CE and Biological Variability

The concept of the CE is distinct from that of biological variability as well as from traditional statistical values relating to intrasubject variability found with repeated sampling (such as Variance and Standard Deviation).

When considering what the CE means, it helps to understand what is going on biologically. There are animals that actually have natural variation, e.g., they have more (or fewer) cells of a certain type than other animals – this is normal biological variability. When we estimate the number of cells in these animals, there is a 50/50 chance that we will overestimate or underestimate the numbers. That means that the methods we use to estimate the total number of cells actually increase the observed variability in the population of animals and, hence, most likely also the variability we observe in the group we investigate.

In addition to the contribution by nature to group variance, this means we now also have to cope with a methodological contribution to variance.

Variance observed around the mean is an element of most statistical comparisons and, together with the number of individuals used (n), the most important one. It is through the CE’s contribution to total variance in the context of statistical comparisons that the CEs of stereological estimators become potentially powerful tools in the retrospective analysis or prospective design of quantitative studies.

A major domain of interest in design-based stereology has been the establishment of criteria for sampling requirements to achieve a specific level of variation in the estimates. Optimal stereological estimates are based on a sampling protocol that yields sufficient data to achieve the desired variation (i.e., precision) of the estimates, while making the work efficient by performing as few measurements as possible to achieve this precision. The main question is what level of variation in the estimates is acceptable. The secondary question is how much work should one do to achieve an acceptable variance in the estimates.

The answer to the first question, of course, depends on the purpose of a study. In an experimental study, the variance introduced by methodology should not result in a total variance that makes it impossible to statistically detect a biologically significant group difference.

Retrospective Analyses

Estimates of the contribution of methodology to group variance are often performed retrospectively, i.e. after the group means have been calculated, and statistical comparisons have been performed. Take the case where statistical differences between groups have been found, this also implicitly means that the CEs have been ‘good enough’. This means however much the methodology contributed to group variance; it was not enough to statistically obscure group differences. In this case, calculating and presenting CEs serves the purpose of allowing other researchers to evaluate a sampling scheme and quality of the data for their own purposes.

The critical point of a retrospective analysis is negative statistical findings. Let us assume that the difference between group means has a size, which by itself would seem biologically significant, but the statistical outcome is negative. Are natural differences between animals responsible, i.e., are our findings really negative for the n we could use? – or was methodology so poor that it resulted in type-1 statistical error, i.e., a negative statistical outcome even though the difference statistically should exist for this n and natural variance?

The “good enough” of the CEs is now determined by the contribution of methodological variance to total variance. If it contributes less than natural variance, the most efficient way to decrease total variance would be to decrease the contribution of natural variance, i.e. to increase n. The CEs are ‘good enough’ in the sense that they are not the weakest link in the experimental setup – adding new animals makes more sense than measuring more precisely in the already available animals.

If methodologically introduced variance contributes more to total variance than natural variance, it may be worthwhile to increase the counting, i.e., do more work, to obtain a higher precision of the estimate. Increasing the work expended on an individual in the sample may not be the only way to increase estimate precision. There may be other factors such as the orientation of the sections, i.e. frontal, sagittal or horizontal, can have a major impact on estimate precision for a fixed workload (see figure 3 in Slomianka, 2005). Exploring the effect of the orientation of sections may be an efficient way to increase estimate precision if it is not critical for the delimitation of the structure of interest or other concerns associated with a study


Slomianka L, West MJ., Neuroscience 2005;136:757.



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developers of Stereo Investigator, the world’s most cited stereology system