A Simple Formula to Accurately Subtract Positivity of Cytometric Negative Controls


Alex Bäcker

Division of Biology, California Institute of Technology, Pasadena, California 91125, USA



Most flow cytometric measurements use a negative control to assess the percentage of positivity in the experimental sample that can be attributed to reasons other than the one which the experimenter wants to quantify. Traditionally, the percentage of positivity in the negative control has been subtracted from that in the experimental sample to yield the desired percentage positive. Previous techniques of histogram subtraction (1,2) for flow cytometric data subtracted the percentage of positive cells in the negative control from the percentage of positive cells in the experimental sample. We show here that this procedure is a priori flawed, and provide a simple formula that yields the true experimental positivity if the positivity in the negative control is statistically independent from the true positivity assayed for in the experimental sample.


Key terms: Flow cytometry, percentage of positive cells, histogram subtraction.




In any flow cytometric experiment, experimental positive cells (EP) can be divided into two classes: true positives (TP) and false positives (FP). True positives are defined as those cells which meet the criterion for positivity due to the reason being assayed in the experiment, i.e. that which is exclusive to the experimental sample with respect to the negative control. False positives are all other cells which exhibit positivity in the sample. For example, in an antibody assay for CD-24 positive cells, true positives are those cells which bind the antibody and express the CD-24 epitope, while false positives might include autofluorescent cells, nonspecific binding of the second (fluorescent) antibody, etc. The aim of the flow cytometric measurement is to determine the percentage of true positives, but the observed variable is EP. Because of this, most experiments include a negative control which differs from the experimental sample only in the determinant of true positivity, e.g. a sample of the cells with the second (staining) antibody but no first antibody (that which binds the molecule being assayed for). Previous procedures subtract the percentage of positive cells in the negative control (NCP) from EP in order to obtain a value for the true positivity, TP. The goal of this paper is to show that such simple subtraction does not yield the desired TP, and to provide a formula to calculate TP under the simplest and a priori most plausible hypothesis.


The key idea in this paper is that the causes of TP and NCP need not be mutually exclusive conditions. For example, returning to our previous example, a cell can both express CD-24 (and thus be stained by CD-24 antibody) and be autofluorescent or bind second antibody nonspecifically. To calculate TP, one must subtract from EP the fraction of EP that is not TP. Traditionally, this fraction has been assumed to be equal to the percentage of positive cells in the negative control. This assumption, however, is tantamount to assuming that TP and NCP are mutually exclusive, and in truth NCP provides an estimate of the percentage of cells in the sample that are positive for nonspecific reasons.


When one realizes that TP and NCP are not mutually exclusive, the question lies in finding the fraction of NCP that is not also TP. In general, this depends on the specific causes of TP and of NCP in each experiment. For any given experiment, however, the most reasonable a priori hypothesis in the absence of evidence to the contrary is that the causes of TP and of NCP are independent.


If the causes of TP and of NCP are independent, the percentage of positive cells in the negative control which are also true positive is the same as the percentage of true positive cells in the experimental sample:




The percentage of positives in the negative control that we wish to discount from EP is then




so that the true positivity is given by




where TP, EP and NCP are expressed as percentages. Solving for TP:




Note that this formula applies regardless of which of the thresholding techniques in the literature (1-2) is used to calculate the difference EP-NCP.


The magnitude of the 100%-NCP correction increases with the positivity of the negative control. In the limit, when there are no positive cells in the negative control, the corrective divisor equals one and there is no correction. But for 15% of positive cells in the negative control, for example, the formula without the correction is off by 18%.


To our knowledge, there is no case in the literature in which the correlation between true positivity and false positivity is known. However, were this known, then the independence assumption can be substituted by the known correlation, replacing expression (1) with the more general




where p(TP|NCP) is the probability that a cell that yields positive in the negative control is a true positive. In this case, then, the more general expression for (4) is given by:






The author thanks Leonardo Fainboim, Adrián Morelli and the rest of the Hospital de Clínicas Immunogenetics Lab for their confidence in an 18-year-old and for countless lessons. Supported by the Consejo Nacional de Ciencia y Técnica de la República Argentina (Conicet). This work is dedicated to the memory of the late Leonardo Satz and Bibiana Achino, whose spirit will animate the life of this and many scientists for years to come.




1.        Overton WR: Modified Histogram Subtraction Technique for Analysis of Flow Cytometry Data. Cytometry 9:619-626, 1988.

2.        Lampariello F: Evaluation of the number of positive cells from flow cytometric immunoassays by mathematical modeling of cellular autofluorescence. Cytometry 15:294-301, 1994.

Figure 1: EP = % positive in experimental sample. NCP = % positive in negative control. TP = desired value of % true positive.


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