Interpretation of the Correlation Coefficient A Basic Review RICHARD TAYLOR EDD RDCS A - PDF document

35Interpretation of the CorrelationCoefficient: A Basic ReviewRICHARD TAYLOR, EDD, RDCSA basic the evaluation of professionalmedical able to understand thestatistical analysis presented. One of morereported statistical methods involvescorrelation analysis where a correlation is 36Correlation analysis most widelyused and reported methods izing and data. In thisanalysis willbe reviewed with emphasis thecorrelationto actualcalculation of this statistical value.It is to determine relationshipexists between different If so, howsignificant or how strong is this the two variables? For example, relationship between the asonographer and scores achieved on the registryexamination? The correlation ris statistic used to measure the degreeor strength of this type Asthis reported in the The following examples serve to illustrate thispoint: 0.94 was notedDoppler-derived transaortic gradientand transaorticgradient in the 30 adult patients with�" ID="I2.25.2"stenosis."3 0.92)hemody-namically mitral statistically significant.4 statistical was used todetermine of betweenclinical was derived echocardiography) compared withinvasively derived data (catheterization) to evaluatevalvular feasibility of strongcorrelation between sets orvariables should be obvious. Intuition and empiricalobservation may indicate that arebut in its most basic sense themeasures towhich related. 5The correlation is tor coefficient.6The correlation r value a a direction of either take on a range from -1 to towhere are absolute and with no units correlationzero no associationthe measured variables. The closerthe approaches of associationmore linear relationship between theof values the by specificequation two variables. The the direction or the sign. Thus,0.90 and r = -0.90 are the degreeof association of the measured variables. A positivecorrelation an first correspond variable, thus implying a directrelationship between the variables. A negativean inverse one variable increases, the second variabledecreases.A graph can the conceptof correlation and to relationshipwhich the variables. For xand and plotted on a graph. Figure 1illustrates some different sets and how theyare summarized by a correlation or coefficient. that situations,y would two beinganalyzed such school GPAversus SAT scores, or blood serumcholesterol disease to determinethe degree la illustrates a perfect positive correlationof r be noted that all data observations 37on a term linear correlationbecomes The correlation inbecause x increases, the same direction. The relationship variables in Figure 1d correlation, negative in direction All data lie on as variable x increases, decreases.An example of a negative correlation might number of to of body fat whereas with body fatpercentage increases, a decrease the ofpull-ups is observed. real are alwaysrandom in aperfect linear coefficient no longer perfect,the correlation remains the datacloser straight line as the datamore from the straight line (Fig.1e).If there is no linear relationship thevirtually the data pointson the corresponding graph scattered and approximate a (Fig. ib). isthat a nonzero no correlationactually exists.5 Also, good to high correlations existeven though less than a perfect of thosecorrelation coefficients other than 1.0?Like any statistical value, the correlationis of importance unless beproperly interpreted. Like all theto interpret.Labeling systems exist values where correlation coefficients (in absolutevalue) which are <_ generally consideredlow or weak to 0.68 to 1.0strong or high correlations with 0.90 very However, describing correlation r = 0.55 asa moderate correlation basic needs to relates of thecorrelation the random ofa r in realcorrelation exists. exist whichdefine what observed beforethe correlation to be statistically for correlation statisticalsignificance vary as sample size used andthe level assumed thatcorrelation of r = previously 35 interval-scaled pairedobservations which were both x and y variables are normally distributed,then that correlation = is significantly zero. words, the relationship existing betweenthese statistically significant. However,to add of correlationinterpretations, is not important one. A statistically coefficient merely the observedprovides ample null that the population parameter (rho) is zero therebyconcluding that the population correlationcoefficient equal to example, givena large sample size (n > 100), a correlationsmall be from a = degree oflinear correlation would have importance as we shall see be that the and given a that the higher the relationship. Although correlationbest known and subject tostatistical the coefficient ofdetermination is more meaningful.8 The coefficientof determination can fully interpretby simply squaring correlationr. coefficient of determination defined of the values variable (y) be"explained" value theindependent technique in a percent makes easier interpret more if a correlation coefficient of 0.20 was observedvariable x variable y, then thecoefficient determination is = 0.04. This meansonly of variation ycan be accounted by invariable x. Therefore, even though = statistically significant 0.05 with in the example noted above, be only 4% variation of 38can be by x. determination technique conservativemeasure relationship twomany statisticians, butreported analyses.true of proce-dures, correlation analysis is limitationsand misinterpretations that can be serious. Inaddition to the limitations mustunderstood although measures closely the approximate a straightit not the strength ofa nonlinear or between variables may also exist.Browner wrote believe a research theP value ularly low prior probability makes aparticular association "unsuspected."They noted finding of P valuedealing with a correlation between and pancreatic cancer .05) not the truth of the research hypothesis; subsequentthe problems such lab bias, or research could reliable conclusions.One of the most frequent and serious misusesof correlation analysis is to interpret a highcorrelation between variables cause-and-effectanalysis relationship or theor its abetween a sizeand handwriting be presump-tuous a riting.6 Statistics do but they sometimesfalse conclusions. Caution exercised this pitfall. might indicate that 99.9% of who diedof cancer drank some water within The but beeasy to into however ridiculous, here.correlation purpose is the closeness of the linear relationship between thedefined variables. The correlation coefficientclosely the data fit a Generally, correlation analysis furtherdefining the theexisting relationship. This procedure asregression mathematical equation isdeveloped for the line of best fit data. From this regression equation, predictionbecomes can bepredicted based on a value other variable.Predicting unknown values (dependent given values (independent variable) and widely used in wellbusiness education.For example, reportedbetween Doppler pressure half-for mitral valve area relativecatheterization good (r =0.85, 84x + 0.17).10 Upon interpreting thiscorrelation statistically significant0.85, P existsbetween evaluation areaand mitral rfall general category labelof would yield (r2) or 72%, meaning mitral valve area measurements by Doppler pressure we have associationof usefulness. pattern ofDoppler comparedwith catheter measurements for assessment by the regression = 0.84x + 0.17 where x = Doppler pressure half-time mitral and y = catheterization-derivedif determine pressure mitral area 1.2cm2, then the catheterization-derived area (y) predicted to be 1.18 cm2 [0.84 Correlation closer the data "fits" the correlation predic-to potential be noted that although correlationanalysis often routinely includes regressionanalysis in the "package," on either correlation coefficients or regressionequations independently.the scope of exist correlation depending the samples. Computer programs areavailable to routinely perform this task. Regardlessof or formula used, the interpretationof the correlation coefficient is basically sameconsumer. 39CONCLUSIONThe interpret research professional literature becomes withoutstatistics. This articleshould shed some light onto statisticalprocedure known as purposeof statistical a the correlation is a summaryvalue of a large the degreeassociation between two measuredvariables. to reduce the largeamounts of data down to a manageable form forsonographers to review. For sonographers must understand the statisticalcoefficient represents and what itThe author wishes to SusanChafin, Betty and Bradytheir gracious assistance in manuscript.REFERENCESN, R, Vered Analysis of in evaluation valves in the mitral and aorticpositions. J Am Soc Echo, 1988;1:211-225.referesher II: of size. JDMS 1989;4:176-183.Stamm Martin RP: gradients across stenotic valves by Doppler J Am 1983;2:707-718.4. Martin Reliabilityreproducibility of of mitralAm J JC, Lamb DR: Statistics in St. Mosby Co, 1970, 222.JW: Sciences. Palo Publishing pp 158-169.Lind DA, Marchal WG: Statistics: AnHarcourt Jovanovich, VE, Real3. Publishing Co, 1983,P valuesThe analogy between and JAMA 1987;257:2459-2463.R, al: Comparativetwo-dimensional echocardiography andDoppler methods severity in mitral stenosis patients prior commissurotomy. Circulation 1986;73:100-107.