Alice Richardson

University of Canberra

Journal of Statistics Education Volume 15, Number 1 (2007), jse.amstat.org/v15n1/richardson.html

Copyright © 2007 by Alice Richardson all rights reserved. This text may be freely shared among individuals, but it may not be republished in any medium without express written consent from the authors and advance notification of the editor.

**Key Words:**Estimation; Proportions; Sampling distribution; Statistical education.

Distance sampling can be seen as a more complex version of transect sampling, where only objects seen on the transect line are counted. The main advantage of distance sampling over the simpler process of transect sampling is that it allows data from beyond the selected line itself to still be used in the estimation process. Distance sampling is also frequently preferred over random sampling because it is easier to implement in rough terrain. The disadvantages of distance sampling are associated with the assumptions required to make it work. These centre on the distribution and visibility of objects in the area and the behaviour of the objects (particularly if they are animals that move around or travel in groups.)

Bishop (1998) produced a tutorial comparing transect sampling to random sampling using an aerial photograph. In this paper, we describe an activity that extends data collection to beyond the transect i.e. carries out distance sampling, and effort is spent on physically observing objects and measuring distances. Otto and Pollock (1990) carried out a similar experiment using beer cans, which we used as the inspiration for developing this activity.

The activity follows the format of Spurrier, Edwards and Thombs (1995), whose activities cover a range of statistical concepts to a similar depth. The first time we used it, the activity followed a lecture by Dr. Ann Cowling on her experiences in applying distance sampling to fish populations in the Great Australian Bight. The lecture and activity appeared in a quantitative literacy course (see Richardson (2000)) as one of a range of statistical estimation techniques that included population size estimation using capture-recapture methods (Scheaffer, Gnandesikan, Watkins and Witmer (1996, p.126)), and estimating the maximum of a set of data (Scheaffer, et al. (1996, p.148)).

Distance sampling is not often presented in introductory statistics courses. We envisage this activity being used in courses where students learn about specialised methods of data collection. For example, the activity could be used in a survey methods course, as an unusual approach to the problem of census undercount. It would also fit in a general quantitative research methods course that covers a variety of survey and experimental designs for data collection. It could also be used in a more specific research methods for biologists course that focuses on the techniques used in biological research, such as quadrat sampling, capture-recapture methods and point transects. Finally, we have successfully included the activity in a very general quantitative literacy course. With the references provided within this paper, the activity is intended to be comprehensible without the need for supporting lectures.

A secondary teaching goal is to bring students face-to-face with the issues involved with collecting data including the appropriate degree of accuracy of recording measurements, the time taken to collect data, and working effectively in groups.

Care is required in placing the beads in the field. Truly random allocation, perhaps by using a map of the area to be used, would take a very long time to arrange. Moving through the field and throwing handfuls of objects is easier but the objects tend to land in clusters. This makes data collection more of a burden and violates a key assumption of the distance sampling method, e.g. independence. We therefore recommend shifting individual beads after an initial scattering, so that not too many beads are too close to each other. A discussion of this method, and whether or not it is truly random, can form part of the discussion towards the end of the activity.

The amount of space allocated to the activity compared to the number of students involved is also a consideration. A large enough area should be chosen so as to avoid crowding. We found that with 15 to 20 students, an area approximately 20 metres by 20 metres works well.

At the end of the activity, students are asked to fan out across the field and slowly move across it in an attempt to pick up all the beads (a strategy known as a emu parade in Australia). In our experience about 10% of the beads were still lost. Hence it is important to have a ready and cheap supply of the objects used in this activity. It is also important to use objects that will not be a nuisance in the grass later, if any do get left behind.

It is important to spend some time at the start explaining what is involved in the activity, as students are often keen to leap into data collection without any clear picture of what they are doing and why they are doing it.

The main goal of this activity is to introduce students to the method of distance sampling. Students will also study sources of variability associated with the method, particularly variability between observers and variability between transects.

As background reading for this activity, tutors could be given copies of Otto and Pollock (1990) and Welsh (2002).

In the Introduction, tutors are expected to lead a discussion on methods for estimating population size. As an initial idea tutors can suggest taking a census, followed by the idea of taking a census in a representative small area and multiplying that value up to become an estimate of the total population size.

The likely loss of beads during the clean-up of the activity also gives tutors an opportunity to discuss the accuracy of a census. The lost beads can be regarded as an example of undercount in a census.

In the Background, tutors are expected to lead a discussion on possible methods for the selection of transects. Tutors can suggest firstly, that a ruler can be dropped at random on a map of the field. That placement can then be transferred to the field. This also shows that transects need not be parallel to one side of the field. Secondly, a numbered grid could be drawn on a map of the field and random numbers could be used to locate a starting point on a random side of the field and an ending point on another. These random starting and ending points are then transferred to the field. An optional point that tutors can explain is that the existence of several methods of choosing random transects is related to the problem of choosing random chords on a circle. Tutors may also be interested to note that this problem is a form of Bertrand’s paradox: see Holbrook and Kim (2000).

On a practical level, it is useful to have string attached to pegs (pencils are a suitable substitute for pegs). These can be used to mark out an area approximately 20 m by 20 m for the activity to take place in, and they can also be hammered into the ground at the beginning and end of a transect.

When we ran this activity, the students decided to peg out one transect for each group of three and leave them fixed, rather than peg out random transects each time. In a class of 15 – 20 students, this usually results in between five and seven transects. Three of these transects were then selected and used by all the other groups in their data collection. The advantages of this selection of fixed transects is that (a) it speeds up the data collection process; (b) it eases crowding in the area of data collection; and (c) it is easy to comment on between-observer variability because each transect is travelled by at least one group.

Anecdotal evidence suggests that students generally enjoy the chance to get out of the classroom, generate some data of their own and then analyse it using statistical software. Some students have commented adversely on the amount of time taken up by data collection. We counter this by pointing out that an introduction to the nature of primary data collection is one of the teaching goals of the activity. Alternatively, a shorter version of the activity is suggested in the next section.

Another variation that would also save time involves spreading the work of walking along transects among groups as well as only recording data within 1 metre of certain transects. For example, if there are six groups, two groups could collect full data from each of the three transects. Then the groups that did transect 1 could split up and one group go to transect 2 and the other to transect 3 to simply count the number of objects within 1 metre. The same applies to the other two transects. This would result in two complete data collections and two data collections within 1 m for each transect, which still allows for variability between groups and transects and data collection methods to be assessed.

Based on the assumptions that the beads are randomly distributed in the field and that every bead in the field is seen, a histogram of distances to beads should be uniform, with every bar the same height as the first one. The probability of seeing a bead at further distances from the transect line is then the ratio of the height of each bar of the histogram to the height of the first bar. For students in advanced classes, these ratios can be used to carry out a logistic regression of the probability of seeing a bead on the distance of the bead from the transect line. However these students would need to undertake the full data collection along transects 2 and 3 in order to carry out such an analysis.

Bishop (1998) describes extensions associated with the use of parallel transects that could be used here as well.

To use distance sampling to estimate the size of a population of objects in an area, the area of interest is first measured and sketched. A line (also known as a transect) that crosses the area is chosen at random. Your tutor will lead a discussion of how a transect could be chosen at random, and the implications of non-random selection of transects. Tutors: see the Guidelines. Observers travel the length of a transect recording distances to objects visible from the transect. An estimate of population size is constructed based on the distances. We will now examine this method in the activity.

Each group of three should peg out a randomly selected transect. Number the transects as you will need to be able to tell them apart later on. Next, each member of the group of three has a specific task in the data collection along the transect: one (the “eyes”) will spot beads, one (the “ruler”) will measure distances and the third (the “writer”) will record the data. The “eyes” should walk along the transect indicated. Whenever he or she sees a bead on either side, he or she should stop and the “ruler” should measure how far the bead is from the transect. The “writer” records the distance measured by the “ruler”. The “ruler” should measure a perpendicular distance i.e. the distance, in metres, from the transect to the beads when the “eyes” are at right angles to the bead. It is not the job in this activity of the “writer” or the “ruler” to spot beads, even though they may see ones that the “eyes” miss. Record your measurements in Table 1. If you need more room, continue on a blank sheet of paper.

Object No. | Distance | Object No. | Distance | Object No. | Distance | Object No. | Distance |
---|---|---|---|---|---|---|---|

1 | 21 | 41 | 61 | ||||

2 | 22 | 42 | 62 | ||||

3 | 23 | 43 | 63 | ||||

4 | 24 | 44 | 64 | ||||

5 | 25 | 45 | 65 | ||||

6 | 26 | 46 | 66 | ||||

7 | 27 | 47 | 67 | ||||

8 | 28 | 48 | 68 | ||||

9 | 29 | 49 | 69 | ||||

10 | 30 | 50 | 70 | ||||

11 | 31 | 51 | 71 | ||||

12 | 32 | 52 | 72 | ||||

13 | 33 | 53 | 73 | ||||

14 | 34 | 54 | 74 | ||||

15 | 35 | 55 | 75 | ||||

16 | 36 | 56 | 76 | ||||

17 | 37 | 57 | 77 | ||||

18 | 38 | 58 | 78 | ||||

19 | 39 | 59 | 79 | ||||

20 | 30 | 60 | 80 |

Choose two other transects pegged out by two other groups. Write down which transects you have selected at the top of Tables 2 and 3. If your tutor says you have time, walk along these transects as before, this time recording your measurements in Tables 2 and 3. Otherwise, simply record the number of objects seen within 1 m of the transects.

An example of a transect and distances to observed objects is shown in Figure 1.

Figure 1. Example transect and distances to observed objects

What shape does the histogram follow? A typical histogram of distances is shown in Figure 2. Notice how the frequency of observations reduces as the distance to the beads increases. If the beads are distributed at random in the field, then this drop-off is not because there are fewer beads at greater distances from the line; it is because the beads are harder to see at those distances.

Figure 2. Histogram of distance to objects

There is no reason to expect that there are fewer objects far away from the observer compared to close by. Thus the skewed appearance of the histogram indicates that the further away an object is, the less likely an observer is to see it.

Assuming that the beads really are distributed at random in the field, then there should be the same number of them within 1 m of the transect line, between 1 and 2 m of the transect line and so on. So if you, the observer had seen every bead up to, say 12 m away, theoretically the histogram above should look like Figure 3.

Figure 3. Histogram of expected distance to objects

You can now use the results of your distance sampling to estimate the population size in the following way.

- Count up the number of objects you saw i.e. the number of objects represented in the histogram in
Figure 2. Call this number
*n*. - Because of the assumption of a uniform distribution of objects, the number of unseen objects is the difference in the
number of objects represented in the histogram in Figure 3, and the number of
objects represented in the histogram in Figure 2. Count up the number of unseen
objects and call this number
*u*. - Call the length of the transect line
*L*and the largest upper class limit in the histogram*w*. - The estimated density of objects i.e. the estimated number of objects per unit area is
density estimate = . - To estimate the number of objects in the whole area, multiply the density by the total area to obtain
population size estimate = density estimate * total area.

Apply the formula above to your value of *w*, *L* and *n*, and arrive at an estimate of the population size.
In the example, *L* = 20, *w* = 12 and *n* = 84, and 16 beads were seen within 1 m of the transect line.
Therefore we estimate that there were (12 * 16) – 84 = 192 – 84 = 108 unseen beads. Thus

Finally, since in the example the total area is 20 m long and 20 m wide with a total area of 400 m^{2},

If you had a short time for this activity and only recorded the number of beads seen within 1 m of certain transects, this
value is used to compute *n* + *u* as follows. If you saw 19 beads within 1 m of the transect line, we estimate
that there were 12 * 19 = 228 beads (seen and unseen). Thus

and

We will now compare the population size estimates from the whole class. Your tutor will ask each group to read out their transect numbers, and the corresponding estimates of population size. Record the transects numbers and estimates in a spreadsheet. Calculate descriptive statistics for these estimates separately for each transect. The amount of variation in any one of these sets of estimates reflects the variation between observers. Your tutor (who laid out the beads) will also now reveal the true population size.

You could conduct a census in a number of small areas within the study area - these small areas are known as quadrats. The population size is then estimated by scaling up the count in the quadrats, in a similar fashion to what is done with the distance sampling estimate we calculated. Point sampling is also sometimes used in preference to distance sampling. An observer stands at a fixed point and measures the distance to all the objects he or she can see from that point. Capture-recapture methods are also used to estimate population size, where a number of objects are caught, tagged and released back into the population. A second capture is carried out, and the proportion of tagged objects in the second capture is used to estimate the total population size.

Another way to attack the distance sampling problem is to find a smooth curve e.g. half a normal distribution, that matches the shape of the histogram in Figure 2. Then use this function to estimate the probability of seeing objects at various distances from the line, and in turn use those probabilities to estimate the population size.

- What were your three estimates of the population size from each transect that you travelled? Show your work in each
case. What was the true population size?
- Calculate the mean, standard deviation and range of the estimates in question 1. How accurate are your estimates? How
precise? The amount of variation in this set of estimates reflects the variation between transects. Is this small or large?
- Select one of the transects. Write down the estimates of the population size from all the other groups who travelled
that transect.
- Calculate the mean, standard deviation and range of the estimates in question 3. How accurate are the class's
estimates? How precise? The amount of variation in this set of estimates reflects the variation between observers. Is this
small or large?
- The estimation of population size on the basis of distances relies on the following assumptions.
- Every object on the line is observed without fail.
- Every object is detected at its initial location.
- Measurements to the objects are exact.
- Objects are distributed at random i.e. they do not form clusters.

Explain whether each assumption has been met in this activity.

- Suppose a conservation group wishes to use distance sampling to estimate the number of whales of a particular species present in the Great Southern Ocean in spring. Explain whether the assumptions in question 5 are likely to be satisfied. What measures could you take to ensure that the assumptions are more likely to be satisfied?

jse.amstat.org/v6n2/bishop.html

Buckland, S.T., Anderson, D.R., Burnham, K.P, and Laake, J.L. (1993), *Distance Sampling: Estimating Abundance of
Biological Populations*, London: Chapman and Hall.

Chen, S.-X. and Cowling, A. (2001), “Measurement errors in line transect surveys where detectability varies with distance
and size,” *Biometrics*, 57, 732 – 742.

Holbrook, J. and Kim, S.S. (2000), “Bertrand’s paradox revisited,” *Mathematical Intelligencer*, 22, 16 – 19.

Otto, M.C. and Pollock, K.H. (1990), “Size bias in line transect sampling: a field test,” *Biometrics*, 46, 239 – 245.

Scheaffer, R.L., Gnanadesikan, M., Watkins, A. and Witmer, J.A. (1996), *Activity-Based Statistics*, New York:
Springer.

Spurrier, J.D., Edwards, D. and Thombs, L.A. (1995), *Elementary Statistics Laboratory Manual*, Belmont, CA: Duxbury
Press.

Welsh, A.H. (2002), “Incomplete enumeration in sample surveys: whither distance sampling?” *Australian and New Zealand
Journal of Statistics* 44, 13 – 22.

Alice Richardson

School of Information Sciences and Engineering

University of Canberra

Canberra ACT 2601

Australia
*Alice.Richardson@canberra.edu.au*

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