Why does Stereology use Systematic Random Sampling?
Systematic sampling refers to a method where all samples have an equal probability of being chosen but the spacing between samples is constant. That is, although each individual section is equally likely to be chosen, the combination of samples is not. The Wikipedia article on systematic sampling provides a good overview.

Stereology refers to the analysis of a three-dimensional structure using two-dimensional slices. Such an approach is common with histological techniques. A block of tissue is sliced into thin sections that allow for examination of the tissue with light microscopy, usually with the aid of stains.

If you sampled purely randomly, by chance, you are likely to obtain "gaps" in your analysis by random chance. For example, imagine you section a 3mm block into 40um sections. This process gives you 75 sections. It is too time-consuming to analyze all 75 sections, so you want to choose just 15 of them.

Let's try generating 15 random numbers from 1 to 75 and look at the maximum gap. I do this by taking the first 15 numbers of a random permutation of 1 to 75. In MATLAB, one can do: x=randperm(75,15), max(diff(x));

Here are some results:

23    26    33    34    35    37    44    54    55    58    60    69    72    73    74


Max diff: 10

12    14    15    16    17    20    22    27    28    34    44    47    58    59    73


Max diff: 14

 8    14    15    18    19    20    29    32    40    41    47    67    70    71    72


Max diff: 20

 3     7    16    19    30    32    33    34    38    39    50    57    61    65    75


Max diff: 11

 3     7    17    18    21    26    29    37    42    46    49    52    60    61    69


Max diff: 10

If we were to instead use systematic sampling, the difference between sampled sections would be fixed at 5: you skip 4 sections to get 15 evenly spaced sections out of 75 total. But with purely random sampling, the largest gaps are bigger, sometimes much bigger! 20 skipped sections means there is a 800um gap with no data! Even worse, I noticed later that in the first draw, the lowest number was 23! So the first 22 sections are completely omitted from analysis! There can be major morphological changes and entire brain nuclei missed within a gap that size, whereas 100um gaps are no problem.

I didn't print out the minimum gaps, but you can look at my examples to see the other side of this problem: in the first set, numbers 33, 34, 35 and 72, 73, 74 are right in sequence! There is likely little or no morphological difference on those short intervals and yet with random sampling one can waste time analyzing those adjacent slices. Because these draws are completely random, you can notice similar adjacent stretches in all of these random draws, even though I made no effort to select particular examples: these are just the first 5 results I got from MATLAB.

References



Gundersen, H. J. G., &amp; Jensen, E. B. (1987). The efficiency of systematic sampling in stereology and its prediction. Journal of microscopy, 147(3), 229-263.

Gundersen, H. J. G., Jensen, E. B. V., Kieu, K., &amp; Nielsen, J. (1999). The efficiency of systematic sampling in stereology—reconsidered. Journal of microscopy, 193(3), 199-211.