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PnCHO","IOLOGY

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Vol. 17. No. 2Pnntcd .. l: .S.A.

The Statistical Analysis of Heart Rate: A ReviewEmphasizing Infancy Data

JOHN E. RICHARDS

DefHl1'1'Mniof Psychology, University of California, Los Angeles

ABSTRACT

Heart rate is a dependent variable used widely in psychological and psychophysiologicalresearch. Several statistical problems arise in the analysis of heart rate data, many of them specificto inCancy research. The present paper discusses the problems of a statistically appropriate cardiacmeasure, the Law of Initial Values, the problem of differential variability in heart rate scores, andthe use oCmultivariate statistical methods in analyzing heart rate data. Special attention is given tothose problems and solutions which have potential application to the analysis of infant heart ratedata. A ftowchart is presented which may guide the researcher in the appropriate use oCthe severalstatistical t~hniques reviewed in this paper.

DESCRIPTORS: Heart rate, Heart rate statistical analysis, Infant heart rate, Statisticalanalys::;, Infancy. '

The statistical analysis of heart rate data is aproblem which has been discussed in relation toadult heart rate research (e.g., Benjamin, 1963,1967;Lacey, 1956;Wilson, 1967).Althoughmen-tioned in some reviews of infant hean rate ex-perimentation(e.g., Crowell, Blurton,Kobayashi,McFarland, & Yang, 1976;Graham & Jackson,1970), a systematic presentationof the range ofstatistical problems for infancy researchers does notexist. This paper summarizes some statistical dif-ficulties which may confront the developmentalpsychologist or developmental psychophysiologistinterested in heart rate research. The areas which arediscussed are the Law of Initial Values, an appro-priate cardiac measure, the problem of differentialvariability, t!me series analysis, and multivariatestatistical methods. Special attention is given to theapplication of these problems to potential experi-ments using infant subjects, although much of thediscussion might apply equally well to adult heartrate research.

The reader should be familiar wiL~the analysis of

The preparation of this paper was panially supported byNIMH Predoctoral Training Grant, 784040.31428-5.

Address requests for reprints to:' John E. Richards. Depart-ment of Psychology. University of California. Los Angeles, CA90024.

variance and covariance, linear regression, andknow of the extension of the analysis of variance to .the case of multiple dependent measures. Through-out the paper the variable X wilI refer to prestimulusheart rate measurements, Y to poststimulus heartrate, YI to poststimulus heart rate in.the ith mea-surement interval, and D to the difference betweenpre- and poststimulus heart rate. Raw hean ratescores are represented by capital letters, while meandeviation scores are represented by lowercase let-ters, e.g., x = X - Mx, where Mx is the mean of Xscores.

The use of a computer and packaged statisticalprograms is helpful (if not necessary) in the analysisof heart rate data. The author has found the SAS(Barr, Goodnight, Sull, & Helwig, 1976;Helwig &Council, 1979), BMD (Dixon, 1970; Dixon,P-Series, 1977), and the SPSS (Nie, Hull, Jenkins,Steinbrenner, & Bent, 1975) statistical packages tobe useful general purpose data analysis systems,and these will be refelTed to frequently as analysis

'techniques since they are so widely available.Ronald Wilson (Note 1) has prepared a set of com;'puter programs especially for hean rate data, andthe present author has compiled a set of computerprograms which perform many of the statisticaltechniques presented in this paper (Richards, Note2).

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154 JOHN E. RICHARDS Vol. /7, No.2It

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1Appropriate Cardiac Measure

Two cardiac scores are commonly used in heartrate research, based on different units of analysis.The most common is heart rate, which is defined asthe inverse of the heart period, usually measured bythe length of the R-R interval in the electrocardio-gram. Heart rate is measured in real-time units suchas seconds, and standardized in terms of beats-per-minute (bpm). The other cardiac score whichmay be used is heart period, defined as the length ofthe R-R interval, and it is measured in cardiac-timeunits (beat-by-beat). Graham and Jackson (1970)and Graham (in press) discuss the relative merits ofthe different units of analysis. They favor real-timeunits, mainly because they are easier to coordinatewith experimental events. Lewis, Bartels, andGoldberg (1967) and Lewis and Spaulding (1967)give some reasons for preferring the beat-by-beatunits of analysis, since they provide a sort of"biological clock" against which cardiac responsescan be evaluated. A method of making the cardiacscore the average of a certain number of extreme(high or low) beats in a certain interval (e.g.,Campos, Langer, & Krowitz, 1970; Kagan, 1971;Kagan & Lewis, 1965; Kinney & Kagan, 1976;Lewis, Kagan, Campbell, & Kalafat, 1966;McCall& Kagan, 1967) is not a good unit of analysis,though popular in infancy research, becausechanges in variance of the poststimulus period favorfinding, results in the hypothesized direction whensuCheffects are not present in the data (Graham &Jackson, 1970). ,

One consideration in the choice of a cardiacmeasure and units of analysis has been demon-strated by Graham (1978b). Real-time units shouldbe used with heart rate, and cardiac-time unitsshould be used with heart period. Otherwise, thearithmeti~ mean of the unit estimates is not equal tothe definitional estimate of the mean based on allunits. In the past it has been assumed that thechoice of time units was independent of the choiceof heart rate or heart period (Graham & Jackson,1970; Jennings, Stringfellow, & M. Graham, 1974;KhachatUrian, Kerr, Kruger, & Schachter, 1972).Graham (1978a) also correctly asserts that argu-ments by Khachaturian et al. (1972) that heartperiod is the' 'raw data' ~ and heart rate only atransfonnation of the raw data are incorrect. Theunits of analysis determine which is the raw data,being heart period for cardiac-time units and heartrate for real-time units.

Statistical considerations concerning the choiceof a carJiac score favor using heart rate as thecardiac s~ore for infancy research. Many parametricstatistical t~chniqu~s, including the analysis of var-iance, assume that the dependent variable is nor-mally distributed and that variances are homogene-

ous across treatment conditions. Although many ofthe statistical techniques are robust to violations ofthese assumptions, a dependent variable whichmeets these assumptions is preferable to one whichdoes not, methodological considerations beingequal. Jennings et al. (1974), Khachaturian et al.(1972), and Graham (1978a) have studied the distri-butions of heart period and heart rate for infant andadult subjects, and the latter two studies also lookedat the homogeneity of variance assumption. Thedata for adult subjects (Jennings et aI., 1974;Graham, 1978a) are inconsistent, perhaps due todifferences in obtaining the cardiac response. Forinfant subjects (KhachatUrian et al., 1972;Graham,1978a) it is clear that heart rate meets the assump-tions better than heart period. In both studies,homogeneity of variances was more likely to befound with heart rate than with heart period.Khachaturian et al. report that the degree of skew-ness (left-right asymmetry) was greater for heartrate than for heart period, but distributions were,more often skewed for heart period than for heartrate. Graham, reporting data over a wide range ofstudies with stimulated and unstimulated heart rilte;found that skewness occurred more often in' infantheart period than in infant heart rate. Kurto~is(excess or deficit in the distribution center) wasequal in the two measures. These data clearly,showthe statistical superiority of heart rate over heartperiod for the infant. Methodological conside~tions may lead the researcher to choose heart period,but in general heart rate should be the preferreqcardiac score. In the remaining sections of the pagerheart rate will almost exclusively be discussed,although heart period could be used in many of the '

statistical techniques as well.

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jAnalysis of Variance of Heart Rate Data

Sincethe reader of this paper is presumedto befamiliarwith the analysis of variance, only a few'comments will be included. Analysis of variance 1~'

the appropriate statistical technique for most ex- ,

perimental research, and heart rate research is not anexception. The dependent variable for the analysismay be heart rate or heart period, whichever theresearcher chooses to be apI'ropriate. Wilson (1967)has given some suggestions specific to the analysisof variance of heart rate data. and the reader isreferred there for further details. Wilson has alsoprepared several computer programs for the statisti-cal analysis of heart rate (Wilson. 1967. 1974. NoteI; Wilson & Scott. 1970). Benjamin (1967)providesstatistical rationale for using the analysis of varianceand covariance for heart rate data. For the generaltechnique of the analysis of variance. the interestedreader may consult one of several texts (Hays, 1973;Keppel, 1973; Winer, 1971).

HEART RATE STATISTICAL ANALYSIS

Law of Initial Values

One of the most prominent issues in the statisticalanalysis of heart rate data is the problem of theinitial value of the prestimulus heart rate duringexperimental situations. The Law of Initial Values(Wilder, 1950, 1956,1958), which applies to awiderange of physiological systems, states that the post-stimulus heart rate level is partially determined bythe prestimulus heart rate level. At high prestimuluslevels the heart rate acceleratory response is smallerthan it is at low prestimulus levels, and the heart ratedeceleratory response is larger than it is at lowprestimulus levels. Poststimulus heart rates, there-fore, contain variance due not only to the experi-mental manipulations but due also to the level of theprestimulus cardiac rate. The statistical problemhere is to remove the variance due to the lawfuldependence in the heart rate, making the post-.stimulus heart rate scores independent of the pre-stimulus cardiac level.

The Law of ~nitial Values is important for theinfancy researcher to consider. Heart rate levelvaries considerably over a range of behavioral stateconditions, and it"changes in the first few months oflife. Harper, Hoppenbrouwers, Sterman, McGinty,and Hodgman (1976) found an overalllineardecIinein heart rate over the first six months of the infant aswell as a cubic trend (Fig. 1). Fig. 1 also showsthat heart rate occurs at different levels in thefour behavioral states monitored. Harper, Hop-penbrouwers, Bennet, Hodgman, Sterman, andMcGinty (1977) found that feeding was followed byhigher heart rate in sleeping periods than in thosesleeping periods not preceded by feeding. It hasbeen suggested (Schmidt, Rose, & Bridger, 1974)that the different heart rates in behavioral conditionsmay be the cause of the observed relationship be-tween behavioral state and the cardiac response tostimulation (e.g., K. M. Berg, W. K. Berg, &

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Graham, 1971; Jackson, Kantowitz, & Graham,1971; Lewis et aI., 1967; Pomerleau-Malcuit &Clifton, 1973). Therefore, when different ages ordifferent behavioral states are used in an experi-ment, different heart rates are likely to be found.Additionally, the normal amount of variability as-sociated with heart rate implies that the cardiaclevels on different prestimulus trials will be dif-ferent.

At least three procedures can be followed to han-dIe the problem of the Law of Initial Values. Anexperimental procedure would be to withhold pres-entation of the experimental stimulus until theheart rate level is within a certain range. Of course,this is not possible for all experiments, and ignoresthe possibility of mean heart rate level differencesamong subjects or conditions. A second strategy tocontrol for significant mean heart rate differencesamong subjects or between conditions is the use of amean deviation cardiac score such as the autonomiclability score suggested by various researchers(Crowell et al., 1976; Lacey, 1956; Wilson, 1967),and discussed in a later section of this paper. How-ever,even with deviation scores, it is likely thatthere will be a relationship between pre- and post-stimulus scores since not all prestimulus deviation'scores will be identical. The analysis of covarianceis the third procedure that can be used to control foreffects due to the Law of Initial Values. It has thewidest acceptance by cardiac researchers as theanswer to the problem (Benjamin, 1963, 1967;Crowell et aI., 1976; Graham & Jackson, 1970;Wilson, 1967).

8111.

Analysis of Covariance

The analysis of covariance removes effects due tothe Law of Initial Values by a linear regressionadjustment of the poststimulus heart rate. Post-stimulus scores are regressed on prestimulus scores,giving the linear regression coefficient, by-x. Thelinear regression coefficient may be used to computescores, y, which are statistically independent of X.Inmanycases,actualYscores are not computed andused in statistical significance tests. For example, inthe analysis of variance, a covariance analysis

. wouldproceedby subtractinga term fromthe sumsof squares which represents variance accounted forby the sum of deviation crossproduct scores relativeto the deviation sum of squares of the prestimulusheart rate. Benjamin (1963, 1967) gives the statisti-cal rationale for using the analysis of covariancewith heart rate data, while one may consult Winer(1970, Hays (1973). Keppel (11)73).or Keriingerand Pcdhazur (1973) for the general method of theanalysis of covariance. The SAS, SPSS. and 8MDcomputer programs have analysis of covarianceroutines.

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JOHN E. RICHARDS Vol. 17, No. ,

The analysis of covariance just mentioned only. adjusts for linear relationshipsbetween pre- and

poststimulus heart rate scores. It is possible, how-ever, that other relationships exist among heart ratescores, which are curvilinear in nature. Graham andJackson (1970) state that in heart rate data there wi1lbe few cases in which a curvilinear regressionadjustment will be necessary, and in the cases inwhich one is done the increase in variance ac-counted for will be slight. The statistical signifi-cance of suspected curvilinear relationships shouldbe tested, and any important curvilinear tenns uSedto adjust poststimulus scores. One may do thisadjustment on the BMD computer program 05R(P-Series, P5R).

A special problem arises in the adjustment ofheart rate scores by the covariance methodology.Most researchers assume homogeneous regressioncoefficients and use a regression coefficient pooledacross all groups, including experimental groups,repeated trials within each experimental treatment,and each measurement made in multiple post-Stimulus intervals (beats or seconds). The assump-tion of homogeneity of linear regression coefficientsis necessary for the use of a pooled regressioncoefficient (Graham & Jackson, 1970; Lacey,1956). Generally this assumption is warranted,since experimental conditions should not contributeto the Law of Initial Values. However, in somecases when covariances are not equal across allgroups, this assumption may be violated. The as-sumption of homogeneity of regression coefficientsshould be tested before adjusting scores, or usingthe analysis of covariance. Details of the test can befound in Hays (1973), Keppel (1973), Winer (1971),Kerlinger and Pedhazur (1973), among others. Thisassumption is routinely tested in the analysis ofcovariance routine ofBMD (PIV and P2V). If thereis heterogeneity among regression coefficients oneshould adjust the Y values for each group by theregression coefficient for that group of scores. Fur-ther statistical analysis must be done separately foreach group. Differences in adjusted heart rate scores- will include treatment effects and effects due toheterogeneous regression coefficients.

An assumption of the analysis of covariance isthat the covariate is reliably measured. It may not bethe case that heart rate scores are a reliable measure,which would underestimate the appropriate valueof by.'x,and thus notcorrect fully for the effectof theinitial prestimulus level. If this underestimation isacceptable to the researcher. one may use theanalysis of covariance with the uncorrected pre-stimulus ~art rate. The value of the alternatives tothe uncorrected heart rate is uncertain. Forexample,measurements hased on several scores are generallymore reliable than measurements based on single

scores. This suggests that a mean value over severalprestimulus intervals should be a better covariatethan the first second preceding the stimulus. How-ever, if the sample of prestimulus scores is largeenough the value will approach the population meanand the covariance adjustment will not be sensitiveto deviations temporally contiguous to the stimuluspresentation. Secondly, if the reliability of the pre-stimulus score is known, one may adjust the by.xupward by an amount proportional to the reliability.However, it is not usually the case that the value of,the reliability of heart rate scores is known or can beestimated. A third approach which one could use tocorrect for this problem is to employ a time seriessmoothing function (Blackman & Tukey, 1959;Box& Jenkins, 1970;Fuller, 1976) which remOvessam-pling error from the scores, leaving the first pre-stimulusheartratevalueacceptable. .

Difference Scores

Difference scores are also subject to the Law ofInitial Values. Graham and Jackson (1970)pO~nt.outthat many researchers use difference scores becauseit is felt that the correlation between the prestimulusscore and the difference score is lower tli4n thecorrelation between the pre- and poststimulusscores. Difference scores may also be used becausethe interpretation of heart dite change is of primaryconcern. Since, D = Y - X, a small derivatiori will

show that, .: )~:..~;"(1)

If the linear regression coefficient, by.x, representsa significant relationship between Y and X; .theregression coefficient, bD.x, will also be of Somevalue. Only when by.x = 1.0 is the dependencybetween D and X nonexistent and adjustment ofdifference scores unnecessary. The adjustment ofdifference scores, like the adjustment of actual Yscores. can be accomplished by adjusting the actualD scores or by adjusting the error and effect terms-inthe analysis of variance. Similarly, the problems ofcurvilinear regressions, homogeneity of regressioncoefficients, and reliability of the measures apply todifference scores.

When Should the Adjustment Be Made?

One final question in the statistical adjustment ofthe effect of the Law of Initial Values is when theadjustment is appropriate. Graham and Jackson(1970) state that when the Law of Initial Values isaffecting the data. by, will be a positive numberless than 1.0. This is bascd on the fact that the.Lawof Initial Values predil:ts a neg:uive relationshipbetween the level of the prestimulus heart rate and.the ditfcrcO\:escore. Since 1.0 = b" - h." (bJua-tion I). a negativc hi)., will require that by.x is-

bD.x= by.x - 1.0

Marrh. I'JHO HEART RATE STATISTICAL A~ALYSIS 157

positive but less than 1.0 (Benjamin, IIJ63, 1967).Benjamin (1967) points out that bnx, bn, and rHare not equivalent statements about the relationshipbetween X and D. She suggests that a considerationof the methodology upon which these scores arebased implies that rXlJis the best indicator of thepresence or absence of the Law of Initial Values.Thus when r~\I)is statistically significant, i.e., dif-ferent from 0.0, covariance adjustments should bemade. It may appear inconsistent to assess theexistence of the Law of Initial Values by a score(rxo) which takes into account the two-way rela-tionship between X and D, and then adjust scores bythe linear regression coefficient. which includes.only the variance of the prestimulus score as astandardizing function. However, the correlationcoefficient may truly reflect the Law of Initial Val-ues' and it still would be correct to remove variancedue to X by the linear regression coefficient, sincethis procedure is the only one which results in ascore, Y, which is statistically independent of X.

However, an easier approach would be to assessthe amount of variance accounted for by the inclu-sion of the covariate, and if this is significant theadjustment should be made. No matter what thevalue of rXDis, if the adjustment does not result in adecrease in variance from Y to y, the adjustment issuperfluous. Statistical rationale for this test may befound in Kerlinger and Pedhazur (1973) and Tat-suoka (l97l). The BMD and SPSS analysis ofcovariance routines provide a test for the signifi-cance of the covariate as standard output.

Differential Variability

Three potential areas of difficulty face the re-searcher with regard to differential variability inheart rate. First, there is likely to be substantialintersubject variation in heart rate data in neonatesand older infants (Porges, 1974). This differentialvariability implies that even though two scores areof the same magnitude they will have a differentprobability of occurrence because they are fromdifferent distributions. Secondly. infant subjectshave been shown to have different levels of heartrate variation at different ages (Harper et a1., 1976)and in different behavioral states (Harper "etaI.,1976, 1977). Finally. there are good reasons forsuspecting that the assumption of homogeneity ofthe covariance matrix is violated with repeatedmeasures designs using cardiac rate as the depen-dent variable.

llttt'r.mbject limn HalL' "arialiol/

Porges and his associates have shown that infanthe.lrt rale v.lriability is an important SlIlJr~'eof var-iance in infant research. h)r ex.ul1ple. Por~es. Ar-nold. and Forbes (1~73) found that infants with hi~h

variability heart rates responded to an auditorystimulus in a pattern similiar to adult subjects, withgreater acceleratory and deceleratory responsesthan infants with low variability heart rates. Stampsand Porges (1975) and Stamps (1977) report thatinfants with high variability in heart rate were moreeasily influenced by a conditioning procedure thaninfants with low variability in heart rate. This re-search demonstrates intersubject variability in heartrate, and suggests that such variability might be animportant index of stimulus responsivity (Porges,1974).

The problem of differential variability betweensubjects' heart rates has been addressed in a solutionby Crowell et aI. (1976) based on Lacey's (1956)autonomic lability score. A similiarscore may befound in Program 3 of Wilson (Note I). Individualheart rate lability and difference in mean tonic levelare adjusted by transforming heart rate scores by astandardization procedure. Each heart rate score inthe poststimulus period is posited as being a combi-nation of the child's lability (reactivity, responsiv-ity, variability), the mean heart rate level. effects ofthe Law of Initial Values. effects due to the cardiacresponse to the treatment condition, and error. Thestandardization procedure removes the effects of thechild's lability and mean heart rate level, and sinceerror is thought to be randomly distributed, only theLaw of Initial Values and the response to thestimulus remain in the scores. If there is a significantcovariance relationship among the standardizedpre- and poststimulus scores, the procedures of theanalysis of covariance discussed in the previoussection will remove that effect. This leaves effectsdue only to the response to the experimentalstimulus in the poststimulus score, which can thenbe analyzed by common statistical techniques. Onemay refer to Crowell et al. (1976) for a presentationof the technique, and an example of its use in aneonatal heart rate classical conditioning study. Aslightly different form of the lability score is pre-sented by Wilson (1967) and can be calculated fromProgram 3 of Wilson (Note 1: Wilson & Scott,1970). A computer program for the Crowell et al.(1976) procedure is available from the present au-thor (Richards. Note 2).

This deviation procedure adjusts for differencesin the subjects' labilities and mean heart rate level.If means are normally distributed (or equal) andvariances are not significantly dill'erent betweensubjects, the analysi~ of heart rate data should bedone on unadjusted score~" Since the deviationprocedure demands multipk lIIeasurement intervalsin the prestimulus pcriod. a repeated measuresanalysis l)f vari.U1~e of prc"'lJlIIlIlus S~'l'res and var-iances would Icsl intersuhj~'d dilh~reIK'es in Iheprestil1lulus r~'ri"ll. A ~rollp'" main drc~.t should not

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JOHN E. RICHARDS Vol. /7. No.2

,be significant sincethe experimental treatmentsshould not affect prestimulus (pretreatment) scores.The subjects sum of squares tested against thegroups by subjects interaction term, if significant,would reveal intersubject differences. If there areno mean or variance differences the deviation pro-cedure should not be used.

Differential Variability in Treatment Conditions

It is likely that the assumption of the analysis ofvariance that variances be homogeneous for alltreatment groups will be violated for heart ratescores. The research of Harper et aI. (1976, 1977)has shown that heart rate variability is affected bybehavioral state and that variability changes in thefirst six months of life (Fig. 2). Differential var-iability in experimental groups may occur in devel-opmental studies of heart rate when age is used as afactor, or in studies of heart rate when behavioralstates vary. This problem occurs when the treatmentfactors contain independent subjects. In a repeatedmeasures design a different problem may occur(next section).

Heterogeneity uf variance in treatment conditionsis not a serious problem for most experimentalresearch. Monte Carlo studies of the violation ofthis assumption with both the F (Box, 1953, 1954;Norton, 1952) and the t ratios (Baker, Hardyck, &Petrinovich, 1966; Boneau, 1960) have shown thatthe violation of the homogeneity of variances as-sumption is not serious when equal sample sizes areused in treatment groups. Therefore, testing forhomogeneity of variances is not necessary if theresearcher assures equal sample sizes in the treat-ment conditions. If unequal sample sizes exist,however, the empirical probability of rejecting the

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null hypothesis is greater than the nominal a level.This may occur in clinical or naturalistic studieswhere equal sample sizes cannot be obtained. In thiscase, one should consult analysis of variancetextbooks for the proper procedure.

Differential Variability of Repeated MeasuresTreatment Conditions

Assumptions in the repeated measures analysis ofvariance are more stringent than in the independentgroups analysis of variance. Repeated measuresanalysis of variance demands distribution normalityand homogeneity of variance in any between-groupfactors. It also has the assumption that subjectsretain their relative standing in the various condi-tions of the repeated measures factor. This is for-mally known as the homogeneity of covariancematrices assumption. It demands that: 1) withineach level of the nonrepeated factors homogeneouscovariances are found among the repeated factors,and 2) the covariance matrices are homogeneousacross all factors.

The consequence of violating the covariancehomogeneity assumption is that statistical tests arebiased positively on the repeated measures factors(Box, 1950; Collier, Baker, Mandeville, & Hayes,1967; Keppel, 1973; Winer, 1971). This means thatthe researcher will reject the null hypothesis of nodifference among repeated factors means moreoften than the nominal a level implies. Collieret aI.(1967) in a Monte Carlo study of the violation of thecovariance homogeneity assumption used a designwith one repeated and one nonrepeated factor. Theyfound that the nonrepeated factor was not affected

. by the violations. The repeated factor, however,was almost always positively biased (includingmain effects and interactions). For example, withthe a = .05 only 2 of 60 times was the empiricaltest size below. OS, and ranged as high as .115.Empirical a levels of this size are unacceptable tothe researcher.

Heterogeneity in the covariance matrix mayoccur in infant heart rate research for several rea-sons. These reasons are based on a dictum thatmeasurements made on the same individual adja-cent. in time are more highly correlated than mea-surements separated in time. This could occur inheart rate research when repeated measurements aremade in several intervals on each trial. It may alsooccur in a longitudinal experimental design. Onewould expect that the correlation between occasionsI and 2 might be higher than the correlation betweenoccasions I and 3. A third potential design in whichthe homogeneity assumption is often violated iswhen heart rate measures arc made in ditt"erenttreatment conditions separated by unequal time per-iods. Ot these three potential causes of covariance

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March. /lJ80 HEART RATE STATISTICAL ANALYSIS 159

homogeneity, the first and last problems are mostunique to heart rate research. This is especiallytrue when multiple measures of heart rate aremade in the poststimulus intervals and treated asa repeated measures factor in the analysis of var-iance.

One can test a set of repeated measures heart ratedata for. homog~neity of covariances by the Box(1949, 1950)test. The Box test is a generalization ofthe Bartlett (1937) test for homogeneity of variancesin independent groups. .One calculates a valuewhich may be compared against )(2 or against Fdistributions. The reader may refer to Box (1949,1950), Cooley and Lohnes (1971), Morrison (1967),or Namboodiri, Carter, and Blalock (1975) for thestatistical rationale of the test and details requiredfor its calculation. The researcher is encouraged touse one of several computer programs to do theactual calculations (Nie et a!., 1975, DISCRIMI-NANT, Statistic #7; Rawlings & Pautler, 1972a;Richards, Note 2; Wilson, Note I, Program 12).

What should' the researcher do if heterogeneityexists in the covariance matrix? The two mostreasonable options are the Greenhouse and Geisser(Geisser & Greenhouse, 1958;Greenhouse & Geis-ser, 1959) correction of the degrees of freedom ofthe critical F ratio or the use of a multivariate test(Bock, 1975; Harris, 1975; McCall & Appelbaum,1973; Morrison, 1967). Both Davidson (1972) andMcCall and Appelbaum (1973) recommend the useof the multivariate method over the Greenhouse andGeissercorrection, claiming the latter is too conser-vative. Wilson (1975) points out that their conclu-sion is based on the maximum conservative value ofthe Greenhouse and Geisser correction. rather thanthe correction value which is proportional to thedegree of heterogeneity in the covariance matrix.Opposing the Davidson and the McCall and Appel-baum conclusion is the study of Collier et a!. (1967)which found that the correction resulted in an empir-ical test size approximately equal to the nominal alevel. Wilson concludes that the repeated measuresanalysis of variance with the Greenhouse and Geis-ser correction is fully protected against bias result-ing from violation of the homogeneity of covarianceassumption. .

Wilson (1975) may be correct in stating that theGreenhouse and Geisser correction protects theanalysis of variance framework against bias whencompared to the uncorrected analysis of variance.However, direct comparison needs to be done be-tween the multivariate pro\.'edure and the Green-house and Geisser \.'orredinn to corre\..tly assessthe relative power and error protection of thetwo methods. There are other reasons that a re-searcher may prefer the multivariate method. It isnot restricted by the homogeneity of covariance

assumption, and so evades the issue entirely. It is aunified statistical technique developed within thetheory of classical statistical testing and is as-sociated with several well-established and exacttesting procedures, The multivariate method can beincorporated-into many analysis routines, includingmean comparisons and polynomial trend compari-sons. In some cases, such as clinical studies inwhich one or more data points are missing forsubjects, the multivariate method is the only appro-priate one (Wilson, 1975). On the other hand, theGreenhouse and Geisser correction method appearsto the present author to be an ad hoc technique withno relationship to the central statistical premises ofthe analysis of variance. The study of Collier et a!.(1967) upon which the recommendation of Wilson(1975) is based studied the correction value only forsome limited designs. It mayor may not have thesame success with other designs. Finally, the easeof computing the Greenhouse and Geisser correc-tion and the associated repeated measures analysisof variance is no longer of concern due to the use ofcomputer programs for multivariate research (e .g.,Wilson, Note I, Program 12), excellent discussionsof the statistical rationale behind the test (Bock,1975; Morrison, 1967), and its application to re-'search (Davidson, 1972; McCall & Appelbaum,1973). One may refer to Wilson (1975), McCall andAppelbaum (1973), and Davidson (1972), for afurther discussion of this topic.

The multivariate test proceeds by transformationof the data by a matrix which represents several posthoc comparisons. One may test such hypotheses asequality among adjacent scores (Morrison, 1967),compare a criterion measurement interval with sev-eral other intervals, or use a set of polynomial trendcontrasts (McCall & Appelbaum, 1973). Followingonly the significant multivariate tests (Hummel& Sligo, 1971; McCall & Appelbaum, 1973;Wilkinson, 1975) each univariate mean compar-ison is tested, McCall and Appelbaum's (1973)presentation of the multivariate method for repeatedmeasures designs is an excellent guide for theconceptual and computational explication of thistechnique. The multivariate test can be done withProgram 12 of Wilson (Note I) or the program ofRawlings and Pautler (1972b). The SAS package(Barr et ai., 1976; Helwig & Council. 1979) hasgood, flexible multivariate programs under theGLM procedure. The program of Finn (1973) iscomprehensive. hut requires some knowledge ofdesign matrices.

Under the appropriate conditions. researchersmay prefer the Greenhousc and Gcisscr correction.In this correction. a value is \.'l)lI1putcdwhidl ismultiplied by the degrees of freedom of the critil:alF value. This value is always less than 1.0. and is

.,

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proportional to the amount of heterogeneity in thecovariance matrix. The greater the heterogeneity ingeneral leads to more positively biased statisticaltests. Thus, the greater the heterogeneity, th6smaller the degrees of freedom of the critical Fvalue after the correction, and the greater the ob-tained F value needed to reject the nun hypothesis(McCan & Appelbaum, 1973). Program 12ofWiI-son (Note 1)computes the correction value, and the.complete analysis of variance with the Greenhouseand Geisser correction can be done with the corn-puterprogramofGames (1975,1976). Both Wilson(1975) and McCall and Appelbaum (1973) giveexamples of its usage.

Two cautions must be observed in the use of theGreenhouse and Geisser correction or the mul-tivariate method. First, these methods are onlyalternatives to the univariate repeated measuresanalysis of variance when there is some indicationof heterogeneity in the covariance matrices. Theuncorrected test is more powerful than either of thetwo alternatives when its assumptions are met.Secondly, only those factors involving repeatedmeasures (i.e., trials, intervals) should be tested bythe alternative method. Collier et. a!. (1967) dem-onstrated that heterogeneity of covariance matricesdoes not affect factors not involving the repeatedmeasures, and the analysis of variance is robust toviolations of the homogeneity of variances assump-tion for independent groups with equal samplesizes. Therefore in the Greenhouse and Geissercorrection the correction is only applied to the maineffects and interactions involving the repeated fac-tors. In the multivariate strategy, the tests for non-repeated factors are identical to conventional testswhile those effects involving repeated measuresemploy the multivariate statistical techniques.

Graham (1970) shows another way in which thehomogeneity of the covariance matrix is importantto consider in heart rate statistical analysis. In manyheart rate experiments one measures several inter-vals on each trial, has several treatment conditions,and may present each condition more than once toeach subject. Recan again that measures made closein time win be correlated higher than measuresseparated in time. In the designs just mentioned, themeasurement of intervals is closer in time thaneither the trials or the treatment factors. Thehomogeneity of covariance assumption is that thecovariance matrices representing both the trials andthe intervals be identical. which will not often be thecase because of the different separations of time.

Graham (1970) states that this type of het-erogeneity demands unpoofed error terms in therepeated measures analysis of variance tests. Poolederror tenns will inflate the F ratio for the intervalvariable since it usually has a small effects variance,

while the pooled error term has taken from it themuch larger variance of the trials factor. Tests forsignificance should use error terms specific to thateffect. This includes tests for trends, which shouldnot be tested against the total interval error term,since it is pooled across all trends, but tested againstthe error term specific to that trend. Researchersshould be cautious in the use of packaged statisticalprograms which do not allow specification of effectand error terms, since they often test every effectagainst a pooled error term. The SAS statisticalpackage (Barret al., 1976;Helwig & Council, 1979)is recommended for this purpose, since the proce-dure GLM allows the specification of error termsand therefore offers the greatest flexibility in the.analysis of repeated measures designs. The'caveatagainst the use of pooled error terms is applicable toboth univariate and multivariate methods of testingrepeated measures designs.

Time Series Analysis .

David Crowen and his associates (Croweh h a1. .

1976: Jones, Crowell, & Kapuniai, 1969~)o'nes.Crowell, Nakagawa, & Kapuniai, 1971)"presentanautoregression technique for the analysis 0[. infaniheart rate data. Hean rate is a time series in whicheach score contains lawful dependence on the pre-vious score. The time series technique of autore-gression removes the sequential dependency.amongheart rate scores and allows one to identifysignifi-cant deviations from the time series.:':,

In the autoregression technique. one calculates anautoregression coefficient, a" which is base4cOntheprestimulus heart rate scores: ".

= Ix,xt+l = CoV XIXt+t (2)at ~ 2 f? 'AR~x, P. ~ .

where x, and X'+t are from prestimulus heart. rate,and are deviation scores at adjacent intervals.Ix,x'+t and IXi2 are the sum of deviation cross-products and the sum of squared deviations'. ofx" respectively, and COVXtXI+1and VARx, are thecovariance and variance associated with thesescores. The autoregression coefficient is actually thelinear regression coefficient between adjacent heanrate scores. It contains information about the se-quential dependency among adjacent heart ratescores. Given the autoregression coefficient and themean of the prestimulus heart rate scores, M\, theautoregression formula.

VI+I = Mx + a, (Yi - M,), (3)

is used to predict poststil11u!usscores giYJ~ntheimmediately preceding heart rate s,,'ore,The predic-tion of the poststimulus s..:orcsgiH'n the unstimuclated sequential relationship can Ocl."olllpareJto theactual score following presentation of the stimulus,

March./980 HEART RATE STATISTICAL ANALYSIS 161

-. .-..

Given the standard deviation of the prestimulu~scores, SDx, one may compute a I value based onpredicted and actual Yi,

I = Ysi>xY' (4)

which is distributed as Student's I with df =d - 2,where d is the number of measurements in theprestimulus interval upon which the autoregressioncoefficient is based. One can thus identify signifi-cant deviations from the time series. Fig. 3 presentsgraphically both a significant and a nonsignificantresponse to an auditory stimulus in a newbornsubject, with 951ii-confidence intervals around thepredicted heart rate scores. Jones et al. (1969)present a Fortran subroutine which may be used forthis entire process. Jones et al. (1971)extend thisbasic analysis to smooth irregularities in heart ratedata due to rapid fluctuations, and develop a modelwhich adaptively updates parameters of Equations2,3, and 4, weighting prestimulus data of the mostrecent stimulus more heavily than past prestimulusdata. Both programs are available from the presentauthor (Richards. Note 2). Examples of the use ofthe autoregression technique can be found in Joneset al. (1969) and Crowell et al. (1976).

The autoregression technique has met with littleusage in infant heart rate research, but is worthconsidering for several types of experimental de-signs. Jones et al. (1969) and Crowell et aI. (1976)use it to detennine the pattern of response in the

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poststimulus period. If one is interested in compar-ing the effects of several stimuli, this technique maybe preferred over factorial design analysis for de-tecting significant responses to stimulation. It maybe used in single-subject experiments or in studieswhere the number of subjects is not as large as thenumber of variables the researcher is interested inmeasuring. The autoregression technique, as wellas several other types of time series analyses, haveproven beneficial in clinical applications when onlyone subject is available and isolation is needed ofsingle critical events such as atrial fibrillation withPVC's (Gersch, Lilly, & Dong, 1975) or cardiacarrhythmias (Gersch, Eddy, & Dong, 1970). Per-haps it may be used in conjunction with discrim-inant analysis for a pattern of responses or eventswhich occur in conjunction with heart rate de-viations. Finally, this technique should prove usefulin programming computers for on-line decisionmaking, since the critical parameters can be esti-mated adaptively as data is collected.

Recent Statistical Applications

A bri~f mention wiH be made "f some statist:caltechniques which have recently been applied to theanalysis of hean rate data. They will not be exten-sively discussed because for the most part the appli-cations to date have been with adult subjects, andlittle discussion exists about their problems or theirapplicability to infant heart rate data. They deservemention, however, because they are potentially

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JOHN E. RICHARDS Vol. /7. So. 1

valuabletools for the analysis of infant heart ratedata. Such a discussionin a review of this typemight encourage their usage and supply researcherswith a better understanding of the techniques and oftheir application to heart rate data analysis.

As mentioned in the preceding section, heart ratecan be thought of as a time series. One property of atime series is the frequency spectrum which under-lies variations of the data (Blackman & Tukey,1959;Box & Jenkins, 1970;Fuller, 1976;Jenkins &Watts, 1968). Spectral analysis of heart rate data hasbeen used recently to examine variability in heartrate data caused by respiration activity. Porges(1976) and Porges and Humphrey (1977) have usedthe cross-spectral correlation between heart rate andrespiration as an indicator of parasympathetic ner-vous system integrity. Porges (1976) speculates thatthe lack of coherence between heart rate and respira-tion in retarded and infant populations may be anindex of the inability of those subjects to engagein sustained attention. Harper , Walter. Leake,Hoffman, Sieck, Sterman, Hoppenbrouwers, andHodgman ~1978) have found that respiratory,:induced variation in heart rate is different in be-havioral states which differ, and increases over thefirst six months of life. These changes may becaused by the increasing influence of the parasym-pathetic branch of the autonomic nervous system inthe first few months of life.

Another statistical technique which may be fruit-fully applied to infant heart rate analysis is factoranalysis. Many readers will be aware of the applica-tion of factor analysis to the structural analysis ofpsychometric tests. In factor analysis, one deter-mines the underlying structure of a set of mul-tivariate dependent variables (Cooley & Lohnes,1971;Harman, 1960; Tucker, 1958, 1966; Van Eg-eren, 1973). Given several dependent variableswhich are observed by the researcher, one usesfactor analysis to reduce these into a few unob-served latent constructs which are thought to be thetrue variables underlying the observed variables.Sinceheart rate whenmeasuredover severalinter-vals may be conceived of as a set of multipledependent measures, factor analysis may be rele-vant in its statistical analysis. A study by Van .

Egeren, Headrick. and Hein (1972) is a good exam-ple of the application of factor analysis to heart ratescores. They analyzed the factor structure of a set ofheart rate scores in intervals 10sec prior to and 3 secafter the presentation of a conditioned stimulus.After detennining the factors. scores lJf each indi-vidual on the factors were cakulated and the groupaverages were pl()tt~..d.They detennined three re-sponse pattents 10 ex.ist in the data. including anacceleratory-deceleratory shift which may representhoth an unconditioned and a conditioned response,

which may be based on.and defensive (accelera-deceleratory -acceleratory

a monophasic responseorienting (deceleration)tion) reactions, and ashift.

One should be cautious in using factor analysis onheart rate scores. Cronbach (]967) shows that thefactor analysis of data in which scores close in timehave higher correlation than scores far apart in timemay result in arbitrary fa~tors determined by theamount of separation of the measurement intervalsrather than by an underlying theoretical const11.1ct.One should check the validity of the factors bycomparing the individual subjects' raw data with theobtained factor pattern (e.g., Van Egeren et a!.,1972, Fig. 3). Another technique to avoid thisproblem would be to cluster subjects according tothe similarity of the factor score patterns. and use'the resulting groups asempirically defined responsepattern groups, doing quantitative analyses on theraw scores of the defined groups (e.g., McCall.Appelbaum, & Hogarty, 1973, 43ff.).

Factor analysis in most cases has been explor-atory, i.e., it is an a posteriori analysis of thestructure inherent in the observed response system.On the other hand, confirmatory factor analysis(Joreskog, 1969) is a model testing system which'can be used to test specific structural models againstobserved data. For example, Porges and his as-sociates (Porges, 1976; Porges & Humphrey, 1977;Walter & Porges, 1976) have speculated that atwo-component attentional system exists which in-cludes reactive and sustained attention, and which

can be indexed by patterns of heart rate responding.Presumably the first few poststimulus seconds rep-resent reactive or reflexive attention which is stim-

ulus dependent, and the next poststimulus secondswhile attention is occurring is sustained attentionwhich is cognitively determined. This predictioncould be tested by specifying two factors under-lying heart rate responding in the poststimulusintervals, one reflected by immediate response pat-terns and one reflected by long-term responses. Thecomputer program COFAMM (Sorbom & Jores-kog, 1976) can be used to do confirmatory factoranalysis. Joreskog has recently extended this type ofanalysis to a general structural model testing systemwhich includes confirmatory factor analysis andcausal modeling (path analysis) in one system(Joreskog, 1970. 1973.1974.1977), and a computerprogram is available for computational aid (Jures-kog & Surbom. 1978).

Factor analysis methodologies an: restricted tl1analyzing single populations. With ml1re than linegroup discriminant analysis can he lI~ed in a similiarfashion. In discriminant analysis a hllll'tilll! is eSlj-maled which is a weighted linear l'l1mhinatil1l1l1f IlK'multiple dependent variables whidl lIIa:>.imallydis-

..I

March. /980 HEART RATE STATISTICAL ANALYSIS 163

criminates between multiple groups by maximizingthe variance of the scores of the predefined groupson the discriminant function (Bock, 1975;Cooley &Lohnes, 1971;Tatsuoka, 1971;Van Egeren, 1973).The discriminant function can then be used toclassify a new set of individuals into one of thepreviously defined response groups. Multivariateanalysis of variance estimates a discriminant func-tion for each effect being tested and may be de-scribed as a univariate analysis of variance on thescores of the groups on the linear discriminantfunction. Given multiple intervals of heart rate intwo or more groups, or in the case of an experimen-tal factorial design, one may use discriminantanalysis to combine heart rate patterns for the bestdiscrimination among the experimental groups.This technique has an important clinical applica-tion. One may be able to classify persons belongingto pathological populations by comparing the sCoreon a discriminant function which was obtainedby discriminating between criterion normal andpathological groups. This technique is especiallyimportant to the heart rate researcher interested inthose diseases which are known to produce charac-teristic heart rate patterns, such as respiratory dis-tress syndrome.

Conclusion and Summary

The present paper has been devoted to the presen-tation of several statistical problems encountered bythe heart rate researcher, emphasizing those whichare encountered in infancy data. Table I presents aseries of steps based on this discussion which theresearcher might follow in choosing the appropriatestatistical techniques for analyzing heart rate data.

TABLEI.

Steps for choosbrK statistical techlliques

I. Choose: the appropriate cardiac measure.

II. Use: a lability score if there are dilferences in variances

and means of each subject. If no dilference, or if meansare normally distributed, use raw scores.

III. Choose: the appropriate statistical analysis for the main

experimental questions. Analysis of variance is usually

appropriate for factorial experimental designs. Use: auto-regression techniques if individual response:s or on-linedecisions are of interest. Could the heart rate response:

pattern be profitably analyzed with a multivariate statisti-

caltechnique?

IV. Should covariance adjustments be made for the Law of

Initial Values? If so, are curvilinear relationships impor-

tant? Are linear regression coefficients homogeneous?

Are the cardiac scores reliable? Adjust raw scores or use:

analysis of covariance.

V. If repeated measures analysis of variance is used, check.

the homogeneity of covariance assumption. If hetero-

geneity is prese:nt, use [he multivariate test or the Green-house: and Geisser correction. Do not use: pooled error

terms in repeated measures factors.

As with all statistical procedures, the indiscriminateapplication of techniques is unwarranted. Statisticalmethods each have assumptions underlying theirusage and characteristics which define their range ofapplication. The researcher is advised to proceedwith caution in the application of these techniques.Hopefully, the existence of this paper will helpeliminate some of these problems and provide theresearcher with the knowledge necessary to applythese techniques.

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JOHN E. RICHARDS Vol. 17. No. 2

REFI::Rf.~n::-.ions

I. Wilson. R. S.CARDIV AR. Available from ASIS. SAPS.~ticrofiche Publications. 305 E. ~th 51.. New York. N. Y

10017. Order document #NAPS 000933. remitting 53.00for photocopies and 51.00 for microfiche. The statisticalmethods are presented in Wilson. 1967: a description of the

programs is in Wilson & SCOII. 1970: examples of the use ofthe programs are given in Wilson. 1974

2. Richards. J. E. CAIHR: Computu una/y.lu of infunt ht'urt

rate. a compilation. Unpublished manuscript. 1978. Univer-

sity of California. Los Angeles. 9002~.

(Manuscript recei\"ed April 20. 1979: accepted for publication August 27. 19791