Partial Correlation Coefficients Approximate the Real IntrasubjectCorrelation Pattern in the Analysis of Interregional Relations of CerebralMetabolic Activity Barry Horwitz and Stanley I. Rapoport Laboratory ofNeurosciences, National Institute on Aging, National Institutes of Health, Bethesda, Maryland Correlation coefficients between pairs of regional glucose metabolic rates have been used to assess patterns of functional associations among brain regions in humans and animals. Partialcorrelationcoefficients(partialingout the globalmetabolicrate)or correlationsbetween reference ratios (regional to global metabolic rate) have been used to remove the distorting Influence of systematic intersubject differences in glucose utilization. Suggesting that partial correlationsmaynot preservethe true (but unknown)intrasubjectcorrelationstructurewithin the brain, others have used a theoretical example to show how an artifaCtUallylarge negative partial correlation coefficient might arise. Here, we show that such an example is highly unlikely when a;@@ to resting cerebral metabolism, and can be identified in expedmental data by testing for a bimodal distribution of partial correlation values. We then show that partial correlations or reference ratio correlations of experimental resting metabolic rates give values which closely approximate the intrasubject correlation coefficients. J Nuci Med 29392—399,1988 ecause the brain is organized into frequently over rCMRglc between pairs ofregions, a correlation matrix lapping neuroanatomic and functional systems (2), sev is obtained that can display the pattern of functional eral efforts have been mounted to determine func associations for the entire brain (6-9). Different pat tional interactions among brain regions in specific terns have been found between healthy young and groups ofsubjects under given experimental conditions. elderly men at rest (13), between demented patients Generally, the functional interactions have been eva! and healthy controls (14,15), between subjects given uated by correlating, across subjects, regional values of pain stimulation and controls (8), and between split a measured quantity related to local neurons! activity. brain and sham-operated rats at rest (16). Early efforts used macroscopic electrical measurements A complication of the correlation matrix method in ofbrain activity (e.g., obtained from electroencephalo metabolic studies is that intersubject differences in graphic or evoked potential studies) in the correlation rCMRglc often are larger than intrasubject differences. analysis (3-5). More recently, we and others correlated This causes essentially all pair-wise correlation coeffi regional cerebral metabolic rates for glucose (rCMRglc) cients to have large values (6). Two procedures have (6-8), obtainedin humansby positronemissionto been used to overcome this difficulty. One uses corre mography (PET), and in animals by quantitative auto lations between reference ratios (the ratio of rCMRglc radiography (9). to some measure of global cerebral metabolism) (7,8). Changes in glucose uptake have been shown in hu The other uses partial correlation coefficients, in which man and animal studies to be related to changes in the controlled parameter is some measure of global cerebral functional activity (10-12). By correlating cerebral metabolism (which we take to be the hemi spheric average cerebral metabolic rate for glucose, CMRg1c) (6,9). The matrix of correlations is then put ReceivedFeb.17,1987;revisionacceptedAug.10,1987. into a form to be examined by retaining only those For reprintscontact Barry Horwitz, MD, Lab. Neurosci., correlations that meet a specified criterion of statistical NIA/N!H,Bldg.10,P.m.6C-103,Bethesda,MD20892. acceptability. For the data sets treated by our laboratory 392 HorwitzandRapoport TheJournalof NudearMedicine @ @rCMRglc + T@X@ (6,9,13,14),inlation there is little difference between the corre vided into 59 regionsof interest (ROl), which are listed matrices obtained using either the reference ratio 1 (seealso Fig. 1)of Horwitzet al. (6), and rCMRglc or partial correlation procedure (Horwitz and Soncrant: foreach regionin the 60 subjectswasdetermined.A measure unpublished observations). of global metabolism, the hemispheric average of glucose Recently, the use of partial correlation coefficients utthZ@tIon,CMRglc,alsowasdetermined.For the 60 subjects, CMR@lcrangedbetween 1.97and 7.87 mg/100 gJmin, with has been criticized by Ford (1). Ford's major objection g/min.on . a mean (±s.d.)of4.88 (±1.10)mg/tOO was that partial correlation coefficients may not reflect,Table correlationpattern.occasion, the underlying intrasubject anRESULTSartifactual,He used a theoretical example to show how statistically significant, negative partial cor Coefficientsbecauserelation coefficient might arise, and concluded that,Objection to Using Partial Correlation ofthe artifacts can occur, “anyresults produced usingWe first summarize Ford's critique (1) of the use partial correlation approach should be treated withpartial correlation coefficients. The following will cx extremepartialof caution―(1). He did not address the likelihoodplain how an artifactual statistically significant andratiosthis event. Ford also suggested that using referencecorrelation coefficient arises in Ford's example, brain.aticto obtain a correlation matrix may be as problem why this situation is unlikely to occur in a real isInas using the partial correlation method.The statistical model for the data used by Ford (1) this paper, we show that Ford's objection to the @correlations “ —@‘÷Pr ÷ 1 use of partial correlation coefficients (or to the use ofx sthresting of reference ratios) does not apply to realwhere X@denotes the measured rCMRglc in the @ton brain metabolic rates. We show that a correla ROI of subject r(r = 1, 2, . , R; s = 1, 2, . , S), theinmatrix, obtained using actual values of rCMRglcrepresents the mean metabolic rate in region s, p is randomapproximately20 healthy men whose global metabolic rates are“subject effect―for the rth subject, and @isthe metabolicpartialedequal (so that CMRglc need not becomponent representing the variability in subject.correlationout), has the same structure as the partialrate within a jton matrix (and, as the reference ratio correla The intersubject correlation between regions i and bymen,matrix) derived from measurements in 60 healthyis given ofwhich the 20 arc a subset. We also explain why @. an artifactual negative partial correlation coefficient p,@= ,-2-- , (2) ‘4'sents arises in Ford's example, show that his example repre a highly unlikely biologic condition, and demon where o@is the ijth matrix element of the S x S inter strate how this condition, if present, can be identified.subject covariance matrix.(1)—For the model of Eq. 2_@ aMF@HODSwhere eli — regionsValues r@is the intrasubject covariance between for rCMRglcusedin this studywereobtainedfrom j, and r@is the (unknown) between-subject van callsexaminationPET scans done as part of the Laboratory of Neurosciences'and ance (1). The quantity of interest, what Ford (1) bymale of aging in healthy men (17-19). SiXtyhealthythe intrasubject correlation, is given ±s.d.:volunteersbetweenthe agesof 20 and 83 yr (mean 48 ±18 yr) were scanned with an ECAT II positron. L@ _ ______ (4)emission Ujj inthe tomograph (Ortec; Life Sciences, Oak Ridge, TN) ‘TjjTjj medium resolutionmode (fullwidth at ha1f@maximum= . assumption of the correlation approach is that r@ 1.7 cm in the image and axial planes) using 2-['8flfiuoro-2- @ deoxy-D-glucose (FDG). Criteria for subject inclusion in the a measure ofthe functional coupling between regions @ studygivesfilledhave been detailedelsewhere(17-19). All subjectsful and j. Combimng Eqs. (2)—(4) stringent medical, neurologic, and psychiatric criteria on + r• @ examination.Data from 40 ofthese subjectswereused in the Pu , 2 2 (5) correlation matrix analysis ofHorwitz et a!. (6).a2 (m&/100 g/min) was measured using the opera If there were no between-subject variance (i.e., r@= 0), tionalsubjectslayequation and constants ofHuang et a!. (20). All the correlation coefficient of the absolute metabolic eyescoveredsupine but awake, in a dark, quiet room, with their values would represent the intrasubject correlation (i.e., 45-mmtimeand their ears plugged with cotton during the Pu On).For the additive model of the data, given by sevenserialinterval following the injection of['8FJFDG. Up to mmabovePET slices, 1.4 cm apart, were obtained 5 to 100 Eq. (1), the source of the between-subject variance is and parallel to the externallydefined inferior orbito the term Pr, the “subjecteffect―.When the between @ meats.!line.Details(IOM) subject variance is large (i.e., re), as it is for real @ beenpresentedofhow the PET data were analyzed also have metabolic data, p@ 1 (i.e., all correlations between elsewhere (6,13,1 7,18). In brief, the brain was di absolute metabolic values are large). Volume29 •Number3 •March1988 393 @ @2 0 UI 0 U- Cl) E £ c=5 n=8 z .@ 4 A c=1O n=4 I.;. a c=20 n=2 N . line of identity UI n=no. of clusters z c=no. of regions/cluster C/) U- FISHER Z-TRANSFORMED 011 FIGURE1 Graph comparing Fisher Z-transformed values of the intrasubject and partial correlation coefficients for the theoretical exampleof Ford(1). Thelineof identityhasa slopeof 1 anda y-interceptequalto 0. They-interceptandslopefor the best-fitregressionlines,respectively,arethe following:for c = 5, —0.13and 1.05;for c = 10,—0.30and 1.11; for c = 20,