Coefficient of Error

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Understanding the Purpose of the Coefficient of Error

The quality of quantitative estimates obtained from design-based stereological methods can itself be estimated. This essentially means we can have some understanding about the accuracy an estimate performed with a stereological procedure. The measure of how good the estimate is, is called the coefficient of error, or CE.

The coefficient of error is a standard statistical value that is used extensively in the stereological literature. The definition of the CE is rather simple. It is defined as the standard error of the mean of repeated estimates divided by the mean.

In practical applications of design-based stereology, the amount of sampling error (the difference between an estimate and the true value) is unknown. Therefore, several methods have been developed to predict the accuracy of a stereological estimate. Different CE formulas have been developed using models based upon different assumptions and with different considerations taken into account, such as the shape of the region of interest, the distribution of objects within the region of interest, and the sampling criteria applied to the examination. References 1-14 below are publications describing different methods for estimating the CE for population estimates. References 15 – 20 below describe different CE methods for volume estimates.

Computer simulations have shown that the methods published for predicting the CE of population estimations obtained with the “N V V Ref” method do not result in adequate predictions of CE21.

Although CE equations may be the most complicated mathematical formulas used in a study, the most “complicated” mathematical expression used is typically a square root. They are easy to implement in a spreadsheet and, for convenience and those who shudder at the mention of the word “formula”, the calculation and presentation of a number of different CE estimates is part of advanced stereology packages such as Stereo Investigator.

It is important to note that the CE has no real biological meaning. Rather, it is most useful for evaluating the precision of stereological estimates.


CE for population estimates using the Optical Fractionator estimator
1. Gundersen HJ, J Microsc 1986;143:3.
2. Gundersen HJ, Jensen EB., J Microsc 1987;147:229.
3. Geiser M, Cruz-Orive LM, Im Hof V, Gehr P., J Microsc 1990;160:75.
4. West MJ, Slomianka L, Gundersen HJ., Anat Rec 1991;231:482.
5. Thioulouse J. Royet JP. Ploye H. Houllier F., J Microsc 1993;172:249
6. Scheaffer HL. Mendenhall W. Ott L, Elementary Survey Sampling, 5th edition, PWS-Kent, Boston, 1996.
7. Glaser EM, Wilson PD., J Microsc 1998;192:163.
8. Larsen JO., J Neurosci Methods 1998;85:107.
9. Schmitz C., Anat Embryol 1998;198:371.
10. Cruz-Orive LM., J Microsc 1999;193:182.
11. Nyengaard JR., J Am Soc Nephrol 1999;10:1100.
12. Glaser, EM., Wilson, PD., Acta Stereol 1999;18:15.
13. Gundersen HJ, Jensen EB, Kieu K, Nielsen J., J Microsc 1999;193:199
14. Schmitz C, Hof PR., J Chem Neuroanat 2000;20:93.

CE for volume estimates using the Cavalieri estimator
15. Gundersen HJ, Jensen EB., J Microsc 1987;147:229.
16. Roberts N, Garden AS, Cruz-Orive LM, Whitehouse GH, Edwards RH., Br J Radiol 1994;67:1067.
17. Geinisman Y, Gundersen HJ, van der Zee E, West MJ., J Neurocytol 1996;25:805.
18. Garcia-Finana M, Cruz-Orive LM, Mackay CE, Pakkenberg B, Roberts N., Neuroimage 2003;18:505.
19. Glaser E., J Microsc 2005;218:1.
20. Cruz-Orive LM, Garcia-Finana, M., J Microsc 2005;218:6.

Computer simulations of CE estimators
21. Schmitz C, Hof PR., J Chem Neuroanat 2000;20:93.



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