It is rarely possible to measure all of the trees in a particular forest. Therefore, a samplemust be taken from the complete populationof all sample units. The most important basic principle is to ensure that the sample is representative. Otherwise, the information obtained will be biased in some way, and the inferences drawn from the data are likely to be invalid. The first step is to divide the forest area to be Choosing a sampling design | 87
surveyed into sampling units, for example by dividing it into grid squares or forest stands or patches. The next step is to decide which sample units to include in the survey, according to an appropriate design. Four basic sampling designs are used in forest surveys: simple random,stratified random,systematic samplingandcluster sam- pling(Figure 3.2). Each of these is considered below.
Whichever method is used, it is important that the survey is as accurate as pos- sible. If a sample is taken from a population, the values of measurements obtained from the sample may differ from those of the population. This difference is referred to as the sampling error, and is expressed as the standard error of the mean.
Sampling error can be reduced by increasing the sample size. Other (non- sampling) errors may arise as a result of mistakes or inaccuracies in data collection, bias in the estimates because of a lack of independence in the sample units, inaccurate production of maps and calculation of areas, and poor data processing or management (Husch et al. 2003). Attempts should be made to document and minimize such sources of error.
3.3.1 Simple random sampling
This method requires that there be an equal chance of selecting every possible com- bination of sampling units. It is important to note that this is not the same as each sampling unit having an equal chance of being selected (Avery and Burkhart 2002). The method involves ensuring that the selection of any unit is completely
1:1 systematic grid
Simple random sample Stratified random sample 1:4 systematic grid
A: 7 plots
B: 5 plots A: 7 plots
B: 5 plots
C: 4 plots C: 4 plots
Fig. 3.2 Different sampling designs, based on four possible arrangements of 16 samples in a population composed of 256 square plots. (From Avery and Burkhart 2002. Forest Measurements, 5th ed. with permission of McGraw-Hill.)
independent of selection of any other unit. This can be achieved by assigning every unit in the population a number, then selecting a sample of these according to ran- domly generated numbers. Alternatively, random numbers can be used to select intersection points on a sample grid. Random numbers can be obtained from stati- stical tables, or can be generated by some pocket calculators as well as some spreadsheets or statistical analysis software. It is important to use numbers that genuinely are randomly generated in this way, and not simply plucked from the air in some arbitrary fashion.
The main problem with this approach is that it may be difficult to accurately locate the selected sample points, and it may be difficult to define the most suitable route between two points, making the process of collecting information less effi- cient than some other methods (Reed and Mroz 1997). It is also important to note that randomly distributed locations will tend to be clustered. As a result, some parts of the surveyed area will not be included in the sample.
3.3.2 Stratified random sampling
In this approach, the forest to be surveyed is first divided into relatively homoge- neous areas (or strata). Sample units are then randomly selected from each stratum (usually at least two from each), using the same approach as for simple random sampling. There are a number of key advantages to this approach. Forests are usu- ally spatially heterogeneous, as a result of variation in environmental variables such as topography, soils, aspect, and altitude, as well as in patterns of natural disturb- ance and previous management. This variation can often be detected on aerial photographs or during preliminary field surveys. Strata can therefore be defined on the basis of such information, in a way that is relevant to the objectives of the sur- vey. For example, it may be known that in different parts of the forest the stands are dominated by different tree species because of variation in soil conditions and drainage. In this case, if the objective is to assess stand structure in each of the dif- ferent kinds of forest stand present, stratified approaches offer an advantage by enabling each type of forest stand to be adequately represented within the sample.
How should the number of sample units in each stratum be determined? There are two main options. Samples can be allocated among strata in proportion to their relative areas (proportional allocation); in other words, more samples are allocated to larger strata than smaller strata. Alternatively, sample units can be allocated to strata taking into account both the size and expected variance of the strata (optimal or Neyman allocation). In other words, more variable strata are sampled more intensively than less variable strata of the same size (Reed and Mroz 1997). The lat- ter method results in a more precise estimate of the population mean, but requires prior estimates of the sample variance within the individual strata. However, opti- mal allocation is generally preferred if stratum areas and variances can be reliably determined (Avery and Burkhart 2002). Both of these methods provide separate estimates of mean values for each of the strata, which might be different forest types or administrative units.
Choosing a sampling design | 89
3.3.3 Systematic sampling
In this method, the first sampling unit is selected randomly or arbitrarily located on the ground, and locations to be sampled are thereafter spaced at uniform inter- vals throughout the area to be surveyed. For example, if 10% of the area were to be sampled, every tenth sampling unit would be selected. Typically, sample units are established on a grid.
This method has been widely used in forest survey because the sampling units are easy to locate on the ground, and because the samples are distributed over the entire area, giving the impression that a representative sample has been obtained.
The main problem is that it is less statistically powerful than random sampling methods. Specifically, it is not possible to obtain a genuinely valid estimate of the sample variance, because the sample units are not truly independent. This makes it difficult to estimate the precision of the measurements taken. This is particularly the case when there is a regular spatial pattern in the forest being surveyed, which may be caused by regular variation in soils, topography, or hydrology. The effects of topography can be minimized by referring to soil maps or by orientating grid lines to be surveyed up and down slope, rather than along the contours (Reed and Mroz 1997). However, in situations where estimations of precision are not required, systematic sampling may be preferred, primarily because it may be more efficient (in terms of information gained per unit effort expended) than random sampling approaches (Avery and Burkhart 2002).
3.3.4 Cluster sampling
A cluster is a group of smaller units (subplots) that taken together make up the sam- pling unit. Clusters can be arranged in many different ways, depending on the num- ber of subplots included, the distance between them, and their spatial relations. As noted earlier, clusters are often used in NFIs, particularly in areas that are remote or difficult to access. This is because time and resources can be saved if information is collected from a number of locations within a particular area. However, cluster sam- pling is also sometimes used in regeneration surveys at a local scale.
Clusters are randomly selected from the population. However, they are not stratified, as in stratified random sampling. Cluster sampling may also be divided into a number of stages. For example, in two-stage clustering, clusters are ran- domly sampled and then, instead of each subplot within the cluster being sur- veyed, a subsample of these is randomly selected for survey (Reed and Mroz 1997).
When implementing cluster sampling, the first step is to specify appropriate clus- ters. Ideally, the number of subplots within a cluster should be relatively small relative to population size, and the number of clusters should be relatively large.
Cluster sampling will be more precise than simple random sampling if variation at the local scale is high relative to variation at the scale of the entire population (Avery and Burkhart 2002). It should be remembered, though, that the subplots within a cluster are not independent of each other, and therefore the independent sample unit is the cluster rather than the subplot.
3.3.5 Choosing sampling intensity
How many samples should be taken? Enough to obtain the level of precision required. It is therefore important to specify an acceptable level of precision before the survey is initiated. For example, in a survey designed to estimate the basal area of a forest stand, the forest manager might require an estimate within5 m2ha–1with a 95% confidence level. This corresponds to achieving estimated mean basal area that is within 5 m2ha–1of the actual value 95% of the time (Reed and Mroz 1997).
There is a trade-off between precision and cost, because both increase as the number of sample locations increases. An index of efficiency can be calculated as the product of the squared standard error, which is a useful measure of precision, and the survey time (or expenditure) required (Avery and Burkhart 2002). This is based on the fact that to halve standard error four times as many sampling units are required. The required sampling intensity for a specified level of precision is given by the following formula:
n(ts/E)2
where nis the number, tis the tvalue (which can be obtained from statistical tables), sis standard deviation, and Eis the desired half-width of the 95% confidence inter- val (Avery and Burkhart 2002). In order to apply this equation, an estimate is required of the expected variance that is likely to be achieved, as indicated by the confidence interval. This can best be obtained by carrying out a preliminary survey before the main investigation. Equations for calculating means, standard devi- ations, and confidence intervals for these different sampling approaches are given by Avery and Burkhart (2002), Cochran (1977), and Reed and Mroz (1997).