Importance of Using Rigorous StatisticalMethods to Analyze Low Energy Laser

Experimental Data: Part TwoDaniel W. Ebert, MS and Cynthia Roberts, PhD*

Biomedical Engineering Center, Ohio State University, Columbus, Ohio

Background and Objective: Numerous authors have reportedsuccessful alteration of peripheral nerve action potential char-acteristics through application of low energy laser irradiation(LELI). The statistical analysis that accompanies many of thesereports frequently does not account for the special nature of thedata generated in typical LELI experiments. The objective ofthis study was to evaluate the application of repeated measureslinear regression techniques to the analysis of this type of data.Issues of analyzing raw versus normalized data, proper ac-counting for correlation between measurements, and discretetime point hypothesis testing were addressed.Study Design/Materials and Methods: The data analyzed in thiswork were generated from an experiment in which in vitro frogsciatic nerves were irradiated with a helium-neon laser using avariety of treatment protocols. Compound action potential(CAP) amplitude, latency, depolarization rate, and repolariza-tion rate were recorded at 1-minute intervals for 135 minutesfor each nerve. Laser-induced changes in CAP parameters wereanalyzed using various repeated measures linear regressionmodels.Results: The findings of statistical significance were highly de-pendent on the rigor of the regression model applied. Applica-tion of the same regression model to raw and normalized dataproduced different findings of significance. Determination ofsignificant contrasts was highly dependent on how well the re-gression model accounted for the correlation between repeatedmeasurements made on the same nerve. In general, models thatfailed to account adequately for this correlation produced morefindings of significant contrasts than increasingly rigorousmodels. Finally, discrete time point hypothesis testing on nor-malized data can suggest improper statistical conclusions if theproper correlation structure is not applied to the data set.Conclusion: Linear regression analysis offers advantages overdiscrete time point hypothesis testing in the analysis of highlycorrelated serial data of this type. Trends in the behavior of themeasured parameters are evident, rigorous accounting for cor-relation between measurements is facilitated, and hypothesistesting is highly flexible. Lasers Surg. Med. 21:42–49, 1997© 1997 Wiley-Liss, Inc.

Key words: biostimulation; HeNe laser; low intensity laser therapy; repeated mea-sures linear regression analysis; statistical methods

*Correspondence to: Cynthia Roberts, 270 Bevis Hall, 1080Carmack Road, Columbus, OH 43210.

Accepted 23 September 1996.

Lasers in Surgery and Medicine 21:42–49 (1997)

© 1997 Wiley-Liss, Inc.

INTRODUCTION

Use of low energy laser irradiation (LELI) toenhance wound healing, pain relief, and nerve re-pair and regeneration has been a subject of inter-est for more than 15 years [see 1–5 for reviews].Reports of LELI-induced alteration of healthy andinjured peripheral nerve function [6–12] offer tre-mendous potential to generate efficacious clinicaltreatments of peripheral nerve injuries and dis-ease if the reported laser-induced effects can bereplicated and verified.

In a companion work [13], we reported theresults of an experiment in which in vitro frogsciatic nerve compound action potential (CAP)characteristics were measured under a range ofLELI conditions. The data set generated in thecourse of that work was typical of those found inother studies of the effect of LELI on peripheralnerve tissue characteristics in vitro and in vivo.Since the study of the effect of laser irradiation onthe function of peripheral nerve tissue often re-quires the collection of repeated electrophysiologi-cal measurements made on the same nerve prepa-ration over time, many LELI studies make use ofserial measurements taken at discrete timepoints preirradiation, during irradiation, andpostirradiation.

Experiments that measure changes in CAPparameters in response to irradiation over min-utes, hours, or days frequently rely on statisticalcomparisons made at discrete time points to dem-onstrate the effect of the laser irradiation on thetissue. Due to the inherent variability of biologicaldata, CAP behavior at any arbitrary time pointmay or may not be indicative of the overall laser-induced trend of that CAP parameter during thatphase of the experiment. If the CAP parametersat the chosen time points happen to be greatlydifferent than the behavior of the parameter atother times during that phase of the experiment,the statistical conclusions drawn may be incorrectand misleading. Furthermore, important statisti-cal dependencies of successive measurements areoften ignored in discrete time point methods. An-other disadvantage to the use of discrete timepoints to describe LELI effects is that trends inlaser-induced alterations of CAP parameters arelost, although this information is necessary to un-derstand completely the influence of LELI on thetissue.

In the companion work [13], repeated mea-sures linear regression analysis was introduced toexamine the data for evidence of laser-induced

changes in the CAP. This statistical method iswell suited to the analysis of repeated measure-ments taken on the same tissue over a definedtime period, a commonly encountered type of dataset in investigations of laser effects in peripheralnerve tissue. Although this form of regressionanalysis has not been previously applied in LELIresearch of this sort, it was found to offer signifi-cant advantages over discrete time point statisti-cal methods encountered in the LELI literature.

In particular, the use of regression modelingin this experiment proved valuable as a means ofcapturing the trends in behavior of the measuredCAP parameters over the various phases of theexperiment. Additionally, this method of analysispermitted rigorous accounting for the strong cor-relation between the repeated measurements.Hypothesis testing was also enhanced by use ofregression analysis since discrete time pointscould be compared while capturing the trend inthe CAP behavior over time and properly account-ing for the strong correlation noted between mea-surements made on the same nerve.

The purpose of the current work was to ex-plore fully the utility of using repeated measureslinear regression analysis to analyze peripheralnerve electrophysiological data collected seriallyin the course of an LELI experimental protocol.Following a general explanation of the applicationof repeated measures linear regression analysisin this context, some important ramifications ofusing linear regression analysis are further inves-tigated. Among these ramifications are differ-ences in the statistical conclusions obtained whenusing raw measurements of CAP parameters ver-sus normalized measurements, the effect of rigor-ously accounting for the strong serial correlationbetween measurements on the same nerve prepa-ration, and the potential advantages of using re-gression analysis over discrete time point hypoth-esis testing in data sets of this type.

MATERIALS AND METHODS

A full description of the materials and meth-ods is provided in the companion work [13] and isnot repeated here. Following is a brief overview ofthe experimental methods employed.

Tissue Preparation, Irradiation, andCAP Recording

The sciatic nerves in each leg of large bull-frogs were harvested and laid over five wire elec-trodes in separate nerve chambers. Following a

Analyzing Low Energy Laser Experimental Data 43

60-minute baseline CAP recording period, 15 min-utes of HeNe laser (632 nm, Spectra PhysicsModel 125) irradiation was delivered to the sur-face of one of the two nerves via a 0.6 mm diam-eter optical fiber at the site in which the nervecontacted either the ground electrode or proximalrecording electrode. Laser output power was var-ied among the treatment groups that were desig-nated according to the combination of mean totalenergy delivered to the tissue and site of irradia-tion (see table in companion work). The contralat-eral nerve in the nonirradiated chamber was as-signed to the control group.

Throughout the experiment, each nerve wasstimulated at 1-minute intervals using a supra-maximal stimulus (0.025 ms, 1.5 V). The resultingCAP was recorded differentially via a MacLabstimulator-A/D converter unit and saved on acomputer for off-line measurement of CAP ampli-tude, latency, rate of depolarization, and rate ofrepolarization. CAP amplitude was measuredfrom peak to peak; latency was measured fromstimulus onset to the negative peak of the CAP.The rates of depolarization and repolarization ofthe evoked CAP were estimated by measuring theslope of the CAP trace on either side of the nega-tive peak.

CAP recording in each nerve was performedin three consecutive phases that totaled 135 min-utes. In phase 1, which required 60 minutes, base-line characteristics of each nerve were establishedby stimulating and recording evoked CAPs onceper minute. Phase 2 consisted of 15 minutes ofirradiation delivered to the nerves assigned to ir-radiated treatment groups. During phase 2, CAPswere simultaneously evoked and recorded in irra-diated and nonirradiated nerves at 1-minute in-tervals. In phase 3, the postirradiation phase,CAPs were stimulated and recorded at 1-minuteintervals in both nerves for an additional 60 min-utes.

Statistical Analysis

a. Optimum linear regression model fornormalized data. Normalized measurementswere computed by determining the mean value ofeach CAP parameter over the 60-minute preirra-diation recording period for each nerve. All sub-sequent raw CAP parameter measurements re-corded in that nerve were divided by the meanpreirradiation value of the appropriate parameterto derive the normalized value of the CAP param-eter corresponding to that time point in the givennerve.

By graphically examining individual re-sponse profiles in each treatment group, it wasdetermined that the optimum way to analyze thedata was by separate linear regression models foreach of the three experimental phases: preirradia-tion (Phase 1), irradiation (Phase 2), and postir-radiation (Phase 3). A random-coefficient linearregression analysis for unbalanced repeated mea-sures was performed on the normalized CAP pa-rameters for each phase-treatment group combi-nation using the SAS computer program. AkaikeInformation Index (AII) and Swartz-BayesianCriteria (SBC) goodness-of-fit indices were usedto determine the relative success of each linearregression model in fitting the data [14,15].Larger AII and SBC indices were indicative of re-gression models that fit the data more closely.

The data strongly suggested that measure-ments in the same phase recorded nearer in timewere more highly correlated than measurementstaken farther apart in time (e.g., normalized am-plitude data yielded a correlation coefficient >0.91). To account for this strong serial correlationbetween measurements made on the same nervein the same phase of a treatment group, a first-order autoregressive covariance structure was in-cluded in the regression model. The autoregres-sive covariance structure, which is commonly ap-plied in repeated measures regression analysis,models the correlation between two measure-ments as a function of the time lapse between themeasurements [14]. In this work, it was assumedthat the correlation between measurements madeon a nerve within the same phase decreased ex-ponentially as the time between measurementsincreased.

The random-coefficient aspect of the linearregression model provided a second competingmechanism to account for correlation between se-rial measurements. In this scheme, the measure-ment correlation was also assumed to be a func-tion of the time lapse between measurements, al-though the exact form of this function was notexponential, but depended on the degree of het-erogeneity of the regression slopes fit to indi-vidual nerves in the treatment group. The func-tion resulted in a maximum correlation whenmeasurements were taken very close to one an-other and rapidly dropped off as the time betweenmeasurements increased.

The covariance structure of the response wastaken as the sum of the covariance of each of thecomponents, the autoregressive structure, andthe random coefficient portion of the model. AII

44 Ebert and Roberts

and SBC goodness of fit indices confirmed thatsimultaneous inclusion of these two methods ofcorrelation modeling optimized the fit of the re-gression model relative to other correlation struc-tures attempted.

Hypothesis testing was conducted on con-trasts that measured differences in regressionline slope between two phases in a treatmentgroup relative to the difference in slopes in thecontrol group between the same two phases: e.g.,is the change in regression line slope betweenphase 1 and phase 3 in Group 1 G different thanthe change in slope between phase 1 and phase 3in the nonirradiated control group? By arrangingcontrasts in this manner, comparisons of laser-induced trends in CAP parameters in a givenphase with trends demonstrated in the controlgroup were possible, simultaneously accountingfor nonlaser-induced changes in regression lineslope due to confounding factors (e.g., decreasingnerve viability over time). The overall thresholdfor significance (a 4 0.05) was adjusted by theBonferroni method to reflect multiple compari-sons [16].

A special shorthand notation was developedto describe concisely the various contrasts tested.Each label included the CAP parameter mea-sured, the treatment group, the site of irradiation,and the two phases being compared. For example,the contrast ‘‘latency, 4 R Phase 3:1’’ denotes thatthe difference in regression line slope betweenphase 3 and phase 1 in treatment group 4 R wascompared to the same difference in the nonirradi-ated control group for CAP latency.

b. Analysis of normalized versus rawdata. One commonly encountered data analysistechnique found in many LELI studies is normal-ization of CAP measurements to some preirradia-tion value, although frequently this preirra-diation value is not explicitly specified. In thissection, the same repeated measures linear re-gression model was fit to both normalized and rawCAP measurements to determine if different sta-tistical conclusions would be generated.

Initial attempts to use the optimum randomcoefficient regression model previously fit to thenormalized data failed when attempting to fit theraw data due to the occasional inability of thecomputer program to converge to unique values ofthe regression equation coefficients. A likely ex-planation for this program convergence failure isthat in some cases the two competing methods ofcorrelation accounting were so similar numeri-cally that it was difficult to determine how much

weight each method was responsible for in thecorrelation structure. In these instances, the simi-larity between the autoregressive structure andthe random coefficient regression was so largethat a ‘‘ridge’’ was created in the maximum like-lihood estimate. On this ridge of possible regres-sion equation coefficients, several choices wereapproximately equally appropriate and the pro-gram was incapable of converging to a true maxi-mum.

Therefore, the regression model that was fitto the raw and normalized CAP measurementswas slightly reduced in scope from the optimummodel described previously. The reduced regres-sion model differed from the optimum model inthat only the autoregressive covariance structurewas utilized to account for the correlation be-tween measurements. Of the competing mecha-nisms of correlation accounting, elimination of therandom coefficient aspect of the model providedthe best fit to the raw CAP measurements. Allother aspects of the model and hypothesis testingwere the same as described previously. Accordingto the AII and SBC goodness of fit indicators, thereduced regression model fit the normalizeddata only slightly less closely than the optimummodel (e.g., for CAP amplitude, AIIfull 4 5,928,AIIreduced 4 5,894; SBCfull 4 5,915, SBCreduced 45,885).

c. Effect of correlation accounting. Re-peated measurements on single nerves are fre-quently encountered in LELI-peripheral nerve in-teraction investigations, as it is important to fol-low irradiation-induced alterations in tissuefunction over time. Serial measurements of thistype tend to be highly correlated and not indepen-dent, necessitating special consideration duringstatistical analysis. In order to determine the im-pact of properly accounting for the correlation be-tween serial measurements, a series of five re-gression models were fit to the normalized data.Each successive regression model contained lessrigorous accounting for the correlation present be-tween measurements.

The first model chosen (Model 1) was the op-timum regression model for the normalized datadiscussed previously. Correlation between mea-surements was accounted for by the combinationof the autoregressive covariance structure as wellas the random coefficient nature of the regressionmodel. Model 2 corresponded to a regressionmodel in which the only the autoregressive covari-ance structure was used to model the measure-ment correlation. In Model 3, a random coefficient

Analyzing Low Energy Laser Experimental Data 45

regression model was fit to the normalized data,but no autoregressive covariance structure wasincluded. Model 4 was constructed so that the cor-relation between measurements made in anyphase was assumed to be uniform and not vary asa function of the time lapse between measure-ments. The final model, Model 5, assumed that allCAP measurements were independent of eachother and demonstrated no correlation. Hypoth-esis testing proceeded as previously described.

d. Discrete time point hypothesis test-ing vs. regression analysis. In order to comparedirectly the statistical conclusions derived fromregression analysis to discrete time point statisti-cal methods, paired Student t-tests were used tocompare irradiated group mean values to controlgroup mean values at each time point (i.e., onceper minute over the 135-minute experiment) at asignificance level of a 4 0.05. Statistical conclu-sions derived based on these t-tests were com-pared to conclusions derived from regression lineslope contrasts determined from application ofthe optimum regression model to normalizeddata.

Student t-tests were also conducted betweenirradiated and control values at the midpoint ineach phase using the fitted regression lines de-rived from the optimum model applied to normal-ized data. For these tests, the overall significancelevel of a 4 0.05 was adjusted by the Bonferronimethod to reflect multiple comparisons. The find-ings of significance generated by these t-testswere compared to those computed from directlytesting the normalized CAP measurements priorto fitting the regression models.

RESULTS

a. Optimum Linear Regression Model forNormalized Data

The effect of HeNe laser irradiation on CAPparameters as determined by the optimum re-gression model fit to normalized data is fully dis-cussed in the companion work [13]. Briefly, laserirradiation under the experimental conditions de-scribed failed to induce any statistically signifi-cant change in CAP amplitude, rate of depolariza-tion, and rate of repolarization at any time in theexperiment. Only one contrast, 7 R Phase 3:1 (i.e.,treatment group 7 R, postirradiation phase) dem-onstrated a statistically significant increase in la-tency compared to the nonirradiated controlgroup over the same time period.

b. Analysis of Normalized vs. Raw Data

The reduced regression model fit to the rawCAP measurements resulted in one finding of sig-nificance, namely, that treatment group 7 R dem-onstrated increased latency in the postirradiationphase relative to the nonirradiated control group(7 R Phase 3:1, P 4 0.0003).

Fitting the same regression model to the nor-malized data also resulted in the finding thattreatment group 7 R demonstrated significantlyincreased latency relative to the control group (7R Phase 3:1, P 4 0.0001). One additional findingof significance was noted, however. Treatmentgroup 4 R latency was determined to be greaterthan the nonirradiated control group latency inthe postirradiation phase (4 R Phase 3:1, P 40.0191). Furthermore, a contrast that narrowlymissed achieving the required level of significancein the raw data (latency, 4 R Phase 2:1, P 40.026) was far from achieving significance whenanalyzed as normalized data (P 4 0.1551).

c. Effect of Correlation Accounting

In order to determine the effect of correlationaccounting on the findings of significance, fiveprogressively less rigorous regression modelswere fit to normalized CAP data. The results ofthis experiment are shown in Table 1. The num-ber of findings of significance was greatest in themodel that assumed uniform correlation betweenmeasurements and did not take into account thefact that measurements taken closer together intime were most highly correlated.

d. Discrete Time Point Hypothesis Testing vs.Regression Analysis

As previously indicated, the only contrast de-termined to be significant under the optimum re-gression model applied to normalized data wasCAP latency, 7 R Phase 3:1. Student t-tests con-ducted on normalized CAP measurements prior tofitting a regression model resulted in findings ofsignificant differences in latency treatment group7 R at 35–37, 39–51, 53, and 55–60 minutes post-irradiation. However, Student t-tests conductedon the fitted regression lines at the midpoint ofeach phase demonstrated no significant differ-ences in any treatment group.

DISCUSSIONAnalysis of Normalized vs. Raw Data

The regression model that produced the op-timum fit to the normalized data, as determined

46 Ebert and Roberts

by the goodness-of-fit indices, suggested that onlyone contrast achieved the required level of signifi-cance, the 7 R Phase 3:1 contrast applied to CAPlatency. When the same model was fit to the rawdata, however, difficulties with program conver-gence forced a modification to reduce the extent ofthe correlation accounting. This reduced regres-sion model, when applied to the raw CAP mea-surements, confirmed the results of the optimummodel by finding only the CAP latency contrast 7R Phase 3:1 to be statistically significant.

Application of the reduced regression modelto the normalized data, however, producedslightly different results. Although the reducedmodel applied to normalized data confirmed thesignificance of the latency contrast 7 R Phase 3:1,a second contrast (latency, 4 R Phase 3:1) was alsodetermined to have achieved significance. Fur-thermore, a contrast that narrowly missed achiev-ing the required level of significance under theraw data (latency, 4 R Phase 2:1) was determinedto be quite far from significance under the nor-malized data.

The fact that use of raw and normalized dataproduces different statistical conclusions shouldnot be surprising. In each case, the parametersmeasured are on different scales. Consequently,one can reject the equality of regression lineslopes on one scale and fail to reject equality onthe other scale. It is possible to construct artificialdata sets in which it can be mathematicallyproven that equality of slopes on one scale in dif-ferent phases of the experiment does not guaran-tee equality of slopes on another, normalizedscale.

In the case of complex data sets such as thoseresulting from most LELI studies, it is not pos-sible to develop a general rule to predict differ-ences in the statistical conclusions expected usingnormalized data versus using raw data. It is morelikely that the phenomenon of producing differentanswers depending on which type of data is usedis dependent on the actual structure of each indi-vidual data set. In situations in which laser-induced differences are very clear between irradi-ated and nonirradiated treatment groups, bothraw and normalized data are likely to provide thesame answers, at least qualitatively. Unfortu-nately, many LELI studies report subtle differ-ences in laser-induced parameters that are, there-fore, subject to the type of data set chosen foranalysis.

Effect of Correlation Accounting

The investigation of the effects of correlationaccounting in the application of repeated mea-sures linear regression analysis to normalizedCAP data produced interesting and unexpectedresults. In general, it was expected that the use ofless rigorous models would fail to capture the cor-related nature of the data and, therefore, lead tomore findings of significance. One method of con-ceptualizing correlation accounting is by notingthat the effective sample size decreases as morerigorous accounting is made for the measurementcorrelation. Smaller effective sample sizes usuallylead to fewer findings of significance, but morehonestly represent the way the experiment wasactually conducted. In contrast, regression modelsthat use a covariance structure that is not as rig-

TABLE 1. Results of Correlation Accounting Experiments*

Model Model assumptions Significant contrasts CAP parameter P value

1 Random coefficients 7 R Phase 3:1 Latency 0.0158Autoregressive cov.

2 Autoregressive cov. 4 R Phase 3:1 Latency 0.01917 R Phase 3:1 Latency 0.0001

3 Random coefficients None — —4 Uniform correlation 1 G Phase 2:1 Latency 0.003

1 G Phase 3:1 Latency 0.00044 R Phase 2:1 Latency 0.00554 R Phase 3:1 Depolar. rate 0.00017 R Phase 3:1 Latency 0.00014 R Phase 3:1 Depolar. rate 0.0001

5 No correlation 4 R Phase 3:1 Latency 0.00017 R Phase 3:1 Latency 0.00014 R Phase 3:1 Depolar. rate 0.0001

*In each treatment group (1G, 4G, 4R, and 7R), four contrasts were tested for each of the four CAP parameters measured.Therefore, a total of 16 contrasts were tested in each treatment group. Only those contrasts that achieved statistical signifi-cance under the five regression models tested are listed.

Analyzing Low Energy Laser Experimental Data 47

orous, such as Model 4, effectively utilize a samplesize on the order of the number of measurementsmade, rather than the number of nerves con-tained in the treatment group. Using an inflatedsample size in this manner typically renders find-ings of statistical significance more likely.

Interestingly, use of Model 4, which assumeduniform correlation between measurements, ledto multiple findings of significance, suggestingthat the laser irradiation was relatively effectivein altering the measured CAP characteristics.When a more rigorous model such as Model 1 is fitto the same data, a completely different conclu-sion is evident regarding the efficacy of the laserirradiation in this tissue. Previous studies thatreport wide-ranging success with using LELI toalter the characteristics of peripheral nerve tissuemust be carefully evaluated, especially with re-gard to the statistical methods utilized to confirmthose claims. It is possible that in many cases,claims of significant laser-induced findings maybe due to improper application of statistical meth-ods, as demonstrated by use of Model 4 on thisdata set.

It is not as straightforward to compare Mod-els 2 and 3, which both included similar methodsof accounting for correlation. In this data set, in-clusion of the autoregressive covariance structureproduced findings similar to the optimum model.The model that utilized the random coefficientmethod of correlation accounting, however, foundno contrasts to be significant. This points out theimportance of proper and thorough correlation ac-counting when analyzing serial data of this type.As the goodness-of-fit indicators demonstrated,the correlation of the data was best accounted forby inclusion of both methods of correlation ac-counting.

Discrete Time Point Hypothesis Testing vs.Regression Analysis

Many LELI studies of laser-peripheral nerveinteraction generate serial data by making re-peated measurements on the same tissue andanalyzing the results via hypothesis testing atdiscrete time points. In this work, comparisonswere made between Student t-tests performed onthe normalized data and the findings of signifi-cance from both the regression line slopes and t-tests conducted at discrete time points using fit-ted values of the optimum regression equations.Student t-tests conducted on discrete time pointsresulted in the general finding of significant dif-ferences in latency in treatment group 7 R during

the postbaseline period. Testing of regression lineslopes demonstrated similar conclusions. How-ever, discrete time point testing of regressionmodel fitted values revealed no contrasts of sig-nificance in any treatment group.

The reason that the discrete time point test-ing following regression differs so dramaticallyfrom testing prior to regression modeling involvesissues discussed previously. Specifically, t-testsconducted directly on the normalized data in noway account for the strong correlation structurepresent in the measurements. The same tests,conducted after fitting the best possible model tothe data, rigorously accounted for this correlationand thus found no significant differences at dis-crete time points. Failure to analyze this data setproperly by use of a rigorous regression modelwould have led to erroneous reports of statisti-cally significant laser-induced changes in themeasured CAP characteristics.

One other point is also important to noteabout discrete time point hypothesis testing. Instudies in which laser irradiation is applied toperipheral nerve tissue over time, it is often im-portant to track trends in the behavior of the elec-trophysiological properties of interest. Use of re-gression analysis enables trends in this behaviorto be captured without the loss of informationthat occurs when using discrete time point hy-pothesis testing. If discrete time point testing isdesired for some reason, the fitted values of theregression equation should be utilized for thispurpose.

CONCLUSIONS

The results of this work confirmed that re-peated measures linear regression analysis pro-vides additional insight into understanding theaction of low energy laser irradiation on periph-eral nerve tissue. In particular, use of regressionanalysis enables full retention of the informationcaptured in all phases of the experiment, since itdoes not rely on an arbitrary choice of a limitednumber of discrete time points as the basis ofcomparison. Rather, the trend in the behavior ofthe treatment group over a given time period isclearly evident and easily quantified.

Linear regression modeling, as applied inthis work, was also found to provide a flexiblehypothesis testing scheme. From measurementscollected in this study, it was possible to test mul-tiple contrasts simultaneously, once the appropri-ate regression model had been fitted. Traditionaluse of discrete time point hypothesis testing is

48 Ebert and Roberts

inevitably limited to comparisons of irradiatedmeans to control means or postirradiation versuspreirradiation comparisons made within a treat-ment group. The flexibility in hypothesis testingand contrast generation offered by regressionanalysis can provide information and statisticalconclusions not readily available from tests con-ducted only at discrete points. If testing of specificcontrasts at discrete time points is desired, how-ever, regression analysis offers the ability to ac-complish this, at the same time, properly account-ing for the correlation structure of the measure-ments.

One of the frequently ignored aspects ofLELI-peripheral nerve interaction studies is cor-relation induced in measurements when multipleCAP recordings are made on the same tissuepreparation over time. In an experiment in whichthe CAP is measured repeatedly from the samenerve at certain time intervals, it is expected thatCAP measurements recorded closer in time de-pend more strongly on one another and thus aremore highly correlated than measurements madefurther apart in time. The strong correlation pat-tern holds true for measurements made over min-utes, hours, or days. Discrete time point hypoth-esis testing techniques often neglect the repeatednature of the measurements and fail to accountfor the strong correlation present from one mea-surement to another, potentially leading to erro-neous findings of significance such as those dem-onstrated in this analysis.

LELI research must be accompanied by theuse of rigorous statistical analysis in order to sup-port claims of laser efficacy. Repeated measureslinear regression analysis techniques offer a pow-erful tool to ensure that suitable statistical rigoris applied to the analysis of LELI experimentaldata so that reported results will be believableand reproducible. Failure to progress on this frontwill continue to hamper the advancement of LELIresearch.

ACKNOWLEDGMENTS

Special thanks to Dr. Rob Leighty, Dr. SusanVolman, Dr. Bradley Clymer, and Ms. PeilingYang for their advice and assistance.

This work was funded by the American So-ciety for Laser Medicine and Surgery (ASLMS) as

part of the Graduate Student Summer ResearchGrant Program. D.W.E. is currently a WhitakerFoundation Doctoral Fellow in Biomedical Engi-neering at Ohio State University.

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