on counting and counting errors
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On Counting and Counting Errors
R.W. GUILLERY*Department of Anatomy, University of Wisconsin School of Medicine, Madison, WI 53706
ABSTRACTCounting objects in histological sections is often a necessary, sometimes an unexpected
part of a research project. The recent literature shows that the subject of counting is ofparticular interest to readers of the Journal of Comparative Neurology but that it is alsocontentious and difficult. Even a brief review of past issues of the Journal shows that thereare many misconceptions about counting and that there remain issues that have receivedlittle or no attention. Counts are subject to many errors. Some reports include readilyrecognizable errors, others fail to include all of the information that is needed for anevaluation of their accuracy. This review is above all a plea for adequate information aboutthe methods used for counts in all publications. It serves to help those who are new toquantitative methods in histology; it considers some of the basic issues arising for anyoneundertaking counts, or reviewing manuscripts that include counts. In particular, it considersrecently introduced or re-introduced counting methods that depend on accurate measuresalong the axis perpendicular to the plane of the sections, and looks at the difficulties inherentin these measures. J. Comp. Neurol. 447:17, 2002. 2002 Wiley-Liss, Inc.
Indexing terms: quantitative histology; optical disector; cell counts; sampling
Yet another article on counting needs justification, andthat is where this one must start. There have been severalarticles about counting methods in the Journal of Com-parative Neurology (Coggeshall and Lekan, 1996; Saper,1996, 1997; Guillery and Herrup, 1997; Geuna, 2000) andalso, quite recently, in several other publications (West,1999, 2001; Benes and Lange, 2001a1; Guillery and Au-gust, 2002; Williams et al., 2001); these represent diverseand sometimes strongly opposing views. There are manypapers published in this Journal that include counts ofone sort or another, and there are good reasons for expect-ing the number to increase. One strong reason for raisingthe subject again is that a significant proportion of thesepublished papers contain either methodological errors or,more commonly, insufficient information for a clear as-sessment of the significance of the results. A second, andpossibly more important reason is that simpler ap-proaches to counting are being introduced; these, whilenaturally appealing, raise new problems that are notwidely recognized and that merit careful evaluation. Inparticular, two of these methods (the optical disector, andthe comparison of counts obtained from two sections of
different but known thicknesses) rely on accurate mea-sures in the z axis (i.e., the axis perpendicular to theplane of the section); measurements that are often difficultto make and whose accuracy commonly cannot be evalu-ated from the information provided. A third reason is thatthe need for counts appears to be on the increase, andmany find that they must produce some quantitative re-sults, but have had little or no training in a subject that
Grant sponsor: National Institutes of Health; Grant numbers: EY 11494and 12936.
*Correspondence to: Dr. R.W. Guillery, Department of Anatomy, Uni-versity of Wisconsin School of Medicine, 1300 University Avenue, MadisonWI 53706. E-mail: email@example.com
Received 5 December 2001; Revised 25 January 2002; Accepted 25 Jan-uary 2002
DOI 10.1002/cne.10221Published online the week of April 1, 2002 in Wiley InterScience (www.
interscience.wiley.com).1The subsequent correspondence in Trends in Neuroscience 2001, 24,
374380, illustrates the extent to which there is significant disagreementon some of the issues.
THE JOURNAL OF COMPARATIVE NEUROLOGY 447:17 (2002)
2002 WILEY-LISS, INC.
appears simple at first encounter but proves to have hid-den and often quite unexpected complexities when thedetailed consideration of results is brought into clearfocus.
The following review is intended to stress two crucialpoints. One is that there is no one best method of count-ing. The most suitable method for any particular problemdepends on the level of accuracy required, on the nature ofthe material that is being studied, and on the type ofobject being counted. The method used should also beallowed to depend on the inclinations of the author, pro-vided that the method is clearly described and producesthe level of accuracy needed. This is the second point, thatthe author is responsible for providing a complete accountof the methods used, one that can serve to inform thereader about exactly what measures were taken, and howthe authors evaluated the relative accuracy of these mea-sures. References to general accounts of a method aregenerally not sufficient.2 Details are essential. There isnow a significant reservoir of numerical results that acritical reader has to assign to a pool labeled: perhaps so,but not sufficient documentation.
The problems that arise in light microscopical studiesare considered. Electron microscopy raises some distinctproblems but these will not be discussed here (see Guilleryand August, 2001 for some of the special issues raised byelectron microscopy).
Why is there a problem?The basic problem faced by anyone counting objects in
sectioned material is presented in Figure 1. This repre-sents a two dimensional view of a part of a block of tissuecut into four equal sections (14). In section 2, if onesimply counts all of the profiles identifiable as sections ofthe objects under studycells, mitochondria, or any otherobjectall of the objects shaded will be counted. This hasbeen called a profile count or a two-dimensional count.As some of these cells will also appear in sections 1 and 3,a straightforward count of all the cells seen in the sectionwill be an overcount. This is sometimes called a doublecount because many cells are counted twice. However,this term has led some, even in quite recent issues of theJournal, to state that if they space the sections for theircounts far enough apart, then no cell will be counted twice,and so there will be no double count. Figure 1 illustratesthat no matter what section spacing is used, each sectionby itself will produce an overcount, and that overcountmust be addressed before a realistic number can be pro-duced. There have been two major methods of dealing withthis problem, and we look at both in what follows. One isto eliminate the error by avoiding the initial overcount(see The Disector, and Comparing Two Sections ofUnequal Thickness); the other is to use profile counts,calculate the size of the error and make an appropriatecorrection (see Profile Counts and Corrections). Eachmethod can be useful; which method should be used de-pends on the nature of the material, on the sections thatcan be made available for the count, and on the informa-tion that is available or can be generated about section
thickness and the size and shape of the objects beingcounted. It may also depend on the level of accuracy thatis required.
BiasMethods that avoid the initial overcount are often de-
scribed as unbiased or assumption-free (e.g., Sterio,1984; Mayhew and Gundersen, 1996; West, 1999). Thisuses the term in a statistical sense to indicate that themethod is designed to produce counts that are evenlyscattered around a true mean, with no need for any cor-rections. The use is unfortunate because observer bias,which cannot be corrected after it has occurred, is notincluded in this consideration. No matter what method ofcounting is used, decisions must be made as to the inclu-sion or non-inclusion of particular objects or object profiles
2Particularly where the reference includes, as in one recent instance, twoquite different approaches, citing Guillery and Herrup, 1997 and West,1993 for the method.
Fig. 1. Schematic, two-dimensional representation of four sections(14) cut through a single block of tissue. Objects to be counted,(mitochondria, nerve cells, glial cells, nuclei, etc) are scatteredthrough the block and all those that would appear in whole or in partwithin section 2 are shaded. The figure illustrates three points con-sidered in the text: (1) A count of all of the profiles seen in section 2produces an overcount of the objects that actually have a (notional)central point, marked by an X, in the section. The size of the overcountdepends on h, the dimension of the objects in the z plane (perpendic-ular to the plane of the section). If all objects had the same h, then thecount of all profiles would represent the number of objects having acentral point in a section Th in thickness. Using h as a mean of thereal dimensions (h1,h2,h3,h4), gives a correction, T/Th). (2) Whenobjects in adjacent sections 1 and 2 are counted, one can count all ofthe objects in section 2, subtract those that also appear in 1, andarrive at a true count of the number of objects in section 2. This is thedisector method. (3) In the central nervous system cells are notevenly distributed. The lower right part of the figure shows a partic-ularly dense crowd of objects (e.g., neurons) such as might be found incentral nucleus or lamina. In order to obtain a randomly distributedsample of sites for counting that will not leave out small areas ofparticularly high or low packing, the method of sampling needs to berelated to the known (and described) spatial distribution of the ob-jects.
2 R.W. GUILLERY
in a count. Where a study aims to produce a single total,observer bias cannot be addressed, although it is likely tobe present. But where comparisons are made, the readeris entitled to assume that observer bias played a role inthe final differences reported unless there is a specificstatement that the observer was blind to the conditionsbeing compared