# Presentation: 914292

## Application of Adaptive Sampling to Biological Populations

- Dave Smith
- USGS – Leetown Science Center

**Notes:**
Mention introduction of technique by Thompson (1990), possibly refer to more historical predecessors.

## Outline

- Adaptive sampling designs
- Overview of Adaptive cluster sampling
- A basic example for adaptive cluster sampling
- Design factors specific to adaptive cluster sampling
- How to know if ACS will be efficient
- How to plan for sample size uncertainty
- How to incorporate detectability

**Notes:**
I will begin by introducing the sampling technique.  There is a basic example that occurs routinely in ACS introductions.  I will use that introductory example and then discuss variations on that basic design.

After the introduction and in keeping with the theme of the symposium I will discuss a series of challenges that one would encounter when applying the technique.  And along with the challenges, I hope to identify some choices or solutions to those challenges.

By efficiency of a design, I mean whether the sampling design will result in an estimate with more precision than other possible design.  So, in other words, when will application of ACS result in a relatively precise population estimate?

A much discussed challenge is presented by the fact that the final sample size is a random variable when adaptively sampling.  There has been a lot of work in this area to come up with design modifications that reduce sample size uncertainty.

Because adaptive sampling takes advantage of spatial clustering, neighboring units are sampled in response to finding what you are looking for.  This has the potential negative consequence of counting animals more than once when mobile species are flushed from an initially sampled unit to neighboring ones.  This is common problem in wildlife studies, for example when applying distance sampling techniques.

## Outline (continued)

- Other useful adaptive designs
- Two-stage sequential sampling
- Adaptive allocation
- Double-sampling for stratification (two-phase)
- Summary and future directions

**Notes:**
Sampling can  destroy habitat or disrupt species behavior.  Adaptive sampling concentrates, to some extent, sampling in the vicinity of where species are located.  Therefore, sensitive species present a special challenge especially when adaptively sampling.

Navigating among sampling units is almost always a challenge.  Adaptive sampling magnifies that challenge especially for large clusters of animals.

Sometimes the objective is to estimate population size or density.  Other times objectives include more complex analyses, such as regression analysis.  Data collected in complex surveys (unequal probability samples), such as ACS, present challenges in complex analyses, which assume data arise from equal probability sampling.

I conclude with a summary and a few comments on future directions.

## Adaptive sampling designs

- Adaptive designs allow sampling effort to adapt to what you find as you sample
- Examples
- Adaptive cluster sampling
- Sequential sampling
- Adaptive allocation
- Double sampling for stratification

## Are you an adaptive sampler?

- Angler
- Hunter
- Bird watcher
- Bargain shopper
- Have you ever played Battleship by Milton BradleyTM?

**Notes:**
Adaptive sampling is an intuitive technique.  One reason for its appeal is that it follows (or mimics) the natural process of finding clusters of things.  Here are a few examples and you can either find yourself among this list or perhaps can think of others who increase effort in response to finding what they are looking for, which is the essence of adaptive sampling.

## Introduction to Adaptive Sampling

- _Adaptive Cluster Sampling (ACS)_

## Since Thompson (1990)

- Adaptive cluster sampling applied to or evaluated for
- Forestry			–  Fisheries hydroacoustics
- Sea urchins		–  Shrimp
- Rare plants		–  Pollution monitoring
- Rockfish			–  Pacific hake larvae
- Polar bear		–  Harbor porpoise
- Waterfowl		–  Freshwater mussels
- Amphibians		–  Agricultural pests
- Bibliography at _[http://](http://www.lsc.usgs.gov/aeb/davids/acs/index.asp)__[www.lsc.usgs.gov/aeb/davids/acs](http://www.lsc.usgs.gov/aeb/davids/acs/index.asp)__[/](http://www.lsc.usgs.gov/aeb/davids/acs/index.asp)_

## Study area: N = 400    Pop. Density:  = 0.085, 2 = 0.1107

- Study area: N = 400    Pop. Density:  = 0.085, 2 = 0.1107
- Variable of interest is count of red dots in each unit: yi

**Notes:**
Even though many of you are already familiar with adaptive cluster sampling as developed by Steve Thompson,  I will step through a basic example to make sure we are all on the same page.

Consider sampling  this idealized and clustered population.  The dimension of the study area is 20 by 20 and contains 400 sampling units.  The population density is 0.085 red dots per unit with a standard deviation of 0.1128.

## Networks that met condition

- Neighborhood:  adaptively sample 4 adjacent units.

**Networks that met condition**

**Notes:**
To apply adaptive sampling we need to define a couple factors prior to sampling.  Namely, we need to define a condition so that if it is satisfied then adaptive sampling would commence.  Also, we need to define a neighborhood that is the set of units that would be adaptively sampled.  Networks are created as a result of these definitions.  An essential characteristic of a network is that if one of its units are sampled then all are sampled.

So the study area is redefined.  Instead of 400 units, the study area can be thought of 373 units that don’t meet the condition and 3 networks containing units that do meet the condition. If we define a unit that doesn’t meet the condition as a network containing 1 unit then our study area can be view as comprising 376 networks – 373 networks that do not meet the condition and 3 that do..  The networks vary in the number of units it contains so in that sense networks vary in size.

## Initial sample design: conduct simple random sampling with initial sample size = 10

**Notes:**
We need to choose a probabilistic sampling design to get the ball rolling.  In this case, let’s take an initial simple random sample of 10 units without replacement. This is an actual SRS, not contrived.  Although there is about a 50:50 chance to miss all units that contain red dots with an initial SRS of size 10, in this case one of the initial units hit a big network.

## Sample the neighborhood

**Notes:**
Because the a unit contains a red dot and meets the condition, we sample its neighborhood, which was previously defined as the 4 adjacent units. Two of the neighboring units also contain red dots so their neighborhoods need to be sampled and so on until we no longer encounter red dots.

## Sampled one cluster, which contained network + edge units.

- Final sample size = 31

**Notes:**
The final outcome is that of 10 units were initially sampled 1 contained a red dot, which lead to sampling 1 of the large networks.  In addition to the large network being adaptively sampled, we also adaptively sampled units around the edge of the network.  Although these did not contain red dots, sampling of the edge units was a by product of sampling neighborhoods.

The initial sample size was 10 and the final sample size was 31.

## Design Factors

- An _initial sample_ of size *n**1* is selected following a probabilistic sampling scheme.
- The _condition_ to initiate adaptive sampling is typically *C* = {*y**i*  *c*}.
- A _neighborhood_ is defined such that if 2 units (*i *and *j*) meet the condition, then if unit *i* is in the neighborhood of unit *j* then unit *j* is also in the neighborhood of unit *i*.

**Notes:**
So, to design a study that uses adaptive sampling you need to determine some factors before hand.

Given a study area partitioned into N sampling units and a value of interest yi, which will typically in fish and wildlife studies be a count of organisms in the ith sampling unit,the study objective is to estimate the total or mean of the variable y, which equates to estimating the total or density of organisms in the study area.

The design factors that must be chosen before sampling are the initial sampling design and sample size, the condition to adapt, and the neighborhood of units that will be adaptively sampled.  The choices are wide open.  

WRT to initial sample, just about any probabilistic sampling scheme can be used to select an initial sample. 
Initial sampling design (Much of ACS literature deals with this aspect.)
Simple random sampling
Unequal probability
Primary and secondary units (includes systematic)
Stratified ACS
Two-stage sampling
Latin square

Choice of a sample size, which is almost always a difficult choice, is somewhat more difficult in the context of ACS because it merely gets the ball rolling.  The final sample size depends on the size and number of networks that are intercepted by the initial sample size.

Since we are going to apply an adaptive sampling procedure, we must set the condition that will trigger adaptive sampling.  Typically, the condition is set up as the count greater than or equal to some value, c, such as zero.  However, this is one area where a good deal of flexibility can be built into the ACS design.  For example, the condition could be to adapt if a count is greater than 10 or 100.  Or the condition could be based on an auxiliary variable, such as a habitat characteristic or a rapid assessment of the variable of interest.  In a multi-species survey, the condition could be based on species richness so that adaptive sampling is triggered if a sufficient number of species were found in a sampling unit.  In this way, sampling effort would be disproportionately allocated to species rich areas.  In a multiple site survey, the condition can differ among sites.  For example, if a rapid assessment of density at a site indicates a high density, then c could be set high – adaptively sample only if the count is greater than 10, say.  But if the rapid assessment indicates low density, then c could be set low – adaptively sampling if the count is greater than 0.

It is best to set the condition before sampling begins.  There are a few exceptions to this rule.  For example, if the condition is satisfied if you count any animals and you find that this leads to too much sampling, you can make the condition more restrictive by raising the bar and adaptively sampling only if you count 10 animals.  This is not a desirable practice because either some of the effort will be wasted by ignoring units that met only the liberal condition or the study area will have to be post-stratified into areas with differing conditions.  You can not go from restrictive to liberal after finding that you aren’t adaptively sampling enough unless you can go back to initially sampled units that would have met the criteria and recover the units that you missed.  Again, this practice is not desirable because it leads to wasted effort or requires post-stratification.

Varying the magnitude affects efficiency, not straightforward nor easy to predict
Could be based on another variable (xi > c*)
Rapid assessment
Multivariate response
Could be based on order statistics

The neighborhood is another design factor that needs to be defined before sampling commences.   Symmetry is required in the definition of a neighborhood.  If a sampling unit 1 is a neighbor of sampling unit 2 then the reverse is true – unit 2 is a neighbor of unit 1.

## Design Factors

- A _neighborhood_ is symmetric, but otherwise flexible.

**Notes:**
The neighborhood, like the condition to adapt, is very flexible.  Possible shapes include those shown here and others can be imagined as long as the neighborhood is symmetric.

Neighborhood
Important that neighborhood is ‘symmetric’
If 2 units (i and j) meet the condition, then if unit i is in the neighborhood of unit j then unit j is also in the neighborhood of unit i.
Larger neighborhoods (e.g., 8 nearest quadrats) includes more edge units
A 4-unit neighborhood compares well to 2 and 8-unit neighborhoods (Brown in prep)
Definition may depend on logistical considerations

## Design Factors

- Neighborhood defined in Roesch (1993)

**Notes:**
Roesch applied ACS to a forest inventory by defining a neighborhood as a fixed radius around the center of a tree, which was the sampling unit.  Roesch also used an unequal probability design to take the initial sample.

## Design Factors

- Study area can be partitioned into _networks_.

- For condition: *y**i*  *c* and a
- ‘cross’-shaped neighborhood,
- there are 376 networks.

**Notes:**
As we saw in our introductory example, the study area can be partitioned into networks as a consequence of defining the condition and neighborhood.  Some networks contain units that meet the condition and the others do not.

## Design Factors

- If initial sampling unit intercepts a network, then the whole network will be sampled plus _edge units_ will be sampled.
- A _cluster_ is comprised of the network plus its edge units.

**Notes:**
And in addition to networks, edge unit will be sampled.  Edge units don’t meet the condition, but are in the neighborhood of a unit (or units) that does meet the condition.  In the introductory example a large network with 10 units was sampled, but in addition 12 edge units were sampled as a result.

A network plus its edge units comprise a cluster.  Although edge units add to the final sample size, the data from edge units are not used to estimate population parameters.

Although edge units require effort to sample, the information gathered is not of immediate use in estimating population parameters.

## Estimators (Thompson 1990)

- The larger a network, the more likely it is to be included in the sample.
- Conventional estimators of population mean and total are biased for ACS.
- Estimators have been constructed for unbiased estimation when adaptively sampling.

## Estimators (Thompson 1990)

- Modified Hansen-Hurwitz (HH) Estimator based on network means
- Modified Horvitz-Thompson (HT) estimator based on intersection probabilities
- Both estimators are unbiased for adaptive sampling
- The modified HT estimator tends to be as efficient or more so (smaller variance), due to “proportionality principle” (Salehi *in press*)
- For rare _and _clustered populations, network total (*y**k****) is proportional to the probability that a network is sampled (*k**),* which leads to small variance in HT type estimators

## Estimators (Thompson 1990)

- Modified Horvitz-Thompson estimator based on intersection probabilities
*y**k**** is the sum of the *y*-values in the *k**th* network
- *k* is the probability that the initial sample intersects the *k**th* network. *jk* is the probability that the initial sample intersects the *j**th* and *k**th* networks
- *k* and *jk* depend on initial sample design
- Edge units are ignored because their inclusion probability is unknown

## IntroductoryExample

- Ten networks sampled
- Network totals (yk*)
- Nine 0’s and one network with an 11
- Intersection probabilities (k)
- For a network with 1 unit, k=0.025
- For a network with 10 units, k=0.226
- Joint intersection probabilities (jk)
- For 2 small networks, jk = 0.00056
- For small and large networks, jk = 0.00515

**Notes:**
For completeness let’s revisit the introductory example.  Ten networks were sampled.  Network totals included nine zeros and one network with an 11.  Intersection probabilities were the same for all networks with one unit.  Intersection probability for the network with 10 units was an order of magnitude larger than for the other sampled networks.  There were two intersection probabilities depending on whether the large network was one of the pair.

## Introductory example

- Thus, density estimate is
- And variance estimate is

**Notes:**
The abundance estimate is a sum of the network totals divided by the intersection probabilities, and the density estimate is abundance divided by study area.

As you can see the only contribution comes from the large network with a network total equal to 11.  All other network totals were equal to zero.

This fact greatly simplifies the variance estimate, which would otherwise be a large summation.

## “Practical efficiency is equal to statistical efficiency times the probability that the method will be used.”

John W. Tukey

*John W. Tukey*

**Notes:**
ACS looks easy enough and appears to have potential, at least in theory.  But what challenges do we face when we put the theory into practice.  The challenges that I have identified are:

How to know when ACS will be an efficient procedure (less variance for fixed cost or fixed variance for less cost).
How to deal with uncertainty in the final sample size.
What special considerations are introduced when applying ACS to mobile, elusive, or sensitive species?
What logistical considerations are introduced when applying ACS to biological populations?
How can data collected by ACS be analyzed?

John Tukey’s quote is a good one to keep in mind as we talk about moving ACS from theory to practice.  Tukey said that ...

## Final sample size = initial sample size + adaptively sampled units

**Notes:**
So sample size must be fixed when comparing adaptive sampling to conventional sampling.  One way to make that comparison is to use the concept of efficiency, which is the ratio variances given fixed cost.  In this case, we make the comparison between SRS and ACS with cost measured by the expected final sample size when ACS starts with an initial SRS of size 10.  For the single replication that we discussed, the final sample size was 31, but over 1000 replications of the sampling procedure the final sample size was 22 units, on average.  Thus, it would be fair to compare expected variance that would result from an ACS with an initial SRS of size 10 and an expected final sample of size 22 to variance that would result from a SRS of size 22.  In other words, we compared variance for a fixed amount of sampling effort or cost.  

Efficiency is defined here as the ratio of SRS variance over ACS variance.  Sometimes you will see the inverse of that in papers.  Nevertheless, as defined here an efficiency > 1 indicates that ACS would be more precise than SRS and an efficiency < 1 indicates that the reverse is true – that SRS would be more precise than ACS.  For ACS with initial SRS of size 10 and an expected final sample size of 22, efficiency is 0.5 indicating you’d be better off taking a SRS.  From the graph we can see that ACS is not efficient until the expected final sample size exceeds 80 or equivalently a 20% sampling fraction.  An expected final sample size of 80 results from an initial sample size of 50.  This pattern can also be seen when comparing CVs.  CV for SRS is smaller than ACS until about a final sample size in excess of 80 units after which CV for ACS is smaller than SRS.

So we can see that sample size (the initial sample size and ultimately the final samples size) are factors that affect efficiency and could determine whether you choose to apply ACS.  We will discuss other factors, but sample size is an important one.

## When is adaptive sampling a good option?

- Technical answer
- Within-network variance is a high proportion of total variance
- Final sample size is close to initial sample size
- Practical answer
- Organisms are clustered
- Clusters are geographically rare
- Additional consideration
- Detection of rare species and increased collection of individuals
- How can cost of sampling edge units be reduced?

**Notes:**
The basic challenge is to decide when it is a good idea to apply ACS.  A simple answer to that is when the method is efficient (smaller variance for fixed cost) and when the uncertainty in final sample size is not too great.

Regarding efficiency...
Conventional cluster sampling tends to be efficient when the clusters capture the variation in the variable of interest. Same is true for adaptive cluster sampling.  In addition it helps when final sample size is close to initial sample size.

From a practical point of view, ACS tends to be efficient when the animals (or plants or whatever you are interested in) are clustered and the clusters are geographically rare.  Let me comment on rarity. Geographic rarity means that the units that are occupied by organisms is a small proportion of all units in the study area.  It is rarity of clusters that is important.  You could have a population that is rare but occurs as individuals. In that case the within-network variance would be zero, all adaptive units would be edge units, and ACS would not appropriate.  Or you could have a population that is clustered but the clusters are numerous.  In that case the final sample size would likely be much larger than the initial sample size and ACS would not appropriate. 

It is not a simple matter to know if population is clustered and rare enough for ACS to be efficient.  Let’s look at some examples.

## A

**A**

**B**

**C**

- Few, large

- Many, small

- Few, small

**Notes:**
Here are three populations with the same abundance, but with different spatial distributions.  

Population A, which is our introductory example, is characterized by a few large networks.  (I am using large and small describe the number of units in a network.)

Population B is characterized by many small networks.  To create population B, I simply broke up the networks in population A.

Population C is characterized by a few small networks.  To create population C, I simply tightened up the networks in population A.

As we saw previously, for our introductory example ACS was efficient for larger sample size.  Variance to mean ratio was 1.3 and within-network variance was 44% of total variance.

ACS is not a good choice for population B; efficiency never made it above 1 and barely above 0.8.  Variance to mean ratio was 1.3 and within-network variance was 45% of total variance.  These measures of spatial distribution were very similar to those for population A.

ACS was a better choice for population C where efficiency was above 1 in all cases.  Variance to mean ratio was 3 and within-network variance comprised 48% of total variance.

Most ecological populations are clustered.  However, it does not follow that ACS is appropriate for most ecological populations.  In addition, choice of condition to adapt and neighborhood affect within-network variance and final sample size and thus affect efficiency.  A practical, cost-effective procedure is needed to identify populations for which ACS is appropriate.

By pilot survey, e.g., Salehi and Seber (1997)
Field experiments across range of conditions/densities for a species
Predict ACS performance from index of dispersion?  Performance not likely to be a simple function of dispersion.
Simulation prior to application

## Adaptive sampling can increase detection of rare species

**Adaptive sampling can increase detection of rare species**

**Notes:**
In addition to estimating density/abundance, a common objective is to find more individuals of a rare species or find more rare species. For example, it is common that measurements of individuals is desired, such as taking blood samples for genetic analysis. ACS is good at that, I think reliably so.

Here is an example from a multi-species survey of freshwater mussels:
ACS applied at 24 sites along the Cacapon River, WV
Prior rapid assessment had indicated low-density (<1 m-2) of mussels present
Sampling unit was 0.25 m2 quadrats
Initial sample systematically placed
Criterion: C = yi  1
Neighborhood was quadrat that met criterion plus 4 adjacent quadrats

The probability of finding uncommon species was much higher in the adaptively sample quadrats.  For example, it took half as many adaptive quadrats to achieve a 50% probability of encountering uncommon species compared to initial systematically-placed quadrats.

So increased yield is an additional reason to apply ACS.

## How to control final sample size(most were suggested originally by Thompson)

**  ****Adjust the condition or neighborhood**** **
**  ****Woodby (1998)**
**  ****Adjust the condition that determines when to adaptively sample.  That is for ***Condition* = {*y**i*   *c*}**, make the condition more restrictive (i.e., make *****c***** a bigger number) so that adaptive sampling is triggered less often.**

**Notes:**
	The uncertainty of the final sample size is another challenge and frankly a significant deterrent to widespread application of ACS.  Researchers have suggested a variety of strategies to cope with this uncertainty and to control the final sample size.
	An obvious thing to do is to adjust the condition so that adaptive sampling is triggered less frequently.  We saw that with the mussel population.  Another is to adjust the neighborhood.  For example, do not sample adjacent units, but ‘leap frog’ over and sample the next unit.  The effectiveness of these strategies will depend very much on the spatial distribution of the population.
	Edge units are a nuisance.  They add to the sample but do not contribute much to the precision of the estimate.  It would help if you can reduce the cost of sampling the edge units.  One way to do that is to base the condition on an auxiliary variable that is related to the less expensive to measure.  For example, the condition could be based on a rapid count or a rapid assessment of habitat.
	Stratification is another strategy.  The study area can be stratified, then ACS could be applied in a stratum to ‘test’ if the condition or neighborhood are good choices.  The condition to adapt or the neighborhood definition could then be adjusted for subsequent strata.
	If you start applying ACS as you move through a study area, but discover that you are sampling too much, you can stop adaptive sampling or make adjustments and then post-stratify the study area to strata where ASC was applied and strata where it was not applied.
	If you can take the initial sample and go back later to take the adaptive sample, then you can base the condition on order statistics.  For example, in an environmental survey where concentrations of pollutants are stable you can use the initial sample to come up with a condition that identifies ‘hot-spots’.  Return to the hot spots to complete the adaptive sample.
	There have been several sampling schemes designed to restrict final sample size.  I will spend a little more time discussing them.

## A: Condition = {yi  1} 

|  |  |  |  |  |  |  | 0 |  |
| --- | --- | --- | --- | --- | --- | --- | --- | --- |
|  |  |  |  |  |  | 0 | 1 | 0 |
|  |  |  |  |  |  | 0 | 1 | 0 |
|  |  |  |  | 0 | 0 | 3 | 2 | 0 |
|  |  |  | 0 | 2 | 0 | 1 | 0 |  |
|  |  | 0 | 1 | 4 | 1 | 2 | 0 |  |
|  |  |  | 0 | 7 | 0 | 3 | 1 | 0 |
|  |  | 0 | 1 | 4 | 0 | 2 | 1 | 0 |
|  |  |  | 0 | 6 | 0 | 0 | 0 |  |
|  |  |  | 0 | 4 | 0 |  |  |  |
|  |  |  | 0 | 3 | 6 | 0 |  |  |
|  | 0 | 0 | 0 | 7 | 1 | 0 |  |  |
| 0 | 1 | 1 | 1 | 1 | 0 |  |  |  |
|  | 0 | 5 | 0 | 1 | 0 |  |  |  |
|  |  | 0 |  | 0 |  |  |  |  |

|  |  |  |  |  |  |  | 0 |  |
| --- | --- | --- | --- | --- | --- | --- | --- | --- |
|  |  |  |  |  |  | 0 | 1 | 0 |
|  |  |  |  |  |  | 0 | 1 | 0 |
|  |  |  |  | 0 | 0 | 3 | 2 | 0 |
|  |  |  | 0 | 2 | 0 | 1 | 0 |  |
|  |  | 0 | 1 | 4 | 1 | 2 | 0 |  |
|  |  |  | 0 | 7 | 0 | 3 | 1 | 0 |
|  |  | 0 | 1 | 4 | 0 | 2 | 1 | 0 |
|  |  |  | 0 | 6 | 0 | 0 | 0 |  |
|  |  |  | 0 | 4 | 0 |  |  |  |
|  |  |  | 0 | 3 | 6 | 0 |  |  |
|  | 0 | 0 | 0 | 7 | 1 | 0 |  |  |
| 0 | 1 | 1 | 1 | 1 | 0 |  |  |  |
|  | 0 | 5 | 0 | 1 | 0 |  |  |  |
|  |  | 0 |  | 0 |  |  |  |  |

**A: ***Condition* = {*y**i*  1}** **

**B: ***Condition* = {*y**i*  2}** **

**Notes:**
For A, adaptive sampling triggered if any mussels found in quadrat – yi>0

For B, adaptive sampling triggered if more than one mussel found in quadrat – yi>1

## Elliptio complanata

*Elliptio complanata*
- Condition: *y**i*  1

**Notes:**
Efficiency is not just affected by population characteristics, but also depends on condition to adapt and neighborhood definition.  For example, as the condition changes so does the partition of the study area into networks and so does the within network variance.

Here is a population of freshwater mussels in a river.  The colors may be a bit confusing at first.  The rust is dry land.  The blue area 0.25 m2 areas where mussels are present; dark blue are areas were the condition is met and as you will see in a moment, light blue are areas where mussels are present but the condition is not met.  Yellow is river bottom with no mussels.

I will show a series slides, each with a different condition to show how the condition affects how the population is partitioned into networks and ultimately how the condition affects within network variance and efficiency.

## Elliptio complanata

*Elliptio complanata*
- Condition: *y**i*  2

**Notes:**
Dark blue shows areas that contain networks where the condition is met.  The muted or light blue shows areas where mussels are present but the condition is not met.

## Elliptio complanata

*Elliptio complanata*
- Condition: *y**i*  3

## Elliptio complanata

*Elliptio complanata*
- Condition: *y**i*  4

## Elliptio complanata

*Elliptio complanata*
- Condition: *y**i*  5

## C: yi > 0

*C: y**i** **>** **0*

*C: y**i** **> 2*

*C: y**i** **>** **4*

**Notes:**
Within-network variance
Final sample size
Coefficient of variation

For c=1 final sample size was as high as 9 times initial sample size and efficiency was as low as 0.2.  Highest efficiency for reasonable sample size and cv was found for c=5, even though within-network variance as a very small proportion of total variance was low (0.02).

## How to control final sample size(most were suggested originally by Thompson)

**Stratification**
**Stratify the area beforehand**
**Post-stratify**
**Restrict final sample size**
**Stopping rule (Brown and Manly 1998)**
**Two-stage sequential sampling (Salehi and Smith 2005)**
**Reduce cost of sampling edge units**** **
**Use ‘inexpensive’ auxiliary variable as condition**
**Auxiliary variable in GIS then edge units do not have to be sampled in the field**

**Notes:**
	The uncertainty of the final sample size is another challenge and frankly a significant deterrent to widespread application of ACS.  Researchers have suggested a variety of strategies to cope with this uncertainty and to control the final sample size.
	An obvious thing to do is to adjust the condition so that adaptive sampling is triggered less frequently.  We saw that with the mussel population.  Another is to adjust the neighborhood.  For example, do not sample adjacent units, but ‘leap frog’ over and sample the next unit.  The effectiveness of these strategies will depend very much on the spatial distribution of the population.
	Edge units are a nuisance.  They add to the sample but do not contribute much to the precision of the estimate.  It would help if you can reduce the cost of sampling the edge units.  One way to do that is to base the condition on an auxiliary variable that is related to the less expensive to measure.  For example, the condition could be based on a rapid count or a rapid assessment of habitat.
	Stratification is another strategy.  The study area can be stratified, then ACS could be applied in a stratum to ‘test’ if the condition or neighborhood are good choices.  The condition to adapt or the neighborhood definition could then be adjusted for subsequent strata.
	If you start applying ACS as you move through a study area, but discover that you are sampling too much, you can stop adaptive sampling or make adjustments and then post-stratify the study area to strata where ASC was applied and strata where it was not applied.
	If you can take the initial sample and go back later to take the adaptive sample, then you can base the condition on order statistics.  For example, in an environmental survey where concentrations of pollutants are stable you can use the initial sample to come up with a condition that identifies ‘hot-spots’.  Return to the hot spots to complete the adaptive sample.
	There have been several sampling schemes designed to restrict final sample size.  I will spend a little more time discussing them.

## Restricted adaptive cluster sampling (Brown and Manly 1998)

- Employs a stopping rule
- Edge units used for stopping rule, but not in estimate
- Unbiased estimator presented by Salehi and Seber (2002)
- Disadvantage – take initial sample in a random sequence

**Notes:**
The basic approach is this... when current sample size exceeds a predetermined number, then stop sampling.

Edge units are used for stopping rule, but not in the estimate.

The sampling scheme is biased when using the Thompson’s ACS estimators.  Typically, restricted ACS results in an overestimate, but there are cases where underestimation is possible.  Salehi and Seber(2002) derived unbiased estimators based on estimators proposed by Murthy (1957).

The main disadvantage is that the initial sample must be taken in random sequence, so you can’t plan an optimal path to take the sample.

## Restricted adaptive cluster sampling (Brown and Manly 1998)

- Condition: count > 0
- Neighborhood: 4 adjacent units
- Initial sample: SRS, n1 = 10
- Stop when sample size  15

- Final sample size = 15
- Only 5 of n1 selected

**Restricted adaptive cluster sampling ****(Brown and Manly 1998)**

**Notes:**
These are counts of blue-winged teal in 25 km2 quadrats in a 5000 km2 study area in central Florida.

Suppose initial sample is a SRS of 10 quadrats, but you only have time to fly 15 quadrats.  So the stopping rule is to stop sampling when your final sample size equals or exceeds 15.

Sampling stops after hitting the first cluster.  As a result 5 out of the 10 initial units were sampled.

## Detectability in adaptive sampling

**Detectability in adaptive sampling**

- Efficiency declines with decreases in detectability because bias and mean square error increases.  The effect on efficiency increases as sample size increases.
- Estimates of detectability (*g*) from an independent study, such as, double sampling or capture recapture, can be used in an unbiased estimator. See Thompson and Seber 1994.

## Introduction to Adaptive Sampling

- _Two-stage sequential sampling_

## Two-stage sequential sampling(Salehi and Smith 2005 JABES)

- Divide study area into
- primary sampling units (PSU) and
- secondary sampling units (SSU)
- Select some (or all) primary units
- Select *n*i1 SSU within each selected PSU
- If condition met in a PSU, then select *n*i2 additional SSU
- No neighborhood or edge units

**Notes:**
To apply two-stage sequential sampling, large primary units are selected at random.  Within each primary unit a sample of secondary units is selected at random.  If the condition is met within a primary unit, then additional secondary units are selected.  Two-stage sampling is an attractive option for large sites, especially where it is expensive to move around the study area.

A significant advantage is that there are No Edge Units!

## Two-stage sequential sampling(Salehi and Smith 2005 JABES)

**Two-stage sequential sampling****(Salehi and Smith 2005 JABES)**

**Notes:**
	These are counts of blue-winged teal in 25 km2 quadrats in a 5000 km2 study area in central Florida.
	We simulated the application of two-stage sequential sampling by dividing the area into 8 PSUs.  Four PSUs were selected.  And within each of those, 2 quadrats were initially sampled.  If the count was greater than 10 in any quadrat then 4 additional quadrats were sampled within its PSU.  
	We simulated a 560 cases and in 99% of those cases two-stage sequential sampling was more efficient than SRS or conventional two-stage sampling.
	A wide range of sampling designs are possible through the choice of a method to select PSUs, condition C, ni1, and ni2. At this time, we recommend use of ni2 s not much larger than ni1 s.  And we recommend sampling a large proportion of the PSUs.
	An interesting aspect of the design is that when all of the PSUs are selected then it becomes a stratified sequential sampling design.  Moreover, two-stage sequential sampling tends to allocate more samples to the primary sample units (strata) with larger standard deviations.  It is in the same line as Neyman allocation for stratified sampling (Cochran 1977). We can therefore expect that two-stage sequential sampling will perform similarly to stratified sampling with optimal allocation. To implement optimal allocation we would need reliable estimates of strata-specific standard deviations, which might not be available.  However, optimal-type allocation is a natural product of the two-stage sequential sampling design. 

## Adaptive Two-stage Sequential Sampling

- Modification to allow greater flexibility in amount of sampling at the sequential phase
*n**i*2 = *g**i* λ is the number of adaptively added units in the *i*th primary sampling unit, where *g**i* is the number of units satisfying the condition and λ is a multiplier.
- For example, more units occupied by rare species in a primary sampling unit result in more sequential sampling effort.
- Efficiency is comparable to two-stage sequential sampling
- Brown et al. (in press) An Adaptive Two-Stage Sequential Design for Sampling Rare and Clustered Populations. Population Ecology

## Introduction to Adaptive Sampling

- _Adaptive Allocation_

## Adaptive Allocation

**Adaptive Allocation**

- Lo, N. C. H., J. R. Hunter and R. Charter. 2001. Fish. Bull 99: 554-571.
- Lo et al. used continuous egg counts to allocate placement of plankton net tows.
- If sardine egg count > 1 egg/min, then tow plankton net at 4 nm intervals.
- Post-stratify into high density (region 1) and low density (region 2)
- Compared to conventional survey, similar CV for estimates of egg production.  However, net tows encountering eggs increased from 11% to 65%

## Introduction to Adaptive Sampling

- _Double Sampling for Stratification_
- _Two-phase Sampling_

## Basic Design

- Initial sample selected to observe auxiliary variable only
- Rapid assessment, catch per unit effort (CPUE), ‘eyeball’ estimate
- Auxiliary variable less expensive to observe than variable of interest
- A subset of the initial sample is selected to observe the variable of interest
- The relationship between the auxiliary variable and the variable of interest is used to estimate totals and means

**Notes:**
Double sampling.  In double sampling, the investigator assesses a response variable in a large number of sampling units using an approximate, inexpensive method, and resurveys a fraction of the units with a precise but costly method.  The design relates the results from the inexpensive method to those from the costly method to achieve precise estimates at less cost than if the costly method was applied alone.  Double sampling is flexible and can be applied at multiple spatial scales.  For example, it can be used to select either sites or units within sites.

## Benefits

- Provides a framework for combining ‘qualitative’ and ‘quantitative’ methods
- Can be used to allocate more effort
- where organisms are and less where they aren’t
- where costs are lower and less where costs are higher
- where uncommon types are found and less where common types are found

## Two-phase Sampling of a River

- Objective to estimate riverwide density of freshwater mussels
- Don’t want to spend much effort at reaches where mussels aren’t present
- Timed search conducted at all sample reaches
- Reaches stratfied based on timed-search results
- Quantitative sampling methods depend on strata

**Notes:**
the 2-phase sampling design is flexible and can be implemented at multiple scales to select sample sites within a river or watershed or to select sample units within a stream reach.  The cost of sampling should be lower because not all sites must be revisited during the 2nd phase of sampling.  A time cost is associated with establishing the frame of sampling (i.e., assigning sites to strata), but the 2-phase design has the advantage that a great deal of prior knowledge of the whole river is not needed before stratifying sites.  The design flexibility can be used to reduce total survey time.  For example, the 1st phase of sampling does not have to be completed at all the sample sites before initiating the 2nd sampling phase.  The decision to conduct a quantitative survey at a site can be made upon completion of the qualitative assessment of the site.  This flexibility can further reduce survey costs by reducing travel time. On the other hand, all qualitative sampling could be completed during one year and all quantitative sampling could be completed during the next. 

## Two-phase Sampling of a River

- Timed search linearly related to density
- Most sites were classified in ‘low’ density stratum
- Adaptive sampling applied in low-density stratum
- Conventional quadrat sampling in high-density stratum

**Notes:**
Relationship between timed search and unbiased density estimates

## Two-phase Sampling of a River

- Minimizing variance and cost leads to optimal allocation
- Best to quantitatively sample a low to moderate proportion of low-density reaches
- And a high proportion of high-density reaches

**Notes:**
variance and cost were minimized when 20 to 40% of low-density sites and > 60% of high-density sites were selected for quantitative (phase 2) sampling 

	The optimal sampling scheme in 2-phase sampling allocates larger sample sizes to the more variable strata (high-density sites) and smaller sample sizes to the less variable (but possibly more difficult to sample) strata (low-density sites).  Stratum-specific sampling allocations can deviate from optimal allocation when specific objectives warrant.  For example, a higher fraction of high-density reaches (even up to 100%) could be selected for quantitative sampling if high-density reaches have special significance.  One caveat is that ≥2 reaches must be sampled per stratum so that stratum-specific variance can be computed.  If no estimate of density is obtained in one of the strata (e.g., a low-density stratum), then density cannot be estimated for the drainage as a whole. 

## Introduction to Adaptive Sampling

- _Summary and Future Directions_

## Pros and Cons of adaptive sampling designs

- Pros
- Flexible
- Design unbiased
- Allocate effort to units that include clustered populations (sample more where they are and less where they aren’t)
- Can be efficient for rare _and _clustered populations

- Cons
- Complex
- Random final sample size
- Efficiency depends on spatial distribution
- Often CV no better than conventional design for equal effort

## Summary and Future Directions

- Adaptive designs are an active area of research
- Future directions (*look for advances in these areas*):
- Methods to identify populations appropriate for adaptive procedures
- Designs that control/limit final sample size
- Continued development on alternative designs
- Software development is important