2.3. How Many Samples to Take?

   There are two major methods for planning the number of samples:
    1. two-step sampling, and
    2. sequential sampling:

Two-step sampling

   The number of samples, N, required to achieve specific accuracy level can be
   estimated from equations for standard error (S.E.) and accuracy, A:

                                     [eqseq3.gif]

   where M is sample mean and S.D. is standard deviation. Here the third equation is
   derived from the first two equations.

   Standard deviation, S.D., is usually not known before sampling. Thus, the first
   step is to take N[1] samples and to estimate N using the equation above. Then, at
   the second step, take N[1] = N - N[1] samples.

   Taking samples in two steps is possible only if population numbers don't change
   between two sampling dates.

Sequential sampling

   The main idea of sequential sampling is to take samples until some condition
   (which is easy to check) is met.

   The first example is the sampling plan targeted at achieving specific accuracy.
   It is based on the Taylor's power law:

                                    [eqtaylor.gif]

   Coefficients a and b can be estimated using linear regression from several pairs
   of M and S.D. estimated in different areas with different average population
   density. Combining two previous equations we get:

                                   N = [eqseq2.gif]

   Mean (M) equals to the total number of recorded individuals (S) in all samples
   divided by the number of samples (N). Now, we substitute M by S/N, and solve this
   equation for S:

                                     [eqseq1.gif]

   Stop-lines for accuracy levels of A = 0.1; 0.07; and 0.05 are plotted below:

                                     [gseqsam.gif]

   The blue line shows the total number of captured individuals in all samples.
   Sampling terminates when this line crosses the stop line for selected accuracy
   level.

   The second example is the sequential sampling plan used for decision-making in
   pest management. This method was developed by Waters (1955; Forest Sci. 1:68-79).
   It is described in Southwood (1978).

                                     [gseqtr.gif]

   Here the blue line again shows the total number of captured individuals in all
   samples. While the blue line is between magenta inclined lines, sampling
   continues. If the blue line crosses the upper magenta line, then sampling stops
   and pesticides are applied against the pest population. If the blue line crosses
   the lower magenta line, then sampling stops and pesticides are not applied.

   Deriving the solution of this problem it is too complicated. Thus, we will
   consider the final result only.

   If the population has a negative binomial distribution (see [1]next lecture),
   then stop lines correspond to the linear equation:

                                     [eqseq4.gif]

   where:

                                     [eqseq5.gif]

                                     [eqseq6.gif]

   [2][back.gif] [3][up.gif] [4][forward.gif]
     ____________________________________________________________________________

   [5]Alexei Sharov 1/12/96

References

   1. http://alexei.nfshost.com/PopEcol/lec3/agregat.html
   2. http://alexei.nfshost.com/PopEcol/lec2/simple.html
   3. http://alexei.nfshost.com/PopEcol/lec2/sampling.html
   4. http://alexei.nfshost.com/PopEcol/lec2/geostat.html
   5. http://alexei.nfshost.com/~sharov/alexei.html