.1 51 Table 15. Relative bias of estimates of density D (males/km2) on the grid based on sighting distances and angles. Transect effort in Table 13 and details of the estimation procedure in Table 14. No estimate of density is given for grid 1* since both estimators failed the Z-test. year survey n mean angle P32.7 P 4 5 estimator used D %cv %bias 1 36 49.3 0.001 0.22 GH 20.7 5.8 -68 1996 1* 11 62.7 0.001 0.03 none 2 9 54.7 0.01 0.16 GH 1.5 0.8 -98 1995 1 11 28.9 0.68 0.98 H 28.0 9.5 13 52 Figure 6. Correlation of the line-transect estimates of density (males/km2) based on sighting distances (DH) VS. the estimates based on perpendicular distances ( D P ) . The data are from'Tables 11 and 14. As explained in Table 14, estimates based on sighting distances are either from the Hayne's or generalized Hayne's estimator. The dotted line shows a 1:1 correlation. The coefficient of correlation r = 0.18. 120 100 + 120 53 4.5 DISCUSSION. 4.5.1 Mobility prior to detection. In some cases, male ptarmigan did move away from the centreline prior to detection during the ground line-transects, but this mobility was not problematic for the line-transect estimator based on perpendicular distances. A visual analysis of the distribution of perpendicular distances does not show any net movement of the birds away from the line (Figure 4). This does not preclude the possibility that movement is sometimes occurring; it only shows that the net movement may be too low to affect significantly the estimate. If evasive movement prior to detection were occurring at a high level, then the estimator would be biased low (Buckland et al. 1993). The underbiases of Table 13 are not large enough to be problematic. Turnock and Quinn (1991) (in Seber 1992) survey this topic and give some methods for extending the theory to handle responsive movements in cases where it becomes a problem. In the field, the mobility of the birds prior to detection could be assessed when possible by looking at the tracks left by the birds in the snow. Male ptarmigan did not consistently try to move away from the line, and the movement rarely exceeded more than about 1 m. It appeared that the movement on the ground was more a way to get ready to flush by getting out of willow bushes and getting on open ground. 4.5.2 Accuracy and precision. With a bias ranging from -7 to -3%, line-transect estimates based on perpendicular distances are consistent and accurate enough for the calibration. The precision seemed to reach a plateau between 15-25% (cv) given at least 20 observations in the area. This limit to the precision might be caused by the clumped distribution of ptarmigan. Given the efficiency of the technique, I recommend its use for surveys of ptarmigan where a total of 60-80 males can be seen at least. This is the minimum sample size recommended by Buckland et al. (1993, p.302). 54 4.5.3 Estimators based on sighting distances and angles. The generalized Hayne's estimator does not seem robust because the estimates of Table 14 based on sighting distances and angles (70% of which were calculated with the generalized Hayne's estimator) correlated poorly with the estimates of Table 11 based on perpendicular distances (r = 0.18) (Figure 6). When I used only Hayne's estimates (even when the assumption that the average sighting angle was of 32.7° had failed) to compare with the estimates based on perpendicular distances, the correlation was excellent (r = 0.98) (Figure 7). The Hayne's estimate were biased by 13% (first survey of 1996 without 29 April), -4% (second survey of 1996) (compared to -98% using the generalized Hayne's estimator), and 13% (1995 survey). Therefore, I do not recommend the generalized Hayne's estimator, but I do recommend the Hayne's estimator as a simple alternative to estimators based on perpendicular distances (e.g. for estimation of density while in the field). Estimators based on perpendicular distances should, however, be preferred in the end since they are more robust (Anderson and Burnham 1996, pers. comm.). The smaller range of bias of the estimates based on perpendicular distances (-3 to -7%) is an indication of it. A drawback of the Hayne's estimator is that many observations (up to 30%) could not be used when birds did not flush. 4.6 CONCLUSION. Using line-transect estimators based on perpendicular distances, I obtained estimates of density in 2 areas in 1995 and 6 areas in 1996 for the calibration of the next chapter. According to the test of the line-transect technique I conducted, estimates should be accurate. The precision seemed to reach a maximum level between 15-25% (cv). I will take into account this imprecision in the calibration. 55 Figure 7. Correlation of the Hayne's line-transect estimates of density (DH) (males/km2) vs. the estimates based on perpendicular distances (DP). The dotted line shows a 1:1 correlation. The coefficient of correlation r = 0.98. 120 100 4-120 56 5. CALIBRATION. 5.11NTRODUCTION. The practical use of the calibration of aerial indices is to predict in the end the density for a given value of the index. A calibration of the index should also detect how the bias of the index varies with changes in density. Osborne (1991) provides a statistical review of calibrations. Recent calibrations of aerial surveys include Dodd and Murphy (1995) and Rodgers et al. (1995). A simple calibration can be done by plotting the index vs. the true density. A constant bias is desired (no bias is even better) since it will then not affect the results of population trends (Cochran 1977, p.379-384). To be considered constant, the bias must not vary over time nor be related to the attribute whose range is being measured (Stopher and Meyburg 1978, p. 18), i.e. in this case the population density. It can be defined quantitatively using the general linear model: xk = a + pu* + £k where Xk = observed measurement value for the population density at time k [Ik = true but unknown value for the population density at time k a and P are parameters that jointly describe the measurement bias for the method (intercept and slope respectively) &k = random error of measurement at time k Jaech (1985) states that: • if a = 0 and P = 1, the method is unbiased • if a * 0 but p = 1, the method has a constant bias • if p * 1, the method has a non-constant bias I can add that if P = constant * 1, the method has a bias linearly related to the population density. 57 Cases of constant bias must be rare in large wildlife surveys as the population density can have a large effect on the bias; however, it is sometimes assumed (e.g. Caughley 1977). Changes in population density can bring changes in the behaviour and distribution of the animals, which can in return affect the proportion of the population visible and the ability of the observers to count accurately (Sinclair 1989). Studies assuming a bias linearly related to the population density (i.e. that a constant proportion of the population is seen) reckon the possible importance of the population density as a factor affecting the bias (e.g. Marsh and Broome 1985). Even this assumption could be far from reality for indices of abundance since the indices may, for example, be non-linearly (e.g. model for call-counts (Eberhardt 1978)) or inversely (e.g. white-tailed deer (Odocoileus virginianus) pellet group counts (Ryel 1971, in Eberhardt and Simmons 1987) correlated to the actual density. A calibration is necessary to establish the relationship between the index and the actual density. 5.2 METHODS. I used ground line-transect estimates (chapter 4) as best estimates of the true density. I did ground counts only in 1995 (2 areas) and 1996 (6 areas), so I based the calibration on the comparison of the index with the density by area instead of by year. I assumed that what was happening at a minor scale in one year was reflective of what was happening across years. The aerial indices used are the means of the replicate surveys per area. This calibration can also be viewed as a double-sample (section 2.4.2) with the exception that I used a regression estimator instead of a ratio (e.g. Eberhardt and Simmons 1987). The use of the geometric mean regression (GMR) (Ricker 1973, 1984, in Krebs 1989, pp.458-464) was necessary for the calibrating model because of errors in both variables: x the ground counts and y the aerial indices. The slope of the central trend line is given by 58 . b lb where v = estimated slope of the GMR b = estimated slope of least-squares regression of y on x r = correlation coefficient between x and y d = estimated slope of the least-squares regression of x on y The standard error of the slope of the GMR is the same as that of the slope of the least-squares regression and with a = 0.05 the confidence limits of the slope are: 95%CL = v±Sit005{2)df The intercept of the regression was set = 0 (at a true density of 0, no ptarmigan can be seen) so that the G M R model be: I = vD where / is the value of the' index and D is the density. Based on the calibration, the sensitivity of the method to detect changes in density is sensitivity = ' v v * \VLCL VUCLj x 100% where vLCL and vVCL are respectively the lower and upper confidence limits of the estimated slope of the GMR. 5.3 RESULTS. Both types of indices, RQ and/, can detect changes in density. They are both positively (v > 0) and linearly related to the density (Figure 8). The outlier was not included in the calibration since its standard error in y is too large. The low precision of this 1995 aerial index can be 59 partially explained by the fact that it was based on only 2 surveys, compared to 5 or 6 for the 1996 indices. The other 1995 datapoint had a low standard error and was included in the calibration. 60 Figure 8. Central trend line of the geometric mean regression (solid line) ± 95% confidence limits (dotted lines) for the calibration of the aerial index per area vs. the density (males/km2). Each datapoint represents a different area and is shown with the standard error. The aerial index is a) RQ the number of ptarmigan seen per km of transect, and b)/the frequency of transects having at least one sighting of a ptarmigan. The regression model is given by index = slope x D, r is the coefficient of correlation. The outlier was not included in the regressions since it was too imprecise. 61 QU - XapiH |BU3V 62 o in in m o ^ d <N o o j - xapuj leuav 63 Because of the linearity of the relationship, the bias is proportional to the density. In other words, even at different densities a constant proportion of the population is detected. The value of this proportion cannot be calculated since the indices and the density are not in the same units. Using the GMR model, the sensitivity of the method to detect changes in density = 61% for RQ and 60% for/(a = 0.05). This is the smallest change in density the index can reliably detect. For this calibration, it did not matter whether I used a standard least-squares regression or a G M R since the coefficients of correlation r were so high (r = 0.96). 5.4 DISCUSSION. 5.4.1 Calibration. Although the calibration showed a positive linear relationship between the aerial indices and the density (Figure 8), there is a flaw. The aerial index, RQ, was significantly lower in 1996 (Table 16) but densities were significantly higher in 2 out of 3 areas (Figure 9). So the calibration cannot be applied to survey indices of previous years. The calibration worked well on a yearly basis, but does not work between years. The slope of the calibration v must vary from year to year and be higher in years where ptarmigan flush more easily. For instance, V95 should be > v% (but this is not testable since there are only 2 datapoints on which to base the 1995 regression). What could explain the lower 1996 index even though the density was higher? The lower index is not due to 1) less visibility (e.g. more snow cover) in 1996 or 2) less experienced observers in 1996 vs. 1995 because a higher proportion of birds were seen on the ground in 1996 (Table 17). Birds on the ground are more difficult to detect than birds that flush. One explanation left is that there was a change in the flushing behaviour of ptarmigan: they flushed less in 1996. Reduced flushing could not be a consequence of increased density (e.g. from stress effect) or of low willow cover (1996 was a late year for the melting of snow), but it could be a 64 consequence of increased avian predation pressure. If increased density was the cause, the slope of the calibration based only on birds that flushed (Figure 10) would have been negative, which is not the case. Low willow cover could not be the cause neither because flushing was not up in the last 1996 survey when only about 25% of snow cover remained on most areas. Actually, the influence of willow cover was hypothesized to be the reverse. When the willow is high enough to provide a good cover, the birds should sit tight rather than flush when the plane flies overhead (Doyle 1995, pers. comm.). However, ptarmigan could flush less because of an increased avian predation pressure. In the study area, golden eagles (Aquila chrysaetos) and northern goshawks (Accipiter gentilis) are the main species of raptors to prey on adult ptarmigan on the breeding grounds. There were more sightings of golden eagles in 1996 than in 1995; there were 4-5 suspected active nests in 1996 and 2 in 1995. At Denali, Alaska, the proportion of the territorial population of golden eagles that lays eggs each year is known to change in response to the numbers of snowshoe hare and willow ptarmigan available in the area (Mclntyre 1996, pers. comm.). So ptarmigan may have adapted their behaviour as a consequence of more avian predators in 1996, or the composition of the ptarmigan population may have changed, the new individuals not tending to flush. The turnover of individuals can be quick in ptarmigan populations with an adult mortality rate between 40 to 60 % per year (Mossop 1987) and an average of 3 offsprings fledged per clutch (Martin et al. 1989). One way or the other, it does not take much of a change to affect the index. Only a small fraction of the population that is seen flushing from the plane per survey (2-7% in 1995; <1% in 1996 [counts in 1996 were standardized to the transect lengths of 1995 for this calculation]). This survey program was based on the assumption that the behaviour of ptarmigan towards the plane does not change from year to year. This is apparently incorrect. This type of assumption is very common in wildlife surveys. For instance, estimates of density of small 65 mammals based on trapping assume an equal catchability from year to year (Krebs 1989), which is often wrong (reference?). If the value of the calibration slope really changes from year to year due to a change in the availability bias, then it is mandatory to use a correction method that corrects for the entire counting bias, not just for the perception bias. The double-sampling method is the only method that can efficiently do so with fixed-winged aircraft sampling. The calibration I did was a kind of double-sampling (except that I used a regression instead of the ratio estimator). However, to get valid yearly corrections the ground surveys would become too costly. The number of areas surveyed per year should be of at least 4-5 so that a regression be possible. For instance, it was not possible to get a correction factor for 1995 based on double-sampling because I surveyed only 2 areas on the ground. The error on such a regression would be too large to permit a reliable correction. An alternative to surveying whole areas on the ground is to census small grids (50-100 ha each), with the grids set up on different areas. The correction is not anymore a true calibration, but simply a ratio of the aerial index for area il density on a grid of area i and is only meaningful for corrections between years (not between areas, unless a few grids are set up at random on each area). An easier solution would be to use helicopters. Helicopter sampling would be a better choice for aerial surveys despite the fact that helicopter time is twice as expensive than airplane time (775$/hr compared to 375$/hr in Kluane - summer 1995) because the value of information provided would outweigh the extra cost. In this case, airplane sampling gave no reliable information on trends in densities, so it was wasteful. Helicopter sampling may provide better information since helicopters can fly very low and very slow. This could result in the following advantages for helicopter sampling over fixed-winged aircraft sampling: • distance sampling possible => generates estimates of density (not only indices) • can also use capture-recapture methods 66 • more birds can be observed • sampling bias can be reduced • detection of birds on the ground becomes easier => variable flushing behaviour has less importance Table 16. Comparison of the means of aerial indices per year between 1995 and 1996 with 2-tailed t-tests. Data from Table 7. Log transformation for ratios and arcsin transformation for frequencies. The ratio index was significantly lower in 1996 than in 1995. The frequency index was not significantly different between the 2 years, but the power was low (0.049). estimator 1995 mean SE 1996 mean SE df t p ratio frequency 0.766 0.058 0.719 0.044 0.333 0.051 0.653 0.045 5 3.092 0.027* 5 0.833 0.443 * significant difference (a = 0.05). Table 17. Comparison of the mean proportions of ptarmigan seen on the ground out of total per survey between 1995 and 1996 with a 2-tailed t-test. The proportions were arcsin transformed. 1996 mean SE 1995 mean SE df t p 44.7 21.0 8.8 12.5 5 2.583 0.049 * * significant difference (a = 0.05). 67 Figure 9. Comparison of the density of ptarmigan (± SE) (#males/km2) between 1995 and 1996 in areas 1 and 2 and on the grid. The comparisons of the line-transect estimates of density of areas 1 and 2 are with 2-tailed t-tests where varo96-D95 = varo96 + varo95- For area 1, t = 0.546, df = 19 and p = 0.591. For area 2, t = 2.154, df = 42 and p = 0.037. The comparison of the total counts on the grid is a comparison of 2 Poisson counts (Zar 1988, p.415)) for which z = 4.25 and p < 0.0001. In 2 out of 3 cases, the density was significantly higher in 1996 than in 1995 (a = 0.05). 68 69 Figure 10. Central trend line of the geometric mean regression (solid line) ± 95% confidence limits (dotted lines) of the aerial index RQ based on the number of ptarmigan flushed/ km of transect vs. the density (males/km2). Same as Figure 6 but using only flushed birds. Each datapoint represents a different area and is shown with the standard error. The regression model is given by index = slope x D, r is the coefficient of correlation. Outlier not shown. 70 OH - xapm AsAjns |euav 71 5.4.2 Ratio estimator vs. frequency estimator for the aerial index. Could the frequency index/be better than the ratio index RQ for the prediction of population trends? Both performed well in the calibration (Figure 8) and gave a clear trend between the aerial index and the density. Nonetheless, only/was not significantly lower in 1996 compared to 1995 (Table 16). RQ was erroneous, showing that the density should have decreased between 1995 and 1996 (Table 16)./could be more robust, but does not necessarily perform better because it did not give a higher index for 1996. The robustness of the/is more of a lack of power, as shown by the test of variance in chapter 3 (Table 9). Its power to detect changes in density is not as high as for RQ. Clearly,/"saturates" at high values and does not have the same capacity as RQ to detect peaks (Figure 11). It seems to perform better at low values where it makes a bigger difference between indices than RQ. However, if there were a random-error on / it would probably show that those big differences at low values are in fact really small. Big differences at low values are most probably due to the discontinuous nature of the estimator. This could lead to incorrect conclusions about low peaks than one would obtain using RQ. RQ may be limited by the preponderance of transects with no observations, but it appears a more reliable choice. 72 Figure 11. Non-linear correlation between the frequency estimates/(% of transects with > 1 ptarmigan) and the ratio estimates RQ (ptarmigan/km of transect) ± SE used as aerial indices. Each datapoint represents a distinct survey. 73 74 5.4.3 Comparison with trends from other techniques. In addition to aerial counts, other techniques were used at Kluane to provide indices of population trends of willow ptarmigan: (i) encounter rate on foot (called foot seen sheet): each field worker recorded the number of sightings of different species per hour in the field on foot. I used only the winter data (November - March). (ii) encounter rate on snowmobile (called snowmobile seen sheet): same as for the foot seen sheet but for the time spent on snowmobile. A more valid type of encounter rate would incorporate the distance travelled instead of the time spent in the field. The encounter rates are ratio estimates. I do not provide the random errors. The one consistency between all the indices is that they dropped by half from 1995 to 1996 (Figure 12). If we remember that the density actually increased from 1995 to 1996, it becomes obvious that none of these indices are able to track changes of density on the breeding grounds through years. A problem with the winter encounter rates at Kluane is that these are in boreal forest in the valley, and that the occupancy of the forest by ptarmigan (probably mostly females as Gruys (1991) showed for the Chilkat Pass) may depend on the weather in the alpine. Doyle (1996, pers. comm.) observed that flocks of ptarmigan could be seen in the valley even in March during snowstorms in the alpine. The proximity of the alpine to the valley bottom (2-5 km) can contribute to this mobile behaviour. In conclusion, the failure of the winter encounter rates to track changes in density could again be attributed to the behaviour of the ptarmigan, this time in reaction to weather conditions in the alpine instead of a reaction to avian predation pressure. 75 Figure 12. Comparison of trends of different indices of density for the Kluane ptarmigan population from 1989-1996. The indices are from different techniques and are ratios of i) aerial counts (# ptarmigan/ km of transect), ii) encounter rates on foot (# ptarmigan/ hour in the field), and iii) encounter rates on snowmobile (# ptarmigan/ hour in the field). The indices from the various techniques were standardized to fit visually on the same scale. Encounter rates for the year x are from Nov.(x-l) - March(x). 76 Aiisuap io xapu| 77 6.5 CONCLUSION. Aerial indices were linearly and positively related to densities. However, the ability to detect differences in density applied only between areas for a given year, and not between years since the density was higher in 1996 than in 1995, but the aerial index was lower. This could be explained by a difference in flushing behaviour between the 2 years. Ptarmigan may have flushed less in 1996 as a response to increased predation pressure from golden eagles. Other types of indices of abundance based on winter encounter rates were no better at detecting changes of breeding density through years, perhaps also because of a variable behaviour of the ptarmigan over years. 78 7. CONCLUSIONS. The empirical evaluation established at first that the Kluane aerial survey technique was repeatable, given the high variability of the indices over years. And in the calibration, the indices were linearly and positively related to densities. However, the ability to detect differences in density applied only between areas for a given year, and not between years since the density was higher in 1996 than in 1995, but the aerial index was lower. This could be explained by a reduced flushing behaviour in 1996. Other types of indices of abundance based on winter encounter rates were no better at detecting changes of breeding density through years. Little can be done with the existing trend data other than to note ptarmigan were continuously present at Kluane from 1990-1996. My analysis showed that surveys should be carefully planned beforehand. The calibration was a good method of correction, but it requires too much effort to be applied on a long term. The simultaneous double-counts correction method was not informative since not enough double-counts were recorded. Helicopter sampling should be tried as an alternative to fixed-winged aircraft sampling for aerial surveys of ptarmigan. For ground surveys, line-transects based on perpendicular distances provided accurate (bias = -3 to -7%) and fairly precise (cv = 15-25%) estimates of density of male ptarmigan per stratum given about at least 20 observations per stratum and 60-80 observations overall. The general protocol of evaluation based on the total error developed here could be applied to many survey estimates. It should not be forgotten that for survey estimates the error is not only made of the counting bias and random error. The estimator used is only one thing that can also affect results, as shown by the comparison off with RQ for the aerial counts and by the comparison of the estimators based on perpendicular distances with the estimators based on sighting distances and angles for the line-transects. Therefore, for any survey estimate I recommend an analysis of the total error using the equation presented here. 79 BIBLIOGRAPHY Anderson, D.R. and K.P. Burnham. 1995. Design and analysis of distance sampling data (a pamphlet presenting the workshop). Colorado cooperative fish and wildlife research unit, Fort Collins, 9 pp. Barnes, A., G.J.E. Hill, and G.R. Wilson. 1986. Correcting for incomplete sighting in aerial surveys of kangaroos. Aust. Wildl. Res. 13:339-348. Bear, G.D., G.C. White, L .H. Carpenter, R.B. Gill, and D.J. Essex. 1989. Evaluation of mark-resighting estimates of elk populations. J. Wildl. Manage. 53: 908-915. Bjerhammar, A. 1973. Theory of errors and generalized matrix inverses. Elsevier, Amsterdam, 420 p. Bodie, W.L., E.O. Garton, E.R. Taylor, and M . McCoy. 1995. A sightability model for bighorn sheep in canyon habitats. J. Wildl. Manage. 59(4): 832-840. Boutin, S., C.J. Krebs, R. Boonstra, M.R.T. Dale, S.J. Hannon, K. Martin, A.R.E. Sinclair, J.N.M. Smith, R. Turkington, M . Blower, A. Byrom, F.I. Doyle, C. Doyle, D. Hik, L. Hofer, A. Hubbs, T. Karels, D.L. Murray, V. Nams, M . O'Donoghue, C. Rohner, and S. Schweiger. 1995. Population changes of the vertebrate community during a snowshoe hare cycle in Canada's boreal forest. Oikos 74(1): 69-80. Bowden, D.C. and R.C. Kufeld. 1995. Generalized mark-sight population size estimation applied to Colorado moose. J. Wildl. Manage. 59(4): 840-851. Broome, L.S. 1985. Sightability as a factor in aerial survey of bird species and communities. Austr. Wildl. Res. 12:57-67. Buckland, S.T., D.R. Anderson, K.P. Burnham and J.L. Laake. 1993. Distance sampling: estimating abundance of biological populations. Chapman & Hall, London, 446 p. Burnham, K.P., D.R. Anderson and J.L Laake. 1980. Estimation of density from line transect sampling of biological populations. Wildl. Monogr. 72. Butler, W.I. Jr., J.I. Hodges, and R.A. Stehn. 1995. Locating waterfowl observations on aerial surveys. Wildl. Soc. Bull. 23(2): 148-154. Caughley, G. 1974. Bias in aerial survey. J. Wildl. Manage. 38(4): 921-933. Caughley, G. 1977a. Sampling in aerial survey. J. Wildl. Manage. 41: 605-615. Caughley, G. 1977b. Analysis of vertebrate populations. Toronto, Wiley, 234 p. Caughley, G. and D. Grice. 1982. A correction factor for counting emus from the air and its application to counts in Western Australia. Aust. Wildl. Res. 9: 253-259. 80 Caughley, G. , R. Sinclair and D. Scott-Kemmis. 1976. Experiments in aerial survey. J. Wildl. Manage. 40(2): 290-300. Choquenot, D. 1995a. Species- and habitat-related visibility bias in helicopter counts of kangaroos. Wildl. Soc. Bull. 23(2): 175-179. Choquenot, D. 1995b. Assessing visibility bias associated with helicopter counts of feral pigs in Australia's semi-arid rangelands. Wildl. Res. 22: 569-578. Cochran, W.G. 1977. Sampling techniques, 3rd edition. New York, Wiley. Conroy, M.J., J.R. Goldsberry, J.E. Hines and D.B. Stotts. 1988. Evaluation of aerial transect surveys for wintering american black ducks. J. Wildl. Manage. 52(4): 694-703. Cook, R.D. and J.O. Jacobson. 1979. A design for estimating visibility bias in aerial surveys. Biometrics 35: 735-742. Dodd, M . G . and T . M . Murphy. 1995. Accuracy and precision of techniques for counting great blue heron nests. J. Wildl. Manage. 59(4): 667-673. Eberhardt, L .L . 1978. Appraising variability in population studies. J. Wildl. Manage. 42(2): 207-238. Eberhardt, L .L . 1982. Calibrating an index by using removal data. J. Wildl. Manage. 46: 734-740. Eberhardt, L .L . 1985. Assessing the dynamics of wild populations. J.Wildl. Manage. 49: 997-1012. Eberhardt, L .L . and M.A. Simmons. 1987. Calibrating population indices by double sampling. J. Wildl. Manage. 51(3): 665-675. Floyd, T.J., L .D. Mech, and M.E. Nelson. 1979. An improved method of surveying deer in deciduous-coniferous forests. J. Wildl. Manage. 43(1): 258-261. Gates, C.E. 1979. Line-transect and related issues, in Sampling biological populations, eds. R.M. Cormack, G.P. Patil, and D.S. Robson. International co-operative publishing house, Fairland, pp. 71-154. Graham, A. and R. Bell. 1989. Investigating observer bias in aerial survey vy simultaneous double-counts. J. Wildl. Manage. 53(4): 1009-1016. Grier, J.W., J .M. Gerrard, G.D. Hamilton and P.A. Gray. 1981. Aerial visibility bias and survey techniques for nesting bald eagles in northwestern Ontario. J. Wildl. Manage. 45(1): 83-92. Gilbert, P.F. and J.R. Grieb. 1957. Comparison of air and ground deer counts in Colorado. J. Wildl. Manage. 21(1): 33-37. Gruys, R.C. 1991. Autumn and winter movements and mortality of willow ptarmigan at Chilkat Pass, B.C.. M.Sc. thesis, Dept. of zoology, Univ. of Alberta, 182 pp. 81 Hannon, S.J. 1983. Spacing and breeding of willow ptarmigan in response to an experimental alteration of sex ratio. J. Anim. Ecol. 52: 807-820. Haramis, G.M. , J.R. Goldsberry, D.G. McAuley and E.L. Derleth. 1985. An aerial photographic survey of Chesapeake Bay and North Carolina canvasbacks. J. Wildl. Manage. 49: 449-454. Hayne, D.W. 1949. An examination of the strip census method for estimating animal populations. J. Wildl. Manage. 13: 145-157. Holt, R.S. and J. Cologne. 1987. Factors affecting line transect estimates of dolphin school density. J. Wildl. Manage. 51(4): 836-843. Jaech, J.L. 1985. Statistical analysis of measurement errors. Exxon monograph, Toronto, 293 p. Jolly, G.M. 1969. The treatment of errors in aerial counts of wildlife populations. East Afr. Agric. For. J. 34: 50-55. Kelley, J.R.Jr. 1996. Line-transect sampling for estimating breeding wood duck density in forested wetlands. Wildl. Soc. Bull. 24(1): 32-36. Kraft, K . M . , D.H. Johnson, J.M. Samuelson and S.H. Allen. 1995. Using known populations of pronghorn to evaluate sampling plans and estimators. J. Wildl. Manage. 59(1): 129-137. Krebs, C.J. 1989. Ecological methodology. Harper Collins, New York, 654 p. Krebs, C.J., S. Boutin, R. Boonstra, A.R.E. Sinclair, J.N.M. Smith, M.R.T. Dale, K. Martin, and R. Turkington. 1995. Impact of food and predation on the snowshoe hare cycle. Science 269: 1112-1115. Krishnaiah, P.R. and C.R. Rao, eds, Handbook of statistics, vol. 6, Elsevier science publishers B.V. (1988) 594 p. Laake, J.L., K.P. Burnham, and D.R. Anderson. 1979. User's manual for program TRANSECT, Utah state university press, Logan. Laake, J.L., S.T. Buckland, D.R. Anderson and K.P. Burnham. 1994. DISTANCE user's guide v2.1. Colorado cooperative fish and wildlife research unit, Fort Collins, 84 pp. LeResche, R.E. and R.A. Rausch. 1974. Accuracy and precision of aerial moose surveying. J. Wildl. Manage. 38(2): 175-182. Marsh, H. and D.F. Sinclair. 1989. Correcting for visibility bias in strip transect aerial surveys of aquatic fauna. J. Wildl. Manage. 53(4): 1017-1024. Martin, K., S.J. Hannon, and R.F. Rockwell. 1989. Clutch size variation and patterns of attrition in fecundity of willow ptarmigan. Ecology 70(6): 1788-1799. 82 Mossop, D.H. 1987. Winter survival and spring breeding strategies of willow ptarmigan. In: Adaptive strategies and population ecology of northern grouse, Bergerud, A.T. and M.W. Gratson, eds., Univ. of Minn. Press, Minneapolis, pp. 330-377. Mossop, D.H. 1994. Trends in Yukon upland gamebird populations from long-term harvest analysis. In: North american gamebirds: developping a management and research agenda for the 21st century. Trans. 59th no. am. wildl. & natur. resour. conf., Wildlife management institute, Washington, p. 449-456. Mourao, G.M. , P. Bayliss, M.E. Coutinho, C.L. Abercrombie, and A.Arruda. 1994. Test of an aerial survey for caiman and other wildlife in the Pantanal, Brazil. Wildl. Soc. Bull. 22: 50-56. Osborne, C. 1991. Statistical calibration: a review. Int. stat. rev. 59(3): 309-336. Packard, J.M., R.C. Summers and L.B. Barnes. 1984. Variation of visibility bias during aerial surveys of manatees. J. Wildl. Manage. 49(2): 347-351. Peterson, R.O. and R.E. Page. 1993. Wildl. Soc. Bull. 21: 80-86. Pojar, T . M . , D.C. Bowden, and R.B. Gill. 1995. Aerial counting experiments to estimate pronghorn density and herd structure. J. Wildl. Manage. 59(1): 117-128. Pollock, K.H. and W.L. Kendall. 1987. Visibility bias in aerial surveys: a review of estimation procedures. J. Wildl. Manage. 51: 502-509. Prenzlow, D . M . and J.R. Lovvorn. 1996.Evaluation of visibility correction factors for waterfowl surveys in Wyoming. J. Wildl. Manage. 60(2): 286-297. Rao, P.S.R.S. 1988. Ratio and regression estimators. In: Handbook of statistics, vol. 6, Krishnaiah, P.R. and C.R. Rao (eds.), Elsevier science publishers B.V., p. 449-468. Robb, L .A. and M.A. Schroeder. 1995. Census methods for upland game birds in British Columbia (draft). Resource inventory committee, Province of British Columbia, 88 p. Rodgers, J.A., S.B. Linda, and S.A. Nesbitt. 1995. Comparing aerial estimates with ground counts of nests in wood stork colonies. J. Wildl. Manage. 59(4): 656-666. Ryel, L .A. 1971. Evaluation of pellet-group surveys for estimating deer populations in Michigan. Ph.D. Thesis, Michigan State Univ., East Lansing. 237 pp. Samuel, M.D. , E.O. Garton, M.W. Schlegel, and R.G. Carson. 1987. Visibility bias during aerial surveys of elk in northcentral Idaho. J. Wildl. Manage. 51(3): 622-630. Seber, G.A.F. 1986. A review of estimating animal abundance. Biometrics 42: 267-292. Seber, G.A.F. 1992. A review of estimating animal abundance II. Int. Stat. Rev. 60(2): 129-166. Sherman, D.E., R.M. Kaminski, and B.D. Leopold. 1995. Winter line-transect surveys of wood ducks and mallards in Mississippi greentree reservoirs. Wildl. Soc. Bull. 23(2): 155-163. 83 Siniff, D.B. and R.O. Skoog. 1964. Aerial surveying of caribou using stratified random sampling. J. Wildl. Manage. 28(2): 391-401. Stopher, P.R. and A . H . Meyburg. 1979. Survey sampling and multivariate analysis for social scientists and engineers. Lexington Books, Toronto, 385 pp. Thompson, B.C. and B.W. Baker. 1981. Helicopter use by wildlife agencies in North America. Wildl. Soc. Bull. 9(4): 319-323. Thompson, S.K. 1992. Sampling. Wiley, New York, 343 pp. Weeden, R.B. 1963a. Grouse and ptarmigan in Alaska: their ecology and management. Alaska department of fish and game, Juneau, 110 pp. Weeden, R.B. 1963b. Management of ptarmigan in North America. J. Wildl. Manage. 27: 672-683. White, G.C. , R .M. Bartmann, L .H. Carpenter and R.A. Garrot. 1989. Evaluation of aerial line transects for estimating mule deer densities. J. Wildl. Manage. 53: 625-635. Wolfe, M . L . and J.F. Kimball. 1989. Comparison of bison population estimates with a total count. J. Wildl. Manage. 53: 593-596. Zar, J.H. 1984. Biostatistical analysis. Prentice-Hall, Toronto, 718 pp. 84 

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