The correlation coefficient gauges linear association between two variables. However, interpreting its value depends on the question at hand. This article argues that relying on the correlation coefficient may be irrelevant for many neuroscience research tasks. When the experimental dataset is contextually suitable for binning-averaging, other indicators of statistical association could prove more suitable

Rusakov, Dmitri A

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UCL Discovery

Trends inTINS 1855 No. of Pages 3Neurosciences OPEN ACCESSForumA misadventure of thecorrelation coefficientDmitri A. Rusakov 1,*The correlation coefficient gaugeslinear association between twovariables. However, interpreting itsvalue depends on the question athand. This article argues that rely-ing on the correlation coefficientmay be irrelevant for many neuro-science research tasks. When theexperimental dataset is contextu-ally suitable for binning-averaging,other indicators of statistical asso-ciation could prove more suitable.Interpreting Pearson’s RThe commonly used Pearson’s correlationcoefficient, Pearson’s R, evaluates linearassociation between two experimentalvariables, say X and Y. It has been widelyaccepted, at least across biological sci-ences, including neurobiology, that R > 0.7indicates a strong linear relationship be-tween X and Y, R < 0.3 weak, and the restis moderate. This rule of thumb normallyreflects the aim to predict the value of Yfrom measuring X, based on the X–Y rela-tionship established for the population.The prediction reliability aspect of Pearson’sR follows from the fact that the R2 value,termed the coefficient of determination,stands for the proportion of the variationin Y that could be predicted from X. Inessence, the correlation coefficient indi-cates how closely the data fit a linear pattern[1,2], which may explain practical usage ofthe foregoing traditional rule of thumb, par-ticularly in clinical studies where predictionreliability is of importance.However, in many documented casesthese cut-off points appear arbitrary and in-consistent [2,3]. The correlation coefficientis of little importance unless it can beproperly interpreted, which has beenconsidered a difficult task for all scale values[1]. In fact, it has long been emphasised thatthe interpretation and usability of R dependon the specific question under study [2,4].The use of Rmay be of particular relevanceif the question falls into one of the followingthree categories: (i) How accurate is the pre-diction? (ii) What is the magnitude of error ina reliability task? (iii) What is the strength ofthe X–Y relationship? [4]. The present articlewill argue that, in many instances, neuro-science research tasks that involve thelinear X–Y relationship fall outside thisscope, so that referring to R appears irrele-vant to the posed question.Cumulative effects arising due to apopulational trendIn many experimental quests, predictingthe Y value from measuring X for individualobjects is not among the relevant, or evenfeasible, objectives. Such cases arise,for instance, when one or both variablesshow high inherent (biological) variabilityand/or when there has been a large mea-surement error. Multiple examples for thiscan be found in neurophysiology. In cellularneurophysiology, for example, a commonobjective is to understand whether cellbehaviour is affected by some cumulativepopulational feature arising acrossthousands of synaptic connections. TheX–Y datasets describing individual synap-ses normally display significant scatters.Such datasets may include, for instance,dendritic spine density against dendritediameter [5]; local numbers of synapticAMPA receptors against distance from thesoma [6]; quantal content against a short-term plasticity parameter for glutamaterelease [7]; and so on. Here, the X–Y rela-tionship among synapses varies toostrongly to reliably predict Y from X for anyindividual synapse, which is duly reflectedin low R values. It can be argued, however,that the focus of these studies is beyondthe scope of questions relevant to R: here,given the unlimited numbers of realisationsor events (synaptic inputs times synapticdischarges), relatively weak (but highlysignificant) positive correlations can have aclear cumulative effect, which is not dulycaptured via the analyses of R. For instance,a weak correlation between synaptic releaseprobability and the distance from thesynapse to the neuronal soma can result ina highly significant shift in synaptic signalsummation leading to cell spiking [8].In this context, the question is therefore notabout the strength of the X–Y relationshipper se but whether the X–Y relationship isstatistically significant (i.e., whether it tran-spires over virtually unlimited sampling).Here, the magnitude of the physiologicaleffect will depend not so much on theadherence of the X–Y scatter to the straightline, but rather on how steep the X–Ydependence is. A conventional estimatorof this steepness is linear regression,where the regression coefficient (algebraicslope of the best-fit straight line) describesthe expected magnitude of change in Yfor a one-unit change in X. In such cases,referring to the R value as an indicator ofthe effect strength could be highly am-biguous, as illustrated later.Data binning-averaging increasesR for the same datasetThe X–Y data scatter shown in Figure 1Ais produced by generating a linear depen-dence between uniformly distributed Xvalues over the 0–1 interval and Y valuesthat follow theGaussian distribution centredaround the slope line (coefficient 0.5). Thisdata scatter is illustrative of the experimentaldatasets mentioned earlier [5–7]: it reflectshigh data variability, low R value, yet highlysignificant regression (Figure 1A; β, linearregression coefficient). However, if we binthis dataset, by averaging Y values overthe regular 0.1-long X intervals, we arrive atthe reduced data scatter that shows an ex-cellent linear relationship, with the stronglyincreased R but virtually unchanged β(Figure 1B). As an experimental examplefrom work by the author and colleagues, aTrends in Neurosciences, Month 2022, Vol. xx, No. xx 1(A) (B)Raw data Binned data0.0 0.2 0.4 0.6 0.8 1.0-2-1012elbairaVYVariable X0.0 0.2 0.4 0.6 0.8 1.0-2-1012elbairaVYVariable X(C) (D)'Raw data' Binned x 2 Binned x 3R = 0.415p < 0.001R = 0.766p < 0.001R = 0.825p < 0.001Channel 1Channel 2'True signal'R = 0.894p < 0.001Trends in NeurosciencesR = 0.213 β = 0.552 p < 0.001  R = 0.903 β = 0.521 p < 0.001 Figure 1. Binning raw data in the X–Y dataset elevates the correlation coefficient. Hypothetical datasets were computer generated to illustrate the effect ofbinning on Pearson’s correlation coefficient. (A) A Monte Carlo generated data set (n = 300 data points; OriginPro random number generator), with X values distributeduniformly randomly over [0; 1] interval and Y values following a Gaussian distribution centred at the slope line Y = βX (β = 0.5), with standard deviation σ = 0.7;unbroken line, best-fit linear regression; R, Pearson’s correlation coefficient; P, confidence level to reject the null-hypothesis ‘zero-slope of linear regression’. (B) Thedataset as in (A), but binned with Y values averaged over 0.1 X intervals; X values shown at bin centres; other notations as in (A). (C) An example of Pearson’s R basedimage colocalisation analysis: computer generated brightness signals in a pair of image channels (green and magenta; Gaussian pixel noise added; ImageJ Fiji Process/Noise). The size of image pixels (binning) increases from left to right column as indicated. The Pearson’s R values for the interchannel pixel brightness colocalisation, andthe statistical significance of the respective linear regression slope (H0-hypothesis rejection level p; ImageJ Fiji Colocalization/Coloc 2), are shown for the respective imagepairs. (D) ‘True signal’: images as in (C) 'Raw data' panel but with all noise removed; notations as in (C).Trends in NeurosciencesOPEN ACCESS2 Trends in Neurosciences, Month 2022, Vol. xx, No. xxTrends in NeurosciencesOPEN ACCESSsimilar increase in R by data averaging canbe shown in a dataset reporting releaseprobability values versus distance from thesynapse to the soma [8]. In the latter case,averaging (binning) synaptic features over adendritic segment length reflects a realisticscenario when the bulk of synapses in thatsegment are activated [9]. Thus, essentiallythe same data sample could generatedifferent R values, as long as the bio-logical question under study permits databinning-averaging.Image binning and Pearson’scorrelationPearson’s R has also been used to evaluatecolocalisation or correlation between dis-tributed signals recorded in two separateimaging channels, either static or dynamic(e.g., [10,11]). Digital images consist ofpixels or voxels, the ‘elementary’ squareor cubic areas over which the intensity ofnative signal sources (diffraction-limitedlight in case of light microscopy) is averaged.Thus, setting the pixel or voxel size bythe imaging system is equivalent to theaveraging-binning of raw data, as dis-cussed earlier. It is no surprise, therefore,that Pearson’s R calculated to assess signalcolocalisation between two images or imagechannels may depend on the pixel size [12].The simulated example in Figure 1C illus-trates a colocalisation test for a pair ofimage channels, each with two brightnessspots representing the signal of interest;Gaussian noise has been added to re-flect typical imaging conditions. Calcu-lating Pearson’s R for signal colocalisationshows that its value increases stronglywith image binning (increasing pixel size;Figure 1C). This increase does not appearspurious because the ‘true signal’ (noiseremoved) corresponds to the highest R(Figure 1D). At the same time, the linear cor-relation slope remains significantly abovezero throughout (P < 0.001), suggestingthat colocalisation is statistically significant.However, quantifying the strength of signalcolocalisation in such cases is a complex,context-dependent task, which is outsidethe scope of this article. Here we simplyargue that relying on Pearson’s R may notbe the most robust approach for establish-ing signal correlation or colocalisation formultiplexed imaging.Concluding remarksThis article outlined some basic consider-ations around the use of Pearson’s R andargues against overinterpreting Pearson’sR as an unequivocal indicator of the‘strength’ or ‘weakness’ of the effect arisingfrom the linear dependence of two variables.In some experimental designs where thetask in question permits data binning oraveraging, as often encountered in neuro-science, a regression analysis should pro-vide more reliable insights. In this context, itmight be particularly ambiguous to rely onR in the case of experimental metrics withinherent binning-averaging, such as variablepixel/voxel values used in cellular or brainimaging.AcknowledgmentsThe study was supported by: Wellcome Trust(212251_Z_18_Z), Medical Research Council (UK) (MR/W019752/1), European Research Council (323113),and European Commission NEUROTWIN (857562).Declaration of interestsThe author declares no competing interests in relationto this work.1UCL Queen Square Institute of Neurology, University CollegeLondon, London, UK*Correspondence:d.rusakov@ucl.ac.uk (D.A. Rusakov).https://doi.org/10.1016/j.tins.2022.09.009© 2022 The Author(s). Published by Elsevier Ltd. This is an openaccess article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).References1. Taylor, R. (1990) Interpretation of the correlation-coefficient -a basic review. J. Diagn. Med. Sonogr. 6, 35–392. Schober, P. et al. (2018) Correlation coefficients: appropriateuse and interpretation. Anesth. Analg. 126, 1763–17683. Mukaka, M.M. (2012) Statistics corner: a guide to appro-priate use of correlation coefficient in medical research.Malawi Med. J. 24, 69–714. Monroe, W.S. and Stuit, D.B. (1933) The interpretation ofthe coefficient of correlation. J. Exp. Educ. 1, 186–2035. Bannister, N.J. and Larkman, A.U. (1995)Dendriticmorphologyof CA1 pyramidal neurones from the rat hippocampus: II. Spinedistributions. J. Comp. Neurol. 360, 161–1716. Andrasfalvy, B.K. and Magee, J.C. (2001) Distance-dependent increase in AMPA receptor number in thedendrites of adult hippocampal CA1 pyramidal neurons.J. Neurosci. 21, 9151–91597. Mendonca, P.R.F. et al. (2022) Asynchronous glutamaterelease is enhanced in low release efficacy synapses anddispersed across the active zone. Nat. Commun. 13, 34978. Jensen, T.P. et al. (2021) Release probability increases to-wards distal dendrites boosting high-frequency signaltransfer in the rodent hippocampus. eLife 10, 625889. Grillo, F.W. et al. (2018) A distance-dependent distribution ofpresynaptic boutons tunes frequency-dependent dendriticintegration. Neuron 99, 275–28210. Dunn, K.W. et al. (2011) A practical guide to evaluatingcolocalization in biological microscopy. Am. J. Physiol.Cell Physiol. 300, C723–C74211. Mann, K. et al. (2017) Whole-brain calcium imaging reveals anintrinsic functional network in Drosophila. Curr. Biol. 27, 238912. Heddleston, J.M. et al. (2021) A guide to accuratereporting in digital image acquisition - can anyone replicateyour microscopy data? J. Cell Sci. 134, jcs254144Trends in Neurosciences, Month 2022, Vol. xx, No. xx 3

A misadventure of the correlation coefficient

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A misadventure of the correlation coefficient

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