Optional line transect and mark-recapture material
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line transect and mark-recapture methods:

density estimate from a systematic sampling scheme   Reference: Burnham, K.P., D.R. Anderson, and J. L. Laake. 1980. Estimation of density from line transect sampling of biological populations. Wildlife Monographs No. 72. Absolute density? YES. Count all in study area? NO Capture or count? COUNT Is population dense? NO Can all animals be detected at a distance? NO How do we deal with incomplete detectability?   d. Line Transects

Fig. 3.2. Diagram of layout for line transect sampling scheme. Transect lines are laid out in some systematic fashion across an area large enough to detect a reasonable sample (>= 40, preferably more) of "objects" (animals, plants). Observer walks the transect lines noting distance and angle to objects detected.
 
 

Fig. 3.3. Diagram of detection events. Objects detected have a perpendicular line connecting them to the transect line. Objects without lines went undetected. Note that objects on the line are always detected, while those at greater distances have lower probability of detection. In this example n (number of objects detected) = 13, including the two objects detected on the transect line itself.

Data that must be collected for line transect sampling:
 

Total length of lines traversed, L (most applications will include replicate lines; these may be parallel to each other, zigzag in random directions or apply a form of stratified randomization)

Number of objects detected, n.
Number of individuals within each object (e.g., birds per covey)

Perpendicular distance, xi from transect line to object i (i = 1 to n). It is usually best also to measure angle and distance from detection point to object (since many objects will be detected before they are directly perpendicular to the transect line).
Maximum distance of detection will be w* (the one-way "width" of the transect detection area). Total "detection" area will then be 2Lw*.

Sample size (n) of number of objects detected should be no less than 40.

 Rationale (why we need this sort of complex sampling/counting scheme). The main point here is to overcome the opposite pitfalls of low accuracy (high bias) on the one hand and low precision (high variance) on the other, while counting animals in a way that allows for non-detection of some. That is, in order to get a reasonably high count of objects detected, we have to accept the fact that we will fail to detect some objects.
 
   
Fig. 3.4.  A schematic diagram for the basic idea behind line transect sampling.  You don't need to be able to read the numbers, just try to grasp the central concept.  The X-axis shows eight distance categories (all objects detected in the first x meter interval, all objects in the second x meter interval, etc., where x will be determined by the species being estimated and the nature of the habitat).   The Y-axis shows the percentage of the total detections represented in each of the given distance intervals.
TOP PANEL.  If a population were uniformly dispersed and we conducted a complete census we should come up with a histogram of objects at intervals from the center line as shown in the top histogram.  That is, we would fully detect all objects, regardless of distance from center line � so all the bars in the histogram would be of equal height,
MIDDLE PANEL.  In practice, however, detection will decrease with increasing distance from the center line, as shown in the middle histogram.  These bars are just one particular example of the general kind of pattern we should expect from actual field data � we should see a "shoulder" of large bars near the center line, and then the bars furthest from the center line should be noticeably smaller.
BOTTOM PANEL.  The challenge is to estimate the "undetected" objects shown by the shading in the bottom diagram.  The method of line transect sampling does just that and allows us to compute an estimate of the total population (shaded plus unshaded portions of the bottom diagram corresponding to the "full" histogram at the top).  One more point: our estimator is f-hat(0).  Think about why that first bar should be a good basis for going from the middle panel (actual field detections) to the top panel (hypothetical perfect knowledge).


Our task is to come up with a good estimator that will allow us to "fill in" the gap between the density (Dw) of objects we detected (n/2Lw*) and a good estimate of the actual density. That "fill-in estimator" will be a function we will call "f-hat(0)", where the hat stands for "estimator".
 


Our end result will be a function to estimate density given as

                                                                                           (Eqn 3.1)


In order to get this "fill-in estimator" we need to make (and meet) some assumptions:
 

1) Objects directly on the transect line will never be missed (Probability of detection =1.0)

2)Objects are fixed at the initial sighting position. That is, they do not avoid the observer before the observer detects them, and they don�t move away in such a way that the observer risks counting them twice.

3)Distances and angles are measured without error

4) Sightings are independent events. If we see one animal it does not affect (positively or negatively) the probability that we will see other animals.

 Examples of violations of these assumptions:   Tropical canopy birds, lizards in rocky terrain, whales that are submerged as observer�s transect line passes over them   Animal moves prior to detection or double counting occurs because it moves far enough that observers don�t realize they have already counted it.   Observers estimate angle and distance poorly   Groups flush   Deriving the basic equations necessary for developing an estimator.
  Let�s start with the most basic equation for population density. If we had absolutely perfect knowledge, we could write.
  D= N/A                                                                         (Eqn 3.2) where D is density, N is population size and A is area. So say we have 100 animals in a 10-hectare plot (a hectare is 100 m by 100 m). Then D = 100/10 = 10/ha. When you start doing equations with mixed units it can sometimes be very important to keep track of your units. If you had physics think back, for example, to units of acceleration. Sometimes too, dimensionless numbers are very useful. In that case, the units cancel.

If we were doing a complete count in a strip transect of width w, this should be number counted/area covered
 

                                                                                           (Eqn 3.3)


where the "hat" over the D means we are estimating D, the small n means it is our count (not the full population total as before), L is the total length of our transects, and w is the one-sided width of the transect.

Develop a "detection function", which we will call g(x), where x is the perpendicular detection distance to objects we detect. The function g(x) tells us the probability that we will detect an object, given that it lies a distance x from the centerline of our transect. What we will do is integrate over the distance from x = 0 to x = °. This gives us, essentially, the sum of the detection probabilities.
 

                                                              (Eqn 3.4)


Now if we divide the original function g(x) by the integral, a, we will have a function that sums to one (we have "normalized" it). This function is formally called a probability density function or pdf.
 

                                                                   (Eqn 3.5)


We have simply rescaled g(x) by the integral a, to give a function, f(x), that sums to one [f(x) has exactly the same shape as g(x)!!]. It tells us what the proportion of the total probability is of detecting an object at a given distance, x occurs at that distance. Say x = 3.5 meters and the original g(x=3.5) was 0.43. How much of the total probability (a number that might be much greater than one) lies at x = 3.5? Again, though the values are different, this new f(x) will have exactly the same shape as our original function g(x).

So, from where we stand on the centerline, we are asking: what proportion of all the objects out there are we seeing over the range of distances? At the centerline we assume that we actually see every object that is there. Given that we see every object on the line (seems reasonable) we can say.
 

g(0) = 1                                                                         (Eqn 3.6)


substitute Eqn 3.6 into Eqn 3.5 to get
 

                                                                  (Eqn 3.7)


Now let�s revisit Eqn 3.3. For a finite w, the probability of detecting an object within the (half)-strip will be
 

                                                                      (Eqn 3.8)


That is, the total probability, a, divided by the half-width of the strip, w. If a total of Nw objects actually occur, the expectation of number of objects we will see will be
 

E(n) = PwNw                                                       (Eqn 3.9)


where E(n) stands for the "expectation of n".  This is simply the probability of detecting an object times the total number of objects there. We will know n, the number of objects we saw, so we can develop an estimator of Nw as
 

                                                                            (Eqn 3.10)


where we have used the right-hand-side of Eqn 3.8 as a substitute for Pw in the last form of the equation. Now substitute nw/a for the N of Eqn 3.2 to get:
 

                                        (Eqn 3.11)


NOTE that the w�s cancel out. This means that we can use an "infinite" w (as far as we can see, or the furthest object detected, called distance w*). One more step and we will have our handle on estimating density. Substitute the left hand of Eqn 3.7 for the 1/a of Eqn 3.11 to get:
 

                                                         (Eqn 3.12 = Eqn 3.1)


Great! If we can just get some estimator f-hat(0), we can estimate density from our line transect data.
I began by reviewing the derivation involved in going from Eqn 3.1 back to itself as Eqn 3.12.
Moving from theory and equations to crunching data and getting an estimate.
I ended last time with Eqn 3.12 (the end result of deriving Eqn 3.1), which I claimed would be our path to estimating density from our field transect data. So� our job is to get a good function, f^(0) for estimating f(0) or its equivalent 1/a.  
Real-world issues: Animals move

Animal response behavior and our ability to detect them may change during the survey (because of changing wind levels and other factors.)

Individual animals may respond differently.
 

 Here are the criteria that Burnham and Anderson (1980) set for a good model of f(x).
  General robustness (flexible enough to fit a range of actual shapes of detection probability).

Robust to pooling (variation in detection probability factors - some unknown)

Shoulder near zero -- near the centerline we expect to see almost all the animals (the "shoulder" but then the detection probability drops off from some moderate distance.

Efficient estimator (low variance, low bias)
 

 The graph below is a histogram of objects detected in various distance intervals from the center line.  It has a "shoulder" near the center line.


 

Fig. 3.5.  Detection distances (x) for a total of 99 objects in five distance intervals (3.8 m per interval) from the transect center line. A detection function g(x) has been fitted to the observed histogram.  Note that the data have been truncated � 6 objects that were detected at distances > 19 m from the center line were omitted from the analysis.  This truncation makes the analysis much easier, and more robust, but has little effect on the final estimate of D.  (The truncation means that w* will be 19 meters).  These are the data for the fully worked Excel spreadsheet example.


Deriving the best estimator goes beyond the scope of this course. Suffice it to say that sharp minds have shown that a Fourier series estimator is robust (works under a variety of conditions) and has reasonable precision and no bias (at least in the limit).

It boils down to an expansion several terms long that is based on a cosine function of the observed detection distances.
 

                                                              (Eqn 3.13)


where the w* is just the furthest distance we actually saw an object at (after throwing out any outliers), the m is the number of expansion terms (indexed across k). The quantity a-hat-sub-k is estimated as follows:
 

                                               (Eqn 3.14)


So... let�s say we wanted to estimate the term for k = 1. Start inside the rounded brackets toward the right hand end. We have 1 (= k) times p, times xi (our first observed distance; i = 1 to n, the total number of objects seen), all over our maximum distance, w*; take the cosine of that. Now comes our summation term. Work through our full set of distances (i = 1 to n) and add all those n terms together. Multiply that sum of n terms by 2/nw*. That�s a-hat-sub-k. Even better, a computer program can do it all for us.

Note for calculations from "heaped" data (like the homework problem and LineTransect.XL).  The above formula works best when each of the sighting distances is recorded separately (the actual measurement) rather than when working with midpoint distances for lumped data.

For lumped data we can rewrite Eqn 3.14 as follows so that the calculation method is more explicit:

                                                  (Eqn 3.14a)

here we see explicitly the term Ntot for ALL the objects detected versus the ni (number of objects detected in distance category i, where i goes from 1 to Dists, the number of "lumps" or distance categories � 5 in the example in LineTransects.XL and 6 in your homework problems).  Rather than go one by one through all the objects we do the Cosine operation on each of the Midpointi distances TIMES the number of objects, ni , in that lump (at that midpoint distance).

CALCULATORS: RADIANS vs. DEGREES.  Excel and some calculators assume the input is in radians, as do the formulas above.  However, your calculator may use degrees instead.  In that case, you would want to use the term:
 

Cos[ (k 180 Midpointi)/ w*].  That is, you would need to multiply the stuff inside the cosine bracket by 180/p before taking the cosine.  (Since p was already in the denominator it cancels out). Alternatively, you may be able to set your calculator to work in radians.  Here's how to check which of the two your calculator is doing.  Take the cosine of 1.0.  If your answer is 0.5403 then you are working in radians and everything is fine.  If your answer is 0.9998 then you are working in degrees. You must either convert values to radians by multiplying by 180/ or switch your calculator to do its calculations in radians. I have run fairly quickly through sets of equations related to line transect sampling. Let�s look back at the high points.   First I used various equations related to density to show that
                                                (= Eqn 3.1 = Eqn 3.12)


I then said that if we could just estimate f(0) we could turn that back into an estimate of density, given the observed number of animals, n., and our transect length, L. The estimator function we will use is based on a Fourier series:
 

        (Eqn 3.15 = Eqns 3.13 + 3.14)


Here is a summary of the real-world issues:

 
Animals move   Animal response behavior and our ability to detect them may change during the survey (wind level etc.)   Individual animals may respond differently.


Here are the criteria that Burnham and Anderson set for a good model of f(x).
 

General robustness (flexible enough to fit a range of actual shapes of detection probability).   Robust to pooling (variation in detection probability factors - some unknown)   Shoulder near zero   Efficient estimator (low variance, low bias)


That ROBUST estimator is the Fourier series function given in Eqns 3.13 and 3.15.

We had a question about the stopping rule for the number of terms in the theoretically infinite number of terms in the Fourier series expansion. That rule is to use the first value of m that fits the following criterion:                                                                              (Eqn 3.16) That is, we look one step ahead (at m+1) and see whether the inequality is met. If so, then we use m; otherwise, we proceed to try m+2. For the interested: that stopping rule is based on a principle called MISE (minimum integrated squared error; it is not something I expect you to remember).

We�ve now covered some of the the theory of line transect sampling and obtained an idea of how we can turn field data in actually calculating a density estimate. We�ll now turn to a brief consideration of possible layouts for the transects. You may remember that I mentioned that one will usually want multiple transect lines that sum to L. Mathematically

                                                                                                                           (Eqn 3.17) That is, we will have q transects whose total summed length is L.

Here are two possible schemes. Which is the more practical? Which is the "ideal"?
 

 
Fig. 3.6. Transect lines (l1 to l7; ·li = L) selected with random starting points and common (but randomly selected) direction. Lines may be of equal or unequal length.  
Fig. 3.7. Transect lines (l1 to l8; ·li = L) each selected with a random starting point and a randomly selected direction. Lines are of unequal length. The bottom case is arguably the ideal, because each of the lines has a randomly selected direction and length. In practice, this sort of design will be exceedingly difficult to carry out, so we might go with the upper case. Even that may not be possible so we may need to go with one of the following.

We could use any of a series of different possible layouts. These could include connected lines with "kinks" that change angle, systematic designs that cover the study area in a regular design, or subdivided plots that partition the study area.

Somehow, though, we should try hard to incorporate some randomization. This could involve a randomly selected starting point on the perimeter, a randomly chosen angle for the parallel transect lines, or some other way of avoiding systematic bias (again, at all costs we want to avoid some obvious source of bias such as paralleling ridge tops.

SUMMARY of considerations in designing the layout:

Define the population of interest

Delimit the population (figure out the logical and logistically feasible boundaries)

Devise a sampling scheme that truly samples the population of interest

Conduct a survey the gathers the required data

Goal of the design: Sample the population of interest in a way that yields an adequate representation of reality
 

 To meet the goal in practice:  
Every member of the population must have an equal probability of being sampled
  (can�t have submerged whales that are not susceptible to sighting)


Must have adequate replication (adequate sample size)

Sample must have adequate spatial dispersion (and possibly temporal dispersion)

Must avoid correlation between transect orientation and particular features of the landscape (such as ridges or valleys). It�s acceptable to have one line (of many) that (by chance) aligns with the environmental feature (road, river), just not that they ALL do so -- that may produce bias.

  Must bring randomization into the design process at some point -- the more the better, but if nothing else then a random starting point or random direction (for parallel or systematic designs).


See the Excel spreadsheet (link below) for a fully worked example of applying the Fourier series estimator and the stopping rule to the set of data given in Fig. 3.5.
 

Go to Excel spreadsheet


Here is a website for the current documentation on line transect sampling using the program DISTANCE
 

http://www.colostate.edu/depts/coopunit/distancebook/download.html Burnham and Anderson Distance manual for software (including line transect analysis) in pdf format


End of material on line transect sampling

 ________________________________________________________________________

Our next topic will be

mark-recapture schemes. The simplest scheme here is the Lincoln-Petersen index. Note that it is not really an index but an estimate. Refer back to the dichotomous key of estimation techniques in Lecture 8 to see where it falls. We have four variables, three of which we will be able to measure -- we can solve one equation in one unknown. N = population size (we won�t know that, normally)

M = initial sample that we mark

n = number recaptured

m = number in recaps that are already marked.

 Here�s the (low bias) formula for the estimated population size:                                                                                                                        (Eqn 3.18) The variance of this estimate is given by the following formula:                                                                                              (Eqn 3.19) From the variance, we can calculate a confidence interval. The confidence interval (abbreviated CI) tells us the limits between which we are 95% sure the true estimate lies. The most common alpha level is 0.05 (leading to a 95% interval) but we could use alpha = 0.10 (leading to 90% CI) or alpha = 0.025 (leading to 97.5% CI). The formula is:                                                                                                                     (Eqn 3.20) where the upper CI is our original estimate plus the term on the right and the lower CI is the original estimate minus the term on the right. The t refers to a value taken from Student�s t-distribution and will depend on our sample size (determining the degrees of freedom abbreviated d.f.) and the alpha value. For a sample with a large degrees of freedom (>30) and an alpha of 0.05, the t-value will be approximately 2. That means that our CI will be the original estimate plus or minus two standard deviations (s.d. = square root of the variance).

Last time we began on mark-recapture schemes. Remember to look back at where it lies on our dichotomous key. Unlike line transect sampling, which is largely restricted to obtaining a density estimate, mark-recapture can actually do several things for us. It can help us estimate one or more of the following:

Type A (can focus just on marked animals)
  Movement of individuals
Growth rate (size/weight) of individuals
Age-specific fertility rates
Age-specific survival rates Type B (requires estimation of proportion unmarked)
  Size of the population
Birth + immigration
Death + emigration
Rate of harvesting The first four "outputs" require only that we be able to identify marked animals. We can ignore the unmarked individuals. For all of the Type A and Type B methods, though, we are assuming that loss of marks either doesn�t occur or is negligible. One way to test that assumption is by a double-banding study. Then we have four kinds of animals Those with both bands
Those with one band (lost one)

Those that lost both bands
Those that were never marked.

 We can identify only those animals in the first two categories. Nevertheless, the data allow us to generate expectations of proportion with one or two bands. If our data differ significantly from the predicted values, then we have reason to suspect that band loss is biasing our results.

To use mark-recapture for estimating density and the other Type B uses, we are interested in both the caught and the uncaught. In that case we have one overriding assumption that we should meet

Equal catchability for all. Why might we fail to meet that assumption? Individuals may differ inherently in their behavior at traps or marking stations

Individuals may learn to seek or avoid traps or marking stations

Spatial dispersion may affect opportunity for capture (can�t catch an animal if the trap is not in its home range)

The last assumption can be addressed by proper randomization. Even with randomization, sexes may differ in catchability (home range size) and may need to be analyzed separately.

The second assumption can be tested by analysis of the pattern of captures and recaptures.

 The simplest scheme here is the Lincoln-Petersen index (or Petersen estimate). Again, the parameters we need for this are: N = population size

M = initial sample that we mark

n = number recaptured

m = number of recaptured individuals (n) that are already marked.

 We start from the premise that the ratio of the population to the marked will equal the ratio of our first capture (n) to the number already marked caught in the second round.                                                                                                                                     (Eqn 3.21) Unfortunately, that simplest estimator has been shown to be biased. Two estimators exist that work on the bias.  The following formula has low bias:                                                                                                                  (Eqn 3.22 = 3.18) from which we can calculate a variance given by:                                                                                       (Eqn 3.23 = 3.19) we can then generate a confidence interval for the population size estimate by using                                                                                                        (Eqn 3.24 = 3.20) for example we might be seeking to say that we are 95% confident our estimate lies between upper limit 1 and lower limit 2.

Last time I gave formulae (Eqns 3.22 to 3.24) for reduced bias estimation of the population size, its variance and a confidence interval. We can use Eqn 3.24/3.20 to calculate a confidence interval on our estimate. The attached Excel spreadsheet provides a concrete example of all the necessary calculations.

Go to Mark-recapture Excel spreadsheet
Say we caught and marked 301 animals on a first round and captured 236 on a second round. If 146 of the animals were previously marked our  would be 485, the variance would be 604.3 and the 95% CI would range from 437 to 534 (= ± 10 % of ).

[That CI corresponds to an alpha level of 0.05 and we use a Student�s t-distribution with the requisite number of degrees of freedom to calculate the interval].

The estimator above is still somewhat biased. We can come up with a completely unbiased estimator by deciding beforehand on a number of recaptures (m). We continue trapping until we catch the required number of already-marked individuals. In that case our estimate is:

                                                                                                                          (Eqn 3.25) and the corresponding variance estimate is:                                                                        (Eqn 3.26) We can use Eqn 3.26 to figure out how many animals we need to mark (M) and then recapture (m) in order to get a standard deviation for N-hat that is acceptable. Say we want to get a s.d. of <10% of . For a total population size (N) between 100 and 1000 we then need the following numbers of M and m. M {100, 200, 250}

m { 55, 80, 90}

 Using our previous example: if we captured and marked 301 (=M) animals on the first round, then set a goal of 151 already marked (= m). Say it takes us 244 captures to reach that goal.

= 491, Var () = 612.9, 95% CI = 442 to 540 (= ± 9.96%).

Notice that in the two previous examples we caught a fairly large percentage of the total population (in fact 40-60%). It may not always be feasible to catch that high a proportion of the population. In that case we can use the slightly more complex multi-capture method. We can use methods developed by Schnabel in the 1930's and modified by Schumacher and Eschmeyer in the 1940's. The applications were developed largely for estimating fish populations. (Method 4 of our dichotomous key)

                                                                                                                         (Eqn 3.27) In the 1960�s G.M. Jolly and G.A.F. Seber began modifying these multi-capture techniques in various ways that allowed them to serve both Type A and Type B estimation purposes. These multi-capture methods have been the subject of intensive analysis since then, and are the basis for numerous software programs that analyze survival (I�ll come back to the software programs in a later lecture). I�m not going to take the time to go into those but extensive references exist for using such methods, including Caughley�s book Analysis of Vertebrate Populations. Look back at where the different options lie on our dichotomous key. Depending on whether we are more interested in knowing about survival rates or are still more interested in estimating abundance, we will wind up at methods 10, 11 or 13 of the key (what does that mean about a fundamental population assumption early in the key?).

References for the interested:

Caughley, G. 1977. Analysis of Vertebrate Populations. Wiley, New York.

Jolly, G.M. 1965. Explicit estimates from capture-recapture data with both death and immigration -- stochastic model. Biometrika 52: 225-247.

Schnabel, Z.E. 1938. The estimation of total fish in a lake. Am. Math. Mon. 348-352.

Schumacher, F.X., and R.W. Eschmeyer. 1943. The estimation of fish populations in lakes and ponds. J. Tenn. Acad. Sci. 18: 228-249.

Seber, G.A.F. 1965. A note on the multiple capture-recapture census. Biometrika 52: 249-259.
 

 Some web-based resources for analysis of populations:
  http://www.stanford.edu/group/CCB/Eco/popest2.htm Stanford U. population estimation page


http://canuck.dnr.cornell.edu/misc/cmr/

Evan Cooch (Cornell) page of mark-recapture software and sources


http://www.cnr.colostate.edu/~gwhite/mark/mark.htm

Gary White (CSU) page of software etc. Interestingly, the Jolly-Seber method is more widely used for our next topic -- estimating survival or mortality rates. Much of the analysis of survival in natural populations is based on marking animals and then using the patterns of recapture to make inferences about survival rates. Obviously when we start focusing on survival rates we can�t be talking about a closed population.

We will start next time with an overview of patterns of mortality.

END of  line transect and mark-recapture

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