Sampling Problems in Estimating Small Mammal Population Size1

Transcription

1 Sampling Problems in Estimating Small Mammal Population Size1 George E. Menkens, Jr.2 and Stanley H. Anderson3 Abstract. -Estimates of population size are influenced by four sources of error: measurement, sampling, missing data, and gross errors. Measurement error can be reduced by using the correct estimator, reducing variation in capture probabilities, and by increasing sample size and trap period length. Sampling error can be decreased by increasing the number of grids trapped. Species conservation and management or analysis of environmental impacts require accurate estimates of population size. Because censusing entire populations is difficult, if not impossible, a sampling program is generally employed to estimate animal abundance. In small mammal studies, sampling is frequently performed using live traps placed in grids. Numerous approaches have been used to estimate animal abundance on trapping grids (eg., catchper-uni t effort, removal methods) but capture-mark-recapture techniques are the most commonly used (Seber 1986). Four sources of error may influence an estimator's bias and precision (Cochran 1977, McDonald 1981 ). Two, missing data and gross errors (e.g., misreading tag numbers) are "human" errors and can be avoided by using careful field and laboratory techniques. The remaining sources, 'Paper presented at symposium, Management of Amphibians, Reptiles, and Small Mammals in North America (Flagstaff, AZ, July 79-21, 1988). 2George E. Menkens, Jr., is a Research Associate with the Wyoming Cooperative Fish and Wldlife Research Unit,4 Laramie, WY Stanley H. Anderson is Leader, Wyoming Cooperative Fish and Wildlife Research Laramie, WY Cooperators in the Wyoming Cooperative Fish and Wildlife Research Unit include: the Department of Zoology and Physiology, University of Wyoming; Wyoming Game and Fish Department; and the U.S. Fish and Wildlife Service. measurement and sampling error, may, in many cases greatly affect an estimate (McDonald 1981). Measurement error is the error resulting from the use of imprecise or biased (or a combination of these) data collection methods (McDonald 1981). In markrecapture studies, measurement error influences the bias and precision of an estimate for any single grid. Sampling variance is considered to be a measurement error in mark-recapture studies (White et al. 1982). Sampling error is error introduced by natural variation between sampling units, i.e., trap grids. Potentially large sources of measurement error in mark-recapture studies may result from capture probability variation and model selection. All mark-recapture estimators make specific assump tions about capture probability variation within and among animals and trapping days. Three factors influencing individual capture probability variation have received attention (Burnham and Overton 1969, Otis et al. 1978, Pollock 1981, Seber 1982) and are time, behavior, and individual heterogeneity. Models assuming time variation allow all animals to have the same capture probability on a given day, but this probability may change between days. Models allowing behavioral responses to trapping assume all animals initially possess identical capture probabilities, but these probabilities may change upon first capture. Capture probabilities may increase (animals become trap happy) or decrease (animals become trap shy) after initial capture. Models assuming that individual heterogeneity is present allow each animal to have a unique capture probability that does not change over time. Combinations of these factors may also occur. For example, an animal's capture probability may be influenced by both time and behavioral effects. Model selection is another source of measurement error. Selection of an inappropriate or incorrect model for data analysis results in estimates with unknown degrees of bias and unacceptably large or unrealistically small standard errors (Otis et al. 1978, White et al. 1982). CAPTURE (Otis et al. 1978) is a widely used computer program for estimating population size using mark-recapture data that also provides an objective method for selecting the correct model when any of the above sources of capture probability variation are present. In this paper, we investigate the effects that variation in capture probabilities due to time, behavior, and individual heterogeneity have on estimates of animal abundance and model selection. We also discuss improvement of an estimate using data pooling. We use these results to show how reducing trap period length influences estimator bias, standard error, and confidence interval coverage rate, and discuss how this may help reduce the number of

2 grids required to detect a given difference between yearly estimates of population sizes. Material and Methods To investigate effects of both capture probability variation and trap period reduction, we used program CAP- TURE (Otis et al. 1978) to randomly generate and analyze data sets with known population characteristics (see Menkens 1987 for details). CAP- TURE contains eight models, five with estimators, for estimating population size for closed populations when capture probabilities do not vary (model M(o)), or when they vary with time (model M(t)), behav- ioral response (model M(b)), individual heterogeneity (model M(h)) or a combination of the behavioral and individual heterogeneity models (model M(bh)). Using CAPTURE, we specified the number of trapping periods, population size, and capture probabilities, and patterns of variation. CAPTURE was then used to analyze each data set. We analyzed the same data sets using Chapman's unbiased version of the Lincoln-Petersen estimator and its variance estimator (Seber 1982). Because the Lincoln-Petersen estimator uses data from only two periods, each data set was split prior to estimation. Thus in. a 5 day trapping study, the first 3 days constituted the marking period, and the second 2 days was the recapture period. In studies 10 days long, the first 5 days were the marking period, the second 5 days the recapture period. Data were generated for a wide range of conditions. We used trap periods of 5 and 10 days, population sizes of 50 and 100 and a wide variety of capture probability patterns (table 1). One thousand data sets were generated for each combination of these conditions. In this paper, we only generated data meeting the assumptions of one of the five models with estimators in CAPTURE. For each data set, CAPTURE was forced to perform the analysis using the correct model. For example, if data were generated under the assumption of time variation, CAPTURE was forced to use model M(t) for the analysis. Simulations were also performed using the same, and additional, capture probabilities (table I), with CAP- TURE being allowed to select an estimator using its model selection procedure. Results Performance of both the Lincoln-Petersen estimator and CAPTURE is dependent upon the size and magnitude of the variation in capture probabilities (table 2). Estimators have lower degrees of bias, smaller standard errors, and higher confidence interval coverage rates when capture probabilities are high and their variation is low (tables 1 and 2) over all population sizes. When capture probability variation is constant, the estimator's bias tends to decrease and confidence interval coverage rates increase with increasing population size (table 2). Although this pattern is evident for standard errors, patterns of change with increasing sample size are not as clear (table 2). In general, the estimator's bias decreases and confidence interval coverage rates increase as trapping period length increases (table 2). This

3 pattern is not as obvious for standard errors, although they do tend to improve with increasing trap period length (table 2). In most cases the magnitude of change in bias is smaller for good capture probabilities than for poor capture probabilities when the trapping period increases (table 2). Although estimated standard errors tend to decrease with lengthening trap periods (more so with good capture probabilities), the magnitude of this change is generally smaller than is change in bias (table 2). As with bias and standard error, confidence interval coverage rates improve as trapping period increases; the magnitude of change tends to be larger when capture probabilities are poor (table 2). Except when data were generated under model M(o), CAPTURE selected the correct model less than 11 % of the time (table 3). The Lincoln-Petersen estimator failed to provide an estimate at most 7% of the time (table 3). Discussion In small mammal studies, measurement errors can significantly influence an estimator's bias and precision. This study shows the importance of both reducing capture probability variation and increasing the size of those probabilities on measurement error. Decreasing capture probability variation reduces the estimate's bias and coefficient of variation, and increases its confidence interval coverage rate. This result has also been stressed by Burnham and Overton (1969), Menkens (19871, Menkens and Anderson (in press), Otis et al. (1978), and White et al. (1982). Of particular significance is the need to reduce variation due to behavioral responses (i.e., trap-happiness and shyness) and individual heterogeneity, especially when these factors act in concert (Menkens and Anderson in press, Otis et al. 1978, White et al.

4 1982). Reduction of time variation, particularly if the Lincoln-Petersen estimator is used, is important, but not as critical (Menkens 1987, Menkens and Anderson in press). Again, reducing variation in capture probabilities leads to estimates that have lower bias and increased precision. Methods for reducing variation in capture probabilities are numerous (see Otis et al. 1978, Seber 1986, White et al. 1982). Behavioral responses may be reduced by the use of different capture and recapture techniques. For example, animals could be captured using live traps and marked, and then "recaptured" visually using spotting scopes (Fagerstone and Biggins 1986). In addition, use of traps not avoided by animals, and use of non-intrusive marking techniques (e.g., ear tags instead of toe clipping) may also help reduce behavioral responses. Use of traps not avoided by animals may help increase capture probabilities. If sample sizes are large, heterogeneity may be reduced by stratifying the data into sex and age groups with separate analyses performed on each group (Otis et al. 1978, White et al. 1982). If data are stratified however, the effects of small sample size on the estimator's properties must be considered. Capture probabilities may be increased and their variation reduced after study completion by pooling individual trap periods into single marking and recapture periods as was done in our simulations (Menkens 1987, Menkens and Anderson in press). When data are pooled in this way and the.lincoln-petersen estimator used, capture probabilities are 20 to 25% higher than those for individual days (Menkens 1987). In most cases, data pooling results in estimates with improved properties. Use of the wrong model for analysis leads to estimates with unknown degrees of bias and unacceptably large or unreasonably small standard errors (Otis et al. 1978, White et al. 1982), thus contributing significantly to measurement error. In this study, we forced CAPTURE to use the correct model for analysis. This, is unrealistic however, in that biologist never know which model is appropriate. CAPTURE provides a objective model selection procedure, however this procedure works poorly with the small sample sizes typically encountered in many field studies (Menkens 1987, Menkens and Anderson in press, Otis et al. 1978, White et al. 1982). In most cases, the Lincoln- Petersen estimator is a valid alternative to CAPTLJRE when sample sizes are small, except when capture probabilities are influenced by severe behavioral responses or large degrees of individual heterogeneity (Menkens 1987, Menkens and Anderson in press). Because use of the most appropriate model is critical, CAP- TURE should be used to determine the type and magnitude of capture probability variation in a data set, and if variation is low, the Lincoln- Petersen estimator should be used in analysis (Menkens and Anderson in press). Many additional factors contribute to measurement error. Eliminating these requires detailed knowledge of species behavior and ecology, and use of compatible techniques. For example, baits identical to, or that closely approximate natural f d items, should be used (Dobson and Kjelgaard 1985). Tags that are easily lost will lead to severe overestimates of population size and should not be used. Other factors that could contribute to measurement error include use of traps or other activities that decrease survival or increase emigration or immigration, and use of improper traps for the species. Sampling error is the error that results from natural variation between sampling units; the larger this variation, the larger the number of units that must be sampled to detect a difference in population size. For example, when environmental impacts are being assessed, sampling error would be decreased by increasing the number of grids in the control and experimental groups. Reducing variation in capture probabilities allows decreasing the number of days each grid is trapped without large increases in bias or standard errors or decreases in confidence interval coverage rates. By reducing the trap period, more grids can be sampled in a shorter period of time, thereby reducing sampling error and improving the estimate of overall population size. Trapping in as short a time interval as possible will also decrease variation caused by temporal population effects. One approach to reducing Sampling error is to reduce intergrid variation by using a stratified Sampling approach. In this case, investigators could stratify the habitat based on some characteristic that is correlated with animal density and trap within these strata. Sample sizes would be estimated for each strata. Conclusions Reduction of capture probability variation and maximizing their mag-

5 nitude are critical to obtaining unbiased and precise estimates of population size, and also allow the selection of the proper model for use in analysis. Although we have concentrated on small mammals, our points concerning reduction of both variation in capture probabilities and of measurement and sampling error, pertain to other studies using mark-recapture techniques (e.g., papers in Ralph et al.). Our conclusions will hopefully force investigators to realize that their techniques, particularly in poorly designed and carelessly performed studies, may not provide as detailed and profound conclusions as they might expect. We reiterate that care in designing a study can minimize many (but not all) of the sources of measurement and sampling errors we have discussed. Acknowledgments We thank Mark Boyce, Marc Evans, Richard Greer, Lyman McDonald, and Brian Miller for comments on earlier versions of this manuscript. The Department of Zoology and Physiology, the University of Wyuming provided the computer time used for the simulations reported here. Literature Cited Burnham, Kenneth P. and William S. Overton A simula tion study of live-trapping and estimation of population size. Oregon State University, Department of Statistics, Technical Report p. and appendix. Cochran, William G Sampling Techniques. John Wiley and Sons, New York. Dobson, F. Stephen and Julia D. Kjelgaard The influence of food resources on population dynamics in Columbian ground squirrels. Canadian Journal of Zoology Fagerstone, Katheleen A. and Dean White, Gary C., David R. Anderson, E. Biggins Comparison of Kenneth P. Burnham, and David capture-recapture and visual L. Otis Capture-recapture count indices of prairie dog densi- and removal methods for samties in black-footed ferret habitat. pling closed populations. Los Great Basin Naturalist Memoirs Alamos National Laboratory, LA- 8: NERP. McDonald, Lyman L Summarizing remarks: Observer variability, In Estimating numbers of birds, Studies in Avian Biology Number 6, p J.M. Scott and C.J. Ralph, editors. The Cooper Ornithological Society. Menkens, George E., Jr Temporal and spatial variation in whitetailed prairie dog (Cynomys leucurus) populations and life histories in Wyoming. Unpublished Ph.D. Dissertation, University of Wyoming, Laramie. Menkens, George E., Jr. and Stanley H. Anderson. in press. Estimating small mammal population sizes. Ecology. Otis, David L., Kenneth h). Burnham, Gary C. White, and David R. Anderson Statistical inference from capture data on closed animal populations. Wildlife Monographs 62:l-135. Pollock, Kenneth H Capturerecapture models: a review of current methods, assumptions and experimental design, In Estimating numbers of birds, Studies in Avian Biology Number 6, p J.M. Scott and C.J. Ralph, editors. The Cooper Ornithological Society. Ralph, C. John and J. Michael Scott (eds) Estimating numbers of birds, Studies in Avian Biology Number 6, p The Cooper Ornithological Ssciety. Seber, G.A.F The estimation of animal. abundance. MacMillan Publishing Company, New York. Seber, G.A.F A review of estimating animal abundance. Biometrics 42: