SECTOR SAMPLING can be used for a wide variety of

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1 A New Type of Sample Plot that Is Particularly Useful for Sampling Small Clusters of Objects Kim Iles and Nicholas J. Smith Abstract: The sampling method presented here uses a new type of plot, with surprising characteristics. Although useful in sampling any grouping of objects, it is illustrated here by correctly sampling small patches of trees remaining after harvest. Small patches are difficult to sample unbiasedly because of the edge effect caused by sampling with fixed or variable plots. By this we do not refer to a biological effect on trees near an edge, but to a common sampling bias that distorts the selection probabilities of those trees. Groups of objects that are small and irregular are very prone to this edge effect bias when using typical sampling systems. Sector plots will allow the user to sample the trees in a small patch, and the method can also be extended to small trees planted outside the patch after harvesting, or to solitary dispersed trees outside the patch but within the harvested area. The method can unbiasedly select sample trees along an irregular border. The method eliminates all bias from edge effect. It can balance the tree selection on opposite sides of the patch, and the pivot point of the plot can be arbitrarily placed by the sampler in any convenient location without introducing any bias to the selection process. This new plot shape might be described as a constrained angle shape with variable area, randomly oriented around an arbitrary sample point, which selects objects with equal probabilities. FOR. SCI. 52(2): Key Words: Sector sample, sampling, variable retention, plot shape, cluster sampling. SECTOR SAMPLING can be used for a wide variety of sampling problems, but was developed by the authors for the sampling of variable retention areas in British Columbia, Canada. Variable retention systems are silvicultural systems designed to leave enough tree canopy to have a forest or residual tree effect over the majority of a harvested area. This silvicultural system is described in Mitchell and Beese (2002) and Beese et al. (2003). Trees are left as groups or patches, usually over 0.25 ha in size, as individual trees dispersed through the remaining harvest area or as a combination of both. Our intent was to devise a sampling system that could monitor the remaining patches after harvest, and the natural or planted regenerating trees outside these patches if that was of interest to a researcher. These small patches presented several sampling challenges for traditional fixed area plots, strip plots, line transects, or variable plot (angle-gauge) sampling on larger areas. The main objectives were to sample 1. With equal probability (or known probability) for individual trees. 2. With a balance of trees on opposing sides of patches (to examine the effects of azimuth, i.e., N, S, E, or W). 3. With a method that could be extended out into the area around the patch of trees, to record initial conditions and monitor new trees as they develop in the area surrounding the patch. 4. In a way suitable to sample for growth over time. 5. In a way suitable to sample for mortality over time, especially from windthrow, which is more likely to occur near the edge of the patch. 6. In a way that eliminates edge effect biases. 7. In a way that did not require knowing the area of the tree patch. The process described here provides for all of these considerations with a single design, and allows a few other advantages. 1. It is possible to put the pivot point (the point from which the sector sides are projected) in an arbitrary location (rather than a random or systematic one) and still maintain exactly the correct probabilities of tree selection. 2. It is possible to simply and correctly select trees along a convoluted and unmapped border of a sampled area. Previous Work The authors are not aware of any previous description of what they have introduced here as a sector plot. It is likely that something similar (such as sampling part of a circle) has been used before, but it seems less likely that these unusual sampling and probability issues have been observed before, or that the sector has been projected to the edge of an arbitrary polygon shape. We have not been able to find any previous descriptions in the statistical literature, and an Kim Iles. Biometrician, Kim Iles & Associates Ltd., 412 Valley Place, Nanaimo, British Columbia V9R 6A6, Canada Phone and Fax: (250) ; kiles@island.net. Nicholas J. Smith, Cascadia Forest Products, 65 Front St., Nanaimo, British Columbia V9R 5H9, Canada. Acknowledgments: The authors acknowledge the partial support of Forest Investment Account funding from the British Columbia government. Manuscript received November 24, 2004, accepted April 14, 2005 Copyright 2006 by the Society of American Foresters 148 Forest Science 52(2) 2006

2 appeal on the Internet and at professional meetings did not result in any responses regarding this topic. The ideas about this plot shape are, as far as we can determine, original. We have been repeatedly surprised at the statistical properties of the method, and are not certain we have explored all the possibilities this new plot type has to offer. The expertise of the authors is largely restricted to forest sampling and a bit of wildlife work, so perhaps another specialized field has developed this method independently of our own work. If readers have any knowledge of previous work of this kind, the authors would appreciate that information so that appropriate credit may be given. Sector Plots An Unusual Plot Shape Assume that there is a single patch of trees left somewhere within a harvest block. A researcher wants to monitor the condition of these trees with an appropriate sampling method. We are proposing what we believe to be a new type of plot shape for this purpose. We call this new plot shape a sector plot. [1] This plot selects any members of the population inside a fixed angle (the sector angle ) projecting from a pivot point toward the edge of the area to be sampled. The area sampled could be within the patch of trees, the area outside the patch but still inside the harvest area, the outside border of the harvest area, or all of these combined. As an example, one might begin with sampling the patch of trees. Specifically, one might consider a design with four symmetric or balanced sectors projecting from the pivot point (see Figure 1). Each of these sectors can be considered to be an individual plot, or the group of four sectors could be considered a single cluster plot. The authors would prefer using a single cluster value derived from the total of the four sectors. A sector plot is constrained in its basic shape, but the area varies as it projects from the pivot point toward the border of the area sampled. As an example, assume a sector angle of 9 in each sector. The total of the 4 sectors is 36, and the total area covered is 1 10 of a circle. Orient one of the sectors randomly from 0 to 360, and the rest are fixed into place at right angles to each other. It is also quite possible to use several independently oriented sectors as a random sample. Doing so would allow the usual estimations of variance for the patch total, as described later. There is no problem when several random sectors overlap, which is analogous to a situation where random fixed plots might overlap. A simple sector plot, such as the one shown here, will give every tree in the sample area exactly the same probability of selection. In addition, it will tend to select sets of trees from opposing sides of the patch. This balanced selection is desirable in examining the effects of light exposure or windthrow, which was a point of interest with the study that generated this sampling scheme. One could also use some other cluster configuration, such as a group of three sectors or simply opposing pairs. When each plot is established, the researcher could use a transit to check the edges of the sector and determine the trees to be included. The authors have found it effective to use a sector centerline, with offsets along that centerline to establish the sector Figure 1. A set of four Balanced Sector Plots. Forest Science 52(2)

3 edges when visibility is a problem due to vegetation or other visual obstructions. The right angle offset amount at horizontal distance D along the sector centerline is simply [tan( /2)*D]. One can control the expected proportion of trees chosen by varying the size of the sector angle. The sector angle should be chosen before the orientation is randomly assigned, and a larger angle will obviously give a larger expected number and proportion of trees. To choose approximately 20 trees from a patch of approximately 120, use a (20/ ) 60 angle, and if a cluster of 4 sectors is to be used, make them individually 1. It is the randomization of the centerline of the sector after the pivot point is established that eliminates bias in the selection probability of the objects to be sampled trees in this case (see the later topic Why is the tree selection equally probable? for further elaboration). Sector plots are in no way restricted to the sampling of patches of objects. It is quite possible to sample inside a more uniform area. With a pivot point near the center of the area, the systematic design can provide a good balance of trees from various aspects within the area and along the border of the area. The central pivot point should be chosen for ease of monumentation, convenience, and clear sighting. One might avoid, for instance, putting the pivot point on a rock outcrop that would be difficult to monument or hard to relocate. No bias will occur because of this avoidance of difficult pivot point locations. No bias will occur if the field crew locates the pivot point incorrectly according to some existing sample plan, as long as the orientation of the sector is chosen after the pivot point is placed. Sampling Outside the Patch of Objects It is also possible to monitor, as an example, smaller trees (and any replanting stock) outside the tree patch. This might be of interest because of the influence that the patch of objects has on the surrounding area. To do this, simply project the sector, and measure any trees out to the limit of the influence zone around that patch, however one might choose to define that zone. One variation of the variable retention method leaves dispersed individual trees instead of tree patches, and sector plots can also be used to select those trees unbiasedly. If the sampler projects the sector all the way to the harvest area border, the sector method will sample any remaining dispersed individual trees with equal probability. Because the sector includes more area as it projects from the pivot point, the field crew may begin to measure too much area and too many objects. Some form of subsampling may be desired. In this case, one can step down the angle used when the width of the sector becomes too large. In Figure 2, the angle has been reduced by one-half at some arbitrary distance from the center point to reduce the number of trees chosen. One can choose to implement this procedure for any reason at all, at any distance from the pivot point. From this point outward a half-angle 2, having the same vertex as the original sector plot, can be used to continue the selection process. Which half of the divided sector to use must be a random choice. The authors experience is that this is simpler for the field crews than choosing a new centerline for the reduced angle within the larger sector, but that could also be done. In this example, any trees within the steppeddown (shaded) area will have one-half the probability of being chosen ( 2 /360 ) compared to the trees in the full sector area. This reduction can be done again at other points along the sector. This is one practical way to monitor the trend all the way from the patch to the border of the harvest area without getting too many measured objects. Other ways to subsample are also possible, but the authors have found this one to be useful in the field. All trees measured within this reduced (shaded) area in Figure 2 would be weighted twice as much as the other trees for computations such as averages or totals. The authors suggestion is to use simple integer relationships for these adjustments, although it can be done with any proportion. The term probability of tree selection as used in this article (P t ) refers to the probability of selection by the initial Figure 2. Reduction to a partial sector to reduce effort. 150 Forest Science 52(2) 2006

4 full sector angle. A stepped-down sector where an initial angle was divided to limit the number of items chosen will obviously be less than this initial probability. The weight w t for any trees in those reduced areas would be ( s / p ), where s is the initial angle beginning the sector and p is the smaller angle used in the partial sector where that tree was actually chosen with probability P p. In this case the weighted probability P t (w t P p ) is equal for all trees. Sampling Multiple Patches If there are relatively few patches, one might decide to put a sector sample into each patch. All trees inside a patch would be chosen with the same probability, and the probability of choosing a tree will also be equal from patch to patch when the same sector angle is used, regardless of the size of that patch. Therefore, all the data on the area can be combined easily for analysis whenever the area involved can be ignored and only the measurements on the trees are an issue. If a different sector angle is used in some of the patches, the tree weights will be proportional to the angle used. If one sector plot uses 9, and another 12, then the tree data could be weighted by 360 /9 and 360 /12, respectively. If several independent sectors are taken within each patch, this allows the patches to be treated as strata, processed as a stratified sample using the standard equations for combined averages, totals, or variability for these statistics. A discussion of these stratified sampling computations is available in most statistical texts. Cochran (1977, p. 89) is an example. For some research needs, only a selection of sample trees is required, regardless of the patch in which they reside. When patches are chosen randomly (with equal probability) and one sector has been established in each patch (each with the same sector angle) the sampler can easily subsample from the combined trees. A list of trees could be made, then a random or systematic sample could be chosen from that list. In the case of a systematic sample, a random start should be used. One patch might have 100 trees, and another patch might have 15 trees, but this is of no consequence. The cumulative probability of initially choosing any tree in each patch would be exactly the same (10% in Figure 1), and therefore the probability of selection in a subsample would also remain equal. Sampling Regions Surrounding Individual Tree Patches When areas are large, it may be appropriate to divide the overall area into sampling regions that tessellate the entire tract. With the border of each sampling region clearly designated, trees will be selected only by shorter sectors originating from a pivot point within their own sampling region and extending only to that region boundary. The size and shape of the sampling region has no bearing on the probability of tree selection, and therefore the boundaries can be arbitrary and convenient. The authors have done this in a practical field study, with one tree patch in each sampling region. If there are six tree patches in an area, then that area can be divided into six sampling regions using convenient and easily distinguished boundaries, such as roads, well defined watercourses, or straight lines between easily visible landmarks. It is also possible to do several independent sector plots within an area that is not designated into sampling regions, and if trees are chosen with independent sector plots they would simply be recorded more than once. The same equations apply whether different or identical pivot points are used. To be more specific, the cumulative probability of any tree being sampled (P t ) inside a particular sampling region, with n s sectors in that particular region, and those sectors having potentially different angles s with which the tree might have been chosen, is p t n s s 1 s. (1) 360 Sampling for Growth and Mortality over Time Mortality is a much bigger problem than growth, which will be relatively more consistent, and much easier to predict. It is probably desirable to sample for growth and mortality separately. The authors believe that the only current way to solve the mortality problem is simply to monitor a large number of trees, perhaps all of them in the patch. This means tagging them initially, and perhaps recording dbh. In the future, all that is needed is to make a quick check of all tagged trees to see which ones have died, and are either blown down or are still standing. This should be relatively inexpensive. At the time individual trees die the researcher can get many of the possible required measurements, including past ones, by felling or boring the tree. Accurate diameter and growth measurements (as well as final decay amounts) for any fallen trees can be gathered at that time. Trees that are cut and removed from the site are obviously a problem for these kinds of retrospective measurements. When there are many trees in the patch, one can use a sector plot to tag only part of them, but the aim would still be to deal with a large number of monitored trees. Growth is a much less variable problem, so one might use a narrow sector to choose just a few trees for growth measurement inside the patch, even if all of them are tagged for mortality. If only some trees are tagged to track mortality using an initial sector angle, a second smaller sector angle can be used to choose growth measurement trees within that larger sector. When a tree dies, the growth of the tree becomes zero, not negative. If very accurate growth is needed (or needed in a short time interval), a few dendrometer bands could be used to record small changes in dbh. If an approximate estimation of future tree growth can be made, the sampler should be able to sample efficiently by using that estimation. Ratio sampling is one such example of this technique (Cochran, 1977, p. 150, 343). Forest Science 52(2)

5 Traditional Plots and Their Edge Effect Problems Edge effect bias is a difficult problem when using ordinary plots. The problem here is not about a biological effect taking place near the edge, it is about an artificial effect that is caused by the sampling procedure itself. When plots are placed either systematically or randomly within the sampling area, some may lie close to the edge and part of these plots can fall beyond the sampling area boundary. When this is the case, the trees near the edge of the sample area are sometimes chosen with reduced frequency. In essence, the plots are smaller than expected because part of them is falling outside the sampled area. The result of having some of the items selected less often creates a bias, which may be positive or negative depending on what calculations are done. When the purpose of the sample is to determine biological differences near the edge, this selection bias can be particularly troublesome. Whenever items are chosen with relative probabilities different than those assumed by the calculations, a bias is virtually certain to exist, and it can be a serious one. This is especially true when sampled patches have a high proportion of edge compared to their area. Without visiting every tree in the area, there has not been a simple and correct sampling solution to this problem. Currently, there are several ways to attempt to equalize the sampling probabilities due to this boundary line slopover, a term suggested in Beers (1966). A fairly complete review of the nature of the problem and the existing correction methods to account for edge effect is available in Iles (2003, Chap. 14). When using the sector plot method, one is not subject to the edge effect biases that plague traditional fixed plots, transects, and variable plot samples. This is because the sector plots do not have a fixed area, and therefore do not reach beyond or fall short of the boundary. The authors believe that a major problem with traditional fixed or variable plots is the difficulty of exactly correcting for edge effect biases within small areas. Sampling Specifically for Trees Along a Border In some applications, a scientist might be interested specifically in examining items along a border of some kind. A border might include trees along the exterior perimeter of the harvest block where the harvested area meets the original forest, or it could be the border of the tree patch. If the research requires equal probability of selection for trees along this border, then there is a simple solution using sector plots. From any pivot point inside the harvest block, simply project the sectors to the border and select any trees along that border that are also inside the sector. This can be done with one or more sectors. It may be wise to balance the selection using sectors in opposite directions. [2] If the area is very contorted, a single sector may include several sections of border as shown in Figure 3. It does not matter if the sector crosses the border several times, the trees along that border will still be chosen with equal probability. The border can also vary in width. Choose the size of the sector angle to give a reasonable expected sample size. In some cases, the location or width definition for the border might be difficult to determine in the field. Once the sector is established, however, the complicated work of defining the exact border need only be done within the sector boundaries. Why Is the Tree Selection Equally Probable? Probability issues are subtle and difficult. They require just the right way to visualize the problem. We believe that the following logic is easy to understand, and quickly illustrates the sampling theory issues. With this approach, it is easy to show that the probability of selection is the same for each tree, and that the location of the center point can be absolutely arbitrary. Imagine a sector plot as shown in Figure 3. The circle more than covers the harvest area. The area in the sectors is 10% of the total circle. Select any position within the harvest area and place a single tree there. Rotate the sectors around the pivot point to a random orientation. What is the probability of including that tree? Obviously, it is 10% ( /360, where is the sector angle in degrees). This is true no matter where the tree location might be, whether it is right next to the center point or near the border. The same probability applies to any other tree. The shape of the border is irrelevant. The area being sampled is irrelevant. Would this probability change if the pivot point of the sectors was located anywhere else, even at the edge of the harvest block or outside of it? No, it would not. Although another location might produce a more variable tree count, it is clear that the probability of selecting a tree is the same no matter where the pivot point of the sector is placed, even if it is placed outside the sampled area. The pivot point placements can be arbitrary (such as the center of the tree patch), random, or systematic. This is demonstrated simply by observing that the probability of tree selection, /360, does not depend on the distance from the pivot point to the tree. If two equal sector angles are used, the cumulative probability of a tree being chosen doubles, no matter where the two pivot points are located. How Should the Pivot Point Be Chosen? The authors certainly do not recommend that pivot points be placed randomly. Any necessary randomization with this system is guaranteed by the random orientation of the sector, after the pivot point is established. The sampler might want a roughly equal representation (by area) in each direction around the pivot point. They might want to roughly balance the data by the aspect if they are doing growth or reaction to change studies. Although any location is acceptable, the most efficient place to put the pivot point is probably approximately in the center of the tree patch. By placing the pivot point where the number of trees would be approximately equal for any orientation of the sector, one would expect the calculated variability of any 152 Forest Science 52(2) 2006

6 Figure 3. Illustrating the probability of selection with a sector plot. Note that a sector might cross the border of a harvest area several times (arrows, lower right quadrant), and would only select trees inside both the sector and the harvest area. totals or averages to be minimized, but this cannot be assured because it depends on the tree spacing and measured variables for those trees. Although the probability of tree selection remains exactly the same with any placement other than the center of a patch, the probability of a well balance set of trees is very much improved. This is the classic advantage of systematic sampling: When the sampler knows something about a population, the sampling scheme can be arranged to take advantage of that to produce balancing sets of observations. In addition, a researcher might want to avoid any pivot point locations that are particularly difficult because of operational issues concerning expense, visibility, safety, trampling of sensitive plants, etc. The authors are not aware of any other plot type that allows this type of discretion by the sampler. Suggestions for Sample Size or Proportions with Growth Studies Because mortality is a large and variable effect, we would suggest that the researcher tag as many trees as possible. For trees inside a small patch, they might tag them all. This process is relatively inexpensive. It may also be feasible to tag all the dispersed trees in the harvest block. If some proportion must be selected, then using the sector approach is a reasonable way to do it, but certainly not the only way. For growth, where high precision is needed (but where the answer is consistent and small sample sizes are sufficient), the authors suggest consideration of the sector plot approach. It could be used for sampling within patches, as well as for sampling dispersed trees. If the size of the sector begins to sample too many dispersed trees, a partial or reduced sector can be used, as explained earlier. The authors have experienced no difficulty dealing with small angles for this purpose. An error of 5% in the angle size would clearly lead to an error of 5% in the tree selection probabilities and sector area, precisely as such an error would bias a fixed area plot. Although there is the possibility of error when the angles are incorrectly determined, the use of offsets from a centerline by the authors has proven simple and reliable. Gradient Studies The selection of trees within a sector offers the same opportunity to observe a gradient response as one would get from a narrow fixed plot (a strip plot) going across the center of the patch. One difference is that the tree numbers are lower near the center point when using a sector plot. Forest Science 52(2)

7 Since there is less area in the center, this may not be a problem for many researchers. Certainly the selection probabilities in a fixed-width strip centered within the patch are not the same for all the trees (or other items), and the distance to those trees is required to properly weight the results when a strip is used. If these distances are not used as weights, averages and other relationships will almost certainly be biased when trees are selected with these strip plots. Calculations There are several ways that one might want to calculate a simple total using one of the sector plot observations. First, without knowing the area enclosed by the sector sides, one can compute the total by a simple expansion estimator because the probability of selection (P t ) is known. Such estimators are discussed in most statistics books, for instance Särndal et al. (1992, p ), and often called a Horvitz-Thompson estimator. Using n selected trees, with individual tree volumes v t as an example, the following equation for estimating the total volume of the area using one simple sector (v s ) is appropriate: n v s 360 t t 1 sector angle n t t (2a) p Put more simply, if the sector angle is 1 10 of a full circle, the volume of all the trees in the sector is simply multiplied by 10 to estimate the patch total. More generally, where trees are chosen with different angles, including the process of subsampling with reduced angles along the same centerline, and with the angle choosing that particular tree denoted as t, the total v s estimated using that sector would be n v s. (2b) v t t t A simple arithmetic average of these total volumes from several sectors can be used to estimate an average total volume of the area (v s ). Several sectors that are each randomly oriented can be measured and the variance and other standard statistics can be easily computed. Such computations are readily available, even on simple spreadsheets like Microsoft EXCEL. The estimation of the standard deviation for total volume by using ns independent sectors within a sample area is Standard Deviation of total volume (v s )is ns v S v S 2 s 1 SD V (3) ns 1 Such a calculated standard deviation is used in the sample size calculations common to any sampling project. A simple example of a sector calculation is as follows, based on the sum of the basal areas for trees inside four independent sectors of a patch of trees. t 1 Angle Sum inside sector of basal area, m Estimated total, m Giving a standard deviation of 5.9 m 2 and an average total of 10.6 m 2 for the entire patch. Conclusion The use of sector plots gives researchers the opportunity to choose samples from small clusters of objects without the edge effect bias typical in other sample plots. In addition, it allows the unbiased selection of objects along borders, even when these borders are complicated. The ability to place the pivot point in a location that is expedient to the sampler is unusual for a sampling system and very convenient. Neither the sector areas nor the sampled region areas are necessary for calculations of totals. The statistical properties and calculations are the same as when the sampler is using any plot that can calculate a total just using the volume of items on that plot, and the authors have illustrated this with a simple expansion estimator. Other calculations are also possible, and the authors are currently developing methods where the sector area is known. [3] The system is well suited to small patches of objects, such as those left behind in variable retention silviculture, or other situations where objects are clusters in small patches. Endnotes [1] This is named after the sector developed by Galileo for use as a computing device. The two legs of that instrument pivot from a central axis in the manner of a compass drawing instrument. [2] The sampler gets a good balance of border trees if the center point of the sectors is roughly near the sampling area center. The selection is still unbiased, no matter where it is put, but with a central location the balance is better. [3] As one reviewer pointed out, using the known area as an auxiliary variable with ratio estimation is one of several such methods. Literature Cited BEERS, T.W The direct correction of boundary-line slopover in horizontal point sampling. Research Progress Report No. 224, Purdue University Ag. Exp. Sta. 8 p. BEESE, W.J., B.G. DUNSWORTH, K. ZIELKE, AND B. BANCROFT Maintaining attributes of old-growth forests in coastal BC through variable retention. For. Chron. 79(3): COCHRAN, W Sampling techniques, 3rd ed. John Wiley and Sons, New York. 428 p. ILES, K A sampler of inventory topics. Kim Iles & Associates Ltd., Nanaimo, B.C., Canada. 869 p. MITCHELL, S.J., AND W.J. BEESE The retention system: Reconciling variable retention with the principles of silvicultural systems. For. Chron. 78(3): SÄRNDAL, C.E., B. SWENSON, AND J. WRETMAN Model assisted survey sampling, 1st ed. Springer-Verlag, New York. 694 p. 154 Forest Science 52(2) 2006