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1993;53:6042-6050.Cancer ResDaniel F. Heitjan, Andrea Manni and Richard J. Santen

Tumor Growth Experimentsin VivoStatistical Analysis of

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( CANCE R R ES EARC H 5 3, 6 04 2- 60 50 . D ec em be r 1 5. 1 99 3]S ta tistica l Analy sis o f in V iv o Tumor Growth Exp erimen ts1Daniel F. H eitjan,2 A ndrea M anni, and Richard J. SantenC en ter fo r B io sia iisiic s a nd E pid em io lo gy [D . F . H . a nd D ep artm en t o f M ed icin e A .M ./, P en nsylv an ia S ta te U niv ersity C olle ge o f M ed icin e, H ersh ey , P en nsylv an ia 1 70 33 , a ndD ep artm en t o f M ed ic in e, W ayn e S ta te U nive rsity , D etro it, M ic hig an 4 82 01 J R. J . S .

ABSTRACTWe review and com pare statistical m ethods for the analysis of in vivo

tum or grow th experim ents. T he m ethods m ost com monly used are deficient in that they have either low pow er or m isleading type I error rates.W e p ro po se a se t o f m ultiv ar ia te s ta tistic al m od elin g m eth od s t ha t c or re ctth ese p rob lem s, illu strating th eir ap plication w ith d ata from a stu dy of th eeffect of a-d iflu orom eth ylor nith in e on grow th of th e li 1-20 h um an b reasttu mor in n ud e m ice . A ll th e m eth od s fin d sign ifican t d iffe ren ces b etw eenth e or-d iflu or om eth ylorn ith in e d ose grou ps, b ut recommen ded sam plesizes for a su bseq uen t stu dy are m uch sm aller w ith th e m ultivariate m etho ds . W e c on clu de th at th e m ultiv ar ia te m eth od s a re p re fe ra ble a nd p re se ntg uid elin es f or t he ir u se .

INTRODUCTIONThe analysis of alterations in in vivo tum or grow th is a pow erfulto ol fo r stu dy in g th e effe cts o f p oten tia l ca nce r tre atm en ts. In a typ ica le xp er im en t, o ne r an domi ze s t umor -b ea ri ng a nim al s i nto v ar io us t re at

m ent groups, periodically observing tum or volum es. T he resultingdataset consists of a series of volumes for each anim al, which oneanalyzes to determ ine w hether and how the treatm ent affects tum orgrowth. In this article we discuss the statistical methods that area va ila ble f or a na ly zi ng s uc h e xp er im en ts .T he sta nda rd w ay of d em on stra tin g tre atm en t e ffe cts is to es ta blishth at in te rg ro up d iffe re nce s a re s ta tistic ally sign ific an t; th us w e fo cu so n s ig ni fic an ce t es ts a nd th ei r p ro pe rti es . C la ss ic al s ta tis ti cs e va lu ate stests in term s of type I error rate and pow er. T he type I error rate is thech anc e o f o btainin g a sig nific an t re su lt w he n th ere is n o e ffec t, a nd th epow er is the chance of obtaining a significant result w hen there trulyis an effect. N o w orthw hile test can be foolproof in the sense of beingalw ays significant w hen there is an effect and never significant w henthere is none. The best one can do is to fix the performance atpreselected levels, conventionally 5% for type I error and 90% forpow er. For a given type I error rate and actual difference, the pow erincreases w ith the sam ple size. T hus a common m ethod for determ ining the sam ple size is to fix the type I error rate, estim ate the size ofthe effect (often from past data), and choose n to be just large enoughfor the power to exceed 90% .Just as there can be m any w ays to m easure a biological param eter,not all of w hich are equally efficient, there can be m any w ays to testa statistical hypothesis, not all of w hich are equally pow erful. A m orepow erful m ethod m ay find significance w hen a less pow erful m ethodd oes n ot; co nse qu en tly , th e m in im um samp le size re qu ire d to ac hie vethe desired pow er is sm aller w ith a m ore pow erful m ethod. Thus thechoice of statistical m ethod, far from being irrelevant, can tangiblyaffe ct th e efficien cy o f e xpe rim en tatio n a nd the c red ib ility of res ults.T o d ete rm in e c ur re nt s ta tis ti ca l p ra ct ic es amo ng in v iv o e xp er im en ters, w e surveyed two sum mer 1992 issues of each of seven leadingjournals that commonly report such studies: B reast C ancer R esearchand Treatm ent, C ancer Research, European Journal of C ancer, In-

Re ce iv ed 3 /9 /9 3; a cc ep te d 1 0/ 12 /9 3.T he costs of publication of this article w ere defrayed in part by th e paym ent of pagec ha rg es . T his ar tic le m us t th ere fo re b e h ere by m ark ed ad vertis em en t in ac co rd an ce w ith1 8 U .S .C . S ec tio n 1 73 4 s ole ly to in dica te th is f ac t.1 S up po rted b y U SP HS G ra nt C A- 40 01 1.2 T o w hom requests for reprints should be addressed, at C enter for B iostatistics andE pid em io lo gy . P en ns yl va ni a S ta te U niv er si ty C ol le ge o f M e di ci ne , H er sh ey , PA 1 70 33 .

ternational Journal of C ancer, International Journal of R adiationOncology, The Prostate, and Radiation Research. W e selected articles that presented analyses of in vivo tumor growth data and rev ie wed the statistica l m eth od s u sed .O ur review revealed that a variety of m ethods are in use. Severalauthors (1-7) did a separate analysis of tum or volum es at each tim ep oin t, in dic atin g a ll th e tim es a t w hich d iffe ren ce s w ere s ig nifica nt.The analysis at each tim e w as either a / test, a M ann-W hitney test, anANOVA3 F te st, o r a K ru ska l-W allis test. O th ers (8 -1 1) e xec uted te stso nly at th e fina l m easu rem en t tim e o r th e fin al tim e w he n a su bstan tia lfraction of the anim als w ere alive. Still others (9, 11-14) analyzedtum or regrow th or doubling, tripling, or quadrupling tim es. T wo others (15, 16) analyzed anim al survival tim es but not tum or volum es. O fthe articles where ANOVA was used, three (3-5) used the Duncanm ultiple-range test (17) to account for m ultiple com parisons. O nearticle (14) fit a Gompertz curve to tum or growth in an untreatedco ntrol gro up , w ith ou t a ny fo rm al sta tistic al te stin g.Table 1 lists these methods along with a brief summary of theiru nd erly ing a ssumption s a nd p rop erties. T he m os t p opu lar m etho d w asto execute a test at each tim e point and report all the tim es w here thetest is sig nifica nt. T his pro ced ure is a ttrac tiv e b ec au se it is s im ple an duses all the data. Its main w eakness is that its type I error rate is notthe conventional 5% . To see this, note that if there truly are notreatm ent differences, there is a 5% chance of significance (a type Ierror) at each tim e point. C onsequently the overall chance of significance, being the chance of significance at any tim e, exceeds 5%. Asecond popular strategy is to analyze only the data from the finalm easurem ent tim e. T his procedure has the conventional type I errorrate of 5% but is deficient in pow er, inasm uch as it com pares only theen ds o f th e c urv es a nd m ay m iss re al d iffere nce s at in te rm ediate tim es.The experim enters who looked at doubling and regrowth timesanalyzed their data by ANOVA or its rank-based analogues. Suchm ethods are not applicable if the tim es are subject to censoring, i.e.,if tum ors m ay fail to double or regrow by the end of the observationperiod. F or this reason it is preferable to use m ethods that explicitlyaccount for censoring, such as the logrank test (18). T hese m ethodsgenerally have correct type I error rates but suboptim al pow er.A fin al c ritic is m th at a pp lie s to all th e m eth od s re vie wed is tha t th eygenerally yield little biological insight; i.e., w ith these m ethods onec an sta te w hich g ro up s are sig nific an tly d iffe re nt a nd p ossibly ran k thegroups, but otherw ise, because no m odeling is being done, it is difficu lt to relate re su lts to u nd erly ing m ec ha nism s.The past three decades have seen the developm ent of classes ofstatistica l m etho ds de sig ne d to av oid th ese c ritic ism s. O ur pu rp ose inthis article is to present a subset of these m ethods that we find bestsuited to the analysis of tumor growth experiments. W e call them eth od s "m ultiv ariate " b eca us e th ey trea t th e series of tu mo r v olu meson an anim al as a single m ultivariate observation. T hey use the entiredata series and perm it detailed m odeling of grow th curves and intra-anim al correlation patterns, thus substantially im proving the effic ien cy o f tes tin g an d re duc ing samp le s iz e re qu irem en ts . T he m etho ds

3 T h e ab brev ia tio ns u sed ar e: A NOVA , a naly sis o f v ar ia nc e; d .f ., d eg re es o f fre ed om ;D FMO , a -d if lu or om et hy lo rn it hi ne ; G EE, g en er al iz ed e st im at in g e qu at io ns ; LR, l ik el ih oo d r at io ; MANOVA, m ul ti va ria te a na ly si s o f v ar ia nc e; R E/ AR , r an dom e ff ec t;gressive6042

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TUMOR GROWTH EXPER IMENTS: STATIST ICAL ANALYS IST ab le 1 M eth od s f or s tatis tica l an aly sis o f in v iv o tu mo r g ro wth d ata

D at a u se d Analysis method Key assumptions CritiqueV olu mes a t a ll tim es

V olu mes a t f in al tim e

Doub li ng t im e s

En ti re c ur ve

ANOVA o r K ru sk al -W al li s, r ep ea t u nt ilsignificanceA NOVA (I tes t)

K ruska l-Wa ll is (Mann -Whi tney)L og ra nk t es t

MANOVA

Mu lt iv ar ia te g row th -c ur ve a na ly si s

R eg re ss io n w it h R E /AR e rr or s

IndependenceNormalityS am e v ar ia nc e i n a ll t re atm en t g ro up sIndependenceIndependencePropo rt ional ha za rdsIndependenceNormalityS am e v ar ian ce in a ll tr ea tm en t g ro up sIndependenceNormalityS am e v ar ian ce in a ll tr ea tm en t g ro up sG r ow th c ur veIndependenceNormalityS am e variance in all g roups and at all tim esRE/AR corre la ti onsG rowt h c ur ve

In fla ted ty pe I e rr or rate

S ub o pt im a l p owe rS en si ti ve t o n o rma li ty

Su bo pt im a l p owe rSu bo pt im a l p owe rS en si ti vi ty t o h az ar ds a ss ump ti onE xc lu de s c as es h av in g m is si ngvaluesSu bo pt im a l p owe rS am e as M AN OV A, but m ore pow erfulw hen g ro wth c urv e is c orre ct

Mo st p ow er fu l, b ut p ot en ti al ly s en si ti veto a ssump tionsI nc lu de s a ll c as es

are not new to statistics or even to cancer research (see R ef. 19, C hap.8), although evidently they are not w ell know n to in vivo experim enters. W e illustrate the m ethods by applying them to previously publish ed d ata o n th e e ffec t o f D FMO , a p oly am in e b io sy nth etic in hibitor,on the growth of BT-20 human breast cancer cells in nude m ice.MATERIALS AND METHODSExper imenta l Methods: The BT-20 Exper iment

T he objec tive of the experim ent (20) w as to determ ine w hether horm one-independent hum an breast ca ncer cells grow ing in nude m ice m an ifest sen sitiv ity to th e p oly am in e-b io sy nth etic in hib ito r D FMO . T um ors f ro m th e B T-2 0c ell line w ere establishe d in 4-6-w eek-old ovariectom ized athy mic N cr-/ium ice (National C ancer Institute, B ethesda, M D) by injecting 5 x IO6 cellsresuspended in 0.25 ml m edium into two mam mary fat pads per mouse. W erandom ized size-m atched mice bearing established tum ors to one of sixD FM O dose groups: 0% (control), 0.5% , 1% , 2% , and 3% in drinking w ater.W e continued treatm ent until the m ice in the 0% group had to be sacrificedbecause of large tumor burden, m easuring tumor volum e on days 0 (baseline), 3, 7, 10, 14, and 16 posttreatm ent. W e m easured the length (/), w idth(w), and height (h) of the tumors with a Jamison caliper and calculatedv olu me fro m th e h em ie llip so id fo rm ula

Y = T Tlw h/6Sta ti st ica l Methods

In this sec tion w e describe th ree m ultivariate m ethods for analyz ing tum orgrow th data (for details see the "T echnical A pp endix" ). A ll th ree correct them ain flaw s of the currently popular m ethods by achieving the ir nom inal typeI erro r rates and usin g the entire volum e series. N ote that the m ethods requirethat the data be norm ally distributed. If the volum es are not norm al they canoften be m ade so by transform ation; w e assum e in this section that the logar it hm i c t rans fo rma ti on i s approp ri at e.The MANOVA Mod el. T he m ultiv ar ia te lin ea r m od el ( 21 ) (s ee "Ap pe nd ixSection A.I") takes the anim al's vector of log tum or volum es to be the unit ofdata. In other w ords, it assum es that the anim als are independent but that theobserv ations w ith in an anim al m ay be c orrelated. T he underlying m ean vector is the same for all animals in a treatment group, and the unknownvariance-covariance m atrix is the sam e for all anim als in the population.M ath em atic ally , th e m od el is

where Y is the matrix of observed log volum es, XM is a design matrix, BM isa m atrix of regression coefficients, and e is a m atrix of random errors, therows of w hich are independent and m ultivariate norm al w ith m ean 0.One can represent a number of tum or growth m odels with Equation A byappropriate selection of XM . The model we consider here is called theMANOVA model. In it, the columns of XM are indicators of dose groupm em bership. The m atrix B M has as m any colum ns as there are m easurem enttim es, w ith eac h c olum n represe nting the m ean log volum e for the five groupsat that tim e. T his m ode l asserts that a separate A NOVA m od el o btains at eachtim e, w ith th e r an dom e rr ors o f m ea su reme nts w ith in th e same a nim al p ossib lycorrelated.If there is a dose effect, be tw een-group differences depe nd on the m easurem ent tim e; e.g., th e difference betw een the 0% m ean and th e 0.5% m ean w ouldbe one thing on day 0, another on day 6 (a pattern of effects called a dose-b y-tim e in te ra ctio n). T o te st th is, w e tr an sla te it in to a lin ea r h yp oth es is a bo utthe coefficient m atrix B M (Appendix Equations A .2 and A.3), w hich w e testusing the H otelling-Law ley trace test (21). H enc eforth w e re fer to this as theM AN OV A dose-by-tim e interaction test. It can be executed in the G LM procedure of the SAS System (SA S Institute, Inc., C ary, NC ) and other com mercial statistical progra ms. Its pow er func tion is com plicated, although a non-central F approxim ation is available (22). A s com monly practiced, e.g., inSA S, the test requires com plete data on all anim als; thus anim als for w hichany volum e m easurem ents are m issing are excluded from the analysis. If thetest is sign ificant one can proceed to un ivariate tests a t each tim e: the requirem ent of a significant M ANO VA pretest guarantees that the type I error rate isprese rved a t 5% .T he M ultivariate G row th C urve M od el. T his m od el (23) (see A pp en dixS ec tio n A .2 ) a ssu me s th at

Y = XrBrP ,: + (B )

Y = XBU e (A )

w here Y is the m atrix of observed log volum e data, X G is a betw een-anim alsdesign m atrix, BG is a m atrix of regression coefficients, PG is a within-individuals design matrix, and e is a matrix of random errors, the rows ofwhich are independent and m ultivariate norm al. It states that in each treatment group the data follow the same kind of curve (specified by PG), although the coefficients (the row s of B G) m ay differ from group to group . T hem atrix XG c on sists o f in dic ato r v ar ia ble s d en otin g g ro up m em be rsh ip , a na logous to the XM matrix in the M ANOVA model of Equation A.The grow th curve m odel is not a special case of the general m ultivariatelinear m odel, but it can be m ade so by transform ing the log volume data. Apopular approa ch is to com pute a set of regression coeffic ients from each animal and analyze these by M ANOVA. In the BT-20 data, for exam ple, sup-

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TU MO R G RO WT H EX PERIM EN TS: STA TISTIC AL A NA LY SISpose that in each dose group the log volum e curve is a straight line. To testfor dose effects, we test whether the slopes are the same. The Hotelling-L aw ley test of this hypothesis reduces to a un ivariate A NOVA F test appliedto the w ithin-anim al slopes. A ny good statistical package can do suc h a test,although com puting the within-anim al slopes m ay require som e effort. Thepower can again be com puted using the noncentral F (22). A s in M ANO VA ,a nim als w ith o bse rv atio ns m iss in g a re ty pic ally d ele te d, a lth ou gh some a da ptations (24) retain a ll the data. S ee " Discussion" for m ore on this point.

Regression w ith Random E ffects/A utoregressive E rrors. T his m odel(25, 26) (see A ppendix Section A 3) asserts that

T ab le 2 K ey as su mp tio ns o f th e R EIA R re gre ss io n m od el

X , =*, (C )where for anim al i, Y s the vector of log tum or volum es, X , is the m atrix ofp re dic to rs, a nd e , is th e v ec to r o f r an dom e rro rs; Ks a re gr essio n c oe ff ic ie ntvector c ommon to all a nim als. T he error is assu med to have a R E/A R variance-covariance matrix, as described below. W e henceforth refer to this as theR E/A R re gre ssio n m od el.In the B T-20 study w e w ant to fit a linear m odel w ith different slopes but acommon interce pt. T he R E/A R m od el, unlike the MANOVA a nd grow th curvem ode ls, can accommoda te this assum ption. T o assess dose effects on e sim plytests the hypothesis that all the slop es are equ al.The RE/AR model assumes that the error term is the sum of an animal-specific random effect a nd a n autoregressive process. B y "ra ndom effect" w emean a random difference between animal / and the mean volume for allanim als; i.e., the tendency for the tum or on one anim al to be always larger ora lw ay s sm alle r th an th e m ea n amo ng a ll a nim als. A n " au to re gr essiv e p ro ce ss"is a random process in w hich the correlation betw een observations de creasesw ith in cre asin g se pa ra tio n in tim e. In tu mo r g ro wth se rie s a nd o th er b io lo gic aldata, w e commo nly observe that the shorter the tim e betw een m easurem ents,th e h ig he r is th eir c or re la tio n (2 7) . F ittin g a n a uto re gre ssiv e e rro r m od el is o new ay to m od el th is p he nome no n.W e have fit the R E/A R regression m odel using a program we have w rittenin F or tra n a nd S -P lu s V er sio n 3 .1 ( Sta tistic al S cie nc es In c., S ea ttle , WA ). It c analso be fit in BMDP program 5V (BM DP, Inc., Los Angeles, CA) and SASpro cedure M IX ED [ava ilable in V ersions 6.07 and later (S AS Institute, Inc.,Gary, NC)]. Hypotheses about the regression param eters can be tested in anum ber of ways; w e have used LR tests, the pow er of w hich we approxim atewith the noncentral \2 (28). The RE/AR model uses all the data, and consequently there is no need to exclude incom pletely observed anim als.

Comparison of the M ultivariate M odels. The M ANO VA model is them ost gen eral of the m ultiva riate m odels in that it plac es no restric tions on theshape of the growth curves or the variance m atrix. The grow th curve m odel issim ilar to MANOVA e xcept that it restricts the m ean grow th curves to be of thesam e param etric type (in our exam ple a straight line), w ith different coefficients in e ach group. T he R E/A R m odel diffe rs from the grow th-curve m odelin allowing the dose groups to have com mon regression coefficients (in ourexam ple the interce pt). U nlike the other m odels, it assum es that the standarddeviation is the sam e at e ach tim e in a ll groups, and correlation w ithin anim alsfo llo ws th e R E/A R p atte rn . In co rp ora tin g th es e m or e re str ic tiv e a ssu mp tio nsm akes th e analysis m ore pow erful, if they are justified. If not, both type I errorra te a nd pow er m ay su ffer.

RESULTSIn this section w e analyze the BT -20 data and com pare the candidate methods in terms of power and type I error rate. To give an ideaof the steps involved in a m ultivariate analysis, w e present detailedresults for the R E/A R m odel.R egression M odeling of the B T-20 D ata. Th e first step in RE/AR

m odeling is to verify that the data reflect the m odel assum ptions, atle as t a pp ro xim ate ly . T he k ey as sump tio ns, ou tlin ed in T ab le 2, a re th at(a) the SDs ar e equal at al l times in al l groups, (b) the er rors aresym metrical, (c) correlations follow the RE /A R m odel, and (d) thegrow th curve is adequately represented by the proposed param etricform (e.g., a straight line, a quadratic or a spline). Note that in

AssumptionEqualSD at all tim esi n a ll g ro up sSymme tr ic al e rr or sRE/AR cova r iance sShape o f the cu rveD iagnos ti cSp read -v s .- le vel

plot29)Symmetryp lo t ( 29 )S emi va ri og ram (26 )S tr ai gh tn es s p lo t ( 29 )Goodness-of-fits ign if i cance t es t sAc tion

toakePowertransformationPower

transformationA dj us t v ar ia nc e m od elPower t ran sfo rma t ionS el ec t b es t- fi ll in g mode lw i th in appropr i at efamily

assumption b we attem pt to assess error sym metry rather than norm ality be ca use it is d ifficu lt to form ally te st n orm ality , a nd symm etryis p resu mab ly th e critica l featu re o f n orm ality .A class of data-analytic techniques (29) is available for assessinga ssump tio ns (i), (ii) an d (iv ). T hese to ols h ig hlig ht de partu re s from th eassumptions and suggest ways to transform the data to make theassum ptions m ore nearly true, as show n in T able 2. A pplication to theB T-2 0 da ta d irecte d u s to a ran ge o f p os sib le tran sfo rm ation s, in cluding the log and the square root. W e chose the log because it has beenselected by m any previous dataseis, and slopes of log-scale data havea s imp le b io lo gi ca l i nt er pr et at io n.Fig. 1 displays boxplots (30) of tum or volum e by tim e for the fivedose groups. T he com parable sizes of the boxes dem onstrate that thelog transform ation has rendered the S Ds nearly equal. T he w hiskersshow that the distributions are roughly symmetrical, although therea re o cc asio nal o utliers (d is pla ye d a s d ots ), m ost often o n th e low sid e.The plot of mean log growth curves (Fig. I/) shows that growth isro ugh ly lo g-lin ea r, w ith slo pe d ec rea sin g as d ose in crea ses.To select an empirical best model we fit a sequence of models,c om parin g th eir fits v ia sig nifica nc e tes ts. A first qu estio n is the b asicshape of curves to use in subsequent m odeling. Solid tum or grow thcurves are often G om pertzian in that they start out nearly log-linear

b ut later flatte n at a lim itin g v olu me . W e h ad e xp ected to se e flatte nedcurves in this experim ent, although we had som e idea that the timesam pling w as so short that the curves w ould be nearly linear. W e thuscompared two key models: a "full linear" model with a commonin te rc ep t an d d ose-s pe cific slop es (E qu atio ns A .1 0-A . 12 ), a nd a "fu llquadratic" m odel w ith a common intercept and dose-specific linearand quadratic terms. Although the quadratic model is not Gom pertzian, it should be a m uch better approxim ation than the linearm odel. T he full m odels are nested (i.e., one obtains the linear from theq uad ratic b y s ettin g th e tim e-sq uared c oefficien ts to 0 ) a nd th us c an becom pared by a LR test. For the BT-20 data the LR x2 statistic is 3.4on 5 d.f. for a P value of 0.64. This suggests that departures fromlinearity are sm all relative to the discrim inating pow er of the data.T o test the R E/A R A ssum ption iii, w e reestim ated the variance andcovariance em pirically using the sem ivariogram (26), a plot of thec ov aria nce o f o bserv ation s 0 u nits p art m in us th e c ov aria nc e at A un itsa pa rt, a s a fu nction of A . F ig . 2 compa res th e em piric al an d b est-fittin gRE/AR semivariograms from the BT-20 data. Agreem ent is good,suggesting that the R E/A R assum ption is adequate for these data. T hesemivariogram estimate of the variance of a single log volume is0.541, in good agreem ent w ith the m odel-based estim ate of 0.536. Ifthe fit had been less encouraging w e could have adopted one of them ore general covariance m odels proposed by D iggle (26).As noted above, Fig. 1 suggests a monotonie dependence of thegrowth rate on the DFM O dose. The slope estimates (Table 3) bearthis out. T o assess this em pirically w e conducted a series of L R tests,first com paring the full linear m odel to a linear m odel w ith a commonslope and intercept, essentially a test for any dose effects. A s Fig. 1suggests, the full m odel fits significantly better (L R = 74.5 on 4 d.f.,P = 3 x I O" 15). H avi ng establ ished that the gr oups di ffer , we sought

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TU MO R G RO WTH EX PER IM EN TS: STA TISTICA L A NA LY SIS

la ) 0 .0% Ib) 0 .5% le ) 1 .0%

-S -

B B0 fl8 -

g oi -s s -

14 16Tu(Days)I d) 2 .0%

03 7 10 14 16T im e (D ay s)

7 10T im e (Da y s)

14 16

le ) 3 .0% If) M ean log grow th curve

8 -a - 8 .7 10

T im e (Da y s)7 10

T im e (D ay s ) T i m e (D ay s )F ig. 1. B oxplots (3 0) of m ean tu mor volum e (log scale) versus tim e since the beginning of D FMO therapy , by dose group (a-e), and m ean log gro wth curves for all five doseg ro up s ( /) .

a sim pler description of the dependence of the slope on the dose. W et rie d s ev era l p os sib ili tie s, in cl ud in g a l in ea r mod el , a q ua dr at ic mod el ,and a linear m odel with an additional parameter for active versusplacebo. N one fit as w ell as the m odel w ith arbitrary slopes. Thus w econcluded that, w ithin the ability of these data to resolve such questions, (a) tum or grow th rates differ betw een dose groups, and (b) therelationship between DFM O dose and growth rate is negative andm onotonie, but (c) a sim ple linear or quadratic function cannot adequately describe the dependence of slope on dose.The slope in a log-linear growth model can be interpreted as theproliferation rate m inus the death rate. The fact that the slope decreases w ith increasing dose suggests that D FMO affects one or botho f th ese rates in a d os e-d ep end en t m an ner. In te resting ly , an aly sis o f ap ar all el e xp erime nt i nv ol vin g t he h ormo ne -d ep en de nt c el l l in e MCF -7showed a nonmonotone dose effect, raising questions about them echanism s of D FMO action in these cell lines.C om p ar ison w ith O th er M eth od s of A nalysis. W e also an alyz edthe B T-20 data using the other m ultivariate m ethods and the m ethodsgleaned from the cancer literature. First w e executed A NO VA andK ruskal-W allis tests com paring the dose groups at each tim e point.T he groups differed significantly at days 10 and beyond by A NO VAand days 7 and beyond by K ruskal-W allis. A logrank test com paringth e do ub lin g tim e d istrib utio ns w as h ig hly sign ific an t, a s w ere s im ilar

tests fo r trip lin g an d q uad ru plin g tim es. T he MANOVA do se -b y-tim einteraction test (F = 3.8 with 24 and 222 d.f.) gave a P value of 5 X10~ 8. T he grow th curve slope A NOVA w as also significant, w ith F =15.1 on 4 and 62 d.f. (P = 1 X IO"8). In short, all the methodsc on clu de th at th e d ose g ro ups are sig nific an tly d iffere nt.C om p ar ison of T y pe I E rr or R at es. For t he m u lt iv ar iat e m et hod sand ANOVA at the final m easurem ent, statistical theory tells us thatthe type I error rates are close to 5% as long as m odel assum ptions aren ea rly c orre ct. W e e stim ated th e erro r ra te s o f th e o th er te sts (ANOVAo r K ru sk al-W allis at a ll m ea su rem en t tim es, an d lo gra nk on d ou blin gtim es) by a M onte Carlo experim ent. W e generated data under theassum ptions that (a) the true, underlying grow th curves are all equal,with intercept and slope equal to estimates from the 0 dose groupunder the RE/AR model (from Table 3) and (b) the true underlyingvariance m atrix is R E/A R (E quation A . 1 3), w ith param eters equal tothe estim ates from the B T-20 data. W e sim ulated 1000 independentdata sets having the design of the BT-20 experiment. Each datasetconsisted of 5 groups of 15 anim als, each anim al having tum or volumes measured on days 0, 3, 7, 10, 14, and 16. We applied all thetests to each sim ulated dataset at each sam ple size, by taking the firstn uni ts fr om each gr oup, n = 2, 3, ..., 15. We esti mated type I er rorrates as the fraction of sim ulated dataseis w here significance w asattained.6045



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TU MO R G RO WTH EX PERIM EN TS: STA TISTIC AL A NA LY SISOO

p Jc i s JI uFig. 2. Empirical and model semivariograms S- p -based on the full regression model. O ^

csp -0

p

EmpiricalModel

10 11 13 14 16Lag

Fig. 3 plots the type I error rates versus the sam ple size per groupn . T he so lid lin es in dica te th e ta rg et ra te (5%) tw o M on te C arlo SE s(SE = VO.05 X 0.95/1000); thus a symbol lying outside the twoo uter so lid lin es differs sig nifica ntly from th e targ et va lu e. A lth ou ghwe expected the logrank test to have an error rate near 5% , we wereconcerned that it w ould be off som ew hat because of the sm all sam plesizes and discreteness in the doubling tim e distribution. F ig. 3 show sthat the error rate of the test is never far from 5% and im proves rapidlyw ith increasing n. O n the other hand, the type 1 e rror rate of testing ateach tim e point (by either A NOVA or K ruskal-W allis) considerablye xc ee ds 5%.Comparison of Powers. Among tests having equal type I errorrates, the m ost pow erful is generally preferable. T hus w e com paredour tests by computing their powers under the BT-20 design forn = 2, ..., 15. This time we assumed that (a) the true, underlyinggrow th curves differed, w ith grow th param eters equal to the R E/A Rparameter estimates for the BT-20 data (Table 3), and (b) the trueunderlying variance m atrix is R E/A R (equation A .13), w ith parameters equal to the estim ates from the R E/A R m odel for the B T-20 data.W e computed pow ers for three tests (AN OVA of data from the finalday, MANOVA, and ANOVA on the slopes) by noncentral F approximation (22). W e computed the power of the LR test in the RE/ARm odel by a noncentral x2 approxim ation (28), and the pow er of thelogrank test on doubling tim es by M onte C arlo sim ulation.Fig. 4 plots the power of each test as a function of n, the samplesize per group. The methods that use all the data and exploit the underlying m odelin this case the R E/A R m odel and the A NO VA onslo pe s ha ve g rea te st p ow er. T he MANOVA d ose-b y-tim e in tera ction test and the logrank test on doubling tim es are less powerful,

T ab le3 E stim ate d s lo pes o f lo g tu mo r g ro wth : B T-2 0 D FMO e xp er im en tDFMOdose(%)0.00.51.02.113.0EstimatedSlope E0.0906

.00640.0723.00600.0445.00580.0411.00580.01370.0061

because, although they use all the data, they do not exploit the linearity of the log-volume curves. The least powerful method isA NO VA on data from the final day, w hich uses only a sm all fractionof the data and totally ignores the shape of the curves. The m inim umsam ple sizes required for 90% pow er reflect these differences: ForRE/AR, the minimum n is 3, for the slope ANOVA it is 4, forM ANOVA and the logrank test it is 7, and for ANOVA on the finalday it is 11.A lthough our calculations suggest that n = 3/group is adequate, inpractice we would not run such a small study. First, the power approxim ation for the R E/A R m odel assum es that the variance parameters are known a priori, which is never the case. The power ofR E/AR is th erefo re so mewh at o verstated , a lth ou gh w e su sp ect th at th eeffect is to underestim ate sam ple size by only one or tw o anim als pergroup. S econd, our com putations assum e that all tum ors grow and noanim als die prem aturely, w hereas in reality such data losses are comm on and need to be provided for. Finally, n = 3/group may not givesufficient pow er for other outcom es of interest. For exam ple, in theBT-20 experiment tum or polyam ine levels were an im portant endpoint. Because these can be measured only at sacrifice, there is noa lte rn ati ve t o a u ni va ri at e a na ly sis f or t his e nd p oin t, a nd c on se qu en tlya larg er sa mp le size is n ec essa ry .These comparisons do not imply that RE/AR is best in everysituation. A lthough our diagnostic analyses suggest that log-lineargrowth and RE/AR covariance are reasonable assum ptions for theBT -20 data, if they w ere not, the pow er advantage of R E/A R could bereduced or even reversed. H ow ever, it is generally true that the use ofd etaile d m od el in fo rm atio n le ad s to m ore po we rfu l tests; th us it is b es tto use as m uch of this inform ation as is available.DISCUSSIONTable 1 summ arizes the methods we have discussed. W e find thead va ntag es o f th e m ultiv ariate m eth ods (co rrec t ty pe I e rro r pro bab ilities, enhanced pow er, and the capacity to m odel data rather than justtest significance) com pelling. Among the m ultivariate m odels, theRE/AR m odel uses the data m ost efficiently but requires the m ostw ork to apply.6046



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TU MO R G RO WTH E XPERIM EN TS: STA TIST ICA L A NA LY SIS

"O

g ent O(O

F ig. 3. T ype I error rate as a fun ction of sam ples ize p er g ro up f or th ree m eth od s o f a naly sis .

q

A+XT arg et R ate (5%) + /- 2S ELog ra nk T es t o n Dou bli ng T im esA NOVA at E ach T im eK ru sk al-Wall is a t E ac h T im e

x.Vx-+ ~ ~~X-~ -.XL__X-~-H+i----r.~-i

....A -A A-'

2 3 4 5 6 7 8 9 10 11 12 13 14 15S amp le S iz e

A lthough w e have concentrated on log-linear m odels, one can, andoften should, use other shapes to describe the basic grow th curve. F orexam ple, cancer treatm ents are often adm inistered in pulses that reduce tum or volum e transiently before a period of regrow th. In suchcases the volum e curves are not m onotone and are better described byspline (piecewise polynom ial) models (31). W e have used this approach to analyze data from an in vivo study of androgen prim ing inp ro st at e c an ce r ( 32 ).T he practical price of the superior properties of the m ultivariatem eth ods is th e g rea ter ex pen se o f a pply in g th em . T ab le 2 s umm ariz esthe assum ptions of the m odels, the diagnostics that address their

adequacy, and the alternatives available w hen there are problem s.S uccessful execution of the diagnostics and the m odeling requires alevel of programming skill and statistical judgm ent beyond w hat onecan acquire in a typical elem entary statistics course. T hus m any inve stig ato rs w ill n eed p ro fe ssio nal statistica l ass istan ce to ap ply th es emodels.M iss in g d ata , u su ally resu ltin g fro m an im al m orta lity or m orb id ity ,is a com mon and potentially serious problem in grow th analyses. Ifthe m issing data are ignorable, in the sense that the probability of theobserved missingness pattern does not depend on the observed oru no bserv ed d ata va lu es, th en it is a ppro pria te to trea t th e m issing d ata

oo0

VO0

Fig. 4 . Power as a function of sam ple size pergroup for five m ethods of analysis w hose type Ie rr or r ate is ex actly o r a pp ro xim ately 5 %.

rj0

pO

Q . A A SE _JH/ F y**-''//''Ja"""* 7'/"'"

/f1 /'// /1

/ :' A"'1 / -r-.L / !/ t-'/

' / '..'/ /'7 ;/: x/+ o/ vtTorrtofD / ri / r f QT \Q 7\larget rowery\)/o)\JtJf\\TA T ', ' Y i. -.l\rAWUVA, rinai/ayV A X T/~ \T T AMAJNUvAinteractionANOVAonlopesRE/ARModelLogranko n Dou bli ng T im es

2 3 4 5 6 7 8 9 10 11 12 13 14 15SampleSize6047



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TU MO R G RO WTH EX PER IM EN TS: STA TISTICA L A NA LY SISas though they were m issing by design. If the m issingness is notig nora ble , th en p ara me te r es tim ate s m ay b e b iase d, an d sig nifica nc etests m ay have type I error rates exceeding their target values (33).Wh en m is si ng ne ss i s ig no ra bl e, s ev er al a na ly sis s tr at eg ie s a re a va ilable. A s has been indicated, im plem entations of the MAN OV A andgrow th curve m odels in the m ajor statistical packages require balanced data, and to obtain it all anim als are deleted for w hich there areany m issing observations. T his can result in a considerable loss ofefficiency if m any anim als have m issing data. V onesh and C arter (24)have proposed a m ethod for fitting these m odels that uses all the data.A s in dic ated ab ov e, R E/AR m od elin g au tom atic ally u se s a ll av ailab ledata.When missingness is not ignorable, one must model both thegrow th data and the m issingness pattern. T wo approaches have beenproposed. In the first (34, 35), one assum es that each anim al has itsow n underlying slope and that the probability that the anim al is lostdepends on its slope. In the second (36), one assum es that the tim e ofdropout can depend on previous and current values of tum or volum e.W hichever m odel one uses, it is necessary to estim ate the growthc ur ve a nd m is sin gn es s p ar am et er s s im ul ta ne ou sl y.U nfortunately, inferences under nonignorable m odels can be sensitive to the assum ed m odel; i.e., erroneous assum ptions about theunderlying distributions, usually very difficult to detect, can causeserious errors in inferences (37). F urther theoretical and em piricalresearch is needed to elucidate the proper m ethods for m odelingin comp leten ess in tum or gro wth d ata .A second problem w ith the analysis of tum or grow th data involvesselec tio n o f th e tra ns form atio n to n orm ality an d th e reg res sio n m od el.S om e statistician s claim th at on e sh ou ld e xplicitly a dju st the a nalys isif th e d ata are u se d to s elec t a tran sfo rm atio n o r m od el, w herea s o th ersargue that this is unnecessary (38). In this study, for example, ourchoice of log-linear m odels w as based on analyses of the sam e data;therefore a m ore rigorous analysis w ould adjust estim ates and tests toaccount for this selection. Yet it is com mon practice to ignore thisproblem, and there exist no practical methods for making such adjustments.The methods we have presented are just a few of the many techniques available for analyzing tum or grow th data. For exam ple, theR E/A R m odel is a special case of the longitudinal-data linear m odelof Laird and W are (39). This model accommodates more generalrandom effects (such as random slopes) and serial correlation structu re s (in clu din g h igh er-o rd er au to re gre ssio ns ). A ltho ug h th ese m oregeneral correlation m odels m ay be valuable in m any applications, w ebelieve that the R E/A R m odel captures the m ost im portant features oftu mo r g ro wth d ata.A ll our m ultivariate m odels assum e norm ality and linearity, butother m ethods are available w hen such assum ptions are restrictive orunw arranted. O ne approach is to base significance tests on the distrib utio n o f m ultiv aria te ra nk sta tistic s (4 0) or the ra ndomiz atio n d istribution (41). T hese tests are reliable under assum ptions m ore generalthan ours, w ith som e loss of efficiency if a norm al m odel is actuallyappropriate. The GEE approach (42) involves specifying only theshape of the grow th curve and the covariance m atrix, not the underlying distribution. G EE is robust to errors in the assum ed correlationstructure and can handle arbitrary patterns of m issing data; like theran k a nd ra ndomiz atio n te sts, it is re lia ble b ut p oten tia lly in efficien t.A nother approach involves m odeling tum or grow th curves w ith nonlin ea r rathe r tha n lin ear m ode ls (4 3^ 6). T his is m ore diffic ult to a pp lythan linear m odeling but can give greater insight into the biologicalprocesses of tumor growth. None of the methods cited in this parag ra ph is a va il ab le in p ro du cti on v er si on s o f m aj or s ta tis ti ca l p ac ka ge s,a lth ou gh some g ood p ro gra ms a re p ub licly a va ila ble.

In su mm ary , sta tistic ia ns ha ve d ev elo ped an a rra y o f m ultiv aria tem ethods that can dram atically im prove the analysis of tum or grow thstudies. C areful application of these m ethods w ill lead to m ore efficient and hum ane experim ents and m ore valid and com prehensiveda ta analyse s.T ECHN ICAL APP ENDIXA.l. The M ANO VA m odel (21) states that

(A.1)w here Y i s a n N X p m atrix of observed log volum e data, XM i s an N X r designm atrix, B M is an r X p m atrix of regression coefficients, and e is an N X p errorm atrix the row s of w hich are in dependent and m ultivariate norm al w ith m ean0 and variance-c ovariance m atrix 'S ,. H ere N refers to the num ber of anim als,p to the number of times each animal is measured, and r to the number ofpredictors in the design m atrix. The /th row of y is the vector of log volum esfor the /th animal, and the /th row of XM is the design for the /th anim al. Thecolum ns of B M a re the m odel coefficients, with one colum n for each of the pme as ur eme nt t im e s.To apply the m ultivariate linear m odel in the B T-20 exam ple, suppose forthe m om ent that there is = 1 a nim al in each of the five dose groups. B ecau seeach anim al's tum or volum e is m easured at six tim es, p = 6; w ith five groupsand one animal per group, N = 5. A sim ple m odel would assume a time effect(i.e., the tum ors grow) and a group effect (the volum e depends on the dose).T here is a dose-by-tim e in tera ction if the tim e e ffec ts differ by group. In term sof model (A.I), XM is the 5X5 identity matrix and BM is the 5X6 matrixw he re th e ro w-//c olu mn -) e le me nt is th e e xp ec te d tu mo r v olu me a t th ey 'th tim efor an anim al in group /. W hen there are anim als per group, X M i s sim ply ncopies of the 5X 5 identity m atrix stacked vertically.To test for a dose-by-tim e interaction, w e express the null hypothesis as ag en er al m ultiv ar ia te lin ea r h yp oth es is CMBMUM= 0 , w he re CM is a " be tw ee n-u nits" c on tra st m atrix a nd UM i s a "with in -u nits" c on tr ast m atr ix . T he h yp othesis of parallel growth curves (i.e., no dose-tim e interaction) has contrastmatrices

Cu =:1 -1 0 0 0N10-1001 0 0-1 01 0 0 0 -1,

(A.2)

and

(A.3)

The Hotelling-Lawley test of this hypothesis (21) can be executed w ith theG LM procedure in the SA S System . The pow er can be approxim ated with then on ce ntra l F (2 2).A .2. The m ultivariate g row th curve m odel (23) states that

Y = XGBGPG + (A.4)where Y is an W X p matrix of observed log volume data, Xc is an N X gb etw ee n-a nim als d esig n m atr ix , BG is a g X q m atr ix o f re gr essio n c oe ff ic ie nts ,P C is a q X p w ithin-individuals design m atrix, and s an N X p error m atrixthe rows of which are independent and multivariate normal with mean 0and variance-covariance m atrix 2. Here N refers to the number of animalsstudied, p to the num ber of times each tum or is measured, g to the number oftreatm ent groups, and q to the n um ber o f predictors in the w ithin-individualdesign m atrix. The /th row of Y is the vector of log volum es for the /th anim aland the /th row of X (1 i s the betw een -anim als design for anim al /. T hey'th rowof B G is the vector of re gression c oefficie nts for anim als in the y'th tre atm entgroup.

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TU MO R G RO WTH EX PERIM EN TS: STA TISTIC AL A NA LY SISIn the B T-20 exam ple, again take n = 1 anim al in each of the five doseg ro up s. Be caus e e ac h animal 's t umo r vol ume i s measur ed on s ix o cc as io ns ,p = 6, and with five groups and one animal per group, N = 5 and g = 5.Ass um in g t umor v olum e i s l og -l in ea r in t im e, q = 2 ; i .e ., t he t umor g rowthcu rve in each g roup isdesc ribed by two pa rame te r s, a s lopeand in te rcep t.Thusin E qu ation A .4, XG i s th e 5 X 5 id en tity m atrix , BG i s th e 5 X 2 m atrix th e/th ro w of w hich is th e in te rce pt an d slo pe fo r gro up j , a nd PG is th e w ith in -a nimal d es ig n, i n t hi s c as e

1111110 3 7 10 14 16 (A.5)W i th n a nim al s p e r g ro up ,XG i s m er ely n c op ie s o f t he 5X5 i de nt it y m at rixstacked vertically.Mo de lA .4 is n ot a s pe cia l c a se o fA . I, b ut it c an b e m ad e s o b y a pp ro pri ater ed uc ti on o f t he v ol ume d at a. Fo r e xampl e, i ns te ad o f a na ly zi ng Y , o n e analy ze s Y "" = YH, w here

The t ran sfo rmed da ta sa ti sfy(A.6)

(A .7 )wher e t he r ows o f S a re i nd ep endent a nd mu lt iv ar ia te n ormal w it h mean 0 andva riance-covar iance ma tr ix F = tP^H.Practical ly, th is means reducing eachan ima l 's da ta f rom a vector o f log vo lumes to an an ima l -spec if ic in te rcep tands lo pe , wh ic h one t he n ana ly ze s u si ng MANOVA.Bec au se i t r ed uc es t he d at athi s tes t invo lvessome losso f info rma tion,bu t depending on the model and ther ed uc ti on t he l os s c an b e smal l.A g en er al l in ea r h yp oth es is f or te st in g e qu al it y o f s lo pe s i s C cBGUG= 0w here CG i s the 4 X 5 co ntra st m atrix

Cr.=

:1 -1 0 0 0N10-100100-101 0 0 0 -1,and t/c is the 2X1 matrix

( A. 8 )

( A. 9 )With these cont ra s t ma tr ice s the Ho tel ling-Lawley te s t r educes to a un iva riateANOVAF tes t o n t he w i th in -a nima l s l op es . On e c an comput e t h e e xa ct p owe rf rom the non cent ra l F ( 22 ).A.3 . The RE/AR reg re ssio n mo del (2 5, 2 6) a sserts tha t

Y, = Xe,- (A.10)whe re f or a nima l i , Y s the co lumn vector o f pog tumor vo lumes, Xs thepX q matrix of predictors , and e, is the vector of prandom errors;Ks acX l r eg re ss io n coe ff ic ie nt c ommon to a ll a nimal s. Th e e rr or t erm i s t h e s umof a random an ima l e f fec tand an au to reg res s ivep rocess; hence we ca ll th is theRE/AR reg res sion model .I n t he BT-20 s t ud y we a ss ume a l in ea rmod e lw i t h d os e- sp ec if ic s lo pe s a nda common int er ce pt .Be c au se t he re a re s ix t umo r vol ume s p er a nimal , p = 6.Th er e a re f iv e t re atmen t g ro up s w i th a c ommon int er ce pt a nd a s ep ar at e s lo pef or e ac h g ro up ; t he re fo re q = 6 onep ar ame te r (0 )s th e in te rc ep t a nd t hef ive o the rs (,'*(d = 0 ,0 .5 ,1 ,2 ,3 ) a re th e s lo pe s:

R (,/01,,'05',/",,'2',1(3')r (A.ll)An an im al in th e do se g rou p D FMO = 2% th ere fo re h as d es ig n m atrix

X, =

' 1 00 0100010001000 (A.12)

T he e rro r te rm in th is m od el is th e sum o f a ra ndom a nim al e ffec t a nd a naut or eg re ss iv e p ro ce ss . We a ss ume tha t t he r an dom e ff ec ts a re n ormal w i thm ea n 0 an d v aria nce T 2. In a n au to re gre ssiv e pro ce ss, th e c orre la tio n o fm ea su reme nts a t a d is ta nc e o f A i t im e u ni ts i s p 'A ", w he re p i s t he a uto co rre la t ion .An autoregressive process is parameterized by p and a scale parameterc . Unde r t he BT-20d e si gn , t he e rr or t erm e , i n Equa ti on A .1 0 i s mu lt iv ar ia teno rma l wi th mean 0 and va riance -cova r iance .

V ar e,) riJ7 + S2/(1 - p2)

(A.13)

where y is a 6 X 1 matrix of Is.We tes t hypo these s abou t the regre ss ion pa rame te rs u s ing l ike lihood- ra tiotes ts . Wi th the model o f Equat ionsA.10-A.12 , the hypo thesi s o f equa l s lopesi s CKK0 , whe re

I P3P7P10P14P16PJ 1

1-1 0 0 0N10-100100-101 0 0 0-1, (A.14)

Th e powe r o f t hi s t es t c an b e app ro ximat ed w i th t he n on cent ra l x 2 ( 28 ).W e fit the R E/A R m odel in a program w e w rote in S -P lus (S tatisticalSc iences , Inc. ). One can a lso f it thi s model in theSAS MIXED procedu re (SASIns ti tu te , I nc .) a nd BMDP p rogr am 5V (BMDP, I nc .) .

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