Equiprobable sampling designs, such as SRS, give each unit an equal probability of being selected. However many species populations are clustered or the species population has a strong spatial autocorrelation over the area. Stratified Random Sampling (StratRS) is a classic sampling design that is used to improve parameter estimation when dealing with those type of populations. To apply StratRS, the study area is partitioned into strata. A SRS sample is selected in each strata.
Stratification can be used for various reasons. If, for example, the stratification is done such that the values of the target variable in each stratum are as homogeneous as possible, thus by grouping those units that are as similar as possible, it is possible to obtain more precise estimates than the estimates of SRS (Cochran, 1977). For example, for a survey on invasive species, stratification may be based on the type of habitat, since the species density can be expected to be more homogeneous within the same type of habitat.
On the other hand, stratified sampling is often used to increase the spatial coverage and the the spatial balance of a sample, compared with for example SRS. The use of StratRS to increase the spatial coverage of a sample will be discussed in more detail in the next section about spatially balanced sampling designs.
Unbiased population estimators are also available for StratRS. The estimated total number of individuals for StratRS when the study area partitioned into L strata is
ˆ YStratRS= L X h=1 ˆ Yh, (3.7)
where ˆYh is the estimated number of individuals within stratum h thus
ˆ Yh = Nh nh Nh X i=1 yihIs(i), (3.8)
where Nh is the number of sampling units in the hth stratum and nh is the selected sample size from that stratum. The variance of the total abundance estimator ˆYStratRS, can be written as
V ( ˆYStratRS) = L X h=1 Nh(Nh− nh) σh2 nh , (3.9)
with σ2h the variance for the hth strata given by
σh2 = 1 Nh− 1 Nh X i=1 (yih− µh)2. (3.10)
An unbiased estimator for this is given by ˆ V ( ˆYStratRS) = L X h=1 Nh(Nh− nh) s2h nh , (3.11)
where s2h is the sample variance for the hth stratum given by
s2h = 1 nh− 1 Nh X i=1 (yih− ¯yh)2Is(i). (3.12)
Proof that these estimators are unbiased can be found in Cochran (1977).
It is shown that StratRS will almost always improve the precision of a survey compared with SRS (Cochran, 1977). This is because the sum of the within strata variances is almost always smaller than the variance obtained by SRS. However, when strata are ly chosen, it is possible that the variance is not reduced (much) by stratification compared with SRS (Gitzen et al., 2012). This is discussed in more detail in Chapter 4. This can happen, for example, when stratification is implemented out of practical convenience instead of as a means to reduce the variation (Gitzen et al., 2012). An example of ‘practical stratification’ is stratification based on geographical borders, when for example each district has only a limited number of field scientists. In that case, the districts borders are the strata boundaries and the number of field scientists within each district could be used to determine the sample size within each district/strata. Practical stratification can also be useful when a guaranteed number of sampling units are required within each stratum. For example, in the case one would like to select ten units in each district. Practical stratification might result in lower financial costs. However, it does not ensure improvement in the precision of the survey design.
Setting the Strata
To increase the efficiency of StratRS when estimating the total number of individuals the sampling area should be partitioned into strata that are as homogeneous as possible (Cochran,
1977). This means that stratification should preferably be done based on the underlying species distribution. For example, the area could be stratified into a low species density strata, a strata of intermediate density and a high density strata. Unfortunately, most of the time the underlying species distribution is the variable of interest and is thus unknown. To circumvent this issue, auxiliary variables which are highly correlation with the species density are often used to base the stratification on (Lohr, 2010; Guisan et al., 2006; Crall et al., 2013). For example, the density of a population of invasive mosquito species could be correlated to the availability of water. Therefore, strata could be defined based on the proximity of water. Note that in the case stratification is used for other purposes than population estimation, for example mapping of a species population, the sampling intensity could be optimized in another way. For example one could focus the sampling intensity in those areas with the highest change in the density of the species.
Several algorithms have been designed to determine the strata boundaries and to determine the sampling allocation based on auxiliary information. Examples of these algorithms include: Cumulative Square Root Algorithm (Dalenius and Hodges, 1957), Ekman Algorithm (Ekman, 1959), Lavall´ee-Hidiroglou Algorithm (Lavall´ee and Hidiroglou, 1988) and the cube method (Till´e, 2011). These methods look for cut-off values in the auxiliary variable such that units can be combined into homogeneous group based on the level of the auxiliary variable. Many of these techniques have their background in disciplines other than ecology such as economics and agricultural sciences (Benedetti et al., 2010). In ecology, species distribution models and dispersal models have been used to set the strata boundaries (Guisan et al., 2006; Albert et al., 2010; Elith and Leathwick, 2009; Peterman et al., 2013). These models use environmental covariates to estimate the species habitat suitability. Most of these studies use the estimated habitat suitability to base the stratification on. These studies often partition the study area into two strata: A high suitability stratum and a low suitability stratum. In Chapter 4 the applications of species distribution models in designing sampling designs will be discussed in greater detail.
Deciding on the number of strata is another issue with stratification. Adding strata can improve the precision, but depending on the situation the addition of new strata must be traded off
against the additional time restraints and financial costs (Caughlan and Oakley, 2001). These time and financial restraints are issues that do not only occur with StratRS but also with almost any other type of sampling design for example for GRTS and BAS as well as for SRS. These practical issues will be outlined when introducing these sampling designs.
Allocation Sampling Effort
Once the study area is partitioned into strata the sampling units need to be allocated over the strata. The three most common sampling allocation schemes are: fixed allocation, proportional allocation and Neyman allocation.
Fixed allocation distributes a fixed proportion of the sample size to each stratum. The allocation is not based on the size of the strata but rather on other, often logistical or other convenient issues. For example, this can be based on the available number of volunteers in each stratum or based on the number of accessible sites in each stratum.
With proportional allocation, the sample size in each stratum is set proportional to the size of the stratum, thus
nh = n
Nh
N . (3.13)
For example, if a stratum covers 40 percent of the study area, then 40 percent of the total sample effort will be allocated to that stratum. This type of stratification is convenient since it spreads the sample more uniformly over the study area compared with SRS (Gitzen et al., 2012). Neyman allocation is an allocation scheme that allocates the sampling effort proportional to the population variance of each stratum (Neyman, 1934). Neyman allocation is given by
nh = n
Nhσh
PL
k=1Nkσk
. (3.14)
More units will be selected in strata with higher variances. Unfortunately, σ2
h for each stratum
is rarely known prior to sampling. A possible solution to obtain σh2 is based on previous surveys or in the format of a small sample pilot study. In the case that these variances are known, or well approximated, Neyman allocation can lead to more precise population estimates. Optimal
allocation includes a cost function such that strata that are more time consuming or more costly to visit, will get allocated a relative smaller portion of the sampling effort.