On counting and counting errors

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<ul><li><p>Commentary</p><p>On Counting and Counting Errors</p><p>R.W. GUILLERY*Department of Anatomy, University of Wisconsin School of Medicine, Madison, WI 53706</p><p>ABSTRACTCounting objects in histological sections is often a necessary, sometimes an unexpected</p><p>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.</p><p>Indexing terms: quantitative histology; optical disector; cell counts; sampling</p><p>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</p><p>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</p><p>Grant sponsor: National Institutes of Health; Grant numbers: EY 11494and 12936.</p><p>*Correspondence to: Dr. R.W. Guillery, Department of Anatomy, Uni-versity of Wisconsin School of Medicine, 1300 University Avenue, MadisonWI 53706. E-mail: r.guiller@facstaff.wisc.edu</p><p>Received 5 December 2001; Revised 25 January 2002; Accepted 25 Jan-uary 2002</p><p>DOI 10.1002/cne.10221Published online the week of April 1, 2002 in Wiley InterScience (www.</p><p>interscience.wiley.com).1The subsequent correspondence in Trends in Neuroscience 2001, 24,</p><p>374380, illustrates the extent to which there is significant disagreementon some of the issues.</p><p>THE JOURNAL OF COMPARATIVE NEUROLOGY 447:17 (2002)</p><p> 2002 WILEY-LISS, INC.</p></li><li><p>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.</p><p>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.</p><p>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).</p><p>Why is there a problem?The basic problem faced by anyone counting objects in</p><p>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</p><p>thickness and the size and shape of the objects beingcounted. It may also depend on the level of accuracy thatis required.</p><p>BiasMethods that avoid the initial overcount are often de-</p><p>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</p><p>2Particularly where the reference includes, as in one recent instance, twoquite different approaches, citing Guillery and Herrup, 1997 and West,1993 for the method.</p><p>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.</p><p>2 R.W. GUILLERY</p></li><li><p>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. Adding the adjective unbiased to a par-ticular method of counting is rather like wearing a badge;it is generally more decorative than informative. It shouldbe discouraged because when the method is fully de-scribed sources of bias (or their absence) should be readilyidentifiable. Bias can be introduced into counts in manydifferent ways, only some of which are addressed by meth-ods labeled unbiased or assumption-free. Some of theseare considered here (see also Guillery and Herrup, 1997;Guillery and August 2001).</p><p>The DisectorThe earliest known method of making a count that</p><p>needed no corrections for overcounting was published in1895 in a study of kidney glomeruli, which are large rel-ative to section thickness. Miller and Carlton (1895) usedthe basic principle of what was later called the disectormethod (Sterio, 1984), and this method has been widelypublicized and discussed more recently (Gundersen et al.,1988; Mayhew and Gundersen, 1996; Coggeshall and Le-kan, 1996; West, 1999). It is particularly useful where theobjects to be counted are large relative to section thick-ness.</p><p>In order to arrive at an accurate estimate of the numberof objects in section 2 of Figure 1, one can, in theory,identify each object by a single, dimensionless point. Thefigure shows Xs that mark an arbitrarily defined centerfor each object, and shows small dots, which define the topof each object. Either will serve the present theoreticaldiscussion although we will see that the tops are far moreuseful in practice. If one counts all of the objects for whichthe chosen point is included within the section (includingpoints that lie on one surface of the section and excludingpoints that lie on the opposite surface), one arrives at anestimate that needs no corrections. However, identifying asingle point that identifies the center of the objects is notusually possible, and it was not until quite recently thatthe top of an object was recognized as a useful practicalmarker, giving each object a single identifier. The use ofthe disector addresses this same issue by using two adja-cent sections (e.g., sections 1 and 2 of Figure 1). All of theobjects visible in section 2 are counted first, and then onthe serially corresponding part of section 1 those objectsthat also appear in section 2 are identified and counted.This count is then subtracted from the total of objectscounted on section 2. If the sections are not adjacent, thenthe distance between them must be less than the height ofthe smallest object being counted. In this way, each objectis only counted for one appearance in a section and theovercounting that characterizes profile counts is avoided.This, the physical disector, is a powerful method and isone that is to be highly recommended. However, it is notalways possible to use it in archival material or in studieswhere more than one method has to be applied to sections,that is, where closely spaced sections may not be avail-able. Further, the process of matching the sections can bedifficult and labor-intensive. For these reasons the opti-cal disector or some variant of this has been introducedand has great appeal (Williams and Rakic, 1988; Bjugn</p><p>and Gundersen, 1993; West, 1999). It is technically verymuch simpler to carry out because instead of using twophysically separate sections one can use two optical sec-tions passing through a single physical section, so that theproblem of matching sections is removed. One can use theoptical disector in exactly the same way as the physicaldisector, or one can simplify the process even further andcount the tops of objects as they come into focus as this ismoved from one optical section plane to the next. Thesemethods have been fully described in the cited references,which should be consulted for more details. They can bothbe used to produce reliable counts, but especially for theoptical disector the reliability of the counts will generallydepend on measurements in the z axis (see Measure-ments Along the z Axis).</p><p>Comparing Two Sections of UnequalThickness</p><p>Abercrombie (1946) suggested that one could obtain pro-file counts that needed no correction by cutting alternatethin and thick sections, counting the profiles in each andthen subtracting the lower counts from the higher countsto obtain a count of the number of profiles in a (notional)section whose thickness equals the difference of the twosection thicknesses. In principle this is very like the dis-ector, but it deals with means of counts and needs nophysical matching of adjacent sections. Williams et al.(2001) have recently written in more detail about thismethod. However, the accuracy of the method dependscritically on reliable information about the thickness ofthe sections used for the counts and the use of frozensections recommended by Williams et al., may lead todifficulties. Measuring the thickness of the frozen sectionsactually used for the counts is not to be recommended,since the thickness is significantly reduced when the sec-tions are dehydrated and cleared, and because measure-ments in the z axis are likely to have an error too large foran accurate comparison (see Measurements Along the zAxis). One needs a series of sections th...</p></li></ul>