Systematic sampling

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General descriptions of systematic sampling

Systematic sampling is a name of a wide class of sampling strategies in which selection of individual elements is following a systematic pattern. Examples are square grids of sample points laid out over an area of interest; or the selection of every 10th tree in an alley; or parallel transects.

Systematic sampling and its applications to forest inventory are best illustrated with square grids of sample points. We may imagine a transparency sheet on which this grid is printed; and this transparency is placed randomly over the map, where randomly means: randomly selected starting point and random orientation. From a sample selection point of view, it is important to state that we have only one independent selection of a sample point; after having selected the first point, all others are fixed. We defined earlier that sample size is the number of independently selected elements; an immediate conclusion is that systematic sampling is obviously a sample of size \(n = 1\). The “plot” that is being laid out then is a large cluster plot consisting of numerous sub-plots – that is, all the sample points on the systematic sample are strictly spoken sub-plots of one single cluster that is spread out over the entire area.

A major question is then whether we can make an unbiased estimation of mean and variance from a random sample of size \(n = 1\). For the estimation of the mean, there is no problem at all: the estimator

\(\bar y=\frac{\sum_{i=1}^n y_i}{n}\,\)

can be calculated and yields the estimation of the mean.

However, when we wish to estimate the variance with the estimator

\(s^2=\frac{\sum_{i=1}^n\left(y_i-\bar y\right)^2}{n-1}\,\),

we see that this is not possible as the denominator is not defined. This is also directly understandable by common sense: one single observation does not contain any information about the variability that is present in the population.

It is important to understand for systematic sampling with but one randomization step:

• there is an unbiased estimator for the mean;
• there is no unbiased estimator for the population variance and hence neither one for the estimation of the error variance.

Figure 1 Illustration of systematic sampling in terms of stratified sampling or cluster sampling. The population of \(N\) elements is arranged here in groups of \(M\) (Kleinn 2007).

For the estimation of the error variance which is the most important characteristic to evaluate the statistical performance of a sampling technique, we, therefore, need to find a solution. However, at first, some further issues regarding systematic sampling are addressed. Systematic sampling is, obviously, a specific sampling technique for its own. Some authors do also refer to it as non-statistical sampling because of the sample size \(n=1\) (and in many cases, no randomization is done at all!) and because of the lack of variance estimators.

When we look at systematic sampling from the point of view of the sampling techniques presented so far, we may express it as a specific case of stratified sampling or as a specific case of cluster sampling. This is illustrated in Figure 1 where a population of \(N\) elements is arranged in groups of \(M\) elements. A systematic sample is, for example, taken by selecting the elements of one column. If we look at one line as one stratum, then systematic sampling would mean here to select exactly one element per stratum from all strata. Of course, this does not allow variance estimation. Or we take one line completely, that is exactly one cluster - and that does neither allow estimating the error variance.

Sample size

Strictly spoken, sample size in systematic sampling is \(n = 1\). However, this does not allow any conclusion about the variances. Therefore, it is common to look at the systematic sample as a sample in which the sub-plots are considered the observation plots.

Figure 2 One and the same grid randomly laid over the same area results in different numbers of sample points inside the forest area (Kleinn 2007).

When a square grid is used, one can calculate the required grid size for a certain number of points to fall into forest. If, for example, we define \(n = 20\) plots to be the size of the sample for a 1000 ha forest, the required square grid size is calculated via the area that each sample point “represents” around it. This area is \(1000 ha/20=50 ha\). If this is the area of the square that each sample point represents, then the side length of the square of 50 ha is the searched distance d between the sample points on the square sample grid, which is here

\(d=\sqrt{\frac{1000ha}{20}}=\sqrt{50ha}=\sqrt{500000m^2}\approx 707.1m.\,\)

However, if this grid is superimposed randomly over the 1000 ha area of interest it is not guaranteed that always exactly the desired number of \(n=20\) sample points falls into the forest area. The number can be slightly higher or lower; that depends mainly on shape and fragmentation structure of the forest area, which is illustrated in Figure 2. What we have calculated here, actually, is not the sample size \(n\) but it is the expected value of the sample size \(E(n)\): on the average we have \(n\) samples in our area when repeating very often a random superimposition of the grid over the forest area.

Some advantages of systematic sampling

Systematic sampling is by far the most frequently applied sampling technique in forest inventory sampling – and there are a number of reasons for that, despite the fact that there is, unfortunately, not yet a design-unbiased estimator for the error variance available.

Among the advantages are:

• The procedure is easily applied in the field or in any other population of interest and it is easily explained to the field crew or those who are supposed to take the samples.
• It is also easy for those who are interested in the results to understand the sampling procedure. Actually, in random sampling, only those who actually did the random selection by themselves know for sure that the selection had been truly done at random. All others need to believe it. Whatever arrangement of sample points results, all can theoretically be random. There are simply very many possibilities for manipulation. In systematic sampling, however, there are much less thus possibilities.

• In practically all cases of forest inventory applications, systematic sampling yields more precise results than simple random sampling with the same number of sample points. This can intuitively be explained because regardless of the randomization of the grid, it will always evenly cover the area or interest. Extreme values are not possible, that may occur in simple random sampling, if, for example, all points fall incidentally at regions with very low values.
  However, it can also be explained by the autocorrelation considerations made earlier: in systematic sampling neighboring sample points have always a minimum distance. It is not possible that neighboring sample points are very close together in a situation where the autocorrelation is expected to be high. That means that the systematic sample collects more and uncorrelated information and is thus more precise.

• In connection to the former point: by systematic sampling it is guaranteed that all parts of the population are covered. It can not happen that for a larger region there is no sample point. In fact, when we use a systematic grid of points, the whole population is evenly covered and if we distinguish different situations in the population (strata, sub-populations) the number of sample points in each such stratum is automatically proportional to the size of such strata (see Figure 3).

References

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