Quadratic form: a robust metric for quantitative comparison of flow cytometric histograms

Comparison of fluorescence distributions is a fundamental part of the analysis of flow cytometric data. This approach is applied to detect differences between control and test sample and thus analyze a biological response. Comparison of standard test samples over time provides an estimate of instrument stability for quality control. However, application of statistical methods of distribution comparison in flow cytometry is difficult owing to instrument noise and the complex shape of intensity distributions. We applied quadratic form (QF) as a mathematical metric for comparison of flow cytometry histograms. QF operates on histograms as vectors and calculates the total distance in an interbin manner using a matrix of distances between single histogram bins. Euclidean interbin distance and histograms normalized to unity were used. Critical values corresponding to 95% significance level were calculated using Monte-Carlo simulation and single-maximum Gaussian distributions populated with several numbers of events. The QF statistic was then validated for non-Gaussian single-maximum distributions and multiple-maxima distributions. We determined that the critical values for Gaussian distributions depended on standard deviations and number of events in the compared histograms. A simple empirical function was constructed to characterize this dependence. Furthermore, it was verified that critical values (corresponding to 95% significance) for non-Gaussian histograms were similar to values for the Gaussian histograms characterized by the same standard deviation. We applied the QF statistic to estimate the differences between histograms of DNA content (ploidy) in cells of old and young leaf tissue of Brassica campestris. Furthermore, we quantified differences in fluorescence intensity in immunostaining of human lymphocytes. Quadratic form (QF) provides a true (mathematical) metric for estimation of distance between flow cytometry histograms of arbitrary shape. QF can be applied as a statistical test for estimation of significance of the distance measure. The respective critical values depend only on the number of events and standard deviations of compared histograms and are not affected by distribution shape. Therefore, applications of QF do not require assumptions concerning distribution shape and can be easily implemented in practice. This notion was confirmed using empirical distributions of DNA content in plant tissue and distributions of immunofluorescence in human cells.